Just a quick recap: in my May newsletter, I stated that math is the path to better problem solving, but that we have a problem with math. If you didn’t read it, I’d recommend going back.
Let’s talk about learning Math. Math is hard. It is a very strict language where the numeric scripts that we create only have a single meaning and are not open to interpretation. The standard of evidence for a mathematical truth is extremely high, which is what makes learning math so difficult.
Broadly speaking, learning anything is a frustrating process. To learn, we have to confront our confusion head-on. We must admit and accept that what we are doing is not working and that there is something that we don’t know or understand. If the goal is worth achieving, then we need to engage in the learning process and, as best as possible, avoid getting frustrated or angry.
If I’m honest, more than one textbook took flight through my dorm room back in my college days. I didn’t know at the time that this is where all learning begins.
I remember observing my son learning to crawl. He’d see a toy or something he wanted. He’d flip on his belly. He’d push himself up with his hands. Then he’d push forward with his hands and… go backwards! He’d get further from his goal and burst into angry tears. As a parent, you just kind of watch him try and try and try. There is not much you can do. In fact, letting him work through that frustration was necessary for the learning process. My son, who is now 21, has moved far beyond learning to crawl, but the process he uses to learn now that he is in college is the same.
There are several lessons here that we must apply to learning math. It is not all going to come together on the first try. It is likely that we will have to make many attempts before we learn how things work. All learning comes down to trial and error. We can read books, listen to lectures, and get coaching, but until we actually apply what we have heard, the information is not internalized. We have not learned it. And even with all of that instruction, our first attempt still likely won’t get it right.
We need feedback. The textbook would take flight because my answer to a question did not match the answer in the back of the book at all. It is actually through failure that the deepest learning occurs.
If my son had been able to push himself forward on the first try, he may not have learned all he needed to know about his motor skills. He needed to understand what didn’t work in order to eventually learn to walk, run, kick a ball, and more. It is by learning what doesn’t work that we begin to understand why something does work and truly appreciate it!
We have to try things and see the results for ourselves. It is this part of learning that makes math so hard. The conceptual nature of math makes it more difficult to get feedback. Much of what we learn is tangible or has room for interpretation. Math does not. It is hard to conceptualize the forces on the beams of a bridge, and the consequences of being wrong are severe. Learning how to imagine in this way, and then test that imagination is powerful. The ability to use math to imagine a bridge design that does not yet exist is what makes math so powerful.
As I stated in the last newsletter, this is what we do in continuous improvement. We imagine a future state and then model it in mathematical terms. When I see a process engineer predict the impact of a change in quantifiable terms, I know they have a strong understanding of both the cause of the problem and the impact of the solution. This means we need to confront our confusion with math in order to arrive at better solutions.
Overcoming this difficulty is no different than learning to crawl. And this is my main point: math is hard, which means you need to work hard to understand it. The process is no different from learning anything else. It requires accepting the confusion and then being persistent as you work towards understanding.
Begin by being uncomfortable with what does not make sense. Instead of avoiding the discomfort, use it as a signal that there is something you need to learn. For me, it meant going back and picking that textbook up off the floor. Work through the problem with the trial-and-error process of learning.
I never understood why we had to do so many exercises. If I didn’t understand it the first time, why would I expect to understand it the second? This was the wrong way for me to think. The truth is that we need to see something several times before we truly get it.
We have to try crawling in a bunch of ways that don’t work in order to figure out how to coordinate our arms and legs in a way that produces the desired result. It takes time, patience, and persistence. We have to live in discomfort for a while.
Working through lots of math problems means that I can see all the ways to get the problem wrong (I was really good at that part), then rewire my brain and my intuition to get the problem right. This is learning, just like anything else, but for many of us, math may require more focus, more persistence, more perseverance.
We imagine a train headed east at 30 miles an hour, 100 miles away from a train headed west at 50 miles an hour. Where do they meet? We have to imagine this. We cannot touch or feel it. And the answer is not open to interpretation. It is very definite.
In Mathematica: A Secret World of Intuition and Curiosity by David Bessis, David makes multiple mentions of great thinkers like Einstein, who claimed to have average intelligence but deep and unending curiosity. I’ve always thought those were humble statements from great thinkers. David suggests they are being honest.
The real trick to being great at anything, math or otherwise, is stoking and maintaining your curiosity. Creating the curiosity needed to try something over and over again until you get it right. When I look at my own journey with math, this appears to be the case.