The main difficulty I’ve seen in tutoring math is that many students panic if they don’t see what to do within five seconds of reading a problem, maybe two seconds for some. A good high school math student may be able to stare at a problem for fifteen seconds without panicking. I suppose students have been trained implicitly to expect to see the next step immediately. Years of rote drill will do that to you.
A good undergraduate math student can think about a problem for a few minutes before getting nervous. A grad student may be able to think about a problem for an hour at a time. Before Andrew Wiles proved Fermat’s Last Theorem, he thought about the problem for seven years.
21 thoughts on “How long can you think about a problem?”
Here’s a video about thinking about problems for a long time: Hammock-driven development
I think becoming a mathematician means becoming more and more humble. The greatest ones have been thinking about specific problems for the whole of their lives…
The thing is: At the end it doesn’t matter if you will ever succeed because you learn so much along the way!
“Before Andrew Wiles proved Fermat’s Last Theorem, he thought about the problem for seven years.”
That’s what I’m telling my teacher next time he collects my exam paper before I’ve finished ;-)
Good points made. After all, thinking about a problem IS doing mathematics.
There is also a substantial practical limit: how long can you get funded to think about a problem.
I think the following aspect of math (and the sciences in general, but especially math) is part of what makes it such an important subject to study, even if you don’t plan on using it much later in life: you learn to be comfortable with feeling dumb at times, and develop confidence that Ah Hah moments will come if you stick with problems. The time spans you mention seem in line with my personal experience, and if nothing else, what I got out of my graduate study was the maturity to sit with things longer.
I echo vonjd. One needs to get over the feeling that someone else smarter would have solved this problem already, otherwise the whole thing becomes not fun.
Yes. When I was in high school, one very talented teacher used to staple shut the solutions at the end of the textbook. Then he refused to help us with the problems until we’d worked at least a couple of hours on each problem and showed him the (incorrect and incomplete) attempts to solve them. His thesis was that you learn more from your failed attempts than you do from the actual solution. A problem that a student can solve immediately is useless for learning.
Recently I’ve been trying to improve the length of time I can think about a problem. My issue isn’t panicking or getting uncomfortable, it’s that my ability to hold a mental model breaks down at a certain level of complexity. I’ve started carrying a notebook everywhere to use as a cache for bits of the mental model which are important but I can’t hold in my head while I’m examining another part of it. Hopefully with sustained exercise and concentration I will improve my ability to hold the entire model in my head without my external cache.
I guess I am the best mathematician there is. I can think about real math problems forever. ;)
I think of computer programming in the same way – I often need to study and think about a solution to problem that a client presents to me, but most of the time they want it solved *now*.
I think this is excellent advice. Its probably the number one thing I would say to my undergrad self. I find just sitting there staring at a problem for a long time even if broken up with daydreaming can be really helpful.
“Years of rote drill will do that to you.”
I agree completely. Of course, the obvious solution is to promote creative, exploratory approaches, such as Vinod Khare shared, or as many educational researchers have been advising for decades. Some countries, like Finland, have actually tried to follow this advice. The U.S., on the other hand, is promoting test-driven accountability which is putting pressure on even the good teachers (who want deeper learning) to teach to the test through rote drill.
John Maynard Keynes had this to say about Isaac Newton:
His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one’s mind and apply all one’s powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret.
See also http://www.brainpickings.org/index.php/2012/09/04/the-ravenous-brain-daniel-bor/ over at Brain Pickings.
Thanks for reminding me of this, John. It’s something I need to remember to say to all of the students I tutor, before we even start. There is nothing more frustrating to me that having students blurt out pure guesses — but it happens all the time. Apparently blurting out a wildly wrong and unmotivated answer is less embarrassing to them than being silent for 10 seconds.
Could an explanation for the “panic” you see be caused by the student being watched by you?
I find this to be very true. I myself can find myself staring at a problem for 5 seconds, then I realize I don’t know what to do, so I stare at it for another 30 seconds without actually thinking. This is easily my biggest waste of time on Tests, Exams, etc. Good point, John.
I call this the Aaargh! response. I try hard and often over my time with a tutee trying to get them to look at what bits they can deal with and to gradually build up an answer. I agree, it is tough going. Sometimes it even works!
This reminds me of my first exam in grad school. It was an analytical mechanics course (I was in the physics program). Forty-five minutes into the exam, I had written my name on the paper. I just kept reading the problems over and over again trying to figure out how to get started on one of them. I eventually got going and did fine, but was approaching that point of panic.
@Alexsis: Does it cost a lot to sit and think? I think we could get that down to about $10 a day. So a grant of $50,000 would get you 13 years!
I’m sure there are many factors to this whole situation I’m unaware of.