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The benefits of math competitions are well known: focusing on goals, dealing with pressure, learning teamwork, and building friendships are among those typically mentioned. Math competitions also allow for the much needed celebration of intellectual achievement the way athletic achievement has always been celebrated. At IMACS, many of our math enrichment students enjoy competing in the American Mathematics Competitions, International Mathematical Olympiad, and MATHCOUNTS among other contests, so we have a very positive view of math competitions, particularly for the kids who thrive in that environment. We’re extremely proud of our numerous students over the years who have performed well in these prestigious contests. We’re equally proud of our numerous students over the years who have thrived in quiet contemplation.

This brings to mind an article published last week by The Wall Street Journal on the challenges encountered by Type A parents raising Type B kids. The article talked about different ways that ambitious, competitive, and hard-driving parents modify their interactions with or expectations of their dreamy, mellow, and seemingly laid-back children to foster healthy parent-child relationships. Whether Type A or not, most parents are cognizant of today’s ultra-competitive global environment, and many feel a sense of urgency to nudge, push, or even pressure their kids to achieve. It’s not a stretch to imagine the parents of a mathematically talented child thinking, “If all the other kids are involved in math and science, then my child should be doing even more.” Here’s the thing: What makes an activity suitable for even more depends on what works for your child and is not necessarily the same activity enjoyed by all the other kids but with more time dedicated to it or with better results.

For parents considering activities for their mathematically talented child, it is important to understand how innate personality factors into the mix that determines whether math contests provide a net positive experience for that child. Just as there are natural-born competitors among mathematically talented students, there are also natural-born dreamers. These kids used to get a bad rap for being unfocused, undisciplined, and even lazy. There was no observable productivity associated with daydreaming so, of course, it had to be a waste of time. Not so fast. In 2009, researchers from the University of British Columbia published a study in the Proceedings of the National Academy of Sciences detailing that brain activity increases when our minds wander. Reporting on this finding, ScienceDaily.com put it well: “[B]rain areas associated with complex problem-solving – previously thought to go dormant when we daydream – are in fact highly active during these episodes.” And from The Wall Street Journal: “These sudden insights … are the culmination of an intense and complex series of brain states that require more neural resources than methodical reasoning.”

If your child is more of a dreamer (or just not drawn to competition), activities that would cultivate his or her talent in and appreciation for mathematics may differ from what is offered through a typical regional, national, or international math contest. Students of this personality type often find more success and satisfaction with math enrichment programs that focus on deep problem-solving over computational prowess. This is not to say that your child shouldn’t at least try participating in some type of competitive math. The experience just might open up a different side of his or her mathematical personality. In fact, our math enrichment classes use game-playing and mini-competitions as teaching tools, and most of our students really enjoy this aspect of the class the best. But if it’s clear that competition does not bring out the best in your child, we encourage you to explore other options including letting your child have more free and unstructured time to let his or her mind ruminate about the wonders of mathematics.

For parents of talented children, the secrets of mathematical success are really not that different from general advice on positive parenting. They include understanding what kind of child you have, knowing what motivates him or her, and fostering an environment that includes the kind of math activities or, quite possibly, freedom from structured activities that align best with that motivation. And if having a dreamer for a child still makes you worry, just think about the great mathematical and scientific discoveries we owe to dreamers of the past. You never know what grand ideas are simmering behind those eyes staring off into the distance.

Dreamer Matching Puzzle
We made a simple matching game out of the Wall Street Journal article that reported on the aforementioned University of British Columbia study. See if you or your kids can match the great thinker with his profound idea and what he was reportedly doing at the time moment of insight. Answers may be found in the article.

Great Thinker: (A) Archimedes, (B) Newton, (C) Einstein, (D) Descartes, and (E) Tesla.

Profound Idea: (1) special relativity, (2) coordinate geometry, (3) alternating electrical currents, (4) calculating the volume of an irregularly shaped object, and (5) law of universal gravitation.

Dreamy Activity: (i) lying in bed watching flies on the ceiling, (ii) taking a bath, (iii) watching an apple fall from a tree in an orchard, (iv) taking a walk, and (v) imagining trains and lightening.

Editor’s note: Regular readers of this blog will notice a slight change going forward. IMACS will be switching to a bi-weekly publishing schedule with our next post appearing on November 10, 2011.

My earliest memory of chess was from when I was around three years old. I remember sitting on the edge of a rug playing with colorful LEGO blocks. When I looked up, I saw my father and my oldest brother playing chess. The board was illuminated so brightly that it looked as if the pieces were shining. I looked back down at my blocks, and they seemed so boring compared to the multitude of pieces and squares and interesting stuff happening on the board above. That was when I was first aware of my desire to play on that board and with those pieces. I would crawl over to where my father and brother were playing, take a few game pieces, and put them on the board too. After “playing” with them a few times in this way, I was henceforth banished from the vicinity of their games. This continued for two years until my mother determined that I had the capacity to understand what was going on. However, nobody believed her, so she taught me the basics herself. It took me about a week to grasp those rudiments of chess, after which I was handed over to my father for further instruction.

My father and I would take a chess set and chess book from our library at home and bike to one of our favorite places. There, my father would set up a position from the book on the board, and we would puzzle it out together. In the beginning, we would actually move the pieces, but after a few weeks my father urged me to work out the entire solution and move the pieces only in my head. We studied tactics, strategy, endgames, etc. I gradually improved to a stage where I could have a “reasonable” game with him (“reasonable” in the sense that I wasn’t losing after the fourth move). A few years later, when I was eight or so, I learned that there was a chess club at a local elementary school, so naturally I went to check it out. There I found good competition with kids my age and with one boy in particular. We were about evenly matched, so when he went to a tournament and won a big trophy, I wanted one too! That’s how I got started with tournaments.

For a few years though, competition took a back seat to other important family events, which had the strange habit of conflicting with tournament schedules. School was also a factor: It would have been difficult to actively study chess and play in tournaments given all the classwork and homework I would have had to make up for the days I missed. Also, tournaments usually run through Sunday, and my Sundays were taken up with IMACS. I loved IMACS so much that I wasn’t willing to skip even one class if I could help it. When I was in third grade, my parents withdrew me from the school I was attending for a variety of reasons, including lack of a challenging curriculum that kept me perpetually bored even though I skipped a grade. I was homeschooled while my parents searched for another school, and I recognized this as an excellent opportunity to have a say in my education. I was on my best behavior for months before I convinced my parents to homeschool me forever! Now that I was homeschooled, I had loads of free time. Also, my IMACS classes were moved to Thursday. So, now that I had the time and was not sacrificing IMACS, I got interested in competitive chess again.

From my friends at tournaments, I found out about the Internet Chess Club (ICC). With my new ICC membership, I played for hours after my studies were done (which at 3rd grade took a grand total of an hour and a half). My parents always encouraged me to play up (i.e., at a higher level), so already in elementary school I was regularly playing up in K-12 events. This wasn’t very good for my rating as I usually lost many games, but as a consolation, from age 8 on, I was Florida’s Top Girl at the K-12 level for several years in a row. At that time, Women’s World Champion Grandmaster, Alexandra Kosteniuk, was handing out the Florida’s Top Girls prize. After a few years of shaking hands with her, my mom asked her to coach me. I improved dramatically under Alexandra’s guidance – I developed an opening repertoire, whereas before I usually made up my own openings (that sometimes didn’t turn out well). As my rating shot up, I made the Susan Polgar’s National Team for Girls and was invited as the representative from Florida to play in her 2009 National Invitational for Girls event. That was my introduction to girls-only events. I took second place, and made new friends. My mom is always checking the Web site for FIDE (the World Chess Federation, or Federation Internationale des Echecs). There she found out about the 2010 North American Youth tournament. The fact that one of my new friends was going to attend helped seal the deal. We were both US representatives, and it felt awesome to tell people that I was representing the United States of America. A series of wins and draws took me to the final round where I ultimately won! And so in my first international tournament I got my first gold medal for the US. As I hadn’t been having great results before that tournament, the win was welcome and kept me studying chess even with my increasing workload. (Studies at my age now aren’t as easy as in 3rd grade!)

My coach, Alexandra, started a family, which meant that she rarely traveled from her home in Russia to the US anymore. Thus, my parents decided to choose a different coach to fill in the gaps – four-time US Champion and Grandmaster Alexander Shabalov. Under his guidance I won the 2011 US Girls Junior (U21) Chess Championship this past August with an undefeated score. Whenever I’m feeling low, there always comes a win to keep me motivated. Chess is an amazing game, and I’m very competitive, so that also always brings me back to competitions. I love playing and seeing how I stand against some of the big names in chess (or more often, those who say they’ve played them). My favorite competitions are the one day tournaments, because I get to go home quickly. The longer 6-7 day tournaments, like the Susan Polgar, North American Youth, and the US Junior Girls leave me homesick toward the end. (For how long can one eat hotel food?) Now I’m in a lull between tournaments, so I practice by playing on ICC and chesscube.com for about an hour each day. My next big tournament will be the World Youth Chess Championship in Caldas Novas, Brazil, this November where I will represent the United States.

Dan Dugger is a professor in the Department of Mathematics at the University of Oregon. He received his Ph.D. in mathematics from the Massachusetts Institute of Technology and his B.A. in mathematics from the University of Michigan. Professor Dugger is a graduate of Project MEGSSS of Broward County, Florida, which used the mathematics textbooks that underlie the EMF Math online program. As a high school senior, he submitted an original research paper entitled "On the Homomorphisms between Finite Abelian Groups" to the 41st International Science and Engineering Fair, winning–among several other honors–the first place award from the American Mathematical Society.

When did you know that you wanted to be a mathematician? Who or what experiences influenced you in that direction?

When I was in elementary school, there was a class for gifted students run by a remarkable teacher. Her idea was that the students should be in charge of their own education, so she had them each pick a subject they wanted to study. At first I studied space exploration, and then I studied whales. The next year I studied computers for a long while (this was in the days before anyone had a computer in their home!), and the year after that I switched to lasers. One day I was reading an explanation of lasers in a physics book when an older kid came along and in a tone of complete awe asked, "Danny, you're studying physics?" I sheepishly explained otherwise, and he went about his business, but I came away with the idea that physics was a difficult and impressive subject! So the next year, in fifth grade, I was determined to study physics. After that, I was convinced I wanted to go to Caltech and become a physicist.

At that point, I honestly had no idea that there were such things as mathematicians. I liked math, and I liked it when my dad showed me mathematical things he knew that were over my head, but I had no idea a person could make a career of it. Then in middle school I went into Project MEGSSS (the precursor to IMACS) and started learning from the Elements of Mathematics curriculum (the basis for the EMF curriculum). At that point I felt the wonderful beauty and poetry in mathematics. I had great teachers like Burt Kaufman and Ed Martin who were encouraging and really made this material accessible, and thanks to them I was hooked. So pretty much from then on I wanted to be a mathematician.

The important point that I want to get across is this: I remember fairly little of elementary school, but I remember every subject I studied in the four years of that gifted class. That strikes me as remarkable. And in a very direct way I see that class as having put me on a path that led me to my eventual career. It's really hard to overestimate the effect that good, caring teachers have on children.

You attended the University of Michigan at Ann Arbor for your undergraduate degree. At what point did you start looking into grad schools? What were you looking for, and how did you decide on MIT?

As an undergraduate, there were times when I was uncertain that I still wanted to go to graduate school, but I also didn't know what I wanted to do. A kind math professor grabbed me one day after I had taken his class and said "Are you planning to go to graduate school? Because you really should!" At his suggestion I participated in a summer research program at Michigan with another great professor. The next fall this professor sat me down and said, "Okay, here are the places you should apply to graduate school: Harvard, MIT, Princeton…" I was pretty shocked! I had no idea I would be able to get into schools like that.

So I applied for graduate school, although I was still not exactly sure that this was what I wanted to do. For a while I was convinced that I wanted to go to University of Chicago, but when I visited there, it didn't quite click for me. I remember quickly calling up MIT afterwards and asking to arrange a visit! Luckily, during that visit something did click. What I mostly remember was the very pleasant atmosphere among the graduate students at MIT. Seven or eight of them took me out to dinner, and I remember they spent the whole evening gossiping about the professors in the department. That doesn't sound like a good reason to choose a graduate school, but somehow it worked.

I think the real lesson here is that visiting schools is very important. It is quite difficult to know, or to predict, when things are going to click or not click. Every school is different, and different things work for different students.

Tell us about your years at MIT. How did you choose your advisor, or is it the advisor who chooses the student? How do you decide on the topic for your thesis?

I came to MIT determined to study a subject called commutative algebra. But once I was there, I found myself more and more in the company of topologists. There weren't many people studying commutative algebra at MIT, but there were numerous topologists–it was a very active and exciting community. My best friend was a topology student, and I always seemed to be hanging around the topologists and their interesting seminars, so at some point it seemed that I should just become one of them!

Unbeknownst to me at the time, other forces were also at work. During my first semester at MIT I took a seminar in topology run by a retired professor, and he decided that I should become a topologist. (He didn't tell me this at the time.) He somehow set things up so that I started talking regularly with a brilliant topologist named Mike Hopkins, and after a while Mike became my advisor. I never formally asked him to be my advisor, and he never formally invited me to be his student; it somehow just came to be. However, this is somewhat atypical. Most students just choose someone who is in the area in which they want to work, possibly having already taken a class from such a person, and simply ask. It's typical for the student to do some independent reading under the professor's guidance to see if things are a good fit before anything is set in stone.

Concerning choosing a thesis problem, most advisors will assign the student a problem based on the student's interests. Usually it is a problem the advisor has some ideas about, so that he can suggest a plan of attack. Some advisors, however, have the students find their own thesis problem. In some ways the latter is a better, albeit more challenging, technique because eventually in his or her career a student will have to learn to do this anyway. In my own case it was somewhere in between these extremes, but closer to the latter.

Were there challenging times when you just hit a wall with the research? How did you keep yourself motivated?

It is not just that sometimes there is a wall to get past; it would be fair to say that almost all of the graduate school experience is a wall. Let me try to explain. Everyone goes into graduate school assuming that when they write their thesis it's going to be fun and beautiful like the mathematics they fell in love with, but that is an idealized view. When you take mathematics courses as an undergraduate, you are presented with a very beautiful, developed theory. But it's important to realize that what you are seeing in these courses has been refined over hundreds of years–you are almost never seeing "raw mathematics" as it comes out of the ground, so to speak. Instead of polishing a beautiful diamond, a working mathematician spends most of his or her time digging in a mine, carving away at the rock. In addition, when you learn mathematics in a course you are carefully protected from all the false trails that mathematicians took to get there. Actual mathematics research has many false trails, or trails that lead a little ways but then peter out. Someone once described their life as a mathematician this way: "I spend my days writing lots of stuff on the blackboard, most of which turns out to be wrong." This description, although exaggerated for a humorous effect, is not that far off.

So in some sense graduate school ends up being largely about making a transition in how you see mathematics. It is not easy to do. And a challenging factor that accompanies this is that the gulf between working mathematicians and students becomes particularly large here. Working mathematicians made this transition long ago in their careers, and many are not quite aware of it anymore. This can make communication between students and their professors somewhat frustrating.

The question of how to keep motivated in the face of all this is the crucial one. It is very difficult, and that difficulty never goes away. Not many people get PhDs in mathematics, and far fewer are still engaged in research ten years afterwards–and I think this is largely because of the motivation issue.

I don't want to give the impression that this is an impossibly difficult career path; obviously, there are new math PhDs awarded every year. I do think it is important to understand that life as a mathematician is very different from life as a mathematics student. When you are a student doing exercises from a book, the distance between what you know and what you are trying to find out is relatively small. In mathematics research, that gap is multiplied by a hundred. It can seem like you are walking from San Francisco to New York, but with the added twist that there are no roads or maps and that you don't really know that New York is even there! Everyone has to find their own answers for how to keep motivated on a journey like that. You can look at it as an insurmountable feat, or you can look at it as an challenging adventure, or somewhere in between. For what it's worth, I will tell you the answers that have worked for me. For myself, it always comes back to the fact that I like teaching; I like understanding neat things and sharing them with people. This is where my motivation comes from, even for my research. I'm always out to find that next "neat thing" to explain.

How did you know when you were ready for your thesis defense, and what was that like?

In most U.S. mathematics departments the thesis defense is, to some extent, a formality. Basically, you are ready when you have proven a new theorem. As long as you can convince three or four professors that it really is a new theorem, and that your proof is correct, you will get your PhD. The defense consists of you giving a lecture about the topic and about what you've done, and as long as you can do this intelligently and answer a few of the professors' questions, you get your PhD.

In my case, I was ready for my defense much earlier than I realized at the time. I had a new result that would have made a perfectly fine thesis, but it wasn't the result I was after—it wasn't the kind of result I wanted my thesis to be about. But looking back, having seen much more of the mathematical world since then, I realize this was pretty silly. I ended up spending about eight more months working on it until I got a result that I was happier with in the end (although still not really the one I wanted!). It wasn't really wasted effort, but it also wasn't necessary to put myself through the extra struggle.

So you officially become Dan Dugger, PhD. What's next in the process?

This is an important question. The first thing you need to realize is that getting a PhD is just the beginning. Someone once described it to me as a "license to practice mathematics." Getting your license doesn't mean you're ready for the Indy 500, or even rush hour on I-95. In mathematics, recent PhDs do not immediately go on to become full colleagues in a math department somewhere. Usually there is an apprenticeship process called a post-doctoral position that lasts two to three years, and, depending on circumstances, a person might end up doing more than one of these. (I did two of them for a total of five years.) During these years a person needs to establish his or her own research program and prove (by writing several papers) that he or she can do research without having an advisor. Assuming you do this, you can get a tenure-track position in a research department. These are highly competitive jobs, so you go where you are lucky enough to get an offer.

And what interesting research are you working on now?

I'm afraid this would take a very long time to explain. It is the curse of being a mathematician: it is very hard to explain your work to people outside the field, in a short amount of time and space.

“Indeed, your conception of failure might not be too far from the average person’s idea of success, so high have you already flown.”
– JK Rowling, Harvard commencement speech, June 2008

“People who have an easy time of things, who get 800s on their SAT’s, I worry that those people get feedback that everything they’re doing is great. And I think as a result, we are actually setting them up for long-term failure.”
– Dominic Randolph, Headmaster, Riverdale Country School, New York Times Magazine, September 14, 2011

Two weeks ago, the science world was abuzz with talk of a report that neutrinos seemed to have traveled faster than light. News of this finding traveled pretty quickly as well, as media outfits worldwide ran headlines that (gasp!) Einstein may have been wrong. News like this would have been dismissed summarily were it not for the fact that the research team involved in this experiment are not exactly a bunch of weekend armchair physicists. They are part of the OPERA Collaboration working at CERN.

So why would these esteemed scientists put themselves out there to be met by the inevitable wave of skepticism, even ridicule? Because they understand that if they made an error, opening their research to scrutiny in order to find and correct the mistake is exactly what will help them advance their work and the work of others. In fact, the official press release announcing these unexpected observations quotes CERN Research Director Sergio Bertolucci as saying, “When an experiment finds an apparently unbelievable result and can find no artefact [sic] of the measurement to account for it, it’s normal procedure to invite broader scrutiny, and this is exactly what the OPERA collaboration is doing, it’s good scientific practice.”

At IMACS, we couldn’t agree more, and would further assert that the same principle applies in mathematics and computer science. Learning to fail well in these subjects is particularly important because of their exacting and objective nature. Your proof is logically consistent or it isn’t. Your computer program compiles or it doesn’t. There is generally no interpretive latitude around whether you’re right or wrong. There is no arguing that the instructor’s subjective judgment based on his or her personal ideology caused your poor grade. Add to that the tendency of talented students to have a strong aversion to failure and you can see that how resilient one is in the face of failure will be a major factor in determining how much success one has going forward.

While we’re not experts in educational psychology at IMACS, we’ve taught thousands of talented children over the years and are parents ourselves, and what we observe is that learning to fail well is a “scaffolding” process. A talented child who is allowed to have small failures early on without harsh consequences and who is involved meaningfully in determining and executing corrective action shows greater resiliency when faced with the next level of failure. This process builds on itself as the child grows older and the circumstances and consequences become more serious. The failures may grow, but so does the child’s ability and confidence to handle them effectively and independently.

So parental readers, take a look at your mathematically or scientifically talented, award-winning, perfect-scoring children and ask yourself, at this age when the consequences of failure are not so great, are they developing the resiliency that will allow them to take the intellectual risks that are necessary for great success but may also lead to major failure? Do they know from experience, not from having you tell them, that they have it in themselves to bounce back? Ask of yourself, do I allow my child to feel safe about having small failures now so that he or she can rise up from a bigger failure later when I’m not always going to be there to pick up the pieces? If not, then perhaps it is time to lead by example and show that you can make adjustments with an eye toward the long term. After all, if your child is someday going to make the next groundbreaking discovery in physics, he or she should get a head start by learning to fail well now.