Entries from CognitiveScience

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.

…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.

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.

(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.

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

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)

homeschool mom with common sense-y

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.

Even worse, math professors are attending them.

inflexible knowledge, flexible knowledge, and expertise

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

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.

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

I said, 'You do?'

'Yeah.'

'How come?'

'They're not stupid.'

No idea what that means.

update

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

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

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.

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.


Steven Pinker



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

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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.

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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.

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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!

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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.

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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

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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.

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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.

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

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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.

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RoadsideFragmentation 31 Jul 2005 - 13:27 CatherineJohnson


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





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

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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.

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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.




It's in my cart!

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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:

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:

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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.





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

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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.

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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.

Plu