Fixing Math Education 2
AKA "Drill-n-Kill" vs "Inquiry Based"
AKA "Traditional" vs "Progressive"
Notice how some terms just naturally sound better than others? Does that have anything to do with the substance of the ideas?
So there's really no general agreement on what constructivism is. Some people argue it's not curriculum or pedagogy, it's brain science. I simply use it to refer to those approaches of teaching mathematics that require students to develop their own mathematics from scratch. It might include "Problem Solving," "Discovery Learning," and "Inquiry-based Learning (IBL)." My apologies to those who will claim that I do violence to their pet definitions here. But there do seem to be two general camps. On the one hand we have those who think students should learn efficient, time-tested methods of solving problems, and learn them to mastery (automation). These I call "traditionalists." On the other hand, we have those who emphasize that students should learn to think creatively, develop strategies to solve novel problems, and develop deep insights into mathematics. These I call "constructivists."
The answer is actually simple. We need both. However, when sacrifices must be made, there is one approach that is essential, and one that is merely desirable. Unfortunately, reasonable people will disagree about which is which. This, however, is my blog, so my opinion is right.
The much-maligned traditional method is essential. We must first realize that there is a great deal of disinformation floating around about the traditional method. Its opponents claim the traditional method teaches rote memorization without understanding or thinking. Except perhaps in some isolated enclaves where stereotypically poor teaching took place, this has never been the case. All the widely used math textbooks of the 19th century, for example, emphasized "mental arithmetic," that is, the ability to think through multiple-step problems "in your head" and give the solution, not only without a calculator but even without a pencil. The kinds of thinking and understanding that were required differed from what is expected today, because the skill set expected of an educated person has changed. So in those days, being able to carry sums in your head was far more important than, say, sketching the graph of an exponential function. In honest debate, we must realize that "Back-to-the-Basics" or "traditional education" does not imply restricting ourselves to the content or objectives of a bygone era.
The primary features of the traditional method include: 1) Understanding a mathematical concept, e.g. "What does it mean to add two numbers?" 2) Memorization of basic facts/definitions/results, e.g. "the Times Tables." 3)Application of memorized knowledge to novel problems and more advanced concepts. 4) Review and maintenance of memorized knowledge.
The traditional method results in efficient learning and provides the foundation necessary for creative thinking, even if it fails to sufficiently address that objective, according to its critics. And yes, even educators from Singapore, whose students smoke the Americans in international tests, are looking with envy at the creativity of some of our students. This demonstrates that a commitment to the essential objectives may not produce all the results that are desirable.
It is well documented in cognitive science that the brain has a limited capacity to manipulate objects in "working memory." It is often said that we cannot process more than seven memory objects at once, which supposedly explains why phone numbers have seven digits (only now they have 10, but that's OK, because the phone remembers all the numbers for us). The working memory is where problem solving and creative activities take place. The working memory can access permanent memory for information it needs. However, any new information that must be taken in to solve a problem must occupy space in the working memory, thus taking away from the space available for creative activity. That is why we quickly become frustrated when trying to follow assembly instructions that include many terms with which we are not familiar. Even if the actual steps in the process are simple, if they involve several terms that are not defined in our permanent memory, those terms require space in our working memory which is then not available for solving the problem. This explains why the pedagogic fad of "learning to learn" is a failure. We do indeed need skills for learning--but such skills are utterly dependent on a reliable bank of information which can be accessed instantly and does not require the use of working memory. The first step in problem solving or creativity must be putting as much relevant information as we can into permanent memory.