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

by Barbara Oakley

Focused-mode thinking is essential for studying math and science. It involves a direct approach to solving problems using rational, sequential, analytical approaches. The focused mode is associated with the concentrating abilities of the brain’s prefrontal cortex, located right behind your forehead.

Diffuse-mode thinking is also essential for learning math and science. It allows us to suddenly gain a new insight on a problem we’ve been struggling with and is associated with “big-picture” perspectives. Diffuse-mode thinking is what happens when you relax your attention and just let your mind wander.

Related to these difficulties in math and science is another challenge. It’s called the Einstellung effect (pronounced EYE-nshtellung). In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.

one significant mistake students sometimes make in learning math and science is jumping into the water before they learn to swim.

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,

To learn about and be creative in math and science, we need to strengthen and use both the focused and diffuse modes.19

But as long as we are consciously focusing on a problem, we are blocking the diffuse mode.

Once you are distracted from the problem at hand, the diffuse mode has access and can begin pinging about in its big-picture way to settle on a solution.

blinking is a vital activity that provides another means of reevaluating a situation.

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.

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

one of the first steps toward gaining expertise in math and science is to create conceptual chunks—mental leaps that unite separate bits of information through meaning.

There is a bottom-up chunking process where practice and repetition can help you both build and strengthen each chunk, so you can easily gain access to it when needed. And there is a top-down “big picture” process that allows you to see where what you are learning fits in.

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

Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective.

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.

Law of Serendipity comes to play: Lady Luck favors the one who tries.23

But you can’t learn mathematics or science without also including a healthy dose of practice and repetition to help you build the chunks that will underpin your expertise.27

In the same amount of time, by simply practicing and recalling the material, students learned far more and at a much deeper level than they did using any other approach,

recalling material when you are outside your usual place of study helps you strengthen your grasp of the material by viewing it from a different perspective.

Interleaving—Doing a Mixture of Different Kinds of Problems—versus Overlearning

Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.

Continuing the study or practice after it is well understood is called overlearning.

You want your brain to become used to the idea that just knowing how to use a particular problem-solving technique isn’t enough—you also need to know when to use it.

We procrastinate about things that make us feel uncomfortable.

Procrastination is a single, monumentally important “keystone” bad habit.

the better you get at something, the more you’ll find you enjoy it.

Habit is an energy saver for us. It allows us to free our mind for other types of activities.

The only place you need to apply willpower is to change your reaction to the cue.

1. The Cue: Recognize what launches you into your zombie, procrastination mode. Cues usually fall into one of the following categories: location, time, how you feel, reactions to other people, or something that just happened.2 Do you look something up on the web and then find yourself web surfing? Does a text message disturb your reverie, taking you ten minutes to get back into the flow of things even when you try to keep yourself on task? The issue with procrastination is that because it’s an automatic habit, you are often unaware that you have begun to procrastinate. Students often find that developing new cues, such as starting homework as soon as they get home from school or right after their first break from class, are helpful. As procrastination expert Piers Steel, author of The Procrastination Equation, points out, “If you protect your routine, eventually it will protect you.”3

The key to rewiring is to have a plan. Developing a new ritual can be helpful.

The Reward:

It’s particularly important to realize that giving yourself even a small “attaboy” or “attagirl” jump-starts the process of rewiring your brain. This rewiring, sometimes called learned industriousness, helps brighten tasks you once thought were boring and uninteresting.

Also remember that the better you get at something, the more enjoyable it can become.

The Belief: The most important part of changing your procrastination habit is the belief that you can do it.

Belief that your new system works is what can get you through.

A powerful approach is mental contrasting.

In mental contrasting, it’s the contrast of where you want to be with where you are now, or where you have been, that makes the difference.

Learn to focus on process, not product.

To prevent procrastination, you want to avoid concentrating on product. Instead, your attention should be on building processes—habits—that coincidentally allow you to do the unpleasant tasks that need to be done.

The product is what triggers the pain that causes you to procrastinate.

By focusing on process rather than product, you allow yourself to back away from judging yourself (Am I getting closer to finishing?) and allow yourself to relax into the flow of the work.

Multitasking is like constantly pulling up a plant. This kind of constant shifting of your attention means that new ideas and concepts have no chance to take root and flourish.

Learning fundamental concepts of math and science can be a lot easier than learning subjects that require a lot of rote memorization.

Generating (that is, recalling) the material helps you learn it much more effectively than simply rereading it.

One of the most important things we can do when we are trying to learn math and science is to bring the abstract ideas to life in our minds.

Take pride in who you are, especially in the qualities that make you “different,” and use them as a secret talisman for success.

TEN RULES OF GOOD STUDYING 1. Use recall. After you read a page, look away and recall the main ideas. Highlight very little, and never highlight anything you haven’t put in your mind first by recalling. Try recalling main ideas when you are walking to class or in a different room from where you originally learned it. An ability to recall—to generate the ideas from inside yourself—is one of the key indicators of good learning. 2. Test yourself. On everything. All the time. Flash cards are your friend. 3. Chunk your problems. Chunking is understanding and practicing with a problem solution so that it can all come to mind in a flash. After you solve a problem, rehearse it. Make sure you can solve it cold—every step. Pretend it’s a song and learn to play it over and over again in your mind, so the information combines into one smooth chunk you can pull up whenever you want. 4. Space your repetition. Spread out your learning in any subject a little every day, just like an athlete. Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time. 5. Alternate different problem-solving techniques during your practice. Never practice too long at any one session using only one problem-solving technique—after a while, you are just mimicking what you did on the previous problem. Mix it up and work on different types of problems. This teaches you both how and when to use a technique. (Books generally are not set up this way, so you’ll need to do thi...