Consider sports, music, and foreign language. Most people grow up being exposed to what these things are, and in particular often observe experts in action: professional athletes and musicians and native speakers of foreign languages. People also often have opportunities to engage in learning these things from a young age, and their education or training can extend to the majority of their lives. While perhaps not as socially valued as “practical” skills or “core” academic subjects, these are generally quite highly valued abilities.
How do people learn sports, music, and foreign language? It is largely through learning by doing. A learner has to attempt kicking a ball, or bowing a violin, or pronouncing a word, and develop an understanding of the connection between what it feels like to do and what the results are. All of these things involve precise control over one’s body, which can only be learned through physical practice. But how can a learner begin to make an attempt in the first place? They must see or hear what it looks or sounds like, or at the very least have the action explained to them. Then, they can attempt to imitate and compare their results against the expert or instructor’s. With sports and music, an instructor may at times physically position the learner’s body in order to show them what they need to do. And of course, the learner must then practice the skill repetitively.
Now, let’s talk about math. “Traditional” math education is somewhat similar to what I described above. To learn a skill, e.g. long division, the teacher would explain the process, demonstrate it by working through a problem (or multiple problems), then students would repetitively practice applying the skill. This is in fact an effective way of memorizing a step-by-step process. So why has it fallen out of favor with so many math educators?
For many people this is a perplexing question. In my experience teaching math at the high school level, there were some students and parents who were very frustrated with the “modern” approach. In the curricula I used, the step-by-step process by which a problem can be solved was often (temporarily) concealed from students. Instead, students were expected to struggle with trying to figure it out themselves or with their classmates. To some, this felt like a waste of time. There is something of a meme or stereotype that parents have become unable to help their kids with math homework, since the homework may not consist of practicing a step-by-step process or may be practicing a different process than the parent is used to.
I think of the term “math” as referring ambiguously to two different things. First is math as a field of inquiry, and second is math as the content of that field. To draw an analogy to science, there is a difference between understanding chemistry and knowing how to do chemistry.
“Traditional” math education focuses on the content or knowing-what. “Modern” math education is attempting to teach students both. From the perspective of teaching the knowing-how, it doesn’t matter too much what specific skills or types of problems are covered. The important part is struggling to understand something and learning how to make connections between seemingly unrelated ideas. There are plenty of math topics we could be teaching in algebra II, for example, and in fact this can be seen in the differences between curricula. Sometimes matrices are introduced, sometimes not. Sometimes complex numbers are introduced, sometimes not, and so on.
When a student asks, “When will I ever use this?” The answer is very often never. Mainstream high school curricula are usually oriented towards preparing students for college. Not everyone goes to college, and even among those who do, math skills can be more of a gatekeeper than something they really need to know for their desired field of study. Even for me, a math major in my undergraduate career, there were many things I learned in high school math but never needed to know how to do. For example, I never divided polynomials in the decade or so between learning it in school and teaching students how to do it.
In my opinion, people (mostly parents) tend to overestimate the importance of learning specific facts and skills in math, even as they personally deem those things practically useless. People simply don’t need to know so much math to be successful in their daily lives and in many careers. Knowing how to do math, on the other hand, I think would be useful to virtually everyone.
The purpose of teaching specific topics (post-algebra), as I see it, is to open up opportunities. The only reason it does open up opportunities, however, is because those are the things people have decided will be gatekeepers to further academic or career progress. What the topics are, specifically, is contingent; in other words, there’s no reason it had to be that way. So “modern” math teaches these specific topics, but also has a goal of teaching the knowing-how part. With precious little time during a semester to cover even everything that is essential, it is impossible to teach knowing-how separately from teaching the essential skills. So, a process that would “traditionally” be shown and explained to students right away now becomes a chance for students to try doing math by struggling to figure something out.
Here’s the problem I have with that: in order for students to learn how to do math, it is important that they not be taught merely to imitate the teacher; however, this does not mean it is necessary for students not to learn by mimicry at all. What should be demonstrated to students is how to do math. That’s a tall order for most math teachers, though, who largely learned their own math skills in the “traditional” way.
So, what is the solution? Let’s turn back to sports, music, and foreign language, and consider what those things have that math doesn’t: people are not exposed to other people doing math in public or on TV. It’s unclear to most people what doing math is actually like, and think instead only of applying step-by-step processes to solve a problem. Math is not something people want to spectate, like sports or music, and it’s not something that comes up in day-to-day life, like language.
In my opinion, the solution is reading. Reading a math paper is not like reading prose, even for the sections that look superficially like prose. It can take a lot of effort to understand what a paper is saying, depending on your familiarity with the subject matter. I think math educators should be producing mathematical texts at grade-appropriate levels and having students read regularly as part of their curriculum. Note that this is nothing like “word problems,” since the goal of a mathematical writing is to explore ideas, not to set up a problem to be solved.
Again, it’s a tall order. Pie in the sky even. Still, I believe this is the direction math education should be going in if our goal is to provide the most benefit to students.
thoughts on mind 