About a year and a half ago, I had a GRE tutoring client who had a reasonably important political job. She was in her mid 30s, sharp (and sometimes cutting), and had appeared in the news quite a few times.

She was also absolutely terrible at math. I mean really, really terrible. She didn’t know how to do anything past 6th grade math, and she was pretty terrified of that.

This was a problem, as the GRE goes way past that. Even to get a mediocre score, like she wanted, she would need to learn a lot more math. And I’d have to be the one to teach her.

So, picture me, in my office, sitting side-by-side with an accomplished woman who I just met. She’s smart, but she’s bad at math. She knows she’s bad at math. She hates math, and has hated it since high school. It’s my job to teach her. How do I do that?

Teaching an adult who never learned math is different than teaching a kid. When teaching kids, the hardest parts are that everything is unfamiliar to them, and that they often lack a lot of the common sense to make seemingly obvious connections. These are compounded by the kids’ lack of motivation, so teaching is a lot of motivation and cajoling.

When I teach adults like this woman, however, they are generally familiar with the material, they have common sense and life experience, and they are very motivated to learn by the prospect of not getting the degree that they want. So, the question is then: why are they bad at math?

The first issue is always anxiety. When that woman said she hated math, what she really felt was a deep and profound sense of anxiety over it. She is smart and motivated, and accomplished many of her professional goals.

Now she’s forced to confront something she can’t do that 14 year olds do without breaking a sweat. Imagine how that feels. Bad, right? It feels like someone rubbing your nose in failure.

As a tutor, therefore, my student’s anxiety is always the first thing I have to deal with. This is not easy for me. I would mostly describe myself as a nerd, with the bluntness that entails. Gentle motivation has been a skill I have had to learn with a lot of hard work.

As a result, I’ve developed stock phrases, like: “each problem you get wrong now is a problem you get right on the test” or “studying for a test is just like practicing for a marathon; it’s about putting the miles on the pavement.” I even just tell them, “The GRE sucks, but it only has to be a big part of your life for a short time.”

Beyond my students’ anxiety, though, there still lies the central question of why, exactly, they never learned math. They are almost always familiar with the material. They likely took the SAT or ACT and were tested on the material. They might have even taken a prep course to refresh themselves.

This is sufficient for many people to develop a passing understanding of math. But it’s not sufficient for these people. What are they missing?

Well, over the course of working with quite a few of these sorts of students, I’ve learned what it usually comes down to: they are really, really bad at recognizing similar math problems via induction.No matter what subtopic in math we are in, they have a ton of trouble adapting solutions from one problem to the next.

When students (or anyone) does a math problem, the first step is to recognize its similarity to problems they’ve done before. Then they recall the general sort of solution before solving the problem.

After all, none of us are inventing solutions out of whole cloth, unless we’re 9 year old Carl Friedrich Gauss. We’re all just doing variations on what we already know.

(Self plug: this essay is on the website of an app, 21st Night, that is literally designed to make it easier to recognize similarities and recall solutions to problems. Ahem.)

But, these students think they have to be Carl Friedrich Gauss every time. And if they can’t invent the solution, then they give up. They see their fellow students doing problems with ease, and assume there is simply a “math gene” that they’re lacking.

And, if they lack the math gene, why bother trying to learn math? It’ll never work.

This was what it was like with my student. She had somehow “learned” at an early age that math is either about inventing, or following a set of arcane steps, like a magic ritual. Lacking the capability for invention, she had only followed arcane steps for *her entire math career.*

It’s exhausting and difficult to memorize without a framework. Imagine trying to memorize the spelling of every word without ever recognizing the connection between phonetics and orthographics (how it sounds and how it’s spelled). Spelling would be, well, an arcane mystery (pun fully intended). Every word would take so much out of you, and you’d avoid spelling whenever possible.

That was her and math. She had memorized her way through high school math and the SAT. She had a miserable time of it, and avoided math because it was exhausting and reminded her of failure. Now she was working a full-time job and needed to take the GRE, and she simply didn’t have the brain space for it.

Whew! Not fun, right? So what can I do?

Well, after clearing her anxiety (or trying to over a long and drawn-out process), I needed to teach her that math was about recognizing patterns. She needed to see that problems were in fact similar to each other, and that the solutions to problems were consistent with one another (i.e. we weren’t inventing new rules every single math problem).

Put that way, it sounds easy. In actuality, this is a painstaking process for all involved. I needed her to unlearn that math was all about memorization, learn that math is about understanding and similarities, and then actually learn the strategies and techniques she needed to do.

With her, as with most of my students, this is generally best done empirically. I can tell my students this, and they will nod and agree. And then they will continue with the same assumptions they’ve had for 20-something years.

So, instead, I tell them that we are going to need to do and analyze a lot of problems together. It will be many hours, and I need their patience. That’s all I say.

To actually teach them the right way to think about these problems, I ask them two questions after each question we do together, whether they get the question right or wrong:

- How did we know to apply that method to that problem? What prompted it? [This teaches them to recognize similarity to other problems that use that method]
- Why did you choose this approach? In other words, what did we want in the end? [This forces them to recognize the purpose of the steps, and understand the logical consistency.]

Answers to these questions go in the explanation in the error log (the app I mentioned), as well as a step-by-step explanation of how they solved the question. Then the app makes sure they recall everything by a spaced repetition system.

I do that over a period of hours. Then I have them do that at home for even more hours. Between me and their homework, I usually have them spend 100+ hours doing and analyzing GRE math problems with those questions and my app.

And 100+ hours, hundreds of math problems, and thousands of words exchanged later, these unteachable adults are finally mediocre at math.