Background

Success in middle school mathematics has been shown to have a considerable impact on future mathematical achievement (Gross, 1993). With the vision of raising middle school mathematics achievement for all students, an NSF-funded math-science partnership (MSP) was formed in one state as a partnership between a state university and seven K-12 school districts. Economically and demographically, these districts represent a cross-section of the state’s schools. The MSP was designed to develop a cadre of teacher leaders by providing opportunities for them to develop their expertise in mathematics, pedagogy, and leadership.

Cohorts of 20-30 mid-career middle school teachers progress through an Institute of seven graduate courses taught by university mathematics and education faculty. A central purpose of the Institute is to provide opportunities for teachers to develop a deep and flexible knowledge of the mathematics they teach. The research discussed in this symposium will describe three aspects of teachers’ mathematical knowledge as observed over the course of their participation.

Guiding Frameworks

We draw from the literature on mathematical knowledge for teaching (MKT) and pedagogical content knowledge (PCK). Ball and colleagues (e.g., Ball & Bass, 2003; Ball, Thames, & Phelps, 2008; Hill, Ball, & Schilling, 2008) characterize MKT as the knowledge needed to carry out the tasks of effective teaching, comprising both subject-matter knowledge (knowledge about mathematical content) and PCK (knowledge about the intersection of mathematics and pedagogical aspects of the classroom, such as student thinking, curriculum, and teaching methods). By comparison, Silverman and Thompson (2008) view MKT as a type of PCK that allows teachers to provide mathematical coherence in instruction. Additional researchers have studied MKT in the classroom (e.g., Cankoy, 2010; Prediger, 2010; Speer & Wagner, 2009). Building on the commonalities of these perspectives, we use MKT to refer to the deep and flexible mathematical and pedagogical knowledge needed for teachers to engage in high-quality instruction. Specifically, we describe teachers’ development in language, justification, and lesson planning as they participate in university/K-12 partnership.

Research Design and Findings

Teachers’ initial understanding and growth during their participation in the Institute was documented by the Learning Mathematics for Teaching content assessments (University of Michigan, 2011), course pre- and post-tests, and written course work. In order to gain qualitative insight into the nature of participants’ MKT growth, we conducted research in three specific areas as described below.

1.      Middle School Teachers’ Use of Language in Mathematical Contexts

The meaning we construct for a mathematical idea is closely entwined with the way we learn to use language and symbolism to reason about the idea (Cobb, Yackel, & McClain, 2000; Pimm, 1995). The purpose of this study is to explore the nature of five middle school teachers’ use of written and spoken language in mathematical contexts throughout their involvement with the MSP. The data for this study was collected over 9 months during which the participants completed the first three MSP courses. For each participant, the data consisted of: two task-based interviews; four classroom observations; observation data and the participant’s written work collected during the three MSP courses. The participants’ use of language over time will be discussed, with focus on the use of symbols (e.g., the equal sign) and precision in spoken language (coding inspired by LMT, 2006). To provide a fuller description of the MSP language environment, the manner in which MSP instructors attended to language during their courses will be discussed as well.

2.      Middle School Teachers’ Justifications of Algebraic Ideas

Justification is a significant part of algebraic reasoning (Blanton & Kaput, 2002). To facilitate algebraic reasoning, teachers must recognize and form mathematically sound justifications. This project investigates how five Algebra I teachers participating in the MSP determine the validity of mathematical arguments and create justifications. Data was collected in three phases. First, each teacher participated in a task-based interview in which he/she was asked to determine the validity of hypothetical students’ conjectures and provide justifications for responses. Second, teachers were observed during the MSP Institute, and written artifacts were collected which illustrated both individual and collaborative algebraic thinking. Third, task-based interviews similar to those in the first phase were repeated. Interviews with mathematician-instructors were also conducted to understand their course goals with respect to justification. Teachers’ justifications were coded inductively using the findings of Stylianides and Stylianides (2009) as a guide. The progression of teachers’ justifications will be discussed.

3.      Middle School Teachers’ Mathematical Knowledge as Evidenced in their Lesson Planning

Choices teachers make are crucial and sometimes complex, including but not limited to examples and tasks chosen during lesson planning (Ball & Bass 2003, Rowland 2008, Stein et. al., 1996). This study examines the planning process itself as an important window into teachers’ understanding of mathematical topics and the connections between them. We examined the lesson planning activities of five MSP teachers with varying levels of MKT as measured by the data collected during the Institute. Additionally, these teachers were interviewed periodically as their content knowledge deepened to varying degrees during the course of the Institute. A semi-structured interview was conducted before each of four audio-taped classroom observations, covering the lesson to be observed, previous or upcoming lessons the teacher identified as being on related material, and his/her general approach to planning lessons. Each of the teachers also planned at least one model lesson as part of their Institute coursework, which provided additional insight into what each teacher would choose to do under fewer constraints.

Session

After a 10-minute introduction to the MSP, each presenter will have 15 minutes to present findings. The discussant will have 15 minutes to respond, leaving 20 minutes for discussion. Questions for discussion might include:

1. How can the effects of teachers’ mathematical knowledge be observed in the middle school classroom?
2. What methodologies might help to illuminate mathematical knowledge that is specifically useful for teachers’ practice?
3. How can teachers manage the practical obstacles (e.g., curriculum, pacing guides) that inhibit classroom implementation of the knowledge and skills learned through a university/K-12 partnership?

References

Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3-14). Edmonton, AB: CMESG/GCEDM.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407. doi:10.1177/0022487108324554

Blanton, M. L., & Kaput, J. J. (2002). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36, 412-446.

Cankoy, O. (2010). Mathematics teachers’ topics-specific pedagogical content knowledge in the context of teaching a⁰, 0!, and a ÷ 0. Educational Sciences: Theory and Practice, 10, 749-769.

Cobb, P., Yackel, E., & McClain, K. (2000). Symbolizing and communicating mathematics classrooms: Perspectives on discourses, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates.

Gross, S. (1993). Early mathematics performance and achievement: Results of a study within a large suburban school system. The Journal of Negro Education, 62, 269-287. doi:10.2307/2295465

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372-400.

Learning Mathematics for Teaching (LMT) (2006). A coding rubric for measuring the quality of mathematics in instruction (Technical Report LMT1.06). Ann Arbor, MI: University of Michigan, School of Education.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. New York: Routledge & Kegan Paul.

Prediger, S. (2010). How to develop mathematics-for-teaching and for understanding: The case of meanings of the equals sign. Journal of Mathematics Teacher Education, 13, 73-93. doi:10.1007/s10857-009-9119-y

Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69, 149-163. doi:10.1007/s10649-008-9148-y

Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499-511. doi:10.1007/s10857-008-9089-5

Speer, N. M., & Wagner, J. F. (2009). Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education, 40, 530-562.

Stein, M. S., Grover, B. W. & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455–488. doi:10.2307/1163292

Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72, 237-253. doi:10.1007/s10649-009-9191-3

University of Michigan. (2011). Learning mathematics for teaching (LMT) project. Retrieved from http://sitemaker.umich.edu/lmt/home