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A central responsibility of skillful teaching involves helping students develop understanding of, and skill with, academic topics and practices. In order to accomplish this, teachers must know and use content in unique ways. This specialized content knowledge (SCK) is only used in teaching, and not by other mathematically competent adults (MCAs e.g. mathematicians or engineers) during the course of their professional work.

In order to measure SCK for secondary teachers, we used a practice-based analyses of teaching (Author and colleagues, 2016) to construct items that are grounded in (a) the mathematical work that teachers do and (b) the mathematical objects that teachers act on (by mathematical object, we mean the objects that teachers interact with and act on during mathematics instruction). For example, when a student produces written work (the mathematical object), teachers must make sense of the student’s written mathematics in order to determine next instructional steps (the teacher’s mathematical work). In particular, we focused item development on SCK of functions and on three mathematical objects on which teachers must act: definitions, student strategies, and student explanations. With respect to each of these objects, we wrote items that require the following mathematical work: determining whether an informal definition is mathematically equivalent to an formal definition, determining the conditions under which a student strategy is generalizable, and evaluating a student explanation for completeness and precision.

This session focuses on our work to pilot and validate these items. Our approach to validation draws on Kane’s (1990) argument-based approach. In particular, we focus on two different kinds of validity arguments: elemental and structural (Schilling & Hill, 2007). Elemental validity is concerned with, at the level of an individual item, the consistency between participants’ reasoning and their selected multiple-choice answer. Elemental validity would be threatened if participants were choosing the correct answer on the basis of faulty thinking, or if participants were choosing incorrect answers but reasoning about the item soundly. Structural validity is concerned with whether the items are capturing the intended domain of mathematical knowledge (SCK). Structural validity would be threatened if non-teachers were able to deploy other forms of knowledge (e.g. test-taking strategies) in order to reason correctly about the items.

In order to garner validity evidence, we conducted cognitive interviews (n = 57) with mathematicians (13/57), MCAs (7/57), novice or pre-service teachers (12/57), and experienced teachers (25/57). Our presentation focuses on evidence related to participants’ understandings of the three areas of mathematical work noted above. In particular, we focus on differences in how participants understood the meaning of (a) mathematically equivalent definitions, (b) generalizable strategies, and (c) mathematically complete explanations. For example, whereas mathematicians generally understood the notion of mathematically equivalent definitions in terms of whether the definitions generated identical sets of objects, many teachers drew on an understanding of equivalent definitions that also anticipated whether their students would understand a given definition. Our presentation will include example items and further findings related to participants’ understandings in the three categories above.