I recently read a very interesting article called “What it feels like to be bad at math”. I could relate to a lot of what the author was describing, and found it very insightful. In today’s post I’m going to talk about my own experience of finding my boundary in mathematics, compare it to something that happens a lot when running a marathon, and describe three solutions that worked for me and could hopefully work for you.
I experienced very similar emotions to the ones the article describes at several occasions in my mathematical career. Perhaps the most memorable was a course on probability theory that I tried taking in graduate school. After four or five lectures I was already pretty lost. There were some objective factors involved – the course was based on measure theory, which I neither fully understood nor particularly liked in my undergraduate years; other students in the class were better prepared than I was in this particular area; and I had not built a good intuition for probability because I never formally studied it, and instead tried to pick it up as I needed it. But these objective factors quickly gave rise to self-doubt, a feeling of inadequacy, and a fear of failure. Thankfully, the drop deadline had not yet passed and I left the course.
But the emotional response I describe here is not at all unique to mathematics. In fact, when I was running my first marathon a few months later, something very similar happened. While I did experience some moderate discomfort in various parts of my body throughout the race, at mile 23, I suddenly started experiencing a strong discomfort in my entire body at once. This is what long-distance runners call “hitting the wall”. I knew that this might happen, but was hoping it was the kind of thing that only happens to others. My body was ready to give up, and in my mind I felt the same influx of self-doubt and fear of failure as I did in the probability course. Thankfully, after taking a break, eating some gel and drinking some water I talked myself into starting to run again, slowly but steadily, and actually finished the race.
So, what was different about the marathon compared to the mathematics course? First, I felt well prepared for the marathon (I had been training for it for months), but not for the course (I had never formally studied probability). Second, I had a lot of people I was accountable to in the marathon (all those who had donated money to the cause I was supporting), while none of my friends were taking the course, so the consequences of dropping out were not as serious. Finally, I think that the biggest difference was the desire – I really wanted to finish the marathon, while taking the course, and doing well in it, was a nice-to-have rather than a must-have for me. Yet the emotions were very similar in both cases. This suggests that you can create the conditions for yourself that will maximize your chances of getting past the wall, by being well-prepared, being accountable to others and having a strong desire to do it.
Fine, you might say, but what if I’m already in that situation and it’s too late to change my preparedness, accountability or desire? What can I do to get past the wall (either mathematical or athletic)? Well, here are three things that worked for me.
First, take a break! This will help you regain some of your presence of mind, and you will be able to make a better decision. In running, this could mean slowing down your pace, switching to walking, or even actually stopping for a bit (which is what I did after I hit the wall at mile 18 during my second marathon). In mathematics, this could mean going for a walk, working on a different project or course, or leaving math behind for a few days. Some of my most satisfying discoveries resulted from coming back to a problem or subject after leaving it behind for a brief period of time.
Second, go back to the basics! Focus on the simple skills you already mastered, and let these get you through the challenging part. In running, this could mean focusing on your breathing, or paying attention to the rhythm of your feet hitting the ground (this is especially helpful for dealing with minor discomfort). In mathematics, this could mean going back to a concept or a related subject that you understood well before, or reviewing some earlier definitions and theorems. Interestingly, reading or seeing something for the second time makes it feel more familiar, and hence easier.
Third, break it down! To paraphrase what one of my mentors, Hillary Rettig, says in her book, “the wall isn’t a monolith; it’s a giant spaghetti snarl with many “strands,” each representing a particular obstacle or trigger.” What strands are in your way? In running, they could be the worn-out soles of your shoes, the pain in your back, and the fear of disappointing someone waiting for you at the finish line. In mathematics, they could be your dislike of messy algebraic manipulations, your relative lack of preparation, and your fear of failure. Whatever they are, dealing with each one individually is easier than dealing with all of them in a monolithic way. Sometimes, understanding the different pieces of the wall will alone be enough to get you past it.
While these simple tips may not always resolve the issue, they will definitely help you put things in perspective. If you then decide to leave the race (or drop the project) that may well be the right decision; there is no shame in failing at a goal that challenged you to find your limits, and you gained some valuable knowledge in the process. And if you decide to continue, good luck – you may be surprised at how far you can actually go. We all have our limits (both physical and mental), and it’s important to accept them, but most of the time, they are much farther than we think!