# What’s the Best Way to Teach Math: Explicit Instruction or Less Guided Learning?

*Nobody who knows the science of learning is actually debating this.*

This might seem like an opinion-based question where the proper answer is “to each their own.”

… except, it’s not.

One of these options is supported by decades of research into the science of learning. The other is not.

Assuming that we define “best” as “produces measurably superior learning outcomes,” then the best way to teach math, hands-down, is **explicit instruction.**

Nobody who knows the science of learning is actually debating this (though unfortunately much of the education world does not know the science of learning, hence the debate).

Need to see it to believe it? Okay, here’s a direct quote from Clark, Kirschner, & Sweller (2012):

*"Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn.*

...

We also have a good deal more experimental evidence [since the 1960s] as to what constitutes effective instruction: controlled experiments almost uniformly indicate that when dealing with novel information, learners should be explicitly shown all relevant information, including what to do and how to do it. We wonder why many teacher educators who are committed to scholarship and research ignore the evidence and continue to encourage minimal guidance when they train new teachers.

After a half century of advocacy associated with instruction using minimal guidance, it appears that there is no body of sound research that supports using the technique with anyone other than the most expert students. Evidence from controlled, experimental (a.k.a. "gold standard") studies almost uniformly supports full and explicit instructional guidance rather than partial or minimal guidance for novice to intermediate learners. These findings and their associated theories suggest teachers should provide their students with clear, explicit instruction rather than merely assisting students in attempting to discover knowledge themselves."

Opponents may question: “But don’t students need to struggle for long periods of time, without too much guidance, to train their general problem-solving ability?”

**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.

In fact, 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 that **the most effective way to improve a student’s problem-solving ability in any domain is simply to equip them with more foundational skills in that domain.**

The way to increase a student’s ability to make mental leaps is not by having them jump further, but by having them build bridges from which to jump.

**There does not seem to be any tangible, empirically-supported reason for a student to struggle with a problem for a long period of time as opposed to using that time to learn more content.**

For instance, in an hour-long training session, a student will make a lot more progress by solving numerous “deliberate practice” problems that each take a small amount of time given their current level of knowledge, than by attempting a single problem that they struggle with for a long period of time.

(To be clear: the deliberate practice problems must be grouped into minimal effective doses, well-scaffolded & increasing in difficulty, across a variety of topics at the edge of the student’s knowledge.)

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."

Plenty of additional quotes/references here for anyone interested in learning more: *Effective Learning Does Not Emulate the Professional Workplace*

You might ask: “What kind of teacher actually believes in having students struggle for long periods of time, without too much guidance, when teaching new material? Is that belief even real?”

Oh, let me tell you a story…

Minimally-guided struggle is a common teaching method in university-level math classes, especially in serious math major courses.

I would not say it’s absent in lower levels of math – but it seems to be the modus operandi in higher math, and I have a story that takes pace in this setting, so I’m going to focus on higher math here.

Anyway, the story:

I tutored a college sophomore last spring whose Real Analysis class did not use a textbook or any form of scaffolded reference material, and each weekly homework set consisted of a small number of very challenging problems.

I’m told that during each class, the professor proved a theorem or two on the board and asked if the students had any questions. No students did, because no students had acquired enough knowledge to formulate a question more specific than “I don’t get any of this.” (The professor interpreted this silence as understanding.)

Unsurprisingly, things were going the way of a train wreck: despite her best efforts, my tutoring student was spinning her wheels on these homework problems and making very little progress.

And not only was she unable to solve the problems, but also, she was not noticeably improving any supporting knowledge by trying and failing to solve them.

To turn the situation around, I engaged her in direct instruction on all of the component knowledge that was being pulled together in each problem.

This created a deliberate practice experience in which she extended her knowledge engaged in mindful repetition at the edge of her knowledge.

Something like this (the following is non-exhaustive):

*Deliberate Practice on Definitions:*I give you a mathematical object and you tell me whether it meets the definition (and why or why not). Repeat over and over again increasing in difficulty. Okay, now suppose we remove some criterion from the definition. What's an object that didn't meet the original definition but does now after dropping that criterion? Repeat over and over dropping different criteria.*Deliberate Practice on Theorems:*I give you a scenario and you tell me whether it meets the assumptions of the theorem. If so, you tell me specifically what else you know is true about the scenario, according to the theorem. Repeat over and over again increasing in difficulty. Okay, if this is a one-way implication, tell me some scenarios where the converse does not hold.

As soon as we took a step back from the homework problems and started doing the practice described above, she started making actual progress on her supporting knowledge.

After enough cycles of direct instruction / deliberate practice, she’d re-attempt the homework problems, often solving them completely or at least getting a lot further before starting to spin her wheels again.

*The result:* her exam performance skyrocketed and she ended up finishing the course with an A.

*What could have happened:* without this direct instruction / deliberate practice intervention, she would have gotten a low grade in the course and possibly even dropped out of the major entirely.

For anyone who has dug into the scientific literature on direct vs unguided instruction, I should point out a common area of confusion.

In some scientific experiments, active learning with minimal guidance has outperformed passive learning with direct instruction.

**These experiments often misinterpreted by the lay audience as providing support for minimal guidance – which is not even remotely true, even though minimal guidance is a part of the more successful experimental condition.**

The thing is, it’s widely known that *active learning is superior to passive learning*. Students who spend a class period actively solving problems, hands-on, learn more than students who sit and passively listen to a lecture.

So, any experimental condition involving active learning will likely outperform any experimental condition involving passive learning, no matter what other second-order strategies are tacked on!

You could take tired students, engage them in active learning, and probably get a better learning outcome than well-rested students experiencing passive learning.

But does that mean being tired is good for learning? Heck no!

**By the same reasoning, if you find that active learning with minimal guidance outperforms passive learning with direct instruction, you cannot conclude that minimal guidance is superior to direct instruction.**

In general, if you change two variables at once and get a better outcome, you can’t conclude anything about either of the individual variables. All you can conclude is that one combination is better than the other combination.

In the case of minimal guidance vs direct instruction, it turns out that when you do both in an active learning setting, direct instruction outperforms minimal guidance.

Here’s the big picture:

**active/direct > active/unguided > passive/direct.**

(I didn’t include passive/unguided here because I’m not sure it’s even possible to create such a combination.)

Now, how do you do direct instruction and active learning simultaneously? Is that even possible?

Well, that’s essentially what **“deliberate practice”** is, and in the academic field of talent development, there’s a mountain of evidence supporting deliberate practice as the most effective training technique across a wide variety of talent domains.

In particular, a key finding is that the volume of accumulated deliberate practice is the single biggest factor responsible for individual differences in performance among elite performers. (The next biggest factor is genetics, and the relative contributions of deliberate practice vs genetics can can vary significantly across talent domains.)

This seems to be a general result across all talent domains: as far as I’m aware, no counterexamples have been found.

Want to spin up on deliberate practice? Here’s a summary I wrote with plenty of scientific references and relevant quotes pulled out of those references: *Deliberate Practice: the Most Effective Form of Active Learning*

Now, here’s an assumption that’s baked into the superiority of direct instruction over less guided learning:

**The goal is to maximize the amount of subject information that you can reproduce and apply to solve problems independently.**

Note that this goal could be stated equivalently as simply *“maximize the degree to which you can independently solve problems in general within the subject,”* since background knowledge is the key to problem-solving.

If a teacher’s goal is different from that… well, of course they may come to different conclusions regarding direct vs less guided learning for their students.

But personally, I’ve always thought the goal of education was to maximally develop students’ talents, to maximize career opportunities & achievement in their area(s) of interest.

I suppose a debate can be had about “why not focus on developing curiosity, patience, courage, etc., while meeting some baseline level of content knowledge too.”

But at least in math, when you consider that

- students spend most of their lives outside the classroom, and
- students seldom learn additional math outside of the narrow window of time occupied by math class,

it seems that the “school of life” is taught enough outside of math class and that math class should focus its severely limited time on imbuing students with as much math knowledge as possible.

Plus, in practice, that “baseline level of content knowledge” is typically not high enough to fully prepare students who aspire to math-heavy careers, and even many of those students still fail to meet the bar.

By the way, I don’t mean to suggest that less guided learning should never ever be used – just that if it is used, it should be treated as a supplement, not a substitute, for deliberate practice (which requires direct instruction).

If your goal is to maximize the amount of subject information that students can reproduce and apply to solve problems independently, then less guided learning is not really going to move the needle.

However, it can sometimes be useful for increasing motivation to engage in more effective practice that does move the needle – just like having a little bit dessert on the weekend after a week of healthy meals.

But here’s the thing: the time spent away from deliberate practice should be small enough to be fairly negligible, just like the portion of your total weekly calories that you spend on dessert.

You ate 20,000 calories of healthy meals and 1000 calories of dessert? No big deal. You earned your dessert, and it’ll help you stick to your healthy dietary habits without burning out.

You spent 20 hours of training time engaged in deliberate practice and 1 hour playing around? Again, no big deal. You earned your playtime, and it’ll help you stick to your intense training regimen without burning out.

But if you overdo it, and less guided learning becomes the default mode of instruction for your class, then it’s like eating dessert for most of your meals.