Recently, several parents of MIST Academy students have asked me to give study advice to their children to help them prepare for the upcoming state MATHCOUNTS competition. For those not familiar with MATHCOUNTS, many educators (myself included) consider MATHCOUNTS to be the premier middle school problem solving program. In fact, I build a considerable number of my Introductory Problem Solving class lessons with the MATHCOUNTS curriculum in mind.

As a student, I enjoyed considerable success at the national MATHCOUNTS competition. Each year I participated I missed no problems or one problem. Yet I still find giving advice as to how to prepare for MATHCOUNTS difficult. Most of the important advice I want to give is the same advice I give during any given class as to how to study mathematics and prepare for anything important in life. I have tried my best to separate advice into a few convenient categories:


1. Work Problems. There is no substitute for practice. But it wouldn’t be fair if I didn’t divide this most important point up into subpoints:

(a) Think about what you did to solve each problem. Mathematics is about exploration. Along the way to solving each problem, you have to explore various concepts, techniques, and aspects of many areas of mathematics. Spend time consciously identifying all of them. Then use your imagination to dig deeper each of them.

(b) Work Harder Problems. Working harder problems forces you to do more of what I recommend in subpoint (a), and is more likely to lead you into areas you need to explore in order to solve the hardest problems that might appear at MATHCOUNTS. I highly recommend AMC 10 problems for students who have high, but not perfect scores at MATHCOUNTS.

(c) Work Fun Math. Work the problems you love the most — the ones that give you the energy to continue moving forward.

(d) Focus On Your Weaknesses. When the problems you hate the most — the ones that give you the most trouble. If you are a student with a chance of making the countdown round, there may be no better way to improve your ranking.

(e) Practice Without Pencil. By this I do not mean just practice mental arithmetic. I mean that you should see how many of the problems you can learn to unravel in your head. With MATHCOUNTS problems, this is nearly always possible when you have mastered the nice way to approach the most common problem types.

(f) Teach Others. If you can’t teach a concept, you don’t know it well enough [to get all the hardest problems]. Teaching forces you to organize ideas that you know only well enough to get right answers. Right answers are not everything. If you want to solve the hardest problems at MATHCOUNTS, you need to know the concepts forward, backward, inside-out, and in your sleep. Perhaps more importantly in life, when you cannot communicate an idea, you cannot make much value out of it.


2. Develop Number Sense. There are educators who dismiss concepts of number sense as nothing but a worthless bunch of tricks. I strongly disagree and would go so far as to say that a great deal of the number theory I know is a result of exploring and developing my own number sense.

In regards to competitive test taking, number theory can help you solve number theory problems. But just as importantly on a timed test, it gives you a significant time advantage, and when your number sense gets good enough, that edge can become dramatic. That extra time allows you to spend more time on the harder problems, and checking over your work. You might not solve many of the five hardest Sprint Round problems if you get to them with three minutes left. But if you get to them with fifteen minutes left, you have a chance at solving them all!

Here are a few of the things I did as a student to help develop my number sense:

(a) Practice factoring integers. As often as possible, I did this without pencil and paper, though serious exploration often does involve pencil and paper. When you’re bored in the carpool line, factor the numbers on the car tags in the parking lot.

(b) Explore algebraic relationships between integers. This one can be at first hard to explain. An example is using algebra to improve your arithmetic skills. For instance,

104 x 107 = (100 + 4)(100 + 7)

= 100(100 + 7) + 4(100 + 7)

= 100 x 100 + 100 x 7 + 4 x 100 + 4 x 7

= 10000 + 700 + 400 + 28 = 11128.

I plan to make a more instructive post about algebraic arithmetic in the future.


3. Learn to pay great attention in your studies to the “messy” problems. If you don’t know what I’m talking about, think about those problems with more than a dozen criss-crossing lines drawn, and you’re asked to find the total number of triangles. There are other kinds as well — problems that don’t fit well into traditional problem types — problems for which you have to invent your own approach on the spot. It is a rare student who gets most of these problems right. It’s easy to miss something, so the key to working through these problems is to find better and better methods of organizing the way you think. Sometimes this means casework, where your success often depends on how well you define the cases.


4. Prioritize. You don’t regard all the problems on a test as equally hard. You know better. As you work through the problems, you know which ones will give you more trouble. These are the problems you need to be more careful with. These are the ones that require something more than the haphazard scratch work that gets you through the easy ones.


5. Incubation — Learn to Skip Problems. I don’t mean that you should skip them and never come back. Work the easy problems first. When you read a problem, and feel less confident that you’re going to find a nice method of attack within a minute, skip over that problem and come back to it later. Scientists tell us that these problems do rattle around in our heads even when we’re not thinking about them. From experience, I know this to be true. Sometimes, a fresh look at a problem that seemed tough before is enough. A fresh pair of eyes may have that flash of insight necessary to find a correct solution.


6. Finally, I feel the need to make this point one last time: Work Problems! Work lots of problems. Work the problems that push your limits.


I hope to create a PDF document out of this post sometime, and add to it from time to time. For now, I hope it is of some benefit to students preparing for the state MATHCOUNTS competition.



Edit: Recently I’ve been putting hundreds of pages of lessons with problems and developed solutions online at Gliya, including a MATHCOUNTS preparation node of the Math Nexus.

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