After my previous post about a recent article in the NEW YORK TIMES about some "earth-shaking" cognitive science research, I was reprimanded on math-teach@mathforum.org by Mark Roberts for not having the decency to read any of the original research. While I don't feel too guilty about my critique of Kenneth Chang's credulous reportage (I would think a science writer would by nature want to be skeptical, but my take on Chang's coverage is that he was not only anything but skeptical, but also inane in his including a ridiculous and irrelevant problem about trains, as if they had ANYTHING to do with the research. It's one thing to swallow the conclusions without questioning in the slightest the experiment itself or the implications being made (well beyond the parameters of the data), but quite another to try to sex up the story by including a tired old example from grandpa's algebra nightmares and pretending it's relevant to the issues. I wonder if Mr. Chang would permit anyone to draw a diagram?

Okay, so I don't nominate the coverage for a Pulitzer. But was the research itself really so bad? Mark Roberts, despite insinuating I'd not been fair and in fact had lacked decency in what I wrote, did provide a useful link to a follow-up article by the same research team, available free on-line, entitled "Do Children Need Concrete Instantiations to Learn an Abstract Concept?"

I have to say, the title seemed ominous. NEED? Who'd ever said anything about concrete instantiations (and what a clunky phrase, if ever there were one, likely intended like so much social science prose to obscure more than to enlighten) being NEEDED? The question is whether they're useful.

Meanwhile, Mark commented:

This research is not about learning the rules of a

silly game as you seem to think it is. Both the sixth

graders and the college students managed to learn the

rules of the silly game in either the `concrete

situation' or the `generic situation' (the very few

who did not were excluded from further research). The

research is about TRANSFER. The question is whether

the students can use what they learned in a novel but

very similar situation. In this piece of research the

students who learned in the `generic situation' did

this much better than the ones who learned in the

`concrete situation'.

I'm skeptical about the concept of transfer of learning between domains, but less so about transfer within the same subject. So the idea that learning a problem solving method would pay dividends when attacking similar problems AND unrelated problems that might still yield to the method (I'm think in broad, Polya-like categories of strategies, rather than very specific techniques like how to factor a quadratic by grouping).

One key question I have about this research is whether there's good reason to suspect that the students who learned with what the authors call "generic" methods, i.e., purely symbolic ones, were advantaged given the tasks. On my view, they were because the so-called "concrete" instantiation quite likely adds both an unnecessary layer of complexity to the learning task, and takes time to apply to the new situation. There's a lot of processing that has to go on, and I am not surprised that the group with the supposedly concrete examples which, if I'm reading the research correctly is not at all concrete, but rather consists of some pretty rudimentary symbols meant to look like cups that are either 1/3, 2/3 or 3/3 full, performed indifferently. What ever happened to empty cups? Wouldn't that be an important question kids would be thinking about? Is an empty cup the same as a full one? How can that be possible? We're talking about youngsters with little exposure, if any, to abstract algebra. If I wanted to confuse young kids, I can't think of much better ways to do it than with this particular approach.

On the other hand, learning rules about a small additive structure, even if you don't recognize it for what it is (in this case, addition modulo 3), is pretty simple, and well within the capacity of many sixth graders, especially if they've been exposed to "clock arithmetic" or any similar concepts and/or problems. I wonder in particular why the researchers chose pictures rather than concrete objects to make their point about, well, concrete objects (and I really believe they had it in mind to MAKE the point, not to find something out that led them, inevitably, to the only possible conclusion, in spite of their utter objectivity prior to the experiment) that such objects may interfere with the transfer of learning math concepts. And further, I wonder why they chose to look at a bit of mathematics that can be concretely represented in ways that seem more related than their cup illustrations communicate (to me, at least). I said it before I read this follow-up and I say it now that I have: the game seems rigged to get the desired result.

That aside, they just about completely give their intentions and biases away with the following:

"Is it possible that concreteness is helpful but only for younger participants who CANNOT acquire an abstract concept otherwise? In particular, children may need a concrete instantiation to begin to grasp an abstract concept. This argument finds support in constructivist theories of development (e.g., Inhelder & Piaget, 1958) that posit that development proceeds from the concrete to the abstract and therefore learning SHOULD do the same." [my emphasis]

There are several problems here. First, I don't hold that we can know in advance that any given child CANNOT acquire any concept without a particular method being applied. I'm not a mind-reader or a fortune teller. I can't make such predictions about my own learning, let alone about that of someone else. The purpose of using concrete models - pictorial, palpable, narrative, electronic, or otherwise - is to help ground understanding of mathematical ideas, rules, and procedures. It's a fait accompli that kids can learn division without a clue as to what they're doing, why it works, or what the results mean. But if one's goal is to impart more than rote knowledge and procedural facility, simply teaching the rules and having students practice them MAY not suffice for any deeper understanding to develop. And so we use models and tools of varying kinds to help kids (and older students) who may be stuck in grasping what's going on and why.

The researchers seem fixated on the notion that these concrete methods will always be where we begin when teaching math to kids or anyone else. And that sort of rigid teaching is foolish. But the goal of the study appears to be to "prove" that since we only have one choice - to either start with the "generic" abstraction or to start with a concrete instantiation. And then we should CLEARLY pick the former, given the results of this research. My sense is that having created a false dichotomy, these researchers are committed to saving us from the evils and shortcomings they believe are perhaps inherent in "concrete" pedagogy. Have they thus provided real evidence, or merely the results of a rigged game?

In fairness, those who think this research is pristine and conclusive need to consider if they'd accept unquestioningly the contrary results from a study which appeared to be slanting matters in advance to give a desired outcome. I agree fully with Mark's comments below. We really do need to see if this study leads us somewhere useful when we look at results from K-5 kids and if the research results are replicable, but I wonder if the research as constructed is actually worth redoing. I think more thought needs to go into the selection of the mathematical task and the "concrete instantiation." Perhaps results will emerge that show that, not unlike light, which exhibits different properties of particle or wave depending on how one tests it, the suitability of particular methods for grounding or illuminating abstract mathematical ideas will vary depending both upon the ideas and the particular methods. I would suspect that further, the individual learner (not to mention teacher) will prove to be an important variable. If I'm in any way representative of people trying to learn something new in mathematics, different approaches may work better depending on a host of issues.

But there's more to consider. One huge question this study fails to address is that of the INSTRUCTION. Who, exactly, constructed the teaching in this little game? Were learners getting excellent, interactive teaching about how to make connections between the concrete objects and the underlying mathematical structure? Or were the manipulatives (okay, the PICTURES of the weird little symbols) supposed to MAGICALLY convey understanding to students? Having apparently not done a bang up job, the concrete objects are blamed for latter confusion. How pat. It's fascinating that the authors reference one of the most important articles on this whole issue, Deborah Ball's "Magical hopes: manipulatives and the reform of mathematics education," and yet manage to COMPLETELY miss her point: the mathematics isn't in the objects, it isn't in the model, and it isn't in the metaphors. Nor is the teaching or the learning. That exists only in the interplay amongst the mathematical concepts, the teacher, and the learner, with the objects or model merely providing tools. No one, least of all Deborah Ball, who really understands teaching mathematics, would ever believe otherwise.

Perhaps most importantly, why is the goal of the researchers to show that we should either always or never use concrete methods (or the "generic," for that matter)? Who is putting guns to our heads and insisting upon all concrete methods, all the time? Of course, it's not hard to find teachers who use all "generic" abstraction, all of the time in math classrooms at various grade levels, or who lean very heavily in that direction. I'm sure that such teachers would be outraged if forced to abandon symbolic methods entirely, or only to be allowed to introduce them after working through concrete ones. And the reverse would be true for those who are inclined to use the concrete or other "not generic" approaches first and/or foremost.

Reasonable people take a less dogmatic, more pragmatic approach, knowing that there are no panaceas, but that one size doesn't fit all, nor should it have to. Mathematics and its teaching and learning are not Procrustean beds, even if some folks prefer that they be so.

Reasonable people take a less dogmatic, more pragmatic approach, knowing that there are no panaceas, but that one size doesn't fit all, nor should it have to. Mathematics and its teaching and learning are not Procrustean beds, even if some folks prefer that they be so.

Mark Roberts said,

There are things about this study that can be

questioned and certainly replication of this study

with a different underlying mathematical concept,

different teaching materials etcetera is needed

before any definitive conclusions should be drawn.

Yes, Mark. Lots to question here. Or to swallow whole if one is so inclined. I, for one, am not. I hope the NEW YORK TIMES hires science and education staff writers with a little more skepticism. Maybe it's time to bring back Richard Rothstein.

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