Today’s *New York Times* (in the Science Times section, the reason my family buys the *Times* every Tuesday) is an interesting article on a very different approach to math learning: Brain Calisthenics Help Break Down Abstract Ideas, Researchers Say. The idea is that frequent presentation of examples and forcing rapid choices forces perceptual learning, even for what are normally seen as highly abstract symbolic tasks.

For once, the journalist (Benedict Carey, to be precise) has provided pointers to the original article, which are freely available):

Philip J. Kellman and Christine Massey.

Integrating Conceptual Foundations in Mathematics

through the Application of Principles of Perceptual Learning

and

Philip J. Kellman, Christine M. Massey, Ji Y. Sona

Perceptual Learning Modules in Mathematics: Enhancing Students’ Pattern Recognition, Structure Extraction, and Fluency

Topics in Cognitive Science(2009) 1–21

DOI: 10.1111/j.1756-8765.2009.01053.x

The first paper looks at a study in which they applied perceptual learning approaches to make “a series of curriculum modules consisting of fully evaluated web‑based Perceptual Learning Modules (PLMs) and accompanying classroom lessons.” The idea was fairly simple: have students practice chopping up and rearranging visual images (the lessons were called “Slice and Clone”).

The softwaregives the learner two onscreen tools—a slicer, which can be used to divide a given quantity into units of adesired size, and a cloner, which can take the unit created by the slicer and output a desired number ofiterations of that unit to create new quantities. On each trial, the learner uses the slicing and cloning tools tosolve a problem of creating a new quantity from a given quantity.

The lessons they used these tools with were intended to give kids a quick, intuitive idea of how to manipulate fractions, without bogging them down in calculation or writing. Their experiment used a tiny sample size (41 students in the intervention group and 21 in the control group), so the results have to be interpreted cautiously, but with that caveat in mind, the durable improvements in fraction understanding (including a post test after the summer break) are pretty impressive.

The second paper is much wordier and full of theoretical bullshit, as it has to be to get into a “cognitive science” journal, but it does describe some experiments displaying a graph and asking students to rapidly identify which of 3 equations produces that graph (or which of 3 graphs corresponded to an equation, or which of three equations corresponded to a word problem, … All six target-options pairs were used.)

Students received a paper-and-pencil pretest and posttest containing two kinds of problems.Four problems required solving word problems involving linear functions. Eight

translation problems involved presentation of a word problem, graph, or equation with thestudent being asked to translate the given target to a new representation—specifically, togenerate an appropriate graph or equation in response. There were four types of translationproblem: equation to graph (EG), graph to equation (GE), word problem to equation (WE),and word problem to graph (WG).

Again, their sample size was small (68 9th and 10th grade students).

I

n a control condition, students were asked to practice the same kinds of translation problemsthat appeared on the assessments. They were given packets with 32 problems includingequal numbers of the four generation problem types, designed to closely resemble the translationproblems on the assessments. Every time students completed a section of the practicepacket, they were given an answer key to check their answers. Feedback stated the correctanswer and offered no further explanations. For both the control learning condition, and theassessments in both conditions, paper-and-pencil tests were used to give students flexibilityin generating graphs and equations. Time on task for the control was matched to the averagetime required by participants in the PLM condition.

The differences in the post tests was striking, but the authors caution that *“The current results do not provide insight regarding every variable that differed between the experimental and control groups.”*

They did another experiment on algebraic transformations (rearranging an equation) where they showed one target equation and 4 putative rearrangements, asking students to rapidly identify the correct rearrangement. For this experiment, their goal was not to improve accuracy (which was pretty high even in the pre-test), but to decrease the time it took students to recognize trivial rearrangements. Again they got a pretty robust response, with times dropping from 30 seconds per problem to 10 seconds a problem, and that speed increase remained after a 2-week delay (I’d have liked to see a larger interval before the delayed post test).

There is a third experiment with a smaller effect (and sloppy controls) on linear measurement, but I’ll leave readers to look at the original paper for that experiment.

If the ideas in these papers work in other teachers’ hands (and are not an experimental artifact), they do suggest some new approaches for teaching math in elementary and secondary schools.

A reader asked me where they could see the Perceptual Learning Modules described in the paper by Kellman, Massey, and Son. I looked at the paper again to see if it had any pointers, but found none. I then checked Kellman’s lab website, and found only that they were working with a company that claims “product purchase – Coming Soon”. So it looks like they are planning to cash in on the NSF-funded research, without releasing anything for free. (Legal, but not designed to endear them to educators and academics, who would like to see some experimentation with decent sample sizes before buying a pig in a poke.)

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