Welcome to Insight Axis, where I make connections between practical philosophy, technology, books, science, and more. I’m Zan - follow me on Twitter (X)and Substack.
Most people find maths boring in school. I wasn't one of them. But even then, some of the things we learn in maths are utterly useless.
I recently read “The Math(s) Fix" by Conrad Wolfram. It's a tedious book to read, but despite this, Wolfram lays out an excellent framework for how we should teach maths. Here's the summary.
0. Setting the stage
All maths in school should be applied to solve fun and meaningful real-world problems. This is not what happens right now. Often, kids are stuck with pointless lists of dull, repetitive exercises instead. We still emphasise the "computation" of equations, rather than true problem solving. But Wolfram says that we should now let computers do the computing, and let the small humans do the thinking instead. He suggests that maths is a 4 step process (define, abstract, compute and interpret). For too long, we've focused on the "compute" step, without teaching the rest of it.
Here's his remedial framework of what the future of maths should look like.
1. Define
First, students need to learn how to define a real world problem. This also means thinking about what assumptions they make about any problem. For example, should they assume friction is negligible when hitting a baseball?
This is the most important step, and it's frequently skipped in school. Defining the problem means you can find other ways to solve it too, even if it doesn't involve maths. So it's a truly multi-disciplinary skill.
2. Abstract
Once the problem has been defined, it's time to abstract it out into mathematical terms. For example, the trajectory of a baseball gets turned into a quadratic equation.
This step is also skipped in school, and it's usually the hardest. Abstracting a problem into "maths speak" is like being a translator. But often, kids are too shaky in their understanding to do this, because maths is still taught as a rote subject, and not a symbolic language.
3. Compute
After abstracting the problem, it's time to compute (or "solve") the problem.
This is the stage that most maths education starts. Students get a list of quadratics to solve. Or they get a bunch of probability questions to churn through. They simply apply the algorithm - over and over again. Boring.
4. Interpret
Now that the problem is solved, it's time to interpret the answer. There are a few questions which you should answer in this step:
Does this answer make sense?
Does it seem wildly wrong?
How does it change when I challenge my assumptions in step 1?
But the most important question is: "what does this answer mean for me, and the real world problem I'm trying to solve?". Pure maths doesn't teach us this. But school maths should.
This final step is completely missed in school. Also, I know there will be people who read this and argue against changing maths education. They'll say: "it's important that kids know how things were solved before calculators and computers".
And my answer is: no, it's not. Not for school kids, anyways.
Why all school maths should be applied maths
Reason 1: for maths to be useful as a tool, kids need to see its utility in solving problems they care about.
This eliminates the problem which most kids have with things like trigonometry. You'll often hear students say: "What's the point of learning trigonometry when I'll never have to use it ever again?" This might be true. But what if we gave a senior-year baseball player the question: "what angle do you need to hit the baseball for it to land in the stands?" Maybe they'll care about the answer. It might even spark their interest in mechanical engineering.
Reason 2: standard schooling should cater for the majority. Those who want to study pure or abstract maths can do it at university.
Schooling should exist to help the next generation be smart, competent and engaged members of society. Maths education in school should reflect those aims. This doesn't mean that we shouldn't pursue pure and abstract maths. It means they should either be extra-curricular or voluntary in school. Of course, university is different. Society has cordoned off university as a place that exists for the sake of intellectual pursuit. In this type of environment, pure and abstract maths is a perfect fit.
Reason 3: applied maths is future-proof
Currently, we focus on the "computation" step of maths. But as we've seen in the last 50 years, the tools of computation have changed. We've moved from simple calculators, to computers, and now sit at the dawn of artificial intelligence.
We should focus on teaching the mathematical process, rather than specific computational tools. We should equip our kids with the critical toolkit to know when it's appropriate to use the given mathematical tools of the day. We should teach them the timeless skill of identifying important problems, choosing how to solve them, and interpreting their meaning.
To summarise
Maths in school should be contextual, to make it more engaging and useful.
Recommended reading from Substack:
👴
distils Seneca’s philosophy.🥱 Bored?
gives a stimulating argument for boredom.🖥️
asks ChatGPT about the relationship between intelligence and optionality.🪄 For a different twist to my readers, I recommend
’s writing: Twilight Zone Meets Alice in Wonderland.✅
gives us a life review checklist. Great spring cleaning for the late summer.🤖
’s excellent as usual AI roundup.
Beyond Substack:
Not reading this week - but an excellent podcast. Taleb rarely does podcasts, so this is a real treat.
If you enjoyed this essay, please like, subscribe, and share it with others.
I learn from my readers, so leave a comment with your thoughts
If you haven't read it, then I Lockhart's Lament is worth reading: http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf
I also like Jo Boaler's approach to school-level mathematics: https://www.youcubed.org/ (the Ideas section is probably the one to look at, although looking at examples of activities may give more of a flavour - the Mindset Mathematics set of books has some great things in it).
I think much comes down to why we teach mathematics at school. Some basic numeracy, geometry and statistics are good for everyday life, but for most people there probably isn't much they need beyond that, so the question is whether we are teaching it as an art or as a tool for the sciences. And there is a large intersection there to muddy the waters. I always suspect too that many people would really enjoy university-level mathematics who never discover that they would because they are only exposed to school-style mathematics. I was lucky that I encountered the more beautiful aspects of mathematics early on - and that I enjoyed them!
You should check out @holymathnerd if you haven't found her on Substack yet.