Lesson Six: Embracing Challenges

Now that my senior project is coming to an end, I have been looking back and reflecting on my mathematical journey, both over the past month and in my secondary education in general. One thing I realized about my experiences was that when I was challenged I did my best and learned the most. This sounds simple, but there's a lot to be gained from this.

Just like what Polya said in his book about the "right" difficulty of problems for a young student, there is a certain range of problems I find ideal. For the purposes of this post I will divide all math problems into three categories.

1) Trivial/Easy
2) Challenging, but solvable
3) Impossible

Guess which one I am going to advocate ;)

Seriously, the majority of math problems I have done in my life have been easy. You apply a formula/theorem, do some rote calculations, and you win. After doing a certain number of near identical problems, there is nothing more to be gained from these types of problems. Fortunately, math curriculum in a school moves along fast enough that you do not spend an entire year doing the Quadratic Formula or applying the Mean Value Theorem. Especially in a topic like calculus, many of the problems are of this nature simply because the AP curriculum focuses on computational calculus much more than the theory behind it.

The other extreme, if you will, of a problem is an impossible problem. While there are few truly impossible problems, there are many problems that are impossible for a given student given his/her background and knowledge. There is a reason why 9th graders aren't taught Calculus, forget something like Differential Equations or Topology. They simply are not prepared for the material. A key aspect of an impossible problem is that the student does not have the facilities to solve it. Even my beloved, mathematically inclined best friend Daniel Heins looks at certain problems and knows he cannot solve them. In Polya's book, he argues that it is the responsibility of teachers to avoid presenting students with problems they have no legitimate chance of succeeding on. He also believes that teachers are responsible for introducing the ideal type of problem, a challenging one!

Describing what exactly a challenging problem consists of is no cake walk. However, there are some key themes.

1) The problem should take more than a minute to solve.
2) The problem should blend already known things with unknowns.
3) The problem's solution should contain some insights into similar problems in general.
4) The problem's solution may require the student to apply previous knowledge in a new way/method.
5) The problem should challenge the student enough to foster ingenuity and creativity without rendering the student helpless in the situation.

If the student has never worked with modular before, then asking what 17 mod 3 equals is pointless. However, after a few mod problems have been reviewed, a challenging yet achievable problem would be converting something in mod 3 to mod 7 or something.

I wish I had openly tackled more of the challenging problems I was given in high school. This past year, in Mr. Markey's calculus class, the last two/three problems tended to be open ended and somewhat challenging. Being the immature student that I am, I toyed with these problems for a minute or two before giving up, as I had already completed the rest of the problem set.

Only through challenging yourself can you find the extent of your knowledge and continue to improve. There's a reason the Lakers made it to the finals, and it wasn't because they played high school basketball teams. They played a lot of teams that challenged them and forced them to improve. Why not challenge yourself in math? Find these challenging problems, try them, fail at them, and try them again. Then talk with a teacher/friend and find out the solution. You often learn just as much by struggling through a problem and not solving it as you learned from solving it, as you can find out where your mind failed you and whether or not your approach was on track or not.

The rest of your life is full of challenges, some of which are too easy, some of which are too hard, but most of which are difficult but achievable. Why should your mathematics experience be any different?