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Home arrow Mathematics arrow Modeling mathematical ideas: developing strategic competence in elementary and middle school


Once teachers have unpacked the mathematics and identified the curricular topic for the lesson study, we engage teachers in problem solving. Selecting a series of rich tasks that “reintroduces” teachers to the big mathematics ideas prepares them for the lesson study and provides them with the experience of being a learner solving the problem. This real experience of going through the process of solving the problem provides them with a better understanding of how to anticipate potential student strategies and potential pitfalls and misconceptions.

Since lesson study is not about reinventing the wheel by creating new problems or lessons, we invite teachers to bring in their curricular materials and resources to present rich problems to consider for the lesson study. Although we have teachers with diverse ranges of experiences, the most challenging part of teaching through problem solving is selecting a rich and worthwhile task. To select a rich task, teachers must focus on the mathematical content, as well as the different ways in which students learn the mathematics. Knowing what their students already know and can do, what they need to work on, and how much they can go beyond their comfort zone is essential in task selection. If tasks are selected well, it also provides the teachers opportunities to learn about their students’ understandings, interests, dispositions, and experiences.

To help teachers with this selection process, we introduce our teachers to the Smith and Stein’s (2012) article Selecting and Creating Mathematical Tasks: From Research

Table 2.2 Focusing on problems that require higher cognitive demand (Smith & Stein, 2012)

Higher-level demands (procedures with connections)

Higher-level demands (doing mathematics)

Focus students' attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematics concepts and ideas.

Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols, and problem situations. Making connections among multiple representations helps develop meaning.

Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.

Require complex and non-algorithmic thinking—a predictable, well-rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example.

Require students to explore and understand the nature of mathematical concepts, processes, or relationships.

Demand self-monitoring or self-regulation of one's own cognitive processes.

Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.

Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.

Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required.

to Practice on cognitive demand and their task analysis framework. Tasks with high levels of cognitive demand are referred to as “procedures with connections” (when procedures are connected meaningfully to concepts) or as “doing mathematics” (which is an inquiry-based approach with no specific pathway prescribed to solve the problem and is quite open).

The lower-level demands are described as “procedures without connections” (which uses procedures, formulas, or algorithms that are not actively connected meaningfully) or “memorization” (which is simply a reproduction of previously memorized facts). The lower levels of memorization and procedures without connections are excluded from consideration for lesson study. We only allow teachers to choose tasks that fall in the higher levels of cognitive demands such as procedures with connections and doing mathematics (see Table 2.2).

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