At university, I took a compulsory algebra class in third year. Up until this point the mathematics we were studying had been relatively easy to digest, provided you put in the hours. One day my friends and I were stuck on a problem which used a particularly tricky theorem (I wish I could remember what), when we asked for some help the professor told me something that always stuck with me.
"Just stare at it a bit longer."
What changed in this class was not just an increase in workload, but also in the level of understanding you needed in order to engage with the material. In computer science, the other side of my dual degree, there really aren't many concepts that are conceptually difficult to get your head around; if you hold all the facts in your head at the same time, then usually the idea will click. In late university level mathematics, you not only have to hold in your head an understanding of the definitions and theorems that define a concept, you have to also understand the concept itself. Here a tutor can only take you so far, there has to be a point where you make the connection. That's where this advice comes in, advice that I would say can be applied to not just pure mathematics, but to any problem and challenge you come across.
"Just stare at it a bit longer."
You might imagine that I found this to be slightly (very) infuriating. I was in this course to be taught! That was the professor's job, and here he was saying that instead of helping me understand, I had to just stare at the problem? How can reading the same thing over and over increase my understanding? To this day I find myself telling this anecdote mockingly when I'm asked what goes on in a maths lecture, thinking it will highlight the hand-wavy, non-sensical reputation which many associate with pure maths. I've since realised that this was one of the best pieces of advice I ever got as a student, and it epitomises, in a phrase, exactly what is so beneficial about studying mathematics.
I've always found it difficult to articulate why studying maths is a good idea. I used to be sure that I would use all of my years of accumulated knowledge of lemmas and theorems in my career, and while I haven't quite given up on that dream, I've realised that this understanding is not the main benefit of a degree in mathematics. Just stare at it a bit longer conveys the fundamental skill that all mathematicians share; problem-solving.
To be honest I hate the term "problem-solving skills", it's most commonly used to describe the learning gained from experiences where you in fact learned nothing, but I hope I can convince you that mathematics is the exception, not the rule.
Problem-solving for complicated problems is a process, of course we don't all solve problems in the same way, but I'm sure many people can relate to something like this. Every problem is made up of sub-problems. For each sub-problem you either know the answer or you don't, if you know the answer for all the sub-problems then congratulations! The problem is solved. But what if you don't know the answer? Well, then we break down the problem again... and again, and again, until eventually we arrive at an impasse. We can't break the problem down, and we don't know the answer, this is what we call being blocked.
Maybe you don't have access to the information you need? Perhaps the definition itself is slightly too complex to grasp? Or maybe it's just a difficult problem. This is the crux, you can't go forward, and there seems to be nothing you can do to progress. And here is where mathematicians excel. We revel in the unknown. In being comfortable with not knowing. It's an amazing feeling when you find yourself here and have the awareness to really notice it. Here's where Algebra professor's advice comes in:
"Just stare at it a bit longer."
The feeling of staring at a problem you don't understand is usually a mix of disgust, boredom (itself a form of disgust), and maybe some frustration. But having awareness of and controlling these emotions allows you to familiarise yourself with the problem further, and see the situation in a new light. You don't need to be Buddha, this isn't a life-hack or cheat code, I really just mean stare at the problem, feel the cogs turn in your head. Take a break, come back stare some more, stare until your head hurts then stare some more. After all, the problem isn't going to solve itself!
You might be saying to yourself, "Isn't staring at a problem the only way to solve it?". Two things on this; one - you would be exactly right. There is no trick to problem-solving, but the skill is in having the awareness to tackle the problem head on. Often when we tell ourselves we are problem-solving we are really just procrastinating by solving unrelated or adjacent problems. Secondly, due to a certain technology which was released right around the time I first heard this advice, the practice of problem-solving has come into a very real decline.
A reliance on ChatGPT for essay writing (a form of problem-solving) has been shown to reduce cognitive ability as well as consistently producing a worse result, described as soulless by some markers. For less "soulful" writing like Facebook posts and marketing emails, overuse of LLMs showed an 11% decrease in motivation and a 20% increase in boredom. Even the content and analyses produced by AI are more narrow.
It is now more important that ever to remember to stare at the problems we don't understand, LLMs provide an easy road out that might seem tempting, but the science is showing what Orwell, Ray Bradbury, and Wall-E have been trying to tell us - reduced critical thinking, increased boredom and narrow-thinking will come. And I'm no AI-sceptic, AI is my career and expertise and I use LLMs daily. The key is in using them responsibly and with awareness, particularly in self-education.
Staring at problems head on is not just a method for problem-solving and essay writing. This is the approach you should take to everything, from daily quarrels to complete life crisis. Staring the gun down the barrel and being at home in the unknown provides an invaluable quiet confidence even in our most uncertain moments. So next time you find yourself stuck; on a problem set, a project, or even a moment in life where nothing seems to make sense; resist the urge to look away. Sit with it. Let the confusion breathe. Discomfort is an unexplored emotion, on the other side of it, somewhere, is the understanding you need.
"Just stare at it a bit longer."