Teaching Programming and Algebra with Scheme

Jan. 10, 2001

Dr. Stephen Bloch, Adelphi University

Introduction

What does computer programming have to do with teaching math at pre-college levels?

In brief, each hand washes the other. On one hand, the best predictor of student success in college computer science courses is success in high school algebra. The two fields share a lot of the same concepts -- variables, operators, nested expressions, functions, domains, ranges, function composition. On the other hand, computers provide a more concrete (and, for many students, more fun) way to illustrate these mathematical concepts than the blackboard. The same design recipes that help students translate a problem description into a working program, with minor modifications, help students translate a word problem into algebraic form. But students can all too easily tell themselves that things like variables and functions are "unreal", the product of their instructor's fevered imagination; it's harder to dismiss variables and functions as "unreal" when you watch them working right in front of you. We can use Scheme to teach many of the New York Regents "key ideas" for K-12 mathematics.

One day in September, I was talking to a late-added student after class, trying to catch her up with her classmates. I wrote the expression "3 + 4 * 5" on the board and asked what its value was. She confidently answered "35", so we talked about order of operations and how to evaluate complex algebraic expressions, illustrating the difference between (3+4) * 5 and 3 + (4*5) with Scheme expressions that she could watch the computer evaluate, one step at a time. I watched light bulbs go on over her head as she exclaimed "I could use Scheme to help with my math homework!" She had a lot of catching up to do, but she earned a B for the semester.

I've done a statistical analysis of the past three years' freshman CS classes at Adelphi. To nobody's surprise, computer science majors are disproportionately male, and computer science majors get better grades in CS courses. But when I switched the beginning language to Scheme, the correlation between gender and first-semester grade dropped sharply, almost to zero. I'm waiting on more data to assess the significance of this; I speculate that it may have to do with an increased emphasis on thinking creatively, and less on memorizing vocabulary and syntax.

What is this workshop about?

This workshop grew out of work done at Houston's Rice University on teaching beginning computer programming. Rice faculty worked out an approach that worked well for their freshmen, then adapted it for use at the high school level and started offering summer workshops for high school math and computer science teachers. Over the past four years, these techniques have indeed been implemented successfully in many high school computer science courses; but just as importantly, they've been put to use at both high and middle school levels in teaching algebra.

Rice University's TeachScheme! Project comprises several components:

a computer language, Scheme, notable for its simple, consistent structure;
a beginner-friendly implementation, DrScheme, which is freely available for Windows, Macintosh, and Unix;
a pedagogical approach based on "design recipes" that help students approach problem-solving in a methodical, rather than random, way; and
a textbook, also freely available on the Web.

The conceptual core of the project is the design recipes: they can in principle be taught without the text, or without the DrScheme platform, or even without Scheme. For example, I translated many of the design recipes to Java for a freshman class two years ago. But the four components support one another: in particular, it's much easier to teach principled, methodical design recipes with a language that doesn't get in the way.

This two-hour workshop won't get very far into the curriculum or the language (albeit much farther than you could get into C, C++ or Java in two hours!), but will demonstrate some ways to use programming to teach mathematical concepts. I hope to lead a more in-depth, full-week workshop this summer.

Why Scheme?

In 1998-99, when I taught beginning programming in Java, I felt that every time I wanted to introduce a new programming principle, I had to waste a day or more of precious class time on semicolons, curly-braces, menu selections, and dialogue boxes before the students could put the principle into practice. This could be a powerful argument against using computer programming to teach mathematics... but we addressed it with DrScheme.

Scheme is a full-fledged programming language, every bit as powerful as C++ or Java, but with a much simpler syntax, so you can spend your time on fundamental concepts rather than memorizing syntax rules. In my first-semester programming course, I teach perhaps half a dozen syntax rules, one at a time. That's fewer than it would take to write the simplest "hello, world!" program in C++, C, or Java, yet by the end of the semester my students are writing recursive functions, traversing linked lists, binary and n-ary trees, and writing functions that operate on other functions as parameters.

Furthermore, the DrScheme software package is designed for educational use: it provides a minimum of options and menus, a "stepper" feature with which students can observe the evaluation of an expression step by step, and a series of instructor-customizable "language levels" that tailor the language to the student's experience level (so a student who accidentally invokes an advanced feature of the language gets an error message rather than inexplicable behavior).

But don't take my word for it. Read what other high- and middle-school teachers have said about the approach.

Supporting documents

The software we're using, DrScheme, is available for free download for Mac, Windows, and Unix.

The textbook I used for my first-semester course, and from which this workshop is excerpted, is How to Design Programs, by Matthias Felleisen et al at Rice University. The book is due to come out in print from MIT Press in a few months, but it will remain available on-line for free.

Although Scheme's syntax is extraordinarily simple for a computer language, it is still a new language, and you'll need to learn the vocabulary. Here's my page on the minimal Scheme language, covering spelling, grammar, vocabulary, and idioms.

I've summarized the Design Recipe(s). covered in this workshop, and some additional ones covered in my first-semester course. Use them!

You might also be interested in Jack Crouch's CS1 Web site. Jack Crouch teaches a 9th-grade course on beginning programming, using Scheme, DrScheme, and How to Design Programs.

I've set up a folder for programming examples, many of them developed by high school teachers.

Last modified:

Stephen Bloch / sbloch@adelphi.edu