The Problem with “Think Really Hard, Struggle for a While, Eventually Solve it or Look Up The Answer” Problems

by Justin Skycak on

Challenge problems are not a good use of time until you've developed the foundational skills that are necessary to grapple with these problems in a productive and timely fashion.

In general, a lot of people fall into the trap of thinking that to train up their math skills, they should be focusing on the hardest problem types (like competition math problems).

But here’s the thing about “think really hard, struggle for a while, eventually solve it or look up the answer” problems. They can be fun (for a certain type of person), but they’re not an efficient way to learn.

Approaching challenging problems without having the procedures down pat is like jumping into a game of basketball without having developed dribbling and shooting skills.

It might feel fun but you’re just going to be whiffing every shot and getting the ball stolen from you. You might make one layup the entire game & feel good about it, but that’s barely any training volume.

It’s like going to the gym to lift weights but only eeking out a single rep over the entire course of your workout. You need to be banging out more reps if you want to get stronger, and the only way you can bang out those reps is by working with a level of weight that’s appropriate for you.

Math is the same way. In an hour-long session, you’re going to make a lot more progress by solving 30 problems that each take 2 minutes given your current level of knowledge, than by attempting a single competition problem that you struggle with for an hour.

(This assumes those 30 problems are grouped into minimal effective doses, well-scaffolded & increasing in difficulty, across a variety of topics at the edge of your knowledge.)

Now, I’m not saying that “challenge problems” are bad. I’m just saying that they’re not a good use of time until you’ve developed the foundational skills that are necessary to grapple with these problems in a productive and timely fashion.

Lastly, I want to point out my claims here are not “philosophy” so much as “science.” They are grounded in decades of research into the cognitive science of learning.

Research indicates that the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain.

In other words: the way you increase your ability to make mental leaps is not by learning to jump further, but by building bridges.

As Sweller, Clark, and Kirschner sum it up in their 2010 article Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics:

  • "Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.
    ...
    In short, the research suggests that we can teach aspiring mathematicians to be effective problem solvers only by providing them with a large store of domain-specific schemas. Mathematical problem-solving skill is acquired through a large number of specific mathematical problem-solving strategies relevant to particular problems. There are no separate, general problem-solving strategies that can be learned."