A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)
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I didn’t realize it then, but my self-portrait as being technically, scientifically, and mathematically incapable was shaping my life.
Patrick
A self-fullfilling prophecy.
Patrick liked this
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But I began to catch on. Part of my original problem, I found, was that I had been putting my effort forth in the wrong way—like trying to lift a piece of lumber when you’re standing on it.
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This former Queen of the Confused in math went on to earn a bachelor’s degree in electrical engineering and then a master’s in electrical and computer engineering. Finally, I earned a doctorate in systems engineering, with a broad background that included thermodynamics, electromagnetics, acoustics, and physical chemistry.
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Basically, we just need to master the lingo and culture.
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These techniques are especially valuable for helping you learn more deeply and effectively in limited time frames.
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For now, keep in mind that when you procrastinate, you are leaving yourself only enough time to do superficial focused-mode learning. You are also increasing your stress level because you know you have to complete what feels like an unpleasant task.
Patrick
Procrastination is fatal.
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1. How would you recognize when you are in the diffuse mode? How does it feel to be in the diffuse mode?
Patrick
Diffuse mode seems to happen when I'm thinking about other things RELATED TO what it is that I'm reading. My mind is making the connections to information already known. Wait.....no....that seems to be FOCUSED-MODE. I think DIFFUSED MODE happens when I'm making notes on a particular passage that I've read and seems interesting enough to jot down. It feels playful, not as serious, feels like I'm getting a better grasp on the information.
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2. When you are consciously thinking of a problem, which mode is active and which is blocked? What can you do to escape this blocking?
Patrick
Focused mode is definitely the one that's active. Diffused mode is blocked. I know the Promodoro method was mentioned in this chapter as a way of escaping this blockage but I'm not a believer in it. Maybe the latter chapters will shed some much needed light on this method.
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3. Recall an episode where you experienced the Einstellung effect. How were you able to change your thinking to get past the preconceived, but erroneous, notion?
Patrick
One episode that occured quite frequently in youth (and less so but still consistently nowadays) was the notion that rote memory is the most effective way to memorize something. Until I read "Moonwalking With Einstein" and realized that the mind memorizes pictures better than words, I would always hammer something until I memorized it. Now, if it's something I really want to approach and get a grasp of, I'll make picture references.
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4. Explain how the focused and diffuse modes might be equated to an adjustable beam on a flashlight. When can you see farther? When can you see more broadly, but less far?
Patrick
FOCUSED AND DIFFUSED. The focused mode of thought is like a sharp beam of light that can see far and can see bright. Whatever and wherever it shines it's light, it puts a bright spot on it. DIFFUSED MODE is like a dim, wide light that reveals many-a-thing but from close up.
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5. Why is procrastination sometimes a special challenge for those who are studying math and science?
Patrick
Procrastination doesn't leave enough time to use BOTH MODES of thinking. Actually, procrastination doesn't leave enough time for anything.
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I don’t think you can tutor someone on how to think—it’s kind of a personal journey.
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Especially in the early stages of college, avoid the temptation to keep up with your peers.
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According to recent research, blinking is a vital activity that provides another means of reevaluating a situation. Closing our eyes seems to provide a micropause that momentarily deactivates our attention and allows us, for the briefest of moments, to refresh and renew our consciousness and perspective.
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Working in the focused mode is like providing the bricks, while working in the diffuse mode is like gradually joining the bricks together with mortar.
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People with strong self-control can have the most difficulty in getting themselves to turn off their focused mode so that the diffuse mode can begin its work.
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You can think of working memory as the mental equivalent of a juggler. The four items only stay in the air—or in working memory—because you keep adding a little energy.
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1. Name some activities you would find helpful for switching from focused to diffuse mode.
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3. Why is it important to use self-control to make yourself stop doing something? Can you think of times outside studying and academics when this skill might also be important?
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4. When you are learning new concepts, you want to review the material within a day so that the initial changes you made in your brain don’t fade away. But your mind often becomes preoccupied with other matters—it’s easy to let several days or more pass before you get around to looking at the material. What kind of action plan could you develop to ensure that you review important new material in a timely fashion?
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Focused attention, in other words, can often help solve problems, but it can also create problems by blocking our ability to see new solutions.
Patrick
This is where diffuse mode comes in handy.
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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.
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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.
Patrick
Focus on the signal, not the noise.
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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.
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just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later.
Patrick
Understanding how a problem is solved only helps with that particular problem. You must get the general concept in order to truly "chunk".
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3. The third step to chunking is gaining context so you see not just how, but also when to use this chunk.
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Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.10
Patrick
Try to recall anything 3 days after you've read it.
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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
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Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective.
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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.
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(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)
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There are two ways to solve problems—first, through sequential, step-by-step reasoning, and second, through more holistic intuition.
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Most difficult problems are solved through intuition, because they make a leap away from what you are familiar with.24
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bottom-up chunking and top-down big-picture
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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,
Patrick
Up until this point in the chapter, I've concluded that both methods (diffused and focused) have their place.
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But if you try to build connections between chunks before the basic chunks are embedded in the brain, it doesn’t work as well.
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The fact is, when learning any new skill or discipline, you need plenty of varied practice with different contexts.
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Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.
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Continuing the study or practice after it is well understood is called overlearning.
Patrick
A waste of time obviously. This form of repetition actually weakens the learning process. It only helps once ALL the basics are known, then you can begin to hammer down.
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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.
Patrick
Flashcards is one solution, another is to do random problems or open a book to a random page, then recall.
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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.
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1. How is a chunk related to a memory trace?
Patrick
They both involve recalling.
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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.
Patrick
The Spanish alphabet. It's quite easy to understand.
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3. What is the difference between top-down and bottom-up approaches to learning? Is one approach preferable to the other?
Patrick
Top down big picture involves getting the general idea. Bottom up chunking involves solidifying concepts. Both are essential.
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4. Is understanding enough to create a chunk? Explain why or why not.
Patrick
No. Understanding means to only know how a problem was solved; it will not create a true "chunking".
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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?
Patrick
I have a lot of Illusions of Competence. The only way to get over these is to grasp different concepts, try them out in different settings, and chunk/recall/repeat 🔂.
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How is it possible to take something so bad for you and stay alive—and even look healthy?
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We procrastinate about things that make us feel uncomfortable.4
Patrick
The anticipation is what kills us most. The thoughts beforehand, the dread; they are will-killers. Willpower is in short supply, it burns bright and fast. I belive anticipation burns up willpower.
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Procrastination is a single, monumentally important “keystone” bad habit.7
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Researchers have found that procrastination can even become a source of pride as well as an excuse for doing poorly.
Patrick
Ex. Cramming just before test time and saying, "Yeah, I aced that cram session".
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