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What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?

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In view of many of the answers to this question, it might help to have in the statement a definition of heuristic as it is applied to mathematics. – Pete L. Clark Apr 26 '10 at 3:56
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In fact, the harmful entity in most answers is not a heuristic at all! – Victor Protsak May 22 '10 at 15:07

32 Answers 32

The excluded middle ( A Law or an Heuristic) .

On a more general level given any closed question: Is it A or B ? , the heuristic says it is one or the other disregarding the option : the question is wrong or stupid or irrelevant or incomplete.

The principle of excluded middle disregards intuitionist logic. And has been harmful in not providing direct (constructive) proofs which are often more clear - yet can be harder to find.

Intuitionism is is also rather natural : being against anti-communists does not means you are a communist.

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Perhaps more proponents of intuitionism should have readily available examples for the glaring question: what are some natural settings where classical logic is faulty next to an (intuitionistic) alternative. Compelling answers to this question are much scarcer than suggestions to consider intuitionism. – AndrewLMarshall Aug 10 '11 at 23:51
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A topology is an example of a Heyting algebra, not a Boolean algebra. How's that? – Todd Trimble Aug 25 '12 at 20:02

"Teach the subject before its applications."

Some important constructions seem quite pointless until you understand the rationale for them. For example, I recall finding the lectures in freshman linear algebra on constructing Jordan Normal Form extremely boring and pointless until JNF came up in the context of solving linear ODEs a year later. "That's what Jordan Normal Form is for!" - I thought - "I wish I knew that a year ago!"

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As a counterpoint, I never understood Jordan normal form until I learned that it was a special case of the classification of finitely generated modules over a PID. In other words, my difficulty with Jordan normal form came from teaching this application of representation theory before the subject! – Vectornaut May 28 '15 at 22:14
    
Both of your points are true. It is a good idea to bring up the Jordan normal form before the theory of modules over a PID, but it is not at all necessary to teach its proof and the algorithm before the general case of a PID. – darij grinberg May 29 '15 at 0:12

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