Helping Students Come to Grips With the Meaning of Division

School Science and Mathematics, Nov 2004 by Aubrecht, Gordon J II

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60-80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a "whole " and a "package, " and connect that meaning with previously learned operational definitions for area and volume.

The physics education research community has begun to succeed in helping physics teachers confront the deficiencies of lectures and change to more active forms of engagement (see, for example, Goldberg, 1997, 2001; Hestenes, 1987, 1992; Mazur, 1997; McDermott, 1991,1993). Physics education researchers have identified systemic patterns of student misunderstandings and miscommunications. One important result of these investigations has been the creation of new curricula (generated by the researchers themselves) using novel ways of addressing common fixed ideas students have before instruction (preconceptions).

Among the curricula now available, the program of research led by Lillian C. McDermott of the University of Washington (McDermott, 1995a, 1995b) has led to a curriculum known as Physics by Inquiry explicitly designed for preservice teachers. The Washington group conducted interviews to determine student ideas (often called misconceptions); devised questions to investigate these conceptions in additional interviews; and created materials that address those conceptions, which are then further tested in the classroom and revised. The Physics by Inquiry books (McDermott et al., 1995a, 1995b), developed this way, are used in the courses I teach (mainly, but not exclusively, to preservice teachers) and constitute the focus here.

McDermott's research program is based on the pioneering work of Arnold Arons, the proponent of confronting student misunderstandings of both mathematics and science. Arons has pointed out that physical models rely on mathematical language to clarify. Mathematically inclined students who try to express scientific thought often rely on mathematical relationships, formulas (which may be totally misunderstood) to the exclusion of actual knowledge of how the world works. They will often search through lists of formulas until they find one that may suit the situation and call it understanding. The less mathematically inclined have no basis except memorization for their understanding of science, which scientists find unsatisfactory. The incomprehension these science students exhibit of the basic mathematical operations multiplication and division need to be addressed in physics classes. Arons has especially pointed to ratio reasoning and fractions as needed foundations for such reasoning (Arons, 1976,1996).

The Physics by Inquiry program was designed for preservice and in-service teachers and extensively tested with in-service teachers during summer programs at the University of Washington. Physics by Inquiry is based on two pillars: Elicit-Confront-Resolve, a theoretical approach to eliminate misconceptions, and hands-on exposure to physical phenomena. It is assumed that many ideas students bring to introductory physics are shared, hard-fixed misconceptions. This may not always be the case, as many others have argued, but it has provided the guiding paradigm for Physics by Inquiry.

I use Physics by Inquiry because many students on my campus plan to become elementary school teachers. With two exceptions over a decade (excepting students taking a second course from me), my students have never before experienced a course in which learners are responsible for their own learning. (I know because I ask this question in the student journal that forms part of the course.) Over the last decade, I have taught several hundred students in Physics by Inquiry courses (three to five such courses per year). Many of my students lack an understanding of mathematical thinking despite having taken mathematics in college prior to taking my course. Since most of my students plan to become teachers and will influence generations of students, it is prudent to attempt to provide ways of understanding. Prior to the changes described in the following section, my students typically performed at the 60% to 80% level on simple density problems after instruction. I was dissatisfied and wished to help them achieve fuller understanding (my goal was consistent 90% or better student scores on such problems).

The following section describes the part of the module in the book developed by the Washington group (McDermott et al. 1995a) that did not seem to be sufficient for full understanding among my students. The reasoning is then extended as developed in the Physics by Inquiry book to what I term whole and package reasoning. Last, the results are discussed.