Hat tip to Darren at Right on the Left Coast for this article, which starts off saying in a plainspoken way:

Here are two of the clues to America’s current mathematics problem:

1.”Student-centered” learning (or “constructivism”)

2.Insufficient practice of basic skills

The article then goes on to say, of constructivism:

In small doses, constructivism can provide flavor to classrooms, but some math professors have told me the approach seems to work better in subjects other than math. That sounds reasonable. The learning of mathematics depends on a logical progression of basic skills. Sixth-graders are not Pythagorus [sic], nor are they math teachers.

That’s right. Constructivism, when used with the right kinds of students and in the right ways, can be quite effective. But it’s important to remember that *not all* students are ready for this, and *not all* material is taught effectively this way. When I teach geometry to junior and senior math majors, it’s almost entirely constructivist, because the process of mathematical investigation and discovery is precisely what I am trying to teach them (through the medium of Euclidean and non-Euclidean geometry). But I’d be crazy to try constructivism at that level on, say, a precalculus class full of students who have little skill in and absolutely no taste for math at all. Those students aren’t dumb, but they need structure and guidance a lot more than they need the supposed thrill of mathematical discovery.

And then, about drill and practice:

Another problem in math classrooms is the lack of practice. Instead of insisting that students practice math skills until they’re second nature, educators have labeled this practice “drill and kill” and thrown it under a bus.

I wish I had a dollar for every time I heard that phrase. It’s a strange, flippant way to dismiss a logical process for learning. Drilling is how anyone learns a skill. […] Everyone drills – athletes, pianists, soldiers, plumbers and doctors. Drilling is necessary.

It isn’t good or bad – it’s simply what must be done.

I’ve said it before here: No human being can do meaningful creative work until they are completely fluent in the rudiments of what they are working with. Musicians, athletes, and skilled workers all know this. For some reason, there’s no outcry among music educators that we need to just hand new musicians a saxophone and try to get them to discover how to play it all by themselves. This fact — that drill and mastery precede creative work — is so painfully obvious that I feel a little embarrassed for my colleagues in math instruction who don’t seem to get it.

Constructivism and drill/practice are pedagogical tools, not religions. You look at your class, your students, and the material to teach, and then choose the right combination of tools for the job. To hear some proponents, and opponents, of constructivism, you’d think that you’re supposed to choose sides and swear undying allegiances instead.

Thank you for this. I have been trying to think of how I could implement constructivism into my classroom. When I couldn’t figure out how to do it in more and bigger ways than the few special cases I have developed, I thought I wasn’t good enough/thinking correctly. Thanks for easing my guilt.

While I don’t want to debate all your points, I don’t agree that, “No human being can do meaningful creative work until they are completely fluent in the rudiments of what they are working with.” I can think of many examples of people young and old who exhibit high levels of creativity in many areas without the level of expertise you infer.

Drilling does have a role but often, even in Mathematics, drilling comes at the expense of understanding and meaning making. My understanding and use of a constructivist approach offers the learner a chance to make meaning. Drill and and practice shouldn’t be abandoned but I’m also not sure it’s a choice between two pedagogies but rather an appropriate blend.

I have a couple of concerns with this – the first is that US schools are doing lousy in math and science: http://www.washingtonpost.com/wp-dyn/content/article/2007/12/04/AR2007120400730.html

So whatever we are doing is really working all that well.

Second, the rote memorization method is great for passing tests – is lousy for creating engineers. I agree that there is a certain amount of memorization that might be helpful but I think that if assignments are created that utilize the basics, students will learn the basics because they want to participate in the basics. Take Latin grammar for instance: I have seen the “drill and kill” method turn off and entire class of 100 students and really get them no where. The same kind of population at another college is given interesting texts and dialogues to engage in and they actually will learn the grammar in context. It was once thought that you had to memorize all the declensions before you could learn the language and really what you have is a student population that is good at memorizing things for tests.

That is really my point – there is this population of students that are “not good at math” or “not good at languages” and maybe they are not good at tolerating the tedium of the current teaching methods.

@Dean Shareski: Yes, my point is exactly that there has to be an appropriate blending of pedagogical strategies, and that blend is unique for each class.

@Geoff Cain: Again, it’s about an appropriate blend of strategies. You wouldn’t give a Latin class a Latin text and ask them to figure out the grammatical rules on their own. Some level of mastery of the grammar, through a perhaps-tedious study of the rules, has to take place before “learning in context” can take place.

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What I’ve witnessed is that because a constructivist approach to learning and in particular Math requires a very different approach, many teachers abandon it too quickly and revert to completely linear, sequential, drill and practice approaches.

Since Math teachers in general did well with that model and likely are “wired” to think that way, it seems like it has to be the best way. I think there is some research out there that would suggest there may be other ways to approach it.

http://mathematicallysane.com/analysis/reformvsbasics.asp

So I agree it’s a blend but rather than making a choice based upon the needs of an entire class, it should be on the needs of the student. While that’s nothing we haven’t heard before, it’s not a reality in most classrooms because differentiated learning is still far from reality but it is no doubt the goal we should be shooting for. Constructivist approaches are likely suitable for everyone to some degree.

Thanks for starting a really important conversation.

Are those the only two methods? It’s a spectrum?

What about just sitting down by yourself at night and just working on a problem until you’ve got it figured out? No teacher, no pedagogy. Just the noodle.

I too think there needs to be a balance. If all we do is practice skills in isolation, student will never see the value, let alone the beauty in mathematics. If it is meaningless what is the point of having a skill that will be forgotten after the test? Yet there needs to be some guidance to the constructivist approach.

Constructivist teaching is difficult (when done well). Deciding what activities are appropriate for the students in each class, deciding when to let them grapple with a problem and when to step in and offer guidance, asking the right questions, structuring the classroom so that no student heads off (for too long) down the wrong path and internalizes misconceptions, … it is hard work. Do we do skill practice? Yes, of course. After we’ve come to a shared understanding of meaning.

I think maybe we haven’t defined the term “constructivist” here very precisely. Robert and the author of the EdNews piece seem to be using the term to mean “discovery learning,” a teaching approach in which students “discover” mathematical ideas that are well known but new to the student through activities carefully planned by the instructor. The Moore method of instruction common in some topology courses is an example of this.

Discovery learning is one way to apply the theory of constructivism, but it’s not the same thing as constructivism and it’s not the only way to apply that theory. From the Web site you linked to in your post, constructivism “says that people construct their own understanding and knowledge of the world, through experiencing things and reflecting on those experiences.” This is a theory of learning, not a particular approach to teaching.

According to this theory, students can’t help but come to understand new information in light of their prior knowledge. They construct new knowledge (in their heads) by making sense of new information in light of how they already understand the world. Constructivism is a theory that helps explain how student learn no matter what techniques are used to teach them.

For example, a student who is drilled on a particular computational technique will naturally start to make sense of that technique in light of her conceptual understanding of the topic and her knowledge of other similar techniques. For certain kinds of sense-making, drilling can be very effective. For other kinds, drilling doesn’t do much to help students make sense of new information.

Student-centered teaching (which is a better term, I think, then student-centered learning) is teaching that acknowledges the theory of constructivism and attempts to design instructional activities that intentionally help students make sense of new information in light of prior knowledge.

In-class, small-group work can do this, for instance, if students are discussing in their groups concepts related to a particular problem. Peer discussion is one way that this kind of sense-making happens, so creating conditions in class that promote peer instruction is one way to practice student-centered teaching.

It’s certainly not the only way, however. As you point, there are a variety of approaches, and the efficacy of an approach depends on the instructor, the students, the learning goals, and other elements of the teaching context.

This comment has run a little long, but I thought it was worth exploring this term a little. Thanks for the chance to do so!

Sorry, this is what I get for blogging when I’m tired… I shouldn’t have said that the Moore method is an example of discovery learning. It is, in a sense, because it involves students discovering the proofs for theorems on their own, but, at least as I experienced the Moore method in grad school, the theorems were given to the students.

In what is usually called discovery learning, the students are given neither the results nor the proofs of those results. They are instead asked to “discover” results and justify those results.

Sorry for any confusion…

Derek,

Yes, one can construct one’s own knowledge from working out a problem as Myrtle suggests, or from drilling type exercises.