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.

Nice post, Dan.

These insights seem to support the idea of highly differentiated classrooms in which students work at their own level and pace. The challenge then seems to be those ongoing individual assessments to measure whether actual learning has happening rather than mimicking what others are doing out of anxiety.

I really liked what one master teacher at my high school in Kentucky did in providing quizzes and tests that we could take once we felt ready, so you sort of “earned your badges” at a natural pace. I think it definitely took the edge off of many students’ anxieties.

Wow, only three times? I must feel this way at some point in nearly every math lecture I attend at MIT. Partially for fear of exposing my slowness, I suppose, but a lot of it is also because no one else is saying anything; you feel like everyone else has got it and you would drag the whole class down if you spoke up. Needless to say, I am not a mathematician.

I feel like this is true for a lot of people, on the occasions where three blackboards’ worth of material have whizzed by, and then someone timidly asks about what exactly happened three boards ago. Most of the time I really can’t tell if I’m the only one who’s lost.

In this regard I feel like events like some of ESP’s “extreme” classes are actually more socially comfortable. Since everyone is expected to be lost, you can ask stupid questions rather fearlessly and aren’t expected to be grounded on the material until later. Of course, usually these are one-time obligation-free events, which is probably completely different from what you’re talking about.

Yeah, I think I should clarify: I have often felt anxious about math, but not to the point where it might have impeded my performance. These three times, I think it genuinely got in the way of learning as opposed to merely making me feel uncomfortable.

I teach developmental math at a community college (things like Algebra I, for example), and I could write pages and pages on math anxiety. In an effort to keep my response short, though, I want to mention one likely difference between the math anxiety that this blog’s readers have likely faced and the math anxiety that some of the rest of the population has faced: many readers of this blog are probably in the 90th and above percentile for mathematical intelligence, and so their experiences with math anxiety are completely different from someone who’s in the 30th percentile, the 20th percentile, or below.

Have you ever cried because math was too hard? Have you ever skipped math class because you’d be exposed as dumb? Have you ever failed a math class? I’ve taught several students who have gone through every single one of these situations, sometimes on multiple occasions. For some of them, elementary and intermediate algebra are the only classes holding them back from transferring to a four year university. Their math anxiety has delayed their graduation by a year (or two, or three, or more, in some cases).

I understand that many MIT students might feel math anxiety when surrounded by very successful math students, and I feel bad for what those students are feeling. But I think it’s important to put that into perspective and to realize that a solution to math anxiety at MIT might not work well for other types of students. Sometimes it’s difficult for high level academics to really understand what struggles are faced by students on the other end of the spectrum, and it’s important to recognize that some problems are bigger and more complex than one person can understand.