Thinking a bit about this metaphor (or whatever it is we’re doing here) makes me think that learning a mathematical idea or skill is much more than knowing the route from one place to the next. It’s knowing your way around a neighborhood — you know good ways to get to the most important places, and you’re never really lost.

I live in Washington Heights, NYC. Here are some ways that we might teach people to get around Washington Heights.

1. Give ’em GPSs and spend a few days just following directions to go place to place.

2. Same thing, but they have to go there with the GPS and go back without it.

3. Give ’em a map and spend a few days asking them to go from place to place on it.

4. Same thing, but there are five important trips that you use a GPS for.

There are certainly people who would find each of these reasonable, but the reasonablest among us wouldn’t dream of doing 1 or 3.

Where does that leave us? I wish these debates happened at the level of units of instruction instead of individual learning activities.

I have completely lost interest in this debate. Good teaching is not possible in a 100% direct instruction mode, or in a 100% discovery mode. Good teaching requires us to figure out how to combine the two given the complexities of different topics, different groups of students, different availability of tools and materials, and so on. See my article titled “Nothing Works”. http://www.mathedpage.org/teaching/nothing.html

I’m interested in whether we can specify the conditions under which this is true, and the situations under which it is not.

This is a false dichotomy in my opinion. You needn’t do one or the other. You are free to do both for almost any topic. Further, within your own classroom, I would hazard that there are some students who would benefit from starting with GPS-like direct instruction followed up with exploration while others would benefit from wandering about first then explicit instruction leading to an algorithm.

In my instruction, I prefer to let them wander a bit first then producing an algorithm together. I don’t see it as an either or proposition.

The first time I go somewhere I’ve never been before, I use the GPS app on my phone. If it’s somewhere I expect to go again, I try to pay attention to details around me. The second time I go, I might tell my phone where I’m going so it can help me out, but I don’t pay as much attention. The third time I go, I’ve learned the route and have started to pay attention to alternate choices. I currently know a good half dozen “basic” ways of how to get from home to work, in case there’s a traffic back-up.

There’s a sandwich shop around the corner from the high school I teach at that all the students know how to get to. I have used that to explain different paths to the same goal, and students have a better understanding of what it means when the math teacher says that what matters most is a consistent approach to a problem, as long as the solution is “correct”, not always following the same steps.

Point being: I think it’s important to show students a clear route, but it’s equally (if not more) important to reinforce to students repeatedly that most math puzzles have multiple routes. The GPS model of, say, WolframAlpha and similar sites reinforces the One True Way (although WA does provide alternate representations of values by default).

Coming from a perspective of inquiry or bust, I think I am now interested in a broader, more inclusive view of instructional approaches.

Some people may never feel the general confidence to wander around a new city without a map or any support at all. Those people may benefit from a GPS and I would not advocate for that to not be an option for them. I would hope to work with them to eventually not need it, but that would take some practice and gradual release. Conversely, there are some people, like Dan Anderson, who benefit from the freedom of discovery and inquiry to develop their own road map and understandings of new surroundings.

In education, I think every student, teacher, and class is different (to butcher another metaphor, unique beautiful snowflakes in the blizzard of life), thus shouldn’t ALL pedagogical strategies be available to all students and teachers at all times?

Need step-by-step explicit instruction to graph linear functions? We got you, here’s a worked example.

Need the freedom of inquiry to develop your own conceptual understanding of variables? Here’s a balance scale with different shapes on it, lets figure out their values!

I guess what I’m asking is, are these concepts mutually exclusive or can they both exist in the same classroom based on the needs of the learners within it?

Our kids are asked to be able to solve quadratics by a variety of methods, in isolation, many times without context or other outside purpose. They see this as the one-off “appointment.” Why should they care about searching around? In an ideal world, I can make connections to other things they know, and show them on “the map” how they connect and are juxtaposed.
Some of this is knowing your audience. I work with students who are “mathematically fragile” so there is a lot of explicit instruction. But my explicit instruction isn’t just divide by 5. We’ve had a conversation that this is doable if the lead coefficient is 1. So what can we do to make that happen? I don’t just say “divide by 5” with no reason given. My kids have a set of generalized steps we’ve “discovered” together, and then we apply to a variety of situations. I guess I’m teaching them to look for landmarks, and then make a decision which direction to go. I tell my students I want them to not only be able to get an answer, but to be able to solve the problem.
As to active learning — I don’t think the active learning starts with this problem 5x^2-4x-2=0. It started with quadratics, and graphs of them, and playing around with what and why they matter. We build the “map” there. Then, by the time we get to this problem, its one more location on the map.

I don’t disagree, Ihor. I’d suggest that we do curriculum in mathematics (and everything else) in an absurd and generally ineffective way, one that only “connects” with a limited percentage of students and serves most poorly (even if they buy into the game of schooling and “studenting.”

For a more organic and meaningful approach, I recommend Marion Brady’s work, much of which can be found at http://www.MarionBrady.com

I fear you have severely over-reached from this GPS study.

“Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.”

Surface structure always comes first. Deep structure comes from lots of surface structure and practice. You cannot go straight into deep structure. This knowledge is now well established in cognitive psychology. Let’s not try and re-invent the wheel here.

“But the GPS offers too little difficulty, with negative consequences for drivers and even worse ones for students.”

How about GPS to start with and then transitioning to maps? Isn’t this the popular idea of “scaffolding”, that has been around since the 50’s? Perhaps that could be an efficient way?

It’s funny you mention GPS. I moved to Germany last year. I tried the whole looking at a route on google maps thing before I drove anywhere, but always ended up getting hopelessly lost and having to pull over and use my phone, dangerously, for about 3 months. I didn’t seem to be learning routes very well, except for the 500m radius around my house.

Then I got a sat nav and never got lost again. I even know quite a few routes off the top of my head now.

It’s also interesting that you use the complete the square example.

My teacher taught me this when I was 17, doing my A-levels. She even taught us a little song “Halve the coefficient of x and take away the square…do de da”.

I have remembered it ever since those 10 minutes of instruction that she gave us, and have used it more times than I can count since then.

I think it is worse than a false dichotomy between GPS and no GPS. It also assumes every student needs the same thing.

I don’t always tell my clever students how to solve a problem step by step. They need to work some out for themselves or they’ll never grow.

I usually give my strugglers much more explicit instructions. Otherwise they are overwhelmed by the question so much that they shut down. To not do so is to hamper their improvement — because they do get better if led carefully.

Interesting to use “closing the square” as an example. Does anyone expect their students to figure out how to do that by themselves in a logical and reliable manner? I bet we all teach that by an explicit process, if only so that they do it in the quickest way, rather than the awkward way that they otherwise find by themselves.

(And I would usually teach my strugglers to go straight to the quadratic formula, since closing the square is something they find quite challenging.)

@Chester writes: “Interesting to use “closing the square” as an example. Does anyone expect their students to figure out how to do that by themselves in a logical and reliable manner? I bet we all teach that by an explicit process, if only so that they do it in the quickest way, rather than the awkward way that they otherwise find by themselves.

(And I would usually teach my strugglers to go straight to the quadratic formula, since closing the square is something they find quite challenging.)”

I guess that “closing” and “completing” the square are the same thing.

Now, first of all, let’s not make “discovery learning” into some insane approach where the goal and method is to demand that students “discover” for themselves (i.e., reinvent an already well-known and long-standing mathematical procedure for themselves from scratch or with little or virtually zero scaffolding before hand) at every turn. That’s simply not what I think is at the heart of any of the alternatives to teacher-centered, generally mini-lecture-based, direct instruction approaches, different teaching approaches that get often get contemptuously lumped together as “constructivist teaching” in some camps, but which comprise in fact lots of pedagogical methods that vary almost as widely as the instructors who employ them. (Of course, any teacher might CHOOSE to teach in that 100% discovery all the time, no scaffolding, no hints way that the parodists like to tell us is what goes on in all non-traditional classrooms. Just as there may well be teacher-centered instructors who never let students speak unless to answer what usually amount to “fill in the computation answer or next step HERE__________” inquiries. It’s possible at either extreme. But how likely? And in the case of the extremely “pure” (and to my mind terribly ignorant) “discovery” teacher, how much of that is fueled by the failure to think through or understand what such methods are about and how to make them work for given students and groups of students?

But I digress. You want to see one way to get at completing the square that I think is rich and deep, and allows for all sorts of great questions from the teacher, thinking along the way to making sense of the method, and which, by the way, leads to many of my students both eschewing the usual way completing the square is accomplished and presented in the US, and to ignoring the quadratic formula for the most part? Try the video series James Tanton offers (free of charge) at http://www.gdaymath.com under “courses.” It’s the second subset of videos in the quadratics minicourse, and the first of the five is just a look at the use of the word “quadratic” to describe 2nd-degree equations, something that evolved exactly because of the method of “quadrangles,” and in particular regular quadrangles, aka, squares.

I’ve used those videos for 2 1/2 years now, with a few hundred students, and most come away vastly enriched by the experience. They wonder why they weren’t given Tanton’s approach in high school. I wonder, too. It’s not “discovery” and it’s not “typical direct instruction,” and it’s not denying ANY student of any ability the chance to actually understand both why completing the square makes total sense, to SEE it geometrically and algebraically in a deeply interconnected way, to get some basic knowledge of elementary number theory (some of the little bonuses aren’t always explicitly in the videos but follow from them and are things I add in by pausing the videos to pose various sorts of questions) and to feel confident about being able to do the calculations and recall the procedures. And while Tanton points out both that the “famous quadratic formula is derived from completing the square,” he also states intriguingly but without going into a major argument that it’s unnecessary to learn that formula. And I add that like most formulas and procedures in my class, students can and likely should memorize less, understand and think more. And that’s what those videos help them do, along with, I believe, my provocative questions and their own – dare I say it? – explorations and discoveries.

The notion that some kids are just too afraid (or stupid, to be blunt) to handle inquiry, discovery, or anything but step-by-step procedures handed to them on a plate to be repeated until “memorized” (at least until the exam) is wrong-headed and objectionable in a host of ways, at least to me. And I’m not working with kids in all-white upper-middle-class suburbs, but with working class students both in the inner city and blue-collar surrounding areas.

There is so much potential variety in teaching mathematics. Some teachers seem to come into their first classes with a very definite way they believe all math teaching is supposed to be, and when that doesn’t actually suit some, most, or nearly all the students, they either get angry and double down on their one-size-fits-all approach (and blame the kids for things going poorly) or they slowly lower the bar in order to wind up not flunking out most of the kids (and then blame the kids – not entirely wrongly – for forcing them to do so).

Either way, it’s never that the teacher hasn’t thought things through from enough different perspectives, hasn’t done enough really hard thinking about teaching and learning math, hasn’t been willing to try non-traditional methods (or if so, tries them once, half-heartedly and with a significant degree of self-sabotaging behavior, so that the self-fulfilling prophecy comes to pass and the teacher feels justified in returning to the tried-and-failed, completely teacher-centered, dull as dishwater model that we could all recite by heart if awakened at 3 AM and forced to immediately deliver).

I can assure you that any horror stories you care to tell, I can probably top them. I’ve worked the majority of my math teaching career in Detroit, Flint, Pontiac and similarly impoverished and deprived communities, in both charter and regular public high schools, and from grades 3 to 12. (Oh, and many of the horror stories I have aren’t about the students). Math teaching, like doing mathematics, is hard work. It’s not for the weak. But my sense of what comprises weakness may not be yours. I think that denying some kids the opportunity to struggle (sure, with whatever particular scaffolding and individual support that one can REASONABLY determine might be necessary in the short-run, as long as one keeps in mind one of the wisest three words ever stated about mathematics teaching: be less helpful).

From the original analogy: “Another group had to construct their route between those points using a map.”

Thinking on this in light of David Didau’s comments, is this really akin to discovery learning (at least, in its extreme)?

Another way to look at the GPS example is in terms of two versions of DI: Straight-up cookbook mathematics (where students are provided a formula for finding the area of a polygon given the apothem length and number of sides, and a separate formula for finding the area given the side length and number of sides, and a separate formula for finding the area given the radius and the number of sides, and so on: For exercises 5-9, refer to example 10-1) and guided examples.

When I was a lad (in the 1970s-80s), I don’t recall the spoon-feeding cookbook model, but mathematics was predominantly DI. We were trained to expand our strategies; in those days, “teaching to the test” consisted of creating strategies for entry points for problems we’d never seen before, not sitting down with the last five years of SAT/ACT practice tests and going over each problem.

It seems to me that if we’re going to take the GPS analogy its full distance, the “cookbook” model is using the GPS instructions each and every time (refer to example 10-1 on p. 415), while the map-and-memorize model is closer to DI modelling where students are expected to apply general concepts to novel problems. After all, the respondents in the GPS study were provided with a map, and presumably tapped into their existing knowledge of how to read that map.

A fully discovery based model would be more like putting someone in a zoo without a map but with the instructions, “The zebras are over there somewhere, close to the lions. If you get lost, call me and I’ll point you in the right direction.”

And there’s call for that, plenty. That’s how new mathematics gets done, after all: By wandering, and exploration, and listening for the lion’s growls. Take a bunch of kids to the zoo, and some will want to just wander around to see what they find, some will want a map to set out a basic route, and some will want step-by-step instructions. If the goal is for everyone to see the zebras, and everyone sees the zebras, there’s room for multiple strategies.

Michael,

I went away and looked at that site of James Tanton’s.

I have no problem with it at all. However I believe is almost purely Explicit Instruction, so no surprise there.

He starts at the beginning. He works his way up in levels of complexity. He gives step by step instructions. He is clear in language and expectation.

Sure, he adds some other bits on, like we all do. But he adds them at the end, when the students have some chance of following him. He doesn’t lead with that material, because he knows that will just confuse.

Whether you need to close the square or use the quadratic formula depends quite a lot on the curriculum you follow. My students need to give surd answers to the likes of 0 = 3x^2 + 17x – 5 and there is no way I think that is wise to do that by anything other than the quadratic formula — which mine are given, by the way, so they don’t have to waste time memorising it. They also don’t ever have multiple choice, so get no prompts if they have blundered.

I think, Chester, that you are jumping to an easy but mistaken conclusion. The Tanton videos are not a class. They are a resource that can be used lots of different ways. So can really bad instructional videos (e.g., the things I’ve watched from Sal Khan, which I find poorly put together, to be polite), but the depth and richness of bad videos is primarily useful for helping teachers who want to see what not to do and why.

If you look back at my previous comment, you might catch the point that I don’t assign my classes to watch these videos on their own for help or enrichment, etc., though that would certainly be a couple of feasible options. Rather, I show them in one class when we’ve worked through some basics of quadratics but haven’t hit the ones that don’t factor, even using the method where you multiply the quadratic coefficient times the constant, then split the middle term and factor by grouping (some texts call that the AB method; I taught from book in the ’80s to community college kids that called it “the Master Product Method,” perhaps after the authors watched TRON). In other words, the ones that need either completing the square or the quadratic formula or something other than factoring (a graphic/computing tool, for example).

And I don’t have students just passively watching the short series of Tanton vids. I pose preliminary questions to getting them thinking about what might work. And I frequently (perhaps annoyingly so to some students) pause the videos to field questions, take comments, and pose more probing questions, e.g., “Why is it okay for him to do________? Why shouldn’t that change the results?”

That’s not quite direct instruction as I experienced it. It’s not quite discovery, inquiry, etc. But it’s an intentional move to help students move along a continuum from passivity to more active learning. That’s the goal of my classroom. And I only have my students for about 8 weeks. So this is one of the better things I get to do and still take them from pre-algebra review through rational algebraic equations and the like.