For the past few days, during my summer break, I have been intensively self-studying mathematics (namely number theory) for several hours each day without having prior experience in theoretical math. The struggle of learning is not unique to mathematics; during my first year of computer engineering (which I had just completed at the time of writing this article) I faced many similar challenges that I continue to face now. There are many difficulties we face that we must overcome in order to succeed. In this article, I will discuss the challenges that I faced, which are likely common to others with varying academic backgrounds, and how I overcame them. I will also discuss a challenge that I have not yet been able to overcome.
Challenge #1: Doubt
The first and by far the most destructive obstacle to self-learning is that of doubt. Whenever I encountered difficult material, I felt like giving up. I had the nagging thought that I wasn’t smart enough to be able to do the content; that I would fail, and that I should stop wasting my time and give up. Unfortunately, there was no easy solution to this problem. However, I feel like this issue existed because I often compared myself to other people who seemed to breeze through the material (despite the fact that many of these people have much more experience than I). To lessen the impact of this challenge on myself, I had to change the way that I thought, and I had to stop comparing myself to other people. I never gave up, and thanks to my persistence, I always figured out whatever troubled me, and when I did, I knew that I could continue to overcome challenges that came my way. Although these feelings may never go away, their impact on you will lesson with time and experience, so as long as you stay persistent.
Challenge #2: Losing Interest
The second challenge that I faced was losing interest with the material that I studied. I felt like this challenge was a direct consequence of the first obstacle to learning: doubt. When I struggled in specific academic topics, or felt as if I wasn’t progressing with the material fast enough, I often became frustrated with myself. This frustration led me lose interest in the material that I studied. I started to rationalize the purpose of learning the areas I struggled in; such as telling myself that number theory was useless as it had no real life application, and that I shouldn’t bother learning it, that I should focus my efforts on more important things. However, thinking like this was destructive and caused me to lose interest in my studies. After overcoming the first learning challenge: doubt, I had found that I started to regain interest in number theory, because I knew that I was capable of doing it. I also started keeping track of my progress throughout my studies, and I began looking forward to what learning number theory will bring me: more mathematical maturity, which I feel is applicable to every aspect in science (specifically theoretical computer science). I feel that one should always make something to look forward to which will result from the topic that they are currently studying, especially if they start losing interested in said topic.
Challenge #3: Time Management and Unrealistic Goals
The third challenge that I faced was time management and unrealistic goals. I had a vision of all of the books that I would have completed by the end of summer, as well as all of the topics I would have completed. Therefore, I set a goal of completing a certain number of pages per day. This forced quota of “pages to complete per day” ironically caused me to become even less productive. When I encountered a difficult proof in my book that I did not understand, I would simply skip it and continue on. However, there would come a point in the future that understanding this proof was critical. This essentially forced me to either continue skipping material, or to go back and actually learn the difficult material. I have now realized that my goals were unrealistic, and vowed to spend as much time as I need. I feel like an important lesson to share is what follows: don’t set unrealistic goals. Know what you can and can’t do. Be satisfied with what you can do, so as long as you vow to put in 100% effort. However, it is important for one to know their limits. Having never written a math proof before in my life nor knowing anything about math logic before I had started studying number theory, I had come to the conclusion that I was not yet ready to continue studying number theory, because I could not satisfy the demanded prerequisite knowledge. I decided it was best to start learning more basic mathematics (namely about proofs, math logic, sets, etc), so that I will be ready to return to studying number theory at some point in the future.
Challenge #4: Understanding Difficult Problems
The fourth challenge that I faced was that of actually understanding difficult problems. I found that the majority of problems I encountered were due to the fact that I was missing some prior knowledge. Whenever I didn’t understand something, I would find out what I struggled with and then return to previous lessons to relearn what they had taught me. In the case of number theory, I had difficulty approaching certain proofs because they had seemed very complicated and long. It was as if these questions had absolutely no relation to anything I had previously learned, and I felt like I had to use some foreign skill to solve these problems. However, I found that I could often break these problems up into little subproblems, and then use the solutions to previous questions to solve these little subproblems, as well as some thinking to glue everything together. This often transformed giant scary problems into problems with laughable difficulty. However, I found that the key to being able to break up problems is to have a firm grasp of prior knowledge and knowing the content that you must know in order to be competent in your studies.
Challenge #5: Lack of Feedback
The last challenge is one that I have not yet been able to overcome. It is the lack of feedback regarding problems I solve. Despite resources being available to ask for help (such as the internet), often the feedback was not available in a timely manner. There is no debugger for math proofs or physics problems, like there is for software. Most of the time, I could not figure out if my proof was incomplete, or even flat out wrong. This caused me difficulty as I didn’t know how to proceed. This is true even in the case of physics or engineering problems when the numerical answer is provided. Since it often takes a very long time to solve these problems and includes numerous steps, if your answer is wrong, you know that you made a mistake. However, you don’t know what kind of mistake you made. Did you make a calculation error, or is your process/logic just plain wrong? It can be very frustrating, especially in engineering when you only have limited time to learn specific topic, due to the fact that you have a very heavy workload.
All in all, overcoming challenges is a vital part of in life, and affects everyone with varying academic backgrounds and experience. I found that in general, to overcome almost every challenge I faced, all I had to do was the following: never give up, constantly have something to look forward to which result from the topic I was currently studying, be satisfied with my limits so as long as I put in 100% effort, know when to retreat and regroup, and have a firm grasp of prior knowledge.
I have just completed my first year of computer engineering at UofT. My interests include: computer science, mathematics, physics, and reverse code engineering.