Authored by Kyle Hovey, Software Developer
Despite the apparent overwhelming stigma against mathematics education, according to a survey of 1,000 students done by Texas Instruments, 46% of students report that they liked learning about mathematics in school. Perhaps the stigma stems from something deeper than preference: practicality. Sixty-eight percent of students responded that they would appreciate mathematics more if they knew where it would be applicable to their own lives. It is odd, then, that no one would question the usefulness of critical thinking in their own lives. Therein lies the current problem of math education: math is viewed as machinery that will provide some immediate and topical utility to your own life.
Much like critical thinking, however, mathematics skills are more abstract in their benefit and the value is impossible to measure directly. The ties between critical thinking and mathematics go deeper than this fact. Mathematics is built upon an ancient edifice of logic, constructed by some of humanity’s most brilliant logicians. In a way, the reasoning employed by mathematics is dual to that used in science. In science, we rely on the principle of uniformity of nature to uphold that empirical evidence will provide enough certainty about the structure of the world around us and how it will continue to behave in the future. Mathematics, however, relies on deductive reasoning from some very simple truths (axioms) that we accept on the basis of faith alone (there exists a number 0, representing nothing etc…). Along with some rules of inference, we grow these axiomatic seeds into ornate theorems powerful enough to develop understanding of even things we could never hope to interact with.
Sadly, this structure is not revealed to most students until graduate-level courses. This can make sense if most students will only need to use the results of mathematics for STEM careers, but perhaps participating in it is what would drive a better understanding of its place in our lives and endow learners with an enhanced ability to synthesize heuristics, and not only recall/compose algorithms.
Procedure or creativity?
In this way, in order to truly appreciate mathematics, you need to partake in it, not simply absorb its results. Ironically, creativity is not often associated with mathematics. More often, curriculum is based around memorization of procedure, and pattern matching to learn when to apply them. It is important to be adept in these procedures, after all, just like it is important to be fluent in a language before you can speak it. Teaching literacy (despite being the main focus of modern math education) is not the problem, giving students agency in their own creative endeavors is. The challenge, then, is to balance this added context with application. For instance, if a teacher were to explain a concept such as fractal dimension, they could approach it by stating the definition:
The proportion that a manifold’s hyper-volume scales when all metrics are adjusted by a scalar amount.
It would be absolutely forgivable to see a class full of bright students glaze over when hearing this, and it could practically be expected that not many would understand. Still, the teacher could proceed with the algorithm of how to calculate this concept given a structure, and the class could pass an exam with flying colors without ever knowing what it was that they were even being tested on. What if, instead, you began with an example:
Fractal dimension is akin to a measure of roughness. If you were to measure the coastline of Britain with a meter-stick, you would measure a shorter distance than if you were to use a foot-long ruler (due to being able to measure more fine details). Notice that instead of measuring with a smaller ruler, you could also just scale Britain up and use the meter-stick to get the same change. Because of this, the coastline of Britain can be said to be more than one-dimensional, but less than two-dimensional. That is to say that if Britain were to be made n times larger, then the coastline would be n^d times larger where d is the fractal dimension of the coastline, and d is more than one, but less than two.
Cultivating an appreciation for math
What if, after explaining this, you let the students converse with one-another and gave them some tools to find other instances of fractal dimension? Perhaps the first assignment could be to simply research this topic, and have students come up with their own thoughts on the matter. Perhaps the teacher could offer an interactive software tool to build fractals and measure their fractal dimension. The next assignment could ask the students to create a fractal with a given fractal dimension. By the time this second assignment was done, maybe the students wouldn’t even need to memorize a formula. They themselves could create the formula themselves, reconstructing it from their total and personal understanding of the concept. Imagine not having to study for a math test, but instead having to come prepared to think creatively about problems and develop solutions based solely on understanding.
I can offer personal testament to this technique. There was a time I struggled with mathematics. I was profoundly fascinated, but always seemed to fall short of the requisite ability to receive the grade I was aiming for. Then, one day, a close friend of mine walked me through one of their favorite theorems. It was one that was not formally taught in school until graduate classes, yet the way it was explained made it easier to understand than finding the focus of a parabola, or computing the determinant of a matrix. I realized that, armed with this theorem, I could easily derive the 30+ expressions I had diligently written down on flashcards in my backpack and had spent hours memorizing. More than that, I understood at a fundamental level how each of those expressions were derived. Still, I couldn’t tell you the exact value of learning that theorem, or exactly where it would be used in my life from that point on. In fact, the man who developed that theorem (Leonard Euler), though being one of the most brilliant minds to ever grace this earth, could not have expected where this knowledge would be used. We now rely on that theorem to build our bridges so that they won’t collapse in a windstorm, so that we can store thousands of songs on an SD card, so that we can edit photos and view them on our devices, so that we can have wireless communication and broadcast signals into space, and so that we may understand the quantum world that we cannot see.
I’m omitting much, but it is complete and simple enough to say this: the value of mathematics education does not lie in the heuristics we associate with it, but rather in the development of organized abstract thought that powers the technological advancement of our society. If students can come home with one belief about mathematics, I hope it can be that one.