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What We Know
Mathematical reasoning involves the:
- examination of patterns;
- noting of regularities;
- justification and use of mathematical generalizations;
- support of statements and reasoning in mathematics.
Together, these thought processes lead to explanations with a deductive basis and the development of mathematical sense, which provides the basis for insight into mathematical problems (Reid, 2002; Russell, 1999).
With appropriate support and structures in place, teachers can improve the quality of students' abilities to think, reason, solve complex problems and communicate mathematically (Brown, Stein, & Forman, 1996). To accomplish this goal, educators need to place an emphasis on mathematical reasoning in the classroom (Russell, 1999). Research has demonstrated that in classrooms where students are constantly asked to explain and support their reasoning, students clarify their thinking, develop better reasoning skills and learn standards for mathematical explanation and reasoning (Collins, Brown, & Newman, 1989; Lampert, 1990; Yackel & Cobb, 1994, 1996).
Malloy (1999) provides an overview of reasoning development that may be helpful to educators planning instruction. In the elementary grades, students can be expected to develop justifications and make sense of mathematics. In the middle grades, students can be expected to reason to make conjectures and to apply inductive and deductive reasoning. High school students will use reasoning to form, validate and prove assertions.
Sternberg, Torff and Grigorenko (1998) propose this common set of thinking processes in which one engages while reasoning mathematically:
- identification of a problem;
- formulating a strategy for solving the problem;
- mentally representing information about a problem;
- allocating resources;
- monitoring and evaluating solutions.
This framework can be useful for teachers planning instruction in mathematical reasoning.
In addition, Battista and Hallenbeck (1997) suggest these kinds of teacher questions to help develop student reasoning:
- Encourage students to share methods: Show me how you did it. Why did you do that? Did someone have another way to solve the problem?
- Invite students to test their thinking: Would your strategy work on this problem?
- Model inquiry behaviors: I am not sure, what would happen if...?
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"Mathematical reasoning and problem solving requires teachers to teach mathematics as the power of thought rather than the power of discrete facts."
--Sutton and Krueger, 2002, p. 12
"One of the outcomes of the development and use of mathematical reasoning is a stronger, more reliable kind of memory - the memory of the essence of fundamental mathematical relationships, what I like to call mathematical memory."
--Russell, 1999, p. 6
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