A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)

by Barbara Oakley

If you are trying to understand or figure out something new, your best bet is to turn off your precision-focused thinking and turn on your “big picture” diffuse mode, long enough to be able to latch on to a new, more fruitful approach.

The bottom line is that problem solving in any discipline often involves an exchange between the two fundamentally different modes. One mode will process the information it receives and then send the result back to the other mode. This volleying of information back and forth as the brain works its way toward a conscious solution appears essential for understanding and solving all but trivial problems and concepts.

when you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning. You are also increasing your stress level because you know you have to complete what feels like an unpleasant task. The resulting neural patterns will be faint and fragmented and will quickly disappear—you’ll be left with a shaky foundation. In math and science in particular, this can create severe problems. If you cram for a test at the last minute or quickly breeze through your homework, you won’t have time for either learning mode to help you tackle the tougher concepts and problems or to help you synthesize the connections in what you are learning. NOW

If you want to apply a more advanced version of this approach, imagine that at the end of the day, you are reflecting on the one most important task that you accomplished that day. What would that task be? Write it down. Then work on it. Try to complete at least three of these twenty-five-minute sessions that day, on whatever task or tasks you think are most important. At the end of your workday, look at what you crossed off your list and savor the feeling of accomplishment. Then write a few key things that you would like to work on the next day. This early preparation will help your diffuse mode begin to think about how you will get those tasks done the next day.

The Einstellung effect refers to getting stuck in solving a problem or understanding a concept as a result of becoming fixated on a flawed approach. Switching modes from focused to diffuse can help free you from this effect.

That twilight, disconnected feeling one experiences while drifting off to sleep was, it seems, part of the magic behind Edison’s extraordinary creativity.

When faced with a difficult problem, instead of focusing intently on it, Edison, according to legend, took a nap.

As he relaxed, his thoughts moved toward free and open diffuse-mode thinking.

Mistakes are inevitable. To work past them, start early on your assignments and, unless you are really enjoying what you are doing, keep your working sessions short. Remember, when you take breaks, your diffuse mode is still working away in the background. It’s the best deal around—you continue to learn while you are taking it easy.

As the days and weeks pass, it’s the distributed practice—the back and forth between focused-mode attention and diffuse-mode relaxation—that does the trick.12 Enlisting the focused mode, which is often what you need to do to first get a problem into your brain, requires your full attention. Studies have shown that we have only so much mental energy—willpower—for this type of thinking.

When your energy flags, sometimes you can take a break by jumping to other focused-type tasks, such as switching from studying math to studying French vocabulary. But the longer you spend in focused mode, the more mental resources you use. It’s like a concentrated, extended set of mental weight lifting. That’s why brief interludes that involve movement or talking with friends, where you don’t have to concentrate intently, can be so refreshing.

Remember, accepting the first idea that comes to mind when you are working on an assignment or test problem can prevent you from finding a better solution.

According to recent research, blinking is a vital activity that provides another means of reevaluating a situation. Closing our eyes seems to provide a micropause that momentarily deactivates our attention and allows us, for the briefest of moments, to refresh and renew our consciousness and perspective.16 So blinking may momentarily disconnect us from our focused-mode perspective. But on the other hand, deliberately closing our eyes may help us focus more deeply—people often look away or close or cover their eyes to

Figuring out a difficult problem or learning a new concept almost always requires one or more periods when you aren’t consciously working on the problem. Each interlude in which you are not directly focused on the problem allows your diffuse mode to look at it in a fresh way.

good rule of thumb, when you are first learning new concepts, is not to let things go untouched for longer than a day.

use a technique called spaced repetition. As you may have guessed, this technique involves repeating what you are trying to retain, like a new vocabulary word or a new problem-solving technique, but spacing this repetition out over a number of days.

Putting a day between bouts of repetition—extending your practice over a number of days—does make a difference.

This is similar to building the brick wall we saw earlier. If you don’t leave time for the mortar to dry (time for the synaptic connections to form and strengthen), you won’t have a very good structure.

It seems that if you go over the material right before taking a nap or going to sleep for the evening, you have an increased chance of dreaming about it. If you go even further and set it in mind that you want to dream about the material, it seems to improve your chances of dreaming about it still further.

Dreaming about what you are studying can substantially enhance your ability to understand—it somehow consolidates your memories into easier-to-grasp chunks.28

you’re tired, it’s often best to just go to sleep and get up a little earlier the next day, so your reading is ...

Focusing your attention to connect parts of the brain is an important part of the focused mode of learning. Interestingly, when you are stressed, your attentional octopus begins to lose the ability to make some of those connections. This is why your brain doesn’t seem to work right when you’re angry, stressed, or afraid.

The one concern about using worked-out examples to form chunks is that it can be all too easy to focus too much on why an individual step works and not on the connection between steps—that is, on why this particular step is the next thing you should do.

“just do as you’re told” mindless approach when following a worked-out solution. It’s more like using a guide to help you when traveling to a new place. Pay attention to what’s going on around you when you’re with the guide, and soon you’ll find yourself able to get there on your own.

The first step in chunking, then, is to simply focus your attention on the information you want to chunk.

When you first begin to learn something, you are making new neural patterns and connecting them with preexisting patterns that are spread through many areas of the brain.7 Your octopus tentacles can’t make connections very well if some of them are off on other thoughts.

The second step in chunking is to understand the basic idea you are trying to chunk,

That said, it’s important to realize that just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later. Do not confuse the “aha!” of a breakthrough in understanding with solid expertise! (That’s part of why you can grasp an idea when a teacher presents it in class, but if you don’t review it fairly soon after you’ve first learned it, it can seem incomprehensible when it comes time to prepare for a test.) Closing the book and testing yourself on how to solve the problems will also speed up your learning at this stage.

The third step to chunking is gaining context so you see not just how, but also when to use this chunk.

Context means going beyond the initial problem and seeing more broadly, repeating and practicing with both related and unrelated problems so you see not only wh...

Understanding the underlying concept makes it easier to detect errors when you make them.

To clarify here—chunking may involve your learning how to use a certain problem-solving technique. Context means learning when to use that technique instead of some other technique. Those are the essential steps to making a chunk and fitting that chunk into a greater conceptual overview of what you are learning.

Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.

When you have the book (or Google!) open right in front of you, it provides the illusion that the material is also in your brain. But it’s not. Because it can be easier to look at the book instead of recalling, students persist in their illusion—studying in a far less productive way. This, indeed, is why just wanting to learn the material, and spending a lot of time with it, doesn’t guarantee you’ll actually learn it.

When marking up the text, train yourself to look for main ideas before making any marks, and keep your text markings to a minimum—one sentence or less per paragraph.13 Words or notes in a margin that synthesize key concepts are a good idea.14

always work through homework problems in math and science on your own. Some textbooks include solutions at the back of the book, but you should look at these only to check your answer. This will help ensure that the material is more deeply rooted in your mind and make it much more accessible when you really need it.

You don’t want to wait too long for the recall practice, so that you have to start the reinforcement of the concept from scratch every time.

The only way I know of to make that jump is to work with the concept until it becomes second nature, so you can begin to use it like a tool.”

If you work a problem by just looking at the solution, and then tell yourself, “Oh yeah, I see why they did that,” then the solution is not really yours—you’ve done almost nothing to knit the concepts into your underlying neurocircuitry. Merely glancing at the solution to a problem and thinking you truly know it yourself is one of the most common illusions of competence in learning.

They have an illusion of competence because the solution is already there.18 Pick a mathematical or scientific concept from your notes or from a page in the book. Read it over, then look away and see what you can recall—working toward understanding what you are recalling at the same time. Then glance back, reread the concept, and try it again.

You must have information persisting in your memory if you are to master the material well enough to do well on tests and think creatively with it.

The ability to combine chunks in novel ways underlies much of historical innovation.

If you have a library of concepts and solutions internalized as chunked patterns, you can more easily skip to the right solution to a problem by listening to the whispers from your diffuse mode. Your diffuse mode can also help you connect two or more chunks together in new ways to solve unusual problems.

In building a chunked library, you are training your brain to recognize not only a specific problem, but different types ˙and classes of problems so that you can automatically know how to quickly solve whatever you encounter.

you don’t understand a method presented in a course you are taking, stop and work backward. Go to the Internet and discover who first figured out the method or some of the earliest people to use it. Try to understand how the creative inventor arrived at the idea and why the idea is used—you can often find a simple explanation that gives a basic sense of why a method is being taught and why you would want to use it.

Incidentally, strengthening an initial learning pattern within a day after you first begin forming it is important.

Without the strengthening, the pattern can quickly fade away.

Also, you can reinforce a “wrong” process by doing the same problems over and over the wrong way. This is why ...

When we retrieve knowledge, we’re not being mindless robots—the retrieval process itself enhances deep learning and helps us begin forming chunks.

also talk about the need to study every day, not necessarily for long periods of time but just enough to keep what you are learning at the tip of your tongue.