The definition of a concept in math has properties that are different from definitions in other subjects:
• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.
How many college math teachers ever explain these things?
I will expand on some of these concepts in future posts.
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I (try to) teach these things when teaching advanced undergraduates (in classes where many students are writing proofs for the first time). I emphasize that the difference between their earlier, computation-based math classes (calculus, etc.) and abstract theoretical classes like abstract algebra has more to do with the centrality of definitions than with proofs. I like to emphasize this by asking for precise, complete definitions of terms on exams. (And if anyone thinks that’s a tool for grade inflation, they should try it themselves.) I don’t spend as much time as would be ideal on these ideas because time is short and there is subject content to get to.
Charles, you may be interested to know that at Case we plan to introduce a transition course in the fall. I hope that, among other things, it will address precisely the points you raise here.