Mathematics and Case-Based Instruction

An Interview with Senior Lecturer Kay Merseth

Katherine K. Merseth, senior lecturer on education and director of the Teacher Education Program led a team of mathematicians, teachers, and teacher educators, and edited the cases in the recently published Windows on Teaching Math: Cases of Middle and Secondary Classrooms (Teachers College Press). Written for both pre-service and in-service teachers, the book includes 11 cases, each with an objective to improve the teaching and understanding of mathematics at the 7th- through 12th-grade levels and to provide opportunities to examine classroom practice and assess student thinking. These materials adopt the highly successful pedagogical approach of case-based instruction used at the Harvard Business School to examine the work of secondary classroom mathematics teachers. What follows is an interview with Merseth on the advantages of case-based pedagogy in mathematics instruction.

Q: What are the necessary skills to teach mathematics?

A: Solid content knowledge, strong knowledge about pedagogical techniques in mathematics, a deep understanding of the epistemology of mathematics, knowledge about how people learn math, and the ability to listen to and engage the learner. Many consider only the content to be important. In part, this is because there are so many math teachers lacking basic knowledge about the content and for whom math was not a major or focus in college. Many of the current mathematics teachers in secondary schools today have not studied the topics they are teaching since they themselves were middle- or high-school students. In my opinion, cases are a good way to add to the discourse on how we can better understand and teach mathematics.

Q: You note that math students in the U.S. are not performing to the standards that they should be, especially in comparison with other countries. To what do you attribute this gap?

A: Societal beliefs about mathematics, the rote, skills and procedural focus of the curriculum in schools, and the preparation of our teachers are three factors to blame. The common conception is that math is rule-oriented knowledge, static, and a difficult subject only mastered by a few. In reality, mathematics is an interactive problem-solving process involving conjectures, hypotheses, and refutations that flow. Much of the typical curriculum in schools is outdated, repetitious, and unrepresentative of the evolution of the field. Where many textbooks stress computation and procedure, mathematics teaching should focus on understanding and sense-making. Finally, only one out of every two math and science teachers possesses adequate pedagogical training, and thus many tend to teach only in the way they were taught. A change in the mathematical education of our young people is a goal our society can and must achieve.

Q: Why do you think case-based instruction is effective for teaching mathematics?

A: Case discussions are a natural mode of learning about reform-oriented teaching, as they elicit critical examinations of various teaching approaches. The complexities of real classrooms are brought to life by highlighting the multiple dimensions of teaching. Cases and case-based instruction are known as a pedagogical approach that helps participants develop analysis and problem-solving skills. Cases bring practice face-to-face with the content of the subject in a realistic way. It is also important to note that cases provide a safe and engaging way to explore, examine, and analyze pedagogical possibilities as well as specific content.

Q: How did you develop the cases? What was the primary objective in developing them?

A: The set of cases were designed using a theoretical framework that includes four dimensions: subject matter knowledge, pedagogy, student language and understandings, and classroom and school context issues. The style and format of the cases in this book build on the expertise developed by the Harvard Business School. The late C. Roland Christensen of the Business School and the Ed School offered inspiration and encouragement for the exploration of the use of cases in the field of professional education. In order to develop pedagogical skills, case materials need to stimulate and encourage the imagination of the case reader, which hopefully leads to various creative responses presented in the case. In addition, the cases in Windows on Teaching Math cover topics considered "hard to teach" and/or "hard to learn," such as rate, ratios, proportions, limits, independence of events in probability and multiple representations. These materials are cutting-edge and provide marvelous opportunities for teachers to engage in lively discussions about content and classroom teaching techniques in mathematics.

Q: How important is the role of the facilitator in case-based instruction?

A: Case facilitators must wear many hats--they are responsible for the content of the case, for the discussion flow and process, for individual students and their sense of safety to join the discussion. In fact, I believe a safe environment, one where participants feel comfortable raising a range of comments and discussing personal beliefs, is critical to case-based learning. The facilitator is important because he or she must encourage participants to respond to each other, not the "instructor." Finally, the physical environment is important. We've found that case discussions flourish when participants are able to observe each other's facial expressions and body language. A U-shape or circle are ideal as they also allow the facilitator to move throughout the group.