
![MS paint drawn thinking emoji meme](/thonker.jpg)

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](https://www.media.mit.edu/publications/your-brain-on-chatgpt/) as well as consistently producing a worse result, [described as soulless](https://time.com/7295195/ai-chatgpt-google-learning-school/) 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](https://hbr.org/2025/05/research-gen-ai-makes-people-more-productive-and-less-motivated). Even the [content and analyses produced by AI are more narrow](https://lile.duke.edu/ai-ethics-learning-toolkit/does-ai-harm-critical-thinking/). 

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*."


Some of the worst advice I ever got that I use every day.

Just stare at it a bit longer



Some notes I've taken over the years on how to focus. I'll definitely update this as I go.

My credentials for this are none other than the fact that I am terrible at being productive. 

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## Deep Work

Deep work is the kind of work that really pushes the limits of what you can do, it requires complete focus and allows us to do what we know we can but rarely manage to. Importantly, we cannot be in deep work all the time, there's a maximum of around 4 hours or less for most people, and yes, you are most people.

### Build the environment

Even the action of mindfully choosing the environment you think works best for you helps to put you in the right mindset. Dedicate a total time for deep work, set the requirements of what you need beforehand. That is, do not change the scenery during deep work—same drink, same music, same place.

### Planning ahead

**Schedule every minute**, but acknowledge that this will not go as planned. The more you do it the better you will get at it. This is super important, no matter how busy, it is always better to have a plan. The point of time blocking isn’t to create a schedule you can stick to perfectly each day. Instead, this exercise forces you to be more intentional with your time and how you direct your attention.

**Focus on the widely important.** Try to keep most important goals at the forefront, dedicate the most time to these goals.

**Breaks are not optional**, they help *unlearn* the wrong solutions—sounds silly but it is absolutely true. Decide how you will spend your breaks. The best things to do are to move and talk. Social interaction is especially useful as it is more distracting than just going on a walk. Although breaks should be no longer than 15 minutes. Be disciplined with this.

### In the mud

**You want this.** When you're *in the mud*, that is, actually working, the motivation should primarily derive from the fact that you want this and you chose this. This is intrinsic motivation.

**Give yourself the satisfaction**, track how many hours you've worked. I like to keep a stopwatch up on my screen at all times, this way I can see how long I've been focused for. This provides a weird recursive loop where I feel good about having focused for X minutes/hours which motivates me to stay focused longer. This also works towards the end of the session where you can tell yourself "only Y minutes left!"

**Remove the option of distraction.** Being distracted is a choice, the sooner you realise this the better. If you don't want to be distracted then remove all distractions. If you keep using your phone then put it in another room or leave it at home. I guarantee the excuse you tell yourself as to why you need to keep your distraction around, keep notifications on etc is just that—an excuse. The best tool I have found for this is Opal. I use this to block every app on my phone and laptop which is not essential to the work I am currently doing.

**Recognise thought patterns.** If you catch yourself thinking about anything that is not the task at hand, notice it and bring your attention back. Identify thought loops—if you find yourself going in circles, make an active decision that will change this and move to the next step.

---

## Myths

**There's no such thing as "I don't have time".** You choose how to spend all of your time. You can make excuses for the gym and for projects but really, if you changed your priorities you wouldn't have to.

**Motivation is not real.** To get from thought to action we only need discipline.

**Choose to not be distracted by removing all distractions.** Tools such as Opal can facilitate this, also you could place your phone in a different room or leave it at home.

---

## Laws

**Parkinson's Law** — Tasks expand to fill the time they are allocated.  
The best way to work around this is the following exercise:
- Make a list of 3–4 long-term tasks you want to do.
- How would you do them if you only had half as long?
- How would you do them if you had one day?

**Pareto Principle** — 80/20 rule: the first 20% of input is probably where all the value came from.  
Identify what 20% of work is deriving 80% of useful output and vice versa—what's taking up 80% of your time and not contributing much to your outcomes?

**Newton's First Law** — Objects in motion stay in motion, objects at rest stay at rest.  
If a task takes less than 2 minutes to do, just do it—don't put it on the to-do list.  
If you don't want to do a task, do 5 minutes of it.  
Lower the activation energy: if you don't want to work, make a cup of coffee and set up the workspace, clean desk, so all you have to do is sit down.

---

## Powers

### Habits

You need to change the view you have of yourself to what matches your goals. If I view myself as a productive person then doing anything unproductive would just feel wrong.

### Productive down time

Use small bits of downtime—like a break at work or taking a shit or commuting—to get things done. Make it easy to do these things by having them accessible.

### Productive procrastination

Is what it says on the tin.

### Fun Factor

Making things fun is the best hack. Make the task social or gamify.

---

## Misc

**The craftsman approach to tool selection**:  
Identify the core factors that determine success and happiness in your professional and personal life. Adopt a tool only if its positive impacts on these factors substantially outweigh its negative impacts. Reddit and Instagram are not such tools.

**Work with my brain not against it.**  
Need to train my brain to keep good habits, it won't happen just because I want it to. I'm so used to bad habits that it needs practice to break them.

Notes i took during my undergrad degree on how to focus.

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