Why Does Mathematics Become More Confusing in College?

To put it bluntly, the purpose of studying mathematics in college is often just to pass exams, and the learning method tends to be rote memorization. When learning advanced mathematics, it is best to have a global perspective. I recommend All the Math You Missed: But Need to Know for Graduate School (Chinese title: 《研究生阶段你错过的数学》). Although this book is designed for graduate students, college freshmen can also read it early on (without reading it as thoroughly as a textbook) to build an overarching framework of mathematical concepts. It helps students get a head start on the concepts they will study later—for example, when they will encounter real analysis or topology in their academic journey.

This book takes a breadth-first approach, connecting the mathematical concepts learned during college by explaining their background and integrating these scattered ideas into a coherent map. It helps students understand where branches like linear algebra, real analysis, complex analysis, abstract algebra, and differential equations fit on the mathematical landscape—and how they relate to one another. Like a panoramic navigation system, it builds an overall understanding of mathematics. The book is also like the HR department in a company: when you start as a newcomer, the HR introduces you to everyone in the organization, so later when you have questions, you know whom to approach. Similarly, this book does not dive deeply into any specific branch of mathematics but guides you toward the right directions.

The author, Thomas A. Garrity, is a university mathematics professor with many years of experience in math education. He believes that students do not differ significantly in intelligence, but if a critical foundational concept is missing, subsequent learning becomes extremely difficult. I once saw a related answer on Zhihu: a high school student who had repeatedly failed math exams hired a tutor for intensive remedial lessons. The tutor started from middle school math concepts, identified a key missing piece, and from that moment on, the student’s math performance improved drastically.

Because it covers such a wide range of mathematics, many students also use this book as a review tool to organize and fill gaps in their knowledge.