As it turns out, one of the first steps toward gaining expertise in math and science is to create conceptual chunks—mental leaps that unite separate bits of information through meaning.4 Chunking the information you deal with helps your brain run more efficiently. Once you chunk an idea or concept, you don’t need to remember all the little underlying details; you’ve got the main idea—the chunk—and that’s enough. It’s like getting dressed in the morning. Usually you just think one simple thought—I’ll get dressed. But it’s amazing when you realize the complex swirl of underlying activities that take place with that one simple chunk of a thought.
When you are studying math and science, then, how do you form a chunk?
Basic Steps to Forming a Chunk
Chunks related to different concepts and procedures can be molded in many different ways. It’s often quite easy. You formed a simple chunk, for example, when you grasped the idea of continental drift. But since this is a book about how to learn math and science in general rather than geology in particular, we’re going to take as our initial, illustrative chunk the ability to understand and work a certain type of math or science problem.
When you are learning new math and science material, you are almost always given sample problems with worked-out solutions. This is because, when you are first trying to understand how to work a problem, you have a heavy cognitive load—so it helps to start out with a fully worked-through example. It’s like using a GPS unit when you are driving on unfamiliar roads in the middle of the night. Most of the details in the worked-out solution are right there, and your task is simply to figure out why the steps are taken the way they are. That can help you see the key features and underlying principles of a problem.
Some instructors do not like to give students extra worked-out problems or old tests, as they think it makes matters too easy. But there is bountiful evidence that having these kinds of resources available helps students learn much more deeply.5 The one concern about using worked-out examples to form chunks is that it can be all too easy to focus too much on why an individual step works and not on the connection between steps—that is, on why this particular step is the next thing you should do. So keep in mind that I’m not talking about a cookie-cutter “just do as you’re told” mindless approach when following a worked-out solution. It’s more like using a guide to help you when traveling to a new place. Pay attention to what’s going on around you when you’re with the guide, and soon you’ll find yourself able to get there on your own. You will even begin to figure out new ways of getting there that the guide didn’t show you.
When you first look at a brand-new concept in science or math, it sometimes doesn’t make much sense, as shown by the puzzle pieces above on the left. Just memorizing a fact (center) without understanding or context doesn’t help you understand what’s really going on, or how the concept fits together with the other concepts you are learning—notice there are no interlocking puzzle edges on the piece to help you fit into other pieces. Chunking (right) is the mental leap that helps you unite bits of information together through meaning. The new logical whole makes the chunk easier to remember, and also makes it easier to fit the chunk into the larger picture of what you are learning.
1. The first step in chunking, then, is to simply focus your attention on the information you want to chunk.6 If you have the television going in the background, or you’re looking up every few minutes to check or answer your phone or computer messages, it means that you’re going to have difficulty making a chunk, because your brain is not really focusing on the chunking. When you first begin to learn something, you are making new neural patterns and connecting them with preexisting patterns that are spread through many areas of the brain.7 Your octopus tentacles can’t make connections very well if some of them are off on other thoughts.
2. The second step in chunking is to understand the basic idea you are trying to chunk, whether it is understanding a concept such as continental drift, the idea that force is proportional to mass, the economic principle of supply and demand, or a particular type of math problem. Although this step of basic understanding—synthesizing the gist of what’s important—was difficult for Solomon Shereshevsky, most students figure out these main ideas naturally. Or at least, they can grasp those ideas if they allow the focused and diffuse modes of thinking to take turns in helping them figure out what’s going on.
Understanding is like a superglue that helps hold the underlying memory traces together. It creates broad, encompassing traces that link to many memory traces.8 Can you create a chunk if you don’t understand? Yes, but it’s a useless chunk that won’t fit in with other material you are learning.
That said, it’s important to realize that just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later. Do not confuse the “aha!” of a breakthrough in understanding with solid expertise! (That’s part of why you can grasp an idea when a teacher presents it in class, but if you don’t review it fairly soon after you’ve first learned it, it can seem incomprehensible when it comes time to prepare for a test.) Closing the book and testing yourself on how to solve the problems will also speed up your learning at this stage.
3. The third step to chunking is gaining context so you see not just how, but also when to use this chunk. Context means going beyond the initial problem and seeing more broadly, repeating and practicing with both related and unrelated problems so you see not only when to use the chunk, but when not to use it. This helps you see how your newly formed chunk fits into the bigger picture. In other words, you may have a tool in your strategy or problem-solving toolbox, but if you don’t know when to use that tool, it’s not going to do you a lot of good. Ultimately, practice helps you broaden the networks of neurons connected to your chunk, ensuring that it is not only firm, but also accessible from many different paths.
There are chunks related to both concepts and procedures that reinforce one another. Solving a lot of math problems provides an opportunity to learn why the procedure works the way it does or why it works at all. Understanding the underlying concept makes it easier to detect errors when you make them. (Trust me, you will make errors, and that’s a good thing.) It also makes it much easier to apply your knowledge to novel problems, a phenomenon called transfer. We’ll talk more about transfer later.
As you can see from the following “top-down, bottom-up” illustration, learning takes place in two ways. There is a bottom-up chunking process where practice and repetition can help you both build and strengthen each chunk, so you can easily gain access to it when needed. And there is a top-down “big picture” process that allows you to see where what you are learning fits in.9 Both processes are vital in gaining mastery over the material. Context is where bottom-up and top-down learning meet. To clarify here—chunking may involve your learning how to use a certain problem-solving technique. Context means learning when to use that technique instead of some other technique.
Those are the essential steps to making a chunk and fitting that chunk into a greater conceptual overview of what you are learning.
But there’s more.
Both top-down, big-picture learning, and bottom-up chunking are important in becoming an expert in math and science.
NOW I LAY ME DOWN TO SLEEP
“I tell my students that internalizing the accounting fundamentals is like internalizing how to type on a keyboard. In fact, as I write this myself, I’m not thinking of the act of typing, but of formulating my thoughts—the typing comes naturally. My mantra at the end of each class is to tell students to look at the Debit and Credit Rules as well as the Accounting Equation just before they tuck themselves in at night. Let those be the last things they repeat to themselves before falling asleep. Well, except meditation or prayers, of course!”
—Debra Gassner Dragone, Accounting Instructor, University of Delaware
Skimming through
a chapter or listening to a very well-organized lecture can allow you to gain a sense of the big picture. This can help you know where to put the chunks you are constructing. Learn the major concepts or points first—these are often the key parts of a good instructor or book chapter’s outline, flow charts, tables, or concept maps. Once you have this done, fill in the details. Even if a few of the puzzle pieces are missing at the end of your studies, you can still see the big picture.
Illusions of Competence and the Importance of Recall
Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.10 Psychologist Jeffrey Karpicke and his colleagues have shown that many students experience illusions of competence when they are studying. Most students, Karpicke found, “repeatedly read their notes or textbook (despite the limited benefits of this strategy), but relatively few engage in self-testing or retrieval practice while studying.”11 When you have the book (or Google!) open right in front of you, it provides the illusion that the material is also in your brain. But it’s not. Because it can be easier to look at the book instead of recalling, students persist in their illusion—studying in a far less productive way.
This, indeed, is why just wanting to learn the material, and spending a lot of time with it, doesn’t guarantee you’ll actually learn it. As Alan Baddeley, a renowned psychologist and expert on memory, notes: “Intention to learn is helpful only if it leads to the use of good learning strategies.”12
You may be surprised to learn that highlighting and underlining must be done carefully—otherwise they can be not only ineffective but also misleading. It’s as if the motion of your hand can fool you into thinking you’ve placed the concept in your brain. When marking up the text, train yourself to look for main ideas before making any marks, and keep your text markings to a minimum—one sentence or less per paragraph.13 Words or notes in a margin that synthesize key concepts are a good idea.14
Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective. The only time rereading text seems to be effective is if you let time pass between rereadings so that it becomes more of an exercise in spaced repetition.15
Along these same lines, always work through homework problems in math and science on your own. Some textbooks include solutions at the back of the book, but you should look at these only to check your answer. This will help ensure that the material is more deeply rooted in your mind and make it much more accessible when you really need it. This is why instructors place so much emphasis on showing your work and giving your reasoning on tests and homework problems. Doing so forces you to think your way through a problem and provides a self-test of your understanding. This additional information about your thinking also gives graders a better opportunity to provide useful feedback.
You don’t want to wait too long for the recall practice, so that you have to start the reinforcement of the concept from scratch every time. Try to touch again on something you’re learning within a day, especially if it’s new and rather challenging. This is why many professors recommend that, if at all possible, you rewrite your notes during the evening after a lecture. This helps to solidify newly forming chunks and reveals the holes in your understanding that professors just love to target on tests. Knowing where the holes are, of course, is the first step toward getting them filled in.
Once you’ve got something down, you can expand the time between “upkeep” repetitions to weeks or months—and eventually it can become close to permanent. (Returning to Russia on a visit, for example, I found myself annoyed by an unscrupulous taxi driver. To my amazement, words I hadn’t thought or used for twenty-five years popped from my mouth—I hadn’t even been consciously aware I knew those words!)
MAKE YOUR KNOWLEDGE SECOND NATURE
“Getting a concept in class versus being able to apply it to a genuine physical problem is the difference between a simple student and a full-blown scientist or engineer. The only way I know of to make that jump is to work with the concept until it becomes second nature, so you can begin to use it like a tool.”
—Thomas Day, Professor of Audio Engineering, McNally Smith College of Music
Later, we’ll discuss useful apps and programs that can help with learning. But for now, it’s worth knowing that well-designed electronic flash card systems, such as Anki, have built into them the appropriate spaced repetition time to optimize the rate of learning new material.
One way to think about this type of learning and recall is shown in the following working-memory illustration. As we mentioned earlier, there are four or so spots in working memory.
When you are first chunking a concept, its pre-chunked parts take up all your working memory, as shown on the left. As you begin to chunk the concept, you will feel it connecting more easily and smoothly in your mind, as shown in the center. Once the concept is chunked, as shown at the right, it takes up only one slot in working memory. It simultaneously becomes one smooth strand that is easy to follow and use to make new connections. The rest of your working memory is left clear. That dangling strand of chunked material has, in some sense, increased the amount of information available to your working memory, as if the slot in working memory is a hyperlink that has been connected to a big webpage.16
When you are first learning how to solve a problem, your entire working memory is involved in the process, as shown by the mad tangle of connections between the four slots of working memory on the left. But once you become smoothly familiar with the concept or method you are learning and have it encapsulated as a single chunk, it’s like having one smooth ribbon of thought, as shown on the right. The chunking, which enlists long-term memory, frees the rest of your working memory to process other information. Whenever you want, you can slip that ribbon (chunk) from long-term memory into your working memory and follow along the strand, smoothly making new connections.
Now you understand why it is key that you are the one doing the problem solving, not whoever wrote the solution manual. If you work a problem by just looking at the solution, and then tell yourself, “Oh yeah, I see why they did that,” then the solution is not really yours—you’ve done almost nothing to knit the concepts into your underlying neurocircuitry. Merely glancing at the solution to a problem and thinking you truly know it yourself is one of the most common illusions of competence in learning.
NOW YOU TRY!
Understanding Illusions of Competence
Anagrams are rearrangements of letters so that one word or phrase can spell something different. Let’s say you have the phrase “Me, radium ace.” Can you rearrange it to spell the last name of a honorific famous physicist?17 It may take you a bit of thought to do it. But if you saw the solution here on the page, your subsequent “aha!” feeling would make you think that your anagram-solving skills are better than they actually are.
Similarly, students often erroneously believe that they are learning by simply rereading material that is on the page in front of them. They have an illusion of competence because the solution is already there.18
Pick a mathematical or scientific concept from your notes or from a page in the book. Read it over, then look away and see what you can recall—working toward understanding what you are recalling at the same time. Then glance back, reread the concept, and try it again.
At the end of this exercise, you will probably be surprised to see how much this simple recall exercise helped improve your understanding of the concept.
You must have information persisting in your memory if you are to master the material well enough to do well on tests and think creatively with it.19 The ability to combine chunks in novel ways underlies much of historical innovation. Steven Johnson, in his brilliant book Where Good Ideas Come From, describes the “slow hunch”—the gentle, years-long simmering of focused and diffuse processes that has resulted in
creative breakthroughs ranging from Darwin’s evolutionary theory to the creation of the World Wide Web.20 Key to the slow hunch is simply having mental access to aspects of an idea. That way, some aspects can tentatively and randomly combine with others until eventually, beautiful novelty can emerge.21 Bill Gates and other industry leaders, Johnson notes, set aside extended, weeklong reading periods so that they can hold many and varied ideas in mind during one time. This fosters their own innovative thinking by allowing fresh-in-mind, not-yet-forgotten ideas to network among themselves. (An important side note here is that a key difference between creative scientists and technically competent but nonimaginative ones is their breadth of interest.22)
The bigger your chunked mental library, the more easily you will be able to solve problems. Also, as you gain more experience in chunking, you will see that the chunks you are able to create are bigger—the ribbons are longer.
You may think there are so many problems and concepts just in a single chapter of the science or math subject you are studying that there’s no way to do them all! This is where the Law of Serendipity comes to play: Lady Luck favors the one who tries.23
Just focus on whatever section you are studying. You’ll find that once you put the first problem or concept in your library, whatever it is, then the second concept will go in a bit more easily. And the third more easily still. Not that all of this is a snap, but it does get easier.
If you have a library of concepts and solutions internalized as chunked patterns, you can more easily skip to the right solution to a problem by listening to the whispers from your diffuse mode. Your diffuse mode can also help you connect two or more chunks together in new ways to solve unusual problems.
A Mind For Numbers Page 6
