Constance Kamii (University of Alabama at Birmingham) and Judith Kysh (San Francisco State University): “The difficulty of “length x width:” Is a square the unit of measurement?” Too often, secondary mathematics teachers make unfounded assumptions about the background knowledge that their students posses, without ever analyzing the validity of their supposition. These assumptions directly influence the way teachers plan their instruction and carry out their assessment. Kamii and Kysh studied fourth, sixth, eighth, and ninth grader’s understanding of area, as it relates to the notion of the unit square. Their research sought to empirically evaluate the following two assumptions: 1) a square is the unit of measurement that students in grades four through eight use to understand area; 2) squares have a “space-covering” characteristic for students in grade eight. The authors then proceed to evaluate their findings in light of another assumption – that the “length x width” interpretation of area is often difficult for students to understand – which they explain under a theoretical framework provided by Piaget. The results of their research show that not only do teachers need to reconsider the ways they teach area to their students, but also that teachers need to reconsider they ways their students themselves While this study directly addresses the implications of unit squares and teaching the concept of area, one can generalize these implications to extend to the broader context of the secondary education classroom. This paper is not strictly about geometry and measurement. Its implications are much more far-reaching, because they address the universal topic of assumptions – specifically, the assumptions teachers make about the knowledge their students bring to the classroom. The article truly moves the field of mathematics education foreword, because it makes teachers realize that they constantly make invalid assumptions about their students’ academic and cognitive backgrounds. The researchers’ primary objective is to have teachers rethink their approach to teaching area; however, a secondary (and seemingly unplanned) effect is this “awareness of assumptions.” Not only must teacher ask themselves “Do students understand the space occupying characteristic of a square,” but they must also ask themselves, “What else do I assume students know?” In this sense, this article strikes at the core of every Teacher Education program, because it encourages self-reflection on one’s own beliefs and practices. |