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2004 NSF K-12 Math, Science, and Technology Curriculum Developers Conference

Raising the Floor: the Algebra Project

Dr. Alan C. Shaw
Instructional Materials for Algebra I

The Algebra Project is targeting high school students who have not been reached by existing materials for Algebra I and the later courses in the college preparatory sequence. It aims to produce materials that will motivate students to stay in college preparatory courses for all four years of high school, and will enable them to begin college mathematics without needing remediation.

With IMD support, the project is developing two modules for Algebra I that reflect the project’s notion of a five-step sequence that will assist students in learning concepts and procedures. This sequence reflects the insights of philosopher W.V.Quine and mathematicians including Gauss, Riemann, Zermelo, and MacLane, applied within an “experiential learning” process (e.g. Dewey, Kolb). Students first

1. experience a mathematically rich, physical event, and then
2. construct a graphical or physical model of the event; and
3. discuss the event in intuitive or natural language (“people talk”). Next, students “mathematicize” the event by
4. structuring their language (“feature talk”) and
5. learning the standard mathematical notation.

The Algebra Project has piloted a unit called the “Road Coloring Problem,” in which the concept of a function is introduced in the context of “roads” between “buildings.” Students attempt to find a synchronizing function for a given directed graph. They use four representations for a function: arrow diagram, ordered pairs, points in a coordinate plane, and a 0-1 stochastic matrix. They learn how to multiply matrices by carefully considering composition of functions. The unit also explores the basic algebraic properties including the associative property, commutative property, existence of an inverse, etc., considered in the context of both matrix addition and multiplication. An end of unit test and end of semester test showed that four classes of students (n=58) were all facile in moving among all four representations and in adding and multiplying matrices similar to those in the materials.

The second unit, "Games for Mathematical Understanding" will focus on the "Relay Race Game." Teams of students stack Unifix cubes in timed intervals and then place them on a "racetrack". The stacks of cubes are represented as vectors in one dimension, and the material explores vector addition and scalar multiplication. Material addressing linear equations, "average velocity", and slope will also be developed within the context of the relay race.

In April 2003, students at Lanier High School, Jackson, MS, who used earlier versions of these materials outperformed their peers at the same school on the state Algebra I test. They scored significantly higher on both total score and on most subscales than the nonAlgebra Project students who used an Algebra I textbook.

This project is supported by the National Science Foundation (Grant No. ESI-0352345). Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.