A Mind For Numbers

by Barbara Oakley, PhD

  —Bill Zettler, Professor of Biology, codiscoverer of several viruses, and winner of the Teacher of the Year Award, University of Florida

  How soon should you start again once you’ve done a Pomodoro? It depends what you’re doing. If you’re trying to get yourself started on something that’s due in many weeks, you may reward yourself with a half hour of guilt-free web surfing. If you’re under stress and have a lot due, a two- to five-minute breather may have to do. You may want to alternate your Pomodoro sessions with working sessions that don’t make use of a timer. If you find yourself lagging and not working with focus, you can put yourself back on the timer.

  In Pomodoro-type timer systems, the process, which involves simple focused effort, moves to the forefront. You disconnect from being stuck on any one item and can get into a state of automaticity without concerns about having to finish anything.9 This automaticity appears to allow you to more easily access diffuse-mode capabilities. 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. This helps prevent the procrastination that can occur not only when you are studying math and science, but when you are doing the writing that is so important for many different college classes.

  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. When you multitask while doing schoolwork, you get tired more quickly. Each tiny shift back and forth of attention siphons off energy. Although each attention switch itself seems tiny, the cumulative result is that you accomplish far less for your effort. You also don’t remember as well, you make more mistakes, and you are less able to transfer what little you do learn into other contexts. A typical negative example of multitasking is that on average, students who allow themselves to multitask while studying or sitting in class have been found to receive consistently lower grades.10

  Procrastination often involves becoming sidetracked on less essential little tasks, such as pencil sharpening, in part because you can still feel the thrill of accomplishment. Your mind is tricking you. That is why keeping an experimental notebook is so important; we’ll talk about that soon.

  NOW YOU TRY!

  Ignorance Is Bliss

  Next time you feel the urge to check your messages, pause and examine the feeling. Acknowledge it. Then ignore it.

  Practice ignoring distractions. It is a far more powerful technique than trying to will yourself to not feel those distractions in the first place.

  SUMMING IT UP

  A little bit of work on something that feels painful can ultimately be very beneficial.

  Habits such as procrastination have four parts:

  The cue

  The routine

  The reward

  The belief

  Change a habit by responding differently to a cue, or even avoiding that cue altogether. Reward and belief make the change long-lasting.

  Focus on the process (the way you spend your time) instead of the product (what you want to accomplish).

  Use the twenty-five-minute Pomodoro to stay productive for brief periods. Then reward yourself after each successful period of focused attention.

  Be sure to schedule free time to nurture your diffuse mode.

  Mental contrasting is a powerful motivating technique—think about the worst aspects of your present or past experiences and contrast these with the upbeat vision of your future.

  Multitasking means that you are not able to make full, rich connections in your thinking, because the part of your brain that helps make connections is constantly being pulled away before neural connections can be firmed up.

  PAUSE AND RECALL

  If you feel muzzy or featherbrained as you’re trying to look away and recall a key idea, or you find yourself rereading the same paragraphs over and over again, try doing a few situps, pushups, or jumping jacks. A little physical exertion can have a surprisingly positive effect on your ability to understand and recall. Try doing something active now, before recalling the ideas of this chapter.

  ENHANCE YOUR LEARNING

  1. Why do you think the zombie-like, habitual part of your brain might prefer process to product? What can you do to encourage a process orientation even two years from now, long after you’ve finished this book?

  2. What kind of subtle change could you make in one of your current habits that could help you avoid procrastination?

  3. What kind of simple and easy new habit could you form that would help you avoid procrastination?

  4. What is one of your most troublesome cues that spins you off into a procrastination response? What could you do to react differently to that cue, or to avoid receiving the cue?

  MATH PROFESSOR ORALDO “BUDDY” SAUCEDO ON HOW FAILURE CAN FUEL SUCCESS

  Oraldo “Buddy” Saucedo is a highly recommended math professor on RateMyProfessors.com; he is a full-time math instructor for the Dallas County Community College District in Texas. One of his teaching mottos is “I offer opportunities for success.” Here, Buddy provides insight into a failure that fueled his success.

  “Every once in a while, a student asks me if I have always been smart—this makes me laugh. I then proceed to tell them about my initial GPA at Texas A&M University.

  “While writing ‘4.0’ on the whiteboard, I say that I was close to having a 4.0 my first semester. ‘Sounds great, right?’ I ask, pausing for their reaction. Then I take my eraser and move the decimal point over to the left. It ends up looking like this: ‘0.4.’

  “Yes. It’s true. I failed miserably and was kicked out of the university. Shocking, right? But I did return and eventually received both my bachelor’s and master’s.

  “There are a lot of failure-to-success types out there with similar stories. If you’ve failed in the past, you may not realize how important that it can be in fueling your success.

  “Here are some of the important lessons I’ve learned in my climb to success:

  You are not your grade; you are better than that. Grades are indicators of time management and a rate of success.

  Bad grades do not mean you are a bad person.

  Procrastination is the death of success.

  Focusing on taking small, manageable steps forward and time management are key.

  Preparation is key to success.

  We all have a failure rate. You will fail. So control your failures. That is why we do homework—to exhaust our failure rate.

  The biggest lie ever is that practice makes perfect. Not true—practice makes you better.

  Practice is where you are supposed to fail.

  Practice at home, in class, anytime and anywhere—except on the TEST!

  Cramming and passing are not success.

  Cramming for tests is the short game with less satisfaction and only temporary results.

  Learning is the long game with life’s biggest rewards.

  We should ALWAYS be perpetual learners. Always in ALL WAYS.

  Embrace failure. Celebrate each failure.

  Thomas Edison renamed his failures: “1,000 ways to NOT create a lightbulb.” Rename your own failures.

  Even zombies get up and try again!

  “They say experience is the best teacher. Instead, it should be that failure is the best teacher. I’ve found that the best learners are the ones who cope best with failure and use it as a learning tool.”

  { 7 }

  chunking versus choking:

  How to Increase Your Expertise and Reduce Anxiety

  New inventions almost never initially appear in their fully formed glory. Rather, they go through many iterations and are constantly being improved. The first “mobile” phones were about as portable a
s bowling balls. The first clumsy refrigerators were cranky devices used by breweries. The earliest engines were overbuilt monstrosities with about as much power as today’s go-karts.

  Enhancements come only after an invention has been out for a while and people have had a chance to mess with it. If you have a working engine on hand, for example, it’s a lot easier to improve any particular feature or add new ones. That’s how clever innovations such as engine turbocharging arose. Engineers realized they could get more power and bang for the buck by stuffing more air and fuel into the combustion chamber. German, Swiss, French, and American engineers, among many others, raced to tweak and improve the basic idea.

  Did you remember to skim ahead and check the questions at the end of the chapter to help you start building chunks of understanding?

  How to Build a Powerful Chunk

  In this chapter, much as with enhancing and refining inventions, we’re going to learn to enhance and refine our chunking skills. Creating a little library of these chunks will help you perform better on tests and solve problems more creatively. These processes will lay the groundwork for you to become an expert at whatever you’re working on.1 (In case you are wondering, our jump in this chapter from procrastination back to chunking is an example of interleaving—varying your learning by hopping back after a break to strengthen an approach you’ve learned earlier.)

  Here’s a key idea: Learning fundamental concepts of math and science can be a lot easier than learning subjects that require a lot of rote memorization. This is not to trivialize the difficulty or importance of memorization. Ask any medical school student preparing for board exams!

  One reason that statement is true is that once you start working on a math or science problem, you’ll notice that each step you complete signals the next step to you. Internalizing problem-solving techniques enhances the neural activity that allows you to more easily hear the whispers of your growing intuition. When you know—really know—how to solve a problem just by looking at it, you’ve created a commanding chunk that sweeps like a song through your mind. A library of these chunks gives you an understanding of fundamental concepts in a way nothing else can.

  So with that, here we go:

  STEPS TO BUILDING A POWERFUL CHUNK

  1. Work a key problem all the way through on paper. (You should have the solution to this problem available, either because you’ve already worked it or because it’s a solved problem from your book. But don’t look at the solution unless you absolutely have to!) As you work through this problem, there should be no cheating, skipping steps, or saying, “Yeah, I’ve got it” before you’ve fully worked it out. Make sure each step makes sense.

  2. Do another repetition of the problem, paying attention to the key processes. If it seems a little odd to work a problem again, keep in mind that you would never learn to play a song on the guitar by playing it only once, or work out by lifting a weight a single time.

  3. Take a break. You can study other aspects of the subject if you need to, but then go do something different. Work at your part-time job, study a different subject,2 or go play basketball. You need to give your diffuse mode time to internalize the problem.

  4. Sleep. Before you go to sleep, work the problem again.3 If you get stuck, listen to the problem. Let your subconscious tell you what to do next.

  5. Do another repetition. As soon as you can the next day, work the problem again. You should see that you are able to solve the problem more quickly now. Your understanding should be deeper. You may even wonder why you ever had any trouble with it. At this point, you can start lightening up on computing each step. Keep your focus on the parts of the problem that are the most difficult for you. This continued focus on the hard stuff is called “deliberate practice.” Although it can sometimes be tiring, it is one of the most important aspects of productive studying. An alternative or supplement at this point is to see whether you can do a similar problem with ease.)

  6. Add a new problem. Pick another key problem and begin working on it in the same way that you did the first problem. The solution to this problem will become the second chunk in your chunked library. Repeat steps one through five on this new problem. And after you become comfortable with that problem, move on to another. You will be surprised how even just a few solid chunks in your library can greatly enhance your mastery of the material and your ability to solve new problems efficiently.

  7. Do “active” repetitions. Mentally review key problem steps in your mind while doing something active, such as walking to the library or exercising. You can also use spare minutes to review as you are waiting for a bus, sitting in the passenger seat of a car, or twiddling your thumbs until a professor arrives in the classroom. This type of active rehearsal helps strengthen your ability to recall key ideas when you are solving homework problems or taking a test.

  That’s it. Those are the key steps to building a chunked library. What you are doing is building and strengthening an increasingly interconnected web of neurons—enriching and strengthening your chunks.4 This makes use of what is known as the generation effect. Generating (that is, recalling) the material helps you learn it much more effectively than simply rereading it.

  This is useful information, but I can already hear what you’re thinking: “I’m spending hours every week just solving all my assigned problems once. How am I supposed to do it four times for one problem?”

  In response, I would ask you: What is your real goal? To turn in homework? Or to perform well on the tests that demonstrate mastery of the material and form the basis for most of your course grade? Remember, just solving a problem with the book open in front of you doesn’t guarantee you could solve something like it again on a test, and, more important, it doesn’t mean that you truly understand the material.

  If you are pressed for time, use this technique on a few key problems as a form of deliberate practice to speed and strengthen your learning and to help you speed your problem-solving skills.

  THE LAW OF SERENDIPITY

  Remember, Lady Luck favors the one who tries. So don’t feel overwhelmed with everything you need to learn about a new subject. Instead, focus on nailing down a few key ideas. You’ll be surprised at how much that simple framework can help.

  The way in which musicians improve their ability to play an instrument can also be applied to learning math in this sense: A master violinist, for example, doesn’t just play a musical piece from beginning to end, over and over again. Instead, she focuses on the hardest parts of the piece—the parts where the fingers fumble and the mind becomes confused.5 You should be like that in your own deliberate practice, focusing and becoming quicker at the hardest parts of the solution procedures you are trying to learn.6

  Remember, research has shown that the more effort you put into recalling material, the deeper it embeds itself into your memory.7 Recall, not simple rereading, is the best form of deliberate practice in study. This strategy is also similar to that used by chess masters. These mental wizards internalize board configurations as chunks associated with the best next moves in their long-term memory. Those mental structures help them select their best option for each move in their current game.8 The difference between lesser-ranked players and grand masters is that grand masters devote far more time to figuring out what their weaknesses are and working to strengthen those areas.9 It’s not as easy as just sitting around and playing chess for fun. But in the end, the results can be far more gratifying.

  Remember, retrieval practice is one of the most powerful forms of learning. It is far more productive than simply rereading material.10 Building a chunked library of ways to solve problems is effective precisely because it is built on methods of retrieval practice. Do not be fooled by illusions of competence. Remember, just staring at material that’s already on the page in front of you can fool you into thinking you know it when you actually don’t.

  When you first start practicing
this way, it may feel awkward—as if you’re a thirty-year-old sitting down for your first piano lesson. But as you practice, you’ll find it gradually coming together more easily and swiftly. Be patient with yourself—as your ease with the material begins to grow, you’ll find yourself enjoying it more and more. Is it work? Sure—and so is learning to play the piano with verve and style. But the payoff is well worth the effort!

  “CHUNK-PUTERS” ARE GREAT!

  “Between being a full-time engineering student and also working a full-time job as an engineering tech, I have too much academic work to keep it all in the forefront of my mind. So my mental trick is to create big chunks for different areas—thermo class, machine design, programming, et cetera. When I need to recall an individual project, I set my current focus aside and reference the desired chunk, which is like a link on my computer desktop. I can either focus in on a specific area or, in diffuse mode, I can look at the complete desktop and find conceptual links between chunks. When I have a clean and organized mental desktop, I can make connections more easily. It increases my mental agility and also allows me to bore deeper into any one topic more easily.”

  —Mike Orrell, junior, electrical engineering

  Hitting the Wall—When Your Knowledge Suddenly Seems to Collapse

  Learning doesn’t progress logically so that each day just adds an additional neat packet to your knowledge shelf. Sometimes you hit a wall in constructing your understanding. Things that made sense before can suddenly seem confusing.11