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CarolynIsGobsmacked 23 Jun 2006 - 13:33 CarolynJohnston


Did you see this chart in Garelick's article that showed the grants that ed departments were given to come up with new math curricula?

ednext20052_garelickfig1.gif

Man, did I ever go into the wrong branch of academia. Five million dollars for Everyday Math! Six for Trailblazers! Fourteen for Contemporary Math in Context!!!!

The path I should have taken is now clear:

  • get a Ph.D. in education from a prestigious department;
  • become a professor and get a HUGE grant to develop a new math curriculum in whatever flavor the government currently prefers;
  • develop it, then sit back and let the government do your marketing for you;
  • then, develop a lucrative side business as a consultant and speaker on the math curriculum you developed, while still enjoying tenure, a light workload, and the envy of your colleagues.

What the heck was I thinking all those years ago? These aren't the only academic rackets I've heard of, by a long shot -- I know of a number of others too (almost any of which beat pure math, in which you work your fanny off for 50K or so in summer research salary, and are glad to be getting it). But I took the road less traveled by!...

I think this math curriculum thing may actually be the sweetest racket of them all.

It also suggests that the push toward constructivist curricula didn't necessarily come from the ed schools themselves. Any school seeking grant money is obviously going to be responsive to the prevailing political winds, which in this case seem to have been emanating from the National Council of Teachers of Mathematics and the National Science Foundation.


MoreBigNumbers
BigNumbers





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





CalBoardOfEdStudyPart2 16 Sep 2006 - 19:59 CatherineJohnson


Carolyn wrote:

I thought I would do a 'mini-series' [on the California Board of Education study]
describing and discussing their results, section by section. Stay tuned.


What a great idea!

I've been wanting to know more about the famous California Board of Ed study.

Here's a terrific factoid about Dixon et al, from The Principal's Guide to Raising Math Achievement by Elaine K. Mc Ewan:

From a total 8,727 published studies of mathematics in
elementary and secondary schools, they identified only
956 articles that satisfied the minimum identification
criteria of being an experimental study of mathematics.
. . . The evaluators then used the evaluative criteria
for experimental research . . . Only 231 of the original
956 studies made it through an initial screening of
construct, internal, and external validity. When the
methodologies of those 231 studies were screened
for internal and external validity, only 110 studies
were deemed to be of high quality.


8,727 "studies."

Of which, 231 were scientifically valid.

231

Parents, teachers, administrators, and Concerned Citizens everywhere should have this figure tattooed to their foreheads.

When textbook publishers and ed school types use the words "research shows," you're looking at maybe a 3% chance they're right about that.

Given the fact that, by law, all research findings have to be replicated before they can be certified as facts, the odds are probably closer to zero.

(OK, I'm kidding. There's no law. Anyone can call anything a fact if they want to. It's a free country.)

CalStateStudyIntro


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





CalStateStudyOfGroupLearning 24 May 2005 - 19:43 CarolynJohnston


Part 2 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

The most surprising thing, to my mind, in the Cal State Study is its strong endorsement of cooperative group learning. The review included eleven studies of group learning, all with positive findings for the use of structured group learning.

The Cal State Study defines "conventional mathematics instruction" as being characterized by teacher explanation of the new material, followed by independent workbook activity. The study makes the strong claim that the conventional approach has absolutely no theoretical support, and is discredited by the totality of the studies in the review that examine cooperative group learning methods.

Actually, this is the one place I've read in the report so far where I felt I might be encountering just a whiff of reviewer bias, particularly in the glowing interpretation of these findings.

But the evidence might really be there. Carefully constructed group learning opportunities beat conventional teaching methods. Go figure.

The phrase 'carefully constructed' is a point to dwell on. Simply sticking kids together in groups to do their homework conferred no benefit. The group learning environment had to be structured, i.e. the nature of the kids' interactions had to be controlled by external reinforcement systems. In particular, higher performing kids had to be motivated to help lower performers.

To some degree, it didn't even matter what the system was, but it had to be present. One study compared cooperative vs. competitive group reinforcement systems: an example of a cooperative system is one in which each kid in a group gets the average of their individual grades, whereas grouping the kids into competitive teams is a competitive reinforcement system. Both reinforcement systems were shown to confer equal benefits above the conventional teaching style.

The reviewers note that "a particularly interesting aspect of these studies on cooperative work is that all of them included students who were at risk in some sense, either by virtue of being inner city urban students, and/or low SES students, and/or students identified as having special learning needs."

While the reviewers seem rather pleased with this study design, to my mind it weakens the broad applicability of the results. I rather wish they'd done these studies with the most typical bunch of learners they could find, since the presence of special needs can really skew a kid's learning style. Kids who are starved for attention, interaction, or positive reinforcement for learning, might benefit much more from learning environments that include these things than a typical learner would.

Also see: CalBoardOfEdStudyPart2, EdResearch


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





EdResearch 18 May 2005 - 15:13 CatherineJohnson




ednext20023_36fig1.gif

graphic from:
The Virtues of Randomness
by Robert Boruch



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





CalStateStudyOnManipulatives 24 May 2005 - 20:01 CarolynJohnston


Part 3 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

Part 2: CalStateStudyOfGroupLearning

Another surprising fact -- about math manipulatives -- comes out of the Cal State Study.

There were only four studies of manipulative use that were of high enough quality to make the Cal State cut. That really isn't enough to draw a conclusion from, especially given the studies' haphazard coverage of the range of instructional possibilities. Still, there are enough results that they suggest a pattern. See if you can detect it ('benefit' implies that kids did significantly better on normalized tests of math achievement than control groups did).

Kindergarten kids learning counting: no benefit conferred by including manipulatives.

Third graders learning multiplication: two different studies show no benefit to the use of manipulatives before teaching formal computation.

Fifth and seventh graders learning fractions: kids benefit from a fractions game played with or without other manipulatives and pictorial representations.

Elementary schoolers using fraction/ratio manipulatives with fraction/ratio instruction: no benefit.

Seventh graders using fraction/ratio manipulatives with fraction/ratio instruction: benefit.

I love what these results suggest because it is so unexpected and counterintuitive. Most of us think of manipulatives as a stepping-stone from the concrete to the abstract, as something to be used only by the very young when they are first introduced to a topic.

But these results suggest that older kids get more benefit out of manipulatives. In a way, now that I think about it, it makes sense; their relative maturity means kids have a conceptual 'hook' on which to hang the insights that the manipulatives give them. They already have half a clue, and that helps them get the point of the manipulatives.

Perhaps to a younger kid, less able to generalize from the concrete to the abstract, the manipulatives are simply toys.

This is all the evidence I need to get the fraction manipulatives out for my soon-to-be sixth grader.

For more information on math manipulatives, see our favorite math supplements for kids and FractionManipulatives

Also see:
EdResearch
CalBoardOfEdStudyPart2
FractionManipulatives
QuickThoughtAboutFractionManipulatives
FractionManipulativesPart2
NewStudyOnManipulatives
New Study on Manipulatives Part 2





CalStateStudyMasteryLearning 24 May 2005 - 19:34 CarolynJohnston


Part 4 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

Part 2: CalStateStudyOfGroupLearning

Part 3: CalStateStudyOnManipulatives

I had never heard the term "mastery learning" before reading this report, but I immediately understood it, because I am so familiar with the use of the opposite strategy in the Everyday Math curriculum.

The identifying feature of mastery learning is that students have to demonstrate mastery of the current material before moving on to new material. Since students in a classroom generally move on in a block, true mastery learning can be difficult to implement in the classroom, since you'll always have a few kids in the group who are lagging. Still, mastery learning is what happened to many of us when we were in school, and is what parents assume is going on in our kids' classrooms.

But it's probably not. In the Everyday Math primary curriculum, the same material is taught every year, with the kids expected to grow in achievement every time they encounter a topic, and to attain mastery at some time in the indefinite future. This 'spiraling' approach is backed up by the use of both pre-tests and post-tests, given before and after a unit is studied, collectively measuring the student's progress toward mastery of the material during the unit.

My objection to this approach is that The Day of Mastery is pushed off into the indefinite future. If your kid doesn't get it yet, you needn't worry; because he'll get another chance at it; it's never time to press the panic button until the kid switches to another curriculum (or goes to high school).

But even in more conventional curricula, topics aren't taught to mastery on the spot. In the Saxon curriculum, after a topic is introduced, the student moves on to another topic the next day; but encounters the first type of problem over and over again in the practice problem set. As with mastery learning, there is no escape from a topic until it's mastered. New topics that depend on the old topic aren't introduced until the student is competent with the original topic.

How good is mastery learning? Two of the review studies looked at mastery learning by itself and with combinations of other curricula, and found that mastery learning by itself produces better results than what was termed 'conventional instruction'. However, mastery learning got its best results when used with other teaching techniques. One study got decent results for "mastery learning with corrective feedback" (meaning -- electric shock? The review didn't say), but got its best results from mastery learning with 'enhanced cues' -- extremely detailed instructions to the students on how to do problems.

Another study found that mastery learning and cooperative learning strongly enhanced each other (note: cooperative learning is structured working-together among students, as opposed to simply being stuck in groups to do your homework together: see part two of this series).

I can see why cooperative learning and mastery learning would enhance each other. If you're a fast learner and the kid next to you is holding you back from getting to the next topic, you're going to be highly motivated to help him get it. This is fine for the average kid, but for the faster ones...? Frustration, possibly. And yet the study claims that achievement was enhanced by cooperative learning for both the helpers and the helpees...

To summarize, I found it hard to draw conclusions about mastery learning from this review. I'd like to see a study of mastery learning compared to a curriculum like Saxon math, which doesn't demand mastery on the spot, but rather draws it out of the kids over time.



MoneyClassSizeMathAchievement 16 Sep 2006 - 20:00 CatherineJohnson


This item made my day.

After our school board announced that budgetary constraints had left them no choice but to increase class size in the 4th and 5th grades (from 19 or 20 students per room up to 23 or 24) parents voted in our second double-digit tax increase in a row.

Our fourth and fifth grade classes will remain small.

I was skeptical.

For one thing, I was aware that Asian math classes are far larger than our own.

For another, I was aware that comparative education researcher James Stigler* actually recommends increasing class size as a means of improving math achievement in America. Larger class size would allow American teachers to meet with colleagues in the lesson study groups that are standard practice in high-achieving countries.

But while I knew all this, I hadn’t quite allowed myself to draw the obvious conclusion.

I hadn’t grokked the possibility that if you’re living in a school district where everyone’s clamoring for small class size, and no one’s clamoring for teacher release time, . . . that might be a problem.

So this afternoon I found this analysis of TIMSS data in Education Next:


When other factors are taken into account, higher
spending and smaller class sizes seem to correspond
to inferior mathematics and science results, though
the overall effect is relatively small.


Well, all I’ve got to say is, thank heavens there’s only a small correspondence between high spending, small class size, and inferior mathematics and science results.

Because if there were a large correspondence we’d be in trouble.


+ + +


I like this chart, too:

ednext20012_69.gif



soucre:
Why Students in Some Countries Do Better
by LUDGER WOESSMAN
EDUCATION NEXT


* James Stigler was one of the investigators in the 1999 TIMSS study and is coauthor of The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom by James W. Stigler, James Hiebert and The Learning Gap: Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education by Harold W. Stevenson, James W. Stigler.




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.



SummerSupplementTime 07 Jul 2005 - 21:25 CatherineJohnson


Too much going on today!

I'm eager to think about 'teacher boredom' and ed reform . . . plus I have a terrific email from a teacher on the subject of summer regression that needs a few identifying details deleted before I can post --- and I have a life beyond this bliki, too, or at least I used to.

But all that can wait!

summer regression

I've just stumbled across what I think may be a good source of information (pdf file) on summer regression.

Tilley, Cox, and Staybrook47 studied summer regression in achievement for students receiving no educational services for three months. They found that most students experience some regression during the summer recess. Cooper et al.48 reviewed 39 such studies and found that achievement test scores do indeed decline over the summer vacation. Their meta-analysis revealed that the summer loss equaled about one month on a grade-level equivalent scale, or one tenth of a standard deviation relative to spring test scores. The effect of summer break was more detrimental for math than for reading and most detrimental for math computation and spelling. Also, middle-class students appeared to gain on grade-level equivalent reading recognition tests over summer while lower-class students lost on them. Possible explanations for the findings included the differential availability of opportunities to practice different academic material over summer (reading is much more easily practiced than mathematics) and differences in the material’s susceptibility to forgetting (factual knowledge is more easily forgotten than conceptual knowledge).

The critical points bear repeating:

  • Summer loss equaled about one month
  • The effect of summer break was more detrimental for math than for reading and most detrimental for math computation and spelling


Think about it.

One month's loss, for kids who are already at least a year behind their peers in high-achieving countries.

I think it's important to keep up your child's math skills in the summer!

(Carolyn and I have been brain-storming ways to use KTM to help-----)

TO BE CONTINUED


FreeWorksheets
TreadingWater

SummerSupplement
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SummerSupplementTimePart5 (resources for preventing summer regression)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

TeachYourChildToTypeThisSummer

BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
HowToSpell
HowToSpellPart2
MoreSpelling
TheSaxonMathOfSpelling

Summer Supplement Time
linking decline in high school scores to elementary school
research on summer regression
the time costs of not teaching to mastery
U.S. fourth graders not doing as well as thought
Phase 4 topic list, grade 6 class
comments thread on pre-algebra as algebra





GreatMindsThinkAlikePart2 08 Jun 2005 - 19:34 CatherineJohnson


I just mentioned $400 hourly fees to masters of direct instruction and today Joanne Jacobs has a link to Zig Engelmann's new web site.

Jacobs also links to a terrifically useful glossary of terms here.



EdSchoolDangers 10 Jun 2005 - 14:03 CatherineJohnson


David Klein:

The field of math education is more or less at the level of medieval medicine.  In those days you might be better off not seeing a physician, because he might bleed you to death trying to cure you from a bad cold.  So it is with today's colleges of education.  



CM100_Chain_Mail_small.gif


(you can click on this guy)



NewComments 07 Jul 2005 - 20:47 CatherineJohnson


SteveH has a new comment about Base 5 & fuzzy math in the CompareAndContrast thread.

update: More from Steve!

Thank you!

I love this, especially:

when my son was born, I told my mother that I wanted 3 things for him in life: 1. To care about other people. 2. To know the value of hard work. and 3. To be happy. Her response was that if he did numbers 1 and 2, then number 3 will take care of itself.

And this:

If Everyday Math (as an example), thinks that doing things in different ways is helpful, then why do they completely avoid the standard algorithms (the best ways)? While doing Singapore Math with my son at home, he ends up doing a number of things in different ways than his EM at school. This can be helpful, or it can be an overload of the brain.

I think SteveH is also the commenter who pointed out that ed school students are taught constructivist teaching methods via direct instruction.

I say that's not fair.

If our kids have to discover math, ed students should have to discover discovery.

Guess and check, guys!

Lots of sharp observations on math & practice, math & creativity, math & solving problems more than one way here: ILikeMath



FreeWorksheets 07 Jul 2005 - 21:26 CatherineJohnson


from SusanS:

Two more sites with free math worksheets (and other free stuff) are edhelpers.com and superkids.com. I do love the free stuff.

Thank you!

our favorite math supplements

We are slowly but surely pulling together the sidebar pages, so you might want to take a look from time to time.

We also need to get a reader recommendation page going.

I'm adding Susan's recommendations to the 'our favorite supplements' page so they'll be where people can find them easily.

I'll also gather together the grammar, spelling, handwriting, etc. book & curriculum recommendations into one place, with links to the original reader comments. These are invaluable, so keep them coming!

Back to online math resources, also remember Carolyn's recommendation:

... These math worksheet generators can come in very handy.... very configurable; you can set the number of columns and rows of problems, and the difficulty of the problem, and the numbers of significant digits in the solution, and so forth....

We especially found the sheets for fraction and decimal long division useful. That's a skill that just takes a lot of practice.


computer learning versus paper-and-pencil

Susan inspired me finally to track down some of my favorite online resources and get them entered on the Our Favorite Supplements page.

But first I should say that I'm leery of online math practice, for 3 reasons:

  • Christopher has never learned well using a computer

  • I've seen research showing a slight decline in student achievement in Israeli schools after the introduction of computers in classrooms



Christopher didn't really get his math facts down cold until we started doing the Saxon fast fact paper-and-pencil worksheets.

He didn't make any headway that I could see using a software math facts program, and I don't think he made much progress using standard flash cards, either.

To be fair, we have problems using materials like flash cards, since I'm constantly having to hide them from Andrew, which of course makes it harder to find them when I need them, which, in turn, makes me tend to use them less than I would if they were easy to get to ...

So I don't know whether anyone should be drawing conclusions from my flashcard experience.

But when it comes to computers-versus-paper and pencil, if you've got time to print out the worksheets Carolyn & Susan have pointed you to, that's probably the better choice.

Online 'worksheets' may be to paper worksheets what fast food is to homemade.

That said, I've eaten plenty of fast food in my day, and so have my kids.

So here's one of the main online resources I've liked thus far.

Saxon Math online problems and math activities

  • I've seen a number of parents around the web recommend this Saxon Math 'fast facts' generator. The page is clean, simple, and visually compelling. You decide which math-fact problems you want to do, how difficult the problems should be, and how many you want to do. You can also do timed or untimed problem sets. That's great, because kids love seeing their timing get faster.

  • Here are the 5th grade activities.
    Apparently the site now tells you which activities to do after which lessons in the book; plus you can download the activities for use when you are not online.

  • Saxon online equivalent fractions These are great. OK, I'm sold. Forget the Israeli kids; we're doing online equivalent fractions this summer.




TreadingWater

SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4
SummerSupplementTimePart5 (resources for preventing summer regression)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

TeachYourChildToTypeThisSummer



And lots more....



FourthGradeSlump 28 Jun 2005 - 16:36 CatherineJohnson


A slump in math gains begins after fourth grade and extends through high school on both national and state tests. The middle-grade slump also appears on the most prominent international test, TIMSS, as the relative ranking of American students falls precipitously after fourth grade. No other country has a sharper drop in math ranking than the United States. A British publication, The Economist, concluded, "The longer children stay in American schools, the worse they seem to get."9 This overstates the case. Older students have made gains in learning. The slump is not in absolute achievement, but in the pace of improvement. It decelerates after fourth grade.

THE BROWN CENTER REPORT ON AMERICAN EDUCATION 2000 How Well Are American Students Learning?


No other country has a sharper drop in math ranking than the United States.


FourthGradeSlumpPart2




FourthGradeSlumpPart2 14 Nov 2005 - 17:03 CatherineJohnson


Percentage comparisons of students who scored in the top 10 percent of fourth-graders among the 26 TIMSS countries also show that the United States is lagging. In math, only 9 percent of U.S. Fourth-graders were among the top 10 percent, compared to Singapore’s 39 percent, Korea’s 26 percent, and Japan’s 23 percent. At the eighth-grade level, only 5 percent of U.S. Students were included in this bracket . . . once again confirming that U.S. students do not fare well in international comparisons and drop in rankings the further along they are in school.(14)

source:
SCHOOL FIGURES: THE DATA BEHIND THE DEBATE BY Hanna Skandera & Richard Sousa


FourthGradeSlump




MathAndLanguage 03 Jul 2005 - 15:25 CarolynJohnston


I was talking to a friend of mine the other day (okay, okay, it was Catherine). She had been to a party at the French Embassy in Washington, and got talking with a French gentleman there about mathematics.

"Mathematics is a language," he told her.

"It is?" she said.

"Furthermore, it is a dead language," he went on (I bet he says this sort of thing to all the women he meets at parties).

Later, Catherine asked me if I think that mathematics is a language. I've been hearing this sort of thing for years, and had never really thought too deeply about it. I don't even know exactly what qualifies something to be a language. But Catherine asked me, and Catherine doesn't ask questions lightly, so I gave it some thought.

What is a language, exactly? I, for one, certainly don't know. But whatever a language is, I thought, it ought to be able to stand on its own. The jargon of a specialized field shouldn't count as a language by itself; it's just jargon. If you take away the English or German or French or Chinese words that support the jargon, the jargon doesn't stand on its own. And that's how it is with mathematics.

You can write out some simple proofs, of course, without using any words at all. But you can only write out the simplest arguments that way, basically those that follow directly from manipulating expressions and equations. I think a language should be able to support all sorts of complicated ideas from all walks of life; if mathematics is a language, then it's a pretty limited one.

"No", I said, "I don't think it is."

When I mentioned this later to Bernie, he pointed this article out to me. These research results suggest that mathematics and language are pretty much independent functions, as far as our brain functioning is concerned.

I would guess that the intersection of language and math occurs at word problems. Word problems are very hard, perhaps because we do have to integrate totally different functions in our brains in order to understand them. But turning word problems into algebraic expressions isn't translation; it's distillation. A lot of the meaning in the original word problem is left behind in the process; the names of the kids who exchanged marbles, the fact that it was marbles that were exchanged, and so forth. You can't work backward from the algebraic expression and uniquely reconstruct the word problem as it was originally.

So I think Catherine's French acquaintance was wrong. If anyone knows any more on the topic of what language is than I do, and if mathematics actually qualifies as a language, I'd like to hear about it.


What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...)

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
"the language of numbers is not language" 7-3-05





UKFrameworkForAlgebraPreparation 08 Jul 2005 - 17:13 CatherineJohnson


Liping Ma says that math teachers should know where their pupils are headed.

What skills will a child most need in the next stage of his education?

Since I had no clue, one year ago, what skills a 5th grader needs for algebra in 8th, I found this UK 'Framework for Teaching Mathematics' document, Laying the foundations for algebra, terrifically helpful.

After I read it, I spent a LOT of time pushing the distributive property.... which, as my friend Debbie says, 'is one useful property.'

e.gif



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





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.



HighTechHeretic 12 Jul 2005 - 18:13 CatherineJohnson


Jeff Boulier just pointed me to High Tech Heretic: Why Computers Don't Belong In the Classroom and Other Reflections by a Computer Contrarian.

This reminds me that I never got around to reading The Cuckoo's Egg: Tracking a Spy Through the Maze of Computer Espionage, so I'm ordering that, too!

I think Clifford is right about computers in classrooms.

The research I've seen makes me think that Computers are Calculators writ large, with many of the same negative effects on learning.

Even if I hadn't seen the research, the fact that we have Mystery NGOs actively promoting the use of computers in classrooms--and being cited as authorities by Steve Leinwand--would make me leery.

I'll get around to posting the studies I've found on this question sooner rather than later, I hope.


update

Oops.

I already did post the Israeli study of computer use in the classroom.



TeacherTrainingInChina 25 Jul 2005 - 00:06 CarolynJohnston


SusanS wrote the following post about teacher training and education on the Martin Gross thread, and it's got me so intensely curious now about the Chinese school system that I've decided to break it off and give it its own thread.

The Chinese system of teacher training and development has garnered a lot of interest because of Liping Ma's incredible book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. The Chinese teachers whup us in understanding and pedagogy, of course, but it's the details of how they whup us, and what they do differently, that are fascinating.

I've long thought that what lies at the base of the difference in "teacher culture" is the difference in our cultures themselves. It appears to me that Chinese elementary math teachers are respected specialists, for one thing. Compare that with teaching in the United States, where teachers are anything but respected (as the Martin Gross column proves).

Here' s Susan's post:

Okay, back from the web... I found an interesting straightforward article concerning China.

"There are three main educational aims for elementary schools in China today. The first is to develop the students moral character by teaching them to love the motherland, the Chinese people, manual labor, socialism and the Chinese Communist Party, and public property. The second goal is to enable students to obtain a fundamental education, develop skills in reading, writing and science, possess social knowledge and cultivate good study habits. The third goal is to enable students to develop physically. At least one hour a day students are required to perform some type of physical exercise."

"Originally, the duration of general middle schools was five years, but now in many places this has been changed to six years. The six years are divided into two levels: junior middle school and senior middle school. There are over 162,000 middle schools in China with over 65,400,000 students, more than sixty times higher than the number in 1949."

(This is kind of interesting....)

"Professional middle schools train middle level personnel for various vocations. Students entering these schools are required to have graduated from junior middle school and have some professional knowledge in a special area. The duration of these schools is from three to four years. Because all professional schools were closed during the Cultural Revolution some students entering these schools now have already graduated from senior middle schools and are completing the professional program in two years."

"There are seven types of professional middle schools: technical, agricultural, forestry, medical, financial and economic, physical education, and art. There are more than 1700 professional schools in China with more than 500,000 students enrolled."

"Teacher training schools are included in these schools. Students are drawn from senior middle school graduates and complete their training in three years. Tuition is free. Elementary school tuition is five yuan a year, middle school is 10 yuan a year. There are over 1,046 teacher training schools in China with an enrollment of more than 360,000 students (29% women)."

http://www.yale.edu/ynhti/curriculum/units/1982/4/82.04.02.x.html

I'm not certain of the validity of the source, but it's a place to start. I do believe that we can learn a lot from the Chinese and the Liping Ma book is absolutely a great book for any parent or teacher. But there are obviously some things that we probably can't do on any official level.

For me, this view into what's going on behind the scenes in the Liping Ma book (and I always had the feeling it was something very different from what goes on here) raises more questions than it answers.

I have no clue what 'middle school' would be the equivalent of in the US or Europe. It sounds like it might be a technical college, or what would once have been called 'Normal School'? (My grandmother, who like many intellectual women of her generation taught school, was educated at a Normal School, and it was a good education).

And are public schools for children called 'elementary schools', all the way through what we would call high school?

And why is there tuition for elementary school and not professional training schools?

And now I wonder whether people are tracked into professions by the government, or are free to choose what they want to do?

Susan raises an excellent point here. What exactly can we do -- what would we really be willing to do -- to have a teaching culture that is more like China's?


Susan S on teacher training in China
how Chinese teachers learn math
teacher release time & Liping Ma & Elaine McEwan's Princepal's Guide





LovelessOnInconclusiveFindings 21 Jul 2005 - 01:06 CatherineJohnson


Tom Loveless is one of my favorite ed writers & researchers.

Here he is on the question of inconclusive findings:


The research on tracking and ability grouping is frequently summarized in one word: inconclusive. This pronouncement is accurate in that nearly a century’s worth of study has failed to quantify the impact of tracking and ability grouping on children’s education. It doesn’t necessarily mean, however, that the gallons of ink spilled on these issues have been much ado about nothing. A non-effect in educational research is quite common. It can mean that the practice under study is truly neutral vis-a-vis a particular outcome. But it can also mean that the practice has off-setting negative and positive effects, that positive effects are produced under some conditions and negative effects under others, or that effects occur that researchers either don’t measure, because they’re measuring something else, or can’t measure, because of inadequate methods or expertise.

Non-findings must be interpreted with great care, especially when looking for policy guidance. In 1966, a federal report was released that many scholars consider the single most famous study in the history of education, Equality of Educational Opportunity, otherwise known as the Coleman Report for its primary author, the famed sociologist James Coleman. The Coleman Report was widely interpreted as finding that schools themselves have no significant effect on student learning. Fortunately, policymakers did not rush out to close schools and turn them into car washes or something else more useful.




ar_TheCarWash.JPG



LovelessOnTracking 13 Nov 2005 - 19:56 CatherineJohnson


Loveless' survey of the research on tracking is interesting, especially given the philosophical opposition to all tracking that seems to be part of constructivist pedagogy:

Slavin’s support largely resting on the benefits uncovered for grouping in mathematics in the upper grades of elementary school.


I'm confused by the phrase upper grades of elementary school.

Does this mean 4th and 5th grade?

Or is middle school considered technically part of elementary school?


Here in Irvington, de-tracking students was part and parcel of bringing in TRAILBLAZERS.

Differentiated instruction is the buzz word.



tracking good for talented students?

Kulik finds that tailoring course content to ability level yields a consistently positive effect on the achievement of high ability students. Academic enrichment programs produce significant gains. Accelerated programs, where students tackle the curriculum of later grades, produce the largest gains of all. Accelerated gifted students dramatically outperform similar students in non-accelerated classes. Slavin omits studies of these programs from his analysis. He argues that the gains, though large, may be an artifact of the programs’ selection procedures, that schools admit the best students into these programs and reject the rest, thereby biasing the results.38

Three things are striking about the Slavin- Kulik debate. First, the disagreement hinges on whether tracking is neutral or beneficial. Neither researcher claims to have evidence that tracking harms achievement, of students generally or of students in any single track. Second, accepting Slavin or Kulik’s position on between-class grouping depends on whether one accepts as legitimate the studies of academically enriched and accelerated programs. Including these studies leads Kulik to the conclusion that tracking promotes achievement. Omitting them leads Slavin to the conclusion that tracking is a non-factor.

Third, in terms of policy, Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake.



So....just a few short paragraphs ago, Loveless has told us that inconclusive findings have to be interpreted with caution.

Is this finding of 'no harm done' a positive finding?

Or is it an inconclusive finding?

And why aren't we told?


we need editors!

Now that I'm reading think-tank & NRC pubications, I have a Firm View on the question of book editors.

Every book needs one.

I don't care how smart the author is.


talented kids need accelerated classes

High School and Beyond (HSB) is a study that began with tenth graders in 1980. The National Education Longitudinal Study (NELS) started with eighth graders in 1988. These two studies followed tens of thousands of students through school, recording their academic achievement, courses taken, and attitudes toward school. The students’ transcripts were analyzed, and their teachers and parents were interviewed. The two massive databases have sustained a steady stream of research on tracking.

Three findings stand out. High track students in HSB learn more than low track students, even with prior achievement and other pertinent influences on achievement statistically controlled. Not surprising, perhaps, but what’s staggering is the magnitude of the difference. On average, the high track advantage outweighs even the achievement difference between the student who stays in school until the senior year and the student who drops out.40



I say again: think tanks need editors.

I believe what he is saying here is that the gap between the high & low track student in the HSB study was larger than the track between high school graduates and high school drop-outs.


it figures

African-American students enjoy a 10% advantage over white students in being assigned to the high track. This contradicts the charge that tracking is racist. Considered in tandem with the high track advantage just described, it also suggests that abolishing high tracks would disproportionately penalize African-American students, especially high achieving African-American students.


A worthy mission for the fuzzies, de-tracking the whole entire country.

Thanks, guys.


tracking & the achievement gap

Moreover, NELS shows that achievement differences between African-American and white students are fully formed by the end of eighth grade. The race gap reaches its widest point right after elmentary and middle school, when students have experienced ability grouping in its mildest forms. The gap remains unchanged in high school, when tracking between classes is most pronounced.41



Sophie's choice

Third, NELS identifies apparent risks in detracking. Low-achieving students seem to learn more in heterogeneous math classes, while high and average achieving students suffer achievement losses—and their combined losses outweigh the low achievers’ gains. In terms of specific courses, eighth graders of all ability levels learn more when they take algebra in tracked classes rather than heterogeneously grouped classes. For survey courses in eighth grade math, heterogeneous classes are better for low achieving students than tracked classes.42

These last findings are important because we don’t know very much about academic achievement in heterogeneous classes. When the campaign against tracking picked up steam in the late 1980s, tracking was essentially universal. Untracked schools didn’t exist in sufficient numbers to evaluate whether abandoning tracking for a full regimen of mixed ability classes actually works. The NELS studies that attempt to evaluate detracked classes, which thus far have been restricted to mathematics, point toward a possible gain for low achieving students and a possible loss for average and above average students, but these findings should be regarded as tentative.43





grouping versus tracking?

The elementary school practices of both within-class and cross-grade ability grouping are supported by research. The tracking research is more ambiguous but not without a few concrete findings.


Will somebody please get the Fordham Foundation an editorial staff?

What is grouping?

What is tracking?

Why aren't these terms defined?

OK, I'm assuming 'grouping' means grouping kids according to ability within the same class, as Christopher's school does for reading.

I'm assuming 'tracking' means creating separate classrooms with separate teachers for kids of differing ability.


Singapore vs U.S.

Assigning students to separate classes by ability and providing them with the same curriculum has no effect on achievement, positive or negative, and the neutral effect holds for high ,middle, and low achievers. When the curriculum is altered, tracking appears to benefit high ability students.

This is exactly what happens in Singpoare--separate classes, same curriculum--but in Singapore this practice has a large positive effect.


race & income

When it comes to race, the disparities are real, but, as just noted, they vanish when students’ prior achievement is considered. A small class effect remains, however. Students from poor families are more likely to be assigned to low tracks than wealthier students with identical achievement scores. This could be due to class discrimination, different amounts of parental influence on track assignments, or other unmeasured factors.44



what do black parents say?

A study conducted by the Public Agenda Foundation found that "opposition to heterogeneous grouping is as strong among African-American parents as among white parents, and support for it is generally weak."45 If tracking harmed African-American students, one would not expect these sentiments.



choose your poison

The public labeling of low track students may cause embarrassment, but the public display of academic deficiencies undoubtedly has a similar effect in heterogeneous classrooms. There, a low ability student’s performance is compared daily to that of high-achieving classmates.46


At our school the tracking-obsession among the kids is brutal. There's a huge amount of taunting; at least, there has been when I've been around. 'I'm a 4!' 'You're a 2!'

Yuck.


jumped the track

A study of transcripts from five Maryland high schools showed 59.9% of students changed math levels during their high school careers, 65.4% in science. A national survey of high school principals reports substantial movement among tracks, especially upward (see Table 7). But an analysis of NELS data found that only 16.5% of students who were in low-ability classes in 8th grade went on to take either geometry or Algebra II by 10th grade (in comparison to 81.0% of 8th graders in high-ability classes).


Which reminds me, I've been meaning to post the strange goings-on with Phase 3 & Phase 4 this spring.....


you don't say

Without a push, a lot of students remain in low tracks who are capable of moving up.



Singapore vs. U.S. redux

It appears that high tracks are taught by better qualified teachers, however, in the sense of having teachers more schooled in content know-ledge.48 High school principals are inclined to assign teachers who know advanced subject matter to teach advanced subjects.


Another glaring difference.... (more on this another day).


Catholic schools

Reba Page’s 1991 study, Lower Track Classrooms, painstakingly reports on the daily activities of eight low track classes, documenting how they often function as caricatures of high tracks, how teachers and students in low tracks make deals to not push each other too hard so that they can cope with their environment. Low tracks may be used as holding tanks for a school’s most severe behavior problems. Even under the best of conditions, low tracks are difficult classrooms.

Intellectually stimulating low track classrooms do exist, however, and researchers have found the most productive of them in Catholic schools. Margaret Camarena and Adam Gamoran have described low track classrooms where good teaching, lively discussions, and ample learning take place. In 1990, Linda Valli published her study of a heavily tracked Catholic high school in an urban community. The school’s course designations publicly proclaimed each student’s track level. Textbooks and instruction were adapted for each track. Yet Valli discovered that "a curriculum of effort" permeated the entire school, even the lowest tracks. The school culture centered around academic progress, and the tracking system was but another facet of the school that served this aim. Students of all abilities were aggressively pushed to learn as much as they could. Every year, low track students were boosted up a level. By the senior year, the lowest track no longer existed. A judicious tracking system teaches low track students what they need to know and moves them out of the low track as quickly as possible.51



I hate like the dickens seeing Catholic schools go out of business.



NoTVsInChildrensBedrooms 25 Jul 2005 - 23:26 CatherineJohnson


“In this study, we found that the household media environment was related to a child’s academic achievement,” said Dina Borzekowski, EdD, lead author of the study and assistant professor in the Department of Population and Family Health Sciences at the Bloomberg School of Public Health. “Among these third graders, we saw that even when controlling for the parent’s education level, the child’s gender and the amount of media used per week, those who had bedroom TV sets scored around 8 points lower on math and language arts tests and 7 points lower on reading tests. A home computer showed the opposite relationship—children with access to a home computer had scores that were around 6 points higher on the math and the language arts test and 4 points higher on the reading test, controlling for the same variables.”

[snip]

The researchers did not find a consistent negative association between test scores and the amount of television watched per week.



Apparently, a child who spends 20 hours a week watching TV in his bedroom scores worse than a child who spends 20 hours a week watching television in his parents' bedroom.

I believe it.

Christopher had a TV in his bedroom for awhile, because Andrew uses it to watch his Barney videos. Christopher had his Playstation hooked up to it.

Christopher and I stopped having any relationship at all. It was as if he'd moved out.

Finally I complained to my neighbor about how I never saw Christopher any more, I needed to limit the Playstation hours, etc., etc.

She said, 'Take the Playstation out of his bedroom.'

Duh.

I did, and problem solved.

Of course now it's practically impossible for us to watch a DVD, since the Playstation is hooked up to our TV, and playing DVDs on the Playstation, while possible, is a chore.

But we have lots more togetherness.

It's possible that more 'face-time' with your child translates directly to higher scores because of the higher levels of interaction.

Or it's possible that parents who insist on lots of face time are also more involved in homework....

Or it's possible that this study belongs to the 33% of highly cited studies that turn out to be wrong.

That's the great thing about studies.

After you've read a new one, you don't know if you've actually learned something new or not.



MathsInEnglandPart2 27 Jul 2005 - 13:45 CatherineJohnson


I just glanced at the British maths report, Where will the next generation of UK mathematicians come from?, (pdf file) and I think I'm going to read the whole document. It reminds me very much of observations both Carolyn and Bernie have made to me, as well as Carolyn's post, Whither American Talent?.

Still, I'd never quite thought of the issue as a 'failure to reproduce,' as the report sees it. I'm not surprised Britain would be thinking of it this way, because of Europe's declining, or soon-to-decline population, which seems to me to have been covered fairly extensively in the European press.

They're right in framing matters this way. For countries and civilizations to grow and thrive, they must reproduce themselves biologically and culturally--which means, I think, that it's not a great idea to allow math talent to dwindle away, as it seems to be doing.

(fyi, I'm having a metacognitive moment here: I'm asking myself, Do I know, for a fact, that any of these statements are true? Answer: no.)

So I'm assuming these things are true, until I learn otherwise. Excerpts from the report:

  • The UK mathematics community now falls far short of “reproducing itself” – as evidenced by the dramatic fall in the number of students taking A level Mathematics and Further Mathematics; the declining number and quality of students entering highly numerate university courses; the lack of qualified mathematics teachers; the shortage of high quality IT specialists; the narrowness of the UK mathematics PhD; and the apparent need to import large numbers of research mathematicians.

  • The most urgent short-term action was identified in the Smith report – namely to increase markedly the number of students taking, and enjoying, a serious A level in mathematics.

  • However, this goal cannot be achieved by simply easing the apparent demands of A level mathematics. In any effective strategy for recovery two key elements must be

    (i) to strengthen the foundations laid at KS3 and KS4 in a way that better nurtures the interest, and raises the aspirations, of more able students;

    (ii) to devise a concerted programme of professional development to ensure that current mathematics teachers appreciate why these stronger foundations matter.

  • The present situation is far more serious than is generally admitted and needs to be addressed as a whole – since many of the most serious weaknesses arise from a failure to recognise, and to deal with, the interplay between the actions of different agencies.

update

Still reading....

The domestic UK supply of mathematically competent manpower is in such decline that in many areas (including teaching, commercial specialist requirements, post-doctoral fellows and appointments to academic positions) we are now dependent on trawling recruits from other countries for “bread-and-butter” appointments (not just for “key” personnel).


I love it!

Nobody can write like the Brits, nobody. They're unbelievable. (I have GOT to go TRAWLING on the UK ed web sites to find out exactly how they do what they do.)

Have you ever in your life seen a government report in the U.S. produce language like this?

The answer is no.


ok, problem spotted

There are serious shortcomings at the level of individual government departments and agencies. But our failure to nurture the home-grown talent we need has been exacerbated by a consistent failure to coordinate policy between different agencies.

They may write better than we do, but thus far the content is just as stupid.

Sorry.

That was harsh.

it gets worse

(i) We have failed to recognise that the effectiveness of curriculum and assessment change (which is the responsibility of QCA) depends on providing appropriate training and support (CPD) for teachers (whichis the responsibility of the DfES, the TTA and the Strategies).

(ii) We have not faced up to the conflicts between

(a) the official goal of improving the career structure for home-grown post-doctoral fellows (which was the apparent reason for increasing research funding as part of the Treasury’s response to the Roberts review);
(b) the effect of EU law (or its current interpretation) on the way the consequent substantial increase in EPSRC funding is being used;
(c) the local pressures on university departments which arise from this more generous EPSRC funding; and
(d) the effective pressures imposed by the HEFCE controlled research assessment exercise and EU employment law on university administrations and on academic appointment practices.

I take it back.

This is much dumber than the stuff we put out.

let me see if I've got this straight

Apparently, the problem with maths education in England is that there've been a number of government inquiries, followed by a number of government reforms, followed by no discernible improvement whatsoever.

How could that be?

These reports and the published government responses, have subsequently led to significant initiatives by government and its agencies. It would be comforting to conclude that “the nature of the problem has been understood and is being robustly tackled”.


And, apparently the reason nothing got better, was that the government inquiries didn't take the whole thing seriously enough:

....the rest of the introduction [of the DfES response to the Smith report] includes a succession of statements (such as that “achievement in mathematics at . . . KS3 is the highest it has ever been”), which indicate that the nature and seriousness of the problem have simply not been grasped (we give clear evidence of this relating to KS3 below). This negative impression is strengthened by such facts as that the flagship policy of establishing a “National Centre for Excellence in Mathematics Teaching” is being “implemented” with an emasculated budget.


OK, so here we have a close reading of the introduction to a response to a report. This thing is a report about the reports.

OK, why don't I just read ahead until I find some actual content.

I do like the scare quotes around the word 'implemented,' though.

this is interesting

....some of Smith’s recommendations (such as the need for a serious reduction in the proportion of mathematics time devoted to “data-handling”, and the urgent need to consider the introduction of “incentives” to increase numbers taking A level mathematics) have not been pursued in the way the mathematics and mathematics education communities had expected.


If I'm understandng this correctly, what we have here is an anti-Trailblazers moment.

Less data-handling.

The whole entire point of Trailblazers is all data-handling all the time; the curriculum was originally titled TIMS, for Teaching Integrated Mathematics and Science. It's just pure data, every step of the way.

Data and investigations.

this is good

The mathematical community constitutes an increasingly important “micro-culture” within modern society. Hence the different parts of this community need to be structured and sustained so that this micro-culture can “reproduce itself” in a routine and orderly way, passing on to the next generation that which is known to be of value, while at the same time facilitating the development and application of new methods and techniques to serve business, management and society in general. Instead the routine reproduction of mathematical culture in the UK has been allowed to decay.

[snip]

In the whole of the UK there were around 85 000 A level mathematics entries in 1989; 66 000 in 2001; and just 54 000 in 2002. This has led to a concomitant decline in the number of competent undergraduates and graduates in highly numerate disciplines, and hence to a shrinking of the basic “pool” from which competent workers in areas that increasingly require serious mathematical skills (including mathematics teachers) can subsequently be drawn.



This is so long I'm going to put the rest on a separate page.

More Maths In England Part 2




maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
another brilliant person who liked getting right answers (scroll down)
Catherine's cousin talks about Everyday Math

Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)





NewStudyOnManipulatives 29 Jul 2005 - 17:27 CatherineJohnson


I want to follow up on Carolyn's post on Congressional math incentives, but before I do that here's a reminder:

Anne Dwyer has posted new notes on her summer math class.

And...quickly checking her page just now, noticed this comment:

So, what have I learned so far?

  • they like games where they compete with one another
  • they prefer pencil and paper exercises
  • they like to figure out puzzles


This is fascinating, because Kevin Killion, of Illinois Loop, just pointed me to a new article in Scientific American showing that manipulatives are less effective than pencil and paper with young children. I'll write a bit more on this later (bike ride time!), but here's the critical passage:

Teachers in preschool and elementary school classrooms around the world use "manipulatives"--blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.

Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: "Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier."

Dual representation also comes into play in many books for young children. A very popular style of book contains a variety of manipulative features designed to encourage children to interact directly with the book itself--flaps that can be lifted to reveal pictures, levers that can be pulled to animate images, and so forth.

Graduate student Cynthia Chiong and I reasoned that these manipulative features might distract children from information presented in the book. Accordingly, we recently used different types of books to teach letters to 30-month-old children. One was a simple, old-fashioned alphabet book, with each letter clearly printed in simple black type accompanied by an appropriate picture--the traditional "A is for apple, B is for boy" type of book. Another book had a variety of manipulative features. The children who had been taught with the plain book subsequently recognized more letters than did those taught with the more complicated book. Presumably, the children could more readily focus their attention with the plain 2-D book, whereas with the other one their attention was drawn to the 3-D activities. Less may be more when it comes to educational books for young children.




This perfectly supports the study Carolyn mentioned way back when, showing that fraction manipulatives are good for middle schoolers.


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





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 beh