Math contests - continued

I feel a need to further P.J.'s comments on math contests, because I think he indadvertedly overlooked the most important part of math competition, and that is practice. Math contests are not sports. It matters not if you win or lose, what matters is that you try your best to do as well as you can. for most students, and teams hopefully, this does NOT involve endless drill and practice, nor does it have anything to do with memorizing endless formulas. Granted some teams do this, and they are often "winners" but they have missed the entire point of having a math team.

Math team is a place where students who like math and are interested in math, and may be talented in math, have a chance to gather together and work on interesting problems together. It is a chance to share creative insights and to learn about things that are not usually taught in the classroom. Math team practice is where I can ask the problems that are too hard to ask in class , and, yes, if no one can solve the particular problem, then take it home and work on it. The message I hope most coaches send is that we are here to learn math, to work on interesting problems and to have fun doing it. The contest itself is the carrot, but by no means is it why we spend so much time doing this. Out hope is that some of our mathletes will find that math is not about grades, nor is it about winning. It is about working on interesting problems with others who also think it is fun to work on interesting problems.

One of my favorite stories about math team, one I have told countless times, and one that convinced me that math contests are wonderful things, involves a young student named Eric Winfree. It was clear that Eric was brilliant but definitely not fast. He could solve the problems but often not within the artificial constraints set up by contests. One week he failed to make the Chicago area ARML team and did not qualify for theAIME. I talked to him on Friday of that week, and expected to find him dejected. Instead, he was cheerful, even robust. I conversed with him and he said I got all of the problems this week. I asked him what he meant, and he said that he had correctly solved all of the ARML tryout problems and all of the AMC problems by Thursday night, and three of them were really cool. Let me show you what I learned.

Eric is now a working scientist, mathematician and CalTech. . He attributes math contests as one of the many important parts of his education. I always tried to get our math team students to understand Eric's attitude about math contests. I hope most math team coaches see it this way, as that is why I spent considerable time and effort during my career promoting math contests, and coaching teams that did well but didn't win very often. I count my time spent with this as some of the most important successful time I spent doing anything, because  many students were drawn into the math fold in part because of math contests.

Why compete?

My friend Cathy has written extensively about why math contests suck, and there's a lot to what she says.  Many--I'd even concede "most"--contests encourage high-speed, single-step problem-solving rather than thoughtful analysis, or the kind of synthetic work that leads to new ideas and big theorems.  They also discourage kids who don't have the particular skill- and mind- sets that lead to success on math contests:

• "Pyrotechnic" problem-solving ability.
• Near-perfect recall for theorems and situations that come up often.
• Ability to calculate by hand, flawlessly (sadly, because of the huge inequities and arms races that can result, many math contests--including my own ARML--don't let participants use calculators).
• Willingness to push forward with a single solution rather than considering all options carefully.

The real misfortune is when kids lacking in these mind/skill-sets "learn", not just that they're not good at math contests, but that they're "not good at math," period. 

So is there a benefit to competition?

I would suggest three:

• The opportunity for talented math students to learn that they have a lot to learn.  Even our strongest students wind up doing not-so-well on contests once in a while, and it's a reminder that math isn't always easy--and won't be.
• The opportunity for talented math students to learn that hard work can pay off.  For many strong math students, the experience of math class is that what they need to learn to do well comes naturally--so they don't learn to connect effort with success in the context of mathematics.  Our arch-rival Whitney Young is currently the second-ranked high school math team in the state, not because their kids are smarter than everyone else's, but because they work extremely hard--an hour every day, several hours virtually every Saturday.  (Our team mostly feels like we've reached an optimal point on the effort-reward curve:  we work pretty hard, practicing 3-4 hours per week, and make it in the top ten of the state; our students have more time and freedom to do other activities.  But I digress...)  They know it; their students know it; and what they've learned is more useful than any theorem or formula.
• The opportunity for students to learn to lick their wounds.  My son plays competitive chess, and one of the hardest parts of chess tournaments is their duration: you play four or five games in a day, and if you lose one, you still have two or three ahead of you.  I've seen Jonah learn to dust himself off after a loss and go back in swinging (okay, that's a metaphor).  And I've seen the same thing in math team: students screw up a contest, and instead of saying "We're dumb" or "the questions were dumb," students can be taught to go back, study the contest, and practice hard for the next one.  That's another powerful lesson.

I wish there were more other ways for kids to learn these lessons--by doing authentic mathematics research at an age-appropriate level, for example (New York and Chicago now have math fairs that do for mathematics what science fair do for science), or just by having that scrape-your-knuckles-and-try-again experience in math classes.  But the lessons I cite are important ones, and important ones to learn about math.

Preparing Students for College

Again this week, I heard the justification for an absurd policy (in this case: if you make up a standards-based quiz, you can't get as high a score as you would have if you had passed it the first time) being that it "prepares students for college, where you don't get second chances."  (I've also heard the same thing said about the real world.)  This may be the stupidest justification for educational malpractice I've heard, for two reasons.

1. In college, and even more, in the real world, people are mostly reasonable.  How many times have you handed in some paperwork late for your job, or forgotten some important thing, and had people essentially say "It's okay, don't do it again" -- or even (gasp) not mention it at all?  In college, I memorably missed a make-up test for my German class because (soooo embarrassing) I mistranslated my teacher's instructions about time (given in German) and showed up an hour late.  I didn't fail, or get a zero, or even points off.  He laughed, said he'd wondered where I was, and then gave me the test anyway.
   
   The world itself is not reasonable; it obeys the laws of physics, which are notoriously amoral.  So the water in the pipe to my outside spigot really will freeze if I forget to drain it before a cold snap.  But if I forget to renew my license plate sticker, I pay a fine and get on with my life.  Sure, there are exceptions, and we love to spread those stories around--they're like fishing stories, only in reverse.  In real life, people get second chances: they're accepted back into colleges, or even elected vice-president (for two terms) after being caught red-handed plagiarizing.
   
   In fact, I've noticed that (at least for me) the circles I inhabit have gotten more reasonable:  while I have many notable frailties, I've learned workarounds over time (email myself any important information, have a phone that gets email, etc.), and I'm good at enough things that the people around me are willing to put up with the things I'm not good at.  I think that's true of most professional people:  we work our way into niches where we get to spend most of our time doing things that we're either pretty good or trying to get better at, and only a small fraction on things that we truly dislike and are abysmal.  Real life is not high school.  To paraphrase Dan Savage, "it gets better."
   
2. EVEN IF college were the one-strike-you're-out system these teachers say it is, it seems obvious to me that the number one way to prepare kids for college is to actually teach them the academic skills and habits they will need to be successful there.  Sure, it's important to get things in on time, and to do well on quizzes the first time you take them.  But "on time" won't save a literary analysis paper that's an incoherent mash of plot summary and personal reflection: you have to be able to read and write, too.  And it's hard to learn calculus without a solid grounding in functions, graphs, and algebra.  So high schools should adopt policies that encourage students to go on developing those core skills, and recovering from their mistakes, rather than telling them that mistakes are insurmountable.  And isn't that exactly what "no late work" or "no retake" policies say?

The point of draconian policies in high school is to discourage kids from making mistakes.  But everyone makes mistakes; the point is to learn from them.  So I'm not saying there should be no penalties, ever, for late work, or for screwing up an assignment the first time (I rather like "you have to show me you can do it right.").  But those penalties should give kids incentives to learn, not teach them the mostly-false lesson that you can't recover from your mistakes.  Because in the real world, people make mistakes all the time, and learn from them.  Wouldn't teaching kids how to do that be the best preparation of all?

At last, an iPad game that engages mathematics in an interesting way!

OK, so that title is a little overstated.  After all, I (and my kids) love WolframAlpha, and especially the WolframAlpha fractals app--basically an interactive dictionary of fractals, but oh-so-cool!

No, what I'm talking about is about the world of "math games" that really just consist of drills of basic facts, sussed up so that adding numbers blasts an alien or something.  Math games, in other words, that practice very low-level skills and that don't work to develop problem-solving, thoughtful application of skills, multiple pathways, etc.... in other words, math games that turn math into a kind of spelling practice.

And then there's Dragon Box.

In Dragon Box, you drag cards around a two-sided playing field (looks like a tennis court, behaves like an _ _ _ _ _ _ _ ) to isolate the "dragon box", which is represented by a picture of a sparkling crate [screenshot from the PC version, because it's easier for me to use with Blogger]:

In the picture above, I've just added a purple fly to the left side to cancel out the black fly -- totally allowed, although unorthodox -- and the computer is prompting me to deal out another purple fly on the right side.  By merging the flies together, I can make a "galaxy" (zero) and isolate the box!

As you go through the levels, two things happen.  First, the tasks get more complicated: you start having to flip the cards to their opposites before adding them (in this example, I was given the opposite card); the cards at the bottom include ones you don't need to use; the game adds representations of multiplication and division.  (Yes, as with addition, the program forces you to distribute correctly, at least in the first 60 or so levels that I've worked.) Second, the program spirals in and out of representations that look much more like traditional algebra:

Notice the c and -c cards?

And this brings me to a big-picture musing.  My son, in third grade (and not yet doing official Algebra) loves this game.  He's learning a set of rules, without worrying too much about the conceptual underpinnings.  In recent decades, progressive teachers have moved away from this model: everything should be explained when it's introduced, so that it feels natural.  And yet, I'm not so sure.  If we wait until after Jonah has played 200+ levels of this game and does the right moves every time to explain the underlying algebra--what have we lost?

My experience in learning mathematics was that often self-contained systems only made sense later.  And I don't think that's a terrible thing.  The sum and difference identities for trig functions?  The first important thing for 99% of students is just to know and be able to apply those identities--and whom do we serve if we spend the first twenty minutes of the period in a lengthy derivation that leaves students angry and confused?  And anyway, some of the proofs I've seen -- using matrix multiplication and rotation formulas, for example, as my own UCSMP book does -- aren't necessarily that convincing.

Of course, I know that it's possible to motivate these formulas in lots of ways.  My own class on sin(A + B), for example, started with the example of sin(A + π/2) ≠ sin A + sin π/2 because of all the cool ways we could see that the "identity" isn't true; then, graphing on the calculator, we came up with a better answer that we could justify using the unit circle.  But then I went ahead and taught the "right" rules, and we spent the rest of the period learning how to apply them.  The following day was when we went back and came up with some proofs (using geometry, actually).  

There are lots of questions, besides "why is it true?", that engage a student in understanding a formula:

• What happens when we swap the variables around or change them in some other way?
• What versions of this formula correspond to cases we already know?  Does this formula yield the results we've come to expect?
• What about extreme cases?

For example, when I teach Hero's formula in Geometry--which, despite the fact that the proof is beautiful, I do without proof, we discuss the following:

• Does changing the order of the sides change how the formula works?  The area of the triangle?
• What if a = b = c, so that the triangle is equilateral?
• What happens if a + b = c, so that the triangle is degenerate?
• Why are there four terms under the square root sign?  What are the units of the variables and of the answer?

For the vast majority of my honors geometry students, these questions provide a lot of thinking--a lot more food for thought than the kneejerk "Why is it true?"

Obviously, we do a terrible disservice to kids when we say "Math is a collection of rules; your job is just to learn them."  The thing most of my strongest students like about math is that there's so little to memorize. But I think it's okay to say "Here, we're going to learn to play with this set of rules for now, and then we'll figure out why they work or what they apply to."  At least once in a while.

Agree?  Disagree?  Let me know.  But if you have kids -- and even if you don't -- get DragonBox.  What a blast!

Homework Problems

Earlier this week, a friend told me about an awesome problem her daughter (third grade) had for math homework.  Loosely paraphrased, it runs like this:

Annabel and her friends share a pan of brownies.  Everyone gets the same number of brownies, except that Annabel gets two brownies less than each of her friends.  If they have 33 brownies, how many friends does she have?

There's so much in this problem:  opportunities to try possible solutions, develop conjectures, figure out possible intermediate stages, practice multiplication and factoring, and think about functions.  There's just one thing: it's not a homework problem.

Why not?  First of all, while it may use skills students have already developed, the context is new.  It's an extension--in several different directions simultaneously--of ideas about products and factoring.  And there's no obvious place to start working the problem, because there are two unknowns--that is, both the number of brownies in each share and the number of friends--and givens from which it's hard to work backwards.

What is homework good for?  In Classroom Instruction that Works, Marzano points out that homework can help achieve three goals:

• Improve learning of specific skills through practice.
• Develop of self-discipline and independence.
• Improve metacognition, in this case, awareness of what the student does or doesn't know.

But these three goals are only served if the task is something on which students can reasonably expected to make progress working independently.  My friend tells me that, in her child's class, every child got help from parents, who were often at least as flummoxed as the child.  In the classroom, I would use this problem to teach problem-solving strategies:  using systematic trial-and-error to develop observations, conjectures, and theorems; solving the given problem; and then developing generalizations of the problem.  But it's not reasonable to expect any of that to happen at home. And so we achieve the opposite of our original goals:

• Because there's little or no repetition on the child's work (which would happen if the child could work on multiple cases), there's little opportunity to practice specific skills.
• Kids learn to rely on their parents to interpret and answer questions on homework.
• Kids don't get a sense of what they should reasonably be able to do on their own; they do learn that complex tasks are impossible for kids to do.

I'm not advocating a return to long worksheets containing dozens of instances of essentially the same exercise--those only kill the joy of mathematics, and teach kids to stop thinking.  In the best-case scenario, such sheets force kids who are pretty good at a skill to practice it so much that it loses all interest.  Of course, if the kids aren't competent at the skill, they just practice it wrong--or learn that there's yet another thing they can't do.  So those old worksheets aren't the answer.  But problem-solving is a collection of skills and habits that have to be developed--before students are asked to produce them on homework.

The key aspect of homework--the one this assignment overlooks--is that it's something kids should be able to do and learn from at home. What can be done at home?  There's no single answer that works for every subject and every class.  But a first cut might be to ask the following questions:

• What skills are required to be successful on the homework?  Have those been taught in class?  What evidence is there that kids can apply them successfully?
• What habits does the homework build on?  What evidence is there that students in the class already have those habits?
• If a student is stuck or discovers he/she is mistaken, what resources would be necessary to unstick the student or work through the misconception?  Are those resources available at home?
• How can students monitor their own success or failure through the assignment?