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.
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Lots of people fall into the trap of thinking that to train up their math skills, they should focus on the hardest problem types right from the start.
But here’s the thing about “think really hard, struggle for a long time, eventually solve it but usually 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 eking 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.
(That said, it’s not uncommon for a teacher to mistakenly think that their students have the foundational skills down, when they actually don’t… so that’s always something to watch out for.)
Lastly, I want to point out my claims here 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.
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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.”
But what about “productive struggle”? Isn’t it true that many highly skilled professionals spend a lot of time solving open-ended problems, and in the process, discovering new knowledge as opposed to obtaining it through direct instruction?
Yes, but that doesn’t mean beginners should do the same. One key empirical result is the expertise reversal effect, a well-known phenomenon that instructional techniques that promote the most learning in experts, promote the least learning in beginners, and vice versa.
The expertise reversal effect suggests that beginners (i.e., students) learn most effectively through direct instruction – and here are some quotes elaborating why:
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*“First, a learner who is having difficulty with many of the components can easily be overwhelmed by the processing demands of the complex task. Second, to the extent that many components are well mastered, the student will waste a great deal of time repeating those mastered components to get an opportunity to practice the few components that need additional practice.
A large body of research in psychology shows that part training is often more effective when the part component is independent, or nearly so, of the larger task… Practicing one’s skills periodically in full context is important to motivation and to learning to practice, but not a reason to make this the principal mechanism of learning.” — Anderson, Reder, & Simon (1998) in Radical Constructivism and Cognitive Psychology*
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“These two facts — that working memory is very limited when dealing with novel information, but that it is not limited when dealing with organized information stored in long-term memory — explain why partially or minimally guided instruction typically is ineffective for novices, but can be effective for experts. When given a problem to solve, novices’ only resource is their very constrained working memory. But experts have both their working memory and all the relevant knowledge and skill stored in long-term memory.” — Clark, Kirschner, & Sweller (2012) in Putting Students on the Path to Learning: The Case for Fully Guided Instruction
And some other references:
- Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching
- Should There Be a Three-Strikes Rule Against Pure Discovery Learning? The Case for Guided Methods of Instruction
Note: If you liked this article, you may be interested in The Pedagogically Optimal Way to Learn Math.
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