Feb. 2011 Problem Solving in Mathematics James L. Kratky In order to teach mathematics through problem solving, teachers may need to consider, review and refine how they think about their teaching practice. In the short commentary that follows, I present three areas on which teachers may focus in order to promote student success with learning mathematics through problem solving. Selection of tasks The starting point for mathematical problem solving is having tasks that are actually problems to solve; teachers must select tasks and pose questions to students that challenge students and are full of appropriate1 and important mathematical concepts (Hiebert & Wearne, 2003; Kahan & Wyberg, 2003; National Council of Teachers of Mathematics, 1991b). Stein, Grover, & Henningsen (1996) and Stein & Smith, (1998) label such tasks as having “high cognitive demand”. These tasks may not be solvable mentally or by simply following a prescribed algorithm or formula. As a result, students will need to invest more time and effort thinking about the goal of the task and what prior knowledge they have to utilize in formulating a plan. This time will also be used for addressing the problem, solving the problem and successfully demonstrating that their answer(s) make sense (Hiebert & Wearne, 2003; National Council of Teachers of Mathematics, 1991b). Teachers must allow this time for students to think and work – in a study involving 144 of these tasks, Stein, Grover & Henningsen (1996) found that students took twenty-­‐four minutes on average (and a range of 10-­‐51 minutes) to work through and justify their solutions to high cognitive tasks. Need for discourse A natural and immediate consequence of posing tasks that engage students in problem solving is that there will be problems! In order to overcome the various issues that arise, students will need to: effectively communicate their thinking, listen to the mathematical thinking of their peers, look across different solution methods and different representations, rely on mathematical justification of solutions, question each other, and come to a consensus on which solution(s) are valid and why (Hiebert & Wearne, 2003; 1 Appropriate tasks are those consistent with the curricular and/or standards-­‐based goals for a given lesson. 1 Feb. 2011 Problem Solving in Mathematics James L. Kratky Kahan & Wyberg, 2003; National Council of Teachers of Mathematics, 1991a). In other words, the simple act of posing high cognitive tasks promotes classroom discourse. As previously noted, such tasks and the ensuing discourse demand time for students to have success. The teacher must consider, plan, and reflect on both students’ roles and their own roles in establishing and maintaining productive discourse. In addition, the teacher must also think about mathematical tools (such as technology, manipulatives, and representations) that may be used in problem-­‐solving investigations, where discourse is a natural consequence (National Council of Teachers of Mathematics, 1991a); factors related to mathematical tools use include strengths and weaknesses of the tools themselves, possible student difficulties in using or understanding how the tools function, student preferences over which tools to use, and physical constraints (size of the classroom, number of students, number of tools, etc.). Students benefit from collaborating in addressing these factors. Thus, planning for and adapting within classroom discourse is a multi-­‐dimensional process in which the teacher must learn from practice (which also takes time). Shift in how teachers perform the work of professional teaching We teachers, who place importance on student problem-­‐solving in mathematics, may need to reevaluate our roles as teachers both inside and outside of the classroom. Engaging students in problem solving tasks is not easy, nor is it straightforward. Inside the classroom, teachers may need to assume a different or new role in which we yield some of our authority. First, we may need to resist the urge to give students the answers quickly or to confirm their answers without pushing for understanding. Stein, Grover, & Henningsen, (1996) found that teachers who resisted helping their students too much were more successful in promoting problem solving by the students and in maintaining a high level of cognitive demand throughout the task phases, as opposed to teachers who did not resist helping too much. Even when students arrive at a correct answer, they may not understand why or how, and they might not be able to explain what they actually did to arrive at their answer (Carter, Gammelgarrd, & Pope, 2006). By developing the culture of discourse described above, teachers are more likely to help students to take authority over the 2 Feb. 2011 Problem Solving in Mathematics James L. Kratky correctness of their own work. As this develops, students will take ownership over the mathematical thinking in the classroom, thus empowering themselves as problem solvers. As we teachers learn to anticipate student responses and attend to student mathematical thinking, our mathematical content knowledge is likely to increase, serving as a regular dose of informal professional development (Mark, Gorman, & Nikula, 2009). Outside of the classroom, teachers may want to think about and plan their lessons in new ways. In a speech given at the fourteenth annual conference of the Association of Mathematics Teacher Educators, Glenda Lappen strongly advocated for the importance of professional development targeted towards helping teachers improve in current reform teaching practices, including placing importance on problem solving (Lappen, McCallum, & Kepner, 2010). One powerful method to promote ongoing teacher development is the method of collaborative lesson study (Fernandez, 2002; Shimizu, 2002). This process, a cultural norm amongst teachers in Japan, involves instructors planning a single lesson together. They select the mathematical topic as well as the pedagogical theme for the lesson study. For example, one theme might be to promote the norm of students justifying their answers amongst themselves2. Also, this process parallels the problem-­‐solving practices that students engage in; teachers practicing lesson study work together to solve the complex problem of planning a lesson that will serve to maximize the learning of an entire classroom of students. This activity relies on collaboration and discourse, so teachers can practice the very roles that they will have their own students perform. Although lesson study is not yet prevalent in the U.S., it has been the object of research and it offers the benefit of increasing the teacher’s ability to adopt a reflective stance towards teaching mathematics (Fernandez, 2002). Developing such perspective of teaching is very useful in a problem-­‐solving classroom. The literature on teaching mathematics with a focus on problem solving is consistent – dramatic changes must occur in order for it to be a successful endeavor! Both teachers and students must be resilient and overcome the challenges that they will face in the process. 2 Shimizu (2002) offers a brief overview of the other structural components of lesson study. 3 Feb. 2011 Problem Solving in Mathematics James L. Kratky References Carter, J., Gammelgarrd, R., & Pope, M. (2006). Navigating the learning curve. In L. R. V. Zoest (Ed.), Teachers Engaged in Research: Inquiry Into Mathematics Classrooms, Grades 9-­12 (pp. 119-­‐134). Greenwich, CT: Information Age Publishing. Fernandez, C. (2002). Learning from Japanese Approaches to Professional Development: The Case of Lesson Study. Journal of Teacher Education, 53(5), 393-­‐405. Hiebert, J., & Wearne, D. (2003). Developing understanding through problem solving. In H. L. Schoen & R. I. Charles (Eds.), Teaching Mathematics Through Problem Solving: Grades 6-­12 (pp. 3-­‐13). Reston, VA: National Council of Teachers of Mathematics. Kahan, J. A., & Wyberg, T. R. (2003). Mathematics as sense making. In H. L. Schoen & R. I. Charles (Eds.), Teaching Mathematics through Problem Solving (pp. 15-­‐25). Reston, VA: National Council of Teachers of Mathematics. Lappen, G., McCallum, W. G., & Kepner, H. S. (2010). General Session: Common Core State Standards. Association of Mathematics Teacher Educators Fourteenth Annual Conference: Irvine, CA. Mark, J., Gorman, J., & Nikula, J. (2009). Keeping Teacher Learning of Mathematics Central in Lesson Study. NCSM Journal of Mathematics Education Leadership, 11(1), 3-­‐11. National Council of Teachers of Mathematics. (1991a). Discourse Professional standards for teaching mathematics (pp. 34-­‐54). Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (1991b). Standard 1: Worthwhile mathematical tasks. Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics. Shimizu, Y. (2002). Lesson Study: What, why, and how? In H. Bass & Z. P. Usiskin (Eds.), Studying Classroom Teaching as a Medium for Professional Development: Proceedings of a U.S.-­Japan Workshop (pp. 53-­‐57). Washington, DC: National Academy Press. Stein, M. K., 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(2), 455-­‐488. Stein, M. K., & Smith, M. S. (1998). Mathematical Tasks as a Framework for Reflection: From Research To Practice. Mathematics Teaching in the Middle School, 3(4), 268-­‐275. 4