Despite the subtitle, the book is actually designed for "both math experts and mathphobes." I myself am somewhere in the middle of that spectrum. I am not a "math expert," since I have a lot to learn (and, in some cases relearn) in math, and I am not your typical "mathphobe" either. I love math. And I certainly never "flunked algebra". But at the same time, there are times I find learning new things scary--so I do understand what it is like to be a mathphobe.
And, in fact, I found myself feeling pretty phobic in my final exam in my graduate class this past Monday. I got impossibly bogged down on what was probably intended to be a simple problem #1 on the exam.
What happened to me on Monday was a classic example of the phenomenon Barbara Oakley refers to by the wonderful German word "Einstellung", which means getting stuck. As she writes:
[t]he Einstellung (pronounced EYE-nshtellung) effect. In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.I was so stuck on a particular approach to the problem (based on a misconception) that I couldn't get the perspective to see I was just wrong and had to abandon it for a different approach.
Part a of problem 1 was actually kind of fun, calculating the Fisher information for a random sample drawn from a normal distribution with unknown mean $\theta$ and variance 1. It looked kind of ugly at first (since it involved taking the second partial derivative of the log of the joint pdf of the random sample), i.e., it was:
$$I(\theta)=-\frac{\partial^2 \log(\prod_{i=1}^n\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(x_i-\theta)^2})}{\partial \theta^2}$$
but it was kind of beautiful that the ugly looking expression above ultimately boiled down to 1, and as soon as I saw that the answer was 1, it made complete sense that 1 would be the answer (for reasons involving the Rao-Cramér lower bound, which I won't go into here.) And I thought,wow, what a great choice for a first problem, because it was one of those "lightbulb going off in my head moments.)
But then came part b of problem 1, where we had to compute the efficiency of an obviously inefficient estimator for the unknown parameter. That meant that I had to compute the variance of the estimator, which should have been an easy piece of cake.
The estimator was $Y=\frac{1}{2}X_1+\frac{1}{3}X_2+\frac{1}{6}X_3$. It was easy for me to prove that $E(X)=\theta$, so I knew the estimator would be unbiased. So then all I needed to do was to compute the variance of the estimator. And I was used to using the following clever trick to find the variance of statistics: Variance(Y)$=E(Y^2)-[E(Y)]^2$. So, I thought to myself, well, I already know $E(Y)=\theta$, so all I need to do is to find $E(Y^2)$. So I wrote:
$$E(Y^2)=\frac{1}{4}E(X^2)+\frac{1}{9}E(X^2) + \frac{1}{36}E(X^2) =\frac{7}{18}(1+\theta^2)$$
But looking at that answer, I realize that could lead to a negative variance if $\theta$ is sufficiently large and I knew THAT couldn't happen, so I must have done something wrong, but my brain was starting to go "deer in the headlights" and freeze up and I was so stuck on my approach to computing the variance as $E(X^2)-[E(X)]^2$ (which can often be a clever approach) that I couldn't see that my equation above was fundamentally flawed.
If my head had been clearer and I could have gotten unstuck, I could have made the problem trivial by writing:
$$Var(Y)=\frac{1}{4}Var(X)+\frac{1}{9}Var(X)+\frac{1}{36}Var(X)=\frac{7}{18}$$
which is the simple, clear, and straightforward approach to solving what was intended to be an easy softball problem. Somehow I had gotten my brain stuck on an unworkable and incorrect approach. Having done that on the first problem, I was then sufficiently rattled that I strongly suspect I messed up some later problems on the exam as well.
Oh well, live and learn. At my age and stage of life, I don't really care about grades, but I was disappointed because I thought the professor had been terrific and that I had learned a lot in the class and I would have loved to have done a better job of showing him what I had learned. Fortunately, there was also a takehome part of the final, which I felt much better about, plus I felt pretty good about all the homeworks I had turned in as well as the midterm.
But the whole thing was really a remarkable learning experience to me. As an economics professor, I often see students do incredibly silly things on exams and I beat myself up, thinking "Haven't I taught these folks *anything*? How could they be so confused." But having seen the incredibly silly thing that *I* did on the exam and how it didn't really reflect my understanding (but more my nervousness under time pressure), it reminded me again that tests are necessarily flawed measures of learning.
But I also realize that maybe if I knew how to study more efficiently and master a subject more deeply than I had, I might have done significantly better on that exam. And I like challenges, and I feel that following some of the ideas in this book may be helpful to me.
So feel free to stick around if you want and follow the future challenges of a sixty-something "Velveteen Mathematician" who wants to become a "real mathematician."
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