There are two ways to solve problems—first, through sequential, step-by-step reasoning, and second, through more holistic intuition. 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 leap away from what you are familiar with.24 Keep in mind that the diffuse mode’s semi-random way of making connections means that the solutions it provides with should be carefully verified using the focused mode. Intuitive insights aren’t always correct!25
In building a chunked library, you are training your brain to recognize not only a specific problem, but different types ˙and classes of problems so that you can automatically know how to quickly solve whatever you encounter. You’ll start to see patterns that simplify problem solving for you and will soon find that different solution techniques are lurking at the edge of your memory. Before midterms or finals, it is easy to brush up and have these solutions at the mental ready.
NOW YOU TRY!
What to Do If You Can’t Grasp It
If you don’t understand a method presented in a course you are taking, stop and work backward. Go to the Internet and discover who first figured out the method or some of the earliest people to use it. Try to understand how the creative inventor arrived at the idea and why the idea is used—you can often find a simple explanation that gives a basic sense of why a method is being taught and why you would want to use it.
Practice Makes Permanent
I’ve already mentioned that just understanding what’s going on is not usually enough to create a chunk. You can get a sense of what I mean by looking at the “brain” picture shown on p. 69. The chunks (loops) shown are really just extended memory traces that have arisen because you have knit together an understanding. A chunk, in other words, is simply a more complex memory trace. At the top is a faint chunk. That chunk is what begins to form after you’ve understood a concept or problem and practiced just a time or two. In the middle, the pattern is darker. This is the stronger neural pattern that results after you’ve practiced a little more and seen the chunk in more contexts. At the bottom, the chunk is very dark. You’ve now got a solid chunk that’s firmly embedded in long-term memory.
Solving problems in math and science is like playing a piece on the piano. The more you practice, the firmer, darker, and stronger your mental patterns become.
Incidentally, strengthening an initial learning pattern within a day after you first begin forming it is important. Without the strengthening, the pattern can quickly fade away. Later, we’ll talk more about the importance of spaced repetition in learning. Also, you can reinforce a “wrong” process by doing the same problems over and over the wrong way. This is why checking things is so important. Even getting the right answer can occasionally mislead you if you get it by using an incorrect procedure.
THE IMPORTANCE OF CHUNKING
“Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.”26
—William Thurston, winner of the Fields Medal, the top award in mathematics
The challenge with repetition and practice, which lie behind the mind’s creation of solid chunks, is that it can be boring. Worse yet, in the hands of a poor instructor, like my old math teacher, Mr. Crotchety, practice can become an unrelenting instrument of torture. Despite its occasional misuse, however, it’s critical. Everybody knows you can’t effectively learn the chunked patterns of chess, language, music, dance—just about anything worthwhile—without repetition. Good instructors can explain why the practice and repetition is worth the trouble.
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.
This reinforces an idea we’ve alluded to already. When we retrieve knowledge, we’re not being mindless robots—the retrieval process itself enhances deep learning and helps us begin forming chunks.29 Even more of a surprise to researchers was that the students themselves predicted that simply reading and recalling the materials wasn’t the best way to learn. They thought concept mapping (drawing diagrams that show the relationship between concepts) would be best. But if you try to build connections between chunks before the basic chunks are embedded in the brain, it doesn’t work as well. It’s like trying to learn advanced strategy in chess before you even understand the basic concepts of how the pieces move.30
Practicing math and science problems and concepts in a variety of situations helps you build chunks—solid neural patterns with deep, contextual richness.31 The fact is, when learning any new skill or discipline, you need plenty of varied practice with different contexts. This helps build the neural patterns you need to make the new skill a comfortable part of your way of thinking.
KEEP YOUR LEARNING AT THE TIP OF YOUR TONGUE
“By chance, I have used many of the learning techniques described in this book. As an undergraduate I took physical chemistry and became fascinated with the derivations. I got into a habit of doing every problem in the book. As a result, I hard-wired my brain to solve problems. By the end of the semester I could look at a problem and know almost immediately how to solve it. I suggest this strategy to my science majors in particular, but also to the nonscientists. I also talk about the need to study every day, not necessarily for long periods of time but just enough to keep what you are learning at the tip of your tongue. I use the example of being bilingual. When I go to France to work, my French takes a few days to kick in, but then it is fine. When I return to the States and a student or colleague asks me something on my first or second day back, I have to search for the English words! When you practice every day the information is just there—you do not have to search for it.”
—Robert R. Gamache, Associate Vice President, Academic Affairs, Student Affairs, and International Relations, University of Massachusetts, Lowell
Recall Material While Outside Your Usual Place of Study: The Value of Walking
Doing something physically active is especially helpful when you have trouble grasping a key idea. As mentioned earlier, stories abound of innovative scientific breakthroughs that occurred when the people who made them were out walking.32
In addition, recalling material when you are outside your usual place of study helps you strengthen your grasp of the material by viewing it from a different perspective. People sometimes lose subconscious cues when they take a test in a room that looks different from where they st
udied. By thinking about the material while you are in various physical environments, you become independent of cues from any one location, which helps you avoid the problem of the test room being different from where you originally learned the material.33
Internalizing math and science concepts can be easier than memorizing a list of Chinese vocabulary words or guitar chords. After all, you’ve got the problem there to speak to you, telling you what you need to do next. In that sense, problem solving in math and science is like dance. In dance, you can feel your body hinting at the next move.
Different types of problems have different review time frames that are specific to your own learning speed and style.34 And of course, you have other obligations in your life besides learning one particular topic. You have to prioritize how much you’re able to do, also keeping in mind that you must schedule some time off to keep your diffuse mode in play. How much internalizing can you do at a stretch? It depends—everyone is different. But, here’s the real beauty of internalizing problem solutions in math and science. The more you do it, the easier it becomes, and the more useful it is.
ORGANIZE, CHUNK—AND SUCCEED
“The first thing I always do with students who are struggling is ask to see how they are organizing their notes from class and reading. We often spend most of the first meeting going over ways they can organize or chunk their information rather than with my explaining concepts. I have them come back the next week with their material already organized, and they are amazed at how much more they retain.”
—Jason Dechant, Ph.D., Course Director, Health Promotion and Development, School of Nursing, University of Pittsburgh
If you don’t practice with your growing chunks, it is harder to put together the big picture—the pieces are simply too faint.
Interleaving—Doing a Mixture of Different Kinds of Problems—versus Overlearning
One last important tip in becoming an equation whisperer is interleaving.35 Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.
When you are learning a new problem-solving approach, either from your teacher or from a book, you tend to learn the new technique and then practice it over and over again during the same study session. Continuing the study or practice after it is well understood is called overlearning. Overlearning can have its place—it can help produce an automaticity that is important when you are executing a serve in tennis or playing a perfect piano concerto. But be wary of repetitive overlearning during a single session in math and science learning—research has shown it can be a waste of valuable learning time.36 (Revisiting the approach mixed with other approaches during a subsequent study session, however, is just fine.)
In summary, then, once you’ve got the basic idea down during a session, continuing to hammer away at it during the same session doesn’t necessarily strengthen the kinds of long-term memory connections you want to have strengthened. Worse yet, focusing on one technique is a little like learning carpentry by only practicing with a hammer. After a while, you think you can fix anything by just bashing it.37
The reality is, mastering a new subject means learning to select and use the proper technique for a problem. The only way to learn that is by practicing with problems that require different techniques. Once you have the basic idea of a technique down during your study session (sort of like learning to ride a bike with training wheels), start interleaving your practice with problems of different types.38 Sometimes this can be a little tough to do. A given section in a book, for example, is often devoted to a specific technique, so when you flip to that section, you already know which technique you’re going to use.39 Still, do what you can to mix up your learning. It can help to look ahead at the more varied problem sets that are sometimes found at the end of chapters. Or you can deliberately try to make yourself occasionally pick out why some problems call for one technique as opposed to another. 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. Another idea is to open the book to a randomly chosen page and work a problem while, as much as possible, hiding from view everything but the problem.
EMPHASIZE INTERLEAVING INSTEAD OF OVERLEARNING
Psychologist Doug Rohrer of the University of South Florida has done considerable research on overlearning and interleaving in math and science. He notes:
“Many people believe overlearning means studying or practicing until mastery is achieved. However, in the research literature, overlearning refers to a learning strategy in which a student continues to study or practice immediately after some criterion has been achieved. An example might be correctly solving a certain kind of math problem and then immediately working several more problems of the same kind. Although working more problems of the same kind (rather than fewer) often boosts scores on a subsequent test, doing too many problems of the same kind in immediate succession provides diminishing returns.
“In the classroom and elsewhere, students should maximize the amount they learn per unit time spent studying or practicing—that is, they should get the most bang for the buck. How can students do this? The scientific literature provides an unequivocal answer: Rather than devote a long session to the study or practice of the same skill or concept so that overlearning occurs, students should divide their effort across several shorter sessions. This doesn’t mean that long study sessions are necessarily a bad idea. Long sessions are fine as long as students don’t devote too much time to any one skill or concept. Once they understand ‘X,’ they should move on to something else and return to ‘X’ on another day.”40
It’s best to write the initial solution, or diagram, or concept, out by hand. There’s evidence that writing by hand helps get the ideas into mind more easily than if you type the answer.41 More than that, it’s often easier to write symbolic material like ∑ or Ω by hand than to search out the symbol and type it (unless you use the symbols often enough to memorize the alt codes).42 But if you then want to photograph or scan the question and your handwritten solution to load it into a flash card program for your smartphone or laptop, that will work just fine. Beware—a common illusion of competence is to continue practicing a technique you know, simply because it’s easy and it feels good to successfully solve problems. Interleaving your studies—making a point to review for a test, for example, by skipping around through problems in the different chapters and materials—can sometimes seem to make your learning more difficult. But in reality, it helps you learn more deeply.
AVOID MIMICKING SOLUTIONS—PRACTICE CHANGING MENTAL GEARS
“When students do homework assignments, they often have ten identical problems in a row. After the second or third problem, they are no longer thinking; they are mimicking what they did on the previous problem. I tell them that, when doing the homework from section 9.4, after doing a few problems, go back and do a problem from section 9.3. Do a couple more 9.4 problems, and then do one from section 9.1. This will give them practice in mentally shifting gears in the same way they’ll need to switch gears on the test.
“I also believe too many students do homework just to get it done. They finish a problem, check their answer in the back of the text, smile, an
d go on to the next problem. I suggest that they insert a step between the smile and going on to the next problem—asking themselves this question: How would I know how to do the problem this way if I saw it on a test mixed together with other problems and I didn’t know it was from this section of the text? Students need to think of every homework problem in terms of test preparation and not as part of a task they are trying to complete.”
—Mike Rosenthal, Senior Instructor of Mathematics, Florida International University
SUMMING IT UP
Practice helps build strong neural patterns—that is, conceptual chunks of understanding.
Practice gives you the mental fluidity and agility you need for tests.
Chunks are best built with:
Focused attention.
Understanding of the basic idea.
Practice to help you gain big-picture context.
Simple recall—trying to remember the key points without looking at the page—is one of the best ways to help the chunking process along.
In some sense, recall helps build neural hooks that you can hang your thinking on.
ENHANCE YOUR LEARNING
1. How is a chunk related to a memory trace?
2. Think of a topic you are passionate about. Describe a chunk involving that topic that was at first difficult for you to grasp but now seems easy.
3. What is the difference between top-down and bottom-up approaches to learning? Is one approach preferable to the other?
4. Is understanding enough to create a chunk? Explain why or why not.
5. What is your own most common illusion of competence in learning? What strategy can you use to help avoid falling for this illusion in the future?
A Mind For Numbers Page 7
