In this series we are looking at ways that math education can be reformed. In contrast to some other math reform efforts, we are not trying to fundamentally rewrite what math education is doing but to simply admit that we can do better and see where that takes us. (See Part 1, Part 2, and Part 3.) Here in Part 4, let’s look at specific content issues that, I will argue, we could improve when we do a curriculum revision.
Mathematics is an old subject. We have inherited quite a bit of mathematical thought. We must educate future generations so as to make sure that this hard-won knowledge is not lost. But one of the biggest impediments to our task is simply the way in which mathematics is presented.
Here is an illustration that may help: In computer programming we sometimes talk about “legacy code.” Legacy code consists of working programs that have been handed down to us, usually from earlier programmers. Oftentimes, as change requests have come in, one programmer after another bolts features into the code. After a while, the bolt-ons start making the code itself confusing. As a result, later programmers have a hard time making sense of how everything fits together.
Eventually, the code must be “re-factored.” This means that we pull the code apart and rebuild it so that it makes a lot more sense to those who are currently using and developing it.
I think the same process is needed for math education. We have added many bolt-ons over the years. But now we need to stop and rethink both what we teach and how we teach it, so as to decrease the cognitive load we are asking students to undertake.
As an example, take the concepts of complex numbers and vectors. Both of these concepts use multiple numbers to represent a single value. However, in complex numbers, the symbol i (the imaginary unit) is used to represent the part of the value that is on the y-axis. In vectors, the symbol î (pronounced “i hat”) is used to represent the part of the value that is on the x-axis. So, we are essentially using an almost identical symbol to represent the y-axis in one set of problems and then we switch it to represent the x-axis in another set.
There are historical reasons why these developments occurred. But, as with computer programming, the confusion they cause is the reason why, after a time, the system needs refactoring. In this particular case, the electrical engineers have already solved the problem: Many electrical engineers use a j for the imaginary unit, which matches the one used to represent the part of the value that is on the y-axis for vectors.
Another case is the Pythagorean theorem. While almost all students are aware of the Pythagorean theorem, I have found that few students are aware of the many other theorems which are based it, such as the distance formula and many trigonometry identities. Requiring students to memorize each of these formulas separately is problematic for two reasons. It adds to the burden of memory work. But, more than that, it robs students of the opportunity to do mathematical thinking.
I would greatly prefer to hear a student tell me how to go from the Pythagorean theorem to the distance formula than for that student to correctly solve distance formula problems a hundred times over. Those problems can be solved by a computer program. But grasping the relationship tells me that the student can think deeply about the material we are covering.
More broadly, students need to be able to generate formulas. As above, if students can generate a formula from basic facts, not only are there fewer things to memorize but—and this is much more important in the long run—they can command mathematics and logic to do their bidding.
A great math problem would be a practical one. For example, ask students to create a formula for a catering budget based on a head count, and then to modify that formula to see, given a particular budget, what the head count would be.
Too often, formulas in mathematics simply seem to fall from the sky and students are merely asked to use and obey. That works well for younger students who just need a thinking tool to begin with. But our goal is to eventually get students to think for themselves and generate their own solutions to their own problems. Math can help with this but only if we train students to think in a logical way as a normal routine.
In short, the problem with math education isn’t the content. It’s both the way that we are presenting the content and the way that we are asking students to think about the content. Refactoring math education—rethinking the way that we sequence, notate, present, and practice—mathematics can significantly improve both how well math is absorbed and retained and how effective it is in helping students make use of their logical training in later life.
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Part 1: How can we really fix the way math is taught? First, we must understand why we teach math in the first place. Math teaches students how to think more clearly in all areas of life but it mostly performs this function silently, invisibly.
Part 2: Straight talk about fitting the math curriculum to the student. We need to avoid pushing too much too soon, lest students come to see themselves as “bad at math” when they are just not ready for it. About math drills: Every algebra teacher I’ve ever met will tell you that instant recall of math facts is the best predictor of algebra success.
Part 3: Helping students see how math benefits them in the long run. To keep them motivated, we need to answer the “Why bother?” question honestly and directly. Most mathematics topics teach a specific logical skill that will help students solve problems on any career path.
See also: Bartlett’s calculus paper reviewed in a mathematics magazine. The paper offers fixes for long-standing flaws in the teaching of elementary calculus.