Sabio Column

The Importance of Solving Hard Problems

By James H. Choi
http://column.SabioAcademy.com
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As in sports, mathematics has two types of exercise:

1) Repetition, which refines your performance on key tasks (equal to practicing free-throws in basketball).  Routine practice like this is familiar to students, whose math homework usually contains many easy problems.

But students aren’t well-prepared to handle hard problems; their assignments typically skip the most challenging problems at the end of chapters.  The net result student expect problems to be solved quickly.  They expect to shoot 100 free throws and then be done.  They expect to do just 30 minutes of exercising.  If they miss a free throw, they can easily correct the mistake.  They will expect all future problems to be similarly easy.

But repetition is not enough to build math skills.

Math’s second second exercise is a lot like weight-lifting in this regard:

2) Challenge.  When you lift easy weights dozens of times a day, you will never build muscle.  To build muscle, you must try to lift something that is beyond your capability.  Only then will your muscles break down and, while you sleep, rebuild themselves stronger.   It is 100-times better to try lifting an anvil than it is to lift your ball-point pen 100 times.

To develop your brain and make it smarter, you must do mental weight-lifting.  School curriculum generally lacks mentally weight-lifting challenges.  So when students begin studying for math competitions, they become fast discouraged by the difficulty of problems.  It’s as if I’ve taken a scrawny boy and told him he must bench-press 200 pounds to win a competition.  He hasn’t been trained.  He will immediately give up.

Some students get personal trainers to help tutor them in learning to solve these math problems.  But if you got a personal trainer for weight-lifting, they must not lift the weight on your behalf.  So the math tutor cannot solve the problems for you.

Students often get discouraged after attempting hard math problems without any perceivable progress.  They want the teacher to solve the problems.  But extend the analogy: When you lift a weight well beyond your capacity, it doesn’t lift off the ground, does it?  Yet your muscles get stronger anyway.  This is the imperceptible gain the students must understand:  Your accomplishment does not come from lifting the weight, or solving the problem.  Your accomplishment comes from exerting your muscles, or brain, and forcing it to strengthen.

When you engage in this type of learning, it’s essential you let your brain recover.  Let it rebuild, reconnect its neurons, process all of what you just attempted.  This is done while you sleep.   Although sleep-deprived students can study rote-memorization topics, this type of brain-building that helps you master challenging concepts can only succeed if you sleep.

Nobody practices for a marathon by running all night the day before a race.  The same goes for you the night before a math test or competition.   You need to sleep well before either of these.

So do not be afraid of hard problems!  Face them.  Struggle.  Do this without any apparent gain.  And then sleep well.  During the process, your brain will become smarter, moving up to a higher IQ level.