If I were to recommend a book to students who are apprehensive about their math and science classes and want to succeed, it would be A Mind for Numbers by Barbara Oakley. Subtitled “How to Excel at Math and Science (even if you flunked algebra)”, it’s full of good advice, practical tips, uplifting anecdotes, and incorporates evidence-based learning theory. Will the students be willing to pick up a 250-page book? I hope so. It’s engaging, not difficult to read, and Oakley aims (I think) at the right level. Given that she flunked math and only became a scientist later in life, she knows what she’s talking about.
Learning science and math requires effort and perseverance. There’s no Matrix connection that can deliver the knowledge straight into your brain. This baseline assumption runs through Oakley’s book. Why is it so effortful? For one thing, we haven’t evolved those skills biologically. And also, these subjects have a higher level of abstraction and encryptedness. Her example: “You can point to a real-life cow chewing its cud in a pasture and equate it with the letters c-o-w… but you can’t point to a real live plus sign that the symbol ‘+’ is modeled after – the idea underlying the plus sign is more abstract. By encryptedness, I mean that one symbol can stand for a number of different operations or ideas, just as the multiplication sign symbolizes repeated addition.”
The practical suggestions in her book will have two underpinning theoretical concepts so it’s worth describing them briefly.
First, our brain is constantly switching between two types of thinking process: the focused mode and diffuse mode. Oakley says: “Focused-mode thinking is essential… [because] it involves a direct approach to solving problems using rational, sequential, analytical approaches… Turn your attention to something and bam – the focused mode is on, like the tight, penetrating beam of a flashlight.” In contrast: “Diffuse-mode thinking is also essential… [because] allows us to suddenly gain a new insight on a problem… and is associated with ‘big-picture’ perspectives. Diffuse-mode thinking is what happens when you relax your attention and let your mind wander… [its] insights often flow from preliminary thinking that’s been done in the focused mode.”
Second is the Einstellung effect. Oakley writes: “In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found… [think of it] as installing a roadblock because of the way you are initially looking at something. This kind of wrong approach is especially easy to do in science because sometimes your initial intuition about what’s happening is misleading… sometimes it’s tough even figuring out where to begin, as when tackling a homework problem. You bumble about… your thoughts far from the actual solution…” I bet students looking at a problem set will find this feeling very familiar. It’s part of the process of learning science. I’ve certainly experienced it in spades.
Expert learners switch back and form efficiently between the two modes and avoid getting stuck. They also take the time to commit basic things to memory so that they can build on those foundational blocks (without having to look them up) and digest more complex material. Oakley devotes a few chapters to why, what and how you should memorize. She also has great tips to help get out of the black hole of procrastination. Memory and self-discipline are muscles, and it’s good to exercise them.
But let’s get to the meat and what students care most about (as if their grade depended on it!): Exams. In Chapter 17, Oakley writes (in bold): “Testing is itself an extraordinarily powerful learning experience. The effort you put into test taking, including the preliminary mini-tests of your recall and your ability to problem-solve during your preparation, is of fundamental importance… [Testing] has a wonderful way of concentrating the mind.”
Oakley provides a “Test Preparation Checklist” (with credit to Richard Felder). It’s a series of Yes/No questions. The instructions read: “Answer ‘Yes’ only if you usually did the things described (as opposed to occasionally or never).” I think this is a crucial distinction! And the more ‘Yes’, the better. Now on to the list. These are mostly verbatim from the book except I left out one repetitive statement and made minor truncations.
1. Did you make a serious effort to understand the text? (Just hunting for relevant worked-out examples doesn’t count.)
2. Did you work with classmates on homework problems, or at least check your solutions with others?
3. Did you attempt to outlined every homework problem solution before working with classmates?
4. Did you consult with the instructor when you were stuck on something?
5. Did you understand ALL of your homework problem solutions when you submitted them?
6. Did you ask for explanations of homework problem solutions that weren’t clear to you?
7. If you had a study guide, did you carefully go through it before the test and convince yourself that you could do everything on it?
8. Did you attempt to outline lots of problem solutions quickly, without spending time on the algebra and calculations?
9. With your classmates, did you go quiz one another on the study guide and problems?
10. If there was a review session, did you attend and ask questions about anything you weren’t sure about?
And the final key one: Did you get a reasonable night’s sleep before the test?
My “Advice for Success” on my course pages covers this ground in the form of statements. I like phrasing them as questions and I might do so next semester.
Yes, I know this post is getting long but I wanted to cover one more thing that Oakley discusses. It’s counter-intuitive. She calls it the Hard-Start-Jump-to-Easy technique. I’ve never used it myself although I do a variant of it. Oakley first describes the ‘classic’ approach: tackle the easiest problems first so that you gain confidence in doing the more difficult ones later. I’ve never given this advice to students because I don’t do it myself. Oakley acknowledges that “this approach works for some people, mostly because anything works for some people. Unfortunately, however, for most people it’s counterproductive. Tough problems often need lots of time, meaning you’d want to start on them first thing on a test. Difficult problems also scream for the creative powers of the diffuse mode.”
Her advice: “start with the hard problems first – but quickly jump to the easy ones.” The first step: “When the test is handed out to you, first take a quick look to get a sense of what it involves.” (I tell the students they should always do this. I did so when I was a student.) Next: “start with what appears to be the hardest one. But steel yourself to pull away within the first minute or two if you get stuck or get a sense that you might not be on the right track.” The trick is to ‘load’ it into your mind, then “switch attention away from it… turn next to an easy problem, and complete or do as much as you can. Then move next to another difficult-looking problem and try to make a bit of progress.” Rinse, Repeat.
When I was a student, I would quickly glance at all pages to get a sense of the length and where the ‘big point’ questions were. Then I’d actually start from the first question and work my way down regardless of perceived difficulty. But I was good at quickly moving to the next one if I got stuck. While I like the logic of picking the hardest problem first, that requires me to waste some time figuring out which it is. And since I was trained to take exams quickly and under high pressure, my method is probably more efficient although not necessarily better. Oakley does point out caveats to her strategy and it’s worth reading Chapter 17 in full. She also discusses how to tackle test anxiety and has some good tips!
I very much liked Oakley’s book. I might have persevered further in math if I read it as a student when I was struggling in my college math classes. As an instructor, she reminded me that I can communicate ‘study strategies’ better to my students. I mostly assume that they will read my “advice” and use it, but the reality is probably different and I should make more of an effort to help students as a coach in this area! (I once asked a former student to write a letter to my G-Chem students, and I provide my P-Chem students the comments of previous students.) If you’re a STEM student and you’re reading this post, do read Oakley’s book!
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