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2010 August 6

Awesome Math Camp

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On Tuesday 3 August 2010, I had an opportunity to observe the Awesome Math Camp at UCSC.

The camp is for high school and middle school students (grades 6–11) who are very good at math and looking for some real challenge.  It is a 3-week residential camp, with about 6 hours of math a day.

The students I saw there were mostly male and mostly Asian (more East Asian than South Asian, I think). I think that this is driven mainly by which parents are willing to pay a fairly substantial summer camp fee for just mathematics.  Many of the students there are hoping to improve their performance on various math contests (such as AMC-10 and AMC-12) or improve their chances of getting onto the US Math Olympiad team.

The snippets of classes I observed were not contest math prep, though, but a variety of different topics in math (strong induction, graph theory, game theory, … ).  They were preparing students more for being math majors in college than for contests, though any sort of math  problem-solving practice can help with contests.  Personally, I think the broader approach they take is better for the kids and more fun.

Here are examples of  the problems the kids were working on while I was visiting that afternoon (less than 1/30th of what the kids were getting).

  1. S_k(n) = \sum_{i=0}^{n} i^k.  The professor was showing the students how to prove that S_k(n) was a polynomial in $n$ with rational coefficients and that its leading term was n^{k+1} /(k+1).  Although the math is beautiful, I was not greatly impressed with the presentation.  It reminded me too much of my grad student days in math, when a professor would step through a proof without motivating any of the steps or letting students know where he was headed.  I saw the students in the class with their heads down furiously scribbling notes, in the hope that they’d be able to reconstruct the argument and make sense of it when the professor was finished.
  2. Another problem was a game  in which there are 5 boxes  initially with 0,10,20,30,40 pieces.  On their turn each player can choose one box and move any number of pieces from that  box one position left (the leftmost box being a black hole that swallows all pieces moved into it). Whoever moves the last piece wins.  The task here is to figure out whether the first player or the second player has a forced win, and what the winning strategy is.  The kids were making some progress on figuring it out, but the professor was running out of time and ended up giving the answer.  The kids were clearly engaged in the lesson, but didn’t quite have the tools or the time needed to guess the solution.  They had no trouble understanding why the solution was correct, once the trick was shown to them.  After the class, the professor told me that the class was usually working on more advanced topics, but that the game exercise had been done to give them a little break and catch their attention again.
  3. Another class involved attempting to prove a theorem about bipartite graphs: if every subset of k nodes on the left is connected to at least k nodes on the right, then there exists a pairing in which each node on the left is paired with a different node on the right.  In this class the students were very actively involved in trying to find the solution.  Even the background chitchat in the room seemed to be about the proof.  Whenever someone had an idea, the professor would invite them up to the board to show their idea, then the professor would try to find counterexamples or other gentle ways to point out the flaws, without either stifling the kids’ desire to participate or tipping his hand too much for the desired solution.  This was masterful teaching, and the kids in this class seemed almost hyper with excitement about the math.  I stopped observing that classroom before they got to a solution, so I don’t know if he coaxed a correct proof out of them, nor how much hinting he had to use.
  4. The lowest level class I saw was one in which the kids were presenting solutions to problems that had worked on earlier.  One kid was presenting his (correct) solution to the problem of measuring 5 liters of water given only a 7-liter jug, an 11-liter jug, a water tap, and a drain.  A few of the other kids were paying attention, but several seemed disconnected:  I couldn’t tell from my few minutes there whether they were bored because they already knew all this, or bored because they didn’t understand most of what was going on: the two are often hard to tell apart.  As I was leaving, the teacher was segueing into a general discussion of what integer amounts you could measure with two integer-sized jugs, probably leading to a proof based on properties of gcd.  The students seemed to be more alert during this lecture portion, so perhaps they just didn’t like “call a student to the front to present his solution” segment of the class.

Overall, I was impressed by the level of the math the middle school and high school students were doing and by some of the math teaching. Although the classes were fairly large, the kids all seemed to want to be there.  Of course, this was the first week of the camp—maybe some of them burn out after 3 weeks of intensive math.

I would recommend the camp to my son, who would have enjoyed 2 or 3 of the classes I observed. I doubt that I can get him to go, though, as the camp conflicts with the teen theater conservatory with the actors from Shakespeare Santa Cruz, and he’d much rather do the theater camp than the math camp.  Frankly, I think that is a wise choice, as he can learn a lot of the math on his own, but theater is inherently a group activity, and the teen conservatory one of the best theater experiences he can have at his current level.

I would also recommend the math camp (in a year or two) for the 6th grader who did well on the math team I coached last year.  I would not recommend it for students who were not at least 2–3 years ahead in math, though, as the pace was fairly fast.  Students at the camp do have to choose their classes carefully,though, as the teaching styles varied enormously, and a mismatch between the student’s learning style and the professor’s presentations could result in a pretty miserable experience.

2010 July 29

Solving an infinite radical chain

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Pat’s Blog had a post about infinite radicals, which looked at 3=\sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}.  The problem is just as easy to solve for x if you replace the 3 with y.

y =\sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}

y^2 = x + \sqrt{x+\sqrt{x + \sqrt{x +\cdots}}}

y^2 = x + y

x = y^2 - y

Note that this technique is almost identical to how one solves an infinite geometric series, so would make a good challenge problem right after introducing geometric series.  This could be appropriate for a math team recreational puzzle.

One can obviously generalize to n^{th} roots:

y =\sqrt[n]{x+\sqrt[n]{x + \sqrt[n]{x +\cdots}}}

x = y^n - y

And of course one would have to discuss when the formal manipulation makes sense.  For example, y=1 leads to the “solution” x=0, which is clearly wrong.  It might be worth some calculator time for the kids to play with different values of x to see when the infinite radical makes sense. The value of y for x=1 should be a familiar number.

One could also solve the quadratic to see what values y takes on for different values of x, and look for a relationship between the parts of the quadratic formula and the solutions to the infinite radical chain.

2010 July 19

Math team

Filed under: Uncategorized — gasstationwithoutpumps @ 20:19
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Last year I wanted to do some volunteer work at my son’s (private) middle school. The school actually is both a middle school and a high school (grades 6-12), which has advantages for academically advanced middle-school students, as they can take a few high-school courses mixed in with their middle-school classes.

The Parent Club did not provide the sort of connection with the school that I was looking for—about all they did was arrange parties for students, faculty, or the “school community”. Other than for sports and theater, parents were pretty actively discouraged from being involved in the school, despite vague statements by the head of school about wanting more parent participation. Since I have absolutely no interest in party planning or sports, I had to find I different way to assist the school. (My son is very into theater, but he did not audition for any of the school plays, because of the homework load.)

What I ended up doing was volunteering to coach a math team once a week. The school did not have a math team, but each year a few middle students have participated in the county math competition. I got a lot of my ideas on what to do from the Math Circles website, particularly their “Math Circle in a Box”.  We ended up meeting once a week at lunch time (which is when most of the school clubs meet) in one of the math teacher’s classrooms.

For the first couple of meetings I did recreational math (mainly from Martin Gardner’s books, but also from Professor Stewart’s Cabinet of Mathematical Curiosities). After that I switched to alternating contest prep and recreational math.

There were 3 contests I was preparing kids for: the AMC-8, AMC-10, and the county math competition.  The AMC-8 and county math competitions are only for middle-school students, but the AMC-10 goes up to 10th grade.  Although I started the year with one 10th grader on the team, he left early in the year, so I ended up with only 5 middle schoolers involved, and some of them came only sporadically.

I rejected the idea of methodically working through the Art of Problem Solving books, and did a more eclectic mix of problems, starting with old AMC-8 tests (the first contest), then old AMC-10 tests (the second contest), finally using the Art of Problem Solving books to prepare for the county contest.

Math Coach: A Parent’s Guide to Helping Children Succeed in Math, by Wayne A. Wickelgren and Ingrid Wickelgren was recommended to me after I’d finished the year of coaching, but I’ve not looked at it.

The kids seemed to have as much fun with the contest prep as with the recreational math, so if I were to do it again, I’d do more like 2 contest prep sessions for each recreational math session.  I’d probably also try to be a bit more methodical in using the AoPS books, as they are really quite well designed, with good puzzles as exercises (many taken from various math contests).

As it turned out, all five kids who participated were pretty good at contest math, and 2 were exceptional: a 6th grader and an 8th grader. On the AMC 8,we had an Honor Roll (top 5%) and Achievement Roll (top 40%, but for a 6th grader).  On the AMC-10 we had a Young Student Certificate of Achievement (score of 90+). The team ended up winning the county 7/8th grade team contest, and the 6th grader and the 8th grader got the individual firsts in their grades.  The individuals could probably have done that without my coaching, but I like to believe that I contributed to getting the 5 kids to work together to do well on the team contest.

My son is going to a different school next year (a public high school), and I’m considering coaching a math team there.  They have not entered the AMC-10 in recent years.

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