I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.

Now this is my problem, there are few mathematics books written in English that are at the level of high school, most are geared towards graduate school.

The book for the school is very dry and basic. I would like to be able to present better problems for my classes. I need help finding good problems, ones with some historical context or from real life events. Example: using a closed Leontief input-output model for linear algebra.

Or something along this line:

The quality of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility S varies as a function of the water temperature T. (a) What is the meaning of the derivatives S'(T)? What are its units? (b) Estimate the value of S'(16) and interpret it.

with graph

  • From Calculus (6th Edition) by James Stewart.

My other problem is that I've got a group of 6 students that are übersmart. I gave them the AMC 10 & 12 problems, they solve them in class, then I gave them the USAMO problems, and they solve 4 problems in an hour (I gave them both days). Should I just give them the Putnam exams? Now, English textbooks cost a lot and I have to buy them out of my own pocket. Could you help me find online resources which are at the level of a high school student that is just learning calculus and which help him/her understand the more advance mathematical topics? Most I have found are just beyond their mathematical understanding.

The other 24 students are of lower ability than those 6 students.

In summary:

1) Where to find more realistic and historical problems for calculus and prob/stats classes? To help engage my students and improve their English. The graphs and data for the problems, don't worry I can solve it for the answers. Problems that bring in the other subjects : history, economics, etc.

2) What to do about the übersmart students? How to structure the class so the übersmart students don't get bored, but I don't leave the rest of the students behind?

3) Are there any books that are really graduate analysis, number theory, etc. however, are written for high school students?

4) What's your favorite word problem?


closed as primarily opinion-based by Joseph Van Name, Sebastian Goette, Lucia, Yemon Choi, Henry.L Oct 12 '17 at 23:45

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Would en.wikipedia.org/wiki/Heat_equation be realistic enough? The idea would be to use en.wikipedia.org/wiki/… as an introduction to Functional Analysis. (That would probably only work for your top 6 students.) $\endgroup$ – user5810 Jan 21 '12 at 8:04
  • $\begingroup$ I think that is great great great question !!! I wish you success in your important work and hopefully there will be some good answers. I also thinks that current math educating books are "dry and basic". However it might be a real problem to find some book showing the beauty of math and its importance for real life... $\endgroup$ – Alexander Chervov Jan 21 '12 at 11:21
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    $\begingroup$ Should this be Community Wiki? He's asking for a list of problems and books, as well as advice about teaching. Seems like you could end up with very different but equally "correct" answers. $\endgroup$ – David White Jan 21 '12 at 14:57
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    $\begingroup$ Also: I retagged. "Word Problem" refers to a famous type of problem in group theory, not to math problems which come from paragraphs of text. Here's a wikipedia link: en.wikipedia.org/wiki/Word_problem_for_groups $\endgroup$ – David White Jan 21 '12 at 14:59

At the risk of self-promotion (is this an MO no-no? if so, I apologize), my book A Friendly Introduction to Number Theory is quite well suited for bright high school students, while at the same time giving them a taste of real mathematics. I know that the book is used in this country at some high schools, because I've had teachers write to me and tell me that they're using it.

Even if you or your students don't want to purchase the book (which is fine), the first six chapters are available for free at


So at least your students can use these 6 chapters to get started and to see if they enjoy studying number theory.

  • $\begingroup$ If promoting yourself is a side effect of answering his question,no harm nor foul. I recommend editing your answer to make it look more that way, either by addressing his whole question (what does your book do about the other 18?), or sympathizing and giving a partial solution (e.g. "I've been there, and decided to write a book for half the students..."). $\endgroup$ – The Masked Avenger Sep 22 '13 at 23:10

with your übersmart kids, you might try a method used in Russia for many years to teach gifted in maths kids: give them problems to solve, which guide them through mathematics. Here you can find some such material, translated into English (I did a bit that translating, actually, when I had a similar 1st year undergraduate class on my hands 6 years ago): http://shenme.de/listki/

Well, you might call this material a bit dry, but it's certainly not too basic.


About 1), maybe you could try http://en.wikipedia.org/wiki/Logistic_function



It might also solve 2) because there are a lot of possible developments so you can structure a problem with easy questions first and a few harder ones, opening on fancy math stuff like chaos, fractals and the like in the end.


2) Give them http://www.amazon.com/Hungarian-Problem-Book-IV-MAA/dp/0883858312/ and previous volumes. If they are done with it, try selected problems from http://www.amazon.com/Contests-Higher-Mathematics-Schweitzer-Competitions/dp/0387945881

3) Try the following book written for (and read by) high school students: http://www.amazon.com/Topics-Theory-Numbers-Paul-Erdos/dp/0387953205


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