A Mind For Numbers
There are some very good practical tips in this book. It’s a short and quick read, so please read it, or give it to your kids to read.
- Actively try to recall the material that you are trying to learn.
- Learning takes time. Intensive focus, followed by relaxed thinking are best for fully understanding the ideas.
- Rest is crucial for learning
We don’t engage in passive rereading because we are dumb or lazy. We do it because we fall prey to a cognitive illusion. When we read material over and over, the material becomes familiar and fluent, meaning it is easy for our minds to process. We then think that this easy processing is a sign that we have learned something well, even though we have not.
If you are trying to understand or figure out something new, your best bet is to turn off your precision-focused thinking and turn on your “big picture” diffuse mode, long enough to be able to latch on to a new, more fruitful approach.
The harder you push your brain to come up with something creative, the less creative your ideas will be. So far, I have not found a single situation where this does not apply. Ultimately, this means that relaxation is an important part of hard work—and good work, for that matter.”
In other words, just using your diffuse mode doesn’t mean you can lollygag around and expect to get anywhere. As the days and weeks pass, it’s the distributed practice—the back and forth between focused-mode attention and diffuse-mode relaxation—that does the trick.
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.
Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.
“Intention to learn is helpful only if it leads to the use of good learning strategies.”
Using recall—mental retrieval of the key ideas—rather than passive rereading will make your study time more focused and effective.
“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.”
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.
In building a chunked library, you are training your brain to recognize not only a specific problem, but different types ˙and classes of problems so that you can automatically know how to quickly solve whatever you encounter. You’ll start to see patterns that simplify problem solving for you and will soon find that different solution techniques are lurking at the edge of your memory. Before midterms or finals, it is easy to brush up and have these solutions at the mental ready.
“Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.”
This reinforces an idea we’ve alluded to already. When we retrieve knowledge, we’re not being mindless robots—the retrieval process itself enhances deep learning and helps us begin forming chunks.
But be wary of repetitive overlearning during a single session in math and science learning—research has shown it can be a waste of valuable learning time.Revisiting the approach mixed with other approaches during a subsequent study session, however, is just fine.
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.
Paul’s Techniques for Limited Study Time 1. Read (but don’t yet solve) assigned homework and practice exams/quizzes. With this initial step I prime my mental pump for learning new concepts—new chunks. 2. Review lecture notes (attend every lecture as much as possible). One hour of lecture is worth two hours reading the book. I learn far more efficiently if I am faithful in attending lectures and taking detailed notes—not just staring at my watch and waiting for it to be over. I review my notes the following day while the subjects are still fresh in my mind. I’ve also found that thirty minutes with a professor asking questions is easily worth three hours reading the book. 3. Rework example problems presented in lecture notes. It never helped me to practice problems given by either the instructor or the textbook that didn’t have solutions to provide feedback. With the example problems I already had a step-by-step solution available if necessary. Reworking helps solidify chunks. I use different-colored pens when I study: blue, green, red—not just black. I found that it helps me focus on reading my notes better; things pop out more, instead of blending together into a confusing collage of inexplicable mathematical chaos on the page. 4. Work assigned homework and practice exam/quiz questions. This builds “muscle memory” chunks for the mind in solving certain types of problems.
Do you like to check your e-mail or Facebook right when you wake up in the morning? Set a timer for ten minutes of work first thing instead—then reward yourself with online time. You will be surprised to see that this tiny exercise in self-control will help empower you over your zombies through the day. Warning: When you first sit down to try this, some of your zombies will scream as if they want to eat your brain. Tune them out! Part of the point of this exercise is learning to laugh at your zombies’ antics as they predictably tell you, “Just this once it’s okay to check Facebook right now.”
Remember, Lady Luck favors the one who tries. So don’t feel overwhelmed with everything you need to learn about a new subject. Instead, focus on nailing down a few key ideas. You’ll be surprised at how much that simple framework can help.
Remember, research has shown that the more effort you put into recalling material, the deeper it embeds itself into your memory.
This type of “knowledge collapse” seems to occur when your mind is restructuring its understanding—building a more solid foundation. In the case of language learners, they experience occasional periods when the foreign language suddenly seems as comprehensible as Klingon. Remember—it takes time to assimilate new knowledge. You will go through some periods when you seem to take an exasperating step backward in your understanding. This a natural phenomenon that means your mind is wrestling deeply with the material. You’ll find that when you emerge from these periods of temporary frustration, your knowledge base will take a surprising step forward.
“A curious peculiarity of our memory is that things are impressed better by active than by passive repetition. I mean that in learning by heart (for example), when we almost know the piece, it pays better to wait and recollect by an effort from within, than to look at the book again. If we recover the words in the former way, we shall probably know them the next time; if in the latter way, we shall very likely need the book once more.”
“I did not go to an elite school when I was growing up. In fact, my school was below average—we didn’t have the proper teachers for many subjects. But I focused on finding something good in whatever teachers came my way, whether it was an excellent memory or simply an easy smile. This kind of positive attitude helped me appreciate my teachers and keep an open-minded approach toward my classes.
“One of my mother’s Golden Rules was that ‘writing is the foundation of learning.’ From grade school through doctoral studies, I have found immense power in systematically understanding and writing each step of what I really wanted to learn.
But once you make a task list, it frees working memory for problem solving. Yay! But remember, you must absolutely trust that you will check your planner-journal. If your subconscious doesn’t trust you to do that, tasks will begin swirling back up, blocking your working memory.
THE FREEDOM OF A SCHEDULE “To combat procrastination, I make a schedule of everything I have to do. For example, I tell myself, ‘Friday, I need to start my paper and then finish it on Saturday. Also, on Saturday, I need to do my math homework. On Sunday, I need to study for my German test.’ It really helps me stay organized and practically stress-free. If I don’t follow my schedule, then I have twice the amount of the work to do the next day, and that’s really not something I look forward to.”
You might think, Well, yeah, but you’re a professor who is past your youthful study days—of course an early quitting time is fine for you! However, one of my most admired study experts, Cal Newport, used a 5:00 P.M. quitting time through most of his student career. He ended up getting his Ph.D. from MIT. In other words, this method, implausible though it may seem for some, can work for undergraduate and graduate students in rigorous academic programs. Time after time, those who are committed to maintaining healthy leisure time along with their hard work outperform those who doggedly pursue an endless treadmill.
This approach works for some people, mostly because anything works for some people. Unfortunately, however, for most people it’s counterproductive. Tough problems often need lots of time, meaning you’d want to start on them first thing on a test. Difficult problems also scream for the creative powers of the diffuse mode. But to access the diffuse mode, you need to not be focusing on what you want so badly to solve!
Then when you start working problems, start first with what appears to be the hardest one. But steel yourself to pull away within the first minute or two if you get stuck or get a sense that you might not be on the right track. This does something exceptionally helpful. “Starting hard” loads the first, most difficult problem in mind, and then switches attention away from it. Both these activities can help allow the diffuse mode to begin its work.
To help keep his mind occupied when his workday ended and anxiety or boredom reared its head, Feynman began a focused effort to peer into people’s deepest, darkest secrets: He began figuring out how to open safes. By knowing the default settings, the locksmith was often able to slip into safes that had been left unchanged since they’d arrived from the manufacturer. Whereas everyone thought that safecracking wizardry was involved, it was a simple understanding of how the device arrived from the manufacturer that was fundamental.