In the Age of LLMs, Mathematics Is Even More Relevant

As a professional programmer for decades, I've noticed many confusions about the importance of this craft, in particular around the need to learn it from people who don't intend to become programmers. I think the main source of confusion is the lack of understanding of the difference between syntax and conceptual problem solving. But here's the truth: any professional programmer can learn new syntax in weeks - this is the easy part. It's crafting problem-solving skills that's a lifelong exercise, and this is hard and important to everyone, regardless of the profession choice. ?

Calculators and computers solved computation long ago. Now, large language models (LLMs) have solved programming syntax. This is fantastic! It allows us to focus on problem-solving, the real art of computer programming. It is funny to see the crowd defending "Let's all learn how to code" getting mute after LLMs. They too thought syntax is equal to problem-solving!

Surprisingly, LLMs technology can assist in problem-solving too, providing ideas and demonstrating specific cases with spectacular performance. They're so effective that we must elevate our skills in communicating with them. This requires us to get better at conceptualising and describing the essence of problems. And guess what tool humankind developed for that? Mathematics.

The shallowness of current mathematical education

Let's look at my 11-year-old son's math worksheets as a proxy for mathematical education. These worksheets, often themed around current news and pop culture, aim to be student-friendly. But they usually feature convoluted wording that, when decoded, leads to simple math problems with tricky decimals. The result? Frustration upon frustration: students struggle to decipher the wording and if they succeed, they will fail trying to perform computations with wild decimal numbers. Where is the dopamine hit?

These exercises rarely offer opportunities for deeper investigation. They're shallow, leaving no room for exploration or discussion. In my view, the wording should be simple, the problem should be deep, not hard, but deep, with several layers.

Misconceptions about Mathematics

We should learn mathematics not just to solve problems at hand, but to equip ourselves for unseen challenges. Mathematics enhances our problem-solving abilities by forming new brain wiring and fostering creative thinking. It's about developing a mindset to navigate and solve complex issues, even those we haven't yet encountered.

As with programming, there is confusion and misconceptions about mathematics:

1. Maths should have always practical applications to be engaging: There's nothing wrong with abstract exercises without practical applications. Mathematics is inherently abstract, and exploring this abstract universe is crucial for achieving higher levels of understanding and reasoning. Typically mathematical concepts are invented/discovered centuries before a practical application is found, or not.

2. Equating mathematics with basic arithmetic. While mastering percentages is valuable, it's as far from true mathematics as paint-by-numbers is from art.

3. Believing we need to learn prescriptive math before exploring creative approaches. Wrong. Alienating students with rules without understanding is a recipe for disengagement. We should always optimise for understanding.

4. Thinking there's only one way to solve a problem. Typically, there are several ways, sometimes overlapping and sometimes completely different.

5. Thinking mathematics is a robotic soulless activity: Mathematics is one of the richest and creative tools humankind ever invented. It is based in pure thought and there are fewer constraints than any other human discipline. It developed a special kind of aesthetics and achieving this platonic beauty is not rarely an important motivation for mathematicians.?

Standing on the Shoulders of Giants

In this age, there's even more pressure to rethink how we approach math education. Let's focus on inspiring curiosity, fostering deeper understanding, and emphasising the beauty of abstract thinking. Although Australia is "okay" in mathematical education worldwide, we can do much better. I don't see young people jumping to learn math as I think they should. Look what is happening in the world!

Fortunately, we have amazing resources. I strongly recommend Numberphile, a wonderful YouTube channel where Brady Haran interviews professional mathematicians. The focus is on understanding, and even sophisticated concepts can be explained to an interested layperson in a 10-20 minute video.

I always watch Numberphile with my 11-year-old son. He loves it. I curate the content for his age, but you'd be surprised how much kids understand when enthusiastic, intelligent adults present their craft. Sometimes, I use a video to elaborate on an idea or create something new. Recently, we watched "Solving Seven" https://www.youtube.com/watch?v=Ki-M1DJIZsk where James Grime explains how to know if a number is divisible by another.

I developed a concept to explore the results mentioned in the video with help from my son, GPT-4o, and Claude. Visit this link https://alexandre-eisenmann.github.io/diophantus/ to explore the concept of the divisor graph James mentioned.

Some project details:

* The team: My son, GPT-4o, Claude, and myself. Although it is not a long project, my son gave me the stamina to actually finalise the concept.?

* The models can't create the concept in one go. We had to guide them, through multiple prompts and iterations.?

* Claude interface is better than GPT. It renders the artefact you are working in one separated panel, reducing confusion and increasing focus. The flow of the chat happens in another panel and your main artefact is replaced every time to an improved version.

* Decide what to offload the models is an art by itself. Too little is just a distraction and you are better off without it. Too much and they will work further of your cognitive horizon and you will spend a lot of time debugging. There is a Goldilocks zone!?

* This Numberphile video is great because it explains how the concept works, why it works, and why old rules (like divisibility by 3 rule of adding the digits) work in the context of divisor graphs.

* I chose this problem for its aesthetically pleasing divisor graph. I hope the images showcase some of this beauty. ?