فصل 11

دوره: ذهنی برای اعداد / درس 12

فصل 11

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more memory tips

Create a Lively Visual Metaphor or Analogy

One of the best things you can do to not only remember but understand concepts in math and science is to create a metaphor or analogy for it—often, the more visual, the better.1 A metaphor is just a way of realizing that one thing is somehow similar to another. Simple ideas like one geography teacher’s description of Syria as shaped like a bowl of cereal and Jordan as a Nike Air Jordan sneaker can stick with a student for decades.

If you’re trying to understand electrical current, it can help to visualize it as water. Similarly, electrical voltage can “feel like” pressure. Voltage helps push the electrical current to where you want it to go, just like a mechanical pump uses physical pressure to push real water. As you climb to a more sophisticated understanding of electricity, or whatever topic you are concentrating on, you can revise your metaphors, or toss them away and create more meaningful ones.

If you are trying to understand the concept of limits in calculus, you might visualize a runner heading for the finish line. The closer the runner gets, the slower he goes. It’s one of those slo-mo camera shots where the runner is never quite able to reach the ribbon, just as we might not quite be able to get to the actual limit. Incidentally, the little book Calculus Made Easy, by Silvanus Thompson, has helped generations of students master the subject. Sometimes textbooks can get so focused on all the details that you lose sight of the most important, big-picture concepts. Little books like Calculus Made Easy are good to dip into because they help us focus in a simple way on the most important issues.

It’s often helpful to pretend you are the concept you are trying to understand. Put yourself in an electron’s warm and fuzzy slippers as it burrows through a slab of copper, or sneak inside the x of an algebraic equation and feel what it’s like to poke your head out of the rabbit hole (just don’t let it get exploded with an inadvertent “divide by zero”).

MOONBEAMS AND SCHOOL DREAMS

“I always study before I go to bed. For some reason, I usually dream about the material I just studied. Most times these ‘school dreams’ are quite strange but helpful. For instance, when I was taking an operations research class, I would dream I was running back and forth between nodes, physically acting out the shortest path algorithm. People think I’m crazy, but I think it’s great; it means I don’t have to study as much as other people do. I guess these dreams involve my subconsciously making metaphors.” —Anthony Sciuto, senior, industrial and systems engineering

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.

Metaphors are never perfect. But then, all scientific models are just metaphors, which means they also break down at some point.2 But never mind that—metaphors (and models!) are vitally important in giving a physical understanding of the central idea behind the mathematical or scientific process or concept that you are trying to understand. Interestingly, metaphors and analogies are useful for getting people out of Einstellung—being blocked by thinking about a problem in the wrong way. For example, telling a simple story of soldiers attacking a fortress from many directions at once can open creative paths for students to intuit how many low-intensity rays can be effectively used to destroy a cancerous tumor.3 Metaphors also help glue an idea in your mind, because they make a connection to neural structures that are already there. It’s like being able to trace a pattern with tracing paper—metaphors at least help you get a sense of what’s going on. If there’s a time when you can’t think of a metaphor, just put a pen or pencil in your hand and a sheet of paper in front of you. Whether using words or pictures, you will often be amazed at what just noodling about for a minute or two will bring.

METAPHORS AND VISUALIZATION IN SCIENCE

Metaphors and visualization—being able to see something in your mind’s eye—have been uniquely powerful in helping the scientific and engineering world move forward.4 In the 1800s, for example, when chemists began to imagine and visualize the miniature world of molecules, dramatic progress began to be made. Here is a delightful illustration of monkeys in a benzene ring from an insider spoof of German academic chemical life, printed in 1886.5 Note the single bonds with the monkeys’ hands and the double bonds with their tiny tails.

Spaced Repetition to Help Lodge Ideas in Memory

Focusing your attention brings something into your temporary working memory. But for that “something” to move from working memory to long-term memory, two things should happen: the idea should be memorable (there’s a gigantic flying mule braying f = ma on my couch!), and it must be repeated. Otherwise, your natural metabolic processes, like tiny vampires, simply suck away faint, newly forming patterns of connections. This vampiric removal of faint patterns is actually a good thing. Much of what goes on around you is basically trivial—if you remembered it all, you’d end up like a hoarder, trapped in an immense collection of useless memories.

If you don’t make a point of repeating what you want to remember, your “metabolic vampires” can suck away the neural pattern related to that memory before it can strengthen and solidify.

Repetition is important; even when you make something memorable, repetition helps get that memorable item firmly lodged in long-term memory. But how many times should you repeat? How long should you wait between repetitions?6 And is there anything you can do to make the repetition process more effective?

Research has given us helpful insight. Let’s take a practical example. Say you want to remember information related to the concept of density—namely that it is symbolized by a funny-looking symbol, ρ, which is pronounced “row,” and that it is measured in standard units of “kilograms per cubic meter.” How can you conveniently and effectively cement this information into memory? (You know now that placing small chunks of information like this in your long-term memory helps gradually build your big-picture understanding of a subject.) You might take an index card and write “ρ” on one side and the remaining information on the other. Writing appears to help you to more deeply encode (that is, convert into neural memory structures) what you are trying to learn. While you are writing out “kilograms per cubic meter,” you might imagine a shadowy kilogram (just feel that mass!) lurking in an oversized piece of luggage that happens to be a meter on each side. The more you can turn what you are trying to remember into something memorable, the easier it will be to recall. You will want to say the word and its meaning aloud, to start setting auditory hooks to the material.

Next, just look at the side of the card with the “ρ” on it and see whether you can remember what’s on the other side of the card. If you can’t, flip it over and remind yourself of what you are supposed to know. If you can remember, put the card away.

Now do something else—perhaps prepare another card and test yourself on it. Once you have several cards together, try running through them all to see if you can remember them. (This helps you interleave your learning.) Don’t be surprised if you struggle a bit. Once you’ve given your cards a good try, put them away. Wait and take them out again before you go to sleep. Remember that sleep is when your mind repeats patterns and pieces together solutions.

Briefly repeat what you want to remember over several days; perhaps for a few minutes each morning or each evening, change the order of your cards sometimes. Gradually extend the times between repetitions as the material firms itself into your mind. By increasing your spacing as you become more certain of mastery, you will lock the material more firmly into place.7 (Great flash card systems like Anki have built in algorithms that repeat on a scale ranging from days to months.) Interestingly, one of the best ways to remember people’s names is to simply try to retrieve the people’s names from memory at increasing time intervals after first learning the name.8 Material that you do not review is more easily discounted or forgotten. Your metabolic vampires suck away the links to the memories. This is why it’s wise to be careful about what you decide to skip when reviewing for tests. Your memory for related but nonreviewed material can become impaired.9 SPACED REPETITION—USEFUL FOR BOTH STUDENTS AND PROFESSORS!

“I have been advising my students to do spaced repetition over days and weeks, not just in my analytic courses, but also in my History of Ancient Engineering course. When memorizing strange names and terms, it’s always best to practice over several days. In fact, that’s precisely what I do when I’m preparing for lectures—repeat the terms out loud over a period of several days, so they roll easily off my tongue when I say them in class.” —Fabian Hadipriono Tan, Professor of Civil Engineering, The Ohio State University

NOW YOU TRY!

Create a Metaphor to Help You Learn

Think of a concept you are learning now. Is there another process or idea in a completely different field that somehow seems similar to what you are studying? See if you can come up with a helpful metaphor. (Bonus points if there’s a touch of silliness!) Create Meaningful Groups

Another key to memorization is to create meaningful groups that simplify the material. Let’s say you wanted to remember four plants that help ward off vampires—garlic, rose, hawthorn, and mustard. The first letters abbreviate to GRHM, so all you need to do is remember the image of a GRAHAM cracker. (Retrieve your cracker from the kitchen table of your memory palace, dust off the vowels, and you’re good to go.) It’s much easier to remember numbers by associating them with memorable events. The year 1965 might be when one of your relatives was born, for example. Or you can associate numbers with a numerical system that you’re familiar with. For example, 11.0 seconds is a good running time for the 100-meter dash. Or 75 might be the number of knitting stitches cast onto a needle for the ski hats you like to make. Personally, I like to associate numbers with the feelings of when I was or will be at a given age. The number 18 is an easy one—that’s when I went out into the world. By age 104, I will be an old but happy great-granny!

Many disciplines use memorable sentences to help students memorize concepts; the first letter of each word in the sentence is also the first letter of each word in a list that needs to be memorized. Medicine, for example, is laden with memorable mnemonics, among the cleaner of which are “Some Lovers Try Positions that They Can’t Handle” (to memorize the names of the carpal bones of the hand) and “Old People from Texas Eat Spiders” (for the cranial bones).

Another example is for the increases-by-ten structure of the decimal system: King Henry died while drinking chocolate milk. This translates to kilo—1,000; hecto—100; deca—10; “while” represents 1; deci—0.1; centi—0.01; milli—0.001.

Time after time, these kinds of memory tricks prove helpful. If you’re memorizing something commonly used, see whether someone’s come up with a particularly memorable memory trick by searching it out online. Otherwise, try coming up with your own.

BEWARE OF MISTAKING A MEMORY TRICK FOR ACTUAL KNOWLEDGE

“In chemistry we have the phrase skit ti vicer man feconi kuzin, which has the cadence of a rap song. It represents the first row of the transition metals on the periodic table (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn). Then, the rest of the transition metals can be placed on a blank periodic table by other memory tricks. For example, students remember to place Ag (silver) and Au (gold) in the same vertical group as Cu (copper) since copper, silver, and gold are all used to make coins.

“Unfortunately, some students come away thinking that’s the reason these metals are in the same vertical column—because they are used to make coins. The real reason has to do with similarities in chemical properties and valences.

“This is an example of how students sometimes mistake a memory trick for actual knowledge. Always be wary of confusing what is truly going on with the metaphor you are using to help your memory.” —William Pietro, Professor of Chemistry, York University, Toronto, Ontario

Create Stories

Notice that the groups mentioned previously often create meaning through story, even though the story might be short. Poor King Henry shouldn’t have drunk that chocolate milk! Storytelling in general has long been a vitally important way of understanding and retaining information. Professor Vera Pavri, a historian of science and technology at York University, tells her students not to think of lectures as lectures but as stories where there is a plot, characters, and overall purpose to the discussion. The best lectures in math and science are often framed like thrillers, opening with an intriguing problem that you just have to figure out. If your instructor or book doesn’t present the material with a question that leaves you wanting to find the answer, see if you can find that question yourself—then set about answering it.10 And don’t forget the value of story as you create memory tricks.

WRITE ON!

“The number one thing I stress when students come to see me is that there is a direct connection between your hand and your brain, and the act of rewriting and organizing your notes is essential to breaking large amounts of information down into smaller digestible chunks. I have many students who prefer to type their notes in a Word document or on slides, and when these students are struggling, the first thing I recommend is to quit typing and start writing. In every case, they perform better on the next section of material.” —Jason Dechant, Ph.D., Course Director, Health Promotion and Development, School of Nursing, University of Pittsburgh Muscle Memory

We’ve already mentioned that writing out a card by hand appears to help cement ideas in the mind. Although there is little research in this area,11 many educators have observed that there seems to be a muscle memory related to writing by hand. For example, when you first stare at an equation, it can appear utterly meaningless. But if you thoughtfully write the equation out several times on a sheet of paper, you will be startled by how the equation will begin to take life and meaning in your mind. In a related vein, some learners find that reading problems or formulas aloud helps them understand better. Just be wary of exercises like writing an equation out a hundred times by hand. The first few times may give you value, but after a while, it simply becomes a rote exercise—the time could be better spent elsewhere.

TALK TO YOURSELF

“I often tell my students to talk to themselves instead of just highlighting and rereading. They look at me quizzically, like I am absolutely insane (which could be true). But I have had many students come back to me later and say that it really works and that this is now one of their study tools.” —Dina Miyoshi, Assistant Professor of Psychology, San Diego Mesa College

Real Muscle Memory

If you really want to boost your memory as well as your general ability to learn, it seems one of the best ways to do it is to exercise. Several recent experiments in both animals and humans have found that regular exercise can make a substantive improvement in your memory and learning abilities. Exercise, it seems, helps create new neurons in areas that relate to memory. It also creates new signaling pathways.12 It seems that different types of exercise—running or walking, for example, versus strength training—may have subtly different molecular effects. But both aerobic and resistance exercise exert similarly powerful results on learning and memory.

Memory Tricks Help You Become an Expert More Quickly

Here’s the bottom line. By using mental pictures instead of words to remember things, you can leap more easily into expert status. In other words, learning to process ideas visually in math and science is a powerful way to become a master of the material.13 And using other memory tricks can greatly enhance your ability to learn and retain the material.

Purists might sniff that using oddball memorization gimmicks isn’t really learning. But research has shown that students who use these types of tricks outperform those who don’t.14 In addition, imaging research on how people become experts shows that such memory tools speed up the acquisition of both chunks and big-picture templates, helping transform novices to semiexperts much more quickly—even in a matter of weeks.15 Memory tricks allow people to expand their working memory with easy access to long term memory.

What’s more, the memorization process itself becomes an exercise in creativity. The more you memorize using these innovative techniques, the more creative you become. This is because you are building wild, unexpected possibilities for future connections early on, even as you are first internalizing the ideas. The more you practice using this type of “memory muscle,” the more easily you will be able to remember. Where at first it may take fifteen minutes to build an evocative image for an equation and embed it in, say, the kitchen sink of your memory palace, it can later take only minutes or seconds to perform a similar task.

You will also realize that as you begin to internalize key aspects of the material, taking a little time to commit the most important points to memory, you come to understand it much more deeply. The formulas will mean far more to you than they would if you simply looked them up in a book. And you’ll be able to sling those formulas around more proficiently on tests and in real-world applications.

One study of how actors memorize their scripts showed that they avoid verbatim memorization. Instead, they depend on an understanding of the characters’ needs and motivations in order to remember their lines.16 Similarly, the most important part of your memorization practices is to understand what the formulas and solution steps really mean. Understanding also helps a lot with the memorization process.

You may object and say that you’re not creative—that an equation or theory could hardly have its own grandiose motivations or persnickety emotional needs to help you understand and remember it. But remember that inner two-year-old. Your childlike creativity is still there—you just need to reach out to it.

MEMORY TRICKS WORK

“On top of working toward my engineering degree, I am in the process of getting my paramedic license (only two months left!) and have to memorize a large selection of drugs and dosages for both adult and pediatric patients. At first, this seemed overwhelming, especially since there will be lives at stake. But I quickly found little tricks that made learning easy. Take, for example, the drug furosemide, also called Lasix, which draws fluid out of the body. The dose I needed to remember was 40 milligrams. This to me was a godsend, as the numbers 4–0 appeared to me in the word (4–0 semide = furosemide). It is things like this that truly can cement ideas and knowledge in our heads. I now don’t ever have to even think twice about it. Truly remarkable.” —William Koehler, sophomore, mechanical engineering

NOW YOU TRY!

Songs to Help You Learn

Make up a song to remember an identity, integral, or scientific formula that you need for class. Having some of these important concepts memorized, through whatever trick you need, will make more complicated problems easier and faster to solve.

SUMMING IT UP

Metaphors can help you learn difficult ideas more quickly.

Repetition is critical in allowing you to firm up what you want to remember before the ideas fade away.

Meaningful groups and abbreviations can allow you to simplify and chunk what you are trying to learn so you can store it more easily in memory.

Stories—even if they are just used as silly memory tricks—can allow you to more easily retain what you are trying to learn.

Writing and saying what you are trying to learn seems to enhance retention.

Exercise is powerfully important in helping your neurons to grow and make new connections.

PAUSE AND RECALL

Remember how important it can be to sometimes think of what you are learning in a place different from where you learned it. Try that technique again as you recall the key ideas of this chapter. People sometimes recollect the feel of the place where they were studying—even the cushiony feel of the armchair, or the particular music or picture on the wall in the coffee shop where they were sitting—to help cue a memory.

ENHANCE YOUR LEARNING

  1. Take a piece of paper and doodle to create a visual or verbal metaphor for a concept you are trying to understand now in math or science.

  2. Look at a chapter in a book you are reading in math or science. Create a question about that material that would make you want to learn more about it.

  3. Just before you to go sleep, review something mentally that you are trying to learn. To boost this process, review it yet again when you first wake up.

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