Why Self-Studying Mathematics Often Fails and How to Persist

For several years I have helped students who were self-studying mathematics. I did this free, without compensation, because I found it an enriching experience. Most who sought help came from physics forums; I usually contacted them to offer assistance and, if they agreed, we proceeded. My help typically consisted of providing resources, supplying extra problems, checking solutions, and giving additional explanations. As a rule, I avoided helping with formal coursework and focused on self-studies.

Most of the self-study projects I engaged in ended up failing, however, for a variety of different reasons. Below I go over the most common ones. Whether you are studying completely on your own or working with someone like me, the following points illustrate how difficult self-studying can be and what common traps to expect.

1) Helper mismatch: wrong level or approach

This must be said: I helped many students, but often I was not the right person to help them. I enjoy pure mathematics and tend to help in that direction; my explanations reflect this interest. By temperament I am also not well suited to teaching much of high-school mathematics. It has been a surprise that knowing high school mathematics well does not mean one can explain it clearly or engagingly to others. In those cases I could not help much.

What to take away: a helper needs to be at the right level and have the right approach for you. If you struggle with algebraic manipulations, a mentor who is far advanced may confuse you. Conversely, if your goal is very pure mathematics, a helper focused only on applications may not be suitable.

A strong rapport between student and helper is also important. Without it, both may be interested but still make little progress together.

2) Students who advance too quickly

This happens most frequently with people new to abstract mathematics, often high‑school students taking their first steps into pure math. It occurred repeatedly that a student would report after two days that they had finished a chapter and understood everything and had solved all exercises. When I asked to see their solutions they were all incorrect. The student thought they understood, but they did not.

In mathematics it is important to take things slowly. Reading a 1,000‑page novel can be much quicker than working through a 50‑page abstract text. I once finished a 1,000‑page fantasy book in a day, yet I struggled to do more than five pages of mathematics in the same time frame.

Every statement in a mathematical text needs careful attention and repeated revisiting. That takes time. There is no realistic expectation of fully understanding an analysis book in two months, particularly when self-studying.

3) People get discouraged

If your progress is at most a few pages a day, you may calculate that finishing a single book will take months. That realization is discouraging, and many people stop as a result.

Mathematics is hard. You will regularly hit walls and encounter exercises you cannot solve. When this happens repeatedly, discouragement builds and some people quit.

Improvements in mathematics are often slow and not visible day to day. Unlike collecting stamps, where the collection grows visibly, mathematical progress is subtle. Keeping detailed records of your work helps: looking back after two or three months will usually show clear growth. Math does not become easy, only familiar: the only easy math is the math you already know.

4) Life gets in the way

Studying mathematics requires concentration and sustained time—often at least an hour per day. That can be possible initially but less so after life events occur: illness, job changes, family demands, and so on. If other responsibilities are higher priority, intensive study becomes difficult. There are only 24 hours in a day, most of which are spent on other activities.

5) You get bored

Many people discover that mathematics is not what they imagined and lose interest. This is common—people try it, decide it is not for them, and move on. There is no shame in that: you learned something about your preferences by trying.

Conclusion

As you can see, self-studying mathematics is difficult, and studying outside a formal classroom is even harder. Many people begin and many stop for the reasons above. Do not be discouraged: you may find a path that works for you, but you will not know until you try.