Book SummaryA Mind for Numbers, by Barbara Oakley

1-Page Summary1-Page Book Summary of A Mind for Numbers

Because the world is increasingly dependent on technology, being competent in technical subjects like math and science is beneficial for both your career and life in general.

Some people find math and science intimidating, and Barbara Oakley, the author of A Mind for Numbers, used to be one of those people. However, she overcame her technophobia, eventually earning a doctorate degree in engineering. Her purpose in writing this book is to help you learn math and science by showing you how you learn and how to build effective study habits.

In this guide, we’ll discuss the principles that Oakley presents, often examining their scientific basis or comparing them to the ideas of other experts. We’ll begin by discussing Oakley’s exposition of how your brain works, which lays a foundation for understanding how you learn. We’ll then discuss Oakley’s strategies for remembering information and developing good habits, both keys to long-term success in learning math and science. Finally, we’ll consider how to overcome the problem of procrastination, which is a special kind of habit, and one that Oakley warns can severely hinder your academic success if it is not dealt with.

Brain Basics: How You Learn

The Two Modes of Thinking

A key theme of A Mind for Numbers is that alternating between modes of thinking can help you learn new things and problem-solve effectively. Oakley explains that your brain naturally alternates between two modes of thinking: focused and diffuse.

According to Oakley, focused-mode thinking occurs when your attention is focused on something, and it allows you to process detailed information. However, she notes that it’s susceptible to the “Einstellung effect,” which occurs when you are unable to solve a problem because the solution is outside the scope of where you are looking for it.

Oakley asserts that diffuse-mode thinking occurs when you relax your focus or let your mind wander. She explains that it continues to subconsciously process information from previous focused-mode thinking, but in a different way: It can circumvent the Einstellung effect by allowing you to mentally step away from detailed problems and see the big picture, or generate creative solutions by making connections between diverse concepts.

Thus, Oakley explains that solving any difficult problem requires an exchange of information between your brain’s focused-mode and diffuse-mode functions. She recommends that you start by deliberately focusing on the problem and then deliberately divert your attention, allowing your brain to switch to diffuse mode. Repeat as needed, alternating between modes until you solve the problem.

Comparing Perspectives on the Two Modes of Thinking

Other writers have used a variety of terms for what Oakley calls “focused-mode thinking” and “diffuse-mode thinking.”

Edward de Bono coined the terms “Lateral Thinking” and “Vertical Thinking” for what Oakley calls diffuse mode and focused mode, respectively. He emphasizes that lateral thinking (diffuse mode) generates new ideas or solutions, while vertical thinking (focused mode) down-selects between possibilities. Thus, in de Bono’s view, you should switch to diffuse mode to generate new ideas, and switch to focused mode to select and implement a solution.

Malcolm Gladwell contrasted “conscious thinking” with “unconscious thinking.” These modes of thinking can be loosely mapped to Oakley’s focused and diffuse mode thinking, respectively. Gladwell presents conscious thinking as a tool for rational decision making and unconscious thinking as a tool for intuitive decision making. He emphasizes that unconscious thinking operates quickly and that trained intuition can be highly accurate. Thus, he advocates using unconscious thinking to solve certain types of problems directly, although he doesn’t directly address the kind of academic problems that Oakley focuses on.

Daniel Kahneman wrote about “System 1” and “System 2” thinking, defining them similarly to Gladwell’s “unconscious” and “conscious” thinking modes, respectively. However, he highlighted the many types of bias that can cause intuition to be wrong to emphasize the importance of System 2 to catch System 1’s errors. As such, he seemed to assume you operate in “System 1” by default, and emphasized the need to invoke “System 2” to analyze the output of “System 1” rationally whenever the consequences of an error are significant. This theme resonates with Oakley’s observation that once you identify a possible solution with diffuse mode, you still have to switch back to focused mode to work out the details and make sure the solution actually works.

How to Switch Between Modes of Thinking

Oakley lists a number of ways you can give your diffuse mode time to operate on problems you are trying to solve:

• Go for a walk or do something athletic.

(Shortform note: This method can be doubly effective, as health psychologist Kelly McGonigal reports that during physical exercise, your muscles secrete chemicals called myokines that stimulate your brain, increasing cognitive performance and alleviating depression.)

• Take care of routine tasks, such as housework or laundry.

(Shortform note: This helps make efficient use of your time while studying, but as an added benefit, a recent study concluded housework is a feasible means for workers with sedentary jobs and limited time to get the health benefits of physical activity.)

• Get some rest. Oakley asserts that your brain cannot function normally without enough...

A Mind for Numbers Summary Shortform Introduction

Because the world is increasingly dependent on technology, being competent in technical subjects like math and science is beneficial both for your career and life in general.

You may not realize it, but your brain has an extraordinary capacity for complex calculations: Tasks like stepping over a garden hose as you walk across your lawn require tremendously complex computations, and yet they seem easy because your brain does them intuitively. Barbara Oakley wrote A Mind for Numbers to help you learn math and science well enough that they, too, become intuitive. In this book, Oakley presents principles about how your brain works and builds strategies for learning and studying upon these principles.

About the Author

Barbara Oakley is a professor of engineering at Oakland University who also teaches a MOOC (massive open online course) called “Learning How to Learn: Powerful Mental Tools to Help You Master Tough Subjects” through Coursera. To date, this class has reached over two million students.

Oakley grew up thinking she was technically inept. From childhood through...

A Mind for Numbers Summary Part 1: Understand How You Think

A key theme of A Mind for Numbers is that alternating between modes of thinking can help you learn new things and problem-solve effectively. To understand how to do this and why it works, you first need to understand a few things about how your brain works. In this section, we’ll first discuss your brain’s two modes of thinking. Then we’ll comment on switching between them to solve problems.

The Two Basic Modes of Thinking

Oakley explains that your brain naturally alternates between two modes of thinking: focused and diffuse.

As an analogy for these two modes of thinking, imagine a camera with a variable lens. If you zoom in on the subject, details are clearly visible, but the surroundings are cut off: This is like focused-mode thinking. If you zoom out, the big picture comes into view, but details are obscured: This is like diffuse-mode thinking.

Oakley notes that in nature, animals must alternate between detail-oriented tasks, such as eating berries off a bush (focused mode), and general awareness, such as scanning their environment for predators (diffuse mode). She suggests that this could explain the origin of the two modes.

(Shortform note: This may be an...

Shortform Exercise: Alternate Between Focused and Diffuse Thinking

Alternating between focused-mode and diffuse-mode thinking is important for stimulating creativity and problem-solving. In this exercise, you’ll optimize your schedule to promote alternating between modes of thinking and reflect on symptoms of the Einstellung effect that can alert you it’s time to switch to diffuse mode.

Of the things on your to-do list this week, which ones will require intensive focus? (For example, doing your taxes.)

Shortform Exercise: Circumvent the Einstellung Effect

You can overcome the Einstellung effect (that is, the inability to solve a problem or to recognize a better solution because the solution lies outside the scope of the ideas you are focusing on) by switching to diffuse-mode thinking. In this exercise, you’ll analyze a recent experience with Einstellung and formulate a plan to overcome it.

Recall the last time you were stumped by a problem (at least temporarily). What were your symptoms of the Einstellung effect? At about the time you stopped making progress, how did you feel? If there were other people around you at the time, did they advise you to take a break or comment about how you looked/acted at the time?

A Mind for Numbers Summary Part 2: Understanding Memory and Information Chunking

Continuing our discussion of your brain’s capabilities, in this section, we’ll focus on memory and the learning process. According to Oakley, you use both your working memory and your long-term memory to learn math and science, so we’ll start by discussing the distinction between them.

Then, we’ll discuss “chunking.” According to Oakley, chunking is the process in which your memories get consolidated into “chunks” of related information in your brain. From Oakley’s writing, we infer that chunking helps lay the foundation for understanding how your memory works because your working and long-term memory depend on your brain’s ability to organize information into chunks. We’ll build practical study habits on this foundation in later parts.

Working Memory Is Your Brain’s Workspace

As Oakley points out, working memory holds the information that your mind is actively processing. You use it to solve problems in math and science when you focus on the problem and think about the principles you would use to solve it. You also use it when you take in new information and try to make sense of it.

While Oakley doesn’t address this explicitly, there seems to be a strong...

A Mind for Numbers Summary Part 3: Making Information More Memorable

In the previous part, we discussed how your memory works and how it depends on “chunked'' information. Oakley explains that the more connections the chunk has, the more memorable it is, and the more often you access it, the stronger the connections become. More connections and stronger connections both make a chunk more accessible. Thus, there are two factors that determine how well a concept will stick in your memory: how memorable it is, and how often you recall it.

In this section, we’ll cover Oakley’s discussion of techniques for making information more memorable by connecting it to more parts of your brain, starting with an overview of the factors that make things more memorable and then presenting some specific techniques for remembering things. Then, in the next part, we’ll discuss recall strategies.

What Makes Information Memorable

From Oakley’s discussion, we can identify a number of factors that determine how intrinsically memorable a piece of information is. In Moonwalking with Einstein, Joshua Foer similarly identifies factors that make information more memorable. The factors that...

Shortform Exercise: Make a Visual Metaphor

The visual metaphor method is a particularly powerful way to make information more memorable. In this exercise, you will create a visual metaphor for yourself.

Think of a fact or principle that you need to remember, and type it in the box below. If you can’t think of one, pick one of the following:

• The circumference of a circle is equal to its diameter multiplied by “pi,” a constant equal to about 3.14.
• The “kinetic energy” of an object is equal to its mass multiplied by one-half the square of its velocity.
• The density of concrete is 2.4 metric tons per cubic meter.

A Mind for Numbers Summary Part 4: Repetition Strategy

We’ve discussed how your memory works and some strategies for making facts more memorable. However, recall that your brain has natural processes for clearing information from your working memory. Thus, Oakley warns that even a memorable fact may soon become unretrievable if you don’t review it. According to Oakley, repetition helps to ensure that your brain recognizes facts as important and stores them in long-term memory. In this section, we’ll present several strategies for effective repetition.

How does your brain differentiate between trivial information to purge and important information to save? At present, we don’t know: As Joshua Foer points out in Moonwalking with Einstein, scientists can measure which parts of the brain are more active during prescribed activities, but they have not yet been able to determine how memories are stored in your brain, or how human thought actually works.

Someday, if scientists solve this mystery, we may discover new and more efficient ways of flagging certain memories for retention and others for deletion. But until then, we must rely on the tried-and-true,...

Shortform Exercise: Practice Intentional Recall

In this chapter, we discussed how you can use the technique of “intentional recall” to solidify concepts in memory. To practice intentional recall on the chapter you’ve just read, try to answer the following questions from memory.

What is spaced repetition?

A Mind for Numbers Summary Part 5: Understanding Habits

According to Oakley, your study habits (good or bad) have a strong impact on your ability to learn math and science. Earlier, we discussed Oakley’s exposition of chunking. In this section, we’ll discuss her explanation of how chunking facilitates habits and the strategies she presents for changing habits.

Anatomy of a Habit

Oakley defines a habit as a pre-programmed activity that you perform without conscious input of mental energy. She explains that habits allow your brain to operate more efficiently, because exercising willpower takes a lot of neural resources, and habits allow you to act without using willpower. Tasks that you can perform automatically also give your brain a chance to operate in diffuse mode.

(Shortform note: In The Power of Habit, Charles Duhigg asserts that more than 40% of your actions each day are habitual. By extension, according to Oakley’s logic, that’s a large chunk of your day that can be dedicated to diffuse mode thinking. Duhigg also agrees with Oakley’s assertion that habits eventually become unconscious actions,...

Shortform Exercise: Analyze a Habit

Think about the last time you caught yourself in a bad habit. In this exercise, you’ll analyze the parts of this habit and what you could do to change it. Recall that a habit has four parts:

1. The cue is what triggers the habit.
2. The routine is what you do when the habit is triggered.
3. The reward is the benefit you derive from the routine. (For bad habits, the benefit is short-lived and the long-term consequences are negative, but your brain only processes the immediate results as part of the habit chunk.)
4. The belief is what makes the habit powerful by linking it to what you believe about reality or your own identity.

What were you doing before the habit took over? What do you think was the cue that triggered the habit?

A Mind for Numbers Summary Part 6: The Problem of Procrastination

According to Oakley, habitual procrastination is often your most significant barrier to learning math and science. In this section, we’ll discuss Oakley’s model of the procrastination habit and her recommended strategies for avoiding procrastination, and compare them to others’ recommendations.

Why Is Procrastination a Problem?

Oakley acknowledges that procrastination, in the strictest sense, is not always a problem: Sometimes you can make better use of your time by deferring certain tasks until later. Rushing into a task with unfounded assumptions may necessitate rework, while pausing to reflect gives you a chance to apply lessons from previous tasks to new projects.

However, Oakley also asserts that habitual procrastination is a problem because you need to make conscious decisions about how to spend your time in order to spend it productively. She cautions that, like any habit, you can slip into procrastination without thinking, or even without realizing you are doing it.

A Corroborating Perspective on Creative and Destructive Procrastination

[In Eat That Frog, Brian Tracy argues that there will always be more things you could do than you have time...

Shortform Exercise: Apply Anti-Procrastination Strategies to Your To-Do List

Think about the tasks on your current to-do list. In this exercise, we will review a few strategies for avoiding procrastination and you will apply them to your tasks.

Eliminate Distractions: Describe one specific distraction that could hinder you from completing the tasks on your list. What could you do to avoid this distraction?

A Mind for Numbers Summary Part 7: How to Take Tests

In school, you periodically have to demonstrate what you have learned by taking an exam. To do well on an exam, you need to know the material. In the preceding parts, we’ve discussed Oakley’s recommendations for studying effectively and remembering information. In this section, we’ll discuss Oakley’s other test-taking tips and techniques.

Aside from studying effectively, we’ve extracted three general rules for test-taking from A Mind for Numbers: get enough sleep, take a strategic approach to solving the test questions, and manage your test anxiety. We’ve already discussed the importance of sleep for memory and brain function in earlier parts, so we’ll devote Part 7 to discussing Oakley’s strategies for test-taking and managing test anxiety.

Strategy for Taking a Test

Oakley recommends using the “hard-start-jump-to-easy” test-taking strategy:

1. First, skim the test and identify which questions look the hardest and which ones look easy.
2. Work the hardest problems first.
3. If you get stuck on a problem, Oakley says to stop working on it and switch to an easy problem.
4. When you finish the easy problem, come back to the problem you got stuck on and try to make...

A Mind for Numbers Summary Part 8: Achieve Your Full Potential

So far, we’ve discussed Oakley’s explanation of how your brain works and strategies for understanding and remembering material, as well as strategies for building good study habits, avoiding procrastination, and taking tests. As we prepare to conclude this guide, let’s examine a few more principles that Oakley says can help you get the most out of your studies.

Beware of Hidden Weaknesses

According to Oakley, identifying its own weaknesses is one thing your brain does not do well. When we think we understand a concept better than we actually do, we are suffering from an “illusion of competence.” She identifies certain study methods that can contribute to illusions of competence:

• According to Oakley, actions like highlighting or underlining key concepts in a textbook make them stand out on the page but don’t embed them in your memory. Instead, write notes in the margin.
• If you can easily retrieve the information you need to work through a problem from your notes, textbooks, or other sources, this makes you feel like you’ve mastered the problem. However, as Oakley points out, if you can only retrieve it from external resources, and not from your memory, then you...