Story and Disclaimers

Ever since I started studying math olympiad, I and friends always had the conversation about how “usual exams” feel boring and unimaginative. Like we do not use our brains, rather we only “let our hands or mind go” and just circle the answer we “believe” is right. I think this is the main reason why we loved advanced maths, we were actually producing thoughts whereas in the usual SAT/AP-style tests (or YKS in Türkiye), you basically memorize every possible type of question that can be asked from the curriculum which is absolutely no different then muscle memory.

So I ask a simple question to myself: do students who solve problems by memorizing the techniques, really produce thoughts at the process?

As with all thought-based posts, these are my own ideas and feelings, you may or may not agree and it is totally fine!

Defining and Comparing Problem Solving and Producing Thoughts

I am not a philosopher nor a neuro-scientist, however, as someone who uses brain-power to advance his knowledge every day for the past 8 years or so, I firmly believe that I qualify to at least attempt to define what problem solving and producing thoughts is. Here are my attempts:

Problem Solving (those asked at usual test-style exams): By mixing and merging the available tools, attempt to brute-force your way through the problem while letting similar problems and their solutions lead the way.

Producing Thoughts: By mixing and merging the available tools, try to expand your knowledge graph by making connections that seems initially unrelated, and once you are satisfied with what you have, attempt to produce a solution to problem.

Two things to notice in the definitions:

1. They start the same way: know your tools. This could be mathematical theorem, software libraries, or electrical components.
2. Problem solving requires you to memorize problems and solutions, while producing thoughts requires you to absorb the solution idea by adding it to your toolset.

Why you don’t need to produce thoughts for at least the first 18 years of your life

We send our children to school so that they learn the concepts of science, literature etc. They sit down to their designated desks, listen the lecturer for hours, return home, and hopefully do their homework. Notice that the only part where children touches the subject, is where they attempt to solve problems for their homework. Unfortunately though, problem solving does not necessitate producing thoughts: they simply try to remember the techniques and information they were presented during the school day.

It is the same during the exams, too. As far as I have observed throughout my education, almost no one care for the actual subject, the intellectual work, or its history; only thing on their mind is what they score out of the text or exam, and how this score reflects their overall grade. Well, you can’t blame them, it is what success is, at least on the surface.

I believe the main answer to the heading is as follows: we are taught solutions, not problems. Why are we interested in derivatives or integrals in the first place? Why do we keep throwing the damn ball up a cliff over and over again? Why do we obsessively calculate the numbers in front of some random substances in an chemical equilibrium that we don’t even know how they look like? Why do we try to name some random bacteria while we don’t even know when the flower in our garden blooms? In other words, who asked the the most intuitive (hence probably the first) question so that we devised these tools, logic, and concepts?

I feel like there is always this larger frame of mind, the initial reasoning behind the most fundamental concepts, the underlying question we are not told. We learn the solutions, “if you encounter this type of question, you should do this and that”. I hate this idea. I hate it when I am required to do anything without properly understanding its background and motivation.

How should the education system adapt

I am sure that I am not eligible to write out a complete solution to this question as I am not an educator (even though I have little experience on teaching during high school :)). However, I would like to mention a concern of mine that I haven’t seen anyone feeling the same.

You see, when we initially listen to concepts during the lecture, we create node(s) in our knowledge graph. Then, we connect the newly created node(s) with the nearest or the most similar part in the graph. If we do not study it after class, the link weakens, eventually breaking apart. The thing is, we don’t even try to expand our knowledge graph on our own, without a text, without a video, without someone teaching. While studying, we (may) connect more nodes, or strengthen existing ones, but we do not come up with new nodes to add to the graph.

Why so? I believe the blame is the common phrase “do not discover America again”. As if we won’t gain new insight while discovering what we are not thought. As if it’ll be a “waste of time”, and you should “just watch X” or “just read Y” or rather a more modern answer, “just ask an LLM”.

I recently discovered Gemini’s guided learning feature. I don’t intend to be a advertiser or some sort, but I believe it is a unique feature amongst other LLM providers. When you activate it and ask it to teach a concept, it provides a brief background and ends the response by asking a question. To solidify, I have used it in a long-form conversation two times: 1. writing a 2D physics engine from scratch and 2. discovering what differential equations offer. In both cases, I asked it to let me do the math and let me observe what is evident. This way, I aimed to create nodes on my own.

An ideal Student’s Perspective

We need to ask a lot, make connections between subjects, sit down and think calmly and comprehensively. Curiosity, it is the key factor. We can (and I think should) inquire about behind the scenes, what is not thought, or what is not asked, that is not wasting time at all. We need to exchange questions rather than answers so that others may create their own nodes.

I especially think that learning science history is crucial for understanding initial questions. We emphasize the names, say broadly what they did. What about how they started in the first place?

Why the statement “Smart people enjoys problem solving.” is wrong

I believe I made myself clear so far, that you may answer this question as “Smart people like producing thoughts” not “Smart people enjoys problem solving.”

Do you know anyone who enjoys solving “YKS Hazırlık Test Kitapları” or “SAT/AP test books”? We all finished at least a dozen of testbooks for every course. Why, because we enjoyed it? No. Because we wanted to cover every possible question that we may get asked. We were afraid of losing the precious time. We did not want to waste both time and resources for producing thoughts, reason thorough and decide the correct answer during the exam, we preferred retriving someone else’s thoughts and techniques from our database of problems. And that, I believe a sick idea of its own.

Moreover, this is also the reason why many “successful” high school students fail their courses when they graduate to university. They are simply good pattern recognizers and problem memorizers, not thought producers. On the other hand, smart people shine: they ask questions, are always on the run for the big picture, expand and connect their knowledge graph. They can compare what they know with what they do not in an objective way.

Ending

Thanks for the reading it through! If you agree on anything I said, please do not use “I like problem solving” in your daily life, but prefer “I like to produce thoughts, turn the cogs inside my brain.”

Feel free to reach out if you think something else.