Looking back, it’s amazing what a perfect specimen I was. I manifested every symptom that I now see in my own students:

• Muddled half-comprehension.
• Fear of asking questions.
• Shyness about getting the teacher’s help.
• Badgering a friend instead.
• Copying homework.
• Excuses; blaming others.
• Procrastination.
• Anxiety about public failure.
• Terror of the teacher’s judgment.
• Feeling incurably stupid.
• Not wanting to admit any of it.

It’s surprisingly hard to write about this, even now. Mathematical failure—much like romantic failure—leaves us raw and vulnerable. It demands excuses.

I tell my story to illustrate that failure isn’t about a lack of “natural intelligence,” whatever that is. Instead, failure is born from a messy combination of bad circumstances: high anxiety, low motivation, gaps in background knowledge. Most of all, we fail because, when the moment comes to confront our shortcomings and open ourselves up to teachers and peers, we panic and deploy our defenses instead. For the same reason that I pushed away topology, struggling students push me away now.

Not understanding topology doesn’t make me stupid. It makes me bad at topology. That’s a difference worth remembering, whether you’re a math prodigy, a struggling student, or a teacher holding your students’ sense of self-worth in the palm of your hand. Failing at math ought to be like any failure, frustrating but ultimately instructive. In the end, I’m grateful for the experience. Just as therapists must undergo therapy as part of their training, no math teacher ought to set foot near human students until they’ve felt the sting of mathematical failure.

Math is one of those things that makes no sense until it starts to make sense.