A Mind For Numbers

by Barbara Oakley, PhD

  —Paul Blowers, University Distinguished Professor (for extraordinary teaching), University of Arizona

  SUMMING IT UP

  The focused mode can allow you to make critical errors even though you feel confident you’ve done everything correctly. Rechecking your work can allow you to get a broader perspective on it, using slightly different neural processes that can allow you to catch blunders.

  Working with others who aren’t afraid to disagree can:

  Help you catch errors in your thinking.

  Make it easier for you to think on your feet and react well in stressful situations.

  Improve your learning by ensuring that you really understand what you are explaining to others and reinforcing what you know.

  Build important career connections and help steer you toward better choices.

  Criticism in your studies, whether you are giving or receiving it, shouldn’t be taken as being about you. It’s about what you are trying to understand.

  It is easiest of all to fool yourself.

  PAUSE AND RECALL

  Close the book and look away. What were the main ideas of this chapter? Try recalling some of these ideas when you are around friends—it will also help your friends to know how valuable their interactions with you actually are!

  ENHANCE YOUR LEARNING

  1. Describe an example of how you were absolutely 100 percent certain of something and were later proven wrong. As a result of this and similar incidents, do you think you are more capable now of accepting criticism of your ideas from others?

  2. How could you make your study sessions with classmates more effective?

  3. How would you handle it if you found yourself in a group that seemed to focus on other issues besides your studies?

  INSIGHTS ON LEARNING FROM PHYSICS PROFESSOR BRAD ROTH, A FELLOW OF THE AMERICAN PHYSICAL SOCIETY AND CO-AUTHOR OF INTERMEDIATE PHYSICS FOR MEDICINE AND BIOLOGY

  Brad Roth and his dog Suki, enjoying the Michigan fall color.

  “One thing I stress in my classes is to think before you calculate. I really hate the ‘plug and chug’ approach that many students use. Also, I find myself constantly reminding students that equations are NOT merely expressions you plug numbers into to get other numbers. Equations tell a story about how the physical world works. For me, the key to understanding an equation in physics is to see the underlying story. A qualitative understanding of an equation is more important than getting quantitatively correct numbers out of it.

  “Here are a few more tips:

  1. “Often, it takes way less time to check your work than to solve a problem. It is a pity to spend twenty minutes solving a problem and then get it wrong because you did not spend two minutes checking it.

  2. “Units of measurement are your friend. If the units don’t match on each side of an equation, your equation is not correct. You can’t add something with units of seconds to something with units of meters. It’s like adding apples and rocks—nothing edible comes of it. You can look back at your work, and if you find the place where the units stop matching, you probably will find your mistake. I have been asked to review research papers that are submitted to professional journals that contain similar unit errors.

  3. “You need to think about what the equation means, so that your math result and your intuition match. If they don’t match, then you have either a mistake in your math or a mistake in your intuition. Either way, you win by figuring out why the two don’t match.

  4. “(Somewhat more advanced) For a complicated expression, take limiting cases where one variable or another goes to zero or infinity, and see if that helps you understand what the equation is saying.”

  { 17 }

  test taking

  We’ve mentioned it earlier, but it’s worth repeating, in bold letters: Testing is itself an extraordinarily powerful learning experience. This means that the effort you put into test taking, including the preliminary mini-tests of your recall and your ability to problem-solve during your preparation, is of fundamental importance. If you compare how much you learn by spending one hour studying versus one hour taking a test on that same material, you will retain and learn far more as a result of the hour you spent taking a test. Testing, it seems, has a wonderful way of concentrating the mind.

  Virtually everything we’ve talked about in this book has been designed to help make the testing process seem straightforward and natural—simply an extension of the normal procedures you use to learn the material. So it’s time now to cut directly to one of the central features of this chapter and the entire book—a checklist you can use to see whether your preparation for test taking is on target.

  TEST PREPARATION CHECKLIST

  Professor Richard Felder is a legend among engineering educators—he has arguably done as much as or more than any educator in this century to help students worldwide to excel in math and science.1 One of the simplest and perhaps most effective techniques Dr. Felder has used to help students is laid out in a memo he wrote to students who have been disappointed with their test grades.2

  “Many of you have told your instructor that you understood the course material much better than your last test grade showed, and some of you asked what you should do to keep the same thing from happening on the next test.

  “Let me ask you some questions about how you prepared for the test. Answer them as honestly as you can. If you answer ‘No’ to many of them, your disappointing test grade should not be too surprising. If there are still a lot of ‘Nos’ after the next test, your disappointing grade on that test should be even less surprising. If your answer to most of these questions is ‘Yes’ and you still got a poor grade, something else must be going on. It might be a good idea for you to meet with your instructor or a counselor to see if you can figure out what it is.

  “You’ll notice that several of the questions presume that you’re working with classmates on the homework—either comparing solutions you first obtained individually or actually getting together to work out the solutions. Either approach is fine. In fact, if you’ve been working entirely by yourself and your test grades are unsatisfactory, I would strongly encourage you to find one or two homework and study partners to work with before the next test. (Be careful about the second approach, however; if what you’re doing is mainly watching others work out solutions, you’re probably doing yourself more harm than good.)

  “The answer to the question ‘How should I prepare for the test?’ becomes clear once you’ve filled out the checklist. You should:

  Do Whatever It Takes to Be Able to Answer “Yes” to Most of the Questions.

  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?

  _Yes _No 11. If there was a review session before the test, did you attend it and ask questions about anything you weren’t sure about?

  _Yes _No 12. Did you get a reasonable night’s sleep before the test? (If your answer is no, your answers to 1–11 may not matter.)

  _Yes _No TOTAL

  The Hard-Start–Jump-to-Easy Technique

  The classic way students are taught to approach tests in math and science is to tackle the easiest problems first. This is based on the notion that by the time you’ve finished the relatively simple problems, you’ll be confident in handling the more difficult.

  This approach works for some people, mostly because anything works for some people. Unfortunately, however, for most people it’s counterproductive. Tough problems often need lots of time, meaning you’d want to start on them first thing on a test. Difficult problems also scream for the creative powers of the diffuse mode. But to access the diffuse mode, you need to not be focusing on what you want so badly to solve!

  What to do? Easy problems first? Or hard?

  The answer is to start with the hard problems—but quickly jump to the easy ones. Here’s what I mean.

  When the test is handed out to you, first take a quick look to get a sense of what it involves. (You should do this in any case.) Keep your eye out for what appears to be the hardest problem.

  Then when you start working problems, start first with what appears to be the hardest one. But steel yourself to pull away within the first minute or two if you get stuck or get a sense that you might not be on the right track.

  This does something exceptionally helpful. “Starting hard” loads the first, most difficult problem in mind, and then switches attention away from it. Both these activities can help allow the diffuse mode to begin its work.

  If your initial work on the first hard problem has unsettled you, turn next to an easy problem, and complete or do as much as you can. Then move next to another difficult-looking problem and try to make a bit of progress. Again, change to something easier as soon as you feel yourself getting bogged down or stuck.

  “With my students, I talk about good worry and bad worry. Good worry helps provide motivation and focus while bad worry simply wastes energy.”

  —Bob Bradshaw, Professor of Math, Ohlone College

  When you return to the more difficult problems, you’ll often be pleased that the next step or steps in the problem will seem more obvious to you. You may not be able to get all the way to the end immediately, but at least you can get further before you switch to something else on which you can make progress.

  In some sense, with this approach to test taking, you’re being like an efficient chef. While you’re waiting for a steak to fry, you can swiftly slice the tomato garnish, then turn to season the soup, and then stir the sizzling onions. The hard-start–jump-to-easy technique may make more efficient use of your brain by allowing different parts of the brain to work simultaneously on different thoughts.3

  Using the hard-start–jump-to-easy technique on tests guarantees you will have at least a little work done on every problem. It is also a valuable technique for helping you avoid Einstellung—getting stuck in the wrong approach—because you have a chance to look at the problems from differing perspectives at different times. All this is particularly important if your instructor gives you partial credit.

  The only challenge with this approach is that you must have the self-discipline to pull yourself off a problem once you find yourself stuck for a minute or two. For most students, it’s easy. For others, it takes discipline and willpower. In any case, by now you are very aware that misplaced persistence can create unnecessary challenges with math and science.

  This may be why test takers sometimes find that the solution pops to mind right as they walk out the door. When they gave up, their attention switched, allowing the diffuse mode the tiny bit of traction it needed to go to work and return the solution. Too late, of course.

  Sometimes people are concerned that starting a problem and then pulling away from it might cause confusion in an examination. This doesn’t seem to be a problem for most people; after all, chefs learn to bring various facets of a dinner together. But if you still have worries about whether this strategy might work for you, try it first on homework problems.

  Be aware of some occasions when hard-start–jump-to-easy might not be appropriate. If the instructor gives only a few points for a really difficult problem (some instructors like to do this), you may wish to concentrate your efforts elsewhere. Some computerized licensure examinations don’t allow for backtracking, so your best bet when facing a tough question is simply to take a deep breath or two from the belly (make sure to breathe out all the way, also) and do your best. And if you haven’t prepared well for the test, then all bets are off. Take what simple points you can.

  DEALING WITH PANIC BEFORE A TEST

  “I tell my students to face your fears. Often, your worst fear is not getting the grades you need for your chosen career. How can you handle this? Simple. Have a plan B for an alternative career. Once you have a plan for the worst contingency, you’ll be surprised to see that the fear will begin to subside.

  “Study hard up until the day of the test, and then let it go. Tell yourself, ‘Oh, well, let me just see how many questions I can get right. I can always pursue my other career choice.’ That helps release stress so you actually do better and get closer to your first career choice.”

  —Tracey Magrann, Professor of Biological Sciences, Saddleback College

  Why Anxiety Can Arise on Tests and How to Deal with It

  If you’re a stressed-out test taker, keep in mind that the body puts out chemicals, such as cortisol, when it is under stress. This can cause sweaty palms, a racing heart, and a knot in the pit of your stomach. But interestingly, research finds that it’s how you interpret those symptoms—the story you tell yourself about why you are stressed—that makes all the difference. If you shift your thinking from “this test has made me afraid” to “this test has got me excited to do my best!” it can make a significant improvement in your performance.4

  Another good tip for panicky test takers is to momentarily turn your attention to your breathing. Relax your stomach, place your hand on it, and slowly draw a deep breath. Your hand should move out, even as your whole chest is moving outward like an expanding barrel.

  By doing this type of deep breathing, you are sending oxygen to critical areas of your brain. This signals that all is well and helps calm you down. But don’t just start this breathing on the day of the test. If you have practiced this breathing technique in the weeks before—just a minute or two here and there is all it takes—you will slide more easily into the breathing pattern during the test. (Remember, practice makes permanent!) It’s particularly helpful to move into the deeper breathing pattern in those final anxious moments before a test is handed out. (And yes, if you’re interested, there are dozens of apps to help you.)

  Another technique involves mindfulness.5 In this technique, you learn to distinguish between a naturally arising thought (I have a big test next week) and an emotional projection that can tag along after that initial thought (If I flunk the test, I will wash out of the program, and I’m not sure what I’ll do then!). These tagalong thoughts, it seems, are projections that arise as glimmers from the diffuse mode. Even a few weeks of simple practice in learning to reframe these thoughts and feelings as simple mental tagalong projections seems to help ease and quiet the mind. Reframing your reaction to such intrusive thoughts works much better than simply tr
ying to suppress them. Students who spent a few weeks practicing with the mindfulness approach performed better on their tests, experiencing fewer distracting thoughts.

  Now you can see why waiting until the end of the test to work on the hardest questions can lead to problems. Just when you are increasingly stressed out because you are running out of time, you are also suddenly facing the toughest problems! As your stress levels soar, you concentrate intently, thinking that focused attention will solve your problems, but of course, your focus instead prevents the diffuse mode from being able to go to work.

  The result? “Paralysis by analysis.”6 The “hard-start–jump-to-easy technique helps prevent this.

  MULTIPLE “GUESS” AND PRACTICE TESTS . . . A FEW TIPS

  “When I give multiple-choice tests, I sometimes find that students fail to fully grasp what the question is asking before they barge ahead, reading the answer options. I advise them to cover up the answers and to try to recall the information so they can answer the question on their own first.

  “When my students complain that the practice test was waaaaay easier than the real one, I ask: What are the confounding variables that make the two situations different? When you took the practice test, were you at home relaxing with tunes on? Taking it with a fellow student? No time limit? Answer key and class materials at hand? These circumstances are not exactly like a crowded classroom with a clock ticking away. I actually encourage those with test anxiety to bring their practice test to another class (big classes where one can slip right in and sit at the back unnoticed) and try taking it there.”