How Much Can Assessment of Secondary Mathematical Knowledge for Teaching (MKT) Learn From Elementary MKT?

• In Event: 70.043 - Investigating Knowledge for Teaching Secondary Mathematics

Abstract

Overview
To date, most assessments of mathematical knowledge for teaching (MKT; e.g., Ball, Thames, & Phelps, 2008) have been developed at the elementary or middle school level, where teachers’ focus on mathematics can be quite distant from adult mathematics. To examine whether design theory developed for elementary MKT assessment transfers to the secondary level, then, work on two research questions is reported:
1. Do the items written to assess secondary level MKT function as designed, where the design theory is an interpretative argument (Kane, 2001, 2004) specifies that the test-taker will apply their mathematical knowledge to a particular task or problem that is encountered in teaching practice?
2. To what extent do teachers use mathematical knowledge that differs from conventional mathematical knowledge — i.e., mathematical knowledge that is specialized to the work of teaching?

Data and methods
This project examines a set of secondary MKT assessment items developed using the assessment design theory behind the elementary MKT instrument LMT (Learning Mathematics for Teaching, 2008); to answer the research questions, think-aloud interviews with these items were conducted with 5 mathematicians, 5 beginning mathematics doctoral students, 10 in-service teachers, and 10 pre-service teachers; the in- and pre-service teachers were selected for strong conventional knowledge as measured by Praxis scores. Interviewees were asked to solve secondary MKT assessment items as well as conventional knowledge related to the items. Interviews were analyzed using methods similar to that of Hill, Dean, & Goffney (2007).

Results and conclusions
We found that strong conventional mathematical knowledge led to correct answers in ways that do not align with intended reasoning, suggesting that the design theory for elementary MKT may not transfer to secondary MKT. In particular, our findings corroborated that observation that differences between elementary MKT and the knowledge needed do to elementary level mathematics may be less dramatic at the secondary level, where students work with mathematics that approaches the mathematics encountered by non-teaching math majors (Rowland, 2012); moreover, differences in the breadth and depth of teachers’ conventional knowledge teachers hold may be of greater significance at the secondary level (Speer, King, & Howell, 2014). Teaching at the secondary level might instead require a kind of mathematical knowledge that is already available to those with a degree in mathematics. However, the application of this knowledge toward a correct answer depended on mathematically interpreting the item in a way that coincided with the needs of the pedagogical context (Lai, Jacobson, & Thames, 2013 ) depicted in the item – even when interviewees did not connect their interpretation to the embedded pedagogical purpose until after the fact or ever at all. Moreover, wWe moreover found that the conventional knowledge used toward correct answers, whether through the intended reasoning or alternative reasoning, drew from conceptual understanding of real analysis and abstract algebra but not necessarily from technical knowledge of these areas; and that the mathematicians and mathematics graduate students were more likely to attribute their knowledge to course-taking whereas teachers were more likely to attribute their knowledge to general mathematical ability or teaching experience.

Authors

• Yvonne Lai, University of Nebraska - Lincoln

• Heather Howell, ETS

• Geoffrey C. Phelps, Educational Testing Service