As some of you may know. Skanda and I have been doing a bit of research on possibilities for mathematics education reform. Today, I came across a particularly interesting paper by Dr. Constance Kamii entitled "The Harmful Effects of Algorithms in Grades 1-4". Kamii is a Professor in the Early Childhood Education Program Department of Curriculum and Instruction at the University of Alabama at Birmingham.
The paper, published in 1998 proposes that our current incarnation of elementary-level math education may be errant in its emphasis on teaching algorithms over proper mathematics. Of course, no elementary school student would ever use the word "algorithm" to describe what they learn in math class, but the fact is that most of what American students are taught in early stages of their math education is little more than arbitrary sets of rules that help to compute the results of basic mathematical operations. It looks like math, and they learn it as math, but really, it's just a bunch of isolated computations. As Kamii puts it:
"an adult can explain an algorithm for summing two digit numbers to a child. However, listening to a teacher explain this does not ensure that the child realizes or establishes a mathematical understanding of how to combine two quantities."
But why is a mathematicical understanding of the concept of summation so important? If we've taught our kids enough to tackle any addition problem that comes their way, isn't that enough? Kamii argues that it's not, pointing out that algorithms are completely arbitrary, while the laws of mathematics are universal. She says that teaching math by way of arithmetic algorithms is based on the false premise that mathematics is something of a cultural inheritance that needs to be communicated to each new generation. It's possible that this very standard puts certain students at odds before math class even begins. When students are taught procedures like two digit addition or long division, they are taught as if the algorithm and the concept are one in the same. If that student then seeks help from a parent who was never introduced to that specific algorithm, the student will almost certainly end up even more bewildered than when they began. In the space of a single day, they will have been told that one mathematical concept is two different things. This may seem like an overly-specific scenario, but the problem is magnified when you consider the fact that it applies to all students whose parents did not recieve a traditional education, therefore marginalizing a group that is likely in greater need of support.
Kamii's suggestion for reform is as follows: drop the algorithms entirely and let children develop their own brand of mathematical thinking. It may sound radical, but her studies have shown that it is actually a more effective method of teaching. Students that learned simple mathematics without algorithms performed better on exams, and made far more reasonable errors. Attempting to compute 6 + 52 + 185, many students who learned through algorithms came up with answers in the 700s. This is part of what leads Kamii to believe that traditional pedagogical methods in mathematics actually cause students to unlearn concepts like place value.
Earlier this year, Skanda blogged about a TED talk given by Conrad Wolfram which followed the same basic concept in its suggestion for mathematics education reform: drop computation from the curriculum in favor of a "purer" math. Wolfram, however, recommends that we place a greater emphasis on the use of computers in math education, which are of course algorithmically driven. Though this may seem to pit Wolfram's advice against Kamii's, the proposals are actually strikingly similar. Both emphasize the need for a sharper line distinguishing the procedural science of computation from mathematics itself, and both would have students develop their own train of algorithmic thought. The challenge, of course, lies in developing the new pedagogical techniques that would be necessary to make such a shift.