Entries from SingaporeMath

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.

Slavin’s support largely resting on the benefits uncovered for grouping in mathematics in the 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.

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!

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 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.

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.

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

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).

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.

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

But that raises the issue of what happens politically and socially when a parent rejects a school's math curriculum.

Good question.

it doesn't have to be a battle

Of course, I wasn’t rejecting a curriculum the school had embraced; I was rejecting a curriculum the school had rejected (SRA Math, which is being replaced by Trailblazers).

Even so, I was using a different curriculum at home, and everyone knew it. The reason they knew it was that I printed out copies of the Table of Contents for Christopher to take in and show his teacher.

She was great. She admired all the lesson headings, and told Christopher, “All the parents should be doing this.” It was incredibly sweet of her.

At one point I sent an email saying I was having trouble getting Christopher to cooperate (that’s an understatement) and asking if she could tell him he needed to do my homework, too.

She did.





When we told her, in January, that our goal was to move him to Phase 4, the accelerated track, she blanched. There were already a number of kids in Phase 4 who were struggling; the class was oversubscribed. One child had just been moved ‘down’ to Phase 3, and it had been upsetting to all concerned.

She’d never thought of Christopher as ‘a Phase 4 kid,’ she said. She didn’t want to see him try Phase 4 and fail. (Neither did we.)

It took her about 2 minutes to decide she probably could think of Christopher as a Phase 4 kid, and the reason she could think of him as a Phase 4 kid was that ‘you’ll give him the support he needs.’ She saw clearly that Christopher’s dad and I would do whatever we needed to do to help him succeed—and she saw that we would be taking responsibility for the move. If it didn’t work out, we weren’t going to be back in the school yelling at people. (True.)

Once she'd turned her point of view around 180 degrees, she told us that if we were going to move him we needed to do it now. Suddenly it was our turn to blanch; my plan was to move him in the fall, after we'd had another summer to work on his math.

She said, in so many words, that my plan was going to be problematic. For years the middle school has been hammering the elementary school about placing too many kids in the accelerated class, giving inflated grades, etc., etc., or so I gather. ('Hammering' is not the word she used or implied.) The middle school had made crystal clear to teachers & to parents that they would be placing fewer kids in Phase 4 come fall, not more. Which meant they probably weren't going to think Christopher, who'd been in Phase 3 from day one, and who'd done badly in 4th grade math, was an obvious candidate for the accelerated track.

As she put it, 'They aren't going to know him the way we do.' If we wanted to do it, she said, we needed to do it now.

We said, OK, then, we'll do it now.

She got Christopher moved to Phase 4 within the week.

Not only did she support us in doing something she didn’t necessarily think was a good idea, she told us how to work the ropes. Then she worked the ropes for us.

using TIMSS

The TIMSS data on U.S. students is a big part of the ‘secret’ to working well with your school district when you object to the math curriculum. The first time I mentioned to our principal, Don, that I thought Christopher maybe ought to move to the accelerated track, he got that tight not-now-not-ever look on his face administrators always get when parents start bugging them to do things they don’t think they ought to do.

I backed off, because in fact Christopher wasn’t ready to move to the accelerated track. But I publicly raised the issue of why the accelerated kids were using a book that was a full year ahead of the rest of the kids without any of us parents having been told.

Naturally it turned out Don hadn’t been told, either; he’s an interim principal. He looked into it, and was obviously pretty dismayed at what he found (not worth going into here).

When we sat down and talked about it, I took the tack that I didn’t think Christopher is Secretly Gifted And Talented In Math; I just wanted him to be on the same track kids are on in high-achieving countries. Which is true. Here are the Magic Words to use with principals, teachers, administrators, & school boards:


In high-achieving countries, students take and master algebra in the 8th grade.


Here in America, only the accelerated kids take and master algebra in the 8th grade.

I told Don: if kids in Germany pass Algebra 1 in 8th grade, I want Christopher to pass Algebra 1 in 8th grade, too.

He had exactly zero problems with that, and the minute Christopher was ready to move to the faster class, he moved him up.

The fact is, our problems in math ed are national, not local, and everybody knows it.

Everybody knows it, but nobody knows how to fix it. Ideological constructivists think they know how to fix it, but your basic principal and/or teacher is living in the real world, facing real children and real parents who blame them when math scores are bad. They’re on the firing line. I don’t think too many principals & teachers truly believe ‘reform math’ is going to be the miracle we’ve been looking for for the last 100 years.

So basically, his feeling was: I’d like to see all our kids learning at the same rate as kids in Singapore. So would I. I don't blame him for our school having the same problems every other school has.

giving respect where respect is due