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.
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?
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.