What Does ‘Generalization’ and ‘Abstraction’ Mean in Math?

I use an Oxford Mathematics Admissions Test question about candy to demonstrate how you can ‘generalize’ or ‘abstract’ a math solution

Keith McNulty

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Photo by JACQUELINE BRANDWAYN on Unsplash

As someone who has taken the journey from a high school math student to working in cutting edge Pure Mathematics research at the postdoctoral level, I want to highlight one key difference in how the brain operates at those different levels of training.

High School students are taught to exercise math by applying their curricular knowledge to specific problems, usually involving specific values, objects or functions. For example, high school students will usually work with integers, or real or complex numbers, or integer polynomials. This hones their basic training in the knowledge and pattern recognition required to become professional mathematicians.

Fully developed professional mathematicians, however, are trained to generalize and abstract their thinking wherever possible. Instead of working with integers or established everyday number structures, they will work with abstracted algebraic structures such as groups, rings, modules or fields, which encapsulate many known, more specific structures, allowing their results to be more powerful because they apply to more general structures and problems.

I’m being abstract here, pardon the pun, so let me show you a really simple example of how taking opportunities to generalize mathematical insights can make them more powerful. In this question from an Oxford Mathematics Admissions Test, the high school student is asked to make a conclusion about a specific case. However, it turns out to be possible to make a much more general conclusion if we take a moment to spot the generalizability of the approach. Let’s start with the question as it appears in the exam paper.

The original MAT question and solution

MAT 2020, Question 5

For my American readers, just a clarification that the King’s English for ‘Candy’ is ‘Sweets’ 😊

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Keith McNulty

Pure and Applied Mathematician. LinkedIn Top Voice in Tech. Expert and Author in Data Science and Statistics. Find me on LinkedIn, Twitter or keithmcnulty.org