Friday, October 2, 2009


to Points of Intersection. Hope to see you there!

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.

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:
  1. Multiply by 2
  2. Add 3
  3. Multiply by -1
  4. Add -2
The inverse operations, in the reverse order, are
  1. Subtract -2
  2. Divide by -1
  3. Subtract 3
  4. Divide by 2
Applying these operations to -5 solves the equation above. Applying the same operations to a variable yields the inverse of the function y=-2-(2x+3)

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):