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 think about area.  These findings can then be generalized, beyond the topic of area, to help teachers challenge similar assumptions they make about their student’s background knowledge. 
            While Kamii and Kysh studies students in grade’s four through nine, this paper will only address their findings for a specific subset of their sample – students in grades seven, eight, and nine – because this subset relates more closely to the content of the course for which this paper is written.  We begin with a detailed summary of the researchers’ questions, methodologies, findings, and self-stated implications.  We then go on to explore these implications (both practical and theoretical) not only in the context of “the area problem,” but also in the wider milieu of the secondary mathematics classroom. 
            Kamii and Kysh introduce their research topic by establishing the fact that many educators believe “the concept of are is often difficult for students to understand, perhaps due to their initial experiences in which it is tied to a formula (such as area=length x width) rather than more conceptual activities such as counting the number of square units it would take to cover a surface.”  After a brief discussion of why students have difficulty understanding area as “length x width,” the authors hypothesize that “counting the number of square units” does not, in fact, provide students with an effective alternative explanation of area.  Their research asked the following questions: 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.  To answer the first question, the researchers had students a) compare two rectangles made on geoboards, with respect to which “would be bigger” if the rectangles were chocolate bars [“Task 1”], and b) state the area of a photocopied “L” shape, with marks drawn to allow students to count squares, dots, or units of length [“Task 2”].  To answer the second question, the researchers asked eighth graders a) to state the area of nine color tiles arranged in two different ways to see if the nine squares were stable space-covering units [“Task 3”], and b) making a straight cut in a strip so that the strip would have the same amount of space as a given rectangle [“Task 4”].  These experiments found that for the majority of students in Regular mathematics classes, the square is not the unit of measurement of area1.  The square is the unity of measurement only for the most advanced eighth and ninth grade students.  Only half of the students counted squares in Task 1.  57% of students did not consider counting squares in Task 2.  In Task 3, which required simple counting of rearranged squares, the student responses varied depending on the arrangement of the nine squares.  Even more shocking, 87% of the students were unable to complete Task 4, because they were unable to adequately manipulate the concept of “area as a sum of unit squares.”  Finally, the research suggests that for most middle school mathematics students, the square does not have a “space occupying characteristic” – a square is merely “a square!”
            The authors explain that, in light of their research, the National Council of Teachers of Mathematics must revise their suggestion that in order to understand the “length x width” formula, teachers should have students fill in the desired area with small, easily-manipulated squares.  They argue that “empirically covering a surface with squares or a grid and counting them is one thing, and being able to think about a square as a unit for area is quite another thing.”  Before teachers have students use squares to understand the empirical formula “length x width,” they must ensure students understand that squares have both a space occupying characteristic and can be used to visualize area.  The researchers also advise that teachers help their students move beyond the “direct comparison” method of comparing the areas of two surfaces, to more “indirect” and empirical methods.  The direct approach emphasizes the comparison of two quantities students can easily place side by side; whereas the indirect approach emphasizes the properties of area that extend to objects that cannot be placed side by side (i.e. the states of NY and CA).  The direct approach appears to have limited students understanding of area.  To help teachers accomplish this, the “Implications Section” provides examples of “direct” and “indirect” comparisons. 
            Kamii and Kysh greatly contribute to the mathematics education literature and the understanding of mathematical pedagogy.  They provide a very thorough analysis of the “topic-specific” implications of there results – that teachers must rethink the way they teach area and the way the help their students to understand the square as a measurement of area.  Their findings move teachers and perspective teachers to think about (if not entirely challenge) the assumption that “all students understand how unit squares relate to area.”  The research sheds light on this misconception, and in doing so points out that for some students, the “unit square as a measurement of area” is a poor method for reaching an understanding of area. 
            Despite the strong contributions Kamii and Kysh make to the mathematics education community, their research does have a few limitations.  They created a relatively small sample size of 220, which may not accurately represent the whole of American middle school students.  The sample was upper-middle class and predominately white; this does not assess the mathematics teaching and learning that occurs in urban schools.  This study should be repeated on a larger scale, and with a more diverse sampling of the population to gain more accurate results.  Additionally, the authors could have grounded their work in a stronger theoretical framework.  They briefly discuss the work of Piaget, but the topics only address why students have difficulty with the empirical formula of area, not the square unit representation of area.  Because the topic of the research is student’s visual understanding of area (with respect to the unit square), there is no need to give such a detailed theory behind the student’s empirical understanding of area.  In fact, doing so weakens their argument because it leaves the reader to wonder if their research lacks any relevant related literature.

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.