Research indicates that problems in students’ understanding of area measurement may be related to their memorization of area formulas (Baturo & Nason, 1996) and because students lack opportunities to develop a strong conceptual base for these formulas (Zacharos 2006).  Most studies focused on the area of rectangular shapes; however, some show students’ difficulties with the area of non-rectangular shapes. Baturo and Nason (1996) found that only 2 out of 13 students were able to make the connection between the area of rectangles and area of triangles.  Students were memorizing the formula as base times height divided by 2 without actually knowing where that 2 came from. Lipka and Adams (2004) also found that students had the weakest performance in area of complex shapes compared to area measurement content.

Purpose

One of the factors contributing to poor learning of measurement is weaknesses in K–8 written curricula (Lehrer, 2003). In this study, I examined the curricular treatment of area of parallelograms and triangles as an opportunity to learn. I described how key concepts indicated in the literature were addressed.

Method

Data

Given the projected detail of the analysis, it was clearly impossible to analyze all elementary school curricula. Everyday Mathematics (EM), Scott Foresman/Addison Wesley's Mathematics (SFAW), and Saxon Math will be analyzed for the purpose of this study. As a preliminary analysis the data for this proposal was gathered from EM for grades 4 and 5.

Curricular Coding Scheme

To develop a coding scheme, I drew from related studies six key ideas (area as a continuous quantity, partitioning, conservation, unit iteration, spatial structuring, and meaning of height) that might support students’ understanding of area of non-rectangular shapes. Conceptualized as the quantification of a region, area is a continuous quantity, rather than just a collection of discrete units. This conceptualization becomes critical in making sense of the operations involved in calculating the area of non-rectangular shapes (Baturo & Nason, 1996). Partitioning, unit iteration, conservation and spatial structuring are listed as important concepts in Stephan and Clements (2003) in understanding area measurement.  Partitioning, conservation, and spatial structuring help students deal with issues of non-array structure that appear in triangles and parallelograms. As students restructure parallelograms and triangles they need to partition the region into sub-units and compose and decompose the region to build an array structure. This “mental operation of constructing an organization or form for an object or set of objects” is called spatial structuring (Battista & Clements, 1998, p. 503) and is important in structuring shapes in a non-classical form. This structuring brings the concept of height into play. Students need to understand the meaning of height and be able to point out the height of a shape. Within all these procedures students need to recognize that area is invariant under these operations and area is quantified by iterating units or sub-units.

I described how key ideas addressed in the lessons by using this coding scheme. I also examined what kinds of procedures are provided the most and described their foci of attention by using key ideas listed in the curricular coding scheme.

Results and Discussion

I found that not all 6 key concepts were addressed in the lessons for area of parallelograms and triangles. The concept of height got the most attention, followed by conservation. Height being perpendicular and height as a distance drawn from vertex to the base were the most typical way of introducing height. The most frequent way of seeing the conservation concept was the idea that area was invariant under partitioning. There was not any mention of conservation of area under different choice of units and conservation of area under change in location. Area as a quantification of a region, and unit iteration were non-existent concepts in these lessons. Partitioning was addressed as a procedure without visiting the underlying principles behind partitioning activities. In partitioning related tasks most of the time students were asked to divide a region into sub-regions without any discussion around sub-units and region being two-dimensional as indicated in Stephan and Clements (2003). Spatial structuring appeared once in these lessons and that was supporting the idea of recognition of relationships between components and composites.

There might be an assumption that some key ideas were already covered before coming to area of parallelograms and triangles; however, in the early grades it might be hard for kids to transfer their learning from rectangular shapes to non-rectangular shapes.

Results provided evidence of a computational focus in the materials analyzed in this study. There were many questions asking students to compute the area of parallelograms and triangles without emphasizing key ideas underlying those procedures.  The most typical tasks presented in this curriculum to explore the area of parallelograms and triangles were measuring length to find area, completing shapes into rectangles to find area and a straightforward computation of area.

The rectangular method could be considered a tool in dealing with area of triangles and parallelograms.  Instead of multiplying two length measures provided in the question students’ understanding of this new concept is connected to their previous understanding in area of rectangular shapes. This might deal with the problem indicated in Baturo and Nason (1996) not knowing where division by 2 comes. However, there was not any opportunity provided for student to question and explore why height times base give the area of parallelograms and triangles instead of length times width.

The analysis of other curriculum materials will inform us about the scope and limitations of the content provided in the resources for teachers and students. With this extensive analysis we might able to find some good tasks that addresses key concepts very well.

In this session, I will share my findings including my analysis of two other curricula and provide some specific examples of highly emphasized content. In the discussion of typical tasks and foci of these tasks I will revisit key ideas described in the coding scheme.