There are countless articles claiming that everything we know about math education is wrong. Everyone seems to have a pet theory, and the more radical the solution, the more attention it gets. But radical new ideas, exciting as they are, frequently come from reactionary positions, and reactionaries over-correct. Thus, I don’t expect that my list will be particularly shocking, as it’s mostly common sense and what I’ve learned from experience.
Too much too fast
Although it seems obvious that we shouldn’t move through lessons too quickly, it’s always a temptation. Pressured by our schedules, we think to ourselves that maybe the confusing bits will get clearer as we move on. This approach may work elsewhere, but leads only to frustration and despair in math. As much as possible, we should approach mathematics with a mastery mindset: a student should proceed to the second lesson only after she’s mastered the first.
Memorization without understanding
When a concept is difficult or has a complex proof, it is easy to force students to memorize the idea without understanding where it comes from. For instance, it’s easy to make our students memorize the quadratic formula without ensuring they understand how it’s derived and why it’s so useful. I don’t mean, however, to disparage rote memorization. Not to know our multiplication tables would be crippling. But in high school, it is especially important that students understand the origins and proofs behind mathematical concepts. Not only does it aid understanding, but it allows students to see the beauty and wonder behind the ink on the page.
Understanding without memorization
It is easy to err by extremes. If we over-emphasize student understanding and never force students to memorize important tables and theorems, we risk stunting their progress. Mathematical fluency, like fluency in any language, requires two things: an understanding of the rules and a good memory. Knowing the grammar of French but none of the words leaves you unable to speak it. Likewise, understanding math principles without memorizing key formulas, patterns, and operations will leave a student in frustration, requiring her to work out, with great difficulty, problems that would otherwise be easy.
Getting lost in the details
Like persons in a corn maze who have no idea how the maze looks from above, students frequently miss the beauty in math by getting lost in the details. There’s something like a narrative in mathematics, but it’s easy to miss if we’re too busy trying to beat solutions out of problems. Many math concepts build on each other like chapters in a novel, and it’s important to periodically take a step back and look at this progression the way we review plot points of a story.
We won’t appreciate mathematical ideas the way we should unless we can see how they answer problems mathematicians struggled with–sometimes for years. They are each grand inventions and discoveries, but if we only ever see them in isolation, as tools that merely allow us to proceed to the next chapter in the textbook, we’ll miss their beauty and brilliance.
For instance, the Pythagorean theorem is a wonder. Unfortunately, many students greet it with yawns rather than gasps of astonishment. But if students’ imaginations are first led to the old maps depicting vast regions of unexplored territory before being offered the theorems that fill in the blank regions, they will find their studies to be a journey of discovery, rather than an endless series of exercises.
The study of any subject is enriched by history. People are, after all, fascinating. To paraphrase Livy’s famous line, in history we find cautionary tales and inspirational figures. Why then do we not spend more time discussing mathematicians in math class? For instance, wouldn’t students be more excited to discuss irrational numbers if they knew that, according to legend, the Pythagorean who discovered them was murdered in an attempt to keep irrational numbers a secret? While discussing biographies takes time away from doing math problems, it is worth it to inspire a love of mathematical learning.
Using patronizing materials
As any observant high school student is keenly aware, textbook publishers tend to be patronizing. Rather than treating students as serious scholars, most textbooks try to make subjects more palatable by formatting chapters like magazine issues, or by adding pictures of teenagers having fun. No high school student is fooled by the cool skateboarder on the cover of his math book; he knows it’s a ruse. More dangerously, however, by using patronizing textbooks, we communicate that the subject itself is so dull that our only hope to make it interesting is to use bright colors and images of smiling teens.
If we wish to instill a real love of learning in our students, we’ll treat them like serious thinkers and let the subject defend itself. In the math classes I teach at Polymath Classical Tutorials, we use primary sources, avoiding textbooks altogether. I’ve read advice columns by teachers who advise never doing this since the material is too dull, difficult, and complex for students. Fiddlesticks. I’ve had many students, some with little talent for mathematical thinking, discover a deep interest in mathematics by exploring primary texts. Nobody likes to be patronized, and every student wants to feel like he’s doing real and valuable work. And it’s even more inspiring if he’s succeeding at what others say is too difficult.