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I need to compile a relatively short "Crash Course" in high school mathematics packet, whose aim is to be a quick learning guide for academic competition (think Aca-dec) for students with mixed mathematical backgrounds.

I know (to some degree) what topics I want to cover in the crash course, but I am not sure what format would be the most useful for learning large amounts in short times (these are meant for prep in about two weeks)*. I was thinking of doing a "Problem-Solution-Explanation" (Henceforth PSE), in order to let the reader become the most accustomed to the Problem-Solution format of the competition. However, I know in standard pedagogical writing this is bad form, and intuition helps solidify concepts more than PSE, and so I am not sure.


*I know from a pedagogical standpoint this is nowhere near the most useful methodology, and I would love to write a whole textbook intuitively explaining high school math, but in this case it is what is necessary.

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I'm not sure if this is helpful, but the Art of Problem Solving Textbooks are written in a style similar to the one you seem to be proposing, and they are excellent.

http://www.artofproblemsolving.com/index.php?mode=books

The Problems are chosen to guide the students to discovering the mathematical principles that the section aims to teach. A set of problems is given, then each problem is solved with a discussion of the important features of the solution. Often a Key Concept is summarized at the end of the solution. There is no other explanation. These are followed by exercises, and more difficult review and challenge problems at the end of the chapter.

These texts are written with gifted math students in mind, and many of the problems are taken from math competitions. The texts are indeed frequently used by students studying for math competitions.

Of course these are complete textbooks and not a "crash course" but my experience with these textbooks is that this problem/solution approach to introducing topics is very effective for the types of students who are typically interested in math competitions.

I would suggest using both simple problems to motivate and illustrate the topic, and then provide some challenging problems that use or extend the topic, so that the students can see how the various topics might get applied creatively in a competition problem.

If you write it, I'd love to see what you come up with!

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In general, look for preparation material for mathemathics olympiads. There are a lot around, most oriented at covering lots of ground quickly, targeted at gifted students and solving hard problems. –  vonbrand Apr 11 at 10:06
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This should be a comment, but it came out too long.


I think that doing several years of math in two weeks is a hopeless endeavor, mainly because students need time to digest the content. Some successful approaches to math workshops that I am aware are as follows:

  • review: the students already learned the material and now we are only refining their understanding;
  • idea-overstuff: the teachers cram as much different ideas into students' minds and then the pupils digest it for the next six months;
  • limited-content: the workshops cover one aspect in detail, like contest math geometry, counting arguments, etc.;
  • training-camp: there is no general idea, the sole purpose is to gather students in a place free of distractions, so they can prepare for the incoming competition;
  • social event: math is secondary, we want the students to have good time and find peers which share similar interests (for example, it's not easy when one lives in a small town).

It seems that none of the above fits your aim. Could you explain the context a bit more? In particular, what is your purpose? What are the prerequisites, what is the desired outcome?

I hope this helps $\ddot\smile$

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It is not really expected of them to learn years of math at once, but I want it to be tool so that any student (whether at algebra one or pre calculus) can learn a few more types of problems to solve so that they can improve their score. –  Juan Sebastian Lozano Muñoz Apr 11 at 12:47
    
You may do better with sample problems of types that typically come up than with "basic" curricular stuff. I have a sheet for middle school math competitions that goes over a few "typical" problem types, with that very goal, to help any student pick a bit of low-hanging fruit and do a bit better on the contests. –  PurpleVermont Apr 11 at 16:55
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