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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

33 Answers 33

Two bad principles that taste worse together: Decimals are the true numbers. Rounding makes no difference.

Since students learn about decimals after they've learned about whole numbers and fractions, they might assume that decimals are always the preferred way to represent real numbers, and so everything should be converted to decimals. Meanwhile, since in generally one cannot be expected to write out an infinite decimal expansion, they might assume that stopping after two decimal places makes no difference.

I'm not saying that approximations are bad. But it's bad to approximate if you have no sense of your error tolerance, or even of the fact that you're introducing an error at all.

Here are two perverse outcomes.

  1. Imagine a problem whose answer is, say, $\pi/4$, and a solution that ends like this: $$\text{blah blah blah} = \pi/4 = 3.14/4 = .785.$$ I'm sure that there are some situations where it's important to know that your answer is between $.78$ and $.79$. But much of the time, conversion to decimals obscures what's going on.
  2. (Small sample size alert!) About half of my calculus students will, on the first day of class, mark the equation $\frac{1}{3} = 0.33$ as ``true''.
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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

"A set is a collection of elements".

Firstly, this does not distinguish sets and classes. Next in ZFC, sets are characterized more by their relation to each other by $\epsilon$, rather than that they contain anything.

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I think this an excellent heuristic. If someone asked you what a set was, what else would you tell them? –  Pete L. Clark Apr 25 '10 at 20:23
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-1, as the ZFC axiom of extensionality says precisely that two sets that have the same elements are indistinguishable. –  Qfwfq May 11 '10 at 12:23
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I agree with Colin. In strictly ZF terms, a class is a collection of elements. A set is an element of a collection. –  Tom Ellis May 31 '10 at 13:42
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@Clark: I think "collection" does indeed give the wrong idea. I tend to explain sets as a concept that links items together by properties they have, even if the property is arbitrary. Instead of thinking "$x\in A$" as "$x$ is in $A$", I try to explain "$x$ has a property $A$ and everything has this property or doesn't." Sometimes the property is meaningful: "$x$ is even", or arbitrary: "I'm tagging $x$ with $A$." So the prop. is "$x$ is tagged by $A$." It helps people understand things like dense/uncountable sets better. Also, it departs from the arbitrary "sets only contain unique items". –  Quadrescence Oct 3 '10 at 2:33

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