[Lim Woong] The mathematics we need

Every summer, I teach an introductory machine learning course in an international summer program. This year, the class brought together students from more than 10 countries. Most were first- or second-year undergraduates curious about how machine-learning algorithms work and how artificial intelligence connects to their own fields of study.

At the end of the course, I ask students to write a short memo for the next cohort. Nearly all say they need to improve their Python skills. Many also leave more determined to study mathematics. The first response is predictable. The second one interests me more.

For many students in Korea, serious engagement with mathematics ends once college admission has been secured. This is understandable. After years of preparing for high-stakes examinations, students often associate mathematics with admission rather than inquiry. It has served its purpose, so they move on. Mathematics has been presented less as a way to explore the world than as a tool for sorting applicants. Although my summer classroom is international, the contrast is especially striking in Korea, where mathematics carries unusual weight in the competition for college admission.

One thing that continues to puzzle me is that students with the strongest examination records do not necessarily find the mathematics of AI the easiest to grasp. This reflects a familiar pattern in Korea, where many high-performing students later struggle in university mathematics despite excelling on examinations.

Studying how artificial intelligence works reveals the limits of an education built around test performance.

Programming may be the visible part of AI, but mathematics gives the code its structure. It determines what an algorithm calculates, which assumptions it makes and why it succeeds in one setting but fails in another. Mathematics is how an abstract idea about "learning" becomes a sequence of operations that a computer can perform on data.

Consider principal component analysis, or PCA. Suppose a dataset describes each person, object or event through dozens of features, many of which contain overlapping information. PCA identifies a smaller set of new directions that captures as much of the data's variation as possible.

To do this, the data is centered and the relationships among the features are summarized in a covariance matrix. PCA then finds the matrix's eigenvectors and eigenvalues. Each eigenvector defines a new direction through the data, while its eigenvalue measures how much variation that direction captures. PCA keeps only the directions with the largest eigenvalues, helping reveal patterns that may be hidden in the original representation.

The notation is compact: Av = lambda v. Here, A is the covariance matrix, v an eigenvector and lambda its eigenvalue. The equation says that multiplying A by v changes the vector's magnitude but not its direction. In PCA, that direction becomes a new axis, while lambda indicates how much information it retains.

Here, the individual math concepts themselves are not rocket science. The difficulty lies in connecting the ideas. Students must move between centering, covariance, vectors, matrices, projection and changes of basis, while linking equations, geometric intuition and working code. Reading the symbols is only the beginning. They must explain what those symbols mean, translate them into code, examine the results and decide whether they make sense.

The students who make the most progress tend to stay with a difficult idea. They ask why an equation has a particular form. They sketch the geometry, write a few lines of code, inspect the output and try again. They do not assume that immediate confusion means they are bad at mathematics. They often return the next day with more questions.

Those are productive mathematical experiences. Students form conjectures, test them, discover where they fail and revise them.

Mathematics becomes something to investigate rather than something already settled. A student who solves a difficult problem in three minutes may be well prepared for an examination; one who spends three days asking why a method works may be better prepared for research, engineering or data science. We need both abilities, but our admissions culture rewards one far more visibly than the other.

A test-driven curriculum can produce students who solve specialized problems with remarkable efficiency. It can also make them reluctant to begin an unfamiliar problem without an approved method. When every question is expected to have one answer, one route and one deadline, students learn to search for the examiner's intention rather than question assumptions or explore alternatives.

This is not an argument for abandoning examinations, calculus or rigorous problem-solving. It is an argument against defining mathematical ability too narrowly.

Students should also be exposed to linear algebra, probability, statistics, optimization and computational thinking. They should use notation to express ideas precisely, visualization to build geometric intuition and code to explore what happens when assumptions change.

We should therefore look past the familiar debate over whether school mathematics is too easy or too difficult. The better question is: mathematics for what purpose? The quantitative reasoning needed for citizenship is not identical to the mathematics needed for engineering, data science or advanced study in pure mathematics. These areas share foundations, but they place different demands on students. Schools and universities should offer flexible pathways that recognize those purposes without assigning students permanently to a track at an early age.

Such reform requires more than revising textbooks or adding a coding unit. It means rethinking what university admissions reward, giving schools room to value explanation and exploration alongside speed, and helping teachers connect mathematics with data, visualization and code. More importantly, it requires a cultural shift. As long as mathematics is seen mainly as a tool for sorting applicants, classrooms will remain focused on rankings and test-taking strategies.

The hardest step may be to stop treating mathematics as a weapon in the competition for college admission. Instead, it should become a discipline through which students learn to investigate uncertainty, model complex systems and make sense of a changing world. Artificial intelligence has reminded us that mathematics is not merely a collection of techniques, but thinking itself. Korea must now decide whether it wants mathematics to remain a sorting tool — or to become an instrument of inquiry.

Lim Woong

Lim Woong is a professor at the Graduate School of Education at Yonsei University in Seoul. The views expressed here are the writer’s own. — Ed.

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