The Illusion of Area: What We Forget to Teach About Mathematics

Sometimes, as mathematics educators, we place too much emphasis on the power of universality and abstraction in solving problems. We praise mathematics as everlasting—that what our students learn in lessons is rooted in knowledge from thousands of years ago, yet remains vital in solving current real-world problems. "The Mother of Science," we proudly proclaim.

All of this is true. However, we often intentionally or unintentionally omit one key feature that makes mathematics everlasting: the process of abstraction. In any branch of mathematical study, we basically isolate a small fraction of an object's features and strip away the rest, allowing us to focus on one thing at a time. These isolated features are then defined with clear terms and logical consequences, providing a common ground for discussion that spans millennia. By following these established logical and operational rules, one does not need to possess an extraordinary mind to understand and solve some of the hardest problems of ancient human civilisation. That is why we want our students to learn mathematics.

But what is equally important is what we omitted: recognising that mathematical descriptions of patterns and behaviours are limited, and have a fine yet significant distinction from the real world. Take the study of area as an example. In elementary schools, students learn how to evaluate the size of closed plane figures. They learn tricks (aka formulae) to calculate the area of shapes like squares, rectangles, triangles, trapeziums, and circles. With these calculated values, students can confidently state whether a square is larger than a circle without even looking at them.

This is a powerful tool, especially when you need to make a decision based on size. Hongkongers are, inherently, concerned about the property market; for generations, buying an apartment has been a major life achievement. With limited living space in such a densely populated city, the primary indicator of an apartment's value is usually its size in square feet, followed by its price. We also use the average cost per square foot as an economic indicator in Hong Kong, which is quite distinct from the UK. The measurement of area seems to have greatly simplified the decision-making process.

Of course, experienced players in the property market have much more to consider than just the area. The shape and layout of the apartment make a massive difference in determining whether it is a sound choice. A well-proportioned apartment may make a far better home than another, even if it is strictly smaller in square footage.

When you look back at the study of area, you realise this is its core limitation. Ancient wisdom developed area as a tool to compare size, intentionally omitting other factors such as shape. (Indeed, thanks to this omission, we have a meaningful way to compare the sizes of entirely different shapes.) The study of area is therefore perfectly universal when we are tackling a real-life problem that involves area alone. But we must bear in mind that this very same study falls short when the real-life problem involves many other variables—which it almost always does.

The solution? In mathematics, we also study shapes, topologies, and many other concepts. They are limited in their own right, but together, they help handle different facets of the same real-life problem. The challenge goes back to how we frame our problems, identify the most relevant aspects, and attack them with a variety of tools to reach a balanced, informed decision.

This is mathematics, and this is more than mathematics.

Disclaimer: Original ideas by the author, with language polished by AI tools.