4) THE EXPLORER IS THE PERSON WHO IS LOST, OR COURAGE IN THE FACE OF STUPIDITY
EXECUTIVE SUMMARY: Learning something really challenging is scary, but everyone has felt that fear, and you can overcome the fear of the unknown to learn more than you first thought you could learn.
In the summer of 2005, my oldest son had the privilege of attending the MathPath program for middle-school-age mathematics learners. Some other parents and I who brought children there were in the audience with the program students for the program's opening lecture. In the Colorado College auditorium where the lecture took place, there was a large blackboard on the speaker's stage. Mathematician Paul Zeitz stepped on to the stage, and wrote an enormous word on the blackboard:
STUPID
Everyone in the room wondered what kind of lecture would follow the lecturer putting that word on the blackboard. Zeitz began his lecture by asking members of the audience to think about how long they had ever worked on a mathematics problem for school. Most of the young people in the room seemed to indicate that they had spent at most an hour or so working on a mathematics problem for school homework. As the lecture continued, the idea came out that most school mathematics problems can be solved in just a few minutes by a bright learner. Maybe an especially tough mathematics problem from school might take as long as an hour. But Zeitz pointed out that professional mathematicians work on problems that can take days, weeks, months, or even years. Some famous mathematical problems have remained unsolved for centuries, with mathematicians working on a problem that was first brought up before they were born and dying before anyone in the world found out the solution. Zeitz said that if a problem is really a problem, and not just an exercise, the first issue is figuring out where to begin. If a problem is sufficiently hard, and many problems confronted by professional mathematicians are that hard, it may take a long time even to figure out how to start. Zeitz said that the way anyone feels while not knowing how to begin is stupid. Feeling stupid can be very discouraging. Especially if a student is used to getting the answer quickly from school mathematics courses, first working on a professional mathematics problem can be very discouraging and seem impossible.
The most important thing for someone working on a hard mathematics problem is not to be afraid while feeling stupid. When you feel stupid, don't say to yourself, "I'm stupid," but rather "This problem is hard. I'll have to work hard on this, try out lots of different ways to figure this out, and not give up too soon." Just because you feel stupid doesn't mean you are stupid. Sometimes you have to be brave and endure feeling stupid for days, weeks, months, or even years, but if you keep working on the problem, you can break through and feel glad that you didn't give up. Feeling stupid is a sign you are working on a hard problem. It's not a reason to give up.
When I showed a first draft of this essay to my wife, a piano teacher, she commented that much the same kind of courage applies to studying music. After thinking about that for a while, I realized that the self-talk that comes to mind for music students who feel stupid when they think a piece is too hard is "I have no musical talent." But I have seen several examples among various performance students who, at first, seemed stuck on a piece for weeks at a time, practicing over and over and over and never sounding musical, and then having a "breakthrough" so that they sounded completely different, much more musical, every time afterward when they played. Musical "talent" can develop after learning plateaus throughout the course of a lifetime of studying music.
I then recalled that another author related the idea of a specific lack of talent, rather than more general "stupidity," to learning plateaus in the study of mathematics. Many gainfully employed adults think they don’t have a brain for mathematics. Mathematician W. W. Sawyer wrote in his book Vision in Elementary Mathematics (original publication by Penguin 1964, reprint by Dover 2003) about parents who spoke to their children about not having talent in mathematics. Sawyer firmly believed, based on his experience as a mathematics teacher in various countries, that anyone can learn more about mathematics if properly taught. So Sawyer wrote, "The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" [Sawyer 1964:5] In other words, don't set up your children for believing that they cannot succeed in mathematics until you have given them a chance to learn from a good teacher. An even more famous mathematician than Sawyer who wrote interesting correspondence study books about secondary school mathematics, first in Russian and then in English, was Israel M. Gelfand (1913-2009). Gelfand wrote, "Students have no shortcomings, they have only peculiarities. The job of a teacher is to turn these peculiarities into advantages." So if you are young, and feel stupid because you don’t know how to begin in learning a hard subject, first of all look for a teacher who believes you can learn the subject, and who can help you make connections between what you already know and what you need to learn.
A new research publication, "The Neurodevelopmental Basis of Math Anxiety,"
http://stanford.edu/group/scsnl/cgi...Neurodevelopmental_Basis_Math_Anxiety_12 suggests, based on data gathered from brain scans in children, that mathematics anxiety is a learned fear much like other specific phobias. From this point of view, the most important thing a teacher can do for a young mathematics learner is dispel fear, encourage boldness, and focus the learner’s attention on problem-solving skills rather than the negative emotions that derive from fear.
I recently checked my memory of Paul Zeitz's lecture based on the word "STUPID" by emailing Professor Zeitz. He was very glad to hear that some of my mathematics pupils have learned to take "stupid" as an inspirational word, writing it on the whiteboard to cheer themselves on before they play mathematical games against me or against one another. They have lost the fear of taking on tough challenges, and now delight in working on problems that are hard.
Professor Zeitz told me that he now sums up the advice he gave in his 2005 lecture with a phrase he takes from Tim Cahill, founding editor of Outside magazine: "The explorer is the person who is lost." Zeitz used that line in the foreword he wrote for the 2011 Math Survey of Stuyvesant High School in New York City, his alma mater, in which he wrote,
"In 1998, when I wrote the first edition of The Art and Craft of Problem Solving (whose introduction specifically mentions Stuyvesant!), I included an epigraph that was a quote from Jaguars Ate My Flesh, a collection of humorous travel essays by Tim Cahill. The quote was:
"The explorer is the person who is lost.
"The meaning, as we mathematicians like to say, is 'clear.' You cannot accomplish anything meaningful without making mistakes. In fact, even if you don’t accomplish much that is meaningful, you need to make mistakes. This is a sometimes counterintuitive idea for many Stuyvesant students, trained and rewarded over many years not to make mistakes.
"One of the hardest things to do, especially for us bright high achievers, is to get used to being not just stupid, but bumblingly stupid. All mathematicians spend most—usually way above 90%—of their waking hours feeling confused at best. That is not a bad feeling, mind you, although it is less good than the rare moments of insight. After all, mathematicians are explorers, and explorers are lost. It’s an existential condition of being an explorer.
"So, learn to enjoy the sensation of being lost. As you encounter mathematical problems—and here I distinguish problems, questions that you do not, at the outset, know how to approach, from exercises, which are the things at the end of the chapter that you do for homework—you have no choice but to go on wild goose chases, most of which lead nowhere. At least nowhere relevant to the solution of the problem at hand.
"But that’s OK. Rarely does it work like this:
• A problem is posed.
• A smart mathematician thinks very hard.
• She solves it!
"More often, what happens is:
• A problem is posed.
• A smart mathematician thinks very hard.
• She gets nowhere.
• Later, sometimes much later, she realizes that "nowhere" actually is
something that a colleague always wanted to know.
• A new problem is solved!"
I love all the stories Paul Zeitz tells about mathematics learning. He is enthusiastic about sharing delight in taking on tough problems--while acknowledging feeling stupid isn't a lot of fun--and encouraging young people to enjoy a sense of adventure and exploration while they learn. I hope that every young person gains opportunities to learn HARD topics that present TOUGH problem-solving situations, so that each learner can take delight in being lost until the learner finds something new, however long that takes. W. W. Sawyer also had encouraging thoughts about what happens to mathematics learners as they outgrow their teachers in their mathematics level. Advanced learners of mathematics may be beyond the teaching level of any teacher in their high school by the time they reach high school age, and much the same happens to advanced music learners as they pursue their study of music. By the late teens, the advanced learners are learning how to learn on their own, while seeking out higher education opportunities that will bring them into a community of experts in their field of study. When a learner who has no teacher at hand to help encounters a tough topic, the learner can still do as Sawyer advised: "We all meet from time to time some particular problem we cannot solve, and we deal with it much as a mediaeval army dealt with an impregnable castle. We go round it and on, no doubt with the hope that it may yield at some time in the future." [Sawyer 1995]
Thinking about Sawyer's image of an army going around a castle reminded me again of Paul Zeitz’s recommendation of Tim Cahill’s line, "The explorer is the one who is lost." And that in turn reminded me that John DeFrancis, author of the first textbook I used for learning Chinese, quoted Robert Louis Stevenson in the preface to the textbook: "To travel hopefully is a better thing than to arrive." When I first read that at age seventeen, I thought that was very off-putting, as the textbook author didn’t even seem to promise that his book would help me learn Chinese. In fact, the book was very helpful indeed, but it was most helpful in building in me an attitude that I couldn’t count on my study of Chinese being easy, as too many of my school lessons had been up till then. As I learned to enjoy the journey, stepping steadily forward day by day to a far-off destination of proficiency in the language, I learned that courage would take me even farther than verbal ability, and my own practice and effort would take me even farther than the best available textbooks--which I had--and miles of audio tapes of the language in the language lab. The same is true of learning mathematics. Mathematicians say that mathematics, like swimming, is not a spectator sport. You have do problems, or get wet, to make progress in mathematics or in swimming. In the study of music, finger exercises and listening to other performers, and playing for other listeners, are all part of steady skill development that may include many frustrating learning plateaus before each new exciting breakthrough in musicality. The main thing is not to be afraid while feeling stupid and lost, but to enjoy the journey and keep stepping forward.
A very interesting popular article about current research on overcoming learning challenges and thriving from them is "The Effort Effect,"
http://www.stanfordalumni.org/news/magazine/2007/marapr/features/dweck.html which reports the work of psychologist Carol Dweck, who shows how "growth mindset" can make learners smarter. This line of research has been further studied in recent years. Current research suggests that learners who are not afraid to admit that they have more to learn, and who acknowledge they can learn more if they pay attention to their own performance, can enjoy the journey even better than the destination, all the while learning more than learners with less courage in self-examination. A post in the Wired Magazine Frontal Cortex blog by Jonah Lehrer, "Why Do Some People Learn Faster?"
http://www.wired.com/wiredscience/2011/10/why-do-some-people-learn-faster-2/ discusses the importance of not being afraid to face mistakes and learn from them. It is exactly the learners who are most courageous about facing their own mistakes in performance and observing them and correcting them who make the most progress in learning difficult subjects. Lehrer's new book Imagine: How Creativity Works suggests a distinction between problem-solving tasks that yield to steady effort, which Lehrer calls "grit," and problems best solved by insight that comes from going around the problem and letting it rest for a while like Sawyer's medieval army going around a castle.
References:
W. W. Sawyer, Vision in Elementary Mathematics page 5 (original publication by Penguin 1964, reprint by Dover 2003).
W. W. Sawyer, "Catering for the Extremes" Mathematics in School March 1995.
