I still remember when I got my smartphone. I’d used some friends’ smartphones to some degree, and so I knew the basics of moving things with my finger, using two fingers to zoom, etc. However, I lacked a deep understanding of smartphones and apps, and hence, I was very nervous when using my phone. I didn’t know what effect installing certain apps would have, nor agreeing to various licenses. I didn’t know what the preinstalled apps did. Essentially, I was scared of doing much with my phone because I was worried that I would mess it up, and I didn’t want to ask for help because I’d be demonstrating my ignorance.
I’ve felt math anxiety exactly three times before. The first time was my first summer at CTY, when I first learned proof by induction. We had morning quizzes, and I had only a surface understanding of the topic; I was worried that I would demonstrate my ignorance to my teachers, whom I greatly respected. The second time was my first summer at Mathcamp. Throughout the entire program I felt less prepared than my peers because I was seeing all this math that I didn’t really understand. I knew the words behind group theory; I could talk about subgroups and say the words “quotient group,” but I didn’t know what they meant. I was terrified that this would become apparent and that my friends would lose respect for me. The final time was taking algebraic topology at MIT; I had failed to do my reading and I was worried again that I would be discovered for not having understood what we were learning.
To me, the experience with the smartphone and my experiences with math anxiety are very similar. In both cases, I knew some words and some surface features of how to work with something, but I knew nothing about what made it tick. I did not understand, if I poked in one spot, which other spots would be bumped out. Without understanding that structure, I couldn’t do anything beyond what had been rigidly prescribed to me, and hence I felt like I was lost in a world that I did not truly understand.
I believe that this is what math anxiety is like. If you do not understand the mathematics, and you are not comfortable repeating a given procedure to get an answer (perhaps because you sense that there should be something more, or perhaps because you are not very good at memorization), then you’ll feel like you’re in a foreign country where you don’t understand the language — or the laws. You’re terrified that you’ll make a mistake and get arrested, but you can’t communicate with anyone to ask what you should do. Thus, you mimic what everyone else is doing and hope that it’s good enough.
If you want to teach someone a foreign language and set of laws, we have an established way of doing so. Start by giving them basic language tools (reading picture books, if you will) and by communicating the basis of the laws (the morality that the people share). The corresponding aspects of mathematics are understanding the precision with which mathematics is communicated, and understanding basic arithmetic and why it works.
If we move students to more advanced mathematics without those basics, then it is that much more difficult to keep understanding, and they are forced to resort to mimicking what they’re taught. They’re lost in a foreign country, scared of what’s happening to them. They’re terrified someone will see them do something wrong, but they won’t really know why it’s wrong.
That’s the disservice that I believe we do to students when we force them to go to more advanced mathematics before they understand the earlier levels. It’s the disservice we do when we try to teach how to do something to those who don’t understand the earlier steps.