Alan, (by my reading, anyway) does a good job of extablishing the fact that we are stunningly ignorant about a large number of questions that are very critical to the fundamental claims made by both "sides" of the so-called "Math Wars". In fact, one is led to wonder whether the fury of the debate surrounding these highly politicized issues might simply be the turbulence induced by a localized vacuum of knowledge, but that only leads to less productive questions about whether knowledge vacuums are ever very localized.
But since our need here is simply to establish ignorance2, let's leave that question for now. What I would like to do is make a guess, a perhaps educated one, as to what some of the answers to these questions might look like if we ever get around to actually figuring them out. In mathematics, some of the most important work that has been done is in response to conjectures (one may argue that all of mathematics is the making and testing of conjectures, but then one would have quite an argument on one's hands). I am going to propose a few conjectures which I hope will prove to be fertile ground at least for discussion if not research.
Conjecture 1: The learning of mathematics is every bit as natural as the learning of language, and is learned best in "natural" ways.
It has often been said that if we attempted to teach children to speak, they would never learn. Children do not need to be taught to speak--at least, not in the sense that we usually think of "teaching". The extremely successful model we have for "teaching" speech is simply to immerse the "subject" in an environment that is rich with speech, including interesting speech examples and highly skilled, readily available speech mentors. That, basically, is the entire "pedagogy". If we applied current mathematics teaching methods to speech, we would keep earplugs in the child's ears most of the time, so as not to confuse them with the gross complexities of adult speech, and then on certain occasions take the plugs out and recite conjugations to them for a few minutes or so. Then the plugs would go back in until the next lesson. Eventually, maybe by the time they were twelve or so, they would be up to things as complex as compound sentences, but you would be able to tell they hated it by all the grunting.
Hopefully the world will never know what the outcome of such an experiment would be--but we have such an experiment going on now with mathematics. To me, a person who loves mathematics, the traditional curriculum appears to have been chosen to deliberately discourage interest in mathematics (and conjecture two will address a particularly bad implication of that). The traditional curriculum is like learning reading and writing but without anything exciting or interesting to read or write about. Sure, a few born writers may get it anyway, but most people are going to do the least they need to get by, and attempt to avoid all future contact with literature.
The conjecture I'm making is that we will find that "the best" (for a sufficiently loose definition of "best") way to "teach" mathematics is to immerse our subjects in a mathematically rich environment and then quite naturally interact with them in increasingly complex mathematical ways as they grow into the ability and demand such interaction. It works astoundingly well for the acquisition of spoken language. Is there any reason to believe that it would fail for mathematics?
Conjecture 2: The single most important factor determining the long term success of a curriculum is the degree to which the curriculum piques the students' curiosity.
The most difficult thing about proving this is convincing yourself that there is anything to prove2. A few moments' reflection convinces me again that what I remember from coursework is precisely that subset of it that was interesting to me. Not only that, but as I observe and speak with other people, I am led more and more strongly to the conclusion that attempting to "teach" someone something that they have no interest in learning is worse than a waste of time. Worse, because in forcing the material on the students, we kill any hope of a future interest in the subject. Worse because the students who do not want to learn dramatically reduce the rate at which the class can progress. Worse because of the moral implications of forcing someone into taking in information that they would not naturally choose to. But to return to simple, pragmatic concerns, there is so little question in my mind that attempting to teach an unwilling learner will have no positive effect that I barely consider it a question at all. Not that that should matter to the rest of the world--but if you think that you are going to show that there are significant long-term benefits from attempting to teach uninterested students, well, good luck, I guess.
For the "back to basics" folks, well, the implications would be less than helpful to their cause. Given that their case generally revolves around content that is easily and repeatably testable (which more or less translates into "dull and lifeless"), and methods that are anything but natural or amenable to student interest, there wouldn't be much point in returning to them. However, all hope is not lost for them--in fact, the numerical literacy that they wish to engender is probably more easily reached by methods enlightened by the conjectures. So, if all they are really interested in is the resulting skills, they should be quite happy.
But the other side would have much to learn as well. Although a fair amount of effort has gone into what could be viewed as the creation of richer mathematical environments, the topics that are introduced and the skills that they intend to develop are generally selected based on purely external criteria. There are notable exceptions--for example, in some cases, topics are introduced specifically to bring the subject alive. Also, a significant amount of effort is made in reform materials to show the students the relevance of the material. However, our second conjecture, if true, would be a strong argument for another approach--rather than selecting the content and skills to conform to some externally dictated agenda, and then attempting to backfit relevance onto this framework, it would make sense to develop the curriculum from the standpoint of what kids are naturally interested in at a given developmental stage. Not only that, but a great deal more latitude for individual exploration would probably have to be built in, given that interest is going to vary widely from student to student.
But the possible differences between what is now being done in the reform efforts and what would hypothetically be done if the conjectures were proven are only part of what's interesting. One of the most fascinating possibilities regarding this is that the reform efforts are basically arriving at the same conclusion independently. I mentioned above that some of what they are doing could be described as creating a more mathematically rich environment. The use of manipulatives, computer visualizations, writing about mathematics, etc are all justifiable based on other considerations--but they can also be seen under the unifying principle of immersion in a mathematically enriched environment. Similarly, a great deal of effort is expended to do things like relate the study of statistics to the graphs and charts in the newspaper. This is done, presumably, to show the students how mathematics applies in the "real world". Now, since kids are not usually all that interested in the real world, I doubt that these efforts alone are going to help much. But they are addressing something similar to what conjecture two would warrant--attempting to show the students the relevance of mathematics. It's really quite a small step (in some ways--small pedagogical steps can correspond to great political leaps, of course) from here to actually choosing the content based on what students will find interesting.
When two approaches or world views appear to be pointing to the same eventual truth, well, it's just plain exciting. Never mind the fact that you might have just found two distinct routes to the same blind alley-- it's neat that you both ended up there. I appreciate the need for and the value of research, but it's important to remember that in something as complexly human as teaching, intuition can be just as important a guide. My intuition tells me that heavily interest-based curriculum design and current educational refom movements are headed to the same, wonderful place. As long as we can keep from sounding like raving lunatics about it, and realize that attempting to drag, rather than entice, people there is a bad idea, it's going to be a great trip.
The author, Michael South, is in charge of vision at fulcrum.org
This seems so obvious to me that I am tempted to invoke the famous
"We hold these truths to be self-evident" cop-out. Of course, it's
usually the "obvious" part that comes back to bite you three hours
into your proof, so let's at least pretend that we need to support
this conjecture.
It is very tempting to invoke anecdotal evidence when 99% of the
population has similar anecdotes. Nearly anyone that you talk to can
tell you about a teacher they had who "made" a subject interesting
(I have an issue with this usage--it seems to imply that a topic is
intrinsically dull until someone magically infuses it with the ability
to fascinate, while the truth seems to be that the topics have plenty of
power to captivate but are robbed of it by exasperatingly dull textbooks,
assignemnts, and presentations), and for some reason that seems to be the
only class that they remember anything from. When people talk about these
teachers their eyes light up and they often volunteer a description of
some of the main topics or a project they did. Even if it has nothing
to do with their daily lives, the career they eventually chose,
or subsequent classes they took, that material stuck with them.
Further evidence is abundant. Watch the tremendous concentration that
a young child will put into putting a lid on a pen, taking it off again,
and trying to replace it. Some study is said to have shown that you get
about one minute of attention span for each year of age. But measure
how long a child will pay attention to an activity of their own
choosing and the results will be off the charts.