In this session, we describe the research work of a targeted Math and Science Partnership funded by the National Science Foundation since 2003. Our partnership is devoted to improving student achievement in mathematics through programs that provide teachers with solid content-based professional development sustained by mathematical learning communities in which mathematicians, educators, administrators, and teachers work together to put mathematics at the core of 5–12 mathematics education. As one part of the work, we are developing, in collaboration with leaders in the field, a long-term research program with the ultimate goal of understanding the connections between secondary (grades 7–12) teachers’ mathematical knowledge for teaching and secondary students’ mathematical understanding and achievement. One important feature of our approach is the core involvement of mathematicians in all aspects of the work.

Many researchers have studied the notion of mathematical knowledge for teaching (Ball, 1991; Ball, Thames, & Phelps, 2008; Heid, 2008; Hill, Sleep, Lewis, & Ball, 2007; Leinhardt & Smith, 1985; Ma, 1999, Stylianides & Ball, 2008), notably in the context of elementary school teachers, how this specialized mathematical knowledge is organized and used in the classroom and its effect on student achievement (Hill et al., 2008; Hill, Rowan, & Ball, 2005). Building on the literature base, and 20 years of our particular experiences as mathematicians engaged in doing mathematics with secondary teachers (Stevens, 2001), our hypothesis is that teachers who not only possess strong content knowledge but also the mathematical habits of mind used by many mathematicians teach in a way that results in increased student learning and achievement. We define mathematical habits of mind (MHoM) to be the specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians (Cuoco, Goldenberg, & Mark, 1996, 2010). Understanding the mathematical practices and general purpose tools that connect the various topics and techniques of high school mathematics content can bring parsimony and coherence to teachers’ mathematical thinking and in turn, to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching at the secondary level, i.e., the knowledge necessary to carry out the work of teaching mathematics (Hill et al, 2005).

Recognizing the need for evidence-based research to refine and test these conjectures, we are engaged in a focused research study centered on the following question: What are the mathematical habits of mind that high school teachers use in their professional lives and how can we measure them? While addressing this question is not an unfamiliar task (Hardy, 1940/1992; Polya, 1954a, 1954b, 1962), what is less familiar is the task of gathering evidence of MHoM indigenous to mathematics and translating them to the work of secondary mathematics teaching and learning.

In this initial phase of research, we are engaged in ongoing work to identify and precisely define MHoM, and to operationalize this framework into paper and pencil assessment problems that accurately and uniquely measure these habits. We propose four categories or constructs for mathematical habits of mind: 1) applying mathematical reasoning, 2) performing purposeful experiments, 3) seeking, using, and describing underlying structure, and 4) using precise mathematical language. At this time, we are putting particular emphasis on structure (3). This emphasis dovetails very well with the Mathematical Practices of the Common Core State Standards (CCSS, 2010).  

We have designed, piloted, and field-tested a paper and pencil assessment of teachers’ use of structure with the goal of investigating how secondary teachers use structure in their own doing of mathematics and to some extent, what role mathematical structure plays in their teaching. We are also developing a classroom observation protocol designed to capture teachers’ use of structure in their professional work.

In this session, we will focus on the development the paper and pencil assessment and rubrics to measure MHoM for teaching. The five central discussion questions for this session are:

1. How do teachers use mathematical habits of mind in their professional work?
2. What does it mean to measure teachers’ mathematical habits of mind?
3. What are the affordances and limitations of our measurement tools?
4. What aspects of MKT are we capturing with the assessment and the protocol? What aspects are we missing?
5. How can we improve our tools?

This interactive session will have four components:

1. We will begin with a brief overview of the project and the conceptualization of MHoM, particularly our emphasis on seeking, using, and describing structure. We will review the theoretical perspective and empirical and practice-based approaches for the construction and validation of the MHoM framework and assessment. We will also share the five central questions for discussion.  (15 minutes)
2. Next, we will share examples and scoring rubrics from the paper and pencil assessment. Assessment items will be provided to small groups and time will be allotted for participants to work on problems and practice using the rubrics. We will ask participants to discuss how the assessment items function as measures of MKT. (30 minutes)
3. We will have a discussion of the construction of the observation protocol and the synergistic nature of the development of the observation protocol and the assessment: the ways in which classroom observations informed the development of the paper and pencil assessment, and the ways in which the development of the paper and pencil assessment (particularly the development of the rubrics) informed the development of the observation protocol. We will give participants an opportunity to try out the protocol on a short piece of classroom video. (30 minutes)
4. The final component of our work session will be a general discussion of the five central questions. (15 minutes)

References 

Ball, D. L. (1991). Research on teaching mathematics: making subject matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching (Vol. 2, 1–47). Greenwich, CT: JAI Press.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Common Core State Standards Initiative. (2010). Common Core State Standards Initiative: Preparing America's students for college and career. Retrieved from http://www.corestandards.org

Cuoco, A., Goldenberg, E., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15(4), 375–402.

Cuoco, A., Goldenberg, E., & Mark, J. (2010). Contemporary curriculum issues: Organizing a curriculum around mathematical habits of mind. Mathematics Teacher, 103(9), 682–688.

Hardy, G. H. (1940/1992). A mathematician’s apology. New York: Cambridge University Press.

Heid, M. K. (2008). Mathematical knowledge for secondary school mathematics teaching. Paper presented at the Study group 27: Mathematical knowledge for teaching at the 11^th International Congress on Mathematical Education (ICME), Monterrey, Mexico.

Hill, H. C., Blunk, M., Charalambous, C., Lewis, J., Phelps, G., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(x), 371–406.

Hill, H. C., Sleep, L., Ball, & D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts? In F. Lester (Ed.) Handbook of research on mathematics teaching and learning (2^nd ed., pp. 111–154). Charlotte, NC: Information Age Publishing.

Leinhardt, G., & Smith, D. A. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77(3), 247–271.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

Polya, G. (1954a). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. 1). Princeton, NJ: Princeton University Press.

Polya, G. (1954b). Mathematics and plausible reasoning: Patterns of plausible inference (Vol. 2). Princeton, NJ: Princeton University Press.

Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (Vol. 1). New York: John Wiley.

Stevens, G. (2001). Learning in the Spirit of Exploration: PROMYS for Teachers. Mathematics Education Reform Forum, 13(3), 7–10. 

Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332.