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Part of learning to do higher dimensional geometry is learning which aspects of your low dimensional intuition are good or not. I can't visualize two smooth $2$-planes meeting transversely at a point, which I know happens in $\mathbb{R}^4$. But I am experienced enough to know that, when thinking about dimensions and transversality, I need to compute, because ...


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The question of when and why less guidance works or doesn't work is far from obvious. Instead of providing my personal opinion, as many have already done in other answers, I will try to collect some pointers to research that has already tried to answer this question, which you can use to critically review your own approach. Andrew Blair in his article ...


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Over all I think the idea, of hands on for something like C/D is an invariant is laudable, but there are a few caveats. First, students at that age are very aware of precision. The answer "the ratio is 3 and a bit" is fine, but they will not accept it is a constant if one group measures 3.1, the second group 3.2, a third group 3.15. Indeed, in that ...


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I agree with the point in the other answers, that the difference is in the approach, how rigorous we would like to be. However, I would like to also mention that this rigour in discussing elementary geometry is not extended to some other parts of teaching mathematics in high school. I do not think that too many high school teachers discuss vectors in a ...


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I can tell about how it went in Russia, presumably, that might have gone the same way in other countries as well. Traditionally, congruent shapes were called "equal". Then, in the 1960's, it was felt that some set-theoretic notions should be introduced into the school geometry, a charge led by leading mathematicians such as Kolmogorov. Since there'...


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A good think is to students pick up several rounded objects (wheels, cans, glasses, etc.) and mesure with rule the length and the diameter and simply divide. And observe if something happens. You can see this idea here (in Catalan - automatic English translation). It is original from Anton Aubanell. Just a note: for measuring the length I think the better is ...


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