Using Elementary School Materials in a Math Content Course for Preservice Teachers

Ronald A. Ward
Western Washington University

At the risk of oversimplification one might describe the typical preservice teacher education program in mathematics content as a "filtration device." The undergraduate courses in mathematics for elementary teacher preparation are often abstract. The theory is that the mathematical ideas learned in a general or theoretical setting will, in the prospective teachers' later experiences, filter down in their minds to the level and to the context in which ideas will be presented to their students. Even if such a course is successful in developing some understanding of formal mathematics [problematic}, the prospective teachers seldom connect this knowledge with the material and concepts of elementary school mathematics.

I would argue, instead, for a process that might be labeled a "percolation device." Begin with a study of carefully selected elementary curriculum materials that a teacher could actually teach to kids, and conduct the preservice class in much the style that the teachers will be encouraged to use later themselves. The mathematical ideas are thus initially met at the level (and couched in the same language) at which the teachers will instruct their students and in a pedagogical setting similar to that in which students will learn. Then, as the course progresses, the mathematical ideas percolate and suggest generalizations, deeper study, and eventually an understanding of elementary ideas from an advanced standpoint. Such a process can also make clear what is often called a "spiral" development of ideas in a curriculum and prepares teachers just as they should teach. Of course, such material must be challenging and interesting to college students. And while most commercial elementary programs are neither challenging nor interesting to college students (kids either!), I have found that certain programs either created at educational laboratories or developed by universities with NSF funding often do meet these criteria. These also tend to be supported by extensive research and evaluation and are pedagogically sound. Clearinghouses such as the National Diffusion Network and the CD-ROM "Mathfinder" can often be used to identify such programs.

I first advocated such an approach in my 1982 address to the Conference for the Advancement of Mathematics Teaching in Austin, Texas, and since then, have developed enough examples and used them successfully in my own classes so that I am confident of the feasibility of such an approach. Basically, I select student lessons that are compatible with the NCTM Curriculum and Evaluation Standards and that can lead to topics identified by the MAA as appropriate for preservice elementary teachers. I then create the necessary teaching sequences to bridge the gap between the two levels. Some of the examples of such materials were well received by participants at the NSF Institute "Preparation of Elementary Mathematics Teachers" at Miami International University in 1994.

Here is an illustration of an elementary pictorial approach to the composition of functions which can them be extended to a consideration of group theory. Given the lack of color and diagrams in the Newsletter, this should be a challenge to your visualization capabilities! The sample is taken fro CSMP/21 [Comprehensive School Mathematics Program for the Twenty-first Century, published by McREL Educational Laboratory]: Elementary students would be given a representation of a function such as a square in which the "sides" are blue arrows moving in a clockwise direction [draw this if you wish], each arrow carrying one vertex to an adjoining corner. Students are told that a blue arrow "followed by" a blue arrow is a green arrow.

Students are then asked to add as many arrows as they can following the rules:

B * B = G, B * B * B = Y(ellow), B * B * B * B = R(ed).

Try to picture the result. [You should have green arrows moving as diagonals, yellow arrows moving in opposite directions from the blue ones, and red loops at the corners of the square.] After noting that five blues is the same as one blue, and six blues is the same as green, the students consider, say 96 blues or 141 blues and note the cyclic nature of the situation. All this is in the elementary lesson. Now, for an extension for the college students, suing the four types of arrows in the picture, compositions of these functions may be considered with the following tabular results:

An examination of the table shows that this set of four functions with composition has the structure of an Abelian group, an appropriate algebraic structure for study by college students.

(Editor's note: Comments are encouraged and should be sent to the newsletter, to Ron Ward, and/or possibly the listserv. What mathematics should be in the mathematics content courses?)

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