^{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 Opfer

^{2}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.