Training vs. Competing12 Dec 2015
Day after day after day I see parents and students who think that the cornerstone of preparation for a math competition is to try again and again to solve past contest papers. In any case, a lot of people think that a worksheet is simply not appropriate for preparation if it is not in the exact format of the competition and does not resemble the competition paper in almost every way. This is one of the fallacies encountered the most frequently on the road for better performance in math competitions. While I agree that past competition papers are excellent training material, I also see in practice how students and parents use them with little benefit.
Let me explain the difference between the training and competing processes, so that we can begin to have an understanding of what constitutes appropriate study materials.
As any tennis coach can tell, one cannot make progress in tennis by playing with the pros. First you must train at the wall, then you must do warm-ups and athletic training, push-ups, roll-ups, running, etc. - all kinds of training that is not tennis per se, but is meant to help one make progress in tennis without getting hurt. In mathematics, for some reason I cannot understand, everyone seems to think that one should train by trying to solve contest papers over and over. Every parent I meet knows better than me what to do and is generally dismayed that I do not re-test over and over from the same past contests. They believe, for some completely inexplicable reason, that whatever we teach that is not directly sourced from past contests is a waste of time and money.
In any sport, you need to train the muscles all the time, regardless of the sport you play. In mathematics, you need to learn theory all the time. And re-learn it. Again. And again. Every time you solve a problem you must have a proof anchored in theory for each decision you make. Theory inspired the author of the problem and it will also inspire you. The author invented a really neat application of the theory and you must invent a really neat solution based on the same theory. Therefore, you must know theory. When do you know if you know theory? When you have a creative understanding of it. Theory is where the author of the problem and the solver meet on common ground. So learning theory is not a waste of time, or a “useless” activity, it is actually the cornerstone of problem solving.
What considerations lead some students and parents to believe that theory is “useless”? I would guess that it’s because theory has already been discovered, found, understood, and proven by others. Students and parents believe that theory can always be found in a book - there is no need to waste time internalizing it. Wrong! Theory is our first and best example of problem solving. Any theorem is the solution to a problem. That problem was so important and so general that people felt it was important to promote it to the status of a fact to be remembered and used. But, to start with, it was a problem and its solution, even if it is well known, can be a great source of inspiration. You cannot be inspired by a fact that is buried inside a book. You need to have it in your memory and you need to have experienced its power in successive applications.
How much theory should a student learn? Now this is a tricky question. The field of mathematics is very, very big. If you attempted to study it from the top and kept on learning facts, your life would end before you had learned even a very small fraction of it. However, mathematics is also very well connected logically, so very many of its facts are consequences of other facts. This means that one can learn a relatively small number of key facts and strategies in order to be able to operate in a comparatively large set of math problems.
This is something that we seem to have forgotten. Consequently, we see a trend towards “memorization and direct application” instead of towards “covering as much as possible by learning how to connect key facts.” What we seem to miss, other than the very essence of mathematics, is the fact that we do not empower the student to create, we merely turn him into a clerk who cannot do more than a robot can do.
Therefore, we need to know: some key facts and how to get from fact to fact using logic. It’s a bit like rock climbing. That’s where your coach comes in handy because, presumably, he knows how to guide you both in the selection of key facts to be internalized and in how to develop the ability to walk the connections between facts.
Training involves learning theory and practicing problem solving using the largest variety of problem statements. In this way, we train the creative response to the problem at hand. When solving problems, a lot of time must be put into clarifying the statement to the utmost, then into connecting it really well to the theory that inspired it, then finally into making the execution of the idea simpler and simpler, more and more efficient. During training, it may very well be necessary to work on a problem for a whole hour - an hour in which a large number of skills are learned and practiced by the student.
Some students (and parents) feel that we should get to the “right answer” faster and move on to a new problem. Yet the “right answer” to the problem is to learn as much as possible from the statement and solution, i.e. walk and re-walk all the connections that lead to and from the problem.
When training we have to explore more territory from each problem outward. We ask questions about how the statement could be changed, why the numbers have been selected the way they have, what happens if we change them, etc.
Training involves re-deriving a lot of theoretical facts, often from different perspectives. We train to acquire mobility through the space of key concepts that we have learned. Parents often equate the “knowledge” of key concepts with the foundation of problem solving. This is probably the reason for which they insist that the student who graduated from an algebra course with an A is fit for competition and merely needs to polish up a little on the types of questions asked. But having the key concepts is like having a skeleton without muscles - it cannot move from place to place! We have to train the mobility in this space, the ability to connect these key facts and to use logic, and nothing but logic, to patch up and invent on the spot minor concepts that we may not have encountered yet. It is not just a little bit of polishing up but a whole new growth phase!
Competing is a different activity. It involves solving quickly, accurately, and independently of possible distractions that may occur in the environment. To compete successfully one has to be a good problem solver but also an experienced test-taker. Your coach can help you by explaining which specific test-taking behaviors need improvement. The complex set of behaviors that control the performance on a test is not directly connected to the level of competence in the subject matter. Generally, the more confident a student is in his/her ability to compete, the better the performance. But confidence is not everything. In practice, we find that experience competing is very important and helps develop time management skills, accuracy in computations, extremely good focus, and stamina. Therefore, we do recommend participation in all the competitions available in one’s geographic area. The goal is not “to win” them, but to train those test taking skills in the field.
Training and competing are, of course, both necessary. But they are different. One has to understand the difference and train properly in order to compete successfully.