Coffee and math ed readings

This summer I've met with a doctoral student of mathematics education a couple of times. Her area of interest is mathematical learning disabilities (MLD). Last time we met in a coffee shop to discuss an article by Geary et. al.¹ on students' placing of numbers on a number line, a topic I've been fascinated by for some time.

As it turns out, number lines constitute an active area of study in cognitive psychology and neuroscience, of theoretical interest

because magnitude representations, including those that support the number line, may be based on a potentially inherent number-magnitude system that is supported by specific areas in the parietal cortices ... (p. 279)

Geary's article cites earlier work by Siegler and Opfer² which suggests that young children use a more or less logarithmic scale when placing numbers. Children tend to perceive the difference between 1 and 2 as being greater than the distance between 89 and 90 in a semi-systematic way (p. 279), so that most numbers get clustered to the left hand side of the number line. This tendency is thought to reflect the postulated "inherent number-magnitude system."

Geary et. al. compared first and second graders' placements of numbers on a blank number line. They found some evidence that mathematically learning disabled students' placements not only failed to conform to the linear pattern at a rate comparable to that of their peers. In addition, their pre-instructional number placements also looked less like the logarithmic placement of non-disabled children:

Even when they made placements consistent with the use of the natural number-magnitude system, the placements of children with [mathematical learning disabilities] and their [low achieving] peers were less precise than those of the [typically achieving] children in first grade, that is, before much if any formal instruction on the number line. The implication is that children with MLD and LA children may begin school with a less precise underlying system of natural-number magnitude reprsentation. (p. 293)

Geary et. al. report correlations between performance on the number line tests with a battery of other cognitive tests. Unfortunately I know neither enough statistics nor enough cognitive psychology to extract terribly much information from these parts. Of rather more immediate interest to me as a teacher is, in any case, the question of how to go about eliciting the kind of cognitive change that's needed here. It certainly is not the case that all kids have the linear scale all figured out by the end of second grade - many of my 9th and 10th graders last year had not. The good news is that for this important topic, instruction tends to work. Cognitive Daily reports on more recent work by Siegler and Opfer showing that second graders responded quickly to some targeted feedback on their number placements, and that

once the linear form is learned, the transformation is quick, and permanent.

In other news, this thing of chatting about research in MLD over a morning coffee has been immensely enjoyable. The readings are demanding enough that I'd be much less likely to work through them if I were studying alone, but I'm awfully glad to be learning some of this.

I'm wondering how much interest there would be for some kind of regular math teacher/ math ed researcher meetups, such as a discussion of a predetermined article over coffee on Saturday mornings. Many new math teachers already have ed classes scheduled at that time, though, and older math teachers typically have family to take care of during weekend mornings, so how many would remain? And would there really be many researchers interested in talking with teachers? Still, given the curious absence of contact between researchers working on math education and math instructors working in schools, even in cases where their buildings are in the same geographical area, it would seem that some thinking should be done on how to afford more "vertical alignment" in Dina Strasser's sense of the term.

1. Geary, D. C., Hoard, M. K., Nugent, L. and Byrd-Craven, J. (2008). Development of Number Line Representations in Children with Mathematical Learning Disability. Developmental Neuropsychology 33:3, 277 - 299

2. Siegler, R. S. and Opfer, J. (2003). THe development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237 - 243.

Have you used Algebra tiles?

I introduced integer tiles to my Algebra classes early last fall, and then quickly gave it up. Several students balked at using such middle school measures for studying math, and my arguments that being able to represent math statements in many different ways, including with concrete objects, failed to persuade. In a class of insecure freshmen still figuring out their relative positions in the class, and in some cases still stinging from having been placed in Algebra rather than in Geometry after the placement test, using materials perceived as childish just wasn't socially acceptable.

I quietly dropped the project, only including a problem on modeling integer subtraction as an extra credit problem on a unit test some time later. Not one student got it right. Later in the year, when a number of students continued to demonstrate confusion about combining signed integers and combining like terms, I sometimes wished I'd stuck with the manipulatives a little longer.

Now I'm trying to make up my mind about whether - and, if so, how - to use tiles in my Algebra classes in the fall. Apart from the probable social issues to deal with, I'm wondering about the efficacy of Algebra tiles. A point I've picked up in passing while reading this summer (I'm sorry I can't recall where!) is that the same students who are likely to have much trouble with elementary Algebra are also likely to have difficulty picking up how to manipulate Algebra tiles.

There is, after all, no magic involved. The rules for representing addition and subtraction of integers with bi-colored tiles are not self-evident or trivial. Even for me, the representation of subtraction problems by adding the necessary number of "zero pairs" came as a bit of a surprise. And while I then found the very idea to be very cool and exciting, that is more than I can take for granted that my students will, even if they aren't unable or unwilling to master the rules of the game.

So, what are your experiences with Algebra tiles? How to you go about changing the image of tiles as being all too elementary? And assuming that you have gotten your young charges to take the tiles seriously, how much do you feel that the students learn this way that they do not learn just as well by simply reiterating the formalism of signed numbers and like terms?