Defining Reasoning and Proof
 Introduction | Deductive Reasoning | Proof by Contradiction | Inductive Reasoning | Proof by Mathematical Induction | The Teachers' Role | Your Journal
 One key basis for mathematical thinking is deductive reasoning. An informal, and oft-sited example of deductive reasoning, borrowed from the study of logic, is an argument expressed in three statements: Socrates is a man (1), All men are mortal (2), therefore, Socrates is mortal (3). If statements (1) and (2) are true, then the truth of (3) is established. To make this simple example mathematical, we could write: Eight is divisible by two (1), Any number divisible by two is an even number (2), therefore, Eight is an even number (3). This is deduction in a nutshell: given a statement to be proven, often called a conjecture or a theorem in mathematics, valid deductive steps are derived and a proof may or may not be established. Deduction is part of all the mathematics we teach and learn and not only when we work with formal proof, such as the proof students were developing in the inscribed triangle problem in the first section of this course. Students are using deductive reasoning when they simplify terms in an Algebra I expression, as surely as they are using it when inspecting a complex trigonometric graph. Likewise, this type of reasoning is embedded in the problem solving practices discussed earlier in this course. When we reformulate a problem or make a diagram, for instance, we are often using deductive reasoning. The fluent use of mathematical procedures outlined in the content standards relies on deduction. In fact, this type of reasoning is so omnipresent in mathematics, it may seem odd to highlight it as something for special attention. The challenge comes when we ask students to see their work as deductive reasoning and to add the requirement that they think independently about the more abstract concept of proof. Students are ready, sometimes even eager, to work with deductive reasoning within the procedural context of a problem with defined terms and structured outcomes. And to be sure, procedural fluency is a goal of mathematics instruction. However, mathematics outside of the classroom often does not come in such neat packages. There the demand isn't to just apply a pre-determined procedure fluently, but to analyze the problem to find what is to be reasoned about, establishing what is known and not known, formulating that into the correct conjecture, and reasoning about that conjecture. This kind of work, by necessity, requires careful and rigorous use of terminology, mathematical definition, justification, and logical principles. And it is the sort of deductive work which students should be able to at least evaluate if not create for themselves independently. This is one argument why this standard is important. Watch the video segment (duration 0:46) at left to hear reflections from educators Larry Davidson and Martha Brown.
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