We will always relearn old facts. Addition and multiplication appear so frequently and so fundamentally that you just can’t avoid them. They chase you around in circles, and much of mathematics is like that. Like addition and multiplication, some topics in mathematics are so fundamental that at any stage of progression or expertise, you will encounter the same things over and over again. It’s well known that repetition can further ingrain certain concepts in your head, but sometimes doing it too much can cause you to implode: if you’ve seen something so many times and understand a concept so well, at what point will seeing it one more time cause you to forget everything I know?

I don’t know the answer to that question. It was meant to be rhetorical, because maybe it doesn’t have an answer at all, or it is specific to each individual mathematician. This leads me to a conversation about the American education system: it doesn’t work. An education system must equally combine cognitive capability with general aptitude with certain precision to produce the best students and inspire self-growth and independent thinking. China has a wonderful education system, in that children are burnt out by middle school and commit suicide with the same frequency with which the earth orbits the sun. America has a wonderful education system, in that it treats everyone equally with very little consideration for differences in capability, instead insisting that everyone is “very good”, neglecting those who suffer and those who excel. Not to mention the frequent school shootings, but that’s a discussion for another time. Both are examples of extremes.

China has it’s own educational problems, but ultimately it produces intelligent, capable, individuals who will without doubt go on to achieve great things. America, perhaps not so much. Especially in regard to mathematics education, educators often reiterate to their most advanced students how great they are. While it can be a good motivator, sometimes a student can be most benefitted when told the truth. The truth is that not everyone is brilliant, and conversely, not everyone is stupid. I believe that every mathematics student should always be humbled by what they encounter. I say this from experience, as I find that I learn the most about both myself and my potential when I am made aware of my lack of knowledge.

While trivialities and repetition are necessary, it is through struggling and not being told how “great” you are that you really push yourself. You should always strive to be better. However, you should also know your own bounds. An example is the field of topology. The value of topology is in its rigour, but to truly digest that rigour, you mustn’t go about it like reading a children’s book and looking at pictures like topology is some mystical field. I admit that right now, I don’t have the background to learn topology. I don’t deserve to learn topology because I would only do the field and myself a disservice: I know nothing of what topology really should be, and to treat it exactly like a picture book would make me hate the subject. Regardless of what anyone says, this is a fact. You must know what you can and cannot do.

So, to change my tone to one of more optimism, maybe some repetition can be beneficial. I have always found helping others with mathematics to be a wonderful opportunity to test your own depth of knowledge, so perhaps when you are familiar with something you can be of assistance to others. Maybe there is truly a teacher in me after all.

It is difficult to quantify what mathematical growth truly means. Personally, mathematical growth is dictated by my ability to read proofs, to understand difficult concepts, and to approach a field with enough confidence and experience to be leisurely curious. In other words, mathematical maturity. However, “growth” suggests that I’m gaining something, that I’m not only applying what I’m knowing but being surprised by little things and large things that I’ve never encountered before that develop my mathematical maturity. While learning old mathematics may solidify my mathematical maturity, it rarely gives me new insights into something I already know (unless it is something I know little about). Perhaps I should find positive in relearning old mathematics and seek out things that I never would have considered? Perhaps I should enter new dialogues on adjacent topics to gain enrichment in another form? I don’t really know. We’ll see.

(30 June, 2024)