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

by Barbara Oakley

As you first begin looking at a chapter or section of a book that teaches concepts of math or science, it helps to take a “picture walk” through the chapter, glancing not only at the graphics, diagrams, and photos, but also at the section headings, summary, and even questions at the end of the chapter, if the book has them. This seems counterintuitive—you haven’t actually read the chapter yet, but it helps prime your mental pump.

I used to do this! But I thought it wasn’t the way you were supposed to read a textbook, so I stopped. Wish I had known better.

The resting times between your focused-mode efforts should be long enough to get your conscious mind completely off the problem you’re working on. Usually a few hours is long enough for the diffuse mode to make significant progress but not so long that its insights fade away before being passed on to the focused mode. A good rule of thumb, when you are first learning new concepts, is not to let things go untouched for longer than a day.

Great advice for students. A short break is okay, but probably not more than a day.

Ultimately, both bottom-up chunking and top-down big-picture approaches are vital if you are to become an expert with the material. We love creativity and the idea of being able to learn by seeing the big picture. 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 Research published in the journal Science provided solid evidence along these lines.28 Students studied a scientific text and then practiced it by recalling as much of the information as they could. Then they restudied the text and recalled it (that is, tried to remember the key ideas) once more. The results? 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, including simply rereading the text a number of times or drawing concept maps that supposedly enriched the relationships in the materials under study. This improved learning comes whether students take a formal test or just informally test themselves.

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.

Consider creating index cards with the problem question on one side, and the question and solution steps on the other. That way you can easily shuffle the cards and be faced with a random variety of techniques you must call to mind. When you first review the cards, you can sit at a desk or table and see how much of the solution you can write on a blank sheet of paper without peeking at the back of the card. Later, when mastery is more certain, you can review your cards anywhere, even while out for a walk. Use the initial question as a cue to bring to mind the steps of the response, and flip the card over if necessary to verify that you’ve got the procedural steps all in mind. You are basically strengthening a new chunk.

If you find yourself avoiding certain tasks because they make you uncomfortable, there is a great way to reframe things: Learn to focus on process, not product. Process means the flow of time and the habits and actions associated with that flow of time—as in, “I’m going to spend twenty minutes working.” Product is an outcome—for example, a homework assignment that you need to finish.

Best Apps and Programs to Keep on Task (free versions available unless otherwise noted) Timers The Pomodoro technique (varied prices and resources): http://pomodorotechnique.com/ Tasks, Planning, and Flash Cards 30/30—combines timers with a task list: http://3030.binaryhammer.com/ StudyBlue—combines flash cards and notes with text messages when it’s time to study again, along with a direct link to the material: http://www.studyblue.com/ Evernote—one of my personal favorites; very popular for noting task lists and random pieces of information (replaces the little notebook writers have long carried to keep track of their ideas): http://evernote.com/ Anki—one of the best pure flash card systems, with an excellent spaced repetition algorithm; many excellent premade decks are available for a variety of disciplines: http://ankisrs.net/ Quizlet.com—allows you to input your own flash cards; you can work with classmates to divide up the duties (free): http://quizlet.com/ Google Tasks and Calendar: http://mail.google.com/mail/help/tasks/ Limiting Your Time on Time-Wasting Websites Freedom—many people swear by this program, available for MacOS, Windows, and Android ($10): http://macfreedom.com/ StayFocusd—for Google Chrome: https://chrome.google.com/webstore/detail/stayfocusd/laankejkbhbdhmipfmgcngdelahlfoji?hl=en LeechBlock—for Firefox: https://addons.mozilla.org/en-us/firefox/addon/leechblock/ MeeTimer—for Firefox; tracks and logs where you spend your time: https://addons.mozilla.o...

Pausing and reflecting are key, not only in stopping procrastination but in math and science problem solving in general. You may be surprised to learn that the difference in the way that math experts (professors and graduate students) and math novices (undergraduate students) solve physics problems is that experts are slower to begin solving a problem.6 Experts took an average of forty-five seconds to figure out how they would categorize a problem according to its underlying physics principles. Undergraduates, on the other hand, simply rushed right in, taking only thirty seconds to determine how they should proceed. Unsurprisingly, the conclusions drawn by the undergraduates were often wrong because their choices were based on superficial appearances rather than underlying principles. It’s as if experts took their time to conclude that broccoli is a vegetable and lemon is a fruit, while novices barged in to say that broccoli is a tiny tree while lemons are clearly eggs. Pausing gives you time to access your library of chunks and allows your brain to make connections between a particular problem and the bigger picture.

Procrastination is such an important topic that this summary includes key takeaway points from all this book’s chapters on overcoming procrastination: Keep a planner-journal so you can easily track when you reach your goals and observe what does and doesn’t work. Commit yourself to certain routines and tasks each day. Write your planned tasks out the night before, so your brain has time to dwell on your goals to help ensure success. Arrange your work into a series of small challenges. Always make sure you (and your zombies!) get lots of rewards. Take a few minutes to savor the feelings of happiness and triumph. Deliberately delay rewards until you have finished a task. Watch for procrastination cues. Put yourself in new surroundings with few procrastination cues, such as the quiet section of a library. Obstacles arise, but don’t make a practice of blaming all your problems on external factors. If everything is always somebody else’s fault, it’s time to start looking in the mirror. Gain trust in your new system. You want to work hard during times of focused concentration—and also trust your system enough that when it comes time to relax, you actually relax without feelings of guilt. Have backup plans for when you still procrastinate. No one is perfect, after all. Eat your frogs first.

In chemistry, compare a cation with a cat that has paws and is therefore “pawsitive,” and an anion with an onion that is negative because it makes you cry.

We can learn a lot about how to do math and science from sports. In baseball, for example, you don’t learn how to hit in one day. Instead, your body perfects your swing from plenty of repetition over a period of years. Smooth repetition creates muscle memory, so that your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

Taking responsibility for your own learning is one of the most important things you can do. Teacher-centered approaches, where the teacher is considered to be the one with the answers, may sometimes inadvertently foster a sense of helplessness about learning among students.6 Surprisingly, teacher evaluation systems may foster the same helplessness—these systems allow you to place the blame for failure on your teacher’s inability to motivate or instruct.7 Student-centered learning, where students are challenged to learn from one another and are expected to be their own drivers toward mastery of the material, is extraordinarily powerful.

Test Preparation Checklist Answer “Yes” only if you usually did the things described (as opposed to occasionally or never). Homework _Yes _No 1. Did you make a serious effort to understand the text? (Just hunting for relevant worked-out examples doesn’t count.) _Yes _No 2. Did you work with classmates on homework problems, or at least check your solutions with others? _Yes _No 3. Did you attempt to outline every homework problem solution before working with classmates? Test Preparation The more “Yes” responses you recorded, the better your preparation for the test. If you recorded two or more “No” responses, think seriously about making some changes in how you prepare for the next test. _Yes _No 4. Did you participate actively in homework group discussions (contributing ideas, asking questions)? _Yes _No 5. Did you consult with the instructor or teaching assistants when you were having trouble with something? _Yes _No 6. Did you understand ALL of your homework problem solutions when they were handed in? _Yes _No 7. Did you ask in class for explanations of homework problem solutions that weren’t clear to you? _Yes _No 8. If you had a study guide, did you carefully go through it before the test and convince yourself that you could do everything on it? _Yes _No 9. Did you attempt to outline lots of problem solutions quickly, without spending time on the algebra and calculations? _Yes _No 10. Did you go over the study guide and problems with classmates and quiz one another? _Y...

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...