Assessing Secondary Teachers' Algebraic Habits of Mind

• In Event: 21.050 - Developing Measures of Mathematical Knowledge for Teaching for Secondary Teachers: Current Research and Open Questions

Abstract

Objective. Describe an instrument to assess secondary teachers’ mathematical habits of mind (MHoM), focusing on habits relating to Algebra, and results from recent use.

Theoretical Framework and Research Question. Many researchers have studied the notion of mathematical knowledge for teaching (Author, 1991; Author and colleagues, 2008; Heid, 2008; Author and colleagues, 2007; Leinhardt & Smith, 1985; Ma, 1999; Author and colleague, 2008), notably in the context of elementary school teachers, and how this specialized mathematical knowledge is organized, used in the classroom, and affects student achievement (Author et al, 2008; Author et al, 2005).

At the secondary level, we hypothesize that Mathematical Habits of Mind (MHoM)—the web of specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians (Cuoco, Goldenberg, & Mark, 1997; Author, 2015)—are a critical components of mathematical knowledge for teaching. Teachers with strong MHoM can foster in their students the development and use of general purpose tools that make connections among various topics and techniques of secondary school mathematics content; MHoM can bring focus and coherence to teachers’ work with students. This is particularly important in high school Algebra, which functions as a foundation for many subsequent courses; Algebra can also serve as a gatekeeper, preventing students from pursuing STEM endeavours.

The central research question we will address is: What MHoM do secondary teachers use, how do they use them, and how can we measure them? To examine this question, we operationalized our MHoM definition and developed two instruments—a paper and pencil assessment and an observation protocol—that measure how teachers use MHoM while doing mathematics on their own and teaching mathematics.

Methods. To build the assessment, we collected responses for each potential item from 200 secondary teachers. We sought recurring regularities and themes in that data (Guba, 1978), and categorized the responses according to the approaches teachers took rather than the “correctness” of the response. A team of advising mathematicians debated and scored the responses relative to alignment with MHoM. Those categories and scores formed the basis for our coding rubrics. In the session, we will share data from two studies, including 0.87 Cronbach’s Alpha and 0.90 Guttman Lambda. Another recent field test (of 196 teachers) also yielded strong results (including 0.749 Cronbach’s Alpha). We have used the assessment with over 500 secondary teachers.

Results. Use of the assessment falls into categories:
Evaluating effectiveness of professional development (Gates et al., 2016);
Relating student classroom experiences and teacher MHoM (Author, 2018; Gates et al.,
2016);
Relating teacher practice and MHoM (Author, 2015; 2016; 2017);
Relating student achievement and MHoM (Author, in preparation).

Scientific Importance of the Work. The importance of this work is two-fold:

In the assessment, we strongly de-emphasize “getting the right answer.” We are interested in teachers’ thinking as they approach problems, consistent with equitable assessment practices (Steele, 2010).
The assessment illuminates the complex relationships between teacher knowledge, teacher practice, and student outcomes. These relationships are understudied at the secondary level.

Authors

• Ryota Matsuura, Saint Olaf College

• Sarah Elizabeth Sword, Education Development Center, Inc.