Let's Start with a Thought Experiment

Imagine your company is hiring an accountant. During the interview, you casually ask three candidates the following question:

What is 1234 × 9876?

The first candidate immediately takes out a calculator, types in the numbers, and gives you the exact answer.

The second candidate takes out a piece of paper, carefully writes down the vertical multiplication steps, and eventually reaches the same answer.

The third candidate pauses for a few seconds and says, “It should be roughly 12 million. The last digit should be 4. Do you need the exact answer?”

How would you rank their performance?

The first candidate did nothing wrong. Accountants should use calculators. In fact, an accountant who refuses to use available tools would be concerning. But after this interview segment, you know very little about the first candidate other than that they know how to use a calculator, which is a very low-level skill in the modern world.

The second candidate showed more. They are probably careful, detail-oriented, and capable of executing a collection of simple tasks accurately and in an organized way. Those are valuable traits. But you might also worry about their decision making. If their first instinct is to execute the task using a time-consuming process, they may be reliable but not necessarily a strong big-picture thinker.

The third candidate is the most impressive. They did not reject computation, nor did they blindly jump into it. They understood the situation. They estimated the magnitude of the answer and identified a quick error-checking signal: the last digit. Then they asked whether exact precision was needed. That is not merely arithmetic skill. That is judgment.

The Importance of Mathematical Judgment

This little scenario captures something important about math education. Machines have been better than humans at computation for decades. Yet we still teach students the multiplication table and long division. Why?

It is not because we expect adults to spend their lives multiplying four-digit numbers by hand. In the real world, we want people to use tools. But companies also want people who can glance at spreadsheets and identify inconsistencies, notice when an answer is off by a factor of ten, and understand how many digits after the decimal point are sufficient for a given purpose.

In other words, the calculator did not make arithmetic understanding obsolete. It reduced the value of quickly and accurately carrying out precise computations, but elevated the value of people who have an overarching understanding of arithmetic. 

Before calculators, the ability to carry out exact computation itself had practical value. A “human computer” could be employed to perform calculations accurately. That role has largely disappeared. But the person who can quickly say that 1234 × 9876 should be around 12 million has demonstrated an internal sense of quantity, which remains valuable, decades after almost everyone has the tool to compute the exact answer instantly.

That internal sense does not come without effort. It is developed through working with numbers, making estimates, performing operations, seeing patterns, making mistakes, and correcting them. At some point, the purpose of practice shifts. We do not practice long division forever. We practice it enough to build intuition and an understanding of the underlying mathematical structure so we can move on to study other and possibly more sophisticated structures.

This distinction matters because we are now entering a similar but much larger transition.

AI is becoming capable of doing much more than computation. It can solve algebra problems, write proofs, generate explanations, produce code, analyze data, and offer strategies. In some areas, it can already perform mathematical tasks at a level that would have seemed extraordinary just a few months ago.

So we face an uncomfortable question: if AI can already solve all the math problems most people will ever encounter in their lives, should students still learn to solve those problems themselves?

The answer is yes. However, the problem solving process should have a clear purpose.

Students should still learn to solve problems by themselves, not because solutions generated manually will always be rewarded as an end in itself, but because independent problem solving is how humans build the internal models needed for judgment and vision.

The future equivalent of number sense is mathematical judgment.

Math Education That Builds on Human Strengths

In the era when AI tools are readily available, the most valuable human will not be the person who can outsolve AI, but the person who can elevate and guide a team’s collective intelligence by asking:

• Which problems are worth solving first? Which can benefit most from a mathematical model?
• What is the right balance between accuracy and simplicity?
• Have we asked enough questions? Do we have the right framework to define and start solving the problem?
• What set of axioms should be used to build a new model or even a new branch of mathematics?

AI makes mathematical power more accessible, but mathematical judgment is what determines whether that power gets aimed well. And like number sense, this judgment requires dedicated practice to develop.

A student who has never discovered their own false assumption will not truly understand why proofs are the foundation of mathematics. A student who has never solved nonroutine problems will have a hard time knowing which strategy is promising. A student who has never discussed solutions with peers will have a harder time understanding how mathematical ideas persuade, surprise, and excite other people. AI disproportionately empowers people who reach the top in a field, but you cannot get there without going through the educational training that builds those internal models.

However, this does not mean math education should remain unchanged. The advancement of AI should force us to think more carefully about which kinds of practice build judgment and which kinds merely train students to perform tasks that machines will do better.

Some manual practice remains essential. Early in learning, students need to build understanding through direct work. They should calculate, manipulate, draw, prove, and solve without outsourcing every difficulty. They need to experience what it feels like to be stuck, to try an approach, to abandon it, to notice a pattern, to form a conjecture, and to finally see the key idea. These experiences are not inefficient obstacles on the way to the answer. They are the mechanism by which mathematical judgment is formed.

At the same time, not all manual work deserves equal protection. Long chains of routine symbolic manipulation, exhaustive casework with little insight, or repeated execution of a known algorithm may deserve less emphasis once students have built the underlying understanding. This is when they should learn to use tools intelligently. They can ask AI for alternative approaches, compare solutions, find flaws, generate examples, and test conjectures. Eventually, they should become people who can move fluidly between independent reasoning and tool-assisted exploration.

Assignments and assessments should also evolve. We should ask students not only to solve problems, but also to recognize patterns, identify hidden assumptions, estimate before solving, generate counterexamples, compare methods, generalize results, and ask better questions, all of which are already baked into every AoPS course and book.

A valuable math class should include prompts such as:

“Before solving, identify upper and lower bounds for the answer.”

“Study the examples and then share and discuss your hypothesis with each other.”

“Explain which method can be used to solve a more general problem.”

“Modify the mathematical statement so it remains true but becomes more elegant.”

These tasks require students to look beyond the immediate answer, reflect on the big picture, and develop the kind of judgment and vision that can eventually elevate them to the frontier of the field.

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