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 and Part 2.) Here in Part 3, we will concentrate on making the curriculum more “conscious” of what students are supposed to be learning in mathematics.
One of the primary complaints students have about higher mathematics is that they don’t see where they are going to use the information later in their jobs. There are a number of ways of answering this question but first I want to emphasize the importance of having an answer to give. While we might wish students would just trust the system, the fact is that a fair number of them don’t trust the system. Failing to give a succinct reason for why we make them do hard things often means they simply won’t try.
But now here are some specific responses: It’s true that students actually won’t need to know most subjects to do their jobs in life well. Daily performance as a middle manager probably isn’t significantly improved by a knowledge of ancient Egypt, Shakespeare, the amphibian life cycle, or the Pythagorean theorem.
However, the goal of education is not solely to make a person better at their job. The goal is intellectual freedom. Freedom often comes from understanding the world we live in. The mathematics learned in high school are the “greatest hits,” the intellectual achievements that undergird all the things that our society is built upon.
The second reason is specific to mathematics. Math gives us the tools and practice of the art of reason. This is hard for students to see because it will be many years before they recognize it in themselves. But the practice and the confidence that comes from rigorously determining solutions to problems, using a sound method, is a gift that will help them throughout their lives.
I tell my students this: Mathematics gives you practice on core reasoning skills with concrete problems that have definite answers in order to achieve mastery of the reasoning process itself. That will allow you to apply reasoning skills to fuzzier problems where answers are not as certain.
On top of that, most mathematics topics teach a specific logical skill. Take factoring a number into prime components, for example. What does it teach? It teaches the practice of examining something to find out what its “core components” are. Then you can manipulate that object based on its core components.
Take the number 120. Its prime factorization is 23·3·5. We can then rearrange this as 15·8, 6·4·5, 60·2, etc. The ability to break something down into its core components and reimagine it is a principal reasoning skill behind almost every “process improvement” a middle manager does.
Learning to do it with numbers is helpful because numbers offer right and wrong answers. The student gets immediate feedback about whether the logical principle has been correctly applied. Flaws in the process can be clearly identified. The student can use this reality check to hone thinking. What the student cannot do in math class is merely substitute glibness for correctness—as is possible in other subjects.
Additionally, it helps students if we use concrete applications. While some students are highly abstract thinkers, other students need connections to reality in order to make sense of things. Oftentimes, the higher concepts (such as complex numbers) are presented so abstractly as to cause extreme confusion. Coming up with concrete ways of thinking about each subject will help a large swathe of students.
If teachers and textbooks are more explicit about why students are asked to learn these things, students will be more motivated to study them. The goal is to help them see their math education at work on these three levels:
● the freedom that comes from knowing how the world works
● the practice of the art of reason
● the specific problem-solving skills that a given topic is teaching
Awareness of the goals can help a student get a longer term perspective on why adults are continually asking them to study things like algebra and geometry—subjects that just don’t seem as exciting as the latest buzz.
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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 4: To fix math education, see it as a program that needs an update. As a computer programmer, I’ve seen this problem in my work: The basic idea is still sound but “fixes” have made it too complex. We need to “refactor” math education, the way programmers must sometimes refactor old code that still works in principle but needs simplifying.
Bartlett’s calculus paper reviewed in mathematics magazine. The paper offers fixes for long-standing flaws in the teaching of elementary calculus.