The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence.
I understood it when you taught it in class. He had not developed any kind of procedural fluency or ability to apply what he thought he understood. There is an interesting connection between learning math and science, and learning a sport.
When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.
At some point, you just know it fluently from memory. If you use the procedure a lot, by doing many different types of problems, you will find that you understand both the why and the how behind the procedure very well indeed. The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.
So I launched directly from high school into the Army. I had loved learning new languages in high school, and the Army seemed to be a place where people could actually get paid for their language study, even as they attended the top-ranked Defense Language Institute—a place that had made language- learning a science. I chose Russian because it was very different from English, but not so difficult that I could study it for a lifetime only to perhaps gain the fluency of a 4-year-old.
Besides, the Iron Curtain was mysteriously appealing—could I somehow use my knowledge of Russian to peer behind it? After leaving the service, I became a translator for the Russians on Soviet trawlers on the Bering Sea.
Working for the Russians was fun and engrossing—but it was also a superficially glamorous form of migrant work. There was pretty much only one other alternative for a Russian language speaker—working for the National Security Agency.
I began to realize that while knowing another language was nice, it was also a skill with limited opportunities and potential. Unless, that is, I was willing to put up with seasickness and sporadic malnutrition out on stinking trawlers in the middle of the Bering Sea.
Their mathematically and scientifically based approach to problem-solving was clearly useful for the real world—far more useful than my youthful misadventures with math had been able to imagine. So, at age 26, as I was leaving the Army and casting about for fresh opportunities, it occurred to me: If I really wanted to try something new, why not tackle something that could open a whole world of new perspectives for me?
Something like engineering? That meant I would be trying to learn another very different language—the language of calculus. With my poor understanding of even the simplest math, my post-Army retraining efforts began with not-for-credit remedial algebra and trigonometry.
This was way below mathematical ground zero for most college students. Trying to reprogram my brain sometimes seemed like a ridiculous idea—especially when I looked at the fresh young faces of my younger classmates and realized that many of them had already dropped their hard math and science classes—and here I was heading right for them.
But in my case, from my experience becoming fluent in Russian as an adult, I suspected—or maybe I just hoped—that there might be aspects to language learning that I might apply to learning in math and science. What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.
In fact, depending on who you ask, up to 60 percent of high school goes straight into your mental recycling bin, and for quite a few of us, that includes our math skills.
It just takes a little practice. Several studies point out the benefits of being good at math. But that can be you, too. Scientists have shown that when it comes to improving your math skills, practice is what matters most —not talent.
The best way to think about math is to search for patterns. So the question is: What type of math do you want to learn, and what do you want to get out of it? You should know the answer to the latter equation is 25 and that the difference between and is four.
Likewise, ask yourself honestly: Have I pushed my brain to its limit trying to solve this problem first? But after that, challenge yourself to go back and work through the problem without looking at your notes and references. Instead of spending long hours in the library, she advises shorter, more frequent study sessions that are spread out over weeks, not days. This kind of slow, deliberate learning allows your brain to get a firm grasp of each concept and more importantly, the connections between them.
You need to deeply understand both how a concept works and when to use it alongside other concepts and operations. To help you gain mastery in the individual mathematical building blocks, you can use references and tools to help you. But ultimately, they can only help you so far. To gain mastery of the underlying concepts you have to get your pencil and paper and solve the hard problems yourself.
So the money question is this: How do you get better at math? Check out how they all go together: Congrats! Tip 3: Review the Underlying Concepts Sometimes though, your understanding is just too shaky for the problem.
Oftentimes, these have step-by-step solutions that show how other people get to their answers: Finally, ask your professor or teacher for help. It will show you step-by-step solutions for all problems in the textbook your math class uses: Regardless of what you choose, make sure you try the problems out yourself afterwards without looking at the solutions.
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