Mathematical Reasoning

Ball and Bass (2003) are clear that "the notion of mathematical understanding [emphasis theirs] is meaningless without a serious emphasis on reasoning. What, after all, would mathematical 'understanding' mean if it were not founded on mathematical reasoning?" (p. 28). In fact, mathematical reasoning stands at the centre of mathematics learning. More than just making sense of mathematical theories and processes, mathematical reasoning involves weaving the strands together until students are capable of complex thought and application across situations. Most importantly, the ability to reason effectively allows students to make generalizations and ultimately rise above the particular instance to consider the general case.

Encouraging students to learn to reason mathematically involves more than encouraging individual sense making. Sense making is when students make sense of mathematical ideas based on their personal convictions; whereas, reasoning mathematically "comprises a set of practices and norms that are collective, not merely individual or idiosyncratic, and rooted in the discipline [of mathematics]" (Ball and Bass 2003, p. 29).

Teachers can help students learn to reason mathematically by requiring them to formulate conjectures, create and use networks of concepts and processes, justify solutions and find proof that their solutions are valid. Students develop mathematical reasoning through solving problems when they are required to negotiate meaning, construct and critique different representations, defend and debate possible solutions, and pose new problems.

Learning to solve…
… good problems builds procedural fluency.
… problems in a variety of ways develops conceptual understanding.
… a variety of problems develops the ability to generalize.