Supporting Mathematical Assertions With Reasoning

Instructor: Michael Gundlach
'Why does it work that way?' This is a question that teachers love and fear. In mathematics, teachers and students need to understand that proof forms the basis for mathematical reasoning.

Why Is College Math So Much Harder?

Many people who do well in grade school mathematics struggle when they reach college because of a shift from an emphasis on calculation to an emphasis on proof and reasoning. A friend we'll call Ashley tried to prove that geometry can't be math after struggling with her collegiate geometry class. Her logic went something like this:

Ashley is good at math. Ashley is not good at geometry. Therefore, geometry is not math.

One reason people like Ashley struggle at the college level is that they often approach school math from an operator's perspective. People often try to get through mathematics classes simply by memorizing formulas and learning how to do specific types of problems. They may even be able to do difficult problems within the realm of school mathematics, but they never learn why the techniques they have learned work. They are like operators of heavy machinery; they may know how to get their machines to do interesting and creative things, but they have no idea how to build, fix, or improve the machine they use.

In order to help students make the transition from the computational mathematics of grade school to collegiate, proof-based mathematics, teachers can help students develop a mechanic's perspective of mathematics. A mechanic knows how the machine works, and thus is able to fix it and better understand why it does certain things while being used. A student with a mechanic's perspective comes to understand not only how mathematics is used, but also how it works. Teachers can help students develop a mechanic's perspective by helping them learn to support mathematical assertions with proper reasoning.

Proof As the Machinery of Mathematics

Learning to support mathematical assertions with proper reasoning requires at least a basic understanding of deductive proof. Deductive proof is a type of reasoning in which results are determined based on general principles or known statements. In mathematics, these principles and statements take the form of axioms, definitions, and previously proven theorems. Deductive proof forms the foundation of mathematics. One comes to know how mathematics works only through proof. Other disciplines use experiments and studies to determine the validity of statements. While such experiments and statements have a place in mathematics, only proof is sufficient to determine what is true and what is not in the realm of mathematics.

Encouraging Deductive Reasoning in Mathematics

In order to help students develop a mechanic's perspective, reasoning and proof need to be seen as integral to the study of mathematics. Because students are more often assessed on their computational skills, they become less concerned with their reasoning skills. When pressed to give justifications for mathematical arguments, they may seek to give justifications by appealing to authority or to examples. However, we need to help students understand that while these are often sufficient explanations in other subject areas, they are not sufficient in mathematics.

We can help students recognize the importance of reasoning by making sure no formula, technique, or concept is presented without some sort of justification. While many students are not ready for a full axiomatic development of mathematics, we can still make sure there are no ''black box'' formulas that are presented. For example, when teaching the quadratic formula, the teacher could walk students through the proof for the quadratic formula even if students are not going to be held responsible for the proof. If a proof for a particular idea is beyond the scope of the class (for example, something like fundamental theorem of arithmetic), referencing the fact that a proof exists can still be powerful. Presenting or referencing proofs, even ones students may not have to recreate later, helps create an expectation of reasoning in the classroom.

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