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In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ordinary background from high school and there is a big difference. actually this gap between them causes some problems in classes. So we thought about this idea of having a class for 'Not Math Medalist students' to teach them how to think mathematically and problem solving strategies and somehow try to fill the gap (for first year students). I was searching for problems that can make them think, not in a special subject, not very hard not very easy. solvable with undergraduate information, and I guess important thing is these problems should be attractive for them to make them think. maybe mathematical modeling or problems without certain answer are good . Do you have any particular suggestion on what problems (or books) can be helpful ?

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    $\begingroup$ I suspect similar questions appear on MSE; one go-to sure to be mentioned is the Art of Problem Solving materials. $\endgroup$ – Benjamin Dickman Apr 14 '15 at 8:31
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    $\begingroup$ Contest-style problems are often wildly different (e.g., contrived and artificial) in contrast to modeling and more real-math problems, so clarity about which direction you want to go is important... $\endgroup$ – paul garrett Dec 26 '15 at 20:34
  • $\begingroup$ You need to define precise objectives. What do you want your students to be able to do at the end of the course? What do they bring as previously learnt? What are their principal troubles? Just the (too common) "math is too hard for me" and giving up before trying? $\endgroup$ – vonbrand Dec 28 '15 at 12:01
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'Thinking Mathematically' by Mason, Burton and Stacey sounds like a good match. It has a large collection of problems/investigations using high-school level maths, and discusses how to go about the thinking process of doing maths. It isn't about contest maths, but my understanding is that contest mathematics is not the same thing as university maths (some people are good at one and not the other, so some of your first group might also benefit from your proposed class).

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I feel that undergraduate mathematics is almost a calculus grind from start to finish for nearly every group of students (that aren't explicitly mathematics majors). Introductory discrete math (sets, maps, etc.) offers so much more for students to grasp than continuous calculus from start to finish (as is the norm).

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  • $\begingroup$ True, but that's not a feature of the subject matter itself, but only of its "use" in the established situations... Calculus=filter. Other entry-level math has not been institutionalized as filter/weeder at this date, at least in the U.S. And, further, calculus itself is significantly subverted (maybe perverted) to function in this "weeder/filter" role, considering that it is one of the more elevating and explanatory discoveries of humans in the last 400+ years... but is turned into a club to beat 20-somethings to filter toward future careers? What does this have to do with science? Ugly... $\endgroup$ – paul garrett Dec 27 '15 at 0:14
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Perhaps the OpenStax textbooks (discovery-based, available for free, made up of independent chapters to mix and match) serve as a starting point.

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  • $\begingroup$ I've investigated the OpenStax books pretty deeply, I like them a lot, and long-term I'm thinking about how to transition my courses to use them. But I would not call the ones that I've seen "discovery-based" (thankfully). $\endgroup$ – Daniel R. Collins Dec 27 '15 at 23:45
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One of my favorite question for this kind of situation is: "what is the triangle with least perimeter among triangles of given area?". There are several ways to attack it, problem reduction is a must, solutions may use geometry, analysis and basic topology, and it has so many follow-ups ("what about quadrilaterals?") that you can feed student's curiosity for years, and more ("what about domains in a 5-dimensional simply connected manifold of non-positive curvature?" is an open research question).

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