| Used as instructional aids, representational formats are often promoted for their particular “visual”, or “manipulative” qualities. However, the visual quality of a representational format does not automatically ensure “better” or “easier” understanding. The purpose of this study was threefold: to examine ninth grade algebra students' strategies on problems involving the multiplication of polynomials; examine the teachers' presentation of lessons using two representational formats; and to evaluate the influence of form sequencing on transfer problems (i.e., factoring of polynomials).;One format incorporated algebraic notation and the FOIL algorithm, and the other was based on geometry. The latter incorporated diagrams of subdivided rectangles to illustrate that area was analogous to the product of two polynomial factors. Both formats were presented in counterbalanced order to two comparable class sections taught by one teacher. Three tests were administered to students. The first was given after the presentation of the first format, completion of homework, and an in-class discussion of the format. Similarly, the second test was given after the second format had been covered. Conceptually identical problems appeared in blocks in both representational formats. A third test (the test of transfer) was given two days later to both classes.;Results on all tests showed effects of problem format. Students were better able to work through problems correctly when they were presented using algebraic notation. On Test 1, advantages of the algebraic format for the class who had the related lesson was not the same for the class who had the geometry-based lesson. Test 2 showed that both classes performed significantly better than on Test 1, at comparable levels. The test of transfer also showed an effect of representational format, but no effect of class; that is, there was no effect of format sequencing. Videotapes of the teacher's lessons showed that while both lessons were coherent within format, there was little discussion devoted to principles underlying multiplication of polynomials (i.e., the distributive law) and connections across formats. Discussion of the results centers on the importance of connecting representational formats to the current knowledge structures of the learner to foster understanding. |
