## Friday, October 2, 2009

## Saturday, May 16, 2009

### Math deprivation as punishment

Yesterday programs at San Quentin resumed after a week and a half of non-activity – the prison had been closed in order to prevent an outbreak of swine flu. I had been wearing a blue shirt to work in the assumption that there would be no class that Friday night, but found a white student uniform polo lying around at school, donned that for the evening, and got to the carpool on time. The guys were happy to be back after two weeks without classes and settled down to study. I mostly worked with E-, who was struggling with translating word problems into algebraic statements; M-, who was getting familiar with the coordinate grid and with slopes of lines; P-, who needed an explanation why 4.7 is a rational number; and with R-, who displayed a thick stack of notes from the work he had done on fractions while classes were cancelled. We got a good chunk of work done, and it was a satisfying start of the weekend.

Math 50 is a is a numeracy and pre-algebra course that is a prerequisite for enrollment in math courses for college credit. Years ago, it was taught as a lecture-based course, but fail rates were abysmal, and now it is a self-paced course structured by a series of quizzes – when the quizzes for a chapter are passed, the student moves on to the next chapter. Three to eight tutors, mostly grad students from UC Berkeley, work with 20-50 students to resolve difficulties with the material and to grade quizzes. Some students need less than a semester to complete Math 50, others need a couple years, some never make it.

The progress made by many of these students is incredibly gratifying. Students who start out unable to multiply a decimal number by 10 end up fluent at long division and knowledgeable about place value. Students who start out studying times tables end up performing operations on fraction with accuracy and understanding. There are always some students who have significant learning disabilities and who seem to forget everything between classes. And every now and again a natural math whiz come around, making up his own algorithms and doing most of the work on his bunk between meetings so that he can quickly move through the quizzes during class time.

San Quentin is the only prison in California that has a college program, and only some 50 of its 5000+ inmates are enrolled in Math 50. Since there is no public funding in the state for prison education beyond the GED or a high school diploma, all instructors are volunteers, and only a prison located so improbably close to a cluster of universities can staff a college program. Working with these students on Friday nights, I wonder at how little it really is they are asking for, how unnecessary it seems that the public will not afford this opportunity to anyone who would make use of it. If a person wants to spend the time from 6 to 8 pm on a Friday night on learning to add fractions, after having been up since 4 or 5 am to work and attend other programs, is it so much to ask that this opportunity be given?

I wonder, also, whether some argument can not be made that restricting access to education at this level amounts to a kind of differential punishment above and beyond that meted out in accordance with law for whatever misdeed was committed. By analogy, suppose that a person with diabetes commits a crime and is imprisoned. If that person in addition to being confined is deprived of necessary medication and medical supervision to control his illness, that would constitute a differential punishment beyond that implied by his sentence – his punishment would in reality be different from and more severe than that given to a healthy person who had committed exactly the same crime. Can something similar be said for undereducated persons who are incarcerated? To the extent that a certain level of education is necessary in order to function outside the institution, and to the extent that learning meets fundamental human needs, I would say yes. If two people commit the same crime, and the one has a master’s in engineering and the other does not know how to distinguish adding from multiplying, then deprivation of opportunities for learning constitutes a differential punishment, a more severe consequence, for the undereducated person. Restricting access to math at a level afforded by most high schools constitutes a consequence beyond that included in the prison sentence.

I would hope that interest groups for diabetics work to ensure that their incarcerated members receive access to necessary medical care and medication. I would argue that if we as educators think of learning as a fundamental human right, then we should fight for citizens’ access to education when they are in prison, too. Of course my argument hinges on math being necessary, like insulin perhaps, for a human to thrive, and I’m sure my (high school) students would look oddly at me if I suggested that math deprivation constitutes a punishment… ☺ And yet, and yet!

Anyways, if you live in the Bay Area and have either a math teaching credential or a master’s in math or science, and if you’d like to spend one evening per week this summer on making math just a little bit more equitably available, you could send a resume and an application to the Prison University Project. During the summer, in particular, when the grad students go home to their faraway families, an influx of math teachers on summer vacation wouldn’t be a bad thing.

Update: Jonathan made me aware of an important error - it should be "there is no public funding in the state for PRISON education beyond the GED or a high school diploma," and I had left out that word (thanks, Jonathan!). However, it used to be the case that federal Pell Grants could be spent on education for low-income citizens even if they were in prison, and this changed in 1995, immediately decimating junior college programs throughout the state.

Math 50 is a is a numeracy and pre-algebra course that is a prerequisite for enrollment in math courses for college credit. Years ago, it was taught as a lecture-based course, but fail rates were abysmal, and now it is a self-paced course structured by a series of quizzes – when the quizzes for a chapter are passed, the student moves on to the next chapter. Three to eight tutors, mostly grad students from UC Berkeley, work with 20-50 students to resolve difficulties with the material and to grade quizzes. Some students need less than a semester to complete Math 50, others need a couple years, some never make it.

The progress made by many of these students is incredibly gratifying. Students who start out unable to multiply a decimal number by 10 end up fluent at long division and knowledgeable about place value. Students who start out studying times tables end up performing operations on fraction with accuracy and understanding. There are always some students who have significant learning disabilities and who seem to forget everything between classes. And every now and again a natural math whiz come around, making up his own algorithms and doing most of the work on his bunk between meetings so that he can quickly move through the quizzes during class time.

San Quentin is the only prison in California that has a college program, and only some 50 of its 5000+ inmates are enrolled in Math 50. Since there is no public funding in the state for prison education beyond the GED or a high school diploma, all instructors are volunteers, and only a prison located so improbably close to a cluster of universities can staff a college program. Working with these students on Friday nights, I wonder at how little it really is they are asking for, how unnecessary it seems that the public will not afford this opportunity to anyone who would make use of it. If a person wants to spend the time from 6 to 8 pm on a Friday night on learning to add fractions, after having been up since 4 or 5 am to work and attend other programs, is it so much to ask that this opportunity be given?

I wonder, also, whether some argument can not be made that restricting access to education at this level amounts to a kind of differential punishment above and beyond that meted out in accordance with law for whatever misdeed was committed. By analogy, suppose that a person with diabetes commits a crime and is imprisoned. If that person in addition to being confined is deprived of necessary medication and medical supervision to control his illness, that would constitute a differential punishment beyond that implied by his sentence – his punishment would in reality be different from and more severe than that given to a healthy person who had committed exactly the same crime. Can something similar be said for undereducated persons who are incarcerated? To the extent that a certain level of education is necessary in order to function outside the institution, and to the extent that learning meets fundamental human needs, I would say yes. If two people commit the same crime, and the one has a master’s in engineering and the other does not know how to distinguish adding from multiplying, then deprivation of opportunities for learning constitutes a differential punishment, a more severe consequence, for the undereducated person. Restricting access to math at a level afforded by most high schools constitutes a consequence beyond that included in the prison sentence.

I would hope that interest groups for diabetics work to ensure that their incarcerated members receive access to necessary medical care and medication. I would argue that if we as educators think of learning as a fundamental human right, then we should fight for citizens’ access to education when they are in prison, too. Of course my argument hinges on math being necessary, like insulin perhaps, for a human to thrive, and I’m sure my (high school) students would look oddly at me if I suggested that math deprivation constitutes a punishment… ☺ And yet, and yet!

Anyways, if you live in the Bay Area and have either a math teaching credential or a master’s in math or science, and if you’d like to spend one evening per week this summer on making math just a little bit more equitably available, you could send a resume and an application to the Prison University Project. During the summer, in particular, when the grad students go home to their faraway families, an influx of math teachers on summer vacation wouldn’t be a bad thing.

Update: Jonathan made me aware of an important error - it should be "there is no public funding in the state for PRISON education beyond the GED or a high school diploma," and I had left out that word (thanks, Jonathan!). However, it used to be the case that federal Pell Grants could be spent on education for low-income citizens even if they were in prison, and this changed in 1995, immediately decimating junior college programs throughout the state.

## Thursday, April 16, 2009

### Writing Inverse Functions

I like Sam Shah’s approach to teaching function inversion. It’s pretty much what I had intended to do this year, but somehow forgot about at some point. Hopefully writing these notes here will increase the odds that I get it right next time around. The original inspiration was Mr. K’s approach to solving two-step equations, and the following will make no sense without having read that entry, so do that first.

I showed my Algebra 2 students Mr. K’s series of boxes and arrows for solving equations early in the year, and they thought it was fun. We did more complicated cases such as 5-3(x+5)=2 and -2-(2x+3)=-5. It was great for reviewing Order of Operations in a novel way. Many students had started out subtracting 3 from 5 in the fist equation, and in the second case very few students were able to identify “multiply by -1” as one of the operations in the sequence. However, after working through a few cases they got quite fluent at writing sequences of operations such as for the equation -2-(2x+3)=-5, where the operations on x are:

My plan was to bring this visual in again, as a recurring Opener problem perhaps, with every new operation we covered. The processes of squaring and taking a square root, of exponentiating and taking the logarithm, could presumably be organized nicely with the same kind of diagram that Mr. K used to organize the process of undoing linear operations on x. In Algebra 2, the reassuring string of little boxes could make for a beautifully transparent structure for inverting a function, with built-in checks at every step. Practicing with numerical input and output values first would likely be helpful.

Maybe I'll try this when reviewing just to see if it works.

I showed my Algebra 2 students Mr. K’s series of boxes and arrows for solving equations early in the year, and they thought it was fun. We did more complicated cases such as 5-3(x+5)=2 and -2-(2x+3)=-5. It was great for reviewing Order of Operations in a novel way. Many students had started out subtracting 3 from 5 in the fist equation, and in the second case very few students were able to identify “multiply by -1” as one of the operations in the sequence. However, after working through a few cases they got quite fluent at writing sequences of operations such as for the equation -2-(2x+3)=-5, where the operations on x are:

- Multiply by 2
- Add 3
- Multiply by -1
- Add -2

- Subtract -2
- Divide by -1
- Subtract 3
- Divide by 2

My plan was to bring this visual in again, as a recurring Opener problem perhaps, with every new operation we covered. The processes of squaring and taking a square root, of exponentiating and taking the logarithm, could presumably be organized nicely with the same kind of diagram that Mr. K used to organize the process of undoing linear operations on x. In Algebra 2, the reassuring string of little boxes could make for a beautifully transparent structure for inverting a function, with built-in checks at every step. Practicing with numerical input and output values first would likely be helpful.

Maybe I'll try this when reviewing just to see if it works.

## Sunday, March 8, 2009

### Practice Problems, Patterned Practice

A characteristic of problem collections in standard textbooks is that each question tends to stand alone, that attempts at varying the problems in a systematic way in order to elicit patterns are typically few and relegated to the introductory section in each chapter, to be dropped after the rules have been listed and boxed. The incentive to add my own problems is usually a wish for more practice of the kind where each item belongs in a set that has some internal logic, and in the section on evaluating powers with negative and non-integer exponents this is particularly useful, I think.

The very idea of exponents that are not counting numbers causes the students a fair amount of puzzlement and headache. The common misconceptions that negative exponents change the sign of the expression, and that a base raised to the power of zero must be zero, seem very resistant to instruction. So far, my main strategy is to repeatedly place such powers in a sequence and to make a big deal of how they MUST have the values they have in order to follow the pattern we see with positive integer exponents. We complete these tables again and again, graphing the results to see how the rules for negative and zero exponents are nicely continuous with the ones the students know intuitively.

This year I’ve spent more time on treating rational powers the same way. We have, for example, evaluated 27^x for sequences of x-values that are multiples of 1/3, and (81)^x for sequences of x-values that are multiples of 1/4. In completing these tables, the powers with whole number exponents become familiar reference points that students can check their rational powers against, and it becomes clearer that the rules for non-integer powers are not random or contrived, but that they are what they are because they have to be that way:

This also, incidentally, reinforces a basic lesson on fractions that continues to be necessary in Algebra 2: that fractions just are numbers, that they belong on particular places on the number line, and that they behave just as you would expect based on what integers they fall between.

My worksheet on graphing exponential functions for Algebra 2 is here. I’ve just started using Sketchpad for the grids, and am finding it tricky to get them to look neat. Oversized dots, tiny numerals and inconvenient scales continue to be issues, and kids complain a bit about these things. I’d appreciate technical advice! My introductory worksheet on exponential functions for Algebra 1 focuses more on the sheer prettiness of the graphs of 2^x and its reflections when they are drawn on the same grid in colored pencil (this particular sample from a freshman of last year who elected to do her Arts project on Escher's work):

The very idea of exponents that are not counting numbers causes the students a fair amount of puzzlement and headache. The common misconceptions that negative exponents change the sign of the expression, and that a base raised to the power of zero must be zero, seem very resistant to instruction. So far, my main strategy is to repeatedly place such powers in a sequence and to make a big deal of how they MUST have the values they have in order to follow the pattern we see with positive integer exponents. We complete these tables again and again, graphing the results to see how the rules for negative and zero exponents are nicely continuous with the ones the students know intuitively.

This year I’ve spent more time on treating rational powers the same way. We have, for example, evaluated 27^x for sequences of x-values that are multiples of 1/3, and (81)^x for sequences of x-values that are multiples of 1/4. In completing these tables, the powers with whole number exponents become familiar reference points that students can check their rational powers against, and it becomes clearer that the rules for non-integer powers are not random or contrived, but that they are what they are because they have to be that way:

This also, incidentally, reinforces a basic lesson on fractions that continues to be necessary in Algebra 2: that fractions just are numbers, that they belong on particular places on the number line, and that they behave just as you would expect based on what integers they fall between.

My worksheet on graphing exponential functions for Algebra 2 is here. I’ve just started using Sketchpad for the grids, and am finding it tricky to get them to look neat. Oversized dots, tiny numerals and inconvenient scales continue to be issues, and kids complain a bit about these things. I’d appreciate technical advice! My introductory worksheet on exponential functions for Algebra 1 focuses more on the sheer prettiness of the graphs of 2^x and its reflections when they are drawn on the same grid in colored pencil (this particular sample from a freshman of last year who elected to do her Arts project on Escher's work):

## Sunday, December 7, 2008

### Notes from the Saturday at Asilomar

“Oh, I get it,” the students say, and I tell them that that's great, and that it is not enough. I tell them that learning math is a lot like learning to play a musical instrument or like learning to dance or to play a sport. Having once successfully played through a piece of music, or having once correctly executed a move, is all very well as a starting point - and only practice and repetition with feedback will ensure that they perform reliably and smoothly. “Getting it” goes only so far, and learning involves something more and other than that.

This is what I tell my students, so I do “get” that insight is not enough, that learning requires changing behaviors and habits, and yet it seems I have not learned this lesson at a behavioral level myself. And so it was that one of the most useful experiences of the Math conference at Asilomar yesterday was cutting the last class to stare at the ocean and think about what I had wanted to do and not do this semester, and about what is actually happening in my classroom. It’s been an unexpectedly difficult semester for private reasons (conditions at school would have predicted the best teaching year ever), with limited occasion for reflection on my work, and the Math conference provided a chance to take stock and regroup.

I “get” that assigning a large number of practice problems has little or no advantage over assigning fewer problems and is likely to even be counterproductive, I see that assigning homework problems that a large number of students won’t be able to do correctly on their own is silly, I know that having the students do more work than I have time to check is a waste of their time as well as mine, and I believe that there is much value in spending some minutes of class time every now and again on activities whose purpose is building relations rather than practicing math skills. Yet, judging from what I’m doing it would seem I disagreed with all of the above. Somehow, all kinds of resolutions made late last year got lost in the flurry of starting up again this year, and I’ve found myself teaching in ways I had made definite decisions not to.

So, what to do? Since it’s not a matter of “getting it,” not a matter of knowledge or conviction about what to do, I’m thinking about what kinds of feedback mechanisms to set up toward changing my behaviors. I could write down what I want to do and keep rewriting it daily to remind myself. I could talk to colleagues about what I want to do and ask them to do random five minute observations and let me know what things are looking like – and I am fortunate enough to have several fantastic colleagues who would be willing to do this. I could tell the students that there’s a cap of a certain number of homework problems and ask them to remind me if I forget my own policy. I’m sure they’d be delighted to oblige.

On another note, it’s been nice to rediscover the online math ed community recently after having neglected these readings for lengthy periods at a time this fall. New ideas and debate are like food and sleep in that they’re easily given lower priority during hectic times, and yet these things are all necessary for teaching and learning well in the long run. Glad for all of you who are still around and writing...

This is what I tell my students, so I do “get” that insight is not enough, that learning requires changing behaviors and habits, and yet it seems I have not learned this lesson at a behavioral level myself. And so it was that one of the most useful experiences of the Math conference at Asilomar yesterday was cutting the last class to stare at the ocean and think about what I had wanted to do and not do this semester, and about what is actually happening in my classroom. It’s been an unexpectedly difficult semester for private reasons (conditions at school would have predicted the best teaching year ever), with limited occasion for reflection on my work, and the Math conference provided a chance to take stock and regroup.

I “get” that assigning a large number of practice problems has little or no advantage over assigning fewer problems and is likely to even be counterproductive, I see that assigning homework problems that a large number of students won’t be able to do correctly on their own is silly, I know that having the students do more work than I have time to check is a waste of their time as well as mine, and I believe that there is much value in spending some minutes of class time every now and again on activities whose purpose is building relations rather than practicing math skills. Yet, judging from what I’m doing it would seem I disagreed with all of the above. Somehow, all kinds of resolutions made late last year got lost in the flurry of starting up again this year, and I’ve found myself teaching in ways I had made definite decisions not to.

So, what to do? Since it’s not a matter of “getting it,” not a matter of knowledge or conviction about what to do, I’m thinking about what kinds of feedback mechanisms to set up toward changing my behaviors. I could write down what I want to do and keep rewriting it daily to remind myself. I could talk to colleagues about what I want to do and ask them to do random five minute observations and let me know what things are looking like – and I am fortunate enough to have several fantastic colleagues who would be willing to do this. I could tell the students that there’s a cap of a certain number of homework problems and ask them to remind me if I forget my own policy. I’m sure they’d be delighted to oblige.

On another note, it’s been nice to rediscover the online math ed community recently after having neglected these readings for lengthy periods at a time this fall. New ideas and debate are like food and sleep in that they’re easily given lower priority during hectic times, and yet these things are all necessary for teaching and learning well in the long run. Glad for all of you who are still around and writing...

## Sunday, August 3, 2008

### Algebra 2 is amorphous and has multiple heads

and defies reduction into compact and self-contained little parts. I am still trying to do just that, however. This continues the discussion about applying Dan Meyer's assessment system for Algebra 2, and anyone not deeply interested in this narrow topic may as well go on to the next item in their Reader.

Some time in June, Glenn Waddel wrote:

Glenn has meanwhile posted a carefully worked out list, and written about the tension between assessing specifically enough without introducing an intimidating number of concepts. Reading his list reminded me that the differences between the various versions of Algebra 2 around are significant, and that our final lists will have to be different to accommodate our respective course specifications and student groups.

In particular, my Intermediate Algebra course leaves conics and discrete math for the Trig/Precal course, which uses the same textbook and picks up where my course leaves off. I do not need to include assessments on these topics, then. On the other hand I must make sure that graphical features of quadratics, polynomials and exponentials are covered carefully, as this will not be repeated in Precal, and this increases the number of skills for these topics beyond what may be needed in Glenn's course. Also, Intermediate Algebra is for the students who do not make it to Honors Algebra 2, and so I need to include a lot of Algebra review. Dan Greene's version of Algebra 2 is similar to mine in the topics it covers, but students arrive directly from Algebra 1 without Geometry between, and so may need somewhat less review. I'm guessing that Sam Shah's course is for relatively advanced students. However, while our lists will need to be different in order to take these things into account, comparing notes could still be very useful.

Dan Greene and I met a few weeks ago, and made some progress on breaking down the chapter on Exponential and Logarithmic Functions, a unit where I am replacing

Finding a convenient format or platform for technical discussion of how to slice a topic into discrete skills and concepts is a challenge of its own, though. A series of blog posts, one for each chapter, seems both clunky and overly time-consuming (and school starts in just over a week over here). Instead, I've stored my work-in-progress on this Google Site, and if you have time and inclination to think some about what are the essential things to test for each topic or anything else, suggestions would be much appreciated.

Some time in June, Glenn Waddel wrote:

I have a rough draft of my skills checklist done right now. I am not sure I am going to post it yet. I am not happy with it. I think I am stuck in the “do I have to assess everything?” mode.More than a month later, that's where I am still... and since time is running out I'm going to post the incomplete work in case that helps accelerate the process.

Glenn has meanwhile posted a carefully worked out list, and written about the tension between assessing specifically enough without introducing an intimidating number of concepts. Reading his list reminded me that the differences between the various versions of Algebra 2 around are significant, and that our final lists will have to be different to accommodate our respective course specifications and student groups.

In particular, my Intermediate Algebra course leaves conics and discrete math for the Trig/Precal course, which uses the same textbook and picks up where my course leaves off. I do not need to include assessments on these topics, then. On the other hand I must make sure that graphical features of quadratics, polynomials and exponentials are covered carefully, as this will not be repeated in Precal, and this increases the number of skills for these topics beyond what may be needed in Glenn's course. Also, Intermediate Algebra is for the students who do not make it to Honors Algebra 2, and so I need to include a lot of Algebra review. Dan Greene's version of Algebra 2 is similar to mine in the topics it covers, but students arrive directly from Algebra 1 without Geometry between, and so may need somewhat less review. I'm guessing that Sam Shah's course is for relatively advanced students. However, while our lists will need to be different in order to take these things into account, comparing notes could still be very useful.

Dan Greene and I met a few weeks ago, and made some progress on breaking down the chapter on Exponential and Logarithmic Functions, a unit where I am replacing

*all*of my concept test items from last year. So far, the unit on Numbers and Functions has been the most demanding unit, I think. There are so many big, abstract and quite unfamiliar ideas there. On the one hand, the process of breaking down this chapter into parts that can be practiced separately may therefore be all the more necessary in order to make it accessible to students. On the other hand, much of the point is for students to recognize an abstract idea, such as the transformation of a function, across pretty different contexts, and this is just hard to assess in a piecemeal way. Or so it seems to me.Finding a convenient format or platform for technical discussion of how to slice a topic into discrete skills and concepts is a challenge of its own, though. A series of blog posts, one for each chapter, seems both clunky and overly time-consuming (and school starts in just over a week over here). Instead, I've stored my work-in-progress on this Google Site, and if you have time and inclination to think some about what are the essential things to test for each topic or anything else, suggestions would be much appreciated.

## Monday, July 28, 2008

### 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.

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

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:

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}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.
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