The main benefit a student should take from a mathematics course is exposure to rigorous reasoning. In no other field is rigor as accessible to students as it is in a mathematics class. But what are the main features of rigorous reasoning? Below I offer five pillars of rigorous reasoning. The pillars do not provide a guide for constructing a rigorous response to a given problem. Instead, the pillars provide a framework to recognize whether a presented work is rigorous.
Five pillars of rigorous reasoning:
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Define all relevant concepts precisely, or provide references that rigorously present the definitions.
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Clearly state which question or problem is under consideration.
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Clearly state the assumptions under which the proposed conclusion holds.
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State the background knowledge you will use in your argument and provide references that rigorously present the background knowledge.
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Present logical reasoning that deduces the conclusion from the assumptions and the background knowledge. Present logical reasoning which demonstrates that the background knowledge is applicable as used in your reasoning.
Some ideas to ensure that your reasoning holds under critical reevaluation:
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Critically evaluate all aspects of your reasoning: Have all relevant concepts been included? Are there some hidden assumptions that are used but not stated? Is your logic solid?
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Sometimes, it is necessary to be vague about background knowledge. Be honest and admit it in your presentation of the background knowledge.
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Be open to improvements. Look for hidden clues that suggest improvements.
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If the presentation is cumbersome, break it into smaller pieces to make it more transparent.
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Illustrate your reasoning with multiple specific examples. Do the examples reveal something that you might have overlooked in your work?
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Support your reasoning with analogies and similar settings to make it easier to understand.