**I.**

I think at this point, if you’re reading a math blog, you probably have an opinion about the place of discovery (or inquiry or guided inquiry or problem solving or whatever) versus fully guided instruction (or direct instruction or Direct Instruction or explicit instruction or Explicit Direct Instruction).

(By the way, Ed Realist does a nice job trying to clarify the terminological situation here.)

But the thing is that it is difficult to talk about this in a way that is clear and accessible. I was thinking about this while reading Jasmine’s latest post, which lays out what cognitive science researchers say on the matter. Jasmine and I are on the same page, and she is faithful to the researchers, but I felt myself inclined to express these views in a slightly different way. Not necessarily even better; just different.

(By the way, Jasmine is a first-year teacher and new blogger. She’s on the blogroll.)

So here is how I would put it:

Every mathematician and scientist, as far as I can tell, is clear about just how messy their research is. I am very fond of this account from mathematician Andrew Wiles:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…

You think it’s true…then it’s not…then you waste a morning trying to prove something that in fact is not true and not strictly necessary for proving the thing you *actually *care about. Then you feel despair, so you take a break and do something else. A week later you come back and you feel stupid — the thing is now obviously true, and you know why — and that feels good! But that’s just Part 1. So on to Part 2…

Here’s the question, and it’s a fundamental one: do you think it’s a good idea to put your students in this situation, or not?

If you say “yeah! kids need to learn how to do this sort of thing” then you will be a fan of discovery and inquiry and problem solving and etc. If you say “wait, no, this doesn’t sound like a good way to make kids feel” then you will strongly dislike discovery and inquiry.

I feel as if that’s almost all there is to say. It pretty much comes down to that.

**II.**

There is of course a bit more, though. It’s probably easiest to present it in terms of a dialogue. Basically, cognitive science has a bunch of counter-arguments to arguments in support of the “yeah!” view above. Here’s how the dialogue goes.

**Q: You don’t like discovery/inquiry/asking students to do math the way research mathematicians and scientists do?**

A: That is right, I do not.

**Q: But how will students learn to do research-y things if you don’t teach them?**

A: “Do research-y things” is not really a skill. Neither is “creatively problem solve” or “think mathematically.” We don’t have evidence that any of these things can be taught to students, except alongside particular mathematical or scientific content. The things you really need to do research-y things that *can *be taught is a tremendous store of flexible, sturdy knowledge. That’s the best thing you can do to give your kids a leg up.

**Q: But that’s demotivating! It’s boring to learn a discipline that way, and the genuine ways of learning are more motivating.**

A: Bad teaching will always be demotivating, but there are lots of examples of the “boring” approaches being highly motivating. One way you see this is when an intervention measures affect, i.e. how kids feel about a thing.

But honestly if kids aren’t motivated, they won’t learn, and we have evidence that the more explicit approaches help kids learn. Shrug.

**Q: So you think that kids never need a chance to apply their knowledge?**

A: Yo, I did not say that.

**Q: Yeah you did.**

A: No, I did not. Here’s what I think. There’s evidence that when a student has less experience with something, they need a lot of explicit instruction about how to do that thing. Worked examples are a really, really sturdy format for people with little experience in a thing. If a student has never learned how to factor quadratic functions, a good way to start can be to show them examples of factoring quadratic functions. Then, ask them to use the example to solve a problem. And then show another example, and then give them some more practice. And then mix-up the practice, or ask them to apply what they’ve learned in a new context.

And then, the next day, do more stuff like that.

And then on the third day, maybe ask them to solve some problems on their own, and see how that goes.

And if it’s going well, who knows! In a week or so, maybe they’ll be ready to apply these skills to a challenging problem in class. Or maybe it’ll take a few weeks. The point is that as kids get more experienced with a set of skills, they are more ready to take on challenges.

**Q: Thank you for saying “challenges,” I like challenges.**

A: No problem. The point is where you *start. *And that’s genuinely controversial! But we believe (see: evidence) that starting with fully explicit instructions like worked examples gives newbies the help that they need. Starting a unit with a vague activity that students aren’t sure how to handle isn’t giving them the help they need.

**Q: Does this take into account motivation?**

A: No and yes.

No, it doesn’t take into account motivation. Do you have some amazing, super-motivating activity that kids love and that will super-charge a unit of study? Do you start a unit on quadratics with this amazing activity that helps the whole thing get started on a great note? No, the evidence does not take this into consideration. It just notes that it’s hard to find clear evidence of a learning benefit of this sort of thing.

But, yes, this does take into account motivation, because in the long-run there isn’t really any evidence that motivation is easily separable from achievement. So ultimately something like a worked example does a lot of good for motivation, because it helps struggling students understand the material and participate as an equal in your classroom.

*Michael Pershan: Can I step in here for a second?*

**Q: Sure.**

*MP: I would only add that though there is no evidence for this, I do think a certain amount of variety is healthy in a classroom. Like, kids do get bored if you do the same thing day after day. But, two things. First, if you’re starting with something like fully worked examples and moving to interesting, challenging practice, your kids are getting variety. Second, go ahead, take a day and do something interesting and different. Variety is good! There are lots of interesting practice formats, though we don’t talk enough about that.*

*That’s all I want to add here.*

**Q: OK, but here’s the thing. I just want kids to be able to think like mathematicians/scientists in school. That’s the goal I care about. That’s what I think is most valuable. And I don’t even necessarily care if it is helping them do that stuff in the future. You tell me that these skills can’t be taught — OK! You also tell me that there is no real benefit to their skills from these kinds of experiences — that’s OK too! All that I want is for kids to be doing something meaningful in school. Yes, I want to make sure their test scores are OK and they can get into college, but beyond that I want kids to do something they care about. **

There are two answers to this last question, which is what I think this discussion sometimes comes down to.

- There comes a point where people just disagree on what they value. It’s hard to know what to say by the time someone gets to this point of clarity about what they care about.
- It’s a mistake to assume that regular, “boring” school is not meaningful. And it’s also a mistake to assume that regular, “boring” learning is not meaningful. As I’ve written in the past, mathematicians ask for help all the time, and a lot of cutting edge work is simply focused on understanding things, not on solving a particular problem.

But I guess I’d point out that there are three things going on.

There’s a certain picture of what research mathematicians and scientists do and what their culture is like.

There’s a view about what is most effective for learning and motivation.

And then there is a view about whether it sounds like a good idea to put kids under the conditions of researchers in class.

And cognitive science research is relevant for the second, the efficacy question. And there is a value question, about what you think is worth doing in school.

But for me the decisive point is that the work of learning skills and knowledge is meaningful, and you can see this also in the culture of mathematicians and scientists. It’s just not right that learning skills isn’t meaningful to students.

I don’t disagree with any of this, per se. My biggest reason for moving away from IM as my primary/only curriculum for 6th and 7th is that I don’t think there’s quite enough explicit instruction/worked examples. But I also think that more often than not, the research isn’t all that helpful in deciding where we’re at, what to do next, and how to do it. Worked examples toward the beginning of a unit, or to address common errors? Sure. Problem-solving once kids have some skills? Sure again. But those extremes represent maybe 25% of the total time I’ll spend on a topic. That still leaves maybe 75% of the lessons somewhere in between–and that’s where I find all the interesting (and to my mind, unresolved) questions.

For one, it’s not at all clear to me exactly when “applying in a new context” is a problem-solving activity appropriate for developing independence and when it crosses the line into asking the kids to do something new–something unfamiliar enough that they’re essentially novices. If you can find the area of a whole circle, is finding the area of ¾ of the circle problem-solving in a new context, or is “finding fractional parts” a new skill entirely?

Another issue–what exactly should those worked examples look like? Should I teach kids the way I would solve the problem? Or should I have kids use a less efficient method–one I think/hope they might eventually outgrow–if I think they’ll understand it better? Lots of teachers value giving kids multiple approaches. Should I offer multiple worked examples, using different approaches? (I typically don’t, but that doesn’t mean I’m right.)

And another thing–are there activities that are “in-between” worked examples and problem-solving? For 6th and 7th, I’ve been trying some (scaffolded) writing problems in between–I think those are consistent with what I’ve read about promoting self-explanation. But while I like how they’ve worked so far. I’m not sure if they are equally effective for all ages and with all topics. And I have to believe there are other sorts of activities–maybe card sorts done well? maybe Craig Barton’s SSDD? maybe inquiry? class discussion? that could promote developing schema/metacognitive approaches… but I feel like I rarely see people from the “research” community talking about much besides worked examples or direct instruction spaced practice.

Also, I wrote about some of this week. The post got exactly one view. I’m pretty sure most people don’t share my view that the mess in the middle is the most interesting part of teaching. Oh well.

1. I have to catch up on your post! Your blog is the best.

2. The middle part is the most interesting, agreed!

3. I would say what you’re saying slightly differently, maybe provocatively differently. What we don’t have enough interest in is the

learning trajectoryor a sequence of skills or (even) the actual piece of curriculum. Because I don’t exactly think of a topic like solving equations as made up of (1) worked examples at the start and (2) a middle part and (3) application problem solving at the end. I think of it as 14 different mini-skills that I’ll teach on 14 different days. And I need to keep that fresh, somehow, while also making a lot of stuff explicit along the way. Icoulddo 14 days of worked examples, and I certainly do a lot of them, including erroneous examples and compare/contrast the examples and incomplete examples.I agree that there isn’t enough interest on the middle parts. But there’s also just not that much interest in stuff that matters, in general.

Hi Rachel, mind posting a link to the blog post you wrote about some of this? I would like to add it to this meal Michael has me chewing on.

(If clicking your name here is supposed to lead me to your blog, embarrassingly I am not figuring it out)

In a unit on solving equations, what does the problem-solving/application look like for you? Does it happen throughout the unit? (And if so, say you’ve taught mini-skills 1-4 of 14. Are the problems you present solvable with only that skill set?)

(I’m also trying to mentally shift, because I’ve been thinking about area and probability more than equations lately. I think equations are really interesting, but I also think they are more of a skill in their own right and there are fewer real problem-solving/application opportunities until you’re using equations to model stuff. I guess what I’m saying is that the flow of my equations unit feels different even than geometry or probability or ratios would.)

That’s a good point, about the difference. And truth be told I don’t *really* think about my unit as having an “application” part anyway.

Here’s how things worked out for my opening 8th Grade unit on solving equations. First, lots of worked examples (and mistaken examples and compare the strategies and faded examples and connecting representations etc.) were used throughout the unit. Frequently I would discover new types of problems that were hard for my kids as I followed my learning trajectory and then teach them explicitly with examples.

I randomly mixed in there some things that could be called “problem solving.” A few NRich activities one day. Not even that much mixed practice beyond the IM practice sets and one of your amazing shticky worksheets with a joke (the cowboy one). A DeltaMath day.

And now…we’re just moving on. We’ll have to come back to solving equations later in the year, but I was getting bored and the kids hit their skill threshold of “three-step equations” on Deltamath. The only thing is that their quizzes on Friday showed they weren’t handling parentheses particularly well, so maybe some examples/practice with those types of problems? Not sure, exactly.

This was a really good read and I’ve been thinking about it a lot. So in the style of above here’s where I’m at:

“Hey Ben you’re basically doing guided inquiry all the time – do you disagree with cog. sci?”

I’m not exactly in the same position as Michael, I can lean on the regular Math class during school to take care of that part of learning and I’m only seeing the kids once a week not daily.

“But what’s the point of doing it even then? Are you just wasting your time on ineffective intellectual befuddling or non permanent learning?”

I don’t think so. People have different goals. Math Class is aiming at functional competence. I’m really going for first and foremost joy. I.e. I want kids to see math as fun and exciting and to understand it as a very broad subject with a lot room for creative discovery. I don’t necessarily need the kids to leave knowing a lot more material (although I certainly hope some things stick) What I want instead is for them to leave excited and interested in pursuing Math more in the long term. What I also think is valuable is experiencing struggle and problems that require more prolonged thought.

“But don’t students already experience this in class. We’re applying cog sci to make this process more efficient but it doesn’t change the overall difficulty of achieving master?”

The answer is maybe. Like all social sciences, cog sci deals in the average behavior and response of students. Its results are most true for the middle of the curve i.e. 80% of students 80% of the time. But there are outliers (one either side) Some concepts are harder than others and students aren’t always in need of a ton of DI for everything. Likewise some students have not yet hit that first wall of conceptual difficulty and really don’t have much experience of struggle yet in a math context. My aim is to push everyone I have in front of me towards that zone.

“Why not wait until H.S. then when kids are getting closer to the end of that 10 year mastery arc?”

Again not all kids are the same, a lot are ready for this process much earlier i.e. here in M.S. Its also really critical to reach some kids during this point while they are still settling their intellectual identities. Yes person X might be closer to mastery in H.S. but by then they may have written math off as boring. There’s value in showing kids the exciting horizons earlier on even if they don’t get there for a few years.