Publication Type: Research Report
Abstract: Research suggests that it is important to leverage students' home and local discourse practices in mathematics classrooms. Gee's (1990) notion of Discourse suggests that this may be difficult to do without first restructuring the Discourse of mathematics. Our research indicates that by perpetuating culturally based distinctions between math and non-mathematical activity the Discourse of mathematics can marginalize alternative mathematical discourses.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: This presentation will feature evidence-based practices that can be integrated into preservice mathematics methods courses. These practices are aligned with the NCLB mandate for evidence-based instruction including monitoring student performance.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: The NCATE/NCTM Program Standards for Programs for Initial Preparation of Mathematics Teachers (2003) state that teachers of mathematics should “demonstrate knowledge of the historical development of [topics] including contributions from diverse cultures” (p. 4). Several manifestations of a history of mathematics course can be designed to provide opportunities for prospective teachers to gain such knowledge and consequently plan for its use in teaching. In this session I will describe the context and foundation for the history of mathematics course, "Using History in the Teaching of Mathematics," required of the secondary mathematics education majors at Florida State University. The course is currently designed to provide pre-service mathematics teachers with opportunities to: (1) work with mathematical topics as they evolved over time, with respect to the mathematics found in secondary school curricula today; (2) study and discuss the historical and cultural influences on and because of the mathematics being developed; and (3) develop the pedagogical knowledge needed to integrate an historical perspective in the teaching of school mathematics. Five course tasks are used to create the opportunities for mathematical, historical, and pedagogical learning: key topic explorations; library assignments; a journal assignment; a “book club” experience; and a model lesson assignment. The session will focus on the five course tasks and pre-service mathematics teachers’ perceptions of the role of the history of mathematics in their learning and the learning of their future students gained via their experience with the course.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: In this report, we examine the affect of a mathematical problem-solving (MPS) course on the abilities of inservice teachers to demonstrate mathematical understanding in problem solving and to use complex strategies in mathematical problem solving.

To be competent problem-solving practitioners, teachers need several types of knowledge: procedural (Eisenhart, Borko, Underhill, Brown, Jones & Agard, 1993), conceptual or content knowledge (Leinhardt, 1988; Sullivan, Clarke, Spandel & Wallbridge, 1992; Eisenhart et al., 1993). Procedural knowledge denotes the rules, procedures, and skills necessary for completing a task. Procedural knowledge may or may not be supported by conceptual knowledge (Hiebert & Lefevre, 1986). Conceptual or content knowledge denotes the ability to understand the concept and connect or apply several different ideas. Conceptual knowledge for mathematics teachers also includes the ability to make generalizations, describe relationships, and demonstrate higher order reasoning skills (Sullivan et al., 1992). In addition, developing problem-solving practitioners need explore the role of metacognition in mathematical thinking or problem solving (Schoenfeld, 1985).

A group of sixteen inservice middle school (n=2) and secondary (n=14) mathematics teachers participated in a yearlong professional development program focused on mathematical problem solving. Initial baseline measures of skill levels, pedagogical strategies, and views about mathematics were established and a multi-part assessment of problem solving levels was administered. Teachers began the program in a course in Discrete Mathematics that focused upon various modes of group learning and leveling concepts about functions, mathematical reasoning, and mathematical inquiry. After the initial forty-five hours of instruction in Discrete Mathematics, the teachers enrolled in a course in mathematical problem solving. In the MPS course, teachers read and reported upon articles in research and practice in problem solving, applied and discussed Polya’s problem solving strategies in the context of their approaches to in-depth problems, revised and refined their own problem solving strategies by incorporating ideas from the research and practitioner literature, and examined rubrics for assessing students’ work in problem solving. Using a guide created by the researcher, they engaged in creating their own in-depth problems to elicit mathematical problem solving behaviors in their students on difficult concepts. The MPS course was followed by a course in probability and statistics with an emphasis on problem-based learning and real-world applications using technology. Data was collected systematically throughout the program in the form of student interviews, journal entries, student class work, pre- and post- tests of baseline skills, and a problem solving assessment that was administered at four different intervals during the yearlong program.

This report focuses primarily on the results from the problem solving test which was designed by the researcher. The work of White and Michelmore (1996) in studying student’s conceptual knowledge in calculus motivated the design of the problem solving tasks and methodology. Four tasks were the focus of the test items. The mathematics needed to solve the tasks did not go beyond a typical second-year algebra high school course. Each task was structured in four versions so that the manipulation (procedural knowledge) required to solve each version was essentially the same. However, the difference in the tasks was that each successive level involved more sophisticated levels of modeling and generalization. Each item was scored using a rubric designed by the researcher that analyzed understanding, strategies, and accuracy on the task.

Teachers were tested on four occasions: before the Discrete Mathematics course, before the MPS course, before the Probability and Statistics course, and at the end of all courses. The teachers were divided four parallel groups of 4. They were not aware which group they were in. Four tests were constructed and each test included four questions: one version of each of the four tasks. Each version of each task occurred on one and only one test, and each test had only one question in each version. A cyclic scheme was used to administer the tests to each of the four groups over the four data collections.

Findings suggest that there was a statistically significant growth in the strategies, understanding, and accuracy of the teachers in problem solving throughout the year. However, the greatest increase in the teachers’ ability to demonstrate effective problem solving strategies and understanding occurred after their MPS course. Excerpts of student work, journal entries, and interviews also support findings. We explore the components of the problem-solving course that contributed to the largest growth in teacher mathematical understanding and strategies in problem solving.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: This paper reports results of an ongoing design-based research project that seeks to both promote and characterize the kind of deep, well-connected and flexible conceptual understanding of mathematics that is advocated for teachers (e.g. Ball & McDiarmid, 1990; CBMS, 2000; Fennema & Franke, 1992; Shulman, 1986). In particular, this paper investigates how students access relevant mathematical information in the context of mathematical problem solving.
Background
A subject’s mathematical background is an important component of problem solving abilities (Schoenfeld, 1985); yet adequate mathematical preparation does not guarantee success. Studies show that undergraduates (Schoenfeld, 1985; 1992), graduate mathematics students (Carlson, 1999), and even some professional mathematicians (DeFranco, 1996) struggle to access the appropriate mathematics needed to solve a particular problem. In their work with mathematicians, Carlson and Bloom (2003) suggest that a well-developed and well-connected conceptual understanding of mathematics facilitated the mathematicians’ problem solving success. Additionally, the mathematicians reported that content knowledge was more useful when solving difficult problems than other factors such as heuristics.
Research into the mathematical understandings of teachers indicate that even when teachers (both preservice and in-service) appear to be adequately prepared, their mathematical knowledge base is shallow and compartmentalized (Ball, 1990; Bloom, 2001; Bryan, 1999; Post, Harel, Behr, & Lesh, 1991; Tirosh & Graeber, 1990). A rich, well-connected knowledge base allows teachers to conduct the kind of inquiry and discourse recommended by NCTM (Ball, 1991; Ball & McDiarmid, 1990; Koency & Swanson, 2000; Ma, 1999; McDiarmid, Ball, & Anderson, 1989; NCTM, 2000) and inadequate mathematical understandings tend to inhibit efforts to implement “reform” curricula (Behr, Khoury, Harel, Post, & Leah, 1997; Koency & Swanson, 2000; Mathematical Sciences Education Board, 2001; Post et al., 1991).
Theoretical Framework
The Multi-Dimensional Problem Solving Framework developed by Carlson and Bloom (2004) provides the means to scrutinize the resources accessed or overlooked in the process of solving mathematical problems. By attending to both the repetitive cycles of orienting, planning, executing and verifying, and the aspects of resources, affect monitoring and heuristics that influence each of the phases, one can tease out the nature of the subject’s knowledge base and its influence during the solution process.
The Study
This study uses Design-based research methods -- the iterative refinement of instructional innovations employing thought revealing activities and artifacts in conjunction with inquiry into theoretical considerations regarding the nature of the development of mathematical knowledge (Kelly & Lesh, 2002; Lesh, 2002). The subjects are preservice high school mathematics teachers who have completed most or all of their required mathematics courses and will student teach within the next year. The study is set in an upper division mathematics education course called Mathematics in the Secondary School. The curriculum that has been developed over several iterations of the experiment provides students with opportunities to revisit the major concepts of school mathematics while at the same time, enriching their knowledge base and improving their problem solving abilities. For instance, challenging problems are selected for their capacity to reveal student thinking and understanding, and to stimulate mathematical discourse. Students also take typical high school mathematics problems and then they generalize them and analyze them for mathematical structure. Students share their work and their reasoning, orally and in written form, with their peers on a regular basis.
Data sources include clinical and task-based interviews, video recordings of classroom sessions, homework, classwork, reflective journals and pre/posttests consisting of NAEP (grade-12) items.
Findings
Careful analyses of the data show that students enter the class exhibiting a compartmentalized, proceduralized knowledge base. When stripped of examples and cues, these students struggle to solve problems from high school materials. When asked to list the concepts used to solve a problem, students tend to be general and vague. For example, when asked find the rate of change at a moment in time, students tended to find the average rate of change over an arbitrary period of time, and list “algebra” as the main concept employed, rather than calculating the derivative at a point. As the semester progresses, they demonstrate an increasing facility with both doing and discussing mathematics. Mathematical discussions and written work become richer and clearer. Students appear to be more comfortable explaining their work, more confident of their work, and can produce multiple solutions. When asked to determine the mathematical ideas involved in a problem situation, students are able to determine both the “big ideas” as well as unpack them to consider the skills and routines that might also come into play.
At the end of the semester, problem-solving sessions are markedly different. Students are able to articulate the mathematics needed and verbalize a solution path. In addition, they report a better understanding of how topics fit together and relate as well as greater confidence in problem solving in general, and the content of high school mathematics in particular. Improved scores on posttest support this evidence. Finally, students report that they feel they are more effective in their tutoring or internships because they don’t have to “see how the book did it” first, and are able to explain concepts in multiple ways.
This ongoing project provides some insights into a process by which teacher educators can successfully enrich the content knowledge base of prospective mathematics teachers.
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