How to Abstract Your Reasoning
I use an Oxford Mathematics Admissions Test question about candy to demonstrate how you can ‘generalize’ or ‘abstract’ mathematical thinking
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.
