11 July 2008

Crazy Constructivists

I’ve been teaching math at the local community college for several semesters now. My first course was a finite math course, with non-math, non-science majors being the primary target. One of the things that immediately struck me about the book was its apparent backwardness: in introducing a new concept, the book would invariably present some examples, then posit a definition of sorts, then enumerate any steps or processes involved. I was extremely puzzled by this order of presentation. Why on earth throw up some examples before defining or even describing whatever it is that you wish to illustrate? I simply didn’t get it.

Since then, I’ve had several other courses: business calc, probability & stats, “quantitative literacy,” math for elementary ed teachers (the latter being a whole ‘nother story in itself). All of these books followed the same order of presentation – bass-ackwards. I couldn’t fathom why, and I merely continued preparing my lectures in a more logical format: definition, example, steps, problem.

Concurrently, I’ve been reading more about the current state of the American educational system. A common theme seems to be how the progressive-constructivist philosophy has by and large supplanted the traditional, instructivist methods in the classroom. This philosophy, in turn, has resulted in less and less actual learning taking place in our schools, a phenomenon I’ve noticed as a community college instructor and as a private tutor.

I didn’t put two and two together, though, until the other day, then it dawned on me: the textbooks for the courses I’ve been teaching have all been written by authors who have imbibed from the constructivist well. They’ve bought into Piaget’s odd little notion that knowledge is not “true” knowledge unless you’ve “constructed” or discovered it yourself. Aha – so that’s why these crazy books obdurately insisted on placing examples before giving any definitions or enumerating any steps. That explains that.

I still don’t get it, though.


Barry Garelick said...

All knowledge is "constructed" in the sense that there is some processing the brain needs to do in order to get the idea into long term memory. Direct data dumps into the brain do not work. What constructivists don't like to hear is that knowledge can be constructed via direct instruction and in fact there are "aha" type discovery moments when information is imparted in such fashion.

I admire your site and your mission.

You may want to read an article I wrote about the href="http://www.hoover.org/publications/ednext/3220616.html">state of K-12 math ed in the US and how it got that way.

Barry Garelick said...

Sorry. The embedded link didn't work. The URL for the article is:


bky said...

Giving an example before giving the definitions and so on is not really a new idea nor is it pedagogically ass-backwards or necessarily tied to constructivism.

Giving an example first is a way of showing the student where they are going. Mathematicians often talk about "motivation". What is the motivation for a certain definition or even a topic area. The motivation can be indicated by showing an example: we can now work problems like this .... [interesting example follows]. A good use of example would be to show how you can do a special case of whatever it is you're trying to do "by hand" or by exploiting the simplifying features of the example. How do you do the general case? Why for that we need the following definition .... or whatever. This is no crazier than writing an introduction to a report.

Niels Henrik Abel said...

Thanks for visiting, barry! Agreed, there generally needs to be some "wrestling" with whatever subject is at hand to get a good grasp on certain subjects. What drives me crazy is the insistence that one can't possibly know anything unless one has first-hand experience. I see this not only with regards to education (i.e., "don't teach kids the long division algorithm - let them (hopefully) figure it out on their own" - a stupid waste of time!), but also in the broader context of life (i.e., "you can't possibly critique Mr. X unless you've read ALL his books, listened to ALL his lectures, and expressed your concerns to him personally over dinner or at least a cup of coffee").

It is this constant insistence of the necessity of the wheel's reinvention that I find irritating. Knowledge can be gained vicariously, and it's a far more efficient method than constant construction.

Niels Henrik Abel said...

Thanks for stopping by, bky! Part of what you raise, I think, depends on the level of student. Advanced students are more likely, I believe, to take a "where am I at now, and where do I want to end up" approach to problem-solving, or study in general. In other words, they grok the motivation for what's being done, and they can tie it all together.

However, I find that most students (at least the non-math ones, who are just taking math because it's a prerequisite for whatever major they have) prefer a cookbook approach - you have this situation and that condition, so that means that you follow step 1, step 2, step 3 (and there'd better not be more steps than 3!). Showing examples before definitions tends to lead to a "can't see the forest for the trees" situation.

rightwingprof said...

"Giving an example first is a way of showing the student where they are going. Mathematicians often talk about "motivation"."

Yes. See here (specifically, the World Series problem).


Derek said...

bky's point about motivation is a good one, as is Niels' point about more advanced students not necessarily needing as much motivation.

The other reason to lead with examples is that many students learn best inductively--making sense of many examples before starting to see the pattern. Starting with a few concrete examples can go a long way to help these students move to more abstract ideas.

I find that many majors are fine with learning deductively (moving from definition to theorem to examples), as are many math professors. I think it's important to remember that our students, however, might not learn the same way we learn. Switching the order of a lecture to put a few examples first can do a world of good for many groups of students.

Niels Henrik Abel said...

Thanks for stopping by and commenting. Of course, one can easily get into trouble when stating generalities, and consequently it's easy to forget that one size doesn't necessarily fit all.

I guess the main reason why I tend to present definition / theorem first and then follow up with examples is because it seems like with the order reversed, most students have a difficult time seeing the forest for all the trees. Perhaps I should think about which topics would lend themselves more to an inductive introduction. I'm thinking, though, that it should be something with a fairly obvious pattern, lest it muddle things - you know, the law of unintended consequences :)