Kitchen Timers and Calculus
A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)
by Barbara Oakley
Barbara Oakley is something of a wonder — almost as impressive, although in a very different way, as her namesake Annie, the sharpshooter.
From high school Ms. Oakley went into the U.S. Army, who turned her into a fluent Russian-speaker — her first degree was in Slavic languages. Leaving the Army with the rank of captain, and seeing few civilian job opportunities in translating or interpreting, she retrained herself in math and science, subjects she had failed in high school. She tells us that "I … only started studying trigonometry — remedial trigonometry — when I was twenty-six years old." The retraining was so successful Oakley is now associate professor of engineering at Oakland University in Michigan. Along the way she somehow found time for curious and unusual adventures. She and her spouse first met in Antarctica, for example; not a thing many couples can claim.
Oakley is now making a name for herself as a writer of popular books about the mind sciences, though the word "popular" needs some qualifying. The 2011 book Pathological Altruism, which she co-edited and to which she contributed, was interesting and original, but not an easy read, and probably not a favorite at airport bookstores. If I were to place writers of popular mind science books on an airport-worthy spectrum with Malcolm Gladwell at one end and Steven Pinker at the other, Barbara Oakley would be a tad to the Pinker side of center. She does not have Pinker's deep background in experimental psychology, nor — to be fair to Pinker — his wit, nor his skill in organizing his material. On the other hand, as a professor of engineering, she surely, unlike Gladwell, knows how to spell "eigenvalue."
Adventurous and intellectually fearless, Barbara Oakley bears watching. In A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) she offers a short course in learning, with particular reference to math and science learning. Drawing on her own experience — which is obviously highly relevant — and on insights from the mind sciences, she teaches her readers the best techniques for acquiring and retaining knowledge, and the worst pitfalls to avoid.
The two fundamental notions Oakley returns to again and again are, first, modes of attention, and, second, "chunking." I shall take these in turn.
There are, Oakley tells us, two modes of attention relevant to learning: focused and diffuse. Both are crucial to learning, and a key learning skill is the ability to toggle from one to the other as appropriate. This is all grounded in good neuroscience.
Focused thinking consists of concentration on particular features of a problem, the brain following well-tried neural pathways in an effort to encompass new ideas. This takes effort, the more so in math and science, whose topics are less "natural" to the brain than social, personal, or spiritual matters. It is also easily frustrated when trying to master new concepts, those well-tried pathways preventing us from seeing things in a new way — the Einstellung effect, in mind-science jargon.
The diffuse mode of thinking is more autonomous, more creative, and ranges more widely. It is the "big picture" style of thinking. Developmental psychologists make the analogy of a lamp shedding light all around versus a flashlight with a focused beam.
Learning, says Oakley, is an interplay between the two modes: focused thinking bringing new facts into our knowledge stock and making us familiar with them, diffuse thinking showing us important connections outside the flashlight beam.
Why do our brains come equipped with these two modes? In the modern style, Oakley turns for an explanation to evolutionary psychology:
A bird, for example, needs to focus carefully so it can pick up tiny pieces of grain as it pecks the ground for food; and at the same time it must scan the horizon for predators like hawks. What's the best way to carry out these two very different tasks? Split things up, of course … If you watch birds, they'll first peck, and then pause to scan the horizon — almost as if they are alternating between focused and diffuse modes. (p. 20)
Oakley's basic model for learning is then spelled out: First use focused mode for a concentrated assault on new knowledge. Then step away and flip to diffuse mode: noodle around with the topic, talk to others, take a walk, or sleep on it — "the ultimate diffuse mode."
The evolutionary explanation breaks down here, surely. I can't see how falling asleep would help that bird with predator-avoidance. (Chaucer seems to agree: His "smale foweles … slepen al the nyght with open ye.") Be that as it may, it certainly is the case, as anyone who has mastered difficult material can attest, that a problem which seemed impossibly hard at supper time is often surprisingly more tractable at breakfast.
And then the second fundamental notion: chunking. This is a term of art in mind sciences for aggregating lesser concepts into greater ones so that a concept with some inner structure can be dealt with as a single thing by the mind. In language learning, for example, phonemes are grouped into words, words into phrases. In math learning, variables and their powers can be chunked into polynomials, thence into rational functions and series. At each step in the chunking process you have a new piece of information that you can treat conceptually as a single named thing, saving mental resources. If you stay stuck at the detail level, no matter how vast your stock of memorized items, you will never be able to organize them into useful concepts.
Once Oakley has clarified these two fundamental notions — modes of attention, and "chunking" — she deals with the obstacles that get in the way of effective learning.
The greatest of these obstacles is procrastination. So Oakley thinks, anyway: she gives over five of her eighteen chapters to the issue. As a chronic procrastinator myself, I don't need persuading. (This review was commissioned six weeks ahead of publication date: I got down to it the day before deadline.)
Why do we procrastinate? In anticipation of pain, say the researchers:
Medical imaging studies have shown that mathphobes, for example, appear to avoid math because even just thinking about it seems to hurt. The pain centers of their brains light up when they contemplate working on math. (p. 85)
However, it was the anticipation that hurt. The pain disappeared when subjects actually did math:
Procrastination expert Rita Emmett explains: "The dread of doing a task uses up more time and energy than doing the task itself." (p. 85)
How, I wonder, does one qualify as a procrastination expert? The jokes write themselves here.
The solution to procrastination is, of course, self-discipline. How to acquire it, though? Oakley prescribes a sort of mental jujitsu: enlisting your own bad habits — she calls them "zombies" to emphasize the sleepwalking ease with which we succumb to them — in aid of a cure. Like any other habit, Oakley tells us, procrastination is triggered by a cue that launches us into a zombie-like routine that eventually delivers a reward. By changing one's reaction to the cue, one can enlist the zombie mode to get embarked on a productive process.
Oakley offers tricks to get the necessary rewiring done. A simple one that I have used myself is the Pomodoro Technique. You get an ordinary kitchen timer, set it to twenty-five minutes, then focus on a task for that time — no breaks! Focus on the routine, the process, not on any finished result; that is how you enlist the zombie. This is surprisingly productive. (Francesco Cirillo, who thought up this technique, used a timer shaped like a tomato — pomodoro in Italian. You can buy an identical timer online.)
Oakley devotes two chapters to memory tricks: metaphors, mnemonics, jingles, and so on. None of these will be new to anyone who has explored this territory even at the popular magazine level, but it's handy to have them all in one place. The "memory palace" is a particularly useful device employing the brain's visuospatial system — its very well-developed abilities to help us find our way around in three-dimensional space. To memorize a chain of linked items, or the words of a too-lengthy mnemonic, you construct an imaginary building with the items keyed to memorably dramatic events at the door, the basement, the loft … Oakley, perhaps wary of lawsuits, warns: "You do not want to use the memory palace technique when you are doing other spatial tasks, such as driving" (p. 161).
Binge-working — being "in the zone" — is good for occasional creative breakthroughs, but more a hindrance than a help to the steady accumulation of understanding:
Math and science demand the development of new neural scaffolds that are different from the social, pictorial, and language-oriented scaffolds that our brains have evolved to excel at. For many people, math- and science-related scaffolds develop slowly, alternating focused-mode and diffuse-mode thinking as the material is absorbed. Especially when it comes to learning math and science, the bingeing excuse, "I do my best work under deadlines," is simply not true. (p. 145)
That is good advice. So are Oakley's recommendations to make daily to-do lists, avoid multitasking, deal with the competitiveness of others, and get a good night's sleep before tests. To an adult reader much of this comes across as mundane; but to the book's target audience — high school students and college undergraduates — it should be very useful.
The only piece of advice missing from A Mind for Numbers is the one I have always given to students embarking on a math degree: Never buy a math textbook that doesn't include lots of exercises. Oakley has also been remiss in telling us about a song-mnemonic for remembering Avogadro's number, to the tune of "Oh, Them Golden Slippers," without supplying the lyrics for the song. These minor quibbles aside, Barbara Oakley has written a useful and engaging book.