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

by Barbara Oakley

neuroscientists have been making profound advances in understanding the two different types of networks that the brain switches between—highly attentive states and more relaxed resting state networks.1 We’ll call the thinking processes related to these two different types of networks the focused mode and diffuse mode,

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.

Basically, the focused mode is used to concentrate on something that’s already tightly connected in your mind, often because you are familiar and comfortable with the underlying concepts.

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.

This is a recipe for sinking. It’s like randomly allowing a thought to pop off in the focused-mode pinball machine without paying any real attention to where the solution truly lies.

difference between focused and diffuse modes is to think of a flashlight. You can set a flashlight so it has a tightly focused beam that can penetrate deeply into a small area.

Or you can set the flashlight onto a more diffuse setting where it casts its light broadly, but not very strongly in any one area.

If you are trying to understand or figure out something new, your best bet is to turn off your precision-focused thinking and tur...

The harder you push your brain to come up with something creative, the less creative your ideas will be.

Ultimately, this means that relaxation is an important part of hard work—and good work, for that matter.”

This is because you’ve already laid down a focused-mode pattern that you have a tendency to follow. It takes an intuitive, diffuse leap to realize that you need to completely rearrange the pieces if you want to form another square,

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

Articulating your question is 80 percent of the battle. By the time you’ve figured out what’s confusing, you’re likely to have answered the question yourself!”

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.

book. I skim first so I know basically what the chapter is trying to get at and then I read it in detail. I read the chapter more than once (but not in a row).

but rather because sometimes hearing a slightly different way of phrasing something can make your mind look at the problem from a different angle and spark understanding.

the diffuse mode has access and can begin pinging about in its big-picture way to settle on a solution.

When faced with a difficult problem, instead of focusing intently on it, Edison, according to legend, took a nap. But he did so while sitting in a lounge chair, holding a ball bearing in his hand above a plate on the floor. As he relaxed, his thoughts moved toward free and open diffuse-mode thinking.

Many people think that there’s only one way to do a problem, but there are often a number of different solutions, if you have the creativity to see them.

We all have the ability to make new neural connections and pull from memory something that was never put there in the first place—what creativity researchers Liane Gabora and Apara Ranjan refer to as “the magic of creativity.”

Know that you are making progress with each mistake you catch when trying to solve a problem—finding errors should give you a sense of satisfaction. Edison himself is said to have noted “I have not failed. I’ve just found 10,000 ways that won’t work.”

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

that I’m not talking about a cookie-cutter “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.

soon you’ll find yourself able to get there on your own. You will even begin to figure out new ways of getting ther...

just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later.

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.

you may have a tool in your strategy or problem-solving toolbox, but if you don’t know when to use that tool, it’s not going to do you a lot of good.

make errors, and that’s a good thing.) It also makes it much easier to apply your knowledge to novel problems, a phenomenon called transfer.

Context is where bottom-up and top-down learning meet.

chunking may involve your learning how to use a certain problem-solving technique. Context means learning when to use that technique

My mantra at the end of each class is to tell students to look at the Debit and Credit Rules as well as the Accounting Equation just before they tuck themselves in at night. Let those be the last things they repeat to themselves before falling asleep.

gain a sense of the big picture. This can help you know where to put the chunks you are constructing.

Learn the major concepts or points first—these are often the key parts of a good instructor or book chapter’s outline, flow charts, tables, or concept maps. Once you have this done, fill in the details. Even if a few of the puzzle pieces are missing at the end of your studies, you can still see the big picture.

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,

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.

Try to touch again on something you’re learning within a day, especially if it’s new and rather challenging.

It simultaneously becomes one smooth strand that is easy to follow and use to make new connections.

Now you understand why it is key that you are the one doing the problem solving, not whoever wrote the solution manual. If you work a problem by just looking at the solution, and then tell yourself, “Oh yeah, I see why they did that,”

you’ve done almost nothing to knit the concepts into your underlying neurocircuitry.

thinking you truly know it yourself is one of the most common illusions of ...

That way, some aspects can tentatively and randomly combine with others until eventually, beautiful novelty can emerge.

Bill Gates and other industry leaders, Johnson notes, set aside extended, weeklong reading periods so that they can hold many and varied ideas in mind during one time. This fosters their own innovative thinking by allowing fresh-in-mind, not-yet-forgotten ideas to network among themselves.

Sequential thinking, where each small step leads deliberately toward the solution, involves the focused mode.

Intuition, on the other hand, often seems to require a creative, diffuse mode linking of several seemingly different focused mode thoughts.

Most difficult problems are solved through intuition, because they make a lea...