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One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics in the future they can develop that skill to use concepts from different areas outside of mathematics to solve global challenges in mathematics that was already achieved by great mathematicians in the past. As an example famous squaring the square problem was solved by applying Kirchhoff's circuit laws in physics .What are the examples you have relevant to fulfill this requirement not in mathematicians level but within the scope of students level ?
When you talk about geometry, you can find problems which can be solved using Geometry, Coordinate geometry, Trigonometry, Vectors and Argand diagram. In combinatorics you can use formula to simplify or combinatorial arguments where students have much freedom to think creatively. What would you suggest as problems to give this exposure to advanced level students ?

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    $\begingroup$ I would hope that every mathematical experience would include multiple approaches and diverse perspectives from the different participants! $\endgroup$ Mar 20, 2023 at 17:39
  • $\begingroup$ Yes of cause, but here I'm specifically concerning about alternative approaches from different areas rather than within the same topic. As an example Pythagoras theorem you can prove in more than 200 different ways but many of those by plane geometry itself. When you use dot product of vectors you have one example for the stated purpose. $\endgroup$ Mar 20, 2023 at 23:46
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    $\begingroup$ What exactly is the "advanced level" you are talking about? $\endgroup$
    – fedja
    Mar 21, 2023 at 5:45
  • $\begingroup$ @fedja may equivalent to high school K 12 level in USA or Advanced level in UK. $\endgroup$ Mar 21, 2023 at 11:06

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It just crossed my mind that I can offer you some option you haven't probably considered yourself.

Once I experimented in my calculus course (which also involved some elements of analytic geometry and such; usually it is called Calculus 3 or Multivariate Calculus) with the idea to give the students some common sense tasks for which mathematical thinking would be useful

  1. There is a brown electric pole right in front of the entrance to math. department building. Estimate its height with decent precision (telling the answer, the approach and the way to bound the error).

  2. There is a pine tree in the back of the math. building. What (approximately) is the number of the needles on it (same format of submission). An easier version of that is to print a few thousand uniformly distributed points on a sheet of paper and ask to estimate their number. The pine tree was not a perfect rectangle or parallelepiped and the branches on it had different lengths and varying distribution density, so the best way to analyze it (IMO) was to take a few snapshots from various positions and do some measurements on the pictures, plus, of course to approach the lowest branch and to investigate it (it was hanging low enough).

  3. Here is a reprint of the photograph of some electric line going straight to the horizon with a shed somewhere near it. a) estimate the shed height (assuming that the photographer took the picture with a hand camera and was an average size adult) b) Are the electric poles on the picture of essentially the same height or there is a noticeable difference in height between some of them? (both with an explanation of how you did it and why you think your approach works)

The problems were assigned in the beginning of the course and were due any time before the semester ends. They were for extra credit (30 points each, with each of 2 midterms worth 100 points and a regular weekly quiz 10 points). Almost nobody took the challenge so I abandoned the idea and tried something else next time I taught the same class, but it still may be a good source of problems allowing (or sometimes even requiring) a choice between or a combination of various approaches and neither of those problems requires anything a smart high school kid would not know or not be able to derive from first principles.

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  • $\begingroup$ Yes we can't expect average students to know the value of this kind of work because they are lacking in sense of mathematics but as educators who have vision of their work , first we need to realize the importance of this. $\endgroup$ Mar 23, 2023 at 11:16
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At the college level, see here, here, here, here, and here. Only the first link is suitable for high school courses.

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  • $\begingroup$ Quite interesting, yes as you mentioned only part of first case is helpful for high school mathematics. $\endgroup$ Mar 23, 2023 at 11:09

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