Who Needs Worked Examples? You, Eventually.

by Justin Skycak (@justinskycak) on

Math gets hard for different students at different levels. If you don't have worked examples to help carry you through once math becomes hard for you, then every problem basically blows up into a "research project" for you. Sometimes people advocate for unguided struggle as a way to improve general problem-solving ability, but this idea lacks empirical support. Worked examples won't prevent you from developing deep understanding (actually, it's the opposite: worked examples can help you quickly layer on more skills, which forces a structural integrity in the lower levels of your knowledge). Even if you decide against using worked examples for now, continually re-evaluate to make sure you're getting enough productive training volume.

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Who needs worked examples? Wouldn’t it be better to try to solve problems without seeing a worked example first? How will you develop general problem-solving skills if you always start out with a worked example?

These are common questions, and they’re good questions.

I’ll break the answer into two parts.

Part 1: You're eventually going to need worked examples, even if you don't need them right now.

When students start out learning math, it sometimes feels very easy to the point that they could solve problems quickly without having to see worked examples. I’m not disagreeing with that. But the problem is that this phase is temporary. For most learners, as the level of math rises, solving problems without worked examples quickly becomes overwhelming and inefficient.

Without worked examples, learners typically reach a point where unguided problem-solving overwhelms their working memory and puts them in a state of cognitive overload where they feel frustrated, confused, and are unable to solve the problem. They just flat-out stop making progress. No more learning happens.

And even before that point, even if a student is able to solve problems successfully without guidance, it typically takes a lot longer, which throttles the volume of practice. (I’ve written more here about how, according to research in talent development, the accumulated volume of action-feedback-improvement cycles is the single biggest factor responsible for individual differences in performance among elite performers across a wide variety of talent domains.)

To re-emphasize: If you feel like you don’t need worked examples, I’m not necessarily disagreeing with your experience. All I’m saying is that, as you progress up the levels of math, this experience is eventually going to change. Math gets hard for different students at different levels – it can be as early as high school algebra or as late as graduate-level Algebraic Topology (or even after that) – but everyone eventually reaches a level where things no longer feel obvious and they can’t figure things out as quickly on the fly, and that’s where worked examples and instructional scaffolding come in to keep them making fast progress.

If you don’t have worked examples and instructional scaffolding to help carry you through once math becomes hard for you, then every problem basically blows up into a “research project” for you. That’s okay if you’re a research mathematician at the edge of your field, but if you’re a student who still has a ways to go before reaching the edge of human mathematical knowledge, then it’s just less efficient (even if you have fun with it).

If you want to maximize how far you get in a talent domain, then you need to grab all the examples & problem-solving experiences in the direction that you’re going, as quickly as possible, and then only once you reach the end of the road with known examples and problem-solving experiences, do you switch over to creative production. Creative production is a way less efficient way of moving forward so you want to save it for the end when it’s the only way to continue moving forward.

(To be clear: this is not just my opinion, this is based on findings from Benjamin Bloom’s research on talent development. This is the same Bloom who created Bloom’s taxonomy, which many educators use as justification for having students spend a lot of time working on project – but that is actually misinterpreting Bloom’s taxonomy in a way that is not aligned with Bloom’s seminal work that came later in his career. I’ve written more about that here.)

Part 2: The idea of improving general problem-solving ability through unguided struggle lacks empirical support.

Sometimes people advocate for unguided struggle as a way to improve general problem-solving ability. In addition to the issues described above, there’s another issue: this idea of “productive struggle” lacks empirical support.

For students (not experts), empirical results point in the opposite direction. One key empirical result is the expertise reversal effect, a well-replicated phenomenon that instructional techniques that promote the most learning in experts, promote the least learning in beginners, and vice versa. It’s 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. But that doesn’t mean beginners should do the same. The expertise reversal effect suggests the opposite – that beginners (i.e., students) learn most effectively through direct instruction.

There’s a mountain of empirical evidence that you can increase the number of examples & problem-solving experiences in a student’s knowledge base – but a lack of evidence that you can increase the student’s ability to generalize from those examples (by doing things other than equipping them with progressively more advanced examples & problem-solving experiences).

In other words, research indicates the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain.

So, there does not seem to be any tangible, empirically-supported reason to struggle with a challenge problem for a long period of time, when you consider that you could be making more educational progress using that time to learn more content. For instance, 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.)

To dip your toe into the scientific literature on all this, I’d recommend the following papers as a starting point:

Follow-Up Question: But how can you unlock deep understanding?

Understanding is not something that’s unlocked. It’s something that’s built. And you build it in layers.

When you continually layer advanced skills on top of existing skills, it forces you to deepen your understanding, really internalizing those existing skills and the ideas behind them. This is basically the idea of “structural integrity” in the context of knowledge.

  • When advanced features are built on top of a system, they sometimes fail in ways that reveal previously-unknown foundational weaknesses in the underlying structure. This forces engineers to fortify the underlying structure so that the system can accommodate new elements without compromising its integrity.
  • Fortifying the underlying structure often requires improving its organization and elegance, which, in the context of student knowledge, produces deep understanding and insight.
  • When the structural integrity of a system is increased, it also becomes easier to add more advanced features in general. In the same way, when the structural integrity of a student’s knowledge is increased, it becomes easier to assimilate new knowledge in general.

When you layer on advanced skills, it forces understanding. If there’s some level of understanding you’re lacking in some component knowledge, you eventually get to a point where the lack of understanding prevents you from successfully learning more advanced skills.

For instance, there’s no way that you can get through a legitimate calculus course without having a real understanding of algebra. In fact, many math students will tell you that calculus is what really deepened their understanding of algebra.

Overall Conclusion: If you don't want to use worked examples, you don't have to -- but continually re-evaluate to make sure you're getting enough productive training volume.

If you really want to practice solving problems without referring to worked examples, you can always skip worked examples and try your hand at solving the corresponding problems without guidance. It can be a fun challenge! You just need to make sure that you’re still solving the problems quickly and accurately – if you start slowing down and/or becoming less accurate (or, more subtly, if you start to doubt yourself and lose interest), then that’s an indication you need to start leveraging those worked examples.

Additionally, in the way of fun challenges, some people enjoy the feeling of grinding on competition math problems and getting lost in thought for long periods of time. While this is not an efficient method of training, this is totally fine and I’m not trying to tell anyone they can’t or shouldn’t do that. For a lot of people, doing well in a competition-style setting (even if it’s just you alone solving competition-style problems) can be a really good motivational tool. It can help you lean into that saying “nothing succeeds like success,” making you feel really good about what all your hard work training has done for you, and making you want to continue training so that you get even better.

All I’m saying is it’s important to realize that these “think really hard, struggle for a while, eventually solve it or look up the answer problems” are NOT the same as, or a substitute for, actual training – just like how there’s a difference between doing daily training with a basketball coach, versus going down to the park to play pick-up games and trick shot competitions some days when you have time after training.


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