Personally, I’ve found that there are two important ingredients to improvement: making goals and reflecting regularly. Goals are the more obvious component, since to improve you must be working towards something. But there is an important nuance in how you construct goals. Perhaps the ol’ reliable mnemonic serves best. Goals should be SMART: specific, measurable, attainable, relevant, and time based. None of these components can be understated, and all work towards preventing an all too common pitfall: setting yourself up for failure with nigh impossible goals. For math, understanding and avoiding this pitfall is essential. Some days you will solve all the problems you face and feel like a mathematical wizard. However, most days are spent being stuck, getting questions wrong, and seemingly making “no progress”. But as a detail oriented mathematician, you might retort and ask: how does one measure “progress”? It is natural, and even built into our education, to measure progress on problems solved and correct answers given. But in math, one can make a tremendous amount of progress while failing to solve a problem. Or, dare I say, getting a problem wrong. In fact, I would argue that is precisely when one makes the most progress. So, if you want to improve at math, I’d suggest constructing goals that embrace failure, getting questions wrong, and being stuck. But this is all a lot easier said than done. This mentality is difficult, to say the least, and is most difficult precisely when it is the most important. So, to remedy this, I suggest making use of a truly critical resource for anyone: community.
Let’s move onto the second ingredient for improvement: reflection. I believe reflection is absolutely crucial to improvement. For without reflection, important lessons and growth are being wasted. For example, let’s say you put a lot of time into working on a problem but made a mistake along the way. That mistake is a clear place for improvement. One should think about why they made that mistake, and more than just the surface level — truly investigate the process of what led to that mistake, and why you didn’t catch it sooner. People do this automatically — it is a natural process — but not with enough depth. As problems get harder and mistakes more intricate, it is important to actually devote time to this process. To spend time thinking about how you can improve and what to focus on. Reflection is making your improvement intentional and actionable, rather than an accidental byproduct of work.