I am wondering if one of the problems that students with a fear of maths and physics have is that their mental concept of what “understanding something” means is not adapted to those subjects.
For example, I had a very intelligent friend in high school who was extremely bad at maths and physics, despite being able to argue extremely logically whenever the topic was not maths or physics. Once she told me that she did not understand what functions were, although this was something we had been using for a while at that point.
A bit shocked, I tried to explain it to her, making drawings, citing examples like “population of the Earth” as a function of “year”, simple functions like mapping x to x+2.
She understood what I was saying, clearly.
Yet she still claimed she did not really understand what functions were.
I have experienced something like that repeatedly when I was teaching. It seemed that people could follow what I was saying step by step, but then they still claimed they didn’t “understand” the big picture.
Maybe they expect to immediately obtain something “more” than understanding the explanation, some sort of intuition that others seem to have, and if it doesn’t happen, they feel like they failed.
Something similar actually has happened to me. When I was 13, geometry freaked me out. I had the impression that people who were good at it, and my teacher, were able to “see” solutions in some mysterious way that I did not have. This was also aided by the snarky teacher who continued finding everything extremely “obvious”.
My grades in geometry were bad, and so, afraid I was going to kicked out of school, I started working like crazy. I worked hours and hours, and yet at test after test, my grades remained bad.
Then, after a long time of trying hard to “see”, I finally lost it. Okay, I thought, I would never “see”. So I decided to try the following protocol: whenever I experienced a new problem, I would remember the about 15 rules of construction we already had learned, apply all that I could, and one would probably lead me closer to the solution. I’m still quite proud that the 13-year old me figured this out by herself, because nobody told me.
And that was it. From that day on, the problem was gone, my grades immediately went up. I still think I do not have a natural talent for geometry, like I have for things like algebra and analysis. But in the end, I even did geometry class at University together with the maths students and got through it fine.
But I still do not “see” in the way that my math teacher was claiming that he and the best boys in our class could “see” solutions, which is perhaps related to me having a hard time rotating 3-d objects in my head. But it turns out this isn’t necessary for solving geometry problems in school.
I suspect that a lot of people may have a similar problem with all of maths and physics. They see people around them solving problems and assume that the way this is done is by deeply seeing through the problem in a way that they can’t and wait for the big insight to happen before they can start solving problems.
However, all physics and maths problems can be solved by nothing else than applying the rules and procedures, until the solution pops up. With time, you will get there faster and not waste as much time as in the beginning, but you will always get there if you try hard enough.
And almost everyone who is willing to put in some effort is able to learn the few and simple rules that govern all of high school maths and physics.
Maybe the problem is the word “understand”. Perhaps we should not use it as much as we do. Gravity is not something to “understand” in the sense like you can “understand” the fact that you might hurt yourself if you walk onto the street with your eyes closed. Gravity just exists. It follows some rules. That’s it; nobody know why gravity is there in the first place. In its core, it is mysterious.
I wonder how it would work to teach with this mindset. Would the students feel unsatisfied? Would they say that they want to learn things that enrich them and give them a feeling of “understanding”, not blindly follow a set of stupid rules? Would they feel like they are just “going through the motions”?
However, on the other hand, if people cannot appreciate the coolness of a physics law like the law of gravity that holds in every corner of the Universe and at every time since the beginning of time, then maybe they will dislike physics anyway and we should not worry about them. To like physics, one has to appreciate rules in some way, but the same is true in any kind of advanced learning.
So maybe, it would be okay to start physics and maths with a speech going like this:
“Students, in the following years of high school, you are going to learn by heart a set of definitions and rules. The set of rules will fit onto 3 A4 pages per subject.
This is all you will need to know to do any problem and any exam that you will encounter ever in our two subjects.
All we are now going to do in the following years is to introduce one rule after the other, and then fill them with life by doing one freaking example after the other, and make all the connections to your daily life that we can, until you are 19 years old and can leave high school. We do not expect you to “see” or “understand” a single thing, nor develop an “intuition”, nor “imagine” anything. If this happens, good for you, if not, no problem.
But remember every rule we teach you, especially in maths, because if you do not know the rules of the first year you will fail every single exam of maths and physics you will ever have. And remember to solve the problems we give you, and not give up until you can solve them well, so that your math and physics muscle can grow and become strong and fun to use. Enjoy!”
Strangely, although this speech is highly uninspiring, and I would not give it to students, it would probably have helped me as a high school student. For me, the fear of geometry, and later of physics, was the main obstacle to overcome and the fear was always that I would not manage to understand or “see”. I would never have obtained this stupid fear if it had been clear to me that physics and maths is a game with rules one can simply learn, and that a feeling of “understanding” might only come later, or never, depending on how high your bar is for saying that you “understand something”, and that this is okay (and, btw, one reason that boys are less afraid of maths and physics because their bar of “understanding something” is often much lower than the bar that girls set for themselves, which due to girls often being raised to be more self-critical than boys).