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I've been trying to teach my sister school maths, and one difficulty I find is, she is unable to state precise formulation of theorems, and sometimes confuse the assumption and the implication. This is not much of a problem because often the theorems in her life are two ways, the implication goes bothways, but generally this is not good.

I too remember personally have these problem when I was young. How can I help her/ other students generally the importance of remembering theorem exactly and also how to remember theorems exactly?


In my personal opinion, I think it maybe rooted from the lack of seriousness when defining the basic terms. For eg, we don't define what an integer or real number is in school, we just give an intuitive explanation which has to be rememberd. So, when suddenly a student reaches a class that they have to memorize exact theorems involving these objects, they take it with the same seriousness as they took the basics of it.

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    $\begingroup$ Give her some funny elementary logic book to read (Smullyan is the best and the funniest, just use your own judgement which one of his masterpieces is most pertinent to the situation). This will certainly teach her how the standard logic works and it is half the way (the other half is actually understanding what the theorems are about and why they are true; memorization of the statements is pointless and inefficient without going over the proofs). $\endgroup$
    – fedja
    Commented Mar 21, 2023 at 2:16
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    $\begingroup$ There is no need to remember theorems exactly. How old is she? Is she enjoying math? That's what's most important, I'd say. $\endgroup$
    – Sue VanHattum
    Commented Mar 21, 2023 at 3:52
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    $\begingroup$ I think your concern is misplaced. $\endgroup$
    – Sue VanHattum
    Commented Mar 21, 2023 at 16:19
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    $\begingroup$ I can't recall a class where I had to exactly memorize a thereom through all of my education as an electrical engineer. But I am not a mathematician. Then again, neither is your 13 year old sister. $\endgroup$
    – DKNguyen
    Commented Mar 22, 2023 at 15:18
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    $\begingroup$ In Indian curriculum, there sometimes come in tests question which ask students to state theorems. Eg in 10th grade, the fundamental theorem of wrthimetic @DKNguyen $\endgroup$
    – Babu
    Commented Mar 22, 2023 at 21:57

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If the error is actually important, than I would gently try to ask a thought provoking question to make her be more precise. So, for example, if she says a square is a shape with four equal sides, than you could draw a rhombus and ask what about it. I would only do that as one part of the discussion though. In other word, not only being critical. Find some other things to do, that are positive and that further the basic concept (not just the definition correction). E.g. "find four squares in your room". And do the positive stuff first.

I would also recommend not to push for precision in wording if it actually doesn't matter, here. I agree with you that removing a bullet from a man's heart will not revive him. But if we are very young and naive, it is probably more important to just learn what the import of the bullet is, not dwell on irreversibility. Learning is a progressive and iterative process. Emphasizing aspects that are not critical now, will distract from learning the initial topic.

There is a mistaken belief that difficulties with rigor (much) later can be corrected by emphasizing it (much) earlier. But this is mostly an error. Complexity is inherently difficult, so it's a false hope that we can avoid all later pain. "I coulda been a contender!" Also, pushing it too early (especially when not pertinent) may be too much medicine and too little sugar and derail the learning process.

See https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&context=hmnj (especially the section called developmental psychology).

P.s. I would also recommend not to dwell purely on definitional rigor and neglect implementation in calculations. Humans learn by practice. And it is very unreasonable to think that having perfect definitions, is enough or will result in skilled calculations and the automaticity of manipulations that is needed for later work.

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