When teachers know what students know: integrating mathematics assessment

Theory Into Practice, Autumn, 2004 by Maryl Gearhart, Geoffrey B. Saxe

The premise of this article is that excellent teaching requires ongoing assessment of student understanding--a commitment to knowing what students know. The Integrating Mathematics Assessment (IMA) program provides" a well-researched example of professional development that helps teachers interpret children's mathematical thinking and guide children toward deeper understanding. The authors describe the features of the program, and provide evidence that it is effective for teachers who wish to use inquiry methods to support students' developing mathematical understandings. The article concludes with a list of recommended resources.

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Excellent teachers know their subject areas well and possess a flexible repertoire of pedagogical strategies. Shulman (1987) labeled these two domains of expertise as "subject matter knowledge" and "pedagogical knowledge," and then proposed a third domain. Arguing that effective instruction lies at the intersection of subject matter and general pedagogy, Shulman proposed the notion of "subject matter knowledge for teaching":

   I include ... the ways of representing and formulating
   the subject that make it comprehensible to others ...
   [and] an understanding of what makes the learning
   of specific topics easy or difficult, of the conceptions
   and preconceptions that students of different
   ages and backgrounds bring with them to the learning....
   (1987, pp. 13-14)

In Shulman's view, excellent teachers are concerned with knowing what students understand and how they learn, so they can help students integrate new ideas and transform prior conceptions.

Shulman's model of teacher knowledge has guided the design of professional development programs across subjects and grade levels as well as research on teacher knowledge (Wilson & Berne, 1999). This article describes one study in this arena, an investigation of a professional development program designed to enhance elementary teachers' pedagogical knowledge in mathematics. Our study was focused on the knowledge needed to teach fractions, measurement, and scale at the upper elementary grades. Our findings demonstrated that building teachers' knowledge--of mathematics, students' mathematical thinking, and methods of assessing student thinking can strengthen classroom practices and increase students' mathematical understanding and skill. Although the professional development program we designed no longer exists (as it was implemented solely for the purpose of our research), the results provide support for exemplary programs and resources that are currently available.

We begin with an overview of the Integrating Mathematics Assessment (IMA) program, its framework and methods, as well as the background research that guided its initial design. We then summarize the results of our research and include citations to our publications for readers interested in further detail. We conclude with reflections as well as recommendations for resources consistent with the 1MA approach.

The Integrating Mathematics Assessment Program

The title of the Integrating Mathematics Assessment (IMA) program captured our focus on formative assessment of children's mathematical thinking. The program was linked to specific curriculum units for fractions, measurement, and scale in the upper elementary grades, because we believed that teachers are more likely to integrate new knowledge and techniques when a professional development program is connected to the curriculum that they are implementing. The curriculum we discuss in this article is Seeing Fractions (Corwin, Russell, & Tierney, 1990), a unit that emphasized the role of visualization in children's construction of rational number concepts. (1) Seeing Fractions was designed to help children build a conceptual foundation, and many tasks were open-ended problems and mathematical investigations. For example, children use linear models when working with fraction strips to measure length, and ratio models when exploring equivalent fractions for strips divided into different fractional units.

The IMA program was guided by research on the development of children's mathematical reasoning. Our goal was to bridge developmental research and practice by helping teachers interpret the ways children make sense of challenging mathematical problems. Teachers cannot effectively guide children toward deeper understanding of challenging mathematical concepts such as part-whole relations, unitizing operations, and multiplicative equivalence without insight into children's mathematical thinking.

We opened the program with scenarios that focused on the challenges teachers may face when interpreting children's thinking. Here is one example.

Mr. Waters is teaching Seeing Fractions lessons to support his students' understanding of pieces as quantities that are fractional parts of wholes. He uses a fair share problem to introduce the notion of equivalence: Six people will share 3 brownies. How much will each person receive it each receives a fair share? He distributes worksheets with blank squares depicting the brownies. As the students work, he monitors children's strategies and asks brief questions to probe students' understandings and scaffold construction of new insights. He is pleased that many solutions suggest that the students understand important aspects of fractions. For example, most students partition the brownies without leaving a remainder, and they distribute parts of brownies equally among people. But he wonders if students can use equivalence concepts to interpret one another's solutions. He brings the class together with a discussion question: "Is 1/2 of one brownie just as fair a share as 1/6 of each brownie (1/6 1/6 1/6)?" As the class discusses the two solutions in Figure 1, Mr. Waters discovers a pattern of understanding that he never anticipated: Some students agree that both solutions show fair shares, but believe that one share is more than the other! Gabriel argues that 1/6 1/6 1/6 is 'more pieces' than one half; he is using whole number concepts to compare '3 pieces' with ' 1 piece.' Naomi argues that 1/2 is a 'bigger piece' than the three little one sixths; she is focusing on the qualitative difference in size of pieces. Neither student is considering the fractional values of the pieces.