It's pretty common knowledge that math is hard. Some people love math, some people hate math, some people get really good at math, and some people stop after high school. All of these people, though, will likely agree that math is hard.
It is important, though, to take a moment and try to understand precisely why math is hard. I think the importance of this question goes beyond philosophical rambling – a proper understanding of why math is hard can help us teach and learn the subject more effectively.
Different Kinds Of Hard
Becoming a skilled soccer player is hard for different reasons than becoming a skilled piano player is. In soccer, one must develop tremendously accurate gross motor skills and gain a deep understanding for the optimal ways for several players to move across a field. For piano, the physical skills are largely confined to one's fingers, and one (usually) must also learn a new written language called "music."
Therefore, a soccer player will train by running a lot and learning different formations, because he or she knows that that's what it takes – this is the "kind of hard" that he or she must endure. Similarly, a piano player will practice scales, transcribe recordings, and work on music theory because this is the "kind of hard" that must be endured.
Even within a single field, like music, there are many different "kinds of hard." A piano player can sit down and almost immediately start making some kind of musical-ish noise. However, a classical upright bass player, or a horn player, might need weeks of work before being able to make any kind of reasonable sound come out of their instrument. The overwhelming majority of the time, teachers of these instruments know these different kinds of hards, and assign work to their students accordingly.
What type of hard is math?
The way I see it, math is hard for two reasons. One reason is relatively small, and the other is huge – and unlike any other kind of "hard" that exists in human endeavors. Here they are:
- Reason 1: We must learn a new language.
- Reason 2: This language is unlike any other language on Earth.
Reason 1 is the much smaller reason for why math is hard, and is very similar to the "hardness" that musicians go through when they learn to read sheet music. There's a whole collection of words, symbols, meaning, and grammar to learn. Like any language, one must learn to read, write, speak, and understand the language of math, and like any language, this is hard. That said, the hardness of math really comes from Reason 2.
This new language that we have to learn – math – is unlike any other language. To see this, let's distinguish between how humans learn a second language, and how they learn their first, mother tongue.
Learning a second language is hard, yes, but it's made much easier by the fact that we have another language (our mother tongue) to map it to. I.e., we can translate. This is a luxury that we do not have in mathematics. The vocabulary and grammar that we have to learn in math does not have analogues in our mother tongue.
Of course, we must distinguish between, for example, "English" and "math expressed in English". We obviously communicate math in some human language (like English) but a word like "symplectic manifold" – an English word to be sure – does not have another "translation" in English. So when I say we can't "translate" math, I mean that we can't translate from "non-math" English to "math" English. The concepts (like "symplectic manifold") that we learn when we learn the language of math simply don't have analogues in our normal human languages.
So, learning "math as a second language" does not have the same luxuries that learning other second languages has. But it's even worse – i.e., harder – than that. If we think about how we learn our mother tongue, we see some massive disparities.
Namely, we come out of the womb bombarded by human language as well as the real-world representations of the concepts that that language conveys. For example, we hear the word "tree" and we can look at a tree. We hear the word "hunger" or "food" and can feel food in our mouth and our belly going from empty to full. As our ears and brains get inundated with the words in our language, our senses similarly get inundated with what those words mean.
However, we never look at a symplectic manifold or taste an elliptic curve. The objects that the language of math describes are things that we can only access in our brains. This is because math is – as discussed here – the study of capital-T Truth. Our physical world – the world that human languages are meant to represent – is not a world of capital-T truth. It's a messy, chaotic, imperfect world full of chemistry, biology, and emotions. And while that's awesome, it's not mathematical.
Therefore, when we're learning this language of mathematics – the language that has no analogues in our everyday languages – we also don't have the luxury of "real-world examples" to associate things with. We therefore have to create these structures in our brains and then use those same brains to name and reason about those structures. This is not only hard – it's a kind of hard that is unique to math.
What not to do
Knowing how uniquely hard math is, we can stop trying to impose pedagogical lessons from fields that are hard in different ways. As you may have gathered by now, we at Cohomologous hate "tips, tricks, and memorization" in math. These things of course have their time and place when it comes to efficiency, but generally we think these are relied on much too heavily in the early years of math education.
This is in part due to the fact that tips, tricks, and memorization are hugely beneficial in most other human endeavors – and many other academic fields. However, math is uniquely difficult, and so we shouldn't necessarily expect these pedagogical approaches to apply.