Academic journal article The Mathematics Enthusiast

Why Defining the Construct Matters: An Examination of Teacher Knowledge Using Different Lenses on One Assessment

Academic journal article The Mathematics Enthusiast

Why Defining the Construct Matters: An Examination of Teacher Knowledge Using Different Lenses on One Assessment

Article excerpt

(ProQuest: ... denotes formulae omitted.)

Introduction

While test performance is generally reported as if the score assigned a participant were the goal of the assessment, the actual interest is the inferences that can be made about a learner based on that score. It is critical to ensure an assessment is measuring what it is intended to measure if such inferences are to be accurate. Thus, assessments must be written in a way that allows accurate inferences to be drawn. If we are to make claims about quantities of or changes in participants' knowledge, alignment between the content and the underlying assumptions of the domain is critical. Further, if instruction is to be impacted in positive ways by assessment data, we need to ensure that scores accurately report knowledge of the intended construct. Thus, defining the construct one is interested in measuring is vital to the assessment process.

One particularly complex domain from a measurement perspective is that of teacher knowledge. This is complex because teacher knowledge is multidimensional. The specialized knowledge teachers need for teaching (SKT) necessarily includes content knowledge, pedagogical knowledge, and understandings of how students learn (e.g., Ball, Thames, & Phelps, 2008; Baumert et al., 2010; Manizade & Mason, 2011; Shulman, 1986; Silverman & Thompson, 2008). Measuring such knowledge requires adherence to a set of beliefs about the specific construct and how it is best tested, for example, using a paper-based assessment or using feedback in a video-based open-response system (e.g., Kersting, Givvin, Sotelo, & Stigler, 2010). Despite the challenges of this complex knowledge domain, if we care about whether a teacher has the knowledge necessary to support student learning, assessments need to be written to address the domain.

In the case of SKT, we are faced with not only the complexity of the domain, but also the ambiguity of what it means in practical terms for a teacher to have or exhibit particular kinds of knowledge. In our work on proportional reasoning, we have chosen to focus on how teachers understand the mathematics of proportions rather than on their pedagogical understandings related to teaching such content. However, we also acknowledge that teachers need to be able to use the content in the process of teaching, thus our interest in assessment focuses on the knowledge teachers need as it is situated in tasks that ask teachers to make sense of student thinking, analyze multiple approaches to problems, or other authentic teaching activities. Fully defining the SKT in which we are interested is outside the domain of this article. However, we rely heavily on the work of Lamon (2007) and Lobato and Ellis (2010) in our definition. For example, we know that teachers need to have the ability to conceptually connect the two values in a ratio and to understand that a third, abstractable quantity results from that connection (e.g., Lobato & Ellis, 2010). Teachers also need to understand the multiplicative relationships inherent in proportional relationships (e.g., the constant of proportionality is the multiplicative relationship of one value in the ratio to another). And, they need to understand that this corresponds to the unit rate. Further, we assert that teachers should understand how proportional reasoning connects to other areas of mathematics such as fractions and geometric similarity (Lamon, 2007; Pitta-Pantazi & Christou, 2011). We rely on research on teacher knowledge that shows that teachers struggle to use unit rate when faced with values less than one (Harel & Behr, 1995; Post, Harel, Behr, & Lesh, 1988). And, we know from a series of small studies (e.g., Riley, 2010; Son, 2010) that teachers tend to rely on cross-multiplication, which seems to obscure the breadth of knowledge they may have about proportional relationships.

Considering only the domain of proportional reasoning for teachers, one could take a number of approaches to measuring different aspects of SKT. …

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