—Kathleen Nolta, Ph.D., Senior Lecturer in Chemistry and recipient of the Golden Apple Award, recognizing excellence in teaching at the University of Michigan
NOW YOU TRY!
Stage a Mental Play
Imagine yourself within the realm of something you are studying—looking at the world from the perspective of the cell, or the electron, or even a mathematical concept. Try staging a mental play with your new friends, imagining how they feel and react.
Transfer—Applying What You’ve Learned in New Contexts
Transfer is the ability to take what you learn in one context and apply it to something else. For example, you may learn one foreign language and then find that you can pick up a second foreign language more easily than the first. That’s because when you learned the first foreign language, you also acquired general language-learning skills, and potentially similar new words and grammatical structures, that transferred to your learning the second foreign language.13
Learning math by applying it only to problems within a specific discipline, such as accounting, engineering, or economics, can be a little like deciding that you are not really going to learn a foreign language after all—you’re just going to stick to one language and just learn a few extra English vocabulary words. Many mathematicians feel that learning math through entirely discipline-specific approaches makes it more difficult for you to use mathematics in a flexible and creative way.
Mathematicians feel that if you learn math the way they teach it, which centers on the abstract, chunked essence without a specific application in mind, you’ve captured skills that are easy for you to transfer to a variety of applications. In other words, you’ll have acquired the equivalent of general language-learning skills. You may be a physics student, for example, but you could use your knowledge of abstract math to quickly grasp how some of that math could apply to very different biological, financial, or even psychological processes.
This is part of why mathematicians like to teach math in an abstract way, without necessarily zooming in on applications. They want you to see the essence of the ideas, which they feel makes it easier to transfer the ideas to a variety of topics.14 It’s as if they don’t want you to learn how to say a specific Albanian or Lithuanian or Icelandic phrase meaning I run but rather to understand the more general idea that there is a category of words called verbs, which you conjugate.
The challenge is that it’s often easier to pick up on a mathematical idea if it is applied directly to a concrete problem—even though that can make it more difficult to transfer the mathematical idea to new areas later. Unsurprisingly, there ends up being a constant tussle between concrete and abstract approaches to learning mathematics. Mathematicians try to hold the high ground by stepping back to make sure that abstract approaches are central to the learning process. In contrast, engineering, business, and many other professions all naturally gravitate toward math that focuses on their specific areas to help build student engagement and avoid the complaint of “When am I ever going to use this?” Concretely applied math also gets around the issue that many “real-world” word problems in mathematics textbooks are simply thinly disguised exercises. In the end, both concrete and abstract approaches have their advantages and disadvantages.
Transfer is beneficial in that it often makes learning easier for students as they advance in their studies of a discipline. As Professor Jason Dechant of the University of Pittsburgh says, “I always tell my students that they will study less as they progress through their nursing programs, and they don’t believe me. They’re actually doing more and more each semester; they just get better at bringing it all together.”
One of the most problematic aspects of procrastination—constantly interrupting your focus to check your phone messages, e-mails, or other updates—is that it interferes with transfer. Students who interrupt their work constantly not only don’t learn as deeply, but also aren’t able to transfer what little they do learn as easily to other topics.15 You may think you’re learning in between checking your phone messages, but in reality, your brain is not focusing long enough to form the solid neural chunks that are central to transferring ideas from one area to another.
TRANSFERRING IDEAS WORKS!
“I took fishing techniques from the Great Lakes and tried using them down in the Florida Keys this past year. Completely different fish, different bait, and a technique that had never been used but it worked great. People thought I was crazy and it was funny to show them that it actually caught fish.”
—Patrick Scoggin, senior, history
SUMMING IT UP
Equations are just ways of abstracting and simplifying concepts. This means that equations contain deeper meaning, similar to the depth of meaning found in poetry.
Your “mind’s eye” is important because it can help you stage plays and personalize what you are learning about.
Transfer is the ability to take what you learn in one context and apply it to something else.
It’s important to grasp the chunked essence of a mathematical concept, because then it’s easier to transfer and apply that idea in new and different ways.
Multitasking during the learning process means you don’t learn as deeply—this can inhibit your ability to transfer what you are learning.
PAUSE AND RECALL
Close the book and look away. What were the main ideas of this chapter? Can you picture some of these ideas with symbols in your mind’s eye?
ENHANCE YOUR LEARNING
1. Write an equation poem—several unfolding lines that provide a sense of what lies beneath a standard equation.
2. Write a paragraph that describes how some concepts you are studying could be visualized in a play. How do you think the actors in your play might realistically feel and react to one another?
3. Take a mathematical concept you have learned and look at a concrete example of how that concept is applied. Then step back and see if you can sense the abstract chunk of an idea underlying the application. Can you think of a completely different way that concept might be used?
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renaissance learning
The Value of Learning on Your Own
People like Charles Darwin, whose theory of evolution has made him one of the most influential figures in human history, are often thought of as natural geniuses. You may be surprised to learn that much like Cajal, Darwin was a poor student. He washed out of medical school and ended up, to his father’s horror, heading out on a round-the-world voyage as the ship’s naturalist. Out on his own, Darwin was able to look with fresh eyes at the data he was collecting.
Persistence is often more important than intelligence.1 Approaching material with a goal of learning it on your own gives you a unique path to mastery. Often, no matter how good your teacher and textbook are, it’s only when you sneak off and look at other books or videos that you begin to see that what you learn through a single teacher or book is a partial version of the full, three-dimensional reality of the subject, which has links to still other fascinating topics that are of your choosing.
Neurosurgeon Ben Carson, winner of the Presidential Medal of Freedom for his pioneering surgical innovations, was initially flunking and gently urged to leave medical school. Carson knew he learned best through books, not in-class lectures. He took a counterintuitive step and stopped attending lectures to give himself time to focus on learning through books. His grades soared and the rest is history. (Note that this technique would not work for everyone—and if you use this story as an excuse to simply stop attending classes, you are courting disaster!)
In the fields of science, math, and technology, many individuals had to carve their own path in learning, either because they had no other way, or because for whatever reason, they’d thrown away previous learning opportunities. Research has shown that students learn best when they themselves are actively engaged
in the subject instead of simply listening to someone else speak.2 A student’s ability to grapple personally with the material, sometimes bouncing it off fellow learners, is key.
Santiago Ramón y Cajal was horrified when he had to learn college calculus as an adult, after he had become serious about becoming a doctor. He’d never paid attention to math in his youth and lacked even a rudimentary understanding of the material. He had to go rummaging back through old books, scratching his head to figure out the basics. Cajal learned all the more deeply, however, because he was driven by his personal goals.
“What a wonderful stimulant it would be for the beginner if his instructor, instead of amazing and dismaying him with the sublimity of great past achievements, would reveal instead the origin of each scientific discovery, the series of errors and missteps that preceded it—information that, from a human perspective, is essential to an accurate explanation of the discovery.”3
—Santiago Ramón y Cajal
Inventor and author William Kamkwamba, born in 1987 in Africa, could not afford to attend school. So he began teaching himself by going to his village’s library, where he stumbled across a book titled Using Energy. But Kamkwamba didn’t just read the book. When he was only fifteen years old, he used the book to guide him in active learning: He built his own windmill. His neighbors called him misala—crazy—but his creation helped begin generating electricity and running water for his village and sparked the growth of grassroots technological innovation in Africa.4
American neuroscientist and pharmacologist Candace Pert had an excellent education, earning a doctorate in pharmacology from Johns Hopkins University. But part of her inspiration and subsequent success arose from an unusual source. Just before entering medical graduate school, she hurt her back in a horseback-riding accident and spent a summer under the influence of deep pain medication.5 Her personal experiences with pain and pain medication drove her scientific research. Ignoring her advisor’s attempts to stop her, she made some of the first key discoveries involving opiate receptors—a major step forward in understanding addiction.
College isn’t the only way to learn. Some of the most powerful and renowned people of our time, including powerhouses Bill Gates, Larry Ellison, Michael Dell, Mark Zuckerberg, James Cameron, Steve Jobs, and Steve Wozniak, dropped out of college. We will continue to see fascinating innovations from people who are able to combine the best aspects of traditional and nontraditional learning with their own self-taught approaches.
Taking responsibility for your own learning is one of the most important things you can do. Teacher-centered approaches, where the teacher is considered to be the one with the answers, may sometimes inadvertently foster a sense of helplessness about learning among students.6 Surprisingly, teacher evaluation systems may foster the same helplessness—these systems allow you to place the blame for failure on your teacher’s inability to motivate or instruct.7 Student-centered learning, where students are challenged to learn from one another and are expected to be their own drivers toward mastery of the material, is extraordinarily powerful.
The Value of Great Teachers
You will also sometimes have a chance to interact with truly special mentors or teachers. When this lucky opportunity arises, seize it. Train yourself to get past the gulp stage and force yourself to reach out and ask questions—real and to-the-point questions, not questions meant to show off what you know. The more you do this, the easier it will become, and the more helpful it will be in ways you never anticipated—a simple sentence growing from their vast experience can change the course of your future. And also be sure to show appreciation for the people guiding you—it is essential to let them know that the help is meaningful.
Be wary, however, of falling into “sticky student” syndrome. Kind teachers, in particular, can become magnets for students whose true needs involve desire for the ego-boosting attention of the instructor far more than answers to the actual questions being posed. Well-meaning teachers can burn out trying to satisfy never-to-be- satisfied desires.
Also avoid the trap of feeling certain your answer is correct, and attempting to force your teacher to follow the tortured steps of your logic when your answer is obviously wrong. Every once in a while, you might ultimately be proven correct, but for many teachers, particularly at more advanced levels in math and science, trying to follow twisted, erroneous thinking is like listening to out-of-tune music—a thankless, painful exercise. It’s generally best to start your thinking afresh and listen to your teacher’s suggestions. When you finally understand the answer, you can go back if you want to debug your previous error. (Often you’ll realize, in one fell swoop, that it’s difficult to even put into words how wrong your previous approach was.) Good teachers and mentors are often very busy people, and you need to use their time wisely.
Truly great teachers make the material seem both simple and profound, set up mechanisms for students to learn from each other, and inspire students to learn on their own. Celso Batalha, for example, a renowned professor of physics at Evergreen Valley College, has set up a popular reading group for his students about learning how to learn. And many professors use “active” and “collaborative teaching” techniques in the classroom that give students a chance to actively engage with the material and with each other.8
One thing has surprised over the years. Some of the greatest teachers I’ve ever met told me that when they were young, they were too shy, too tongue-tied in front of audiences, and too intellectually incapable to ever dream of becoming a teacher. They were ultimately surprised to discover that the qualities they saw as disadvantages helped propel them into being the thoughtful, attentive, creative instructors and professors they became. It seemed their introversion made them more thoughtful and sensitive to others, and their humble awareness of their past failings gave them patience and kept them from becoming aloof know-it-alls.
The Other Reason for Learning on Your Own—Quirky Test Questions
Let’s return to the world of traditional learning in high school and college, where a little insider knowledge will help you succeed. One secret of math and science teachers is that they often take quiz and test questions from books that aren’t in the assigned reading for the course. After all, it’s hard to come up with new test questions each semester. This means that test questions often have slight differences in terminology or approach that can throw you off your game even if you are comfortable with your textbook and your teacher’s lectures. You can end up thinking that you don’t have talent for math and science, when all you really needed to do was look at the material through different lenses as you were studying throughout the semester.
Beware of Intellectual Snipers
Santiago Ramón y Cajal had a deep understanding not only of how to conduct science, but also of how people interact with one another. He warned fellow learners that there will always be those who criticize or attempt to undermine any effort or achievement you make. This happens to everyone, not just Nobel Prize winners. If you do well in your studies, the people around you can feel threatened. The greater your achievement, the more other people will sometimes attack and demean your efforts.
On the other hand, if you flunk a test, you may also encounter critics who throw more barbs, saying you don’t have what it takes. Failure is not so terrible. Analyze what you did wrong and use it to correct yourself to do better in the future. Failures are better teachers than successes because they cause you to rethink your approach.
Some “slower” students struggle with math and science because they can’t seem to understand ideas that others find obvious. These students unfortunately sometimes think of themselves as not very bright, but the reality is that their slower way of thinking can allow them to see confusing subtleties that others aren’t aware of. It’s the equivalent of a hiker who notices the scent of pine and small-animal paths in the woods, as opposed to the oblivious motorist who’s whizzing by at seventy miles an hour.
Sadly, some instructors feel threatened by the deceptively simple questions that seemingly pedestrian students can pose. Rather than acknowledging how perceptive these questions are, instructors attack the questioner with brusque, brushoff answers that equate to “just do as you’re told like everybody else does.” This leaves the questioner feeling foolish and only deepens the confusion. (Keep in mind that instructors sometimes can’t tell whether you are thinking deeply about the material, or whether you’re having trouble taking ownership of your own role in understanding simple matters, as was the case with my truculent behavior in high school.)
In any case, if you find yourself struggling with the “obvious,” don’t despair. Look to your classmates or the Internet for help. One useful trick is to try to find another instructor—one with nice evaluations—who occasionally teaches the same class. These instructors often understand what you are experiencing and are sometimes willing to help if you don’t overuse them as a resource. Remind yourself that this situation is only temporary, and no circumstance is truly as overwhelming as it might seem at the time.
As you will find when you reach the work world (if you haven’t already), many individuals are far more interested in affirming their own ideas and making themselves look good than they are in helping you. In this kind of situation, there can be a fine line between keeping yourself open to constructive explanation and criticism, versus being closed to commentary or criticism that is couched as constructive but is actually simply spiteful. Whatever the criticism, if you feel a strong wash of emotion or certainty (“But I’m right!”), it may be a clue that you’re correct—or alternatively (and perhaps even more likely, because of your telltale emotions), it may be that you need to go back and reexamine matters using a more objective perspective.
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