What Does it Take to Develop Assessments of Mathematical Knowledge for Teaching? Unpacking the Mathematical Work of Teaching
IntroductionBroad consensus exists about the importance of teachers' mathematical knowledge (Adler Venkat, 2014; Ball, Lubienski, Mewborn, 2001; Baumert et al. 2010; Dohrmann, Kaiser, Blomeke, 2014) Studies have linked mathematical knowledge for teaching to the quality of teachers' mathematics instruction (Eisenhart, Borko, Underhill, Brown, Jones, Agard, 1993; Hill et al. 2008) Mathematical knowledge for teaching has also been linked to student achievement gains in the elementary grades (Hill, Rowan, and Ball, 2005) However, many U.S. teachers lack the deep, nuanced, and specialized mathematical knowledge needed for responsible teaching. This finding is persistent over time, grade levels, and both national and international contexts (e.g. Hill Ball, 2004; Ma, 1999; Tatto et al. 2008) Simultaneously, the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices, 2010) which have been adopted by 47 states and territories, have set out rigorous standards for K 12 mathematics learning that consequently increase the mathematical demands of teaching. To ensure that teachers are well positioned to help students meet these more challenging learning goals, it is now more than ever critically important to focus on developing their mathematical knowledge for teaching.To investigate and grow teachers' mathematical knowledge for teaching (MKT) it is crucial to be able to measure and track the development and uses of MKT. Most work to develop measures of MKT has typically been done by groups of experts in relevant fields, such as mathematicians, mathematics educators, and teachers who have worked together to draft and revise assessment items (Hill, Schilling, Ball, 2004) The early work in this area was focused on developing and refining the construct of mathematical knowledge for teaching while simultaneously and iteratively developing measures of the construct. The process of item development was therefore often time consuming and challenging. Because of the promising results of these earlier efforts, there is now a broad need for assessments of MKT. Building tests at scale means, however, that people who are not deeply immersed in research on MKT will have to be able to write valid MKT items. This will require detailed supports to help test developers understand the nuances of the construct of MKT and ways to assess it. In this paper, we present a framework that identifies the different ways that teachers make use of mathematical knowledge as they go about the work of teaching and provides support to assessment developers. We begin by articulating and specifying what we mean by mathematical knowledge for teaching and its relationship with the mathematical work of teaching that arises in everyday practice.Theoretical FramingConceptualizing Mathematical Knowledge for TeachingBuilding supports for assessment development of mathematical knowledge for teaching (MKT) rests on a clear conceptualization of what we mean by MKT, how MKT is drawn upon in practice, and the specific areas of the work of teaching that we seek to assess. Scholars of mathematical knowledge have examined such knowledge in action as it is used in the practice of teaching (Ball Bass, 2002; Rowland, Huckstep, Thwaites, 2005).Our work builds on a particular practice based perspective on mathematical knowledge for teaching that begins with the premise that, to understand the specific knowledge of mathematics needed in teaching, one must first examine the mathematical work that arises in the context of teachers' instruction in classrooms, a form of job analysis (Ball Bass, 2002) Through detailed analysis of instruction in a 3rd grade classroom over an entire year, Ball and her colleagues identified mathematical problems that teachers regularly encounter and must solve while teaching, such as "interpreting and evaluating students' non standard mathematical ideas" (Ball Bass, 2002, p.