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Entries from CognitiveScience


PracticeAndOverlearningPart1 23 Jun 2006 - 13:29 CatherineJohnson


Carolyn and I have both been using Saxon Math Homeschool Edition with our kids.

Here is Saxon's explanation of the curriculum:

Saxon Math . . . systematically distributes instruction and practice and assessment throughout the academic year as opposed to concentrating, or massing, the instruction, practice and assessment of related concepts into a short period of time -- usually within a unit or chapter.

I can vouch for this.

SAXON 6/5 has 120 lessons in all, plus 12 'Investigations' & 3 Appendix lessons, and when you get to Lesson 120 you're still practicing the stuff you learned back in Lesson 1.

There are 100 or more problems and computations in each of the 120 lessons: Fast Facts, Mental Math, Problem Solving, Lesson Practice, and, finally, Mixed Practice.

This is what we call drill and kill.

Cognitive psychologists call it automaticity:

Practice Makes Perfect But Only If You Overlearn Ask the Cognitive Scientist: How We Learn by Daniel T. Willingham

review


GoodReadingPart1 23 Jun 2006 - 13:29 CatherineJohnson


Just posted to Our Favorite Math Ed Articles:

Daniel T. Willingham's 'Ask the Cognitive Scientist' columns for AMERICAN EDUCATOR (wonderful)

William Schmidt, et al's phenomenally helpful 'A Coherent Curriculum: The Case of Mathematics' (Schmidt headed the Third International Mathematics and Science Study (TIMSS), and summarizes his findings here.)

Specific Learning Disabilities: Finding Common Ground. A Report Developed by the Ten Organizations Participating in the Learning Disabilities Roundtable. This is the American Institutes of Research 2002 consensus report: what findings, hypotheses, and theories do 10 different organizations and insitutions, including the Department of Education and the Learning Disabilities Association of American, agree to be true of 'specific learning disabilities.' (I haven't read this yet.)

See also: PracticeAndOverlearningPart1


CalStateStudyIntro 23 Jun 2006 - 13:36 CarolynJohnston


Part 1 in a mini-series on a review of quality math ed research articles.

In 1998, the California State Board of Education contracted with a group of education researchers from the University of Oregon to conduct a review of high-quality mathematics education research papers. The resulting 100-page report is available here.

Their task was simply to search out all the mathematics education research that had been performed and published within a specified period, cull out the stuff that was of dubious quality (meaning it had unsound experimental underpinnings, or was performed in a setting that was not like a classroom, or had one of a number of other flaws), and see what the remaining studies had to say about mathematics achievement (that is, they avoided papers that did not measure study outcomes quantitatively, using tests of achievement; so studies measuring the impacts of changes in teaching methodology on students' confidence, for example, weren't included).

The results are surprising to me in places. There were studies on the use of manipulatives, studies on kids working with their peers, studies on the use of computers, calculators and technology, studies on motivational methods, and studies on the design of instruction. The researchers seem to have avoided bias, and to be genuinely searching out high quality research. I thought I would do a 'mini-series' describing and discussing their results, section by section. Stay tuned.



California study intro
California state study of group learning
California Board of Ed study part 2
education research - peer reviewed studies - chart




LikePracticingTheViola 08 Jul 2005 - 00:49 CarolynJohnston


The question of the new-new math decade: how do we resolve the need to develop math fluency in children, without sacrificing their ability to think creatively?

This question presupposes that you believe the two to be in opposition - I don't. Math fluency is developed through practice, of the drill and kill variety; it's harder to say how mathematical creativity is developed (and yes, creativity is of immense value in mathematical research -- we don't just sit around thinking about the Really Big Numbers, as one of my grandmothers thought).

But the two really do coexist -- they have to. Mathematical creativity is hard to express when you have to go back to first principles every time you add fractions. But drilling algorithms can be pretty boring. How does the tedium of drilling algorithms coexist with creativity in solving word problems or engineering problems or Fermat's last theorem?

I think learning math is a lot like practicing the viola, which I could never stand to do.

I personally think the tedium of practicing computations is nothing compared to that of practicing viola, or any other instrument, but that's just me. Still, noone doubts that all violists, even the great ones -- especially the great ones -- have had to put in thousands of hours of practice, and probably noone would argue that they weren't necessary.

And how does the need for practice coexist with creativity and inspiration in playing the viola?

Well, pretty much everyone who practices the viola hard, over a number of years, is going to be a competent violist. The concert violists are going to be some subset of those who practiced their fannies off -- in fact, in terms of hours spent practicing, really inspired instrumentalists beat out their merely awesomely competent competitors. That's how you get to Carnegie Hall, after all, and here's a chart to prove it.

cogsci.jpg

How do we deal with the fact that musical practice is boring for most of us? Well, if we don't like to practice, we don't have to play. We opt out if we don't like the arrangement - as I did long ago, and as Ben did this year (although the instrument he is spurning, after a perfect record of non-practice in fifth grade, is actually the cello).

The problem with math is that nobody can opt out of learning it: we all need to be competent at it. An understanding of quantities and numbers and rates and growth are the basis of a lot of thinking in our society. It would be nice if there were a royal road to mathematical fluency, but there isn't one that we've yet found; it takes years for even the most mathematically able child to pick up all the mathematics they'll need as an adult.

Even a merely competent violist has pushed his knowledge of the mechanics of his instrument down out of his conscious brain and into his fingers. This has to happen before a violist can even dream of being creative, because if it hasn't, then his conscious brain is still working on mechanics.

Here is what I saw in my college algebra and calculus classes: people still struggling with the mechanics of math, years after they ought to have had the basic moves down. They didn't practice long and hard enough, and if they ever had the moves down, they'd lost them by then.

So how do you get your kid to practice? You get him into the habit. You provide carrots in the form of praise, trips to Chuck E Cheese, movies, video game time, whatever turns him on. You also provide a stick if necessary. You do what it takes to ensure that your kid does this thing that he needs to do, even if you have to fight with him (this is what Bernie calls being a brick wall, and what Catherine calls being your kid's frontal lobes). You clear out his schedule, if necessary, to ensure he has the time he needs to practice.

And you try to make sure he is taking a line of study that isn't going to let him down in the end.


PracticePracticePractice 10 Oct 2006 - 02:02 CatherineJohnson


I have to do something today besides sit around thinking and writing about math . . .

But all that other stuff can wait!

I'm going to be quick, which means this is off the top of my head:


1. Carolyn's friend Gerry on multiplication

For what it's worth, I think he's dead right about the value of mental multiplication.

I've mentioned that I taught a little after-school class in Singapore Math this winter. In every class I had the kids do mental math.

We did a lot of mental multiplication with the explicit purpose of implanting the distributive property inside everyone's heads.

I'm constantly pushing Christopher to do mental multiplication for this very reason.

He now 'knows' the distributive property; I think he can actually write it out in its 'letter form,' i.e. a(b + c) = ab + ac. (I think.)

He also, I think, knows -- and understands -- that the multiplication algorithm is based on the distributive property.

He knows that when you're doing a problem like:

21
x23

(sorry for the funky alignment; neither Carolyn nor I has been able to figure out how to insert extra spaces in the text thus far . . . )

. . . anyway . . . Christopher knows that when you take the 3 times the 2 you are multiplying 3 x 20; he knows that you are splitting the problem up into smaller multiplication problems and then adding the products together, which you can do because of the distributive property.

But even though he knows all this, I swear he's not as good at mental multiplication as the kids in my Singapore Math class (which Christopher boycotted). Nor does he seem to understand mental multiplication.

He didn't get the practice my Singapore Math kids did, and he's still not really making the connection that the same thing that lets you do the standard multiplication algorithm can be used to multiply numbers in your head or to very quickly multiply numbers horizontally.

His knowledge is still inflexible; he's not generalizing it to other situations and contexts. He's not seeing the connections.

This brings me to --


2. Carolyn's post on practice

This is a HUGE subject, but here are my first thoughts.

I've found that practice per se isn't such a hard thing to get kids to do.

My Singapore Math kids loved the timed worksheets I gave them. (I used the 'Fast Facts' worksheets from Saxon Math.) They used to ask to do more of them, because they made it into a competition. They were revved!

I'd have my timer out, and the kids would call out Done! when they finished the sheet; then I'd call their time & they'd subtract it from the starting time of 5 minutes and write it down on their score sheets.

(I gave each child his own 'Singapore Math' notebook with a Saxon score sheet in the front. So each week they could compare their new score to their previous scores.)

Now, you'd think this could go seriously awry, with the slow kids feeling defeated. I was worried about this myself, since I had kids ranging all the way from a fourth grader who may have been classified with some level of special needs (I have no idea--the parent seemed to indicate this) to a fifth grader whose parents immigrated from China and who's probably one of the best math students in the school.

That's a range.

But nobody's ego got crushed. Exactly the opposite.

Since they all had their own score sheets, they were competing against themselves as well as against the class. They also did different worksheets, depending on whether they'd hit the 5-minute mark on the worksheet from the week before.

As soon as somebody could do the 'Fast Facts' addition sheet, he or she moved on to the 'Fast Facts' subtraction sheet. So the faster kids were doing harder worksheets, and the slower kids were doing easier worksheets.

I guess that's like handicapping in golf, right? (I don't play golf, so I don't know.)

Let's just say that levelled the field considerably, and no one seemed to feel remotely humiliated because they were still doing subtraction when someone else was doing multiplication. They just liked the race.

And they all picked up speed incredibly quickly; I was amazed.

I had one child who, the first time he did a 5-minute addition worksheet, took -- gosh, I don't know -- upwards of 8 or even 10 minutes to get through it.

This child has perfect handwriting and is painstaking when he writes numbers, which was slowing him down, so the second day I actually wrote the answers for him so he wouldn't lose time just on penmanship.

But here's the miracle.

This kid did zero practicing in between classes, and yet by the third class he was coming in under the 5-minute deadline.

I couldn't believe it, and I don't know how he did it. He just . . . got faster. They all did.

They were achieving personal bests every week.

This gets back to Carolyn's post on group learning and Wichita Boy's post about competition: under the right circumstances, practice is fun.

I think the problem for Christopher & Ben is that they're sitting at a table with their mom who is forcing them to do math.

If they were sitting at a table with their friends, and everyone was doing math, it would be different. I happen to know for a fact that this is true, because a couple of times Christopher's friends Drew & Marc, who are fraternal twins, have done a Saxon Math lesson with us. Their mother told them they had to, so they did.

When the three of them are doing Saxon Math together, they peddle.

I've been thinking about group learning ever since Carolyn wrote about it, and I'm turning into a believer.

But more on that later.


+ + +


I see I've gotten off-track.

I meant to talk about Carolyn's observations on practice and expertise.

I'll have to do that later, but in the meantime the single best article I've seen on this subject is here.


+ + +


I wonder if you could get kids to practice the viola if you put 3 of them in a room together and set the timer.


ATeachersStory
CompareAndContrast
FromAReader
PracticePracticePractice
BarModelingVsGraphing (interesting comments from a KTM reader)


InflexibleKnowledge 08 Jul 2005 - 00:51 CarolynJohnston


In HowNotToTeachMath, Catherine posted an example of a fourth grade Everyday Math homework problem:

Homer's is selling roller blades at 25 percent off the regular price of $52.00. Martin's is selling them for one-third off the regular price of $60. Which store is offering the better buy?

I remember this sort of problem from last year, when Ben was in fourth grade. There were a whole series of such problems, more or less just like this. They were the sort of word problems you'd more typically see in a 7th-grade pre-algebra class; fortunately, they were all more or less the same. There was only one way to teach them, and that was to train the kids to do this sort of problem, step-by-step; what you might call by rote. I'm pretty sure this defeated the intention of the Everyday Math curriculum designers, who were trying to get the kids to think creatively about real world problems.

That's the idea behind many of the new-new math curricula. We can skip the tedium of teaching the standard algorithms, and emphasize estimation instead; we can skip teaching algebraic symbol manipulation independently, and teach algebra in the context of the word problems that adults really have to solve. Adults have to work with data, and so in the Everyday Math curriculum, there is enormous emphasis on statistics; kids start learning the median, mode and range before they are even capable of calculating the average. Calculating statistical landmarks is a topic that my son's classes have 'spiraled back to' any number of times in the two years my son has been doing Everyday Math.

And I don't think Everyday Math is even the most extreme of the new curricula: noone gets out of Everyday Math without at least knowing something about how to do multiplication and long division. I credit my son's teachers with taking the extra time needed to ensure that this was the case.

The intent of Everyday Math is to teach kids how to think flexibly about mathematics from the get-go. It's a laudable goal. But apparently it's a misguided one, because that's simply not how people learn new material.

When we're learning something completely new to us, we go through a phase where we understand the new material only in a very inflexible way; we can't generalize it very well, and we find it difficult to apply to new situations.

And that's okay. It's the way our minds work, apparently; we start out with inflexible knowledge, that we can gradually apply more flexibly as we gain more familiarity with it. That's why beginning violinists play stiffly, and why kids learning to read read small words, slowly. Inflexible knowledge isn't the same as rote knowledge, which leads nowhere; it's a necessary precursor to expertise.

This is something Catherine and I will harp on, over and over, because it's really important to understand this hard fact about how humans learn if you want to teach your little humans how to do math, or anything else.

This article from American Educator on inflexible learning, and its relation to expertise, is a must-read.


ImportantQuestionPart2 12 Jul 2005 - 00:29 CarolynJohnston


A question from Catherine's most recent post:

How do you deal with your kid's fuzzy math curriculum while simultaneously working with him on math at home?

This is what I was doing these past two years. The school my son goes to switched, at the end of his thrird grade year, from Saxon Math to the Everyday Math curriculum. My son, who had been doing well in Saxon Math, immediately began to struggle.

It was impossible to help him with his homework; all that ever came home was a "Student Math Journal", with an incoherent, constantly churning set of problems (there was also a reference manual, as it turns out, but it bore no relation to the sequence of topics and was generally useless).

At the end of fourth grade, I told his teachers that I was on the verge of taking Ben out of the regular math class to teach him myself, just for math. Noone told me I couldn't do this, by the way. Homeschool laws vary from state to state. Ben also has an IEP (Individualized Education Program), which gives his parents and teachers a lot of latitude to determine and implement a curriculum that is tailored for him.

So I could have taken him out of class for math if I'd wanted to; but I didn't want to. Math is his strongest subject, and I wanted him to have the experience of being in the class with the other kids, and being one of the stronger kids. We decided to keep him in regular 5th grade math, which unfortunately meant Everyday Math.

Anyway, that's one of your options; see if you can take your child out for homeschooling in the one topic. If he only has enough room in his life for one math curriculum, I would do that sooner than use, exclusively, a crummy math curriculum that won't prepare him for higher level math.

In the fall, I began supplementing at home from Saxon Math, while Ben was also doing Everyday Math at school. We did both curricula at the same time, and neither one wholeheartedly; we definitely had one foot in each world. By contrast, when Catherine was first working with Christopher (before he was placed in the most advanced math class, at midyear), she would do the regular math homework for him in order to enable him to focus on his Saxon math. That took more courage than I had; but Christopher was quickly doing much better in his regular SRA Math class than he had been, which was encouraging, and also what you'd expect under normal circumstances when a kid is being supplemented with a good curriculum.

Learning multi-digit multiplication in 4 different ways, ironically, means a child is spending more time learning to multiply in Everyday Math than he would if he were learning to do it in the traditional way. This time is taken out of other topics that are important, but that are largely left by the wayside -- like fraction division (fraction multiplication is only briefly touched on). I taught Ben only the standard algorithms, and he used them exclusively (though he had rather taken to the lattice method, and used that for awhile). We were lucky that his teachers weren't Everyday Math zealots, wedded to the idea that every kid should learn 4 different ways to multiply; they just wanted every kid to know at least one way to do the problems.

I think that it's worth trying to get dispensation from the teacher for the child to learn, and apply, only one algorithm, especially if trying to learn more than one is confusing. You could argue that after all, the notion of multiple intelligences (very beloved in modern education) dictates that kids shouldn't be forced to learn in ways that aren't suited to them; and so shouldn't kids be allowed to pick, and stick with, the multiplication algorithm that best suits them, instead of having to learn all the others as well?

Having to learn only one algorithm for multiplication and division (and making it the standard ones, taught in advance) frees up a lot of time to learn math at home, while the rest of the class is learning the lattice method.

In fifth grade, the Everyday Math curriculum was never so meaty that we couldn't deal with both that and our Saxon supplementation. We had times when we had to work pretty hard on it; for example, those double pan-balance problems were a bit over the top, but that was a short-lived unit.

If we'd had to, we would have punted on Everyday Math. I would have withdrawn him for exclusive homeschooling.

Failing that -- if it's illegal to do that in your state -- my final suggestion is this: focus on the supplementation curriculum, and if necessary take lower grades in the fuzzy math curriculum. First of all, it's unlikely that a kid receiving one-on-one intensive supplementation is going to go belly up in his regular class, no matter how fuzzy it is. Secondly, in elementary school it doesn't matter what grades your kid gets; it doesn't impact his future until he gets to high school, at which point it's too late to go back and supplement all the work he missed when he was younger.

The stakes in this game are really pretty high, so it's okay.


ImportantQuestionFromJoanneCobaskoOfSocmm
OutsmartingTheTests

BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
SummerSupplementTimePart2
SundaySchool
ILikeMath
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ConversationsWithKids




CalculusProfessorEmailExchangeWithParent 01 Aug 2006 - 19:53 CatherineJohnson


Number 2 Pencil links to this exchange of emails between a math professor and one of his students, who is flunking calculus.

The professor is using a pedagogy known as Process Guided-Inquiry Learning, or POGIL:

I mentioned in an earlier post that one of the more controversial -- and to me, appealing -- aspects of POGIL instruction is that the instructor is not seen as a source of knowledge but rather as a facilitator of learning. In non-eduspeak, this means that the instructor is there to observe and to guide, rather than to tell students what to do or think.

I also mentioned, and one commenter pointed out, that students HATE this (at least at first). Typically, students -- even upper-level students in the major who have been around the block -- just want to get the darn problem done, and when people like me insist on students asking the right questions rather than just forking over the answers, things can get heated.



In practical terms, POGIL means that when 'Pat' comes to his professor's office for help, the professor refuses all requests for one-on-one demonstration of the problems being taught in class:

…when Pat would ask me a question such as, "Can you tell me how to do problem 7?", I would say: Let's start by asking the right questions. What are you being asked to do in this problem? What information is given to you in the problem statement? And what do you know from the course, your reading, or your work on other exercises that will help get you to the goal? I made it a point to NEVER give Pat explicit help on content unless it was a last resort -- Pat absolutely HAD to cut the apron-strings from me an learn how to approach, analyze, and solve a problem alone, or else Pat's chances for success in a future career or even making it through college didn't look good.


This goes nowhere.

Finally 'Pat' sends an email explaining that he requires direct instruction in order to learn.

The professor tells him he is wrong.

Pat sent me an email just after midterms that said something like: I now understand why I am not doing well in your class. My learning style is such that you have to show me exactly what to do, or else I can't do it. But you always answer my questions with more questions, which isn't showing me exactly what to do. So from now on, please show me exactly what to do first, and then I should be able to do it. My response was something like: Pat, we've been doing this every day in class -- I work a few problems at the board all the way through during lecture, and then I give you exercises that are based on the stuff you've seen. So you are seeing me show you what to do, and yet you're still having difficulty solving problems on your own. So perhaps your assessment wasn't quite right, and we should be working on your problem-solving skills in office hours.


Then Pat's mother gets into the picture.

(via email): I know [Pat] tried to explain to you that when [Pat] asks questions [Pat] needs answers not another question. We had [Pat] tested at [a local university] in January through the suggestion of [an academic counselor at my college].

During this testing we found out [Pat] has a learning disability. [Pat] does better with visual explanations then being asked another question. [Pat] needs to see how to physically work a problem so he can comprehend it. [Pat] is a slow reader which also frustrates [Pat]. If it is a word problem [Pat] has problems figuring out what are the essential parts of the question to find the answer.



This infuriates the professor, and, subsequently, all of his commenters as well, who pretty much stomp mom to death in the comments thread.

Pat fails the class.

The commenters are united in their view that Pat is a lucky guy to have experienced POGIL calculus, and he had no business hosing the course.

Memo to self: the time to begin instilling core take no course from professor who blogs principle in 10-year old son is now.


POGIL

POGIL, POGIL, POGIL

This does not sound good, POGIL.

I should reserve judgment.

I should, but every one of the little-red-light thingies on my Constructivist Nightmare Detector is flashing wildly, and all the sirens are going off—

So I’m not doing a very good job of reserving judgment.

POGIL is a student-centered method of instruction that is based on recent developments in cognitive learning theory and results from classroom research that suggest [sic] most students experience improved learning when they are actively engaged, working together, and given the opportunity to construct their own understanding. POGIL emphasizes that learning is an interactive process of thinking carefully, discussing ideas, refining understanding, practicing skills, reflecting on progress, and assessing performance. In a POGIL classroom or laboratory, students work on specially designed guided-inquiry materials in small self-managed groups. The instructor serves as facilitator of learning rather than as a source of information. The objective is to develop skills as well as mastery of discipline-specific content simultaneously. (Emphasis added)


OK, that does not sound good.


homeschool mom with common sense-y

I'll get to the professor’s various posts on POGIL as soon as I can.

I do want to read them.

But in the meantime, there's one homeschooling mom on the thread (son has LD) whose comments make sense to me:

Pat says clearly that your Socratic style doesn't work for [him]. Why do you then believe that it does work? You (rightly) want [him] to learn problem solving, but just because your method of teaching ... works for others doesn't mean it works for [him]. Maybe [he] needs repetition, repetition, repetition of the underlying content before [he's] ready for process. Other students may grasp the process after going through the underlying solutions three times, or six times, but maybe [he] needs thirty times.

You model the problem solving that you want [him] to do-- Where have you seen a problem like this? What rule did we use?-- but in a sense that's no better than modeling the actual rule for Problem 13. You want [him] to intuit that [he] is supposed to be asking [himself] those questions. But what if, as seems to be the case, [he] isn't intuiting that? Then [he's] not learning anything.



The bad news here is that, clearly, constructivists are giving lots of workshops to math professors.

Even worse, math professors are attending them.


inflexible knowledge, flexible knowledge, and expertise

One of the problems with constructivism (and, apparently, with POGILism) is that it tries to teach higher-order problem solving first, instead of second.

That option probably isn't on the menu.

According to Daniel Willingham, knowledge is always inflexible before it's flexible. You can't hopscotch over the inflexible stage by teaching process, or asking students to discover addition.

Problem solving and critical thinking seem to grow out of extensive practice of surface, shallow, inflexible knowledge.

I’d like to know more about how this happens.

At a minimum I’d like to know what cognitive psychologists (psych department cognitive psychologists, I mean) understand about the process at this point.


And I hope that Robert, who writes the brightMystery blog, will join us at KTM once in awhile to think about these things.

update

WelcomeRobertTalbert



SingaporeWordProblemSampler3 14 Jul 2005 - 03:20 CarolynJohnston


Note: solutions to the problems from SingaporeWordProblemSampler2 have been posted here.

So here's a whole new set of problems!

Primary 3: Margo has 3 times as many pears as apples. If she has 84 pears and apples altogether, how many pears does she have?

Primary 4: A cake was cut into 12 equal pieces. Jim ate two pieces and Tom ate four pieces. What fraction of the cake was left?

Primary 5: A bag of potatoes weighs 7/8 kg.. A bag of yams weighs 4/5 as much as the bag of potatoes. Find the total weight of the bag of potatoes and the bag of yams.

Primary 6: Eric has 75% as much money as Joshua. Carl has 60% as much money as Eric and Joshua have together. If Eric has 36 dollars less than Carl, how much money does Joshua have?

I don't know about you all, but I think I perceive these Singapore math problems becoming markedly harder at level 5.


ChristopherOnSingaporeMath 08 Jul 2005 - 00:02 CatherineJohnson


Christopher managed to bargain me down today.

Instead of doing:

  • Megawords 2, Worksheet 10-J
  • Saxon Math 8/7 Lesson 11 Mixed Practice
  • Saxon Math 8/7 Lesson 12 Warm Up
  • Saxon Math 8/7 Lesson 12 Lesson
  • Saxon Math 8/7 Lesson 12 Lesson Practice
  • Math Olympiads: 1 problem

he's doing:

  • Saxon Math 8/7 Lesson 12 Mental Math
  • Primary Mathematics 3A Workbook, problems 8, 9, & 10

So maybe he has a future as an agent.


He just looked up from his bar modeling and said, 'I like the problems in Singapore Math.'

I said, 'You do?'

'Yeah.'

'How come?'

'They're not stupid.'

No idea what that means.


update

Christopher got all 3 of his bar model problems right today. (ummm....no, he didn't. He flubbed the arithmetic on the first one, but he got the bar model almost exactly right.)

I checked his answers & models, and when we got to the 3rd problem, he said confidently, 'This one's a two-parter.'

I was happy to hear that.

I think this signals a new category inside his mind.

  • one-part problems
  • two-part problems

He can tell the difference!

what bar models do for your brain

I'm trying to figure out how to write about bar models and what I think they do for my 'math brain.'

It's incredibly difficult to articulate, and will involve printing out sample bar models, scanning them back into iPhoto, and reducing the image size...so it will be awhile.

But I'll get there.

For the time being, I'll say that I could do the 3-variable problem from Primary 6 that Carolyn posted using algebra.

But I couldn't do it using a bar model.

There's a reason for that, but I'm going to need visuals to express it.

OTOH, once I'd done the problem algebraically, I realized how to interpret the (correct) bar model I'd drawn--thanks to the Math Olympiads problems I did this weekend.

So today's hypothesis is that the perfect 'problem-solving' curriculum for me would be an amalgam of PRIMARY MATHEMATICS & MATH OLYMPIADS.

math-heads & word-heads

Carolyn has mentioned that mathematicians think facility with geometry may be a good indicator of mathematical talent.

I wouldn't be remotely surprised to find out that's true, if only because of the connection between spatial-visual ability & maths. (I've decided I like 'maths' better than 'math.' fyi)

I don't remember having trouble with any of the high school math I took. (Maths!) It may have been an easy curriculum, I don't know.

But I do remember having lots of fun with algebra. The X's and the Y's and all the neatly stacked-up linear equations....it all just felt right.

I could still solve a two-variable equation 30 years later, without even having to think about it.

This has made me wonder if there is something 'word-like' about standard algebra.

Temple, btw, absolutely could not learn algebra.

She's a brilliant person, but algebra was out.

'I couldn't make a mental picture of it,' she told me. 'It was too abstract.'

I have to remember to ask how she did with geometry the next time we talk.


PatternLearningPart2 08 Jul 2005 - 00:12 CarolynJohnston


My favorite book about Asperger's Syndrome is Helping a Child With Nonverbal Learning Disorder or Asperger's Syndrome, by Kathryn Stewart.

It was one of the few books I've ever encountered on this topic that really felt like its recommendations might apply to my son, even though I've never felt that either diagnosis really fit him very well. In this parenting business, though, you take good advice wherever you can get it.

Tonight I was looking for any advice it had to offer on teaching math, and I came across this tidbit in a section on pattern learning (Catherine and I have already written about pattern learning a bit).

A problem seen in both NLD and Asperger's students is their overreliance on learning patterns. This style of learning is often seen as a strength that the student relies upon for skill development. Teachers and parents have used this strength to help the child develop success in playing sports, memorizing facts, and learning the routine for the day.

Unfortunately, this strength brings problems when the child relies solely on the pattern without learning the concept or recognizing the overall point of an activity...

Many NLD and AS students experience difficulty with math, especially fractions. Well-meaning teachers often teach these children the pattern of converting fractions to decimals to make adding, subtracting, multiplying and dividing fractions easier.

My first reaction was: who the heck does this?

My second was: surely they don't think multiplying and dividing fractions is harder than multiplying and dividing decimals?

However, to continue:

This method may be useful in the short run: there is less stress, and the child gets the right answer. Yet they have no idea of what a fraction is; the concept still eludes them. When they get to algebra and formulas are presented in fraction format as part of equations, they don't know what to do.

In short, having learned a pattern for turning fractions into decimals does you little good if the problem you're faced with is:

1/(1+x) = 4/(3-x).

Normal kids pattern-learn too to some degree, especially in learning skills that should be automatic or nearly-automatic, like riding a bicycle or doing a fraction problem. Kids don't know what the big picture is, at first: all they see is the small bit that we are teaching them, and they trust us to lead them wisely. When we teach them fraction manipulation in 5th grade, they don't know they'll use it again, at a more abstract level, in algebra. We're letting them down if we teach them reliance on a method that only works sometimes, or doesn't generalize as fully as it ought to when it's time for them to do algebra.

I hope noone is really doing this. I hope Dr. Stewart made it up.

(To sum up, I didn't find much in Dr. Stewart's book that is specific to learning math, or to any other one subject. However, if you have a kid with NLD or similar problems, the general advice she gives on how to help a kid with AS or NLD be successful in school is the best I've encountered. This is a really terrific book.)


PatternLearning (format shock)
PatternTraining




StevenPinkerOnLearningMath 10 Jul 2005 - 14:46 CatherineJohnson


David Klein sent this excerpt from Steven Pinker's How The Mind Works.

(And, thanks to Carolyn's heroic Creation Of Many Topic Threads last night, I have been able to enter this post in the Cognitive Science category! After I'm done with this, I think I'll go enter it under educational research, too!)


HOW THE MIND WORKS

by Steven Pinker (Linguistics department, MIT)
W.W. Norton & Company, Copyright 1997
page 341

The...way to get to mathematical competence is similar to the way to get to Carnegie Hall: practice. Mathematical concepts come from snapping together old concepts in a useful new arrangement. But those old concepts are assemblies of still older concepts. Each subassembly hangs together by the mental rivets called chunking and automaticity: with copious practice, concepts adhere into larger concepts, and sequences of steps are compiled into a single step. Just as bicycles are assembled out of frames and wheels, not tubes and spokes, and recipes say how to make sauces, not how to grasp spoons and open jars, mathematics is learned by fitting together overlearned routines. Calculus teachers lament that students find the subject difficult not because derivatives and integrals are abstruse concepts--they're just rate and accumulation--but because you can't do calculus unless algebraic operations are second nature, and most students enter the course without having learned the algebra properly and need to concentrate every drop of mental energy on that. Mathematics is ruthlessly cumulative, all the way back to counting to ten.

Evolutionary psychology has implications for pedagogy which are particularly clear in the teaching of mathematics. American children are among the worst performers in the industrialized world on tests of mathematical achievement. They are not born dunces; the problem is that the educational establishment is ignorant of evolution. The ascendant philosophy of mathematical education in the United States is constructivism, a mixture of Piaget's psychology with counterculture and postmodernist ideology. Children must actively construct mathematical knowledge for themselves in a social enterprise driven by disagreements about the meanings of concepts. The teacher provides the materials and the social milieu but does not lecture or guide the discussion. Drill and practice, the routes to automaticity, are called "mechanistic" and seen as detrimental to understanding. As one pedagogue lucidly explained, "A zone of potential construction of a specific mathematical concept is determined by the modifications of the concept children might make in, or as a result of, interactive communications in the mathematical learning environment." The result, another declared, is that "it is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve."

As Geary points out, constructivism has merit when it comes to the intuitions of small numbers and simple arithmetic that arise naturally in all children. But it ignores the difference between our factory-installed equipment and the accessories that civilization bolts on afterward. Setting our mental modules to work on material they were not designed for is hard. Children do not spontaneously see a string of beads a elements in a set, or points on a line as numbers. If you give them a bunch of blocks and tell them to do something together, they will exercise their intuitive psychology for all they're worth, but not necessarily their intuitive sense of number. (The better curricula explicitly point out connections across ways of knowing. Children might be told to do every arithmetic problem three different ways: by counting, by drawing diagrams, and by moving segments along a number line.) And without practice that compiles a halting sequence of steps into a mental reflex, a learner will always be building mathematical structures out of the tiniest nuts and bolts, like the watchmaker who never made subassemblies and had to start from scratch every time he put down a watch to answer the phone.

Mathematics is deeply satisfying, but it is a reward for hard work that is not itself always pleasurable. Without the esteem for hard-won mathematical skills that is common in other cultures, the mastery is unlikely to blossom. Sadly, the same story is being played out in American reading instruction. In the dominant technique, called "whole language," the insight that language is a naturally developing human instinct has been garbled into the evolutionary improbable claim that reading is a naturally developing human instinct. Old-fashioned practice at connecting letters to sounds is replaced by immersion in a text-rich social environment, and the children don't learn to read. Without an understanding of what the mind was designed to do in the environment in which we evolved, the unnatural activity called formal education is unlikely to succeed.

pinker.100.jpg
Steven Pinker



see also:
TheLanguageOfNumbersIsNotLanguage
Children's Mathematical Development: Research and Practical Applications
DavidKleinAtAEI




WillinghamOnRavitch 12 Jul 2005 - 00:34 CatherineJohnson


I've just discovered a Daniel Willingham review of Diane Ravitch's Left Back: A Century of Battles over School Reform:

What makes this book so interesting is Ravitch's documentation that "Progressive" education has been progressing in the same direction for over 100 years. The same ideas are rediscovered again and again, and those seeking to reform American schools have been fighting the same bogeymen (drilling, teacher as "sage on the stage") with the same rhetoric (teach the student, not the subject) for just as long. The book is at its best in showing that these ideas have been recycled numerous times.



The long history of progressive education in this country tells me that we simply must take matters into our own hands.

The math wars aren't going to be won; at least, not by us.

The math wars will go on and on, and will always be new, like an episode of The Twilight Zone.

We have to teach our kids ourselves.

And we have to find, or invent, the resources that will help us do it.


CanChildrenMakeUpForLostTime 11 Jul 2005 - 18:06 CatherineJohnson


I'd like to put this question out to readers of ktm:

Can children make up for lost time?



I ask, because I've now read at least 5 personal stories of children or young adults struggling to make up ground they lost to bad curricula.

Some of the most hair-raising stories I heard from Carolyn were about college kids who simply could not learn algebra because they didn't get what they needed in grade school mathematics.

Carolyn made me wonder whether there might be a critical period for learning math the way there is for speaking a foreign language without an accent.

I've come to think there isn't, mainly because I find it possible (and pleasurable) to learn math as an adult, and I don't think I'm unique.


I started thinking about this because last night I did an impromptu interview with my cousin who, it turns out, pulled her daughter from public school because of a wretched experience with Everyday Math. (I'll post it shortly.)

Her daughter used Everyday Math for 3 years, from 2nd to 4th grade.

Then it took her 'several years' to make up the lost ground.

She just finished her freshman year in high school, and is doing great in high school math. (Her private school used Saxon.)


I talked to another woman who pulled her son out of the Tribeca schools because they use TERC.

He's now high school age and still doesn't have rapid fluency with his math facts. (She spent a lot of time working with flash cards, too. Another flash card failure.)

How can we remediate kids who've fallen behind because of constructivist math?


two immediate thoughts

To me, it seems like it has to be possible to make up lost ground more quickly than this.

At least, I hope so.

Here are my first thoughts:

  • remediation has to mean doing timed worksheets every day, day in and day out, until the child or young adult has his calculations down cold

  • remediation also means doing story problems every day, day in and day out (probably a coherent sequence of story problems, like those in the Singapore Math Challenging Word Problems books) [I have no idea how many story problems to do per day]

  • finally, remediation may mean that you need to back up to the beginning of math, or close to: back up to content well before the point where the child became lost--and move quickly through a coherent 1st, 2nd, or 3rd grade curriculum, regardless of the fact that the child or young adult already 'knows' most of the material


I'd love to hear people's thoughts.


WillinghamOnQuestionOfDifferentLearners 13 Jul 2005 - 20:31 CatherineJohnson


What a day!

First the crocheting mathematician, now a brand-new column from Daniel Willingham!

Do Visual, Auditory, and Kinesthetic Learners Need Visual, Auditory, and Kinesthetic Instruction?


short answer:

no


And see Willingham's deconstruction of Howard Gardner in Education Next.


And remember, Daniel Willingham, like our own Barry Garelick, is prominently featured in the ktm Pantheon!


WillinghamOnLearningModalities 22 Jul 2005 - 20:14 CarolynJohnston


From Daniel Willingham on learning modality theory, an explanation of why learning modality theory might make sense from a teacher's viewpoint:

There are two ways that a teacher might see what looks like evidence for modality theory in the classroom. First, a teacher who believes the theory may interpret ambiguous situations as support for the theory. For example, a teacher might verbally explain to a student - several times - the idea of borrowing in subtraction without success. Then the teacher draws a diagram that more explicitly represents that the 3 in the tens place really represents 30. Suddenly, the concept clicks for the student. The teacher thinks "Aha. He's a visual learner. Once I drew the diagram, he understood."

But the more likely explanation is that the diagram would have helped any student because it is a good way to represent a difficult concept. The teacher interprets the student's success in terms of modality theory because she has been told the theory is correct and because it seems to explain her experience.

Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality.

But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?

I ran into that situation recently, when teaching Ben how to do simple problems by adding and subtracting constants on both sides of an equation. Actually, trying to help Ben get the hang of this has taken quite a bit of effort this week, and I don't think it's a hard idea. I've got kinesthetic, visual, and auditory ways of teaching it, too. I could even sing it, though that's getting a bit ridiculous.

For the kinesthetic learner, you could get out a balancing scale or use Bornstein manipulatives. You could draw pictures of pan balances for a visual learner. You can explain verbally, as I did repeatedly, that what you're doing to solve the problem x + 4 = 13 is to 'undo the addition' of the 4 on the left hand side of the equation. If none of this works, what do you do then?

Try each modality over again, I suppose. Round 2: in case he was a kinesthetic learner, I had him copy each step I made in his own handwriting (laugh, if you will, but it works for me when I do it). In case he was visual, I drew pan balances again, next to the equivalent equation: no dice. "Subtracting the 4 is applying the inverse operation to get the x by itself," I said, auditory-like, but that didn't help either.

All this time, of course, he was able to do the problems by repeating the steps I made; he is a fabulous rote learner (is 'rote' a modality? If not, it should be). But I could tell he wasn't really getting the gist of it. Finally, in exasperation, I said, "Look, Ben, what's the opposite of adding 4?

"Subtracting 4."

"Good! And what's the opposite of subtracting 13?"

"Adding 13."

"Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"

And that's the sound I love to hear.

So, knowing Ben's best learning modality didn't help, and wouldn't have helped. I wish teaching, and learning, were so predictable that all you needed to do to teach a whole class reliably was to know what each kid's best learning style was. But I think that learning is inherently unpredictable. The trick is to be able to hit the teaching problem from a bunch of different angles, and you need to know lots of different ways to present the information. The more, the better (by the way, this is a major part of what Liping Ma's Chinese elementary math experts do with their release time; sit around together, thinking up new ways to teach problems to tough cases).

As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable). As a person on the autism spectrum, he's supposed to be a visual learner; this is accepted theory to such a degree that teachers will assume he needs to learn visually, but it's not always the right approach.

What Ben really is, is an unpredictable learner. You never know what's going to be easy, where he'll get stuck, and what will unstick him. He's the kind of kid who keeps a teacher on her toes.


MorganOnLearningModalities 18 Jul 2005 - 00:30 CarolynJohnston


This is a comment by Carolyn Morgan on the WillinghamOnLearningModalities thread: it's a beauty, so I'm posting it.

Things I note about her teaching approach:

  • It's actually multimodal! She's talking, she's getting the kid to write AND draw his own pictures, she's doing whatever it takes to get the kid on track.
  • It's feedback-driven: she switches gears in midstream if the kid isn't getting it.

Carolyn's post:

SteveH is correct. Whatever you do, you must bring all students to UNDERSTANDING. And this is what students really do want. "Understanding" or "not understanding" are the reasons students "hate" or "love" math.

A good math teacher learns how to "approach" a student having difficulty. The teacher has all of these ideas (hopefully) stored away back there some place, ready to be pulled out when needed. But a teacher's most important job is determining where the student's understanding fell apart, identifying where there might be gaps in reasoning, and knowing how to bridge those gaps. This is where choosing the right approach comes in. It might involve reteaching, reviewing a step that is being omitted, or helping a student reason through a difficult story problem.

So a hand goes up, and a student says, "I need help."

(Those are my favorite times of the math hour because it means I get to find the puzzle piece that is needed to make this all fit together in his/her head and give understanding to what I've just taught or to what's needed to solve this problem.)

So I have some choices, but I always look to see what the student has already done or tried. That tells me what to do next.

I then start by having the student read the problem to me (if it is a word problem).

Then I make a choice:

I might say, "OK, draw Bill's house. Now write 'B' on it for Bill. Now, draw the schoolhouse; now write 'S' on it. Now, draw the road from the house to the school. Now, look at the problem again to see how far it is to the school (and the student answers outloud 4 1/2 miles). OK, write that number on the map you've just drawn."

I could have drawn that little map for the student, and might do it under certain conditions, but having a student draw the map involves his sight and his movement (and mouth from speaking and ears from hearing his own voice) and it involves more importantly his BRAIN. (I've got to make sure his BRAIN is working and focused on the problem so he can "understand".)

So I would continue, "Start at Bill's house with your pencil; now walk to school. OK how far did Bill just walk? ('4 1/2 miles') OK, write that down. Now, he's at school, but he wants to come home, so have Bill walk back home. How far did Bill walk to get home? OK, write that down under the first number. How would you know how much he walked to school and back on that one day? ('add the two numbers together.') Good, do that. OK, but that is just one day. Now, let's read the problem one more time and let's see what the questiion was. (How far does Bill walk in a week going to school and back?) OK, now how could we figure that out?"

Many times the problem just works itself out in the student's brain as they begin to draw out a picture of the problem.

Or, if I've checked over his work and seen that he's added 4 1/2 miles 5 times for the 5 days of the week, I can see that he's overlooked a part of the question. So I have the student reread the question. Many times, the student will catch his own mistake when he hears his voice repeat the words "to school AND BACK". If not, I have him read just that part again.

Something really important: for some students it's just a matter of not knowing "how or where to get started". There are gaps in processing the information and gaps in understanding.

Not only does he need to know where to start, but he needs to know that where he is starting will get him going in the right direction and will help him get the right answer. This is very important to a student's confidence. If a student doesn't know "where to start" or isn't sure "if he's going about solving it properly", a teacher's trying to find the right modality isn't necessarily the answer.

This is where an "constructivist" approach is so devastating to the student. That student wants to be able to KNOW what to do to get the right answer.

It's terribly upsetting and deflating to a student not to know "where to start" and "if they're taking proper steps to solve the problem correctly." To leave this student to come up with his own idea isn't helping him.

Hopefully, though, a teacher will NOT just repeat the instructions that were given initially (if any were given). If a student didn't get it the first time, at least try a different approach.

One of my former students said of another teacher, "Why should I ask her for help? She always just repeats the same instructions that didn't make sense the first time." This student, smart as a cookie, just wanted to understand the entire process and to know how to work to get the right answer.




Carolyn Morgan On Conceptual Gaps
Congratulations Carolyn Morgan




CognitiveHoles 19 Jul 2005 - 16:27 CarolynJohnston


Bernie and I were talking tonight, and he told me a story that worried me a bit.

Ben came to visit us at work the other day, and wanted to get a snack from the vending machine. So he went into his dad's office and asked for some money. Bernie gave him a few coins, and Ben went into the snack room, picked out what he wanted, and put his money into the machine; but he didn't have enough. So he came in and asked for more; but he couldn't tell Bernie how much more he needed. He didn't seem to have much sense of how much more he needed, either.

Well, it wouldn't be the first time we came across this sort of gap in his understanding. We have a sort of a family byword for these things, very much like Catherine and Ed's no-common-sense-y; we call Ben's gaps his Cognitive Holes. They are located in unexpected places -- they're generally about something, like handling coins, that you think is very easy by comparison with other things he can do, like long division. And they tend to be very big gaping holes in his knowledge, and at first they were very frightening. But we come across them less often now than we used to, and we've found that once we know they are there, we can remediate them pretty quickly.

So I thought this was another run-of-the-mill Cognitive Hole.

Well, you tackle these by filling them in. Ben and I were ready for a change from what we've been doing lately, anyway (introductory equations, solved by adding and subtracting). We've been doing them all week, and struggling, and we finally got a 'click' a couple of nights ago (those babies are practically audible, aren't they?), and last night when he took his section test he got a 100. So tonight, when it was math time, instead of doing algebra, I got out some coins.

I had 3 quarters, and a dime. "OK, you're at our work, and you want a snack, and these are the coins I have", I told him. "The snack you want costs 60 cents. Which coins do you take?"

He went for two of the three quarters, and the dime. Good. "How much do I get back from the machine?" I asked. Nothing: good.

"OK, your snack costs 40 cents". He goes for the two quarters: he tells me the machine returns a dime.

"The snack costs 80 cents." He takes all the coins, and tells me the machine returns 5 cents.

In short, he passed my common sense test with flying colors, and Math Time was fun and a breeze for once. So what the heck was happening the other day? In short, what part of this Cognitive Hole we think we've uncovered am I not mapping correctly?

Tomorrow, we try it a little differently; we'll simulate the precise problem we had the other day with the snack machine at work. I'll give him too little money, tell him the snack costs a certain amount, and get him to tell me how much more he needs.

There may in fact be no Cognitive Hole, this time, just some situational rigidity. This is the deal with smart people on the autism spectrum; sometimes they know what they need to know, they just stiffen up when it comes time to apply it in the real world.

physical_check.jpg


NewStudyOnManipulativesPart2 28 Jul 2005 - 20:15 CatherineJohnson


I'm reading the Scientific American article about manipulatives & symbolic representation now:

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.

The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.



here's the anti-constructivist moment:

[The] capacity [to] create and manipulate a wide variety of symbolic representations .... enables us to transmit information from one generation to another, making culture possible, and to learn vast amounts without having direct experience--we all know about dinosaurs despite never having met one. Because of the fundamental role of symbolization in almost everything we do, perhaps no aspect of human development is more important than becoming symbol-minded.


symbols aren't 'natural'

The first type of symbolic object infants and young children master is pictures. No symbols seem simpler to adults, but my colleagues and I have discovered that infants initially find pictures perplexing. The problem stems from the duality inherent in all symbolic objects: they are real in and of themselves and, at the same time, representations of something else. To understand them, the viewer must achieve dual representation: he or she must mentally represent the object as well as the relation between it and what it stands for.

A few years ago I became intrigued by anecdotes suggesting that infants do not appreciate the dual nature of pictures.

[snip]

.... the Beng babies, who had almost certainly never seen a picture before, manually explored the depicted objects just as the American babies had.

The confusion seems to be conceptual, not perceptual. Infants can perfectly well perceive the difference between objects and pictures. Given a choice between the two, infants choose the real thing. But they do not yet fully understand what pictures are and how they differ from the things depicted (the "referents") and so they explore: some actually lean over and put their lips on the nipple in a photograph of a bottle, for instance. They only do so, however, when the depicted object is highly similar to the object it represents, as in color photographs....

[snip]

it takes several years for the nature of pictures to be completely understood. John H. Flavell of Stanford University and his colleagues have found, for example, that until the age of four, many children think that turning a picture of a bowl of popcorn upside down will result in the depicted popcorn falling out of the bowl.



Andrew makes Barney fly

A couple of weeks ago Andrew (10, autistic, nonverbal) brought me Christopher's yellow plastic airplane, on top of which he'd mounted one of his Barney's, and handed the whole big package to me with an urgent look on his face. He was on a mission.

Martine came in and said, 'He wants you to make Barney fly.' She'd been sitting in the family room when Andrew had put his Barney on top of the plane, and then flung plane & Barney up into the air, apparently thinking Barney would fly around the room.

Andrew had been very unhappy with the outcome, and was now appealing to me. Clearly he believed that making Barney fly was one of those things, like operating the TIVO, only adults know how to do.

I was flattered, but also dumbfounded. What goes on inside this child's head? was my exact thought.

I was thinking....does he not understand gravity?

Does he not understand toys?

What's with this kid???!!

The Scientific American article makes me think that Andrew, although he can read, hasn't completely figured out the dual nature of symbolic representation.

He probably couldn't understand the plastic airplane as being TWO THINGS:

  • an airplane

    AND

  • a symbolic representation of an airplane

What I'd like to know is: what does he think about Barney?

is this a shoe?

Here's a little guy trying to put his foot inside a picture of a shoe.

000ACE3F-007E-12DC-807E83414B7F0000_1.jpg

lost in translation

I constantly have the experience of reading constructivist texts, noticing that the ideas they're advocating are good ones or at least not obviously bad ones.....and then, five seconds later, finding that they've taken a sound idea and just completely gummed it up in the application.

Assuming this work on manipulatives & symbolic representation is correct, the constructivist obsession with manipulatives looks to be another instance of a good idea lost in translation. Constructivism is majorly obsessed with manipulatives, that's for sure. I understand that the TERC curriculum is basically just a huge box of manipulatives, with no textbook or 'consumables'--workbooks--at all.

Following in Piaget's footsteps, constructivists believe children don't reach the stage of 'formal operations' until age 11; from 7 to 11 they're in the Period of Concrete Operations. (Often you'll see the word 'developmental' used to designate constructivist curricula. Apparently that's a reference to Piaget.)

Wayne Wickelgren says this is nonsense; children can handle abstract concepts long before age 11. But constructivists are the people time forgot, and they're still basing their pedagogy on work done in the 1950s.

That's bad enough in itself, seeing as how the field of cognitive science was just getting started around that time, and Piaget's work hasn't fared so well over the past 60 years.

But the more glaring misstep, it appears, is that they failed to grasp the nature of the concrete.

The reason constructivists think children should spend their grade school years working with manipulatives is that manipulatives are concrete. But they're not. Manipulatives are symbolic objects that require the child to have mastered the concept of dual representation.

Skinnies and bits are not concrete. They are symbolic representations of the Hindu-Arabic numeral system. Worse yet, they are more intellectually demanding, and hence more confusing, symbolic representations than pencil marks on paper.

They're harder to understand, not easier.

Lost in translation.


question

I hope I'll get a chance to talk to these researchers at some point.

My question is: why should pencil and paper be less challenging than manipulatives?

I can see why pencil and paper wouldn't be any more challenging than manipulatives, but why should pencil and paper be easier? Do pencil marks somehow not involve dual representation? That's what the authors seem to imply, but they don't say so directly.


CA state study on manipulatives
Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
NewStudyOnManipulatives
New Study on Manipulatives Part 2




JDFisherOnTextbookFragmentation 01 Aug 2005 - 16:33 CatherineJohnson


J.D. Fisher of MathandText left a comment today that reminded me I'd wanted to point people to his post on textbook fragmentation, which is a HUGE, documented factor in bad math ed here in the U.S.

One reason publishers maintain a great deal of fragmentation in elementary basal mathematics texts, for instance, is that such a structure allows adoption committees and other, similar decision-making bodies, to quickly judge, with great confidence, that a text has indeed covered all of the requisite state standards.

But this structure also has the effect of 'un-prioritizing' content. Simple ideas and less relevant topics are given the same priority and the same space as more robust, more relevant topics.



And check out his excerpt of a 2005 math textbook's TOC:

1 Place Value Through Hundred Thousands
2 Place Value and Exponents
3 Place Value Through Hundred Billions
4 Compare, Order, and Round Whole Numbers
5 Place Value Through Thousandths
6 Problem-Solving Strategy: Find a Pattern
7 Compare, Order, and Round Decimals

The blue lessons (with the possible exception of Lesson 2) represent the exact same concept applied to larger and larger--and then much smaller--numbers. The red lessons are also closely related, but are separated by two somewhat unrelated lessons.



writing is organizing

People tell you writing is rewriting, which is true, but the main reason for all the rewriting is that what writing really is, is organizing. Ed had this insight today when I read him a line from a terrific critique of constructivism by two cognitive scientists, and it was a Brand New Thought for both of us. More on this later.

In the meantime, I can tell you that I've had a visceral understanding of just how dangerous unprioritized content is ever since I listened to Temple's stories about what happens to animals in a meatpacking plant once the employees have lost sight of the difference between the big stuff and the small stuff.

More on that later, too.


PowerPoint makes you dumb

(although, in the case of dimensional analysis, I am going to be relying on PowerPoint to make me smart)

I have zero time at this moment (or possibly ever) to read Edward Tufte's discussion of Boeing's PowerPoint presentation on the space shuttle Columbia, but I'm hoping maybe J.D. will take a look and fill us in. As I understand it, Tufte argues that PowerPoint's built-in bulleting structure equalized or even 'unprioritized' the 'possible tile damage.' That's my impression.

Whether or not I've got the jist, I can easily imagine a poorly structured, unprioritizing report resulting in catastophic failure. Easily.

Getting back to children and math, a severely fragmented textbook is going to be at a bare minimum a catastrophic obstacle to learning.

Of that, I'm sure.


update

I've just tracked down Edwart Tufte's long essay, The Cognitive Style of PowerPoint.

And a blog called The Talent Show has a lengthy excerpt from the TIMES article on PowerPoint's role in the Columbia disaster that's worth quoting in full here, too:

In August, the Columbia Accident Investigation Board at NASA released Volume 1 of its report on why the space shuttle crashed. As expected, the ship's foam insulation was the main cause of the disaster. But the board also fingered another unusual culprit: PowerPoint, Microsoft's well-known ''slideware'' program.

NASA, the board argued, had become too reliant on presenting complex information via PowerPoint, instead of by means of traditional ink-and-paper technical reports. When NASA engineers assessed possible wing damage during the mission, they presented the findings in a confusing PowerPoint slide -- so crammed with nested bullet points and irregular short forms that it was nearly impossible to untangle. ''It is easy to understand how a senior manager might read this PowerPoint slide and not realize that it addresses a life-threatening situation,'' the board sternly noted.

PowerPoint is the world's most popular tool for presenting information. There are 400 million copies in circulation, and almost no corporate decision takes place without it. But what if PowerPoint is actually making us stupider?

This year, Edward Tufte -- the famous theorist of information presentation -- made precisely that argument in a blistering screed called The Cognitive Style of PowerPoint. In his slim 28-page pamphlet, Tufte claimed that Microsoft's ubiquitous software forces people to mutilate data beyond comprehension. For example, the low resolution of a PowerPoint slide means that it usually contains only about 40 words, or barely eight seconds of reading. PowerPoint also encourages users to rely on bulleted lists, a ''faux analytical'' technique, Tufte wrote, that dodges the speaker's responsibility to tie his information together. And perhaps worst of all is how PowerPoint renders charts. Charts in newspapers like The Wall Street Journal contain up to 120 elements on average, allowing readers to compare large groupings of data. But, as Tufte found, PowerPoint users typically produce charts with only 12 elements. Ultimately, Tufte concluded, PowerPoint is infused with ''an attitude of commercialism that turns everything into a sales pitch.''

(btw, these are the same problems we face writing for the web....


update 2

I'm pulling J.D.'s comment up front:

Mr. Tufte butters his bread by analyzing, among other things, the contexts under which information is presented. He is likely correct in his critique of Powerpoint as a tool for information sharing.

For anyone, even the TIMES, to suggest or insinuate that this technology was responsible for seven deaths is, I think, irresponsible.

I agree, and I certainly don't want to be seen to be blaming PowerPoint for 7 deaths.

I'll also add that the 'PowerPoint makes you stupid' heading is a joke! (That's the heading used by the blog I mentioned.) I don't remotely feel that bulleted points make people stupid, and as a matter of fact I do feel that bulleted points frequently make people much more clear.

It would be extremely difficult to make sense on the web without them.


screenplays are structure, fyi

Back when I first started out, I thought writing was good sentences if you were writing nonfiction, and good dialogue if you were writing fiction.

Wrong.

SCREENPLAYS ARE STRUCTURE," shouts William Goldman in Adventures in the Screen Trade. "The essential opening labor a screenwriter must execute is, of course, deciding what the proper structure should be for the particular screenplay you are writing."

This, he believes, is "the single most important lesson to be learned about writing for films... Yes, nifty dialog helps one hell of a lot; sure, it's nice if you can bring your characters to life. But you can have terrific characters spouting just swell talk to each other, and if the structure is unsound, forget it."

Real Craft

He's right.

And, on the same page, here's Syd Field:

In The Screenwriter's Workbook, Syd Field seconds Goldman. "Structure is the most important element in the screenplay. It is the force that holds everything together; it is the skeleton, the spine, the foundation."

People look at Saxon Math and think it's prosaic, obvious, behaviorist.

But what's brilliant about Saxon is mostly invisible.

It's the structure.


RoadsideFragmentation 31 Jul 2005 - 13:27 CatherineJohnson


Check out the cool image J.D. found to illustration fragmentation:


0_my_photographs_italy_-_tuscany_road_signs_1nq21_large.jpg


I feel a traffic mishap coming on just looking at this.


CognitiveScienceTreasureTrove 01 Aug 2005 - 14:16 CatherineJohnson




What a find!

10 classic articles from the journal Cognitive Science.

I'm starting with number 1:


1. Johnson-Laird, P. N. (1980). Mental models in cognitive science. Cognitive Science, 4, 71-115.

This article postulates that mental models differ from visual images and from propositional representations, and it presents evidence that corroborates the differences. It argues that reasoners use propositional representations of, say, spatial descriptions to construct mental models. It also argues that mental models rather than formal logic underlie syllogistic inference, e.g., some of the parents are drivers, all of the drivers are scientists, therefore, some of the parents are scientists. The article was the first in a journal to present a case for mental models as the end result of comprehension and as the starting point of deductive reasoning. This idea led to many subsequent investigations (see the mental models Website).




I've been struggling with the question of:

What is conceptual understanding of mathematics, anyway?

For some reason, I've gotten the sense that conceptual understanding = visual representation. This notion has tripped me up, because I'm pretty much maxed out on visual models of mathematical knowledge at this point. It's easy to understand addition and subtraction by looking at a visual model, but I don't readily grasp the one visual model I've seen of the multiplication of fractions, and I can't even imagine a visual model for multiplying-by-the-reciprocal (although Dan K has left a possibility in a Comments thread).

fyi, I was just talking to Carolyn about this, and it turns out she doesn't have a visual representation inside her head of dividing a fraction by multiplying by the reciprocal.

I can't tell you how liberating that is!

If Carolyn isn't walking around with a multiply-by-the-reciprocal picture inside her head, I certainly am not going to devote one second more trying to come up with one myself. Forget it!

Anyway, I've been feeling that it's time for me to 'move on,' and that in fact I am moving on. But I haven't trusted the feeling, because I'm stuck on, & stumped by, visuals.

Reading this abstract, I had a moment of recognition. Mental representations sound like what I've been reaching for, and like what I'm starting to develop. (And the new 'knowledge' I'm developing is not at all like a 'proposition' from which I can 'deduce' follow-up truths & principles. It's much more.....'holistic' than that?? Is that the word? I don't know, but early on Carolyn told me that her knowledge of math is a 'seamless whole,' and I'm starting to have a bit of that feeling. Some kind of 'knowing' seems to be taking form inside my mind that isn't a picture and isn't a logical proposition, either.)

So I'm looking forward to reading this article. I suspect it's going to give me confidence that whatever it is that's taking shape inside my head is real; that I'm making progress, not just stalling out in 5th grade math.


BernieOnVisualVersusSymbolicUnderstanding 03 Aug 2005 - 00:40 CatherineJohnson


Bernie Johnston (you may recognize the last name) left this comment in response to my post about visual versus symbolic understanding of math:

Not all mathematical understanding is visual. If it were, all mathematics would be geometry. That's essentially the view the ancient Greeks had, but the development of symbolic computation and an efficient number system have rendered that view obsolete by allowing us to approach mathematical understanding from other quite different directions.

Various mathematicians think in various ways. I've known strong algebraists who literally cannot draw a 3-dimensional cube and understand it. On the other hand, I've known topologists who could see 4-dimensional space in their mind's eye the way I can see a cube.

I prefer to think of different mathematical approaches as analogous to taking pictures of a house. You can take a picture from the front or from the sides or from the back, but none of these ipso facto captures the entire house. And even after viewing the pictures you might want to do a walk-through before buying.

So, while I don't know the definitive answer to the question of exactly what conceptual mathematics is, I'm certain that it need not be a visual representation alone.




I was relieved to learn this.

I think one problem for me, in trying to learn & re-learn math, is that I really have no idea what it looks like when a person has learned math.

All I really know is....they're good at math, majored in math, took graduate degrees in math, and now do or do not work in jobs requiring all kinds of math I don't understand. Makes it difficult to have any sense at all of where I'm headed, and how far I've come.


084930301X.01._AA240_SCLZZZZZZZ_.jpg

It's in my cart!


GiftedAndTalented 18 Jan 2006 - 14:21 CatherineJohnson


The Sunday TIMES had an Education Life section with a number of good articles that will be available online free for 7 days.

One was this interview with James T. Webb, who has a new book out called Misdiagnosis And Dual Diagnoses Of Gifted Children And Adults: Adhd, Bipolar, Ocd, Asperger's, Depression, And Other Disorders:

Q. Parents throw the word "gifted" around. What does it mean, really?

A. Gifted comes in different forms and degrees. Gifted children excel in such areas as general intellectual ability, specific aptitudes like math, creative thinking, visual or performing arts. Most have I.Q. scores between 130 and 155. Above that range are the profoundly gifted - a tiny fraction of the group. Over all, the gifted represent about 3 percent of our population.


Q. Why would gifted children be tagged as having psychological disorders?

A. Behaviors of many gifted children can resemble those of, say, attention deficit/hyperactivity disorder. Most teachers, pediatricians and psychologists aren't trained to distinguish between the two. Most gifted kids are very intense, pursuing interests excessively. This often leads to power struggles, perfectionism, impatience, fierce emotions and trouble with peers. Many gifted kids have varied interests, skipping from one to the other - a trait often misinterpreted as A.D.H.D.


I found the next section, on bipolar diagnoses & giftedness, especially intriguing. The bipolar diagnosis seems to be one of those suddenly soaring categories. I'm reading & hearing that there are teenagers all over the country being defined as bipolar and given Depakote.

Another trend has been psychiatrists prescribing Depakote to bipolar-ish patients taking antidepressants. It's become conventional wisdom that antidepressants can trigger manic episodes in susceptible patients (this would be anyone showing the slightest signs of hyperactvity along with mild depression), so psychiatrists are prescribing preventive Depakote.

I have an opinion on this practice. I'm against it.

I'm against it partly because I am the recipient of glossy brochures for psychopharmacology conferences at which at least one panel on the subject of 'Prophylactic use of valproic acid as an adjunct to antidepressant medication' or some such will be listed as 'sponsored by' Abbott Laboratories, which gets my goat. I'm not remotely anti-big Pharma; I pretty much owe my kids' lives to Big Pharma. So to the pharmaceutical industry I say: Live long and prosper.

But sponsored panels on the miracle of prophylactic valproic acid get my goat anyway. Especially since they seem to have been such a blinding success.

I spent a lot of time writing and thinking about bipolar disorder when I was working on Shadow Syndromes with John Ratey. Bipolar disorder is connected to creativity (Kay Jamison's work is probably still the definitive source on this), as well as to high socioeconomic standing and other good things. As I recall, it's the one severe mental illness in which you see families of the afflicted person move up the ladder instead of down. (I haven't taken the time to fact-check this, but I think I've got it right.)

Q. You write that these misdiagnoses are common.

A. About a quarter of gifted children have their giftedness misinterpreted as a disorder and aren't recognized as gifted. Even when flagged as gifted, another 20 percent are misdiagnosed. Among children referred to me with a bipolar diagnosis, almost 100 percent have been misdiagnosed, as are 70 percent of those with obsessive-compulsive diagnoses and 55 percent of those with A.D.H.D.


I've spent a lot of time here in middle age being defined as bipolar-ish and/or ADHD-ish, which serves me right, seeing as how SHADOW SYNDROMES was my idea in the first place.

Since writing the book I've more or less assumed that I am bipolarish or ADHDish or something in there, that that's where my creativity came from.

But these terms never quite fit, and from time to time I'll have a flash of, 'This is just the way I am.'

James Webb is the first person I've heard say that a person like me might actually be a person like me, not just a milder variant of a whole different kind of person, if that makes sense.

I'm going to have to sit with this one for awhile.

But before I do that, I'm going to finish reading this book:

0743243447.01._AA240_SCLZZZZZZZ_.jpg

The funny thing about Gartner's book, of course, is that while I'm pretty hypomanic, I'm not remotely hypomanic the way Gartner's Wall Street Geniuses are hypomanic.

So basically, if I'd learned lots more math when I was 20, and been just a little bit crazier......I'd be rich!

OK, time to settle down.

Here's Webb:

0910707642.01._AA240_SCLZZZZZZZ_.jpg


MetacognitionAndMath 03 Aug 2005 - 02:44 CatherineJohnson


Steve H raised a question about overconfidence the other day (in this case, college kids assuming they know stuff they don't), which coincided with my having discovered the concept of metacognition in math ed.

I haven't had time to write anything about it yet, but metacognition is a hot topic in radical constructivism and non-radical constructivism, and it's a terrifically useful concept for me, too.

Metacognition at its simplist means knowing what you don't know.

That's probably not how most researchers would define it, but it's how I define it, and how I have defined it for a number of years now. I think the ability to know what I don't know is one of my most important skills as a journalist.

Math has me stumped. I have just about zero metacognition when it comes to math. I don't know what I don't know; I barely know what I do know.

No, that's not it; I do know what I know, I just don't know if any of it's right. In other words, I may know what I know, but should I know it?

Or should I forget it right this minute, because it's stupid, illogical, and wrong?

I don't know.

Here's Bernie Johnston:

I don't think you can ever "learn math". There's just way way too much of it. As a matter of fact, I'm certain even a professional mathematician can't learn the names of all the things that are being produced today within mathematics, let alone understand them.

The real question is: when do you know you've learned a piece of mathematics? For example, when do you know you've understood dividing fractions? As a matter of fact, you may never have "completely" understood it. That's because a successive generation may come along with a new idea which sees dividing fractions from a completely different point of view.

[For example, although the ancient Pythagoreans knew that the square root of 2 is irrational, and suspected that pi was irrational, they had no way of knowing that the former is algebraic (which means "not too irrational"), while the latter is not. They couldn't even have formulated the terms.]

I guess all that we can really aim for then is having the experience of understanding a single mathematical idea using a single point of view. At some point the additional insight gained from seeking different points of view is no longer worth the additional effort.

Mathematics is a journey, not a destination. (What can I say? I had to say it. I'm a child of the Sixties.) I guess the only sensible guideposts we have are reports from those who have already travelled down that particular path.



update: the poetry of Donald Rumsfeld

OK, first of all, as I mentioned earlier, this is a nonpartisan site.

So, yes, I see that I have typed the words DONALD RUMSFELD here in my editing window, but this is NOT to be construed as an invitation to speak of GEORGE BUSH, the IRAQ WAR, or the GLOBAL WAR ON TERROR aka the GLOBAL STRUGGLE AGAINST VIOLENT EXTREMISM.

No.

We are speaking of METACOGNITION. Still. I've typed the words Donald Rumsfeld into my edit window because it just so happens that Donald Rumsfeld is the author of my favorite poem about metacognition.


Rumsfeld.gif


source: The Poetry of D. H. Rumsfeld: Recent Works by the Secretary of Defense



InterestingDiscussionOfLearningStyles 09 Aug 2005 - 03:35 CarolynJohnston


There's been an interesting discussion of learning style differences going on over at Tall Dark and Mysterious. Everyone seems to agree that, all too often, learning styles are used as an excuse for a student's failure.


AgainstTexbooks 19 Aug 2005 - 02:00 CatherineJohnson


I'll be interested to hear what J.D. (Math and Text) has to say about this.

I was thinking about my reaction to the new Saxon Math web site by Harcourt Achieve, which bought Saxon Math a couple of years ago.

I was very upset by the web site, and I found my reaction curious. I felt like I was overreacting (something I rarely do, by the way.)

Suddenly it came to me. I've mentioned my loathing--not too strong a word--for the look and feel of American textbooks more than once. Normally I explain the problem as a gratuitously increased cognitive load on the student, because it takes mental energy to filter out distracting stimuli. When you try to learn math from a wildly over-illustrated textbook, you have to devote mental resources both to seeing the math and to not seeing all the other crud strewn all over the page.

I'm sure my increased cognitive load theory is true, as far as it goes. I've also mentioned that my sister, when she taught elementary school math, would actually Xerox the relevant pages from the textbook, cut out all the gratuitous illustrations, glue what was left onto a white piece of paper, and then re-Xerox. She gave her kids the clean, crisp, black-and-white pages. That way they had a fighting chance of actually learning some math.

Gratuitously increased cognitive load is a HUGE problem in American textbooks.

After I saw the stock photographs of Beaming Learners on Saxon-Harcourt Achieve, I realized there's another problem, too.

It's the Cable TV problem, the experience of being stuck watching one single, low-budget ad that's repeated over and over and over and over again on every commercial break. Trapped.

That's what American textbooks are like, for me. They are all exactly the same, and they are all awful to look at. Every publisher buys the same stock photos of the same stock learners and plasters them all over the same stock page layouts of the same stock books.

This week my neighbor brought over the English language arts book Christopher will be using two weeks from now, and if it didn't have lots of words & no numbers on the pages I wouldn't be able to tell it apart from the math book.

Looking at these books, it's hard for me to see any content at all. It all looks like the same content, the same book over and over.

What was that David Byrne line?

Same as it ever was?

TRAILBLAZERS has terrific design

I forgot to mention: TRAILBLAZERS looks completely different from all other textbooks.

No photographs; all original art work. Quite a lot of white space on the pages.

Plus the font selection is interesting & fun.

TRAILBLAZERS is the only contemporary textbook--mainstream textbook--I've ever seen that's inviting.

update

I can't find images of the TRAILBLAZER 'Student Guide' online, but I did find quite a few copies of the various Implementation Guides, Teacher Guides, TIMS Tutors, etc.

This one is very nicely laid out:

Math Facts (pdf file)


TallDarkAndMysteriousThread 27 Aug 2005 - 10:58 CatherineJohnson




Interesting thread at Tall Dark & Mysterious sparked by Daniel Willingham's article on different learning styles.

One of the commenters there, Meep, has this to say:

I think it’s best, the younger the student and the more important the concept, to teach the content in every possible relevant way….it’s also a matter of students realizing the different ways to relate to a given subject. When there are more handles on a particular subject, you’re more likely to remember it.

Teaching the same content in more than one way has become a guiding principle for me, whether I'm teaching Christopher or me.

TD&M replies by pointing out that students hate being taught the same material more than one way:

My weakest students hate this. Hate, hate, hate. They want to know the ONE TRUE WAY to understand and approach a problem, and they get very very cranky when I don’t oblige them. (One true way => one-size-fits-all formulas, whereas different approaches => need to think about problems.) I’m remembering one student who asked me about intercepts, and I sketched a graph and explained about setting variables to zero. “So which one is it?” she asked when I was finished.


She's right. Kids hate this.

But when you're talking about young kids, at least, all kids hate it, no matter what their skills. I myself find it quite uncomfortable, once I've understood and solved a problem, to go back and start all over again trying to solve the problem a whole different way--or to work through and understand a different solution offered by the book.

My neighbor and I spent some time trying to figure out why this should be so. Why is doing-it-a-different-way so unpleasant? I think this passage from W.W. Sawyer's Prelude to Mathematics explains it:

patternSawyer.gif

I think it's unnatural to throw out the pattern you've just discovered & go off to find a whole new one. It goes against the grain.

I think that's the source of the aversion children, 'weak students,' and adult students like me feel to doing it.

it's not the same thing for us

Here's my other thought.

Around the blogosphere I see an awful lot of complaining about weak students and lazy students and recalcitrant students and students who think their learning style matters.

I'm sympathetic, to a point.

That point is here, asking myself why a 'weak student' would not enjoy having his teacher tell him to 'do it another way.'

I strongly suspect that the practice of doing a problem 'another way'--or simply perceiving that two or more different ways of doing a problem exist--is a different thing for the expert than it is for the novice.

I say this because of my own experience. Slowly, I'm turning myself into what is called, I believe, a talented novice. (I'll check the phrase.)

A talented novice is neither fish nor fowl, neither expert nor beginner.

For that very reason, a talented novice can be a terrific teacher.

As I move into talented novice territory, I find that 'doing a problem more than one way' is becoming a whole different thing. It's starting to be fun. It means making connections--extending the pattern--rather than throwing the pattern out and starting again.

back to school with Steven Pinker

Learning math is hard. Children do not spontaneously see a string of beads as elements in a set, or points on a line as numbers. If you give them a bunch of blocks and tell them to do something together, they will exercise their intuitive psychology for all they're worth, but not necessarily their intuitive sense of number. (The better curricula explicitly point out connections across ways of knowing. Children might be told to do every arithmetic problem three different ways: by counting, by drawing diagrams, and by moving segments along a number line.) And without practice that compiles a halting sequence of steps into a mental reflex, a learner will always be building mathematical structures out of the tiniest nuts and bolts, like the watchmaker who never made subassemblies and had to start from scratch every time he put down a watch to answer the phone.


This is my answer to asking a child to do the same problem more than one way.

Yes, he's going to have to do a problem he's already solved all over again, solving it a different way.

BUT I'm asking him to use the same methods he used yesterday & the day before. It's not chaos; there's a predictable pattern to the two or three different ways he's going to solve the problem.

I'm trying to give him a stable structure he can hold onto while he does (and I do) the hard work of learning math.


LipingMa 24 Aug 2005 - 20:09 CatherineJohnson


Here's Liping Ma:

Multiple Approaches to a Computational Procedure: Flexibility Rooted in Conceptual Understanding

Although proofs and explanations should be rigorous, mathematics is not rigid.... Dowker (1992)asked 44 professional mathematicians to estimate mentally the results of products and quotients of 10 multiplication and division problems involving whole numbers and decimals. The most striking result of her investigation "was the number and variety of specific estimation strategies used by the mathematicians." "The mathematicians tended to use strategies involving the understanding of arithmetical properties and relationships" and "rarely the strategy of 'Proceeding algorithmically.'"

"To solve a problem in multiple ways" is also an attitude of Chinese teachers. For all four topics, they discussed alternative as well as standard approaches. For the topic of subtraction, they described at least three ways of regrouping, including the regrouping of subtrahends. [they also talked about which approach worked best in which situation] For the topic of multidigit multiplication, they mentioned at least two explanations of the algorithm. One teacher showed six ways of lining up the partial products. For the division with fractions topic the Chinese teachers demonstrated at least four ways to prove the standard algorithm and three alternative methods of computation.

For all the arithmetic topics, the Chinese teachers indicated that although a standard algorithm may be used in all cases, it may not be the best method for every case. Applying an algorithm flexibly allows one to get the best solution for a given case. For example, the Chinese teachers pointed out that there are several ways to compute 1 3/4 dividedby 1/2. Using decimals, the distributive law, or other mathematical ideas, all the alternatives were faster and easier than the standard algorithm. Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations--the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics.


SchozLearning 25 Aug 2005 - 16:31 CarolynJohnston


Instructivist cracks me up.

Today he's found that a new learning modality has been added to the expanding list of learning styles: olfactory.

He makes fun of it, of course, but he's too quick to belittle the idea. I think there's something to it; in fact, I think I am an olfactory learner myself. As such, I can see real possibilities for really getting through to kids who are olfactory learners. They are even in line with letting the content determine the modality. Here are some ideas:

1. Science classes in which olfactory learners learn about putrefaction by getting a solid whiff of an uncorked test tube of a bit of rotting road kill.

2. Aiding rote memorization tasks, such as geography drill, by associating each country with a special smell. Examples: Italy - tomato sauce, Switzerland - chocolate, Romania- the way your grandma who uses Oil of Olay smells, and so forth. Think about it. Everyone knows smells are deeply linked with memory; we could potentially ensure that every kid who ever smells watermelon from now on involuntarily remembers all the states that the Mississippi River passes through on its way to the Gulf of Mexico.

3. Health class experiments in which half the class abstains from bathing for a couple of weeks (the others continue bathing as experimental controls).

4. 'Olfactory writing' assignments (with report titles such as "An imagined olfactory tour through the Amazon rainforest" -- a theme which has a nice tie-in with the Naturalist Intelligence).

5. Scratch-n-sniff history textbooks.

6. In a purely behavioral vein -- waking up inattentive olfactory learners with sudden blasts of smelling salts. This would have worked for me, I feel sure.

I am sure there are lots of other possibilities. Please add your ideas.


WillinghamOnOverlearning 26 Aug 2005 - 19:51 CatherineJohnson


Thinking about the girl who learned 1 and a half year's math over the summer, I looked up Willingham's article, Practice Makes Perfect But Only If Your Practice Beyond the Point of Perfection.

If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.



learning a year of math in 2 months
overlearning
Terminator




CalculusInHighSchool 23 Jan 2006 - 16:36 CatherineJohnson


From an interesting page by this title at Math Forum:

What causes students to experience difficulty once they are taking calculus?

  • Many aspects of algebra and pre-calculus are self-contained. A student's success will not necessarily depend upon his or her proficiency with material in a previous chapter. Calculus, on the other hand, is entirely new material that builds upon itself continuously.

  • Calculus students cannot always assimilate the material quickly enough.

  • Calculus students can fall behind and find it difficult to catch up.

  • Some students have difficulty with the large number of word/application problems.

  • Some students do not possess the cognitive flexibility to switch strategies that can be required to solve specific calculus problems.

  • Some calculus students have difficulty with active working memory and consequently with manipulating all aspects of a problem without getting lost.

  • Some calculus students do not remember all of the necessary material, especially formulas, from algebra and pre-calculus.

  • Calculus students are challenged to learn inverse operations in close succession; consequently, they can confuse one type of a problem with its exact opposite.


This list is especially interesting in terms of what cognitive science tells us about learning & remembering.

I mentioned that I need a book that tells me exactly what formulas, concepts, math facts, etc. a person has to know cold in order to take the next course up.

This list reinforces my feeling that we need such a book, along with a workbook that would structure an ongoing sequence of practice.

I think the RUSSIAN MATH book does an amazing job of fending off the last item on the list.

One last note: I suspect this list is probably a good rundown of why students of any age in any level of math have trouble learning it.

cram school

I had an interesting experience yesterday that illustrated the importance of a student having time for math to 'sink in.'

I was trying to teach Christopher the Saxon 8/7 lesson about subtraction of fractions with borrowing (or regrouping).

He couldn't do it at all.

Then his friends Drew & Marc, who've been in Phase 4 since the beginning, came over. Both of them could not only subtract fractions using regrouping, they could do a darn good job explaining what they were doing & why.

They told me they'd learned fraction subtraction with regrouping in 4th grade, back when Christopher was learning basically nothing.

Then they learned it again last year, in 5th grade. I know this, because Christopher & Drew were in the same class by the time Mrs. Woeckener taught the subject.

The difference between Drew & Marc, who've had a year and a half to know what fraction subtraction with regrouping is & how it works, and Christopher, who's had a huge amount of Intensive Math Intervention but didn't learn this topic in the 4th grade, was stark.

He did pick it up almost immediately, after Drew & Marc showed him how to do it. ('Drew and Marc are better teachers than you!') So that's something.

But this is extremely new & fragile knowledge in his head. Drew & Marc have a far sturdier base on which to build.


AnneDwyerOnSingaporeMath 27 Aug 2005 - 00:06 CatherineJohnson


You might want to check out the discussion on teaching things more than one way.

I started by saying that my principle has become 'teach things more than one way.'

Carolyn, Bernie, Chris & others objected.

I have to say that while 'teaching things more than one way' is a core principle for me at this point, whether rightly or wrongly, I don't really know what I mean by that.

In practice, what I've been doing so far is to teach bar models each and every day, along with, each and every day, the standard American 'symbolic' approach. I had Christopher start with the very first word problem in Primary Mathematics Book 3A, which is the first semester of 3rd grade in Singapore, & do one word problem a day, drawing a bar model to illustrate the problem set-up, and then doing the math using the standard algorithms.

And that's it. Each problem takes him a couple of minutes (a little more when he was starting out).

His 'real' math lesson obviously takes a lot longer.

Another example. A couple of days ago a Saxon 8/7 lesson taught two different ways of prime factoring a number. I threw out one of them, and substituted the RUSSIAN MATH approach, which I insisted he learn, almost entirely because when I learned it I found it incredibly fun to do. Christopher ended up liking it as much as I did.

Then yesterday, after Drew & Marc taught Christopher how to subtract-a-fraction-with-borrowing, I forced him to sit with me and watch while I subtracted the same fraction without borrowing, ending up with a whole number and a negative fraction. Then I subtracted the negative fraction from the positive whole number and voila. Fraction subtracted without borrowing.

I didn't make him do the subtraction-problem-without-borrowing himself, but only because he was in a MOOD. If he hadn't been in a MOOD, I would have insisted he do one or two such problems.

Now, I wouldn't insist he practice this approach to mastery, because it's Clunky, and forcing a child to practice Clunky Subtraction would be Wrong. IMHO. It's wrong because math isn't clunky, or shouldn't be.

The only reason I'd insist he work a couple of Clunky Subtraction problems is to make sure he really saw that the reason we borrow or regroup is that regrouping is an elegant, mathematically powerful way to do things--NOT because we can't subtract a bigger number from a smaller number! I know for a fact that a lot of kids think the reason you borrow-or-regroup is that you can't subtract a larger number from a smaller. Well, I don't want Christopher thinking that.

(I actually vividly remember the day, just this year, when my neighbor showed me that YES YOU CAN subtract 17 from 25 without borrowing. She's a statistician, and yet even she was puzzled for a moment when I asked her, 'Why do you have to borrow?')

The point is that I'm feeling my way, basing a lot of what I do on my own experience of relearning math, and on what I read in Liping Ma or see in the PRIMARY MATH series. I have no idea whether & when what I'm doing is a good idea, and whether or when it's a waste of time.

Here is Anne on PRIMARY MATHEMATICS:

I have been studying the Singapore math textbooks and workbooks. This is what Dr. Ma says the math teachers in China do.

When a new topic is introduced for the first time, there is an illustration which visually explains the topic. It is very simple and straight forward and ties into all the other illustrations that have been used in the book. There is usually a short English explanation and an equation if appropriate. For example, in 1B on the topic of comparing numbers: the illustration is comparing the number of stamps. The first illustration has 3 stamps. There is a cartoon of a child saying the number 3. The second illustration has 4 stamps, but 3 of them are exactly the same as in the first picture. The exact same cartoon child is saying the number 4.

Then, there are more illustrations but with all different types of things. For example, when learning about ten and ones, sometimes the illustration is bundles of sticks, sometimes blocks of ten etc.

Finally, there is a set of problems by themselves with no illustration.

Then, the workbook has all different exercises for the same type of problems. For example, Daniel is working on equivalent fractions in 3B. There are about 5 different exercises on this subject, some with illustrations to help and some without.

Since topics are always introduced in the same way with the same type of illustrations, you can tie back to what was learned before.

Additionally, word problems are very uniform also. For subtraction word problems for one, two and three digit numbers, there will always be one that uses the words more than, one that will use how many left, and one that will be how many of one type of thing.

Also, Singapore math introduces the first multistep problems in 2A, but only in the textbook.

So in Singapore math, the student is introduced to the concept first by visual illustration and then the procedure. And he has learned to do problems in several different ways right from the beginning. No one asks him to do the same problem in a different way but different exercises in the workbook show him how to do different problems in different ways for the same concept.

As for Everyday Math...well, I've been studying that too for comparison. I won't bore you with my rantings here. I have just one example that I think sums things up:

In the Everyday Math journal that students use in class, there are pages of Math Boxes that are review. In the first semester second grade, there are 120 Math Box pages with 6 problems on each page.

In one particular box, there was a problem to count back by 5s starting with 45. And there were spaces to put in the numbers. Then underneath is said, "Can you keep going?" And had this: 0, , .

Well, of course, my daughter had left this blank. Her teacher filled it in for her with -5 and -10.

What possible sense does it make to throw in negative numbers in a problem in second grade?

But that is Everyday Math.




fraction subtraction without borrowing


fractionsubtraction.gif


SusanOnMadMinutes 13 Nov 2005 - 18:34 CatherineJohnson


Susan used Mad Minutes with both kids--

I agree about the "mad minute" approach for all levels of math ability. I used it with both kids and I'm glad I did. It just helps with the speed and proficiency.

Gifted kids are notorious for not wanting to memorize anything. It's too boring. I had to make mine do speed drills all through first and second grade. I'm still doing it with my LD 8th grader.

I remember when math kid was learning multiplication in the first grade. He informed me that it was stupid to memorize them because he could always just count the the groups. I said, "Quick, what's 7 X 8?" His eyes went up looking for those groups, and after a couple of seconds of his looking for them and then trying to skip count by 7, I said, "That's why you memorize them."

I think it helps in the confidence department, as well, to get that speed going earlier than if it just came through practice. And if practice is the key, like "exposure" to lots of words was to the whole language crowd, then what exactly happens to the kids who don't get enough of that practice? Where is the place where enough practice crystalizes into math fact proficiency, especially with these "spiraling" curriculums that keep pushing mastery on down the road?

Trailblazers, as well as other NCTM curriculums, never seem to have a plan for the ones left behind. And I guess you can't ever know who is accountable since mastery was never the goal to begin with.

Interestingly, both kids do love the minute drills where they try to beat their last best score, so I never have any trouble giving them to them.

Well, that's two.

I have to say....I simply can't see any reason on the planet why worksheets would be bad (though TRAILBLAZERS explicitly states that worksheets are destructive!)

Given that there's no (apparent) downside, and given that they've worked for other kids, I'm certainly going to be using them with any child whose math education I'm involved in. (I'm starting to pick up a few! It's incredibly fun. More on that later.)

modifying worksheets

That reminds me.

One of the kids who took my Singapore Math class just could not get through a Saxon work sheet--and he didn't improve any over time, either. All the other kids got fast fast. (It really was remarkable.)

This particular boy is BOUNCY; he is one high-energy kid. He just can't stand the thought of a 5-minute worksheet; he probably takes one look at those sheets and sees what I would see if I were contemplating singlehandedly painting a two-story house.

I told his mom: try having him do ONE LINE of the worksheet as fast as he can.

I don't know if she'll get to it, but when they get back from Italy in January, I think I'll experiment with him, and see how he does. Just click the stop watch and tell him to call 'TIME!' when he hits the end of one line.

I bet that will work.


MathFluency 10 Sep 2005 - 16:03 CatherineJohnson


I spoke too soon, and I shouldn't be picking on teachers anyway, even when I've never met them and they're featured in news stories that make them sound dumb. (OK, probably especially not when they're quoted in news stories that make them sound dumb.)

Fourth-graders at Columbia Elementary School in Burbank increased their math scores by nearly 20 percentage points.

Fourth-grade teacher Erin Bennett said much of the growth is because of a new strategy called math fluency. Teachers give a short math assignment every night, and then go over it in class the next day. The assignments revisit the same concepts over and over again, to help children really get it.



I was wondering whether 'math fluency' could possibly mean actual math fluency, and it appears that it does. Here's what looks like a terrific short summary of math fluency and the 4 stages of learning over at Illinois Loop:

The second stage of learning is the fluency stage where the learner acquires the information at an automatic level.

[snip]

Research shows that to be fluent children should be able to accurately solve math facts at a rate of one per every 2 seconds. Naturally, if the child has poor fine motor skills or is younger, that has to be taken into account on any written timed test. One of the biggest teaching mistakes in math is when teachers don't stick with this part of instruction with children who have more difficulty. I'll give my son as an example. Not only was Justin one of the slowest learners of addition math facts I had ever worked with, but once he finally knew them he had absolutely no fluency. It could take him hours to complete 50 addition math fact problems. (I waited him out once.) Fortunately, his teacher wouldn't let him move on until he was fluent with them and I started to work with him on fluency every evening for ten minutes. Now many an educator would have said, "He has an attention deficit disorder and just doesn't have the attention span to do a timed math test." I was not willing to put this limitation on my son in second/third grade. To work on fluency, every night I set aside a time and gave him a sheet with all his addition math facts. I then set the timer and his goal was to complete one more problem than he had answered the night before. I think when he started he could answer 4 or 5 problems in the ten minutes. He literally progressed problem by problem. If he didn't beat his goal, we would practice saying the answers and then set the timer again. Fortunately once he could do the addition, the other facts came much easier. By fifth grade Justin was the fastest student to complete the once-a-year check-up math timed tests, and not only will he be studying algebra in eighth grade, but he can take any timed math achievement test and score around the 90th percentile. If we hadn't focused on the fluency, none of this would have been possible.

This directly contradicts the stated policy of TRAILBLAZERS, which is that math facts 'aren't gatekeepers.'

Good.


And notice: this mother brought her son to fluency in 10 minutes a night.

This is something I've been thinking about. So far, it seems to me that you don't have to put vast hours of time into homework, classwork, Saturday work, summer vacation work, and on and on and on in order to learn math.

It seems, based in what I've seen, that shorter bursts of effort repeated every day are incredibly effective. It's the consistency and the repetition that are the magic, at least some of the time (I've seen this with math facts specifically).

I'm hoping to find some research on this, but I'm not optimistic that 'efficiency' of learning has been an important focus of investigation. Most of us think of studying as work, and of work as good. I certainly do. The question of 'how little you can get away with' is uncomfortably close to the question of 'Will it be on the test?'

So I'm guessing we don't know too much about this. But we'll see.


update

Another very nice statement of the cognitive science supporting math fluency:

Grover Whitehurst, the Director of the Institute for Educational Sciences (IES), noted this research during the launch of the federal Math Summit in 2003: “Cognitive psychologists have discovered that humans have fixed limits on the attention and memory that can be used to solve problems. One way around these limits is to have certain components of a task become so routine and over-learned that they become automatic.” Whitehurst, 2003)


update

The Grover Whitehurst quote comes from a 'promotional white paper,' the kind of marketing document publishers and vendors are producing in response to NCLB's requirement (if that's the correct word) for research-based textbooks & teaching methods (and possibly before). In this case, the product being sold is a software program designed to help students achieve fluency with math facts.

I find these promotional materials incredibly helpful, so long as you bear in mind that they are not literature reviews; i.e., you're not going to hear the contravening evidence.

Just wanted you to know--

Here is the whole paper, which is almost certianly worth skimming. Research Foundation & Evidence of Effectiveness for FASTT Math (pdf file)


fuzzy math in WA state




CargoCultResearch 12 Sep 2005 - 14:03 CarolynJohnston


I read E.D. Hirsch's article on Classroom Research and Cargo Cults last night. It's a short easy read, and a worthwhile one, I think.

I believe, like practically everyone else, that ed research is crummy. I think that's undisputed; everyone is claiming that education research supports their point of view, because everyone can, because ed research results are all over the board. They add to the confusion, rather than clearing the fog. I thought that that was probably because the field is backward, in its scientific infancy, and that they don't understand things like statistics and random sampling and longitudinal studies and experimental design in general.

But Hirsch claims that the statistical methods used in education research these days are of good quality (and he credits a recent article, by Thomas Cook and Monique Payne, for bringing good experimental design into the mainstream of educational research practice). He believes that the problem with classroom research is that it cannot be sufficiently controlled to eliminate extraneous influences that throw the results off (one such influence, for example, would be the effect of the teacher's knowledge and personality, which is, as any parent knows, enormous). This problem, he thinks, results in classroom results being irreproducible and therefore unreliable for guiding education decisions. They are valid as far as they go, he feels, but we shouldn't regard them as science, or try to make policy from them.

He believes that better inferences for education policy can be made from highly controlled research on cognitive science. Cognitive science is actually converging on a consensus about how people learn, and what practices increase learning efficiency. These principles/practices are:

Prior knowledge is a prerequisite to effective learning. Bernie and I used to say, when Ben was little, that it was hard for him to learn things because, having had a condition that made it hard to attend to his environment, he had few 'hooks' on which to hang new knowledge. By hooks, we meant prior knowledge that we could draw analogies to. This principle is just that: that learning is improved when hooks in the form of prior knowledge are present. A novice will learn less than an expert from a new scenario, even though he is a beginner and has more to learn, because he has less context to base new learning on.

The right mix of generalization and example is critical. Good teaching goes from an illustrative set of examples to the general case. For example, you wouldn't just demonstrate the distributive property formula:

a(b+c) = ab+ac

to a bunch of first graders. Instead, we work for several years on examples, learn the multidigit multiplication formula, and so on, before introducing the distributive property in its full generality. We need to do this because the abstract concept and the specific example are inextricably linked in people's minds.

Attention determines learning. Surprisingly, motivation isn't a prerequisite to learning, but attention is. If attention is paid to something, and if we have a 'hook' to hang the knowledge on, we'll learn it, plain and simple.

Rehearsal is usually necessary for retention. How long something is remembered depends on how long it's been attended to. There is a "sweet spot" of practice past which things are permanently remembered, and practice that is spread out in time ("distributed practice") is much better than cramming ("massed practice").

Automaticity (through rehearsal) is essential to building higher skills. Our working memory (our mental scratch space) is extremely limited. Practicing a skill to complete automaticity frees up working memory.

Implicit instruction of beginners is usually less effective. Cognitive scientists actually give a complex answer to the question of whether explicit instruction (in early reading, for example, this would be 'phonics' instruction) or implicit instruction ("whole language") is more effective. The consensus is that both are required. In tennis coaching, for example, drills that isolate skills are desirable, but actual games must also be played. The mix should grow more implicit, and less explicit, as expertise grows.

What I want to know now is this: is there a good book yet on what cognitive science has to say about learning? Willingham's series of articles on cognitive science is excellent, but I'd like to know whether there's something more in-depth that talks about the specific studies that support these principles.

And: are cognitive science studies admissible as research evidence in support of NCLB's goals? If not, how can we get them admitted, and test the degree to which a given curriculum adheres to them?


InnumeracyPart2 13 Nov 2005 - 19:59 CatherineJohnson


A section of the innumeracy article Carolyn linked to caught my eye:

Wieman says getting students comfortable with math as a way of describing the natural world is a nut he has had trouble cracking. He said methods such as those developed by his Physics Education Technology program can give students without science backgrounds a deep understanding of scientific concepts, "yet when something involves a simple arithmetic calculation, their brains click into this totally different mode."

Steven Pollack, a CU physics professor whose research focus is improving the education of physicists, says the problem also happens in reverse. Physics students often can't conceptualize or explain the results of the equations they so breezily manipulate.



This is something I've been wondering about.

This may sound crazy, but, as a kid, I was reasonably good at math. I got straight A's, I had no trouble learning whatever I was supposed to learn (my one bad moment, in 2nd grade, WHICH I REMEMBER TO THIS DAY, involved--guess what?--fractions).

I took my SATs cold, with no practice, a year after I'd looked at my last math book and got a 620, which put me way up in the top percentile of the nation's 17 year olds at the time. (IIRC, I may have been in the top 95th percentile for girls.)

So....I was reasonably good at math.

That's why when Christopher came home with his 39 on Unit 6 it never crossed my mind I couldn't simply sit down and teach him what he'd missed.


no can do

You all know the end of that story. I discovered I knew practically NOTHING about math.....which is an exaggeration, but is sure the way I felt. I've been obsessively re-teaching myself elementary mathematics ever since, and intend to go on to trig & calculus & and a bit beyond.

So what does it mean to say that I was 'reasonably good' at math?

It means I could set up two-variable algebra problems and solve them in a jif. Thirty years later, I could still do it. Easily.

But I had no idea why setting up two equations (or 3 or 4) worked.

This is why I'm such a fan of the Singapore Math bar models (one of the reasons); it was a bar model that first explained to me what subtraction really meant.


the difference between two numbers


barmodeldifference.gif


I had a Helen-Keller-at-the-water-pump moment the first time I drew this bar model. I had simply never noticed that the 'number' of boys and girls in Mrs. Johnston's class, up to the number 10, is the same number. The 'extra' five boys are the difference.

For my entire life I had heard the word 'difference' used to name the number you end up with when you subtract one number from another, but I had never, ever realized that 'difference' actually did mean difference. It wasn't just some random word that had gotten attached to the operation somewhere back in the mists of time.

I now point this out to any kids I teach--and they all seem to find it extremely cool, too. I say, and then I repeat frequently, Subtraction is finding the DIFFERENCE between two numbers.

Then I point out that, if you're subtracting 3 from 5, 3 and 5 are the same number until you get past the 3-that-is-inside-the-5.


quick question re: number partition theory

The article on Everyday Math that I linked to yesterday, Weighing the Factors says this is number partition theory.

Is it?


odd man out

Teaching the how-many-boys-and-girls problem to kids, I also point out what would happen in Mrs. Johnston's class if you paired each girl with a boy.

You would have five boys left over.

That seems to make enormous sense to grade school kids, perhaps because they spend their grade school years being assigned partners or buddies to walk in lines with, or go to the bathroom with, etc.

This is obviously the way to teach the concept of even and odd, too. If you have an odd number of kids, there's going to be one child left over when you assign them to teams.

AND this image works great for teaching the idea that you always get an even sum if you add two even numbers or two odd numbers, but you get an odd sum if you add an even & an odd. In my experience so far, kids can instantly see that, when you add two odd numbers, you get two 'odd men out'--and now those two finally have a partner!


why don't numbers & concepts connect more easily?

This brings me back to my original point: somehow, you can learn numerical manipulations, including more advanced numerical manipulations that require you to set up equations and solve them, and not have a clue what it all means.

I don't understand this.

I don't understand how I could have so much fun setting up equations & solving them, and never gain the slightest idea why what I was doing worked.


keywords: conceptual understanding & bar model difference between two numbers comparison of numbers subtraction as comparison subtraction has two meanings


partial product division in Everyday Math
fighting innumeracy at CO
subtraction as the difference between 2 numbers
study sheet: subtracting integers & absolute value
notes on integer, subtraction, & absolute value study sheet




CriticalThinkingVsRegurgitation 21 Sep 2005 - 20:49 CatherineJohnson


I was at a meeting today where a teacher described a student as being good at regurgitating knowledge.

I took umbrage.

From now on I'm going to be using the expression DOMAIN KNOWLEDGE a lot.

A whole lot.


experts & novices

Daniel Willingham's Cognition: The Thinking Animal came today!

Soon, all will be clear.

Here's Willingham on the difference between and expert and a novice:

By definition, an expert is someone who is very good at solving problems in a particular domain, such as chess, physics, or baking.

[snip]

...experts differ from novices chiefly in their amount of knowledge about the domain.



Experts know stuff. That's what makes them good at solving problems.




more on this later



CreativityInMath 22 Sep 2005 - 15:13 CarolynJohnston


Catherine wrote here:

Steve, if you ever care to write something about what creativity is in math, I'd love to post it.

I used to read books & articles about creativity, but then I stopped, because no one knows what it is. (Not sure whether cognitive science has made more headway, but I haven't seen it so far.)

I've now got something of an idea, which I could probably put into words, about what creativity looks like in nonfiction writing.

But I wouldn't be able to describe it in any mathematics fields, apart from, maybe, economics where you have people like Steven Leavitt & Caroline Hoxby.

If creativity can be thought of as a solution to a problem that is wholly different than anything that came before -- some new approach that just makes you wonder, how the heck did he think of that? -- then it's everywhere in math, every time someone proves a new theorem.

When I was young and stuck on problems in math, my Dad used to tell me to work like the devil on them, then sleep on it. The answer, he said, would well up from my subconscious once I was rested and went back to think on the problem some more. I don't remember it working every time, but by now I'm familiar with the feeling of being frustrated and not having a clue, and then having the answer just come to me.

Here's how my thesis went. My thesis was in harmonic analysis, which is like fancied-up Fourier theory -- representing functions as sums of sines and cosines. Every electrical engineer knows Fourier theory thoroughly. My thesis was about a problem in which I was trying to show that every example of a certain type of function had to have places where it was equal to zero. I racked my brain and made no progress for what seemed a long time.

I had a friend -- one of the professors -- whose area was algebraic topology (which is quite unrelated to Fourier theory). I learned a little algebraic topology just from talking with him. Then, one day, I suddenly realized that algebraic topology had the key to my problem, even though it was a completely unrelated field The rest of my thesis was working out the details from that one insight.

The definition of creativity certainly has to include being able to bring in knowledge from a completely unrelated field in order to solve a problem, doesn't it? In math it happens constantly.

Have you heard of the Black and Scholes option pricing model? Black and Scholes won the Nobel prize in economics for it. One of them was an economist, and one was an applied mathematician, and in talking together, they had the insight that the model for option pricing in financial markets should be analogous to the heat diffusion differential equation in mathematics. The rest of their work was following up on that insight.

I claim that every kid, learning how (for example) to solve a mixture word problem for the very first time, is being creative in mathematics. They are doing something completely new and unfamiliar to them, taking pieces of other things that they know and putting them together to solve a totally new problem.

I don't think, by the way, that Asians are less creative than we are by nature. From having worked closely with a few of them, I think the difference is less due to a 'creativity gene' than to fear. Fear of what, I couldn't tell you; but it seems to me that our creativity is related to our not being a consensus culture. In fact, we Americans are all trying as hard as we can to distinguish ourselves for fear of blending in. So it seems to me.


PageSplatterPart2 22 Sep 2005 - 20:03 CatherineJohnson


Speaking of page splatter, here is an article from Cognitive Neuroscience that is directly relevant to the question of whether ransom note typography in textbooks is good, bad, or neither. (Assuming I understand the abstract, that is.)

Distracted and confused?: Selective attention under load
by Nilli Lavie
Volume 9, Issue 2 , February 2005, Pages 75-82:

The ability to remain focused on goal-relevant stimuli in the presence of potentially interfering distractors is crucial for any coherent cognitive function. However, simply instructing people to ignore goal-irrelevant stimuli is not sufficient for preventing their processing. Recent research reveals that distractor processing depends critically on the level and type of load involved in the processing of goal-relevant information. Whereas high perceptual load can eliminate distractor processing, high load on ‘frontal’ cognitive control processes increases distractor processing. These findings provide a resolution to the long-standing early and late selection debate within a load theory of attention that accommodates behavioural and neuroimaging data within a framework that integrates attention research with executive function.


Roughly, I believe that this paragraph says two things:

  • Ignoring perceptual distractors (like extraneous noise, I assume) gets easier the more intense the perceptual 'item' or element you're paying attention to. If you're sitting in a quiet room and a couple of people are whispering behind you, it's harder to ignore them than if you're sitting in rock concert and the same two people are whispering behind you.

  • BUT, when the item or element you're paying attention to involves the frontal lobes--i.e. when it is more 'cognitive' in nature, as in the case of mathematics--the harder the material, the greater your likelihood of 'processing the distractor.'



If I'm reading this correctly--Daniel Willingham may be willing to tell me if I've got it right--this is, to me, revolutionary.

I don't need cognitive science to tell me that American textbooks are horrifically distracting. I can barely extract meaning from Prentice Hall Pre-Algebra, and I don't think the teacher can, either. When I mentioned the integer tiles PHPA uses ON THE FIRST PAGE she had no idea they were there, in the book.

Although I read the PHPA section on adding & subtracting integers carefully (I thought), I did not manage to notice that the text formally defines subtraction of a number as addition of the number's opposite.

This definition was there, on the page, in a green box no less, but I didn't take it in. I had to come up with the principal on my own, as I was trying to create simple, readable, attendable lesson review sheets for Christopher.

This is one of those issues where I'm simply going to go with my own experience, no matter what the scientific consensus or non-consensus may be.

Page splatter obstructs learning.


page splatter really obstructs math learning

However, it had never occurred to me that the more difficult the material you're trying to master the more harmful page splatter becomes.

I just thought distraction was distraction.

But when I think about it, this abstract captures my own experience of textbook design. I loathe American math books. I feel a kind of repulsion just looking at them, and the reason I feel that way is that I have to put out incredible energy to stay on track.

Interestingly, I feel a corresponding love for clean design in math texts. To this day I remember the simple beauty and elegance of the brand-new math textbooks we were handed in 2nd grade. I can still summon up a picture of those books; I remember the shine of the elegant pages.

They were the most beautiful books I had ever seen.

Same story with Russian Math. It's a lovely book, and I 'had to' read it. The design is pristine, sober, and respectful, and I felt compelled to open the book and begin.



OK, I better knock this off until I find out whether I've interpreted the abstract correctly.....

Because if I didn't, I'm going to have to take this whole post back.


Willingham recommends TRENDS IN COGNITIVE SCIENCE

I asked Daniel WIllingham which one cog sci journal I should order, and his answer was TRENDS, because it carries review articles summarizing trends & questions in the field. From the web site:

Trends in Cognitive Sciences provides concise reviews, summaries, opinions and discussion of the most exciting current research in all aspects of cognition, the mind and the brain. Internationally renowned scientists from cognitive neuroscience, psychology, linguistics, social cognition, artificial intelligence, neural computation, and philosophy regularly contribute to the journal.

Trends in Cognitive Sciences features succinct, lively, and up-to-date Review and Opinion articles and discussion of the latest developments in the primary literature in Research Focus articles. Together with stimulating Book Reviews, Trends in Cognitive Sciences provides an essential overview of the latest thinking for both experts and newcomers to this rapidly expanding, multidisciplinary field.

Most articles are commissioned by the Editor and all Review an Opinion articles are peer-reviewed.



He's right; this is exactly what I need.

Haven't checked the price yet.


OK, I did it

Price for a one-year subscription: $198.

sigh


update update

Hey!

I just realized.

This is another case of It's always worse than you think!

I should start a collection.


Glencoe page splatter
Doug Sundseth on ransom note typography
Tom Friedman piles on
distance tutors & mathematicallycorrect review Glencoe
page splatter and the frontal lobes
page splatter redux
pagesplatter




AlgebraicSymbolsHardForStudents 27 Sep 2005 - 21:22 CatherineJohnson


Another interesting comment from a joannejacobs thread on new research about children's abstract understanding of math:

Imagine what a man like Archimedes could have accomplished if he had had the benefits of Saxon math. It is true that we all have some mathematical aptitude and that certain simple skills develop naturally, but this is far from enough mathematics to function at even the minimum wage level in our world.

I have never met a student who could flawlessly manipulate symbols according to the rules of algebra but had trouble with the deeper concepts of mathematics. Most of my students find poor algebra skills to be an almost insurmountable barrier to deep understanding.

Of course the foundations for success in algebra are those tedious skill sheets we "abuse" our children with in primary school.

Posted by: CRW at September 27, 2005 03:42 AM



hmm.

Now that I re-read this, I'm not sure what he or she is saying....is the point that a student who excels at writing & interpreting algebraic expressions can always also understand algebra?



AdhdIqClassroomPerformanceScatter 28 Sep 2005 - 02:23 CatherineJohnson



I wanted to bring this up front for a couple of reasons:

  • first, most people (I find) aren't aware that a normal IQ does not show a large gap between 'Verbal' and 'nonverbal' or 'Performance.' I'm still hazy on the distinction between verbal and performance myself, but if you think of it as verbal & spatial you're on solid ground, I think.

    This is important to understand, because we're accustomed to thinking talents are separate & opposed; if you're good at 'A' you're bad at 'B.' I have no idea how this idea plays out later in life (I'm guessing it starts to be significantly true in high school.) But, certainly, a normal child's IQ scores on Verbal and Performance will be close. When the scores are not close, that's often a problem - and it's a hallmark of autism, which is the only reason I know about it. (Autism isn't the only cause of a large gap between verbal and performance.)

  • IQ scores, Susan says, predict classroom success quite well


fyi: I used the word 'scatter' in the title of this post, which also comes from autism. Autistic kids will show 'scatter,' or 'islands of ability.' It can be incredibly strange. Donna Williams once wrote that her scores on the various subscales of IQ tests ranged all the way from mental retardation to genius.

That's scatter.


Here's Susan:

Bright children with ADD can get through grade school fairly easily. It's down the road that the problems of attention and/or hyperactivity start to become a serious problem.

I don't know the percentage of states that do this, but in IL some kind of an IQ test is done by third grade. (Here, I believe they did the Otis-Lennin one, which usually corresponds prettly close to the standard WISC III (or whatever WISC they're on.) When the parents are meeting with everyone at the end of 3rd grade/beginning of 4th, they may or may not know that the teachers and administrators have a clear view of what some intelligence test showed on their child. They often won't explain its meaning unless specifically asked for by the parent.

I mention this because in the case of ADD, an IQ score is usually going to be an underestimate of the child's true ability. At the very least, children with ADD or LDs often have wider spreads in their numbers than other children.

In other words, in most IQ type tests the numbers should cluster together. When there is a spread, or when it is below what parents or teachers think it should be, it indicates a possible problem.

Remember that IQ is not an actual measure of intelligence. There is no true measure of that. What it really does is tell you how well a child will do in a standard American classroom setting, and in that regard it is quite reliable. So, the outer edges of the bell curve are really just "falling out of the curriculum" by up to 2 years or more.

So, I'll give you an example of a kid I know who showed ADHD symptoms and had been frustrated quite a bit during his early grade school career. The kid takes the required Otis-Lennin in the third grade and bombs, showing a very low score. The parents are shocked because the kid reads years ahead of his classmates. They demand another test, so they give him the latest WISC (III or IV) and he comes out with a Performance IQ of 146, yet a Verbal of 106, with the overall IQ being around 130. Up to this point, teachers had been treating him as a disorganized, but above-average kid, nothing special. But what they realized they had was a kid with a superior intellect who had a deficiency. While a verbal score of 108 is still considered "above average," the fact that it was in such contrast to the other score told his teachers that he was in need of a completely different approach. They quietly provided the parents with an IEP and made him a part of the gifted program.

So, intelligence testing can often be a useful tool to a parent when dealing with schools. Although, I still don't think many of them really understand ADD, either, or its impact.





OnHavingAMathBrain 29 Sep 2005 - 16:20 CarolynJohnston


(Note -- I've modified this post slightly from its original form. It's surprisingly one of the toughest posts I've tried to write -- Carolyn).

During our recent discussion about whether there are "math kids" -- the consensus seems to be that there are -- you might have noticed that I was quiet. There are definitely kids who find it easier than others -- but that's not the only condition for success in math.

I don't think math is outright easy for anyone. Sooner or later, you're going to hit the wall if you keep up with it. I don't think having a "math brain" that makes math easy for you is what's necessary for success in math.

I do think that some of us, for various reasons, are better equipped to enjoy the work that goes with math than others, because of a combination of their personalities, and the circumstances in which they learn and do math.

For some people math problems are like puzzles (I think that Catherine is discovering this trait within herself these days); for others, they are just a hard slog, a source of pain, and no fun at all. The people who enjoy math problems as puzzles -- as outlets for their obsessiveness, shall we say -- might not find them easy, but they aren't exactly suffering, either.

When I work a problem and get a solution, I check it over and over, obsessively, because I like to be sure that I'm right. I like being absorbed into a problem or a derivation; it beats obsessing over the worldly problems in my head that I absolutely can't solve. It feels as absorbing to me as a crossword puzzle might to you, and it also feels meaningful to be doing it. So it's innately reinforcing to me.

I also, perhaps, am more reinforced than most people by the right answer that comes at the end of all that work. And I feel empowered when I've learned something new and cool. And I've also been reinforced and rewarded by other people for my interest in math. It all adds up to a lot of positive reinforcement.

But there's a wall for everybody if you push it far enough. I hit my own personal math wall in differential geometry -- although I passed it, I took it twice, by choice, and I still don't really get it in my bones (although I do have occasional moments where some missing piece falls into place, presumably because I spent all that time obsessing about it). Differential geometry is abstract, arcane, and mixes badly with my spatial disability (I'm one of those people who can't tell left from right without thinking, and whose intuition about where she is is wrong more often than chance). If a good, tough puzzle didn't suit me, I'd have run from it.

So I don't think you have to have a 'math brain' to go far as an economist or an engineer or a scientist; you can successfully utilize math at a high level in your life without it. You do have to be unafraid of work, because it's just a question of when (not if) you'll first encounter something that's hard to understand. The important thing is that you find it rewarding, not punishing, to do the work most of the time.

That's something we have, as parents, some (though not total) control over. It feels good for a kid to be in advanced math; just being there is reinforcement for doing math. It feels good to see that look in your parent's eyes that says they're proud of you for doing well in math.

And it feels so darn good to get the one right answer.


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent




WickelgrenOnCreativity 03 Oct 2005 - 12:57 CatherineJohnson


Ken's & Carolyn's posts, Tour de force and On having a math brain, are companion pieces, and, reading them back-to-back, I see that it's time for me to get Wayne Wickelgren's work posted.

I'll try to get that done today, and am re-reading Math Coach now. But I'm stopping to post this observation:

Creativity is an outgrowth of learning, and a lot of it. The past twenty-five years of cognitive psychology research has shown that the more a person knows about a subject, the more creative he or she can be in it. No question an adult poses is considered creative if someone else has already asked it. Thus, an adult must know what has come before to ask creative questions.


There's more:
This is true more generally as well. A student's ability to be creative in any area of knowledge increases with his or her knowledge of that area. Knowledge forms the fodder for creative new ideas.



sterling advice

Oh, gosh.

I can feel my day getting Sucked Up in pursuit of Wickelgrenisms. Here's another, from his section on how to evaluate your school's math curriculum:

To check that your child's teacher is focusing on math fundamentals, look at your child's homework. It should include lots of math problems to which there are single, correct answers, and no artwork or writing.


Amen to that.


all Wickelgren, all the time!

If your child is now struggling with math or scoring only in the average range in his or her classes, focus on helping your child master the math taught in each grade. Mastery means getting As or the equivalent in math.



Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors




WickelgrenOnYoungChildrenAndMath 17 Sep 2006 - 01:14 CatherineJohnson


back story:

My neighbor, the statistician, showed me her copy of Math Coach: A Parent's Guide to Helping Children Succeed in Math quite awhile back, before either of our kids had had any trouble in math class. I ordered a copy just because I order lots of copies of books I'd like to read but then don't.

So the book was sitting there on my shelf when Christopher came home with his 39 on the Unit 6 test & I subsequently failed to teach him fractions using SRA Math. I needed help.

It was the right book at the right time. A page-turner.

Most of what I believed to be true of math ed & math achievement, I discovered, was wrong. Severely wrong. I had been operating on the basis of sheer ignorance, naivete, and boneheaded cliche.

This is the observation that probably shocked me the most. It appears in Wickelgren's chapter on finding a school for your child:

There are schools with even less structure than Eastside. Take the Sudbury Valley School, a private K-12 school in a Boston suburb. This school gives each child complete freedom to choose how they spend their time at school. There are no classes except those specifically requested by a group of students. Children learn largely on their own, reading books, talking to each other and to teachers or outside experts, solving problems, playing games and sports, practicing musical instruments, doing arts and crafts, and anything else that can be done on the school grounds.

While you can read at length about the school's strengths on its web site, one of its biggest potential benefits is that every child can proceed at his or her own pace, in math and in other subjects as well.

There are also potential drawbacks. Since young children are not generally highly motivated to learn math, they may choose not to study much of it.



I was bowled over.

I had always thought kids want to learn things they're good at. Christopher is good at social studies, and he wants to learn it. At night he'll bug his dad to 'give me trivia questions.' (Give me superficial facts, Daddy!) Ed finally refused to do it anymore, because he ran out of trivia.

Christopher also has a collection of geography trivia books that he reads, and when he was 7 I read all of the first volume in the History of US series out loud to him as his bedtime story.

That was the book he wanted to hear.

So...I assumed kids wanted to learn subjects they had a talent for.

According to Wayne Wickelgren, this is not the case with math.

Or, at least, not generally. Math talent doesn't (necessarily) manifest itself in an obvious desire to learn the multiplication tables. (Or to write essays on My Special Number.)


late bloomers

That one observation pretty much changed my life. I decided, then and there, that I didn't know whether Christopher had any talent for math or not, or what his eventual level of interest in the subject might be--or, more importantly--could be, given a decent education K-12.

I also knew he had good general intelligence, which meant he had the ability to learn a whole lot of math whether he was going to end up in a math-related career or not.

I decided right then and there that that was what was going to happen. Christopher was going to learn math, lots of it, and learn it well.

We were going to keep the doors open.

When Christopher reached college, he would be in a position to decide to pursue a math-related career or not. That decision would not have been made for him in 3rd grade, when he got sorted into Phase 3.

It wasn't too long after this that I met Carolyn and heard her story: flunked algebra in high school (right?), didn't decide to major in math until senior year in college, then got a Ph.D. In math. Another wake up call.


more late bloomers

Two more stories.

One comes from Christopher's 4th grade teacher. Her daughter was reaching the end of high school, and it was time to do SAT prep.

So her mom hired a tutor, and within a couple of weeks the guy was reporting that her daughter had strong talent in math.

She had no idea. Neither she nor her daughter had the first clue that this kid had a knack for math. Now, working one-on-one with a tutor who, IIRC, had a Ph.D. in math (or engineering, possibly) she was flying.

I have no idea where that girl will end up, what she'll major in, or which job or career she'll pursue.

It doesn't matter. The point is: she's good at math, and she went through 11 years of formal education thinking she wasn't.


you can't predict the future, or even the past

Story number two comes from a friend of ours. As a boy he had two or three chums who sat by each other in class & were bright kids. They were the kind of kids who could learn whatever you threw at them, and they got As in all their subjects & went to good colleges & universities. They got As in math, too, of course, but none of them was a whiz. Our friend became a lawyer.

One of the gang shocked everyone by growing up to become a world-famous econometrician.

No one can understand how this happened. This kid never showed any special talent for or interest in math. He was just a smart kid, like the rest of them. Our friend said that to this day, whenever any of them get together, they always ask each other how that friend could turn out to be not only an econometrician, but a world-famous one.

Go figure.

What I like about this story is the fact that not only could this boy's future as World Famous Econometrician not be predicted when he was 8, it can't be back-predicted now, when he's 40.


Barbara Oakley's bio

I just remembered: Barbara Oakley is in the same category. Here's her bio:

I started studying engineering much later than many engineering students, because my original intention had been to become a linguist. I enlisted in the U.S. Army right after high school and spent a year studying Russian at the Defense Language Institute in Monterey California. The Army eventually sent me to the University of Washington, where I received my first degree–a B.A. in Slavic Languages and Literature. Eventually, I served four years in Germany as a Signal Officer, and rose to become a Captain. After my commitment ended, I decided to leave the Army and study engineering so that I could better understand the communications equipment I had been working with.


Barbara sent me an email that I won't quote without her permission (I'm WAY behind on email). But her story inside an email is more dramatic than her story here, though no different in outline. Barbara is a person who earned an entire B.A. degree in a humanties field and served a full stint in the Army before figuring out she wanted to major in engineering.

And the reason she decided to study engineering is pretty similar to the reason I've suddenly decided to study math; she got tired of not understanding the stuff she was working on. In her case, that was communications equipment; in my case it's K-12 math.

Obviously, Steve H is right, we simply cannoy be assigning grade school kids to our two Standing Committees: math whiz & math's not his thing.


all English Language Arts all the time

from The Learning Gap by Harold Stevenson and James Stigler:

....American teachers like to teach reading; Asian teachers like to teach mathematics. When we asked teachers in Beijing, nearly all of whom were women, the subject they most liked to teach, 62 percent said mathematics, 29 percent said language arts. The reverse was found in Chicago: 33 percent mentioned mathematics and 47 percent mentioned language arts. There is more to the story than preference, however. Americans simply emphasize reading more than mathematics. Despite the large amount of time already spent in reading instruction, more than 40 percent of the suggestions made by Minneapolis mothers who wanted an increased emphasis on academic subjects said they thought that the subject should be reading. Fewer than 20 percent mentioned mathematics.

These data lead to the obvious conclusion that American children do less well in mathematics than do Chinese and japanese children partly because they spend less time studying mathematics....Conversely, American children may fare better in reading, relatively speaking, because they spend more time on this sujbect.



I mentioned yesterday: it's a commonplace for people to say, 'I was never any good at math.'

No one says, 'I was never any good at reading.'


English Language Arts in Irvington

I've seen this here in Irvington.

My sense is that Irvington does a good job teaching reading. Not that I know what I'm talking about, but that's my sense. (fyi, after trying to teach out of the SRA Math book myself, I also think our grade school teachers are near-geniuses at teaching math, too.....& I'm not kidding about that. It was tough.)

Christopher's 6th grade schedule includes:

  • 2 periods of English language arts, one for reading & one for writing
  • 1 period of social studies, taught by a teacher who told us, on back to school night, "I am an English language arts teacher at heart"
  • 1 period of drama

That's 4 periods out of 8, half his day devoted to English language arts. He has 1 period for math, 1 period for science, and that's it. The other 2 periods are specials: study skills, music, art, drama, P.E., technology. Technology will mean creating an online 'portfolio' of his best work in 6th grade, not learning how to program. Study skills is about reading & taking notes, not doing problem sets.

And, on back to school night, the math teacher told us the kids would be keeping a math journal, because a lot of kids in accelerated math probably aren't as strong in ELA, so 'we try to help them with English language arts.'

Thus far she has done nothing of the sort, thank heavens, and she's stopped grading the kids' math tests on spelling, which she did last year. I gather she had a lot of complaints about it, and I made a point of asking her, in front of the other parents, whether she would be grading spelling this year, too. (This is what we call a warning shot.) So she told the kids she wouldn't, and she hasn't. otoh, Christopher is now spelling parenthesis parenthies, so be careful what you wish for.


another story

This last story pretty much sums it up, I think.

I know I've mentioned the fact that we were clueless back when Christopher was in his early elementary years.

So, unbeknownst to us, he was placed in Phase 3 ELA as well as Phase 3 math. Actually, we're still clueless; I have no idea what kind of sorting & phasing they do with ELA. All I know is that in K-5 they divide the kids up into ability groups within the classroom, rather than separating them into different classes taught by different teachers, as they do with math.

In the hall outside Christopher's 4th grade class, after the year was over, I happened to run into his teacher and we fell into conversation, which led to the subject of Christopher's progress that year. I remember I was expressing gratitude for some especially good teaching she'd done, but I don't remember the details. It was probably about English language arts, since she taught him every subject but math.

One thing led to another, and suddenly I heard her saying, "Oh, I could see when he came into my class he wasn't a 3. He was much better than that. Sometimes you just have to ignore the tests."

Christopher had taught himself to read in Kindergarten, had tested two years above grade level in reading back in the 2nd grade, and had just received 4s on both the ELA & the math sections of the NY state tests. He'd been in the advanced reading group all year long as far as we knew.

So when was he a 3?

It took me a moment to recover, but I managed to keep her talking. "I pushed him," she said. "I knew he could do it." And, again: "You can't believe the tests."

Wow.

Think about the implications.

Here we have your dufus mom, completely out of the loop about tests, 3s, & 4s. And it doesn't matter; it doesn't hurt the kid. The teacher steps up to the plate, checks out the kid, decides for herself 'he's not a 3,' then sees to it he stops being a 3, and becomes a 4.

No extra reward, no extra praise, no extra payment or promotion. She just does it, because it's her job, and because she's good at it.

Perfect.

(And yes, I know; I'm tired of 3s and 4s, too. But 3s and 4s are a kind of shorthand, and a useful one.)


The point is: I have never heard this story told about a Phase 3 kid in math. Never.

Until this fall (that's another story), only a tiny handful of kids had ever moved from Phase 3 to 4. Maybe one 1 per year.

I've talked to the Chair of the middle school program about this issue, to one of the guidance counselors, to our 4-5 principal, and to numerous other teachers & parents.

Not one of them has mentioned the school or a teacher pushing a kid out of 3 and into 4. Whenever a move is made, the impetus has come from the parent, not the school. And the school resents it. (I've mentioned this before. We have a meta-narrative about pushy parents pressuring the school to put their kids in Phase 4 math when they don't belong there. Everyone subscribes to this narrative, including aides & other parents.)


The lesson I take away from this is that we really do have some major talent in some schools in this country, in the teaching of English Language Arts. I'm lucky to have my own kids in one such school district.

We need the same kind of teachers, with the same kind of know-how and confidence, in elementary mathematics.


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors




MoreOnMathBrains 30 Sep 2005 - 20:32 CarolynJohnston


Now that we've got the "math brain" notion on the run, let's catch it and pound it mercilessly, shall we?

Here's some input from Bernie, author of a real math book and therefore someone who might lay claim to having a Math Brain if one existed.

He's reminded me of my own first math wall, which also occurred in 4th grade -- I could not, could not understand the multidigit multiplication algorithm for quite a long while during that year. I struggled massively with it and, in the end, achieved nothing better than procedural knowledge of it. Imagine if even that had been taken from me, as the constructivists would now have it? Conceptual knowledge of that algorithm only crept in over the next few years of use.

From Bernie:

I agree that math has walls all the way down. The first one I remember clearly was in 4th grade. I was required to memorize the multiplication tables. I hate memorization. I don't think I even got the concept of memorization. In that particular class we had to take our seat based on the test score we received on the last test. The "A" students sat at the front. It was a linear ranking. I had to move from the front to the very back because I didn't know that answers to the multiplication questions. They threatened to kick me out of class; my parents got involved and forced me to memorize the multiplication tables using flash cards; I moved back to the front. First wall hit and overcome, first trauma endured.

I had many more. Maybe an advantage I had over Catherine is that I never supposed there wouldn't be walls.

I agree completely with Doug that there are walls in all fields. But I think math is different because for the most part it is nothing but walls.

It's a large mistake to believe that there's a group of people over there who get it all. There isn't. Math is really like a set of mountain peaks, and I think this applies all the way down to grade school. You climb one--it takes a lot of work, but you do it--but that doesn't do anything for all the other peaks out there. They all have to be climbed one by one. Some people have climbed several, a few people have climbed a hundred, but nobody could possibly climb them all. And not just because they're lacking time, but because they're lacking talent. There are all sorts of different kinds of mathematics which require different talents of various sorts. No one has all the talents.

The whole concept that there are "math people" who can get it on the one side, and then the rest of us on the other who can't, is incredibly debilitating. It lets kids off the hook for being lazy when they should have continued on and persevered. It's a horrible concept and completely wrong.

And it lets the more mathematically talented off the hook because they think that just because they have some mathematical talent they don't have to work anymore. I've seen a lot of those, and they were all lying by the wayside. Everybody who does math or physics or engineering seriously has to work just as hard as Ed. Grothendieck was probably the best mathematician of the latter Twentieth Century and he was famous for working all the time.

(Grothendieck invented modern algebraic geometry pretty much singlehandedly. He came from nowhere, and ultimately vanished. Think Good Will Hunting when you think of Grothendieck; he was the rarest sort of bird).


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent




WickelgrenOnMathTalent 30 Sep 2005 - 13:29 CatherineJohnson


Differentiating children by their abilities and skills is a controversial subject, but math aptitude can vary greatly among children, just as children differ in their ability to run, jump, give speeches, draw, sing, comfort others, tell jokes, or lead a group. And though it's generally impolite to speak of such differences, it is important to recognize that they exist--and for parents to have a sense of where their children rank among others.

Having a sense of your child's math ability can help you set realistic goals for your child in math. It can help you decide whether your child is progressing in math as fast as he or she can or whether you need to push a little harder or do something different--such as provide supplementary math education. For example, you probably want to supplement our child's math education if

  • Your child has a very high ability in math--that is, appears to be among the best 10 percent of students in his or her class

  • Your child is at least average in math and has career ambitions in math or a math related field

  • Your child scores in the range of college-bound kids, but his or her school math training is inadequate for admission to the college-prep math track or for scoring at the child's ability on the math portion of the Scholastic Aptitude Test (SAT)

  • Your child is of normal intelligence, but isn't mastering the basics of arithmetic in school--a necessity for independent living

One way to assess your child's math ability is to have him or her take an IQ test, but a simpler and probably more accurate method is to observe his or her learning rate relative to that of other children. If your child catches on to math concepts quickly compared to others the same age, he or she is probably a fast learner. Similarly if lots of other kids seem to catch on to math concepts more quickly than your child does, he or she may be of only average ability.

My daughter Ingrid says she was well aware of the differences in math ability between herself and her brother Abe, from the time I began teaching them at home at ages eight (Ingrid) and six (Abe). Ingrid remembers times when I would pose a problem to the two of them, and Abe would soon begin scribbling on his paper to come up with the correct answer. Ingrid, meanwhile, would look puzzled, wondering how Abe had figured it out so quickly. She knew she wasn't stupid, but it was obvious to her that her younger brother had a gift for numbers she could not claim.

Still, Ingrid took algebra in the seventh grade (when she was just eleven) without much difficulty. She learned plenty of math to complete a fast-paced college physics sequence and the curriculum required for a biology major. She is now a successful writer specializing in scientific and medical topics.

Meanwhile, Abe wanted to become a professional basketball player as a child and for years worked incredibly hard at improving his basketball skills. After first grade he played in a league every winter and went to as many as three basketball camps in the summer. Year round, he practiced basketball for an hour a day or more. As a result he was named the starting point guard for his seventh-grade team. But after that, his tiny stature and limited natural ability made him less attractive to coaches than kids who were bigger and learned new skills much faster. While his accomplishments in basketball diminished--he did not make the high school basketball team--his success in math continued unabated. He excelled in his advanced classes and remained five years ahead of grade-level in math.

It was clear from watching this drama unfold that Abe's natural talents lay more in math (and other school subjects) than in playing basketball. Similarly, it should quickly become apparent to you from your child's experiences whether your child is very talented at math or has lesser abilities in this area. But remember: Even if your child's natural talents in math do not suggest he or she should become a mathematician, your child could still use his or her math skills to become very successful-perhaps as an engineer [ed: yikes] or a financial manager. My son did, after all, develop a helluva basketball game, making him a top player in the adult recreational leagues he later joined.

[snip]

Your child's talents may or may not be in math. So do all you can to motivate your child to learn math and provide the best teaching possible, but as only one part of a well-rounded life. When you've done that, you've done your best and should accept your child's progress in math at school.

Despite what my dad told me, it is not true that you can do anything if you work hard enough. But if you temper your ambition with realism, you can derive enormous satisfaction from the truly spectacular results of hard work coupled with excellent instruction.




I love this man.


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Math Coach


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How to Solve Mathematical Problems


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors




WickelgrenOnPrealgebra 16 Jul 2006 - 20:48 CatherineJohnson




Gulp.


A student can learn a year of pre-algebra math in three to six months studying three to ten hours per week, depending on the child's math aptitutde.



I'm gonna have to pick up the pace around here.

I've been working my way through Mathematics 6 since the beginning of June.

It is now the beginning of October.

RUSSIAN MATH has, estimating conservatively, 10,000 problems. At least 10,000. I have now worked 8000. In the process, I've learned a huge amount, although, sadly, even Enn Nurk & Aksel Telgmaa have not been able to dissuade me from the conviction that 7 x 6 = 43. If they can't do it, probably no one can.

I've just begun the last of RM's six chapters, and I was getting excited about starting algebra next. I can't wait.

So last night I took Saxon Math's placement test (pdf file) for algebra 1.

I got a 72.


conclusion number one:

I am going to stop expressing reservations about the Saxon math series until I can actually take and pass a Saxon math test.


conclusion number two:

wow

There are a boatload of topics I still don't know after doing 8000 complicated Russian computation, geometry, & word problems.

They are:

  • using four 'unit multipliers' to convert 630 square yards to square inches: I have no idea what a unit multiplier is, or how to use it
  • what a decimal part of a number is (I got the answer right, but only because I made a blind guess as to what a decimal part would be)
  • negative exponents
  • how to find the volume of a cylinder
  • 'the method of cut and try' to find the square root of 20: to my knowledge, I have never heard the words 'cut and try' in my lifetime
  • how to use a straightedge (what's a straightedge? I still don't know) and a compass to copy an angle
  • how to find the area of a triangle (all I remember is: hypotenuse)
  • how to find the probability that a die will first roll a 6 and then roll a 2, in that sequence
  • base 2
  • update 7-16-2006: I know all these things now, and will finish Lesson 81 (of 120) in Saxon Algebra 1 today.


So my first reaction, in Western polarizing fashion, was: I know nothing.

I know nothing, and I need to work through all 857 pages of Saxon Math 8/7 with Pre-Algebra before I can even think about setting foot inside a real algebra textbook.

I was depressed.

But then I calmed down a little and thought, mmmmm....maybe not.

Maybe I can just go through Saxon 8/7 and do every single lesson & every single problem related to these 9 topics.

Is that wrong?

update 7-16-2006: I ended up working through the entire book. Every lesson, every problem, every test. Then I took the Saxon placement test and placed into Algebra 2, but decided to start with Algebra 1. I'm glad I did.

Christopher began teaching himself Saxon Algebra 1/2 this summer (he starts 7th grade in th fall) so I'm reading through those lessons to make sure I didn't skip anything I need to practice - and just for the joy of encountering John Saxon's take on topics I already know.

Algebra 1 integrates algebra and geometry, though without proofs. I'll start Algebra 2 in September.

In one year I will have worked through:

  • final chapter of Russian Math
  • all of Saxon Math 8/7
  • all of Saxon Algebra 1

That pace seems OK to me.



MathmanOnPractice 01 Oct 2005 - 15:03 CatherineJohnson


from mathman:

So how many exercises should I assign? I can't possibly grade them all. This is not an easy question to answer.

It's much easier to say how many exercises the student should do although most students won't care for what I have to say. The student should work as many exercises as it takes to be able to do them correctly most of the time as fast as he can physically write out a complete solution. When informed that he has made a mistake, he should be able to find and correct his error quickly. When it counts, given time to review his work carefully, he should be able come up with the correct solution every time.

This level of mastery opens the door to calculus, differential equations, linear algebra and the quantitative elements of any science.




I'm going to print this out, ask Christopher to read it out loud to me, and then post it above the dining room table. (We're still waiting on delivery of the Ikea desk I ordered a couple of week ago.)


Willingham on overlearning

I re-read Practice Makes Perfect--But Only If You Practice Beyond the Point of Perfection every few months.



WhyAutomaticity 11 Oct 2005 - 16:21 CarolynJohnston


The short answer: because completely automatic tasks are cost-free.

Every math problem a kid does - any bit of work that anybody does -- takes something out of him. As a kid gets older, the problems get harder and involve more steps. It's automation of the earlier, simpler steps that keep the complexity of a task from getting too big to handle.

Think about it in the context of reading. When you, as an adult, with your long-term expertise in reading, contemplate the work involved in learning something new, such as Roman history, do you consider the cost of the mechanical act of reading the words on the page? Of course you don't. In fact, given a page of text to read, you almost certainly can't help but read it. The act of reading costs you nothing.

Any component of a problem that a kid has mastered to automaticity will be discounted; in other words, completely automatic (rote!) tasks are not a drain on a kid's intellectual energy budget. The kid can focus on what he's intended to learn in that lesson, and can go further before he has to quit.

an example

In order to get specific, let's analyze what needs to be done in order to do an algebra word problem of a type that most of us don't remember fondly; mixture problems.

You have two lemonade mixtures. Mixture A is 5 parts water to 1 part lemonade powder, and mixture B is 2 parts water to 1 part lemonade powder. How much of mixture A and mixture B should you mix tin order to get a quart of a mixture that is 3 parts water to 1 part lemonade powder?

Nightmares are made of this stuff, but let's look at the steps you must take to do this problem. There are different ways to do this problem, obviously, but I would guess that they boil down to the same set of steps, more or less.

Step 1. First, you must figure out what fraction of each mixture is powder vs. water. This involves converting ratios -- such as 1 part lemonade powder to 5 parts water -- into equivalent fractional parts: i.e., 1/6 of this mixture consists of lemonade powder. Mixture A is 1/6 lemonade powder, mixture B is 1/3 powder, and the mixture to be created is 1/4 powder.

Step 2. You have to identify what you want to find; in this case, the unknown is the number of quarts of mixture A (once you know how much mixture A you need, the remainder needed to make a full quart is mixture B). You have to give this quantity a symbol, say x, with an associated unit, say quarts. This step seems trivial, but it's far from it (see the endnote).

Step 3. You must derive an algebraic relationship between the amount of mixtures A and B that you can solve for the unknown. Most reasoning methods will lead you to a conclusion similar to this one: if you have x quarts of mixture A, then you have 1-x quarts of mixture B, and the resulting mixture will have a proportion of lemonade powder that is expressed as:

1/6 x + 1/3 (1-x).

The correct value of x will have to satisfy:

1/6 x + 1/3 (1-x) = 1/4.

Step 4. Manipulate the above equation until you have isolated the variable x and obtained x =1/2.

This involves first multiplying out the terms in the above equation, then isolating x on one side of the equation, then solving. Isolating x correctly will give you the equation

1/3 - 1/4 = x (1/3 - 1/6).

You must then perform the fraction computations, and finally solve for x.

Step 5. Interpret this solution correctly to yield: 1/2 quart of each type of mixture is needed.

So, solving such a problem involves at least 5 separately identifiable steps. The "deepest" one -- the one involving the most insight and the least plug-and-crunching -- is step 3, in which the student derives the relationships among the given elements of the problem, and figures out what must be done to finish out the problem. That's the part of the problem that one would hope would take most of a student's effort and energy.

However, I've taught a lot of kids (in 'college algebra' classes) how to do this sort of problem, and step 3 is not the step that really flattens the kids. It's mainly step 4 that does that; the manipulation of the symbols in the equation, and the addition and subtraction of the simple fractions involved. Not far behind step 4 in difficulty is step 1, conversion of the ratios of the mixture's components to the fractional part that's lemonade powder. They haven't learned this stuff to the point of automaticity.

What does a student see when he looks at the steps involved in doing this problem? If he knows he can perform step 1 -- the conversion from ratios to fractional parts -- then that task shrinks to a point. The student knows he can do it without any effort, and discounts it from his 'energy budget' -- the effort he knows he'll have to expend to solve the problem. If he knows he has mastery of the algebraic symbol manipulations and fraction calculations involved in step 4, then that step also becomes one that has no cost for him. He realizes that most of the cost of doing the problem will come in step 3.

Step 3 really cannot be completely automated, as every problem is unique, so that step will always impact a kid's cognitive energy budget. The other steps should have no cost for a student; they are completely automatable. They should be the easy stuff.

Now imagine that you are a kid facing ten such problems for homework. You know that the equivalent of 'step 3' is going to be a challenge for every single problem. For the kid who has practiced the other skills to automaticity, that's the only challenge he'll have to face; and that's as it ought to be, since it's presumably that higher-level 'step 3' functioning he's trying to learn in this lesson.

But if you are a kid for whom each of these steps demands high-level mental energy, you are going to run out of steam sooner, get much less out of the current lesson, and remember (in all probability) not what you learned on this occasion, but how hard and painful each and every 5-step problem was. With memories of mathematics like that, no wonder you run from math at the first opportunity.

This is why drilling these procedural skills to the point of automaticity is so critical.

endnote

Step 2, identifying the unknown, is also a real hurdle for many students. I discovered when I was teaching algebra that it takes intellectual gumption to select an unknown and give it a variable name, and that sheer timidity made this necessary step difficult for practically all of my algebra students. Practice in identifying and naming unknowns seemed to help here -- simply having "permission" from the teacher to take this step seemed to help as well. If I were teaching now, I'd probably spend some time just drilling this one step for a number of types of equations.


LearnersAreFragile 12 Oct 2005 - 01:47 CarolynJohnston


Our discussion of this math problem, and its value as a problem weighed against the effort of doing it, got me wondering whether we shouldn't be stepping more carefully than we do when we give a learner a math problem. People are tender, and discourage easily. Teachers have to fight to keep learners absorbed and engaged and moving forward; we need to anticipate demons, and head them off.

misconceptions

Learners pick up misconceptions so easily. There have always been many ways to acquire misconceptions, but technology offers some new ones.

Bernie was telling me last night that when he was a student in trigonometry in high school, he had a good teacher. When it came time to learn about sine and cosine, she drew a unit circle on a coordinate plane, put a point on it, and drew the radius from the origin out to the point. She then labeled the angle.

x is sine of this angle, and y is cosine of the angle, she said, and labeled them that way. No! Wait!, she said; it's the other way around! She fixed the labels so that they were correct.

"She only had the wrong thing up for a minute," he said, "but ever since then I've had it stuck in my head that way and I have to think about it to be sure I've got it right."

too much work, too little payoff

He told me another story, of a more recent vintage, about a brilliant coworker of ours. This guy is a brilliant hardware engineer from Caltech, who worked at the Jet Propulsion Lab and was involved in the development of the capture system for the Magellan mission's synthetic aperture radar. This guy is always learning things -- he is one of the most intellectually lively people you could ever hope to meet.

This coworker had a copy of Bernie's book, *Numbers and Symmetry", and was working through some of the problems. One day he came up to Bernie and said, "Is this problem hard?"

As it happened, it was a problem that Bernie and Fred threw into the mix of problems, as they were writing the book, without really test-driving it first (I was there, and I can attest that while they took huge care with the text, the problems were something the editors encouraged them to add at the end of the process). "This problem was harder than I expected it to be," Bernie told me last night. "It was a challenge problem, but I just tossed it in with the others. It should have had some kind of warning on it, two stars or something."

"I think it is hard," Bernie told our friend. "It's been a while since I thought about it."

"I've been working on it for two days," he said, "and I haven't gotten it yet."

After that, according to Bernie, our friend put the book down and has not picked it up again. Bernie feels badly about it. "If I've driven him off," he said, "then there's no question about it: I've messed up. I've failed in my mission with that book. But I was younger when I wrote it, and much more insouciant."

"Yeah," I said, "I was a more insouciant teacher then too."

Insouciant says it, and I think a lot of teachers -- and textbook writers -- are insouciant. Not that teachers can't ever make mistakes; we don't have to be perfect, but we have to understand how delicate even the brightest learners are, and we have to step very carefully. Liping Ma says that teachers in China spend a lot of time discussing the most common student misconceptions with each other, and building up a large knowledge base of multiple approaches to teaching misconception-prone ideas. I really think that's where teacher's efforts ought to be.

I think one intention of constructivists is to try to take that element of fragility out of the learning process, but I don't think it can be done. Inflexible knowledge is a natural stage in the process of learning, and inflexible means brittle. We need to be paying enough attention to nip misconceptions before they flower, and we need to be watching for signs of discouragement.


GrandmastersAndTheNumber7 10 Oct 2005 - 23:27 CarolynJohnston


I had a beer (okay, a margarita) last night with a few work buddies, including my brilliant hardware guy from a recent post, and it turned out someone in the group had read Kitchen Table Math recently and seen KdeRosa's Terminator Essay, and was confused by all the references in the essay to the number 7.

7, of course, plus or minus 2, is the subject of this classic essay on the limits of our ability to separately manage objects in working memory. We humans have various tricks for getting around this serious limitation on our "RAM" (I'm calling it that because of its resemblance to RAM in your computer, which is the space available for storing information that the CPU is working on).

However, if you want to really feel the number 7 in action in your brain, and get a sense for how we work around it, consider this: most people would find the act of having to memorize a random ten-digit number onerous on the face of it. Yet, most of us do it all the time; we memorize phone numbers, with their area codes. This is because we 'chunk' the data; we recode the area code as a single quantity, and possibly also the exchange (the first 3 numbersin the 7-digit local phone number).

Here's another trick to try. When you look at a small set of pennies on the floor, say 2,3, or 4 of them, you can instantly take in the quantity of them without having to count, just from looking at all of them as a whole. This is called 'subitizing'. Most people quit subitizing at right around 7 objects, and have to start counting.

OK, enough digression: I'm getting to my point, gradually.

Hardware Guy is an avid chess player, and of course he immediately began thinking about this limit on working memory as it pertains to chess. I said I figured there must be some neat trick for 'chunking' scenarios in the mind that allows chess players to get past those limitations; either that or they have miraculous, computer-like powers to analyze a game several steps into the future.

He told me that they've done brain scans on highly competent chess players versus grand masters. The frontal lobes of the competent chess players light up, obviously, when the game is hot; they're analyzing. But they found that, in the same situation, nearly every part of a grand master's mind lights up, because not only are they analyzing, they are also tasking their memories heavily.

A grand master, he said, who is playing a game and lands in a certain configuration, is pulling up memories of every game he's ever seen played that was like the current scenario at some point, and remembering what was done, and whether the strategy was successful. In fact, a big difference between the grand masters and that next level down is the vastness of the grand master's database. He claims that his own memory is highly specialized to pull out and store this information: "If I were to meet a guy I played a chess match with once, 5 years ago," he said, "I couldn't tell you his name and maybe not even recognize his face. But I could easily associate him with the chess match I played with him. I could tell you what we did, what the major offenses were, and who won."

Thus, even in chess -- which is, ironically, generally considered the domain of some of the world's purest and most analytical thinkers -- permanently retained domain knowledge is necessary in order to achieve mastery. I don't want to call this sort of knowledge 'rote', because that implies that it's information that is held without being understood; but it's not procedural knowledge, either. I would rather call it 'static knowledge', because it's valuable reference information that doesn't change, and I believe it's an important component of any form of expertise.


Update

E. D. Hirsch got there first and said it better in this Education Next article.

keywords: terminator magical number plus or minus 2


grandmasters and the magical number 7
KDeRosa's tour de force




DeclarativeProceduralConceptualKnowledge 09 Oct 2005 - 13:43 CatherineJohnson



Having just read Carolyn's post about chess players, I wanted to drop this passage in, which is drawn from an article I came across this week:

More than 20 years ago, Ginsburg (1977) explained that mathematics learning and teaching should emphasize building relationships among ... declarative knowledge (facts about mathematics), procedural knowledge (rules, algorithms, procedures to solve mathematics tasks), and conceptual knowledge (connected web of information).

source:
Effects of consistency and adequacy of language information on understanding elementary mathematics word problems
Leong, Che Kan
http://www.findarticles.com/p/articles/mi_qa3809/is_200101/ai_n8951457/print


I find this a very helpful way to conceptualize the different kinds of memory & learning involved in learning math (& in learning reading, spelling, & writing, I assume).


get your mnemonic device right here

Here's how I remember the term declarative knowledge.

Declarative knowledge is knowledge you can declare.

I declare that 2 + 2 = 4
I have that sentence in my head. It works.


I don't have trouble remembering procedural knowledge, because of the word procedure. The classic example of procedural knowledge is riding a bike. You never forget how. Same way with me and math; I acquired a sturdy procedural knowledge that has never left me. When I first tried to teach Christopher fractions I hadn't multiplied a fraction in 30 years, probably. I still knew how to do it. I did have to check up on fraction division. I thought I was supposed to invert and multiply, but I wasn't 100% sure.

It took me about 2 seconds to check--and I could check simply by trying it out on a simple fraction problem and seeing if I got the right answer--then I had it. Procedural knowledge is sturdy stuff.

So I always remember what procedural knowledge is.

But I came across a definition I'd never heard before over at Wikipedia; procedural knowledge is know-how. I love that.

Know-how is a useful term in more ways than one, I think. Your Core American respects know-how. Even more importantly, your Core American does not respect eggheads & pointy-headed intellectuals.

I say we let radical constructivists yammer on about critical thinking skills.

We're talking math know-how.


consigning 'rote memory' to the Banned Words & Phrases bin

We pro-content types should also make a solemn vow not to use the word 'rote' again, ever.

No more In defense of 'Mindless Rote' essays. Please.

What distinguishes our position from radical constructivism is not that we believe in rote knowledge. We do not. (I do not, make that.)

We (I!) believe in:

  • domain knowledge, aka declarative knowledge (A cognitive scientist probably would not equate these two, for reasons I'll get to in another post. However, in general, I'm going to tend to use these terms as rough synonyms.)

and

  • procedural knowledge (more qualifications: I can't tell whether cognitive scientists universally use the term procedural knowledge or procedural memory to apply to intellectual procedures, like adding & subtracting. Normally you see procedural memory applied to motor skills, like knowing how to ride a bicycle. I'm going to check around. Still, Liping Ma uses the term procedural to describe fluency with the algorithms, and that's good enough for me.



Willingham on rote memory

What is Rote Knowledge?
Much of what is commonly taken to be rote knowledge is in fact not rote knowledge. Rather, what we often think of as rote is, instead, inflexible knowledge, which is a normal product of learning and a common part of the journey toward expertise.

In his book Anguished English, Richard Lederer reports that one student provided this definition of "equator": "A managerie lion running around the Earth through Africa." How has the student so grossly misunderstood the definition? And how fragmented and disjointed must the remainder of the student’s knowledge of planetary science be if he or she doesn’t notice that this "fact" doesn’t seem to fit into the other material learned?

All teachers occasionally see this sort of answer, and they are probably fairly confident that they know what has happened. The definition of "equator" has been memorized as rote knowledge. An informal definition of rote knowledge might be "memorizing form in the absence of meaning." This student didn’t even memorize words: The student took the memorization down to the level of sounds and so "imaginary line" became "managerie lion."

"Rote knowledge" has become a bogeyman of education, and with good reason. We rightly want students to understand; we seek to train creative problem solvers, not parrots. Insofar as we can prevent students from absorbing knowledge in a rote form, we should do so. I will address what we know about this problem, and how to avert it, in a future column.

But a more benign cousin to rote knowledge is what I would call "inflexible" knowledge. On the surface it may appear rote, but it’s not. And, it’s absolutely vital to students’ education: Inflexible knowledge seems to be the unavoidable foundation of expertise, including that part of expertise that enables individuals to solve novel problems by applying existing knowledge to new situations--sometimes known popularly as "problem-solving" skills.




In this passage, Willingham uses the term inflexible knowledge to cover using the algorithms and knowing one's math facts. I'll check his text to see how he defines procedural and declarative.


more t/k




JeanPiagetAndDiscoveryLearning 10 Oct 2005 - 23:34 CatherineJohnson



Piagetsmall.jpg



I'm old enough to remember the days when every 20 year old feminist on the planet was on a mission from God to debunk Freud. HUGE quantities of energy went into lambasting the concept of, just to draw a random example out of a hat, penis envy.

We should have been debunking Piaget.


update

Ed just read this and said, 'Oh, Piaget is guilty of multiple sins. He's one of the founding fathers of structuralism.'

It's always worse than you think.

(Ed: 'He wrote a famous little book called Structuralism.)

Structuralism is available for $.073 from Amazon.


here's Ed on constructivism as psychoanalysis




and here's Ed

in case you're curious



WayneWickelgrenOnWhyMathIsConfusing 12 Oct 2005 - 03:21 CatherineJohnson



Carolyn's post on teaching Ben unit conversions reminded me of a passage in Math Coach:

One of the reasons math is so difficult for children is not the number of facts to be learned, but that many of those facts are very similar to one another. To see this, let's first look at what your child has to learn. It's all in a table with the numbers 0 to 10, the numbers to be added or multiplied, on each side....In such a table, there are just 11 x 11 = 121 facts. This isn't much. Everyday, a child learns far more than 121 facts of this size about other life events well enough to correctly answer a question about them the next day. To remember most facts longer than a day requires just repeating them maybe a day, a week, a month, and a year later. It doesn't require the ten to thirty minutes of daily studying over many weeks that children need to learn 121 addition or multiplication facts. What's the difficulty here?



spaced repetition

I think this is so important, I'm going to pause & repeat the basic principles:

  • Everyday, a child learns far more than 121 facts...about...life events well enough to correctly answer a question about them the next day.

  • To remember most facts longer than a day requires just repeating them maybe a day, a week, a month, and a year later.

  • [Remembering ordinary, non-math facts] doesn't require the ten to thirty minutes of daily studying over many weeks that children need to learn 121 addition or multiplication facts.



why is remembering what you've learned about math hard?

Wickelgren:

It's the similarity between the facts. That is, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3's; they both contain +'s, and they both contain single-digit numbers....

Thus, to a child beginning to learn such facts, the facts overlap in the brain, creating a blur that makes it easy to confuse them and difficult to remember any single answer. In cognitive psychology, this "blur" is called associative interference, which occurs when one idea, A, is linked in the mind to two or more other ideas. It's like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station's music or speech. When the child sees 3 + 5 = ___, all the facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.

Indeed, such interference is probably the main reason why all mathematics is harder to learn than other subjects. In every area of math, unlike nonmathematical subjects, a relatively small number of basic concepts are used to express a large number of facts or more advanced concepts. This situation creates interference because each basic concept activates many other facts or concepts, which in turn interfere with one another.




another pause

  • [Math] facts overlap in the brain, creating a blur....called associative interference.

  • When the child sees 3 + 5 = ___, all the facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.

  • [Such] interference is probably the main reason why all mathematics is harder to learn than other subjects. In every area of math, unlike nonmathematical subjects, a relatively small number of basic concepts are used to express a large number of facts or more advanced concepts.

  • [Each] basic concept activates many other facts or concepts, which in turn interfere with one another.



Ben's associative interference

Back to Carolyn:

Well, Ben has been consistently getting his unit conversions backward. He'll convert, say, 13 meters to .013 millimeters, or 12 meters to 12000 kilometers.

Classic.

First of all, the numbers 13 and .013 are extremely similar. Think how much gunk is getting activated in your brain when you contemplates the numbers 13 & .013. Not just math facts, but Friday the 13th, horror films--I'd hate to see the CT scan of a kid staring at 13 and .013.

Second, Ben's right: in the past small numbers like .013 (as opposed to 13) have been associated with small measures like milliimeters (as opposed to meters). And now, all of a sudden, .013 meters is supposed to convert to 13,000 millimeters (is that even right? I'm getting confused myself), yet another number with a big fat 13 in it......who came up with this system, anyway?

Math stinks.

OK, I didn't say that.

Math doesn't stink.

But math is hard, and that, I think, is a core truth about math & about teaching math, which tells me we should constantly be looking for ways to do exactly what Carolyn did:

[Unit conversion] is a reliable procedure that will get him through these problems, and hopefully work around that rut that was forming. We're going to practice it to automaticity.

And as he goes through school, tricks like this -- using dimensional analysis -- will get him through a lot more than unit conversions, too.




back to the magical number 7 plus or minus 2

Needless to say, Wickelgren is a huge fan of practicing to automaticity.

He is also a fan of chunking:

The human mind possesses a way around this problem, namely to create a new idea that binds together a set of constituent ideas or facts into "chunks"--making them hang together in logical ways, like notes in a song.

I love that!

notes in a song: lovely

That's going to be my conscious goal from now on. I want Christopher's math facts, & my own, to hold together the way the notes in a song hold together.

back to Wickelgren:

Students who are very skilled at math may often do this without explicit instruction, but all students can benefit from instruction that helps them do this.

Using my training in learning and memory, I decided to find a way to help children create chunks in math--to glue mathematical facts together in a manner that creates a kind of mathematical melody that is much easier to remember than a sequence of disconnected notes. The key is forming as many connections between the ideas and facts as possible.

source:
Math Coach by Wayne Wickelgren, page 84


procedural memory: another kind of glue?

Early on, Carolyn wrote a couple of posts that have stayed with me: Swoop and Swoop and The Craft of Math.

Here is Carolyn on teaching Ben the cross-multiplication algorithm:

And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.

What Carolyn is talking about here, I think, is procedural memory, which one usually sees defined as motor skills memory. (I'm not sure that's all it involves, but that's what you see. I'll check Willingham's text.)

Cognitive scientists and their fans always invoke the concepts of overlearning and automaticity when discussing math ed.

But Carolyn is the first person I've seen talk about the importance of procedural memory.


CAD students can't draw

I think Carolyn may have finally solved a problem I've been puzzling over, which has to do with something Temple has told me many, many times. Temple says that students in her architectural drawing class who've never learned to make scale drawings by hand, but only on the computer, can't do it.

I've spent quite a bit of time musing over this. It made instant sense to me, but why?

Why should a person who never learned to draw by hand not be able to draw on CAD?

I've come up with different answers at different times. Mainly, these fall into two categories.

I've thought, Montessori. Multiple learning modes, multisensory this-and-that.....

I've also thought, lack of conceptual knowledge, which is undoubtedly correct; Temple's students don't seem to understand what a scale drawing is. They'll click on a 'door' icon on CAD, and not realize that the door is opening into the chute the cattle have to go through, and that, furthermore, the door is wider than the chute itself, and if a chute is so narrow a normal-sized door can't open up all the way, then a 2000-pound cow isn't going to get through it, either. The drawing looks fine on the computer screen, but would be ludicrous in real life.

Now, reading Carolyn's posts again back-to-back with Wickelgren, I'm thinking: procedural memory. Carolyn is enlisting Ben's procedural memory to teach him math, and, perhaps even more importantly, to remember math. Remembering math is hard.

Procedural memory is incredibly sturdy. No one forgets how to ride a bicycle. Which, when you stop to think about it, is pretty astounding. I bought a bike a couple of years ago. I hadn't ridden one in.....20 years?

I hadn't forgotten how. (Though I was rusty, that's for sure, and I felt unstable & even scared when I first got on the thing. That was strange, like being transported back to childhood.)

So....can you get math into motor memory?

I don't know.

But I'm going to assume you can.

One more reason not to rely on computer programs for math facts practice.


update

I've checked Cognition, The Thinking Animal, & it contains only one brief reference to procedural memory. Along with gazillions of references to concepts that sound exactly like procedural memory.

This is the horror of cognitive science.

There are a zillion different terms for the same thing. It's as if every cognitive psychologist who ever lived re-discovered the same concept everyone else discovered, and named it something new.

Cognitive science is the exact opposite of math. In this respect.

Temple and I used to have nervous breakdowns over this back when we were writing Animals in Translation: Using the Mysteries of Autism to Decode Animal Behavior. Temple would actually comb through books & articles and draw up Equivalence Charts. Implicit memory is the same thing as cognitive unconscious is the same thing as incidental learning etc. (I no longer remember whether that particular equivalence is correct; I'd have to do endless checking to see. Which I'm not going to do.)

Anyway, it looks like the answer is yes, procedural memory does cover skills like knowing how to use an algorithm in math.

What I can't tell is whether the 'kind' of procedural memory you use in learning how to use an algorithm has the same sturdiness as the 'kind' of procedural memory you use in learning to ride a bike.


an illustration of my suffering

Here's a great passage from Willingham's textbook that perfectly captures what Temple and I were dealing with:

An early and important distinction was between procedural and declarative memory (Cohen & Squire, 1980). Procedural memory is memory for skills and is often called "knowing how" memory. For example, if you know how to ride a bicycle, that ability is supported by procedural memory. Declarative memory supports memory for facts and events and is often called "knowing that" memory, such as knowing that George Washington was the first U.S. President.

These hypothetical memory systems are closely identified with the implicit versus explicit distinction, [ed.: "closely"? how close? are they the same thing? not the same thing? or what?] which is a distinction of tasks (Graf & Schacter, 1985). [ed: oh, I get it! closely associated with tasks! not exactly memory per se, but memory for tasks, which isn't exactly the same as knowing-how, but is close...] Explicit tasks are those that directly query memory ("Who was the first President?"), and usually they are supported by the declarative memory system. [ed: usually? but not always?] Implicit tasks do not directly query memory ("Ride this bicycle"); rather, memory is inferred from the participant's performance. [ed: memory is inferred?] Implicit tasks usually are supported by procedural memory. [ed: again with the 'usually'...]

This kind of thing was basically MY LIFE for about two years there.

Don't get me wrong.

I love cognitive science; I majored in cognitive psych as un undergrad. I probably think about Daniel Willingham's American Educator articles once a day at a minimum.

But I don't share E.D. Hirsch's conviction that cognitive science is going to save us any time soon.


procedural versus declarative, and a web site about math & memory

These two pages are short, to the point, and useful:

This page has a Temple-like list of memory definitions & experiments:

Course syllabus & textbook here:

And here's a working document on "Mathematical Memory" by some professors trying to figure out the connection between procedural & declarative memory in math ed.


4834440.gif



oh, swell

The Death of Implicit Memory by Daniel B. Willingham & Laura Preuss (question: is Daniel B. Willingham the same person as Daniel T. Willingham?)

answer: yes


Abstract
The thesis of this article is that implicit memory does not exist.....


It's going to be many years before cognitive science knows enough about implicit memory to help with math ed.


keywords: Wayne Wickelgren associative interference math facts why math is confusing


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
grandmasters and the magical number 7
math brain debunked (by Carolyn)
math professors versus computer science professors




MathIsHard 16 Oct 2005 - 21:07 CatherineJohnson



I just found this comment, left by Vlorbik:


math is hard to understand.
but it's easier to understand than anything else.



I love that, though I'm not sure I know what V means by it.

For me, this puts into words something I've been struggling with.

Most non-math types, I'd put money on it, assume math is easy for people who are good at it. I was stunned when I read Carolyn's On having a math brain post, in which she said that everyone 'hits the wall' in math at some point.

On having a math brain is one of my send-outs now, one of the essays & commentaries I keep on hand to give to people. I'll probably be sending it around our school district shortly.

The reason it's important for non-math types to read On having a math brain, or something like it, is that (IMO) a huge number of Life Decisions are getting made on the basis of wrong data, namely that if math is hard for you, then you're not good at it and you should find a different career (or a different math track if you happen to be in 3rd grade).

I keep mentioning Ed's experience as a freshman at Princeton.

He had taken all the hardest math courses at his high school, and it sounds like they really were demanding and serious.

When he got to Princeton freshman year, the guidance counselor looked over his grades & coursework, and told him he belonged in the advanced calculus course for engineering majors.

So that's what he took.

It just about killed him. He didn't understand what was going on, the course moved too fast, all the other students were too smart, etc.

He went to the professor's office constantly, and the professor, a good teacher, spent a great deal of time explaining concepts directly, one-on-one.

Ed ended up with a C.

He concluded from this experience that he didn't have a math brain.

Today, having hung out with math-types here at ktm for awhile, I'm horrified by that story.

A C in advanced calculus for engineering students at Princeton. And this was back before grade inflation.

A C in advanced calculus for engineering students at Princeton told him he was no good at math!

Later on he drew the same conclusion about economics.

He took economics, which he was keenly interested in, found it difficult (not as difficult as advanced calculus for engineering majors), and concluded he wasn't good at economics.

This has nothing to do with work ethic. Ed is an insanely hard worker; he did four hours of homework every single day of his high school career. (That is something I can't even imagine.)

It has to do with wrong assumptions.

Ed assumed that a person who was 'good at math' found math easy. Ditto for economics. A person 'cut out' to be an economist is a person for whom economics courses come naturally. This wasn't as extreme as I'm making it sound. Ed had gone to school with a kid who sounds like a bona fide genius; I think this boy had graduated high school and earned his doctorate in physics by age 20 or something. (I'll check.)

That kid was Ed's standard for: do I have a math brain?


when math is hard for kids

I was on track to make the exact same mistake on Christopher's behalf until the moment I read Wayne Wickelgren on children and math.

Obviously, math wasn't easy for Christopher. It didn't come naturally. He was a 3.

When I read Wickelgren, who said that most children aren't particularly motivated to study math* no matter what their level of innate talent for the subject, the scales fell from my eyes.

On that day I stopped using 'how easy is math for Christopher to learn?' as my standard of judgment.

These days I judge by international standards. If an average child in Singapore studies and masters algebra in 8th grade, then Christopher should study and master algebra in 8th grade, too.

Simple.


back to Vlorbik

Vlorbik's koan (if that's the word) captures Ed's experience, for me.

I've been trying to think why it is I would say 'math is hard' but 'history is easy.'

History isn't easy; history is impossible. (So is writing.)

But taking a history course -- learning history -- isn't hard; at least it wasn't hard for Ed.

Math seems to be hard even at the learning stage, while the soft sciences & humanities become hard at the 'producing' stage, the point at which you yourself are going to try to write history or non-fiction books, or whatever it is you've chosen.

That brings me back to V. For me it's true: math is hard to understand, but once I do understand it (leaving aside the fact that you never 'finish' understanding math) it's easier to understand than most other things.

Reality is opaque. When I try to think about why things happen the way they happen, what the 'big picture' is, why constructivism has been winning for 100 years....it's hopeless.

Why and how a fraction can mean and be four different things--that I have a shot at.

Why and how we have school districts teaching lattice multiplication; forget it.

I'm never going to figure that one out.


late bloomers in math


* I think it was Andy Joy (or Anne Dwyer?) who left a comment saying that she'd loved math as a child, and liked doing multiplication worksheets. I think Wickelgren, in saying most children aren't motivated to study math, is talking about children not being motivated to push themselves through a math textbook at top speed. He was describing an alternative school where children were free to choose whatever they wanted to study. Apparently just about none of the children chose to study math intensively (or even non-intensively, he may have been saying).



MathBrainInLevittown 19 Oct 2005 - 15:43 CatherineJohnson



Speaking of Math Brains, I asked Ed about the kid he went to high school with, the one who convinced him math is easy for Math Brains.

This kid went to Yale at age 17; then, at age 18, sophomore year, entered the Ph.D. program in physics.

In high school he got an 800 on the SAT chemistry test without taking chemistry. He bought a college chem textbook the summer before the test and read it.

He had a 790 on Verbal, too.

Good thing he didn't double-major in history. Ed would have had to be an electrician.



JDFisherOnUnderstandingAndFluency 27 Oct 2005 - 22:40 CatherineJohnson



Understanding and Fluency

Of particular interest to me is the question of whether or not conceptual understanding should be connected with procedural fluency to be considered “conceptual understanding.

This is something I've mused over quite a bit.

Sometimes I'll do a certain procedure or problem so many times that it starts to 'feel' as if it makes sense. Procedural fluency feels like conceptual understanding.

This isn't always the case.

There are some procedures for which I have Major Fluency (multiplying fractions, say) and continue to experience myself as not having conceptual understanding.

As a result, I'm confused. (Yes, that would be something new and different, wouldn't it.)

Half the time, when I think I understand something, I have little confidence that I actually do.

I can see why radical constructivists decided everyone had to explain everything in words all the time.....it really is difficult, in math, to read your own mind.


yes, it's Math Aphasia!

Here's an example.

In my previous post I asked why Christopher's teacher was having the kids 'do the same thing' to both sides of the equation.

I was pretty sure she was right to insist that the kids solve the problems in her much more cumbersome fashion, even though they were apparently all in open revolt.

But in fact, I couldn't say why I felt this until Doug helped me put in words the distinction between 'using an inverse operation' to simplify an expression and 'doing the same thing to both sides' to simplify an expression.

This is a case where in fact I did understand both concepts, and I understood that they were distinct.

Yet I couldn't verbalize the reason why.

Trying to teach math to Christopher is making me feel like a character in an Oliver Sacks book.


more gorgeous graphics

56038789_c7d3142482.jpg


J.D. explains integer tiles!

(Hadn't read the post when I put up the link...)

btw, I'm almost embarassed to say that I came across a Virtual Manipulative of Integer Tiles that I actually liked; in fact, I found it kind of riveting. I don't know why.

I think it does probably have some teaching value, though, because the tiles are labeled with + and -

That's obviously what needs to happen if you're going to use integer tiles.

from J.D.:
56038791_aacfdc15bd_o.jpg



CognitiveScienceMenWomenRedGreen 08 Nov 2005 - 22:45 CatherineJohnson



Several studies presented at the Vision Sciences Society Conference (Sarasota, USA, May 2001) revealed new findings on how we use vision to judge gender. Michael J. Tarr and his colleagues showed that, because of the reflectance properties of hair and skin, adult faces differ in colour according to gender, with male faces containing relatively more red and female faces more green. An optimal model using colour alone can discriminate the gender of a face with 75% accuracy.

source: TRENDS IN COGNITIVE SCIENCE Vol. 5 June 2001, p. 223



Now that is something I didn't know.



FingerLength 30 Oct 2005 - 02:42 CatherineJohnson



Step 1: Take out your centimeter rulers

Step 2: Measure your ring finger from the crease nearest your palm to the fingertip

Step 3: Measure your index finger from the crease nearest your palm to the fingertip

Step 4: Compare

Step 5: Report your results here


ringindexfinger135.jpg



LearningCurvesMathBrain 02 Nov 2005 - 00:55 CatherineJohnson



Each year I come to realize more and more that very few of my students are like me. This even goes for the good students, and I need to stop teaching the type of course where I excelled.

The main differences, I think, stem from my experiences in math classes in sixth through twelfth grades. Don't think that this is shaping up to be an anti-calculator rant. It's not....

Middle School Math
I didn't take middle school math. My parents were absolutely furious when I was in sixth grade. The math teacher had a system by which before the beginning of each unit, we could take the unit test, and if we scored at least 90%, we didn't need to sit through the class. Instead we were given packets to work through (independently) while sitting in the back of the classroom. I learned all sorts of things: the difference between accuracy and precision, all manner of tests for divisibility, combinations and permutations, vectors. Most importantly I learned how to ignore someone yapping in front of a chalkboard and how to learn math by reading a textbook and doing problems until I understood the material. As an added bonus, sixth grade math was the time when most students were indoctrinated that "taking notes in math class" equalled "copying every glyph on the chalkboard onto a sheet of paper." This set the stage for years and years of listening in class instead of taking notes. About halfway through sixth grade the school district was sick of my mom's complaining, so they put a bunch of us in an enrichment class where we flew kites to learn about right triangle trig; in seventh grade they enrolled us in algebra.

Math Homework
From eighth grade through twelfth grade I took seven different math classes, and in none of them did anyone ever check my homework. Homework was assigned, and we were supposed to do it, but no one ever collected it or even walked around the room to verify its existence. (One class also had occasional "problem sets" that were turned in for a grade.) If the problems were interesting, I did the homework. If the material was too difficult to learn just by sitting and listening in class (like max-min problems), I did the homework. If I failed a test (integration by trig subs), I would go back afterwards and do the homework. Sure I made some stupid choices (like doing close to ZERO math homework in all of tenth grade), but in the end I knew what I needed to, and I had learned what I needed to in order to learn math.

My students, on the other hand, seem to prefer being told what to do. Do all these problems by this day. Write this down. Memorize this. Show up in this room at 8am on MWF. And I can't get through to them that they wouldn't need to wake up early and trek across campus in the cold and dark to come to my class if they would just read the textbook and do the problems. That's all it takes: read the book, do the problems. Aside from setting the pace and verifying achievement, I'm inessential. The two things that my students are most reluctant to do (read the book and do the problems) are the keys to learning the material in my class.




Of course I read this and I'm identifying with the mom.


It Seemed Like a Bad Idea at the Time




AbductedByAliens 02 Nov 2005 - 18:27 CatherineJohnson





Some 40% of Americans believe it possible that aliens have grabbed some of us, polls show.


source:
For Space Travelers, Logic Seems to Be A Truly Alien Concept
By SHARON BEGLEY
WSJ October 21, 2005; Page B1



JumpPhilosophy 09 Nov 2005 - 22:50 CatherineJohnson



At JUMP, we believe that children - especially young children - are much more alike in potential than we're currently led to believe, that special abilities can be fostered in most children and that intelligence is much more plastic than psychologists generally allow. We have found this to be true also of young adults.

We have already begun to demonstrate that we can raise the level of the weakest students to the point where they can all be good scientists or mathematicians. At this point, sheer intelligence is almost secondary. In the sciences, factors such as passion, confidence, creativity, diligence, luck and artistic flair are as important as the speed and sharpness of one's mind. Einstein was not a great mathematician technically, but he had a deep sense of beauty and a willingness to question conventional wisdom.

Since the JUMP program was developed to help children who have fallen behind to catch up quickly, we would never claim that it is the only way to teach mathematics, or the best: programs based on manipulatives, or which introduce concepts in an order different from the order in the manual, might work as well, or better. We would claim, however, that whatever method is used, the teacher should never assume that a student who fails to understand an explanation is incapable of progressing.

We believe that one day people will find methods to overcome even the most severe learning disabilities, methods that go far beyond the few simple principles we have observed to be effective at JUMP.



Einstein is widely believed to have been autistic within the autism community. (I believe he was.) So Einstein isn't the best example here, because with autism you can get all kinds of strange savant abilities no normal person could develop even with years of training & practice.

The point being, no one knows what's going on with autistic spectrum disorders.

So I'm not sure about the 'strong form' of this philosophy ('sheer intelligence is secondary').

However, I know the 'weak form' is true, because I've seen it myself.

I've now taught kids who have a clear talent for math that isn't recognized or even perceived by the school & its teachers. These are kids who've been borderline special ed, with various academic services & interventions rendered.

For most people, the concept of 'borderline special ed' and 'talented at math' simply do not go together, ever.

But I've seen it, and I believe it.



FormativeAssessnent 19 Dec 2005 - 01:30 CatherineJohnson



Doug's comment reminded me that I'd pulled an OECD article on formative assessment to post:

Formative assessment – the frequent assessments of student progress to identify learning needs and shape teaching – has become a prominent issue in education reform. In fact, Studies have shown it to be among the most effective educational interventions ever reported.

Between 2002 and 2004, CERI examined exemplary practice of teaching and formative assessment in secondary schools in eight OECD countries – Australia (Queensland), Canada, Denmark, England, Finland, Italy, New Zealand and Scotland – and brought together literature reviews from English, French and German research traditions, relating all this to the broader current policy environment.

The resulting publication, Formative Assessment: Improving Learning in Secondary Classrooms, combines those elements to clarify the concept of, and approaches to, formative assessment and its relation to teaching strategies. The culmination of this study was a major international conference organised by CERI in Paris, on 2-4 February 2005. The conference highlighted international research and case study evidence from the CERI study.

CERI will co-sponsor a regional conference on formative assessment in Budapest, on 29 – 30 September 2005....

Beginning in 2005, the project has just started to look at assessment strategies for adult learners. The study will highlight the issues of why, what and how institutions should assess adult students, and implications for policy.



I think this may be the web site that assured me 'adult learners' don't remotely learn the way young learners do, a fact I decided not to learn.

Being an adult learner, not learning that I can't learn was easy.


update

ah-hah

yes, indeed, I have done a bang-up job of not learning the bit about adult learners not learning, because the CERI web site, far from being the bearer of bad tidings about adult learners, is in fact the bearer of the Certain-To-Be-Correct observation that one can learn at any age. (pdf file)

In recent years, brain science has captured the interest of policymakers and educators. Many believe that new discoveries about the brain yield new insights into early childhood and adolescent learning. However, most of the brain science policymakers and educators cite is not new and even this “old” brain science tends to be oversimplified and misinterpreted in policy and educational contexts. Contrary to popular understandings about the brain, most learning is not limited to early critical periods in development. Furthermore, there is no simple relation between the number of neural connections in the brain and rate or ease of learning. What we do know, from psychological studies of the mind, is that rate and ease of learning depend critically on what one already knows, not on one’s age. We should attempt to use what we do know about learning across the lifespan to provide optimal learning environments for all our citizens.


Does that sound like domain knowledge to anyone else?


oops

Nope, wrong again.

This is the web site with the bad news about adult learners, a fact I seem to have learned in spite of the many obstacles created by my advanced age.

Here's the Good Word from Manfred Spitzer, Psychiatric Hospital, University of Ulm, Germany (pdf file):

You cannot train 15 year olds and 50-year olds in the same way, as the younger ones will perform better.

I'm going to forget that now.


what does this mean?

Spitzer recently attended a meeting on the retraining of employees where he said he noted that the official dogma of every learning institute for retraining of employees stated emphatically that age does not matter. However, he says you cannot train 15-year olds and 50-year olds in the same way as the younger ones will perform better, and that this causes anxiety in the older subjects. But this is not officially recognised, and so when Spitzer told them about the declining learning rate and what the consequences should be for educational programmes it was evident that they were doing exactly the opposite. He explained his theory of a more cost-benefit effect: if this type of retraining was more focussed on split groups according to age decline, it would ultimately produce a curve effect, and in turn produce a cost benefit effect. He says when you start to think about such issues it becomes evident that there is an endless list of possibilities of things you can do, and this is what he will now be exploring in his new Transfer Center.


I wonder if the author of this passage is too old to learn to express himself clearly?

Surely not.


KUMON & formative assessment




OlderLearners 09 Nov 2005 - 18:04 CatherineJohnson



I have completely forgotten the subject of my previous post.



SteinbergOnAdolescentBrain 16 Nov 2005 - 01:40 CatherineJohnson



Laurence Steinberg (Beyond the Classroom) is an expert on adolescent development. Here's the abstract from his February article in Trends in the Cognitive Sciences:

Questions about the nature of normative and atypical development in adolescence have taken on special significance in the last few years, as scientists have begun to recast old portraits of adolescent behavior in the light of new knowledge about brain development. Adolescence is often a period of especially heightened vulnerability as a consequence of potential disjunctions between developing brain, behavioral and cognitive systems that mature along different timetables and under the control of both common and independent biological processes. Taken together, these developments reinforce the emerging understanding of adolescence as a critical or sensitive period for a reorganization of regulatory systems, a reorganization that is fraught with both risks and opportunities.


That sounds about right.

Also by Steinberg: You and Your Adolescent, Revised Edition: A Parents Guide for Ages 10-20

So I'm looking at another 9 years of this?


from the article:


...there is growing evidence that maturational brain processes are continuing well through adolescence. Even relatively simple structural measures, such as the ratio of whiteto- gray matter in the brain, demonstrate large-scale changes into the late teen-age years [6–8]. The impact of this continued maturation on emotional, intellectual and behavioral development has yet to be thoroughly studied, but there is considerable evidence that the second decade of life is a period of great activity with respect to changes in brain structure and function, especially in regions and systems associated with response inhibition, the calibration of risk and reward, and emotion regulation. Contrary to earlier beliefs about brain maturation in adolescence, this activity is not limited to the early adolescent period, nor is it invariably linked to processes of pubertal maturation



a car without a driver

To the extent that the changes in arousal and motivation precede the development of regulatory competence – a reasonable speculation, but one that has yet to be confirmed – the developments of early adolescence may well create a situation in which one is starting an engine without yet having a skilled driver behind the wheel.



smarter

Whatever the underlying processes, during early adolescence, individuals show marked improvements in reasoning (especially deductive reasoning), information processing (in both efficiency and capacity), and expertise.



in case you were doubting the Power of Peer Culture

peercultureSteinbergsmaller.jpg


please note:

That red RISK-TAKING LINE?

It only shoots up like that in the presence of peers.


biology out of synch

the development of anintegrated and consciously controlled ‘executive suite' of regulatory capacities is a lengthy process. Yet, adolescents confront major, emotionally laden life dilemmas from a relatively early age – an age that has become progressively younger over historic time due to the decline in the age of pubertal onset and in the age at which a wide range of choices are thrust upon young people, as well as a decline in the active monitoring of adolescents by parents as a result of changes in family composition and labor force participation.


Latch-key teens.

A bad idea whose time has come.


source:
Trends in the Cognitive Sciences, Volume 9, Issue 2, February 2005, Pages 69-74



SleepAndMathematicalCreativity 15 Nov 2005 - 23:47 CatherineJohnson



TRENDS has all kinds of goodies—

The development of mathematical insight, the knack for discovering novel solutions to mathematical problems, might be one of the most erudite forms of learning that we can hope to achieve. However, Wagner and his colleague now report that a night of sleep after being exposed to a class of mathematical problems more than doubles the likelihood of discovering just such a novel solution.


source:
To sleep, perchance to gain creative insight? Robert Stickgold and Matthew Walker, TRENDS in the Cognitive Sciences, Vol.8 No.5 May 2004



ProceduralMathInWikipedia 18 Nov 2005 - 03:44 CatherineJohnson




here



This is interesting. A few weeks ago, when I looked up 'procedural knowledge' on Wikipedia, I found a generic article on procedural knowledge. Nothing about math.

Now a math teacher has written an entry. It's a little unclear, but it's interesting—interesting just to see that a teacher sat down and did it.


ah-hah

Here's the page I found earlier: procedural memory

Procedural memory, not procedural knowledge.



ProdigiesInNewYorkTimes 26 Nov 2005 - 21:24 CatherineJohnson



The New York Times Magazine has an article on child prodigies this week:

The Prodigy Puzzle

Haven't looked at it yet....



WhyDoPeopleHaveMathAnxiety 30 Nov 2005 - 19:28 CatherineJohnson



I was trying to find the name of a math teacher here in NYC who specializes in remedial college students for Rudbiecka Hirta.

So I searched Amazon for the words "math anxiety," and I came up with 16 books.

And suddenly it struck me.

We don't just have a problem with learning math. We have a problem with Fearing and Loathing math.

Of course, I already knew this; everyone knows it. But I'd never really thought about it.

Nobody's learning how to write expository essays, either, or how to spell, but you won't find 16 books at Amazon on Fear of Spelling.

You can find dozens and dozens of books on fear of writing. But these are all written for people who want to write—often want desperately to write—and find themselves blocked.

Math anxiety books seem to be geared toward people who'd just as soon have nothing at all to do with math ever, but have to learn it to get through the Math Gate to wherever it is they really want to go.

inchworm.gif

One of the titles, in particular, struck me as Wickelgrenian:

Where Do I Put the Decimal Point?: How to Conquer Math Anxiety and Increase Your Facility With Numbers
by Elisabeth Ruedy, Sue Nirenberg


That's Christopher with 'the minus sign.'

I got it right!

I just forgot to put in the minus sign!


from Russia with love

I think (not sure) the name I was looking for is Sheila Tobias.

This passage is from her book Overcoming Math Anxiety:

One day soon after we began working with people who avoid mathematics, a visiting Russian mathematician stopped in at the Math Clinic. It took a while to explain to him what the clinic had been designed to do, how people who have long ago given up on their ability to learn math need to be recaptrued and motivated to try again. Finally he seemed to understand. Then he began to laugh and with a big, generous smile he commented: "You Americans are all the same. You think everybody has to know everything."

Tobias notes that Russian children apparently have as much trouble learning math as American children do:

A collection of essays on teaching arithmetic in the former Soviet Union reports one child's strategy for getting the right answer as follows: "I add, subtract, multiply, and then divide until I get the answer that is in the back of the book."


Russian kids may have as much trouble learning math as we do, but my question is: Do they have math anxiety?

That kid doesn't sound very anxious to me.

And if they don't have math anxiety, how come we do?


hmmmm

According to Sheila Tobias's homepage at the University of Colorado, Colorado Springs, she is a founding member of NOW.

This can't be the person I remember. I thought she worked in the City College system here in NYC.

I'll have to keep looking.



SusanOnMathAnxiety 30 Nov 2005 - 17:46 CatherineJohnson




As the resident math phobe I can tell you why math anxiety happens:

  • Fear of appearing stupid

  • Fear of being stupid

  • Fear that the stupid person next to you doing math quicker is actually smarter than you

  • Confusion as to how you got that way and why others who don't seem any smarter are more capable at it

  • Massive fear of fractions, percentages and decimals

  • Fear of other's expressions when you can't remember basic computation

  • Massive fear of the teacher's expression if you ask a question about a math problem (This only needs to happen once, but you check with most math phobes it probably happened a lot more

  • Angst during the entire hour of math class that you are going to be called to the board and made a fool of thereby preventing you from learning anything that day about math

  • Angst that if you are called to the board you won't be able to strategically position yourself directly behind the teacher who has his book opened to the answers in the back so that you may directly copy them on the board (Hey, a girl's gotta do what a girl's gotta do.)

And on and on. There's a lot more where that came from, but I think I covered most of the biggies.

blueline.jpg



Steve

I have a sister-in-law who is math phobic. She doesn't talk about it much, so it's hard to trace. In my days of teaching college math, I saw some that definitely got anxious.

My guess is that it is based on not having a good foundation in math; there are gaps in knowledge and skills. It's almost like not being able to read, although with reading, it's clearly defined and recognizable. With math, the gaps can come anywhere along the line, but the result is a person that doesn't quite know what is going on and is feeling very uncomfortable. In my classes, these were the students who wanted to do all problems by pattern recognition; they could do the problem only if it was (almost) exactly like another one they had seen. This is a common survival tactic. Unfortunately, many will come to feel that there is something wrong with them; that they are just not good in math.

Perhaps a student did very well in the early grades and could multiply and divide with the best. Then, maybe he/she got a clunker of a teacher and wound up never really understanding fractions. When I taught college algebra, it was hard to figure out the underlying reason why a student was having a problem. I would have to sit with them one-on-one and ask lots of questions.

Once you fail to master one area of math, it affects everything else. For writing, you may have trouble with run-on sentences or saying exactly what you want, but it's still writing. For math, it's just wrong. Kind of like not knowing how to even begin writing a sentence.

A lot of this would be cleared up with quality K-8 math curricula that focus on mastery of basic skills and do not let the student progress until mastery is achieved. A student might not like math, but at least they won't be anxious about it. They will just say that they hate math. I can do algebra, but I hate it. That is properly-based opinion. That's a good thing.

blueline.jpg



back to Susan

....What Steve said.

If I could impart any wisdom to any of you who teaches or tutors a kid (or adult) with what you suspect to be math anxiety, the greatest gift you could give that person is to remain easygoing calm while identifying the gap "Hey, you seem to be a little confused about...." Followed by, "That's a really common problem." And later, "We'll just go back and get those skills and you'll be fine."

This is what my girlfriend said to me in college (She majored in Education). I remember thinking that it couldn't be that easy, this had plagued me all of my school career. But it worked and I made my first A's in math class since grade school.

One of the key components of the "baggage" of math anxiety is the fear that something is wrong with you (maybe a learning disability) and that no one can help you. Once the tension of all that dissapates you can finally focus on the math. I remember actually being able to understand the teacher for the first time in years when he went off on a tangent. It was an excellent life lesson for me.
inchworm.gif


One of things that helped me a lot was that I was an adult (22) before I had to really face this problem down. That makes a huge difference. I also had to take the course to get my college degree because I had managed to switch from being a BFA to a BA. I have no idea why I did that, but I realized that I was stuck with college algebra. It was horrifying. Looking back, it was a great life lesson

In high school, my parents realized there was a serious problem around junior year. Between chemistry and Algebra 2 it all caught up with me. My parents hired student tutors, but I realize now that that was a mistake. At that point, I really needed a good teacher to diagnose the problem. However, in the swirl of high school angst I'm not sure I would have listened.

It is true, the longer it goes on the more you think you have a learning disability. One of things I realized in college was that I never practided. I had internalized some notion that if I didn't get something immediately, then I would never get it. I never did homework and I never connected failure with practice. I was the victim of my own immaturity, but my teachers never really saw what was going on, and of course, I hid.

To go from failure to A's was quite an epiphany. I realized that I didn't have a learning disability. (I was convinced I did). I realized that my attitude had been all screwed up. However, when I walked out of the final that day I never looked back, that is, until I started hanging out with you guys.

My friend in college (who planned on being a math teacher) was very kind when she said something like, "Susan, you can't do Algebra 2 because you don't have your Algebra 1 skills. We'll just go back and get them, that's all." She was being kind, though. We went back to fractions and even cleaned up a lot of basic aritmetic confusion. Once I started doing math daily, it became easy.

I did tell her not to laugh at any question I had, no matter how stupid. I was often confused by math language because I think of 5 different possible meanings of the word instead of locking down on how the term was being used. I doubt I'm alone. I now really pound away at both boys about what things mean.

blueline.jpg



RH on 3 groups of remedial students in college

I think that with college students that the trickiest part would be to get them started with the right attitude.

As a companion to our freshman-level courses we have self-paced, worksheet based supplements (graded by computer because of the hundreds and hundreds of students involved). One of our biggest problems is with the buy-in: the most frequest negative comment is that the skills in the practice sessions are from earlier courses (such as basic algebra, the sort of stuff that Christopher is struggling with with th edistributive law, etc.) and not more practice on the exact same material taught in the regular course.

We have a widely mixed set of problems to contend with. While some of our students have math anxiety, some of them have too much math self-esteem. Some of our low-level students have gotten straight As in hgih school math, even in courses with names like "advanced math" and sometiems even courses called "calculus." These students (and they are a very sizeable minority) are extremely frustrated by not only being put into a low-level course but also failing it.

And then there are the students from hard situations. Many have had enormous family responsibilities growing up (raising younger siblings, etc) that limited the amount of time/effort that could be devoted to school work. Now they have gaps in their backgrounds AND still have their responsibilties AND are now also working full time.

There are probably almost as many stories as there are students. That's what makes it so hard to be successful with these classes (especially when we often teach them in large classes). And we only have 2100 minutes of classroom time with them (2000 when yoou subtract out the absenses the Wednesday before Thanksgiving and the wednesday before fall break.)

Catherine here: so college professors are looking at:

  • anxious students

  • over-confident students

  • students with extremely challenging personal lives who are working full-time and attending college

blueline.jpg



Steve & Carolyn on teaching math to college students

Steve: The problem with fixing math problems in college is that most colleges aren't set up for that and most students aren't interested. They have chosen their academic path and the algebra course is in the way. They just want to get past it. It's hard to convince college students otherwise. Many of the students in my college algebra class were future nurses and teachers. Outside of plenty of office hours, the course was not structured for diagnosing and solving math problems, and students won't come to office hours unless they are close to failing. Perhaps there are some college remedial courses now that focus on solving these issues, but I think that most of these students are interested only in getting through the course one way or another. It is easy to see why many of these future K-8 teachers are open to teaching math differently.

"My thought about math anxiety is this (tell me if you think I'm right): you either can't remediate it at all (which is not something I want to believe), or you remediate it by giving the person repeated experiences of success in their encounters with math. Which, in all probability, a semester-long crash course in "college algebra" is not going to provide."

The longer it goes on, the harder it is. Their reaction might be that they just want to get through the course (many of my students did, barely), not to prepare for a different field of study. Many students (we used to talk about this in our math department meetings) chose their major based on how much math they didn't have to do. At least it was the other department (like Nursing) that decided on the highest level of math they had to take (Trig, if I recall correctly). I wanted to stay out of that battle.

blueline.jpg



Carolyn: I taught kids in college who had a lot of anxiety around math... a lot of Baggage, as RH put it. It was pretty clear that they needed me to be calm and methodical, but I never really understood it, or the roots of it, and deep down I wasn't really very sympathetic.

I think Steve's take on it is right (and I've heard similar things from Bernie); it comes from not having the basics down, and never feeling that you're standing on firm ground with fundamental operations like fractions.

My thought about math anxiety is this (tell me if you think I'm right): you either can't remediate it at all (which is not something I want to believe), or you remediate it by giving the person repeated experiences of success in their encounters with math. Which, in all probability, a semester-long crash course in "college algebra" is not going to provide.

blueline.jpg



Catherine

This Comments thread gave me a jolt, because I hadn't connected the phrase 'math anxiety' with what I know about the neuropsychiatry of fear.

Now I have.

The glaring fact about fear is:

fear cannot be unlearned

Once you've been in a bad car crash, you've always been in a car crash; you never go back to being a person who's sanguine about cars.

Obviously, fear doesn't have to run your life, and for most of us fear doesn't run our lives.

But it doesn't go away. Fear can only be inhibited, not forgotten. Any fear can be reawakened at any time, and easily so.

Moreover, I assume, though I don't know, that the mental effort it takes to inhibit math anxiety draws resources away from the mental effort required to learn and do mathematics. That's the case with other forms of mental inhibition, so I don't know why inhibiting math anxiety would be different.

We do know that consciously experienced math anxiety interferes with working memory.

...people's intrusive worries about math temporarily disrupt mental processes needed for doing arithmetic and drag down math competence, report Mark H. Ashcraft and Elizabeth P. Kirk, both psychologists at Cleveland (Ohio) State University.

Math anxiety exerts this effect by making it difficult to hold new information in mind while simultaneously manipulating it, the researchers hold. Psychologists regard this capacity, known as working memory, as crucial for dealing with numbers.

"Math anxiety soaks up working-memory resources and makes it harder to learn mathematics, probably beginning in middle school," Ashcraft says.

[snip]

In a series of trials, students first saw a set of letters to be remembered. They were then timed as they performed a mental addition problem. After solving it, volunteers tried to recall the letters they had seen.

High-math-anxiety students scored poorly on both tasks but especially on the mental addition. Their performance hit bottom on problems that involved carrying numbers, such as 47 + 18. However, when permitted to use pencil and paper during trials, they did as well as students without math worries did, indicating an underlying math competence.

The third experiment found that high math anxiety translates into poorer performance on an unconventional number-manipulation task that also taxed working memory. In some trials, for instance, students had to add 7 to each of four numbers that they briefly viewed, one at a time, and then verbally report the transformations in the proper order.

Earlier studies have found that math anxiety temporarily boosts heart rate and other physical indicators of worry, notes psychologist David C. Geary of the University of Missouri in Columbia. Psychological therapies that reduce math worries improve math performance, he adds.*

"Ashcraft's study is the first solid evidence that math-anxious people have working-memory problems as they do math," Geary says.



Having put two and two together, so to speak, I'm even more appalled by the curricular carelessness on display at most American schools.

Engelmann is right. The focus should be on curricula; all curricula should be extensively field-tested before being imposed on young children; and all children should be regularly assessed to make sure that a curriculum is working.

"Is working" means "is being taught to mastery."

There's no excuse for our schools to be turning out huge numbers of math phobias.

And I do blame the schools. No child goes into school terrified of math.

But many children come out of school fearing and even hating math.


*This brings to mind desensitization therapy, the kind of thing you do for people with spider phobias and fear of flying.....I suppose you could have math-phobes sit in a room and just imagine a quadratic equation lying out on a table at a safe distance somewhere, in the next galaxy, say. Then work up from there.




TheExpertStudent 09 Dec 2005 - 03:36 CatherineJohnson



Our curriculum committee is reading this article (pdf file).
Abstract

This article suggests that conventional methods of teaching may, at best, create pseudo-experts—students whose expertise, to the extent they have it, does not mirror the expertise needed for realworld thinking inside or outside of the academic disciplines schools normally teach. It is suggested that teaching for “successful intelligence” may help in the creation of future experts. It is further suggested that we may wish to start teaching students to think wisely, not just well.


Robert Sternberg, btw, is an empirical psychologist at Yale who specializes in 'practical intelligence.' I haven't read this article, yet, but I did send links to all of Willingham's American Educator articles to our Assistant Superintendent of Curriculum as suggested reading. Just in case Sternberg says something that needs countering.





While I'm posting links, Ken left this last night:

Constructivism in the Classroom: Epistemology, History, and Empirical Evidence (pdf file).

I started reading before falling asleep, and discovered that constructivism is linked to deconstruction, Derrida, postmodernism, and all the rest.

Over the previous two decades the emergence of post-modernist thought (i.e., radical constructivism, social constructivism, deconstructivsm, post-structuralism, and the like) on the American intellectual landscape has presented a number of challenges to various fields of intellectual endeavor (i.e., literature, natural science, and social science) (Matthews, 1998; in press). Nowhere is this challenge more evident and therefore more problematic than in the application of post-modernism (in the form of constructivist teaching) to the classroom. Employing constructivist teaching practices is problematic at two levels: (1) there is an absence of empirical evidence of effectiveness; and (2) employing this approach for which there is a lack of evidential support, means not employing instructional practices for which there is empirical support. The purpose of this article is to present an overview and critique of constructivist teaching practices, followed by a brief review of evidenced-based practices in teaching.

I had no idea.

I studied all those folks in film school, then left the field because I rejected the entire realm of postmodern thought and scholarship. All of it, kit and caboodle.

I thought I was never going to see that stuff again.


the bad news

If Matthews is correct, if radical constructivism is in fact linked to post-structuralism, things are worse than I thought. (Which should be comforting in its way, since things are always worse than you think, so this is just one more orderly, predictable illustration of the Basic Principle.)

The entire post-modernist project is based in the notion that there is no empirical evidence, for anything.

Whether stated explicitly, or as more often the case, implicitly, the implications of an epistemological view that contends there is no objective reality has a profound effect on how the process or education in the classroom is approached. An important and necessary question in the educational process must be,“How does one establish and evaluate knowledge?” In order to answer this question, we inherently assume that: (1) there is some correspondence between language and reality; (2) our propositions about our observations are logically coherent; and (3) there is a reliable and systematic method of testing our observations. If there is no reality other than that constructed by language and our narrative lacks internal coherence then the two criteria for verifying any observation have been eliminated and one is left with a relativistic nihilism.


My position on this was always: if there's no objective reality, how come you keep hitting the light switch every time you walk into the room?


key words: Piaget Dewey Vygotsky Derrida



HowToAssessKnowledgeFlexibility 19 May 2006 - 16:05 CatherineJohnson



Tracy just reminded me I haven't found out what showing your work would mean when comparing two numbers.

(I suspect I do know. She didn't take off points for not showing work, so I think she was probably just reminding him to compare each digit starting from the left and then underline the first digit that was different. In any case, I've sent an email asking.)

Anyway, looking at his test again, I found myself staring into the Gaping Maw of inflexible knowledge.

hoo boy

Christopher can compare two negative whole numbers.

He cannot compare two negative decimal numbers.

You have to have nerves of steel to deal with this stuff.


showwork.jpg



The principal told us to set up an apointment with the math teacher and ask her to do more formative assessment, more guided practice, etc.

I was surprised by that. To me it feels like stepping over a boundary.

But if it's not stepping over a boundary, great. I do know a fair amount about teaching math at this point; at least, I know a fair amount about practice, overlearning, and and flexible and inflexible knowledge.

I've already sent her the Carol Gambill method.

Here's my question for all of you.

Do you have any ideas about how Ed and I can assess the flexibility/inflexibility of Christopher's math knowledge?

The new timed practices we did for the quiz seemed to work great. (We'll see.)

Speed and accuracy tell you something.

I'm also writing timed practice sheets that combine separate skills. That's where all the problems seem to start.



can you lose skills?

Carolyn and I were talking about this the other night.

Another mom in town told me that TRAILBLAZERS is confusing her son so badly that he's losing the knowledge he came in with.

I feel like I'm seeing that with Christopher.

Skills he seemed pretty strong on, like comparing decimal number size, are crumbling.

The way it seems to work is that he's learned a skill pretty well; at least, he can do it quickly and accurately in isolation.

Then suddenly he has to put a gazillion different things together, and the whole edifice collapses in a heap.

That's about as specific as I can be.

I remember feeling this way myself from time to time.

There've been moments where I felt like nothing made a lick of sense.

And I know I have overlearning on basic algorithims and skills.

Every once in awhile—especially if I'm tired—I'll look at something like a percent problem and think, What is that?

That's probably a different issue.....but on the other hand, maybe not. Maybe Christopher's having brain freeze.

If you have thoughts—either about the issue of math regression or about how to assess math regression & math progression here at home, let me know.



regression in autism

I realize the idea of math regression may sound silly to most of you.

However, regression is a huge issue in autism. Huge and painful. You can have a child who's coming along pretty well suddenly lose everything, months of learning gone.

Generally speaking, the things that happen to autistic people also happen to normal people, in milder forms.

So I'm wondering whether math regression might be real.



ThankYouKtmContributors 17 Dec 2005 - 22:23 CatherineJohnson



I was telling Carolyn the other night how rhetorically powerful it had been to open our meeting with the principal by contrasting IMS's Grade Contract to the contracts posted by Ken and Smartest Tractor.

Those documents anchored & defined the encounter.

Carolyn said, 'Well, you've been putting a huge amount of energy into Kitchen Table Math [true], and it's coming back to you."

It sure is.

If I remember to do it, I'll start taking notes on my daily experience reading Contributors' notes. Offhand, I'd say I learn something new every day. Frequently, I find a new way of seeing something I already know, which is exactly what I need, and what I'm looking for.

By the way, I've given everyone a promotion from Commenter to Contributor. Congratulations! (ummm....Carolyn, OK with you?)

I'm guessing the answer is yes, but since I have a Rule against making Unilateral Decisions, I should say that in my own mind I've given everyone a promotion from Commenter to Contributor. I'm an expert on this kind of promtion, btw. This is the kind of promotion where you get to do even more work for the same amount of zero-money you were already pulling down to do the work you were already doing. (Have I mentioned I spent 7 years as a trustee of the National Alliance for Autism Research?)

In other words, this is the kind of promotion your basic PTSA-Mom spends a lot of time getting, the difference being that you guys get way more appreciation than PTSA moms ever get. Trust me. The world of Volunteer Moms is the world of No Good Deed Goes Unpunished. Like, um, agreeing to teach Singapore Math because the chair of the After-school program wants her son to take the course and then having the Superintendent tell the president of the PTSA that you're undermining TRAILBLAZERS.

oops!

off-topic!

A tiny riplet of suppurating rage escaped me there.

Sorry.

I'm joking. (I really am. Today is a very good day.)



Rudbeckia and Susan J on the greater-than and less-than signs

It just happened again.

I asked for thoughts about assessming flexible/inflexible knowledge at home, and Rudbeckia and Susan J left these comments:

I'm wondering if Christopher is having trouble with the comparisons (deciding which is bigger / smaller) or with the notation (which symbol to use).

[and then:]

I'm asking if he has trouble deciding which symbol to use while simultaneously remembering which number he had decided was larger.

[here's Susan J:]

I'm with Rudbeckia. My sister is an experienced OT and a generally sane and competent person who is required to use > and < in her reports of patient progress. She cannot remember which one means which and she starts screaming if I try to tell her how to remember. She's solved this problem with a sign over her desk with the these two symbols matched to what they mean in words.

By the way, if I call them "left angle bracket" and "right angle bracket" when I'm talking about some computer terminology, she has no problem at all telling them apart.




I'm blown away by both these observations.

Seriously.

I've known what the greater & less than symbols mean for so long, they're devoid of meaning, mystery, or possibility. (Though I expect that to change as I move into algebra.)

The idea that anyone could see them as anything other than mundane and obvious is new to me.

My favorite definition of art, btw, came from the Russian constructivist (any relation to our current constructivists? I have no idea).

They said art was the familiar made strange.

I often have that experience reading Contributors' posts.

I love it.



greater-than, less-than, the mnemonic

In case there's anyone who doesn't know this (I didn't), these days they teach kids that the 'big' part of the greater-than/less-than signs always points toward the big number, while the point of the sign points to the small number.

2 < 3
3 > 2

That has worked well for Christopher, though now Rudbeckia's got me wondering....



learned something new, too

Susan J also said:

BTW, being a programmer, I'm not at all sure whether 0.9066 and 0.906600 are equal or not. They may look equal on a print-out but not be equal inside the computer.

[here's Doug:]

Whether 0.9066 and 0.906600 are equal or not, they aren't equivalent. The second implies two orders of magnitude better precision. I'd probably use ≅.

Of course, in a math class, numbers are all presumed to be precise unless explicitly noted otherwise, so = is the correct symbol here.



Cool.



TracyOnDyspraxia 20 Dec 2005 - 13:04 CatherineJohnson



This is hilarious:


Dyspraxia - clumsy child syndrome. Except it doesn't go away when you get older.

Basically the bit of your brain that deals with movement isn't that good at it. Muscles are fine, decision-making is fine (e.g. should I run over there?), something just goes wrong inbetween. I have a mild case, which mostly shows up in speech - I can hop and do cursive writing and type and drive and etc, and I learnt most of how to talk without any intervention. Some kids are much worse off.

I think what happens is that my brain directs extra neurons to the job of telling my muscles what to do, so when something mentally demanding is going on I just can't manage everything.

I was extremely lucky:

  • When I was little, in NZ there was Plunkett. Plunkett was an organisation designed to ensure the welfare of NZ kids. One of the ways it did that was by monitoring children's progress. So the Plunkett nurse would ask Mum if I was jumping, or hopping, etc, and Mum would lie her head off and say I was, and then that night Mum & Dad would figure out how to teach me to jump or whatever.

  • When I was a bit older and in daycare, the childcarer noticed I had a speech problem which was bad enough for the government to spend money fixing it.

  • They didn't have a diagnosis of dyspraxia then. Their diagnosis was that I was lazy. Luckily I didn't believe I had any speech problem and thus didn't believe I was lazy.

  • Mum and the speech therapist bullied me into learning how to pronounce difficult sounds despite my firm belief I didn't have a problem.

  • When I got to school I was good at maths and reading. Lots of kids with dyspraxia get diagnosed as stupid or lazy. If you regularly get over 90% in maths test, there's a whole sterotype about absent-minded geniuses. So no matter how often I lost a battle of wits with a pencil sharpner, I was still evaluated as smart.

  • I was diagnosed about age 14, when Mum ran into my old speech therapist at the supermarket and she said "Oh, this condition has been recognised, and I think Tracy may have it, send her to this expert."

Apparently it's a diagnosis that's gotten popular in the UK - Joanne Jacobs mentioned a case once of a woman whose son was bad at maths and his teacher told the woman that "He's got dyspraxia - he'll only be able to have a career as a labourer." Which made steam come out of my ears. Firstly there was the immense stupidity of the teacher diagnosing dyspraxia without any real testing, let alone referring to an expert. Then there was the completely wrong career advice - she may as well have told the parents of a blind kid that "She'll have to give up her dreams of being a lawyer and resign herself to being a graphic designer". Talk about playing to your weakness.

Apparently the son's maths problems cleared up once he was properly taught.




We're constantly dealing with this stuff around here. Our two autistic kids can't even write their own names (well, Jimmy can do it; Andrew can't), and I have no idea why or what to do about it, although I have had numerous parent conferences with numerous occupational therapists.

Sadly, I don't know what they're talking about. I find occupational therapy to be as great a mystery as, say, modular arithmetic, which makes me feel stupid, and also makes me feel like a bad mother.

So I'm glad I read Tracy's dyspraxia story.

I feel better now.


oh and by the way—

If Susan J's occupational therapist sister feels like educating us on her profession, my life would IMPROVE.


update

I realize from the Comments thread that this post sounds as if I'm debunking occupational therapy.

That's wrong!

I think occupational therapy is a vastly important field, and I've seen it work. I just have not, ever, been able to understand the concepts. I don't know why. As a result, I've spent years feeling confused and guilty. I've ordered all kinds of OT paraphernalia, then discovered Andrew hates it, or he likes it but I don't use it....I just haven't been able to 'get it' about OT. My latest OT-ish debacle was vision therapy. Not only could I not understand a word out of our vision therapist's mouth, but he told me constantly that the reason I couldn't understand him was that I had visual processing problems. (I've never had any other occupational therapist take this tack, btw.)

Tracy's piece is well-written and funny, and when it comes to OT humor is my only defense. That's why her post makes me feel better—not because I think OT is foolish or overrated. I don't.



IfTheStudentHasntLearned 23 Dec 2005 - 22:16 CatherineJohnson





ktmTee3.png



revision

From Catherine:

Our new pretend-shirt specifically says "If the student hasn't learned, the school hasn't taught," not 'the teacher hasn't taught'.

No more thoughtless (and unintended) teacher-bashing.

Seriously. I'm the last person to want to make teachers feel blamed and bashed, seeing as how half my relatives have been or are currently teachers. I'm sure I'll be one again at some point, too.

The problem is that, when you talk about schools, it's the teachers who are visible. They're in the trenches, so they get the blame. (I realize I'm not telling teachers anything they don't know.) I know better than that, but I've been sounding like I don't.

Time for a course correction.

From Carolyn:

Hey, my entire family on my mother's side were also teachers, every man and woman Jack of them. I've been a teacher too; so has Catherine.

My observation is that policy flows downhill in a school, and the buck stops with the teachers. They get the responsibility, but not the authority; policy changes really have to start with upper management.

We're here to put the pressure on upper management, and support the teachers in doing what they know how to do.


TheLearningBrain 21 Dec 2005 - 14:13 CatherineJohnson



wow!

My new copy of Trends in Cognitive Neuroscience just arrived, with a review of this book:

blakemore140.jpg


The Learning Brain: Lessons for Education by SARAH-JAYNE BLAKEMORE (cognitive neuroscience, University College, London) and UTA FRITH (cognitive development, University College, London).

Frith may be the most important autism researcher we have; she'd certainly rank in the top 5. (Carolyn?)


table of contents

1. Introduction

2. The Developing Brain

3. Words and Numbers in Early Childhood

4. The Mathematical Brain

5. The Literate Brain

6. Learning to Read and its Difficulties

7. Disorders of Social-Emotional Development

8. The Adolescent Brain

9. Life Long Learning

10. Learning and Remembering

11. Different Ways of Learning

12. Harnessing the Learning Powers of the Brain

Appendix

Glossary

References

Further Reading

Index


The Introduction (pdf file) is posted online. If it's half as good as I expect it to be, I'm ordering the book today.

I have to get to Andrew's field trip, so I'll post the TRENDS review later. Looks like it's very positive.


politics, eduation, & cognitive science

At the time, the Early Years Education subcommittee was holding an inquiry into the appropriate care and education of children between birth and six years. The subcommittee had been bombarded with letters, reports, and manifestos from early years charities, schools, psychologists, and educators, many of whom cited research on brain development as grounds for changing early years education in the UK. Some of the arguments put forward contradicted each other. On the one hand, some argued that formal education should not start until six or seven years old because the brain is not ready to learn until this age. On the other hand, others argued that it was clear from research on brain development that children should be “hothoused”—taught as much as possible as early as possible. What were the Members of Parliament on the subcommittee to make of the conflicting evidence?

Both authors were engaged in these kinds of debates when, in June 2000, we compiled a report for the Economic and Social Research Council (ESRC) to indicate whether insights from neuroscience could inform the research agenda in education.





ProceduralLearning 20 Dec 2005 - 15:39 CatherineJohnson




basalganglia.jpg



AgingBrain 27 Dec 2005 - 23:40 CatherineJohnson




agingbrainlg.jpg


But can the aging brain learn calculus.

That's the question.



OrganizedStudentWakeUpCall 18 Jan 2006 - 13:50 CatherineJohnson



OrganizedStudent.jpg


This is what I don't get.

This child goes clear through 6th grade turning in no homework.

His mom gets the Call in......May?


source:
The Organized Student




TracysFamilyRules 08 Jan 2006 - 23:24 CatherineJohnson



from Tracy

The formulation my family uses is:

Unconscious Incompetence Or don't know what you don't know. E.g. when you start skiing you're falling all over the place and don't know why.

Conscious Incompetence You know that the reason you're falling over all the time is that your skis keep crossing, but that knowledge doesn't stop it happening.

Conscious Competence Your skis don't cross but you have to concentrate on it.

Unconscious Competence You don't think about skiing. You think "Hmmm, I'm going to ski over to that point there" and then you do.




I love this.

I look forward to one day achieving unconscious competence in.....um.....ANYTHING AT ALL.

Ever since Christopher starting flunking math, I've been IMMERSED in UNCONSCIOUS INCOMPETENCE.

I'm falling all over the place.

I don't know why.

I don't like it.



on experiencing the Peter principle in the privacy of your own home

The horror is:

NOW I HAVE TO TEACH MY DISORGANIZED KID TO BE ORGANIZED.

I have now officially risen to the level of my incompetence.

I can teach math without knowing any math.

I can't teach organization.


out of the mouths of babes

Last night Christopher asked me where his KUMON sheets were.

When I rapidly located his KUMON box on my desk (not where it belongs) and pulled out the sheets, he said, 'How'd you get so organized?'

You probably have to have a specific learning disability in organizational skills to think I'm an organized person.

Seriously.



AccordionFileForTheOrganizedStudent 09 Jan 2006 - 20:28 CatherineJohnson



971_large.jpg


This is one of the systems Donna Goldberg recommends for middle school kids.

She says middle school is getting so much more complicated than it was just 10 years ago that a lot of kids are switching from her preferred system, the zippered binder, to the accordion file.

Christopher's zippered binder comes with a small accordion file of its very own.


Mead%202in%205star%20binder%20from%20Target.jpg

(this isn't it. this is the zippered binder linked to on MrsKsPlace.)

UPDATE 7-23-2006: We've given up on zippered binders. They explode in two weeks. We're using the Globe-Weis Fabric Poly Files now and they work great. Christopher managed to get through 4 months of school with one file. His friend M. has one, too, and all is well.




ThreeHolePunchForPacketWorld 10 Jan 2006 - 02:10 CatherineJohnson




3-holepunch2.jpg


Swingline® LightTouch™ Desktop Hole Punch
12-sheet capacity
Item 506360
Model A7074026



Middle school these days is all about PACKETS.

I don't know why.

If J.D., Charles, Greta, or Carolyn Morgan are around, maybe they can fill us in.

I didn't have a gazillion PACKETS to keep track of when I was a kid.

Of course, my school didn't have a Xerox machine, either. There's probably only so many mimeographs a teacher can stand to run off before she gives up and just teaches the stuff that's in the book.

Our new system around here, thanks to The Organized Student, is that each night Christopher or I will 3-hole punch that day's PACKETS, and put them in his zippered-binder where he can find them.



breach of copyright

Actually, I do know one reason why we have so many PACKETS.

Schools are Xeroxing old copies of 'consumables' (workbooks) instead of buying new copies as they're legally obligated to do.

Yet another reason why schools should get out of the character education business.



a fancy math packet cover


mathpacketsmall.jpg


Don't say I never gave you anything.


compare and contrast

Interesting.

The reason this math packet cover was produced by a PTA is that the Renton Park Elementary PTA sponsors all kinds of educational activities, such as reading programs & summer math.

That's as opposed to the Irvington PTSA, which shuts down its Singapore Math course the moment the Superintendent levels anonymous charges against the parent-teacher, a long-time PTSA member and volunteer — and agrees, furthermore, that the PTSA will henceforth offer no after-school courses that cover the same subjects already covered by the school. ("For instance," the president said, "we might offer Chinese, because Main Street School doesn't offer Chinese. But we wouldn't offer Spanish.")

Perhaps our upcoming PTSA Forum should be apprised of Renton Park's involvement in math & reading.

I'm thinking.....why, yes!

I think this is something people need to know!

This is something parents need to know!



PrufrockPress 13 Jan 2006 - 00:53 CatherineJohnson



What do people know about Prufrock Press? Apparently it's an entire press devoted to gifted children.

And do the activities in this book look worthwhile?


700.jpg



I have a sneaking suspicion this is what differentiated instruction means for gifted children in the slow track.

Maybe I'm wrong.


from the book:
magic squares

simple closed curves

Fibonacci numbers



BayesianRules 16 Jan 2006 - 14:33 CarolynJohnston


I've been a Bayesian ever since I understood enough probability to know the difference between a Bayesian and a frequentist (these are two different schools of thought about probability and statistics).

Last August, I convinced Catherine that she is a Bayesian too.

Now it turns out that we're all Bayesians. This week's Economist has an article on some cognitive science research that's going to be published this year that claims to prove it.

For a little background before I whet your appetite with this idea, the core idea of Bayesian reasoning is that we reason by updating our preexisting mental 'likelihoods' of events with new information.

A simple example: when you meet a new person, you generally have a low expectation that he has chronic financial problems. Your expectation is based on your knowledge that in the general population, not many people have severe financial problems. However, if you then discover that he was laid off two months ago and had only $200 in the bank at the time, you mentally raise your estimate of his likelihood of having chronic money problems.

This much is pretty obvious. What the new research reveals is that we do perform real-time updates of our initial estimates of probabilities, in our minds, and that the probabilities we form are remarkably accurate. In short, the mathematical formalism of Bayes' formula is part of our innate mental structure, and we use it every day.

In research to be published later this year in Psychological Science, Thomas Griffiths of Brown University in Rhode Island and Joshua Tenenbaum of the Massachusetts Institute of Technology put the idea of a Bayesian brain to a quotidian test. They found that it passes with flying colours...

Dr Griffiths and Dr Tenenbaum conducted their experiment by giving individual nuggets of information to each of the participants in their study (of which they had, in an ironically frequentist way of doing things, a total of 350), and asking them to draw a general conclusion. For example, many of the participants were told the amount of money that a film had supposedly earned since its release, and asked to estimate what its total gross would be, even though they were not told for how long it had been on release so far.

Besides the returns on films, the participants were asked about things as diverse as the number of lines in a poem (given how far into the poem a single line is), the time it takes to bake a cake (given how long it has already been in the oven), and the total length of the term that would be served by an American congressman (given how long he has already been in the House of Representatives). All of these things have well-established probability distributions, and all of them, together with three other items on the list -- an individual's lifespan given his current age, the run-time of a film, and the amount of time spent on hold in a telephone queuing system -- were predicted accurately by the participants from lone pieces of data.

It turns out that we are so good at doing Bayesian analysis in our minds that Tenenbaum and Griffiths think it may be possible to determine the distributions of events in the real world by checking it against our innate Bayesian calculating machinery.

Here's a link to the article -- you need to either pay-per-view, or be a subscriber to access it.

Joshua Tenenbaum

josh.gif

Here's a picture of Dr. Josh Tenenbaum from MIT. He looks young enough to be my son.


Bayes statistics & false positives
does human mind use Bayesian reasoning?
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale

Bayesianprobability



TheAnswerToAllOfDougsProblems 09 Jan 2006 - 17:12 CatherineJohnson




Doug Sundseth wrote:

My branch of the company I work for is shifting focus pretty dramatically right now. The new work is nothing like what we have been doing. We need to document the new stuff.

Right now, we don't know exactly what it is that we will be doing. We don't know what the customer documentation needs to include.

We need an estimate of how many man-hours it will take to complete this documentation. It is supposed to be correct within 25%.

They want this today.



Doug, for challenges of this type, Donna Goldberg recommends the Time Timer:


CU497.jpg



I'm getting one today.


some books that have changed my life
the answer to all of Doug's problems
productivity question
what is an hour? Time Timers
Steve & Susan J & Doug on spiralling curricula
my Time Timer came - how long is a nap?
Time Timer says no!




WhatsTheMatterWithKidsToday 30 Jun 2006 - 11:04 CatherineJohnson



more good news:

SOMETHING IS ROTTEN in suburbia. On average, teenagers who live with wealthy, highly educated parents in tony neighborhoods are more troubled than other teens, even those living in inner-city poverty. Suburban teens smoke, drink and use drugs more than their urban peers and have higher levels of anxiety and depression. Upper-class suburban girls are three times as likely to suffer depression compared with other adolescent girls.

Drug and alcohol abuse often go hand in hand with emotional problems in suburbs. "The implication is that these children are self-medicating," says Columbia University psychologist Suniya S. Luthar, whose study appeared in Current Directions in Psychological Science.




guess whose fault?

"We have a cultural assumption that parents who make more money are more affable, more available to their children than parents in dire poverty." Isolation from parents--both literal and emotional--puts affluent kids at risk.

The study suggested a simple antidote: family dinner. Kids who usually eat with at least one parent have better grades and fewer emotional problems than kids who dine on their own.



Family dinner. Right. We have lots of fun sitting around the dinner table fighting about math.

Obviously suburban schools where kids take 10 AP courses a year if they can get into the course in the first place have nothing to do with it.


Abstract

Current Directions in Psychological Science
Volume 14 Page 49 - February 2005
doi:10.1111/j.0963-7214.2005.00333.x
Volume 14 Issue 1

Children of the Affluent
Challenges to Well-Being
Suniya S. Luthar1 and Shawn J. Latendresse1

Growing up in the culture of affluence can connote various psychosocial risks. Studies have shown that upper-class children can manifest elevated disturbance in several areas—such as substance use, anxiety, and depression—and that two sets of factors seem to be implicated, that is, excessive pressures to achieve and isolation from parents (both literal and emotional). Whereas stereotypically, affluent youth and poor youth are respectively thought of as being at "low risk" and "high risk," comparative studies have revealed more similarities than differences in their adjustment patterns and socialization processes. In the years ahead, psychologists must correct the long-standing neglect of a group of youngsters treated, thus far, as not needing their attention. Family wealth does not automatically confer either wisdom in parenting or equanimity of spirit; whereas children rendered atypical by virtue of their parents' wealth are undoubtedly privileged in many respects, there is also, clearly, the potential for some nontrivial threats to their psychological well-being.



Sure glad we're seeing no signs of anxiety and depression around here.

Actually, looking at these two versions, I see that the researcher has implicated excessive pressures to achieve, which, presumably, could be coming from the schools.

I have to hope she's right family dinners are armor enough to get a kid through pre-algebra in one piece.


did suburbia used to be more fun?

Just asking.



TestMonster 13 Jan 2006 - 03:41 CatherineJohnson



can this possibly be a good idea?

I don't think so.

But maybe I'm wrong.

10test.1841.jpg

This green guy is a test monster:

Stressed. Scared. Nauseous. Sick. These were some of the words that the 9- and 10-year-olds at Public School 3 in Brooklyn used on Friday to describe how they felt about the state fourth-grade reading test that they will take over three days beginning today.

But that was before social workers introduced them to a Test Monster, an art project designed to exorcise fears of standardized tests. Markese Taylor, 9, took one look at the Test Monster he was given - an outline on paper of a beast that looks like a cross between Bart Simpson and a Muppet - and - brandishing a purple marker, declared, "Ooooh, I am going to hurt you!"





oops, now I'm scared

And while many local districts, including New York City's, previously gave their own reading tests in third grade, most of those were strictly multiple-choice. The new state test will include essay questions in every grade.


I told Christopher he has to do some practice writing before he takes the test next week. Also improve his handwriting and learn to spell.



it's a jungle out there

"Oftentimes you have kids who just fall apart during the test; they just start crying or having a temper tantrum," said Barbara Cavallo, clinical director for Partnership with Children, a nonprofit group that works in the city schools.

Ms. Cavallo, who created the Test Monster in the 1990's, said that interest in the program among school officials had increased recently. "Through the years there has gotten to be much more pressure on the children, and there has been lots more pressure placed on schools to show performance," she said.

That is true across the nation, as officials seek to reverse decades of lackluster results in schools by setting higher standards, as measured on tests, and by imposing penalties that get more severe over time if schools keep falling short.

"Certainly every teacher that we talk to, every principal, is screaming that it's getting worse," said Rollin McCraty, the research director of the Institute of HeartMath, a California-based research and education group that recently completed a survey of test anxiety among students in eight states.



Yeah, well, this is what happens when you do ZERO formative assessment for the entire school year.

Can the children read?

Can the children write?

Who knows?

It's all a big mystery!

I say we turn the whole entire Department of Ed over to Seigfried Engelmann.



Mr. Kadamus said that in prior years, teachers in grades other then fourth and eighth might not have felt responsible for test results. With exams in all those grades, he said, teachers must pull together.

"It's up to the teachers to say we have got a coordinated program across every grade level," he said.



I would love to see that happen.

The way things stand now, 4th grade has been very intense, and then 5th grade is a stoll in the park (or so people tell me).



it's all a big mystery, part 2

At P.S. 3 in Bedford-Stuyvesant, the students in Class 4-321 were offered a "special friend" on Friday to help ease their fears: the Test Monster. Using the outline of the beast, students drew in features, often ferocious and ugly, and wrote down their fears. Then they crumpled the drawing and locked it in a cardboard box.

Lamar Butler, 10, drew red eyes, a purple tongue and dark green fangs. On the monster's belly, he wrote: "scared." Shakima Daniels, 9, drew a butterfly in her monster's stomach.

Their teacher, Erin Dempster, said she had urged the students to close their eyes and visualize getting the highest score on the test. She said many students were worried about having to attend summer school, and that she was worried for them.

"I need a Test Monster, too," she said, "because there's so much pressure."



Yeah, well.....again: if a school is doing formative assessment systematically and continuously throughout the year, people wouldn't be CLUELESS about the academic level on which students are functioning.

10test.2.184.jpg


caption:
Kashaya Miller, a student at P.S. 3
in Brooklyn, discarded her Test Monster
on Friday at the end of the exercise.


I would bet the ranch that giving children terrifying images of a Test Monster is a very bad idea.

Standardized tests are not The Enemy.

Standardized tests are a pain in the tukhus (IS THAT THE WAY YOU SPELL TUKHUS?)

End of story.



StupidInAmerica 12 Jan 2006 - 17:55 CatherineJohnson



Ken left a link to John Stossel's special 'Stupid in America' tomorrow night at 10. (January 13, 2006)

Jan. 9, 2006 — American students fizzle in international comparisons, placing 18th in reading, 22nd in science and 28th in math - behind countries like Poland, Australia and Korea. But why? Are American kids less intelligent? John Stossel looks at the ways the U.S. public education system cheats students out of a quality education in "Stupid in America: How We Cheat Our Kids," airing this Friday at 10 p.m.

"We're not stupid. & But we could do better," one high school student tells Stossel. Another says, "I think it has to be something with the school, 'cause I don't think we're stupider."

That's the question Stossel examines in his special report: What is it that's going wrong in public schools?

There are many factors that contribute to failure in school. A major factor, Stossel finds, is the government's monopoly over the school system. Parents don't get to choose where to send their children. In other countries, choice brings competition, and competition improves performance.

Stossel questions government officials, union leaders, parents and students and learns some surprising things about what's happening in U.S. schools. He also examines how the educational system can be improved upon and reports on innovative programs across the country.

"Stupid In America: How We Cheat Our Kids" with John Stossel airs Jan. 13, at 10 p.m.



I'm setting up the TIVO.



SteveAndSusanJOnSpiralCurricula 16 Sep 2006 - 20:10 CatherineJohnson



from Steve:
Spiraling

"... nothing will be taught to mastery ..."

Mastery requires grade-level expectations.

Mastery requires practice.

Mastery requires testing.

Spiraling (as it is applied in these cases) is used to avoid mastery (a.k.a. Drill and Kill)

Spiraling is used as a pedagogical excuse for social promotion. (no tracking and no holding kids back either) This is OK, because they think that there is no linkage between mastery and understanding. They think that everything will work out in the end. They want their pedagogy and eat it too.

Please note that this is not what I would call spiraling, which is a fine technique for both learning and solving problems. Sometimes, when a design project is very large (building, bridge, car, ship), you cannot start at point A and go to point B to finish the design. You start with a conceptual design phase, spiral around through a preliminary design phase (same analysis, but with more details), and then go on to a detailed design phase. It's called a design spiral. (each step of which could involve a complex calculation that is done using only one BEST algorithm) Problem solving in the real world is so much more complex than any silly talk of only one answer or many ways of doing things.

I think that educators have "mastered" the art of saying whatever sounds good just to do whatever they want. They argue with generalities, but they get to define the details. They do not want you to see the details!





and, from Susan J —

Spiral learning isn't over-learning, it is just repeated under-learning.

I'm going to be quoting that a lot.



update: it's not spiraling, it's painting a room

Ed says the real metaphor should be painting a room.

Constructivists think it's like putting on several coats of paint.

The first coat is thin and everything shows through; the second coat covers better; the third coat is final and the room looks right.



from Doug:

Under-learned spiralling is like painting a room with a thin coat of paint that isn't washable. Then living in the room for a year, while washing the walls regularly. Then painting another coat of non-washable paint on the walls.

After a few repetitions, there might be a few places where the paint is thick enough to cover, but in most places it's been washed off enough that you can still see the 1930s wallpaper underneath.





my thoughts exactly

Actually, this is something I've been thinking about for a full two days now.

I'd like to know what the actual time-cost is in a spiraling (spiralling?) curriculum.

Engelmann talks about teachers placing kids nearly half a year behind in the sequence each fall.

The meta-analysis of research on summer regression found 1 month loss.

I'm betting that in actual practice Engelmann is closer to the mark. With a spiraling curriculum very little is taught to mastery, and no formative assessment is done, which means teachers down the line have no idea which students have mastered what prerequisite skills — and which probably also means that while most students have managed to master something, what that something is will vary.

Basically, you have Prerequisite Chaos (except for the fact that a math teacher can count on nobody knowing a thing about fractions).

Sounds like there's a multiplier effect in there somewhere.

A couple of them.


Mike Feinberg of KIPP on spiral curricula
Steve and Susan J on spiral curricula
acceleration versus remediation
parents' stories about spiralling curricula

some books that have changed my life
the answer to all of Doug's problems
productivity question
what is an hour? Time Timers
my Time Timer came - how long is a nap?
Time Timer says no!




TeachersStuckOnMastery 16 Sep 2006 - 20:08 CatherineJohnson


from Becky C, a smoking gun:

Investigationsmastery.jpg


Getting stuck in a unit because you are teaching to mastery is a bad thing.

TERC teachers aren't supposed to do it.

Because they'll be revisiting the concept later.

Note: visit.

Not teach.

Not learn.

Not study.

Not practice.

And not master.


This language doesn't happen by accident.



KIPP on the spiral

You know, talk about curriculum, if I put in front of you a fifth, sixth, seventh, and eighth grade textbook in math and opened up to page 200 and I jumbled them up, and said, “order them from fifth through eighth grade in order,” you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of “we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.”....[W]e have a different math strategy and a different math philosophy.


Maybe that's why KIPP Academy 8th graders pass Regents A at twice the rate Irvington students do.


key words: teach to mastery teach to coverage teach to exposure spiraling direct instruction



DataWarehousing 07 Oct 2006 - 22:10 CatherineJohnson



Our school district is now using 'data warehousing.'

The couple who came to dinner Friday night — both employed in math-related fields — were highly unenthusiastic about this development.

My neighbor, the statistician, had the same reaction when she read about it.

The Friday-night-couple said data warehousing is the same thing as data mining.....which I think I favor.

Is that wrong?

I'm certain they're right, though, that data mining will allow the district to flummox parents with whatever statistics they decide to pull out.

Although.....so far district efforts to flummmox parents, namely me, have been unimpressive to say the least. These efforts consist of the Assistant Superintendent sending me one letter and one email telling me 'scores have gone up' since we purchased TRAILBLAZERS.

I pointed out that scores went up all over the state and that, furthermore, 'scores went up' is raw data, and we left it at that.

Color me Not flummoxed.

Then they shut down my Singapore Math course.



not flummoxed now & don't plan to be in the future

What do I need to start learning in order to not get flummoxed down the line?

Apart from real knowledge, comprehension, & procedural skills, I could use some lingo, just so I sound like I know what I'm talking about.

If the District is going to blow smoke-with-data, I need to be able to blow my own smoke, which I can do just through language. (Have I mentioned how ruthless I am lately?)



whose data is it, anyway?

What I fear — because we've hit this brick wall many, many times in special ed — is that parents won't get to see data because parents seeing data will represent an invasion of other parents' privacy.

Maybe things won't go that way, but seeing as how they've always gone that way for us in the past, and seeing as how Bush & c. had to pass a huge, major, revolutionary law just to get schools to disaggregate and publish their data some place where parents could find it, tells my Bayesian mind to count on it.

So maybe I should be familiarizing myself with the FOIA, right?



Wal-Mart has a warehouse for data, too

No idea whether this book would be useful or not.


6556819.gif



-- CatherineJohnson - 16 Jan 2006


IsMiddleSchoolBadForKids 08 Oct 2006 - 22:41 CatherineJohnson



1101050808_400.jpg



Carolyn's post yesterday made me realize I've been 'blaming the student' myself.

I've been thinking that the awful way Christopher and his friends treat each other is developmental, just part and parcel of being 11.

Reading Carolyn's post, I realized that the principle that applies to adults — lots more squabbling when the stress heats up — applies to kids, too.

These children are under immense pressure few-to-none of them are equipped to handle, and what do we see?

We see Christopher's closest friends calling him 'fat.'

Calling him a 'BOCY.' (That's BOCY meaning BOCES, short for Board of Cooperative Educational Services. Primarily a vocational ed outfit, but the kids here all know it as special ed.)

I can't work up a head of steam about the BOCY slam; these kids are still so young and out-of-it they don't even know how BOCES is pronounced.

NOTE to IMS 6th graders:

It's BOCES-with-an-S.

BOCEEZ

Not: BOCEE

What do we see in Christopher?

We see him calling the friend who called him 'fat' an 'anorexic midget.' That is a horrible thing to say.

We see him putting WWE wrestling moves on the friend who called him a BOCEE and making the friend cry.* (I don't have a problem with that. That's what I call Feedback. Call a kid who has two autistic brothers a BOCY, you're gonna get a wrestling move laid on you. It's good to learn these things early. As of last Friday, they'd patched things up.)

The other day, after Christopher had told me maybe 3 times that, 'So-and-so called me such-and-such, so I extended my legs under the table into his shin' I finally said, 'You mean you kicked him?'

Answer: yes.

Just last week, 2 of Christopher's friends got into a fight so bad both have been expelled for 5 days. These 2 boys are enemies dating bck to last year, when one boy began ragging on the other about his sister, who has Down's syndrome. That's what the fight was about this time, too.



aside: moral equivalence alert

For the record, I disagree strongly with the principal's decision to give both kids the same punishment. (Of course, I'm taking Christopher's word for it that they both had the same punishment, so if he got the story wrong I'll have to come back and REVISE.)

I applaud a boy who stands up for his disabled sister. Obviously, a school principal can't endorse punching kids out when they call your sister names, but the consequence for the boy defending his sister should have been far milder than the consequence for the boy who slammed her. When you mete out the exact same punishment to both, you endorse the principle of moral equivalence, and I'm against it. (My feelings on moral equivalence can be summed up in one word: blech.)

The equal punishments business (assuming it was equal!) is yet another reason why I don't particularly relish the prospect of the school teaching 'character.' As far as I'm concerned, the kid who threw the punch demonstrated excellent character. Good for him.



back on topic

The kids just pound each other, verbally or physically, every day. It never ends. Lord only knows what's going on amongst the girls; it's probably a nightmare. Apparently there was some huge girl-bullying ring in 7th grade last year (this is highly secondhand) — and the child who made the two bomb threats this fall turned out to be a girl.

I'd be amazed if IMS is worse than any other middle school. It could easily be better. The principal is a lovely guy (I use 'lovely' in this context because he really is a sensitive soul who doesn't want to see children suffer). [update: we no longer feel this way] And the teachers we met at the team meeting are all friendly, caring people. All but one of them was far too young to be burned out, and the one teacher who is technically old enough to never want to lay eyes on another 11-year old boy as long as she lives obviously loves her job & likes the boys just fine.

It's not the people. [update 5-15-2006: again, we no longer feel this way. The culture of Irvington Middle School is negative.]

I think, as Carolyn does, that the problem with middle school is the structure &mdsah; the structure, and the demands.

I don't think the way I see Christopher and his friends acting is normal, natural, or developmental.

I think the way I see Christopher and his friends acting is a response to stress. They're overwhelmed, they're powerless to affect their environments, and they're turning on each other.

That's my hypothesis.

Do homeschooled 11-year old friends rip each other apart every day?

Doesn't seem like it.

If I could, I would move 6th grade back to elementary school tomorrow. I would also consider establishing an 'elemiddle' school encompassing grades K-8 as other communities are doing. (Not sure about this one, but I'd sure want to look into it.)

I would absolutely establish a policy of teaching to mastery. I would make the school — not the student & not the parents — responsible for knowing whether each student has mastered the material being taught.

And I would make the school — not the parents — responsible for re-teaching material to kids who haven't mastered it.

This would take enormous pressure off the children, who wouldn't have the threat of bad grades and negative Interim Reports constantly hanging over their heads. Every day, they'd be able to see whether they've learned the material; they'd know how they're doing & they'd know exactly what they needed to do next.

More importantly, they'd be doing well, because the school would make sure they were learning.

They would be succeeding.

I would replace the sink-or-swim environment middle schools are today with an environment in which students experience success not due to grade inflation, but due to having learned subjects to mastery.



where achievement goes to die?

We've talked about this before, but it bears repeating.

There are 2 schools of thought about middle schools:

1. Middle schools are the place where achievement goes to die. This is the fourth grade slump hypothesis, which I believe originated with Jeanne Chall.

2. Middle schools are the place where the gap first becomes obvious.

Until the issue is settled, I'm voting for Door Number 2.

Ed came up with a terrific analogy this weekend (I'm sure it's not original, but it was the first time I'd heard it): the achievement gap between American kids & kids elsewhere is like a race, where everyone starts from the same place, and in the early stages of the race, everyone is clustered pretty closely together. It's not till later on that the winers start to pull ahead.

It's not til the end that you see lots of space between the runners.

For now, that's what I believe.

I think middle schools take in kids who are already behind, but not obviously so.

By the time the kids graduate, the gap is obvious.

So the middle school takes the rap.



one last thing

McEwan talks about a TIMSS study finding no gain in math knowledge between the 7th and 8th grades.

While I'm opting for Door Number 2, it's entirely possible that achievement slows in middle school.

I've been thinking about this.

When you don't teach to mastery — when you teach a spiraling curriculum — the kids end up with gaps.

But they probably don't all end up with the same gaps. (Except for the fraction/decimal/percent gap, which is universal. That should be my new life. Set up shop teaching fractions, decimals & percents to a Grateful Nation.)

Seriously, though, think what a middle school math teacher is up against.

Think what Ms. Kahl is up against.

She's got to teach a cram course to kids who have (mostly) not been taught to mastery. (You probably remember that one of the teachers at the PTSA forum said some topics are taught to mastery.) In theory, each kid could have a different weakness, and each kid is going to stumble over new material that depends on the old material he or she doesn't know.

It's Gap Anarchy.

It seems logical that the further you go, and the more gaps you accumulate, the slower the learning curve is going to be, until finally you hit the wall.



still looking for info on KeyMath

I know our school uses Key Math to test special ed kids.

But do we use it to test the regular kids?

Would Key Math tell you exactly where a particular kid's gaps are?

I wonder whether Smartest Tractor or Carolyn Morgan know.

AND WHY IS ALL THIS INFORMATION SO HARD FOR PARENTS TO FIND OUT???

WHY DON'T PARENTS JUST NATURALLY KNOW THAT KEY MATH EXISTS, WHAT IT DOES, & HOW IT COULD BE HELPFUL TO THEIR CHILDREN???

This is my question.



*I just asked Christopher what he did, more specifically. He says he used a move called the 'DDT.' Now we know.


-- CatherineJohnson - 17 Jan 2006


ImCollectingStoriesAboutGaps 22 Jan 2006 - 16:02 CatherineJohnson



Engelmann's Student-Program Alignment and Teaching to Mastery is still rumbling through my Hebbian networks, toppling every domino in its path.

It's kind of fun. I'm experiencing my very own Paradigm Shift.

I don't know where I'll be when things calm down, but one thing I do know: I'm never going to see 'gaps' the same way.



killer Gaps

We're constantly hearing about Gaps, of course. Achievement gaps, learning gaps, teacher gaps — everywhere you turn, there's another Gap.

I've read so much about Gaps I never really stopped to think what a gap actually is, or might be.

I guess I've thought of gaps as static and predictable. All the gaps seem to grow wider over time, until they look like an ice cream cone lying on its side in a PowerPoint slide.

That was then.

Suddenly, gaps seem dynamic, dark, and entirely unpredictable — more properly a phenomenon belonging to Chaos Theory (does anyone talk about Chaos Theory any more?), not Excel charts.



Anne on diagnosing Gaps

What I've noticed with my tutoring students is this: if they don't understand something in math class, they try to find a procedure or "trick" that works everytime.

Since they don't really understand it, when they have to go back and do it on a test or later, they don't remember the "trick" exactly and their answers are consistent, but wrong.

For example, I was tutoring a student in basic math. He didn't really understand that a whole number has an implied decimal after the number (e.g. 3 is really 3. for a decimal problem)

When he first learned to divide decimals and he was following the teachers examples, he was doing the problems right: So if he was dividing .045 into 15, he moved the decimal over three places for the .045 and three places for the 15. He even managed to get it right on the first test.

But he did them wrong on every test after that. When we were studying for the final, I was able to watch him do the problems.

Since he really didn't understand, he made up his own "trick". In the problem above, he would move the decimal over for the .045 correctly, but he put the decimal point in front of any number inside the divisor sign. So .045 into 15 became 45 into 150 instead of 15,000. And, because he had taught himself this trick, he ignored all decimal points inside the divisor sign. So even .045 into 1.5 became 45 into 150.

Needless to say, it took a while to find the problem and then to correct it.





Susan J on diagnosing Gaps

I think it is very, very hard because it is so personal and unique to the student.

I'm 65 and a computational scientist and I still remember odd and embarrassing gaps that had huge negative effects even in graduate school. Even when you get to the point where you are in charge of your own learning, you can miss these things.

For the mathematicians on the site, I'll admit that it took me more time than it should have to understand that when one solves a differential equation, one is solving for an unknown function rather than a variable.

I still remember puzzling over a textbook diagram of a simple mercury barometer when I was a freshman in college. The difficulty (for me) was that the diagram was simplified and didn't show the support stand for the glass tube with its closed end up and its bottom end part-way submerged in a dish of mercury. So I could never figure out why the tube simply didn't fall over!





here's what I'm wondering

Although I believe that the gap between our kids and kids in high achieving countries starts in first grade or thereabouts, I do trust research showing that achievement slows in middle school. (This finding may not be confirmed, but at the moment I take it as probably true.)

Here's what I wonder.

When you don't teach to mastery — when you teach a spiraling curriculum — kids end up with gaps.

That much we know.

But kids probably don't all end up with the same gaps, except for the Universal American Fraction-Decimal-And-Percent gap.

So think what a middle- or high-school math teacher is up against. Ninety or more kids, each with different gaps affecting different areas of the new content they're supposed to be learning and/or spiraling.

It's Gap Anarchy.

At the moment, it seems logical that the further you go, and the more gaps you accumulate, the slower your learning curve is going to be, until finally you hit the wall.

I don't know whether that's true, but it seems logical.

More than logical.

It seems inevitable.



what do we know about learning gaps & how they work?

Here's Engelmann:

When students are not taught to mastery, they often mislearn the skills and concepts the teacher attempts to teach. For instance, they may learn to guess at words in sentences. Reteaching them requires many more trials and much more work than that required to teach them to mastery initially. Initial teaching may require only 10 or fewer trials on some skills. Reteaching the same skill after students have mislearned it and have practiced inappropriate strategies for years may require several hundred trials.


Here he's talking about the case of a student having learned the wrong thing, rather than merely having failed to learn the right thing. The news is bad.

What else do we know about gaps?

Or about reteaching?

And what have your own experiences been?

I'd love to hear.


key worsd: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator





-- CatherineJohnson - 17 Jan 2006


BoyTrouble 23 Jan 2006 - 20:00 CatherineJohnson



via eduwonk, Boy Trouble (free registration required)

I haven't posted the various articles I've read on this subject, but color me concerned. Very concerned. A 60-40 ratio of boys to girls in college strikes me as a very bad idea.

I've believed for years — since before I had kids myself — that school is a female-dominated & female-friendly environment. (And btw, this isn't an 'impression'; it's an opinion based in years of data collected by researchers).

That's why I jumped on the issue at our middle school, after receiving a negative Interim Report that sounded like your Generic Middle School Boy Report to me.

I'm serious about the Generic Middle School Boy Report. If you took Christopher's name off his Interim Report, and handed it out to 10 random people, and asked them, 'Was this report sent to the parents of a girl or a boy?' I'd lay odds most or all of those people would have guessed 'boy.'

So I jumped on it.

I jumped on it, because I want my own school district officials to be formally on guard against dismissing boys.

The fact that our principal told us, and I quote, 'Everyone knows girls do better than boys in middle school,' is, for me, problematic.

It would be radically unacceptable to say such a thing about girls or blacks or Hispanics or disadvantaged kids in general.

But you can say it openly about boys.

I'll add that I dislike political correctness intensely; I appreciate our principal's honesty.

But I do want him and his staff to be consciously thinking about what it means that 'everyone knows boys do worse.'

Why is that OK?

How is that different from holding lower expectations for individuals based on their membership in a low-performing group?

In their view, it's OK to say that everyone knows boys do worse than girls, because 'boys catch up in high school.' Our principal said that, too.

That doesn't work for me.

The data I've seen shows that in fact boys don't catch up. (And how do boys manage this feat, anyway? Closing a gap once a gap has been opened takes hard work. Are 'boys-as-a-group' doing that hard work?) Along with a black-white achievement gap, we have a gender gap. I've seen estimates that boys finish high school 1 1/2 years behind girls in literacy skills. I'll fact-check this and revise if I've remembered incorrectly.

From the article:

What's most worrisome are not long-standing gender differences but recent plunges in boys' relative performance. Between 1992 and 2002, the gap by which high school girls outperformed boys on tests in both reading and writing--especially writing--widened significantly. Given the reading and writing demands of today's college curriculum, that means a lot of boys out there are falling well short of being considered "college material." Which is why women now significantly outnumber men on college campuses, a phenomenon familiar enough to any sorority sister seeking a date to the next formal. This June, nearly six out of ten bachelor's degrees awarded will go to women. If the Department of Education's report is any indication, in coming years, this gender gap will grow even larger.

The report illustrates a dramatic and unsolved mystery: At some point in the early '80s, boys' relative academic records and aspirations took a downward turn. So far, no one has come up with a good explanation for this trend, but it's a story that affects millions of boys and their families. And yet, according to LexisNexis?, the report was cited by name in only five newspaper and magazine articles.

Not only has there been little media attention to this crisis in boys' education, but there has been surprisingly little research. And the conventional wisdom offered up to explain the problem--boys play too many video games and listen to too much hip-hop music--can't explain a gender slide that's affecting not just the United States but much of the developed West. It also can't explain why boys in a few schools manage to duck the gender gap. But promising new answers have begun to surface--and from some very unlikely places.



how hard is it to close a gap this wide?

The state [of Maryland] has been breaking out its test-score data by gender since 1992, which is why Maryland Superintendent of Schools Nancy Grasmick is dismayed by the gender gaps she sees--72 percent of girls read at a proficient or advanced level by eighth grade, compared with 61 percent of boys.


Sounds like a pretty big gap to me.



the magical child-rearing skills of the upper-middle class parent

The author claims that boys in upper middle class families are doing fine, but I've read elsewhere — specifically in USA Today (I'll find the passage and drop it in later) — that upper middle class boys are affected, too. (I have a CHART! At least, I THINK I have a chart.)

I know for a fact this author is wrong that 'elite' colleges have a 50-50 ratio:

If your father reads, it's not viewed as a sissy thing, as it's seen by many blue-collar students. Not only would that explain why the verbal gap doesn't widen for boys in the wealthiest districts, but it would also explain why the Harvards and Princetons and Stanfords have no trouble drawing talented men. Those schools run close to a 50-50 gender balance among undergraduates.


NYU is getting more selective by the moment, and the ratio there is 60/40 girl/boy.

My niece is at Emory; same story.

My best friend's son is at Occidental; same story.

Telling us that the Ivies still have a 50-50 ratio tells us nothing. IMO.

We're just back to the idea that the 'upper-middle class' somehow, magically, CONVEYS NECESSARY INFORMATION AND SKILLS TO ITS CHILDREN VIA SUPERB CHILD-REARING SKILLS.

On the subject of the Magic Child Rearing Skills of Rich People, that was my one beef with Patricia Clark Kenschaft's article on teaching math to teachers:


It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal “home schooling” of children.


Informal?

3-hour a night Helicopter Parenting isn't informal.

While I'm on the subject of Things That Annoy Me, I have a memory of reading some authority saying upper middle class families 'talk about math at the dinner table.'

Number one, I've never heard an upper middle class family besides mine talk about math at the dinner table. And I talk about math to complain about the curriculum and lament the general state of Bad Math Knowledge in the U.S. I'm exercising my obsessions, not teaching.

And number two, talking about math isn't the same thing as teaching math.

You can't learn math 'informally.'

You have to be directly taught math, and you have to practice.

Reading skills may be different; maybe upper middle class kids pick up more of these skills incidentally, by living in houses where the adults are reading. I don't know.

But I'm suspicious of any line of reasoning based in the twin assumptions that a) upper middle class boys haven't been affected by the decline in literacy skills affecting less well-off boys, and b) that's because affluent boys have dads who read.



does your school use this book?

Last spring, Scientific American summed up the best gender and brain research, including a study demonstrating that women have greater neuron density in the temporal lobe cortex, the region of the brain associated with verbal skills. Now we've reached the heart of the mystery. Girls have genetic advantages that make them better readers, especially early in life. And, now, society is favoring verbal skills. Even in math, the emphasis has shifted away from guy-friendly problems involving quick calculations to word and logic problems.

Increasingly, teachers ask students to keep written journals, even as early as kindergarten. What gets written isn't polished prose, but it is important training, say teachers, some of whom rely on the book Kid Writing, which advocates the use of writing to teach children basic skills in a host of subjects.



032206435X.01._AA240_SCLZZZZZZZ_.jpg


hoo boy

I just read the cover

I wish I had 100 hands so I kode writ and writ and writ.

Speaking as one who does writ and writ and writ, Blecch.



[pause]



I hope that's not somebody's kid I just dissed.

sigh



[pause]



Just in case it is somebody's kid, I think she's adorable.

I do.

I just don't think the book is adorable.

That's all.



I love it!

Basing grades on turning in homework on time guarantees lower grades for boys. Studies consistently show boys have more trouble than girls turning in homework on time.


I'm gonna be sending copies of this article to the middle school.

Heh.



this is interesting

Some educators and parents explain this by saying that many boys simply forget or decline to turn in completed homework. Here's the boy-thinking: If I answered the homework question to my satisfaction, the task is done. Why turn it in? If you're the parent of a girl, that may sound bizarre. It isn't.


Is that true?

Is that the way lots of boys think?



wow

This article is a lot of fun:

The Education Trust, a Washington-based education reform group that looks after the education interests of less privileged students, scoured the nation for gender success stories and turned up Indian River School District in rural Delaware. Indian River's Frankford Elementary appears to be an unlikely candidate for achieving any sort of academic success, let alone overcoming the gap between boys' and girls' achievement: 76 percent of the students qualify for subsidized lunches, 22 percent land in special education, and 64 percent are either Latino or black. Most of the Latinos are sons and daughters of Mexican agricultural workers who have limited English skills.

And, yet, here's Frankford's 2004 state report card for fifth-graders: 100 percent of boys and 95 percent of girls meet state reading standards. When I contacted them, school leaders expressed pride at their success in educating poor and minority students but appeared bewildered when told they had conquered the gender gap. Turns out their education strategy had nothing to do with getting boys in touch with their feelings or eliminating late-homework penalties. Rather, the strategy was a roll-up-your-sleeves effort initially sparked by a state campaign to improve literacy skills. Students whose problems were identified early received extra help from teachers. A special eye was kept on black boys. Most important, no excuses were accepted--when boys fell behind, teachers weren't allowed to consider that the norm.





research on the wealthiest schools

Hilton's research on the wealthiest schools is revealing. Girls still do better in verbal skills in those districts. But Hilton discovered an important distinction. When the wealthy boys enter middle school, they don't lose ground. And that holds steady through high school.

Why the smaller verbal gender gaps in upper-income families? Hilton can only feel his way on this one, in part by drawing lessons from his own family, which teems with educators. At nights and on weekends, Hilton saw his father reading, just as the boys hitting puberty in the wealthiest districts see their well-educated fathers reading. If your father reads, it's not viewed as a sissy thing, as it's seen by many blue-collar students.



This, I believe.

I haven't posted about this topic at all, but Carolyn and I have been talking about the change we see in both our boys, who are, suddenly, obsessed with their dads.

Until just a couple of weeks ago, I had been handling all of the afterschooling.

Christopher and I were locked in brutal battles, and Ed's idea of how to handle this was to decree that Christopher would no longer be required to do school work later than 8 pm, because he was tired.

This meant all Christopher had to do was play out the clock 'til his dad got home and freed him.

Then 2 things happened: 1) brutal battle between Ed & Catherine,* and 2) two Ds on English papers & 1 D on a chapter test in math.

Things changed.

Now Ed is handling the math. He's handling most of the afterschooling. (Actually, he's doing the direct one-on-one re-teaching. I'm doing the management: knowing what chapter Christopher is in, pulling the worksheets, knowing the schedule, etc.)

And Christopher is back on track.

I've been thinking a lot about fatherless boys, and what they're up against. (Christian didn't have a dad around when he was in middle school & high school. He had a very tough mom, but no dad.)

A tough mom isn't enough.

I have the will and stamina to force Christopher to learn math.

But he doesn't need a mom forcing him to learn math.

He needs a dad forcing him to learn math.

He needs a dad who's in charge, who's in a position of authority, and who's telling him: you're learning math, you're learning English, you're learning science, and you're learning history.

He's got one.



update from Ken

Ken's school uses KID WRITING. He's looked into it:

1. No field testing or research base.
2. The curriculum is based on the authors' grad school thesis. And, they wrote a book about it.
3. It's out of print.
4. It's "Evidence Based." This means that they looked at the Kids' Writing curriculum and noticed that it teaches phonics (the wrong kind mind you) then they looked at the extant research and noticed that successful ELA programs also taught phonics (nevermind that most of the reseach was based on DI and this program looks nothing like DI). Therefore, this curriculum is based on research. Disturbing.
5. The parents are getting up in arms because aren't learning how to read on schedule.
6. Kids make lots of errors spelling words in this curiculum. That's bad.
7. Almost none of those errors are corrected by the teacher. This is even worse.


That's about what I'd expect.

Most kids in my son's K class are not reading. Those that are, have been taught at home. They spend a good portion of each 2 hour day doing Kids' Writing. Based on what they've learned in school, I can't see how any kids can be reading yet if they relied solely on what's being taught in school. Now contrast this with the 75 lessons my son has done in the Ordinary Parent's Guide to Teaching reading. He's reading independently already.



*Yet another unsung bonus of the middle school years: way more family blow-ups. Something to look forward to.


USA Today report on 135:100 boys:girls ratio in college
sexism in Everyday Math
invisible boys
boy trouble (New Republic on boys)
slacker boys, middle school, & forbidden positive images of boys in textbooks
throw rocks at them
please remain seated at all times
Ann Althouse thread sums up classroom change
 cooperative vs. competitive learning
the girl show (8th grade graduation awards)
the boy show (character ed)
the other boy show
Where the Boys Aren't

letter from Robert Lerner, former commissioner NCES
Tom Mortenson's research
The Boys Project board
for every 100 girls —


-- CatherineJohnson - 18 Jan 2006


BenCalvinOnKthru8 20 Jan 2006 - 03:33 CatherineJohnson


Ben Calvin left this Comment in the Elemiddle thread:

We choose a K-8 school for our son Jack (now in Kindergarten).

We knew we wanted K-8, rather than a middle-school system, based on our bad Jr. High experiences.

What has impressed me now that he is in his small, Catholic school is how the 8th graders are used to mentor the lower grades, socially as much as academically.

It makes the 8th graders feel responsible, and the Kindergarteners have a high level of respect for these older, sophisticated students.

So when for example one of them told Jack he shouldn't play Halo (a violent Xbox game he had seen at a vacation rental), it made a far bigger impression than if I told him.

When he does reach the upper grades, our hope is that being in an environment, that he knows and where everyone knows him will minimize some of the pressures of early adolescence.



I was thrilled to hear this, because this is exactly what I thought should happen....

I may have mentioned that Temple (Grandin) once told me that, with social animals, trouble happens when rivals are too evenly matched.

Herds and packs are stable when there is a clear leader no one wants to challenge.

I'm thinking it's unreasonable to put 11- to 13-year olds alone together in the same school. They're too close to being peers. They don't have little kids around to look out for; they don't have big kids around to look out for them.

Way too much equality.

Which, of course, appears to have been the genesis of the middle school movement in the first place.

Forced egalitarianism.



Andy Joy has seen the same thing

A similar thing happens at my 7-12 private school. The seniors strive to be role models for the junior highers, rather than writing them off as immature. The junior highers are less squirrelly and immature because they want to earn the respect of the older teens.




This We Believe

According to This We Believe, a seminal document generated from the National Middle School Association (1982, 1995, 2003), the middle school “philosophy” provides a clear set of guiding characteristics for successful, developmentally responsive middle schools to embody. Among these are [bullets added]:

  • a shared vision;

  • educators committed to young adolescents;

  • a positive school climate; an adult advocate for every child;

  • family and community partnerships; and

  • high expectations for all students, buttressed by

  • an integrative, exploratory curriculum.
source:
Monitoring the Middle School Movement: Are Teachers In Step?


Something to look forward to, for all you folks just starting K-5.



The War Against Excellence
by Cheri Pierson Yecke

9836758.gif




Yecke's new report on middle schoolism

from the Fordham Institute:

American middle schools have become the places "where academic achievement goes to die." So says Cheri Yecke, K-12 Education Chancellor of Florida and author of the new Fordham report Mayhem in the Middle: How middle schools have failed America, and how to make them work. Today's middle schools have succumbed to a concept of "middle schoolism" in which a strong academic curriculum is traded for one that focuses more on emotional and social development, and less on learning the basics. And the achievement data reflects "middle schoolism's" results. In 1999, U.S. eighth graders scored nine points below average on the TIMSS assessment of math. What's more, these same eighth graders had outperformed the average by 28 points as fourth graders in 1995! According to Fordham President Chester E. Finn, Jr., "Trying to fix high schools while ignoring middle schools is like bandaging a wound before treating it for infection."



random factoids:

  • RAND reports that U.S. middle school students manifest depression, disengagement, fear for physical safety, a desire to drop out, and boredom with schoolwork at rates that exceed those of every industrial nation except Israel. [ed.: Israel?]

  • Middle schools are overrepresented on the list of failing schools as defined by the No Child Left Behind Act: In 2004-05, they comprised only 14 percent of all Title I schools, but 37 percent of Title I schools identified for improvement.

  • Middle schoolism is partially based on the now-discredited theory of “brain periodization,” which holds that “the brain virtually ceases to grow” in children ages 12 to 14 and that teaching complex material during that period will have damaging effects. [ed.: wrong again ]

  • Schools, states, and districts are returning to the K-8 model of education, the dominant model in the U.S. well into the 20th century. Though some middle schools are high-performing, research from three cities—Milwaukee, Philadelphia, and Baltimore—indicates that the traditional K-8 model may produce better outcomes.

  • Students in K-8 Milwaukee schools had higher academic achievement, especially in math. They also had higher levels of participation in extracurricular activities, demonstrated greater leadership skills, and were less likely to be victimized than those in the elementary/middle school setting.

  • In Philadelphia, analysts showed that students in K-8 schools had higher academic achievement than pupils in middle schools. Their academic gains also surpassed those of middle school students in reading, science, and math. Once in high school, their grade point average was higher than that of their peers who had attended middle schools.

  • Baltimore researchers found that students in K-8 schools scored significantly higher than their middle school counterparts on standardized achievement measures in reading, language arts, and math. Students in K-8 schools were also more likely to pass statewide math tests.

  • Middle schools can be high-performing educational institutions, and the author describes two such examples. The essential problem with middle schoolism is not grade configuration but educational ideology. However a school is structured, in the era of standards and accountability, it must focus first and foremost on students’ acquisition of essential academic skills and knowledge.

That means middle schoolism must end.

Yecke's 81-page report is available online.

Her website is here.

Twin Cities Pioneer Press column about Yecke's ouster in MN; interview with Yecke; anti-Yecke press; Yecke on MN politics.



from Chester Finn's foreword

She is superbly qualified to tackle this topic, having served, among other things, as a senior federal Education Department official, as Secretary of Education in Virginia—a state widely praised for the quality of its academic standards—and, for a brief but astonishingly fruitful period, as Commissioner of Education in Minnesota. As we go to press, Florida Governor Jeb Bush has just named her that state’s new chancellor for K-12 education. She also authored the fine 2003 book, The War Against Excellence, which simultaneously exposed the shortcomings of U.S. middle school education and the country’s strange and dysfunctional animus toward “giftedness.”


Of course, this is not a universal view.



Shared Beliefs of Gifted Education and Middle School Education

First, when it comes to articulated beliefs about what constitutes appropriate instruction for early adolescents, both [advocates for the gifted and advocates for middle schools] are proponents of instruction that:

(1) is theme based,

(2) is interdisciplinary,

(3) fosters student self-direction and independence,

(4) promotes self-understanding,

(5) incorporates basic skills,

(6) is relevant to the learner and thus based on study of significant problems,

(7) is student-centered,

(8) promotes student discovery,

(9) values group interaction,

(10) is built upon student interest,

(11) encourages critical and creative exploration of ideas, and

(12) promotes student self-evaluation (e.g., Currier, 1986; Kaplan, 1979; Maker & Nielson, 1995; Stevenson, 1992).



All I can say is, it's a good thing I don't have a gifted child, because I'm so not down with this stuff.


-- CatherineJohnson - 19 Jan 2006


BayesianBrainRunAmok 22 Jan 2006 - 15:43 CatherineJohnson



middleschoolfaith1.jpg



-- CatherineJohnson - 20 Jan 2006


TheOverAchieversClub 23 Jan 2006 - 18:38 CatherineJohnson



This is funny.

I just took the Overachievers Quiz and came out as a "Happy Medium," continuing my unbroken record of scoring Dead Center on all quizzes on all subjects.


Good luck fitting all of your achievements and activities on your college applications! Just make sure that everything you do is because you really want to. Working at a frenzied pace to make other people happy is a quick road to burnout. You're one of the lucky few who know how to balance homework and life, school and socializing, family and friends. You know that there's more to being a happy person than a 4.0 GPA and a varsity letter. Rock on.


Thank you!

I believe I will rock on!



Becky C on ability and effort

I was inspired to take the Overachievers' quiz, because I've just this morning finally read Becky's incredible post on ability versus effort. (Don't know how I missed it in the first place...)

It's one of the best things we've posted, and I wish to heck I'd written it.

Everyone!

Go read it again!

One of the best moments in my marriage happened the day Ed and I talked to Christopher's Phase 3 teacher about moving him to Phase 4.

That conversation ended with Mrs. Panitz saying that if Christopher was going to move, he should move sooner rather than later — sooner meaning within the next two weeks. She would handle it. (I love Mrs. Panitz.)

UPDATE 10-18-2006: The implication was that if we did not make the move from Phase 3 to Phase 4 now, the middle school would block it. I didn't post this in January. I'm posting it now.

I left feeling happy, but intensely nervous & stressed. I'd been counting on having another 6 months to get Christopher ready; everything I was doing was based on that calendar. [ed.: I think it's pretty obvious we could have used those 6 months....]

UPDATE 10-19-2006: Given the nature of the middle school Phase 4 course, homework assis not graded another 6 months would have done nothing. You can't prepare a child adequately for a bad course.

Ed refused to be nervous.

He said: "Our position is that we want Christopher to be an overachiever."

Of course, if you sorted through the 5 gazillion books & articles I'd been reading on math & math ed, that was my position, too, but I hadn't managed to hone it down to one line.

Our position is that we want Christopher to be an overachiever.

I'm hoping to overachieve in math myself.



Achievement beyond IQ

Meanwhile, I stumbled across James Flynn's book (Flynn as in 'Flynn effect') book on Asian overachievement, published back in 1991.

I may have to read it.


0805811109.01.LZZZZZZZ.jpg


-- CatherineJohnson - 22 Jan 2006


GoodLightingRedux 08 Oct 2006 - 22:16 CatherineJohnson



This is just bizarre.

I mentioned the other day that I've discovered I can't do my KUMON worksheets in (relatively) dim light.

Just now I sat down to do the worksheets, first turning on the Halogen lamp next to the desk.

I was going along OK until I came to the 4th sheet, where I missed 5 out of 16 problems. Everything was wrong; there was red ink everywhere.

All of a sudden I realized I had my left arm propped up on a stack of algebra books (more cramming for Ms. Kahl), putting the worksheet in shadow.

I took my arm off the books, put full light onto the sheets, and got a 100% on the next one, which was the last in the bunch.

What's so strange about this is that I have no sense at all that I'm not seeing the sheets right. None. I don't feel like I'm 'working in the dark'; I don't perceive eyestrain — nothing like that.

The first time I realize that the light is dim is when I grade the sheets and get 5 out of 15 wrong.

sheesh

Anyway, the One Lesson I draw from this is that we should be monitoring our kids' work conditions just in case they're no better at realizing the environment is interfereing with performance than I am.



a lighting needs quiz in French!

So I guess this woman is having some problems with her lighting.

Boy.

I can't say this photo makes me want to take the quiz.

question: does this photo tell us French people have gained weight?

answer: no

It tells us Canadian people weigh as much as Americans.


good_lighting.jpg




I need paper
good lighting redux



-- CatherineJohnson - 22 Jan 2006


DontKnowWhatWeDontKnow 22 Jan 2006 - 23:04 CatherineJohnson



I just stumbled across the Edge Foundation's Annual Question: "What is your most dangerous idea?"

There's some fun stuff, including Richard Nisbett, author of The Geography of Thought.



Telling More Than We Can Know

Do you know why you hired your most recent employee over the runner-up? Do you know why you bought your last pair of pajamas? Do you know what makes you happy and unhappy?

Don't be too sure. The most important thing that social psychologists have discovered over the last 50 years is that people are very unreliable informants about why they behaved as they did, made the judgment they did, or liked or disliked something. [ed.: What, if anything, does this tell us about metacognition in education?] In short, we don't know nearly as much about what goes on in our heads as we think. In fact, for a shocking range of things, we don't know the answer to "Why did I?" any better than an observer.

The first inkling that social psychologists had about just how ignorant we are about our thinking processes came from the study of cognitive dissonance beginning in the late 1950s. When our behavior is insufficiently justified, we move our beliefs into line with the behavior so as to avoid the cognitive dissonance we would otherwise experience. But we are usually quite unaware that we have done that, and when it is pointed out to us we recruit phantom reasons for the change in attitude.

[ed.: in contrast, the cognitive unconscious is shockingly accurate]

In the 1970s social psychologists began asking whether people could be accurate about why they make truly simple judgments and decisions — such as why they like a person or an article of clothing.

For example, in one study experimenters videotaped a Belgian responding in one of two modes to questions about his philosophy as a teacher: he either came across as an ogre or a saint. They then showed subjects one of the two tapes and asked them how much they liked the teacher. Furthermore, they asked some of them whether the teacher's accent had affected how much they liked him and asked others whether how much they liked the teacher influenced how much they liked his accent. Subjects who saw the ogre naturally disliked him a great deal, and they were quite sure that his grating accent was one of the reasons. Subjects who saw the saint realized that one of the reasons they were so fond of him was his charming accent. Subjects who were asked if their liking for the teacher could have influenced their judgment of his accent were insulted by the question.

Does familiarity breed contempt? On the contrary, it breeds liking. [ed.: This claim generates a Testable Hypothesis: Christopher will grow up to love math. I think it's possible. After all, why did I start liking math as much as I do? Maybe because I was doing math all the time?] In the 1980s, social psychologists began showing people such stimuli as Turkish words and Chinese ideographs and asking them how much they liked them. They would show a given stimulus somewhere between one and twenty-five times. The more the subjects saw the stimulus the more they liked it. Needless to say, their subjects did not find it plausible that the mere number of times they had seen a stimulus could have affected their liking for it. (You're probably wondering if white rats are susceptible to the mere familiarity effect.

The study has been done. Rats brought up listening to music by Mozart prefer to move to the side of the cage that trips a switch allowing them to listen to Mozart rather than Schoenberg. Rats raised on Schoenberg prefer to be on the Schoenberg side. The rats were not asked the reasons for their musical preferences.)



We should all stop paying any attention to whatever it is we're saying and just.....watch what we're actually doing, I guess.

That might work.

Pinker's there, too, on the subject of group differences. Reasonable and succinct as always, though I don't trust his predictions about the future any more than I trust most people's.


nisbett100.jpg




yes, it's a cognitive science blog!

I've just discovered mixingmemory.blogspot; no idea whether I'll dive in reading for the next several years of my life, or not.

I think I will put the time into reading this post, on automaticity.

Soon, I hope.



more synchronicity

ah-hah

Speaking of Steven Pinker, and we were speaking of Steven Pinker, mixingmemory isn't a fan:

Pinker has a nasty habit of speaking authoratatively about topics on which he is anything but an authority (like, say, gender differences in mathematical ability). And Fido also links to this very informative forum on race and genomics, titled "Is Race 'Real?'" Like Pinker, I'm not an expert in genomics, or anything remotely related to genetics, but unlike Pinker, I'm not going to comment on the issues discussed in the forum as though I am an expert.


This reminds me of a course being taught at Harvard a few years ago....it was a team effort, with 3 or 4 famous Harvard Guys including Stephen J. Gould and, IIRC, Alan Dershowitz. Can't remember who else.

Here it is. It was called Thinking about thinking.

The students called it 'Talking about talking.'


how Asians and Westerners think differently
describe this picture
how Asians and Westerners think differently, part 2
Harold Stevens, RIP
how Asians and Westerners think differently, part 3
creativity gap, part 2

don't know what we don't know (cognitive science)
synchronicity on 9/11
the 'normal' distribution isn't normal
a science of the divine



-- CatherineJohnson - 22 Jan 2006


NewZealand 23 Jan 2006 - 23:11 CatherineJohnson



I keep saying I'm going to post this passage from about New Zealand and hypomania for Tracy's.

Today I'm doing it:

A HYPOMANIC NATION?

Energy, drive, cockeyed optimism, entrepreneurial and religious zeal, Yankee ingenuity, messianism, and arrogance — these traits have long been attributed to an "American character." But given how closely they overlap with the hypomanic profile, they might be better understood as expressions of an American temperament, shaped in large part by our rich concentration of hypomanic genes.....

A small empirical literature suggests that there are elevated rates of manic-depressive disorder among immigrants, regardless of what country they are moving from or to. [ed.: no kidding] America, a nation of immigrants, has higher rates of mania than every other country studied (with the possible exception of New Zealand, which topped the United States in one study). In fact, the top three countries with the most manics — America, New Zealand, and Canada — are all nations of immigrants. [ed.: Canada? apparently, Canada needs more sunshine. hypomanics like light.] Asian countries such as Taiwan and South Korea, which have absorbed very few immigrants, have the lowest rates of bipolar disorder. [ed.: leading one to doubt that enhanced Asian creativity will result from adoption of constructivist curricula ] Europe is in the middle, in both its rate of immigrant absorption and its rate of mania. As expected, the percentage of immigrants in a population correlates with the percentage of manics in their gene pool.

While we have no cross-cultural studies of hypomania, we can infer that we would find increased levels of hypomania among immigrant-rich nations like America, since mania and hypomania run together in the same families. Hypomanics are ideally suited by temperament to become immigrants. If you are an impulsive, optimistic, high-energy risk taker, you are more likely to undertake a project that requires a lot of energy, entails a lot of risk, and might seem daunting if you thought about it too much. America has drawn hypomanics like a magnet. This wide-open land with seemingly infinite horizons has been a giant Rorschach on which they could project their oversized fantasies of success, an irresistible attraction for restless, ambitious people feeling hemmed in by native lands with comparatively fewer opportunities.

source:
The Hypomanic Edge: The Link between (A Little) Craziness and (A Lot of) Success in America




John (Ratey) said this over 10 years ago. I'm getting his new book.


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sociology 101

My line about New York versus Los Angeles:

The manics went to Los Angeles, the depressives stayed in New York.

It happens to be true.



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-- CatherineJohnson - 23 Jan 2006


BoyTroublePart2 16 Sep 2006 - 20:24 CatherineJohnson



update

I almost forgot.

Here's the link to the New Hampshire Commission on the Status of Men.



Karen A left links to two articles about boys and elementary school:

Do Teachers Dislike Boys?

I have two boys and neither one has ever had a teacher who I thought disliked him, or who made him feel bad about being a boy. [ed.: we've had at least one teacher - a P.E. teacher - who specifically made boys feel bad about being boys, or at least tried to]

However, I have come to believe that elementary school is a very female-centric environment, [ed.: I'll say] one that does not suit many young boys very well. My older son went all the way through elementary school without once having a male teacher, [ed.: ditto] and the younger one did not have a male teacher until fifth grade.

Akira, my older son, was bored and frustrated by an endless parade of worksheets in the first grade, when he was having a hard time sitting at a desk and writing for long periods of time. I was also concerned about the common practice at his school of keeping kids in from recess if they had misbehaved in class. [ed.: ditto]

My feeling is that an active young child who gets into trouble because he cannot sit still needs more time running around outside, not less.

I have come to believe that schools need to do much more to adapt to the way boys learn. This belief has been bolstered by the stories of other parents, who tell me that they are being pushed to put their active young sons on Ritalin. "Being a boy is not a disease," one parent writes.

[snip]

My feelings about boys and learning have been influenced by the book Real Boys by William Pollack, Ph.D. Pollack is a clinical psychologist and the codirector of the Center for Men at McLean Hospital/Harvard Medical School.

[snip]

Read Pollack's book, in particular the chapter "Schools: The Blackboard Jumble," for a detailed analysis of how he thinks public coed schools are failing boys. His most compelling arguments are simply numbers: Research shows that most of the students at the bottom of the class are boys, most of the students in remedial classes are boys, most of the students suspended are boys, fewer boys than girls go to college, and many more boys than girls have serious difficulties with reading and writing.

"These statistics show that there are many more boys at the lowest rungs of the ladder of academic achievement than we had ever imagined or been led to believe," he writes.

One answer, Pollack suggests, may be all-boys schools or all-boys classes within coed schools. It's an intriguing suggestion, one I've certainly never considered for my children. But it has proved to be the right answer for some.



My best friend, Cindy, sent her son to an all-boys' Catholic high school.

She said she absolutely did not get it - it was a completely foreign culture to her - but 'those teachers loved those boys.'

He's in great shape, while a number of the college-age boys in our circle aren't.



slacker boys

I keep hearing the same story.

Our friends' college-age girls are great. They're smart, confident, pulled-together, focused, etc. (With exceptions, of course.)

But the boys worry me.

They're not quite getting off the dime. One couple we talked to, while we were in L.A., said that their college-age son was probably going to have to drop out for awhile. He has a good therapist, so they're hoping the therapist will help him get on his feet.

Another friend said her son wanted to have fun and spend money, but didn't want to get a job.

Another told me a story about her cousin's family. The daughter is the usual family superstar: taking AP calculus, finishing high school, touring colleges, Bright Future Ahead, etc.

The son, who is a couple of years older, 'isn't like that.'

I'm hearing these stories too often — and I feel as if I'm watching this process unfold in some of the boys around me here.

They start out bright-eyed and bushy-tailed.

But by the time they reach 7th or 8th grade, they're not looking so good. The parents wonder where their bright little boy went, and the boys must wonder, too.

Here's a narrative I've heard more than once:

"He was one of those boys who loved math. When they'd be driving around in the car he'd make them give him math problems. Then he got to middle school and his grades weren't good. He was sloppy, he made careless errors. The school told his mom to sit with him when he does his homework, so she does. She enjoys it. But she says his bad grades are his own fault. He's sloppy."

I've heard a variant of this more than once, about more than one boy. I suspect there are more than a few children in this category, if only because the 'Disappointment Narrative' fits so well with the other Master Narrative, which has to do with aggressive Irvington parents thinking their kids are geniuses when they're not.

PAUSE: Let me say that NO Irvington teacher or administrator would say, flat-out, Irvington parents are aggressive people who think their kids are geniuses when they're not.

Instead, this feeling is simply there, present in many, many exchanges. I can't tell you how many times I've been told, by people at all levels of the district — and by other parents — that 'pushy parents' got their kids into Phase 4 when they didn't belong. It's a shared narrative.

Last but not least: I have no idea how often girls are the subject of these narratives. The Pushy Parents In Phase 4 meme could be an equal opportunity storyline for all I know.

Still, I get the feeling that girls don't get as much grief for being girls as boys do.



what goes unsaid

What goes unsaid is that this isn't just about boys being hyper and girls being able to sit still.

There's a political problem.

From Day One, elementary schools stress the existence of oppressed groups, and tell their sad stories.

Always, the oppressors are white men. Always, always, always.

OK, the oppressors are white men. I don't have a problem with that! My problem is: there's no reason in the world for a 7-year old to feel that he is personally resonsible for slaughtering native populations around the world.

Each and every year, there's a women's history month & a black history month. These Months are faithfully observed and celebrated in the schools.

There's no longer a Take Our Daughter to Work Day, because somebody sued, but apparently its official replacement — Take Our Sons and Daughters To Work — is just as bad. (I can't remember if Irvington does TOS&DTW or not, but if so there's zero propaganda involved. A serious Plus in the Irvington column.)

When he was little, every time women's history month rolled around Christopher would ask me why there wasn't 'men's history?'

To him, it only seemed fair that there should be a history month for his group, too.

When black history month rolled around, he'd ask me why there wasn't white history.

Exactly how verboten is that question?

ANSWER: VERY VERBOTEN.

So there I'd be, trying to make him understand that he could not under any circumstances suggest a White History Month at school, and I'd be trying to do this without making him feel he'd just said something shameful and repellant.

For his part, Ed would explain, reasonably, that historians didn't used to write about women & blacks very much, so women's history month and black history month existed for that reason only. Ed would also tell him he didn't believe in women's history month & black history month. That was the right thing to say, but it added fuel to the fire. If his dad didn't think there should be women's history month and black history month, and his dad was a historian, then why did they have women's history month and black history month?

Then Christopher would want to know, constantly, how come on TV the boys were always the stupid, weak ones who lost. I'm serious about this. On Nickelodeon, according to Christopher, the girl characters are not only smarter, they're physically stronger. When they play football with the boys, or fight with the boys, they win.

A friend's son, in 6th grade, asked her this:

How come 'feminist' means 'hates men' and it's good, but 'misogynist' means 'hates women' and it's bad?

They are a liberal Democratic family, and this boy has heard nothing but good things about feminism. I assume his mother considers herself a feminist.

Yet her son believes that 'feminist' means 'hates men.' (She doesn't hate or dislike men, and has certainly never said such a thing to her son.)

Then there's the 'feminization' of content at school. While technically we don't have book banning in the U.S., you don't see a lot of kids reading The Matchlock Gun. (Which is a FANTASTIC novel, btw. Riveting.)



Lionel Tiger on male original sin

Meanwhile, the publicly financed educational system is at least 20% better at producing successful female students than male, yet hardly anyone sees this as remarkable gender discrimination. While there is a vigorous national program to equalize male and female rates of success in science and math, there is not a shred of equivalent attention to the far more central practical impact of the sharp deficit males face in reading and writing.



Here he is at the Independent Women's Forum

We've been through the First World Sex War. For about 40 years there has been a genuine war between men and women ideologically and symbolically. And males have been defined as having "male original sin." For any problem that exists, it's the male's fault. The males are the principle movers of behaviors that are seen as opposed to the interests of females.


This is true.

How do I know it's true?

I know it's true, because I used to be in the war. Then I came to my senses & quit. From there it was a short step to wondering what it meant that it was OK to say terrible things about men — all men — in polite company.

That's when I wrote my magazine article about boys and elementary schools.



do textbooks hurt boys? does school?

This list of prohibited 'positive stereotypes' gives me the chills every time I read it:

imagesofboysgif.gif


source:
Banned Words, Images, and Topics: A Glossary that Runs from the Offensive to the Trivial



this is fun

From what this interviewer has gathered, Dr. Tiger is not your average academic. Throughout his career he has stood for his convictions and not embraced whatever pseudo-scholarly fads happened to come along. He tells the story of his academic travails in the engaging essay, “My Life in the Human Nature Wars,” [1] which, unfortunately, is not available online.

I also encourage our readers to examine Dr. Tiger’s 1999 interview/debate with uberfeminist Barbara Ehrenreich. To say he holds up his own end is an understatement as (in my biased view) he bests her throughout. This is by far my favorite part of their exchange:

EHRENREICH: You certainly got away from the issue of how you feel about it. See, I'm willing to say how I feel.

TIGER: I'm wholly uninterested in your feelings.

How many times in life does one yearn to make such a statement?

source:
Interview with Lionel Tiger




VEERING OFF ON A TANGENT: That reminds me of the time Ed and I became whistleblowers at a school our autistic kids were attending. (MEMO to ktm readers: NEVER become a whistleblower.) In the middle of a parent meeting, as Ed was making a point, the leader of the Enemy Dads shouted, "Shut the f*** up!"

Later on I was telling a friend of ours about this & I said, 'Shut the f*** up! How often do you hear that at a parent meeting?'

Our friend said, 'Never.'

Then he said, 'You think it all the time.'

That cracked me up.

Even though I personally never, ever, think STFU at parent meetings, or any other meetings.



do teachers dislike boys?

I think teachers like boys just fine. Some teachers like boys very much.

Probably many teachers find boys more taxing than girls. I'm in that category, and I love boys. There's no question: BOYS ARE ROWDY. More rowdy than girls.

Once again, teachers are the face of the problem.......so the issue gets formulated in terms of teachers.

The problem isn't teachers, it's institutional structures. The Sitting Still requirements, the Women's History month, the Personal Writing assignments, the journaling, the ban on all forms of violent play including pretend violent play, the being graded-on-handing-homework-in-on-time, the being graded on neatness, the being graded on attractive-artwork-on-the-cover-of-your-report, the chronic Character Education....it's the Whole Package.



is middle school the place where boys stall out?

This is what's worrying me.

Apparently it's universally known amongst educators that boys do worse in middle school than girls, but then 'catch up' in high school.

What concerns me is that middle school is just as female-dominated an environment as elementary school, but all of a sudden, in middle school, they lower the boom.

They 'get tough.'

They give grades, and they 'raise expectations.'

The problem is, who's doing all the getting tough and lowering the boom and raising of expectations?

Women.

I've only heard of one — maybe 2 — male teachers in our middle school. oh, and the P.E. teacher. There are 2 P.E. teachers, and one is a guy.

Thank God the principal's a man.



equal time!

That reminds me.

Christopher went to his second dance Friday night. Ed ran into Scott (the principal) when he went to pick him up, and they got to talking about middle school & middle schoolers.

Scott said that, socially, the middle school years are even harder on the girls than on the boys. Several girls had run off crying to the bathroom that night alone. The boys seem to have got through unscathed.

Christopher's evening was eventful, but he's getting old enough that he'd be mortified if I wrote about it on the web.

He's already mortified that I've said he screams and yells about math....but he's got bigger fish to fry these days, so he's not worrying about his mom's math confessions.


9.jpg

You can buy the Boys Are Stupid book here.

There's a Boys Are Stupid personal journal, too.

The Amazon readers are none too happy.

I love Amazon.


USA Today report on 135:100 boys:girls ratio in college
sexism in Everyday Math
invisible boys
boy trouble (New Republic on boys)
slacker boys, middle school, & forbidden positive images of boys in textbooks
throw rocks at them
please remain seated at all times
Ann Althouse thread sums up classroom change
 cooperative vs. competitive learning
the girl show (8th grade graduation awards)
the boy show (character ed)
the other boy show
Where the Boys Aren't

letter from Robert Lerner, former commissioner NCES
Tom Mortenson's research
The Boys Project board
for every 100 girls —

positivestereotypes



-- CatherineJohnson - 23 Jan 2006


IWantToBeBionic 24 Jan 2006 - 17:43 CatherineJohnson



I do.

I always have, ever since The 6 Million Dollar Man & The Bionic Woman were on TV.

Here's Stanislaus Dehaene on the prospect of that happening sometime in the not-too-distant future (scroll down):

Brain-computer interfaces are already around the corner. They are currently being developed for therapeutic purposes. Soon, cortical implants will allow paralyzed patients to move equipment by direct cerebral command. Will such devices later be applied to the normal human brain, in the hopes of extending our memory span or the speed of our access to information?


As far as I'm concerned, the day of high-quality neuro-gear can't come soon enough.

Every single person in this family needs high-quality neuro-gear, and plenty of it.



using culture to push the limits of biology

Dehaene's got interesting things to say about the brain basis of reading and the use of cultural inventions to 'push the limits' of biology:

As we gain knowledge of brain plasticity, a major application of cognitive neuroscience research should be the improvement of life-long education, with the goal of optimizing this transformation of our brains. Consider reading. We now understand much better how this cultural capacity is laid down. A posterior brain network, initially evolved to recognize objects and faces, gets partially recycled for the shapes of letters and words, and learns to connect these shapes to other temporal areas for sounds and words. Cultural evolution has modified the shapes of letters so that they are easily learnable by this brain network. But, the system remains amazingly imperfect. Reading still has to go through the lopsided design of the retina, where the blood vessels are put in front of the photoreceptors, and where only a small region of the fovea has enough resolution to recognize small print. Furthermore, both the design of writing systems and the way in which they are taught are perfectible. In the end, after years of training, we can only read at an appalling speed of perhaps 10 words per second, a baud rate surpassed by any present-day modem.

Nevertheless, this cultural invention has radically changed our cognitive abilities, doubling our verbal working memory for instance. Who knows what other cultural inventions might lie ahead of us, and might allow us to further push the limits of our brain biology?



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The Number Sense


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sigh

It's impossible to get a straight story on anything.

There are no stable facts.


Dehaene on language and math

Dehaene on high quality neuro-gear
telling more than we can know (cognitive science)
the 'normal' distribution isn't normal
synchronicity on 9/11
a science of the divine



-- CatherineJohnson - 24 Jan 2006


HowClassroomsHaveChanged 16 Sep 2006 - 20:33 CatherineJohnson



Ann Althouse has a terrific thread on the subject of boys & girls in school.

Here's one Commenter's take on how classrooms have changed:

"[E]very decade the industrial classroom becomes more and more protective of the female learning style and harsher on the male.

He goes on to cite evidence of better achievement by girls, which is fine, but I'm curious how the classroom has actually changed.

I wish I could point to data, but all I can do is compare the classroom environment now that my kids are experiencing with my own experiences 25-30 years ago. I would list the following:

  • de-emphasis on competition [ed.: check]

  • greater emphasis on group projects [ed.: check]

  • greater emphasis on daily homework (which puts a premium on clerical skills, organization and compliance with rules & procedures) and a corresponding de-emphasis on tests & quizzes. [ed.: quizzes! I remember quizzes! whatever happened to quizzes?]

  • turning math and science classes into something akin to subdisciplines of english and social studies ('constructivist math', 'writing across the curriculum' programs) [ed.: check]

  • behavior issues now addressed by grade deductions (in my kids school, any unexecused absence means a loss of 1% of the straight-scale semester grade and a zero on any work due that day). [ed.: check]

  • greater emphasis in college admissions on GPA (where girls do better) than on standardized tests (where boys do better -- or at least equally well).

  • greater percentage of female teachers (even in 7-12 math/science) [ed.: check]

  • endless 'you go girl', 'take your daughter to work day', kinds of messages in an out of school. [ed.: check]

  • considering the poor peformance of boys in schools as currently set up as evidence of boys inherent unsuitability for education rather than our education system's unsuitability for boys. (Even the article under discussion strays somewhat from the 'why our schools are badly designed for boys' into 'why boys are inherently defective' territory). [ed.: check & double check]

11:21 AM, December 04, 2005




Brilliant.

Although....I'm not sure colleges are placing greater emphasis on GPA (isn't it the reverse?)

And it's not clear that homework has increased; Loveless says it hasn't. Although I wouldn't be surprised to find homework has increased over what it was 30 years ago. I don't remember doing any homework ever in junior high.



of course, boys are nuts

I love this comment:

Newsflash! Boys ain't girls and no amount of socialization will change that. Give a boy a Barbie doll and he'll turn it into a gun. Actually happened when my little guy was playing with some neighborhood girls. Said gun-crazed maniac is now a pillar of the community and a father of four.
via joannejacobs


Cathy Young on boys & girls

In a 1990 survey commissioned by the AAUW, children were asked whom teachers considered smarter and liked better; the vast majority of boys and girls alike said "girls." Journalist Kathleen Parker recalls that her son, now a teenager, had a grade school teacher who openly said she liked girls more and singled out boys for verbal abuse-such as telling a student who had his feet up on the desk, "Put your feet down; I don't want to look at your genitalia."

I'm pretty sure the AAUW suppressed this finding at the time — wasn't this the poll on which they based their big 'Girls At Risk' report?

I think so.

Haven't fact-checked.

(oops — wrong: A few years later, it effectively hushed up a study it had commissioned-The Influence of School Climate on Gender Differences in the Achievement and Engagement of Young Adolescents, by University of Michigan psychologist Valerie Lee and her associates-when the findings failed to support the shortchanged-girls premise.)

I like this passage:

Traditional schoolmarmish distaste for unruly young males may be amplified by modern gender politics. Some educators clearly see boys as budding sexists and predators in need of re-education. Some classrooms become forums for diatribes about the sins of white males, and some boys may be hit with absurd charges of misconduct-such as Jonathan Prevette, the Lexington, North Carolina, first-grader punished with a one-day suspension in 1996 for kissing a girl on the cheek.


This is the problem (well, maybe it's the problem).

In any case, this is the problem for me.

I'm not a schoolmarm, and I like boys. Nevertheless, boys in the classroom are tough to deal with.

Boys in the FAMILY are tough to deal with. I'm ready to fire my own son. Also my husband. (Another Core Meltdown over tests/study habits/homework last night. This is getting old.)

Female teachers being impatient with boy students would be fine (probably) if it didn't happen in a context of male original sin.

"If you listen to 10- or 11-year-old boys, you will hear that school is not a very happy place for them," says Bret Burkholder, a counselor at Pierce College in Puyallup, Washington, who also works with younger boys as a baseball coach. "It's a place where they're consistently made to feel stupid, where girls can walk around in T-shirts that say 'Girls rule, boys drool,' but if a boy makes a negative comment about girls he'll have the book thrown at him."

Even apart from feminism, some "progressive" trends in education may have been detrimental to boys. For example, British researchers have found that "whole language" reading instruction, based on word recognition by shapes, pictures, and contextual clues rather than knowledge of letters, is particularly ineffective with male students.

Early "school turnoff" may cause many boys to develop an anti-learning mindset the British have labeled "laddism" — a mirror image of the prefeminist notion that it isn't cool for a girl to be too bright. "The boys become oppositional and band together in the belief that manly culture doesn't include grade grubbing," observes University of Alaska psychologist Judith Kleinfeld. For black boys, this attitude may be exacerbated by the notion that learning is a "white thing."



This is what concerns me. (eek! That makes me a concernocrat)!

There's only so much guff a child will take.

Two summers ago, when I started reteaching Christopher math, he'd developed a major case of laddism when it came to math.

Math is for geeks.

Math is for nerds.

I'm not Asian.

etc.

Once he started succeeding in math again, thanks to Saxon, all of that talk went away & I was hearing 'I like math.'

That's what I want to hear.

I want to hear, 'I like math.'

Also: 'I like school.'

I don't think anyone knows what's actually going on, but I do think it's safe to say that public schools aren't causing boys to feel more school-friendly.


USA Today report on 135:100 boys:girls ratio in college
sexism in Everyday Math
invisible boys
boy trouble (New Republic on boys)
slacker boys, middle school, & forbidden positive images of boys in textbooks
throw rocks at them
please remain seated at all times
Ann Althouse thread sums up classroom change
 cooperative vs. competitive learning
the girl show (8th grade graduation awards)
the boy show (character ed)
the other boy show
Where the Boys Aren't

letter from Robert Lerner, former commissioner NCES
Tom Mortenson's research
The Boys Project board
for every 100 girls —


-- CatherineJohnson - 24 Jan 2006


TheBoysProject 16 Sep 2006 - 20:35 CatherineJohnson



This is great!

One thing led to another, and I discovered a blog written by an policy expert on postsecondary education:

I have now been pounding away on the problems of boys in education (especially higher education) since 1995 and I have nothing to show for it. Clearly talking about the scarcity of boys in college accomplishes little more than making people aware that it exists.

For several years reporters (usually women, who like to write about this issue) have been challenging me: Okay, so what do we do about the problem? What do you recommend be done? I just don't know. As one who studies demography I can see that there is a serious problem. I only know that affirmative action for boys in college admissions could diminish opportunities for better prepared and motivated women. I oppose affirmative action for males because it addresses symptoms and not causes--although I am not sure what the causes are.

So, after a fruitless decade where males continue to fall ever farther behind females, a messiah steps forward and agrees to lead a national effort to do something based on real science. And sure enough, as I had long suspected, it is a woman: Prof. Judith Kleinfeld of the University of Alaska at Fairbanks. Dr. Kleinfeld has written on the subject of males in education in the past. She is now organizing a national boys project and is gathering the kind of scientific talent that we might expect to provide answers to the question: Okay, so what should we do about the problem?

This boys project will begin at the beginning: How are little boys different from little girls, and what does this mean for the educational experience we design for each? At last I can see a way to make progress on this terribly important issue.





good news, possibly

First, the back story, from his post on Male Shares of Undergraduates by Family Income

The scarcity of males in higher education has strong class-based roots: males are under-represented compared to females by the largest margin at the lowest family income levels. As income rises the gap narrows. In this analysis we used data from five National Postsecondary Student Aid Studies (NPSAS) to examine the male shares of various undergraduate enrollments. The NPSAS studies used were for 1990, 1993, 1996, 2000 and 2004. Remember that males are about 51% of the college-age population.

Among dependent undergraduates (students less than age 24) males were 47.0% of all undergraduate students in 2004. They were 48.3% in 1990, 48.6% in 1993, 47.4% in 1996 and 46.7% in 2000. By quartiles of parental income the male shares in 2004 were: 44.0% in the bottom quartile ($0 to $34,288), 45.3% in the second quartile ($34,289 to $62,240), 47.6% in the third quartile ($62,241 to $95,006), and 51.7% in the top quartile ($95,007 and over). Between 1990 and 2004 the male share of undergraduate enrollment declined by 1.5% in the bottom parental income quartile, by 2.3% in the second quartile, by 2.2% in the third quartile and by 0.8 percent in the top quartile. [ed.: I've read that the share of male college students in the top income quartile is decreasing, but maybe not]



Now the good news:

The only good news in these data is that the male share of black dependent undergraduate enrollments rose by 4.5% between 1990 and 2004. This was the only racial/ethnic group that experienced an increase and this increase occurred in all four quartiles of parental income. If blacks are the canaries in the coal mine on this issue then the turn around for dependent black males is a good omen since they led the original decline in male shares of undergraduate enrollments.




oh!

These passages don't just come from 'some education policy analyst in Iowa.'

This is Tom Mortenson's blog.

I thought the name sounded familiar.



more from Mortenson on the gap

The National Center for Education Statistics has recently shared with me some as yet unpublished data on higher education degree awards for 2003-04 by degree level and state. These data continue to show women far outpacing men in bachelor's degrees: 804,117 for women compared to 595,425 for men.

However, these new data suggest that since 2000 the boys may finally be waking up to the need to get a college education. The women continue to make extraordinary year-to-year gains in bachelor's and other degrees received. But since 2000, at last, the men seem to be making nearly comparable gains year-to-year. Between 2000 and 2004 the number of bachelor's degrees awarded to women increased by 96,609 (13.7%), while the number of bachelor's degrees awarded to men increased by 65,058 (12.2%). This may not look like progress. But between 1970 and 2000 the number of bachelor's degrees awarded to women increased by 366,289 (107.3%) while the number awarded to men increased by 79,270 (17.6%).



...between 1970 and 2000 the number of bachelor's degrees awarded to women increased by 366,289 (107.3%) while the number awarded to men increased by 79,270 (17.6%)

Now I need someone to tell me how much the population of college-aged people increased during that time period.



Business Week interview with Mortenson

This is exactly what I've been thinking:

Q: About 20 years ago, there was a famous article in Newsweek about how women could pretty much kiss marriage goodbye if they hadn't walked down the altar by the age of 30. Of course, that turned out to be completely false. Much of the research the story was based on was discredited. But you believe that women could be in for store for a marriage squeeze -- a real one. Why?

A: Black women are really the canaries in the coalmine on this. Put simply, I believe white women are headed to where black women are today. If white women want to see the future of what will happen if men aren't brought along through the educational system with them, they should listen to the problems among black women today.

When I make presentations, I can see 95% of the women in the audience nodding to along to this, agreeing with me. I don't think some women -- and some gender feminists -- have fully thought through the idea of what it means to leave a generation of boys behind. And by the time this gender imbalance really hits whites, it will be too late. We're stuck back in the 1960s in terms of producing college-educated men.




USA Today report on 135:100 boys:girls ratio in college
sexism in Everyday Math
invisible boys
boy trouble (New Republic on boys)
slacker boys, middle school, & forbidden positive images of boys in textbooks
throw rocks at them
please remain seated at all times
Ann Althouse thread sums up classroom change
 cooperative vs. competitive learning
the girl show (8th grade graduation awards)
the boy show (character ed)
the other boy show
Where the Boys Aren't

letter from Robert Lerner, former commissioner NCES
Tom Mortenson's research
The Boys Project board
for every 100 girls —


-- CatherineJohnson - 24 Jan 2006


ParentsMentoringChildrenInSingapore 16 Sep 2006 - 20:27 CatherineJohnson



OK, I know I'm supposed to be typing Steve's algebra lessons into Equation Editor so mere mortals like me can read them, and also doing the same for my own Comment for Carolyn's Sticking Points post.

Plus I've probably got a civil servant or two who need bullying this afternoon (where does David Allen say to put 'bully civil servants' on the Master List?)

But all of that can wait!

BECAUSE FIRST I HAVE TO WRITE MY SLAVE PARENTS IN SINGAPORE POST!