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The Snickers Challenge

Some of my fondest memories of middle and high school math were the puzzles. In between memorizing algorithms and vocabulary in class, there was an occasional puzzle thrown in which I invariably found more interesting than whatever I was supposed to be learning on the lesson plan.

This is still true today; an example that comes to mind is last Thursday night (early Friday morning, really), as I lay in bed after preparing my little talk for class on Friday. I couldn't help mulling over some of the sequences of numbers I had prepared for class (this is a recurring theme for me in algebra class; I come in with 5 numbers that correspond to a pattern, and I ask the students to try to deduce the sixth). This particular day, I had the first five numbers of several sequences: the triangle numbers (1,3,6,10,15,...), the perfect squares, the cubes, the primes, and a special sequence called the "highly composite numbers" (look it up!).

I always like to try to think of different patterns the kids might see in these sequences, because there's often more than one way to get to the right answer, and I try to avoid calling any answer wrong, instead just pointing out where (usually) their perceived pattern breaks down. In this case, I knew that the triangle numbers can also be seen as the recursive sequence a_n=a_(n-1)+n, or in simpler terms you add 2, then 3, then 4, etc, running up through the positive integers. Similarly, for the squares you successively add odd integers +3, +5, +7, etc. I wondered if there was a pattern for the cubes? Well, it was 2 am but I still spent a few minutes pondering in that half-awake state trying to see what the pattern was, and I realized that

1^3 + 3 + 4 = 2^3
2^3 + 9 + 10 = 3^3
3^3 + 18 + 19 = 4^3
4^3 + 30 + 31 = 5^3
5^3 + 45 + 46 = 6^3

There's a couple of patterns here -- you could say that the first number added is 3 times the triangle numbers we just saw above, or equivalently that the first of the two sum numbers forms a pattern of +6, +9, +12, ... A deeper pattern is that the difference is 3*n^2+3*n+1, which is mathematically the difference between two successive cubes, but that's kind of cheating. There are actually more, and I think just before I fell asleep I had an even better one, but.. where was I going again? Oh, right.

The point of this tangent is that puzzles are often way more fun than what you're supposed to be learning. Especially if there is some sort of prize or at least some bragging rights for solving them. With that in mind, I created the Snickers Challenge, a collection of math puzzles for the algebra and stats classes.

Each time I go into class, I also try to create a new puzzle for the students. Sometimes, stats and algebra get the same puzzle, other times I give separate puzzles owing to the different skill sets of the classes. I usually try to present the puzzles in class, but since kids often forget the details between classes or just plain aren't paying attention, I thought a website would be a good companion to the in-class mentions, and also perhaps be a way of integrating technology into the classroom.

The website URL is mathisnofun.blogspot.com; this is purposely a tongue-in-cheek domain because I have secret ambitions of using reverse psychology on my students. The first post lays out the ground rules, but basically the first person to present a solution to a challenge gets a Snickers bar (or other candy of their choice; I had several students who were not Snickers fans). I am not above bribing students to think about math with 79 cent candybars -- so sue me.

Note to other TFs and higher-ups: I purposely created a new blogspot username for this other blog, and have not linked it in any way to the TF blogs. This is on purpose, since I would prefer the TF community of blogs to remain as private as such things can be on the internet. Since we often discuss our students, even under aliases, and since I feel it would be difficult to speak frankly about our experiences knowing that our students will read our words, I also ask that no one become a "follower" this blog, as this would also provide a link back to the TF blog community. I may link to the Snickers Challenge from this blog, but only once I make sure that leaves no trace on the receiving end of the link.

Three Goals

This is one of those things I should have published long ago. After talking with George, I've come up with three goals for me during my teaching fellowship:

1) Make sure students are actively involved in at least some small part of every talk. A great example of this was when I had the stats classes all flip coins for our normal distributions discussion. I also recently had students in the algebra class list their ideal careers on the blackboard and vote on which career we'd discuss in detail that day (being sure to fit in math along the way). Weaker examples of class involvement would be just asking students questions and getting them to answer, but I prefer active involvement with some sort of motion or task to be completed.

2) Get feedback from George after each class to see where he thought I was losing attention, needed to speed up / slow down, or if I missed any opportunities to work in class material.

3) Follow along in each class' textbook between visits to look for ways to relate discussions to class concepts. My best example of this was when I had the students flip coins and build bell curves just before the class talked about normal distributions.

Update / Status Overview

Apologies for the very long delay between postings. It has been over a month, which is completely unacceptable! Between running my first set of solo experiments as a grad student, trying to write my first conference paper and taking a long-planned week vacation to DC, I find I really need to catch up on my blog posts.

A brief overview first, before I go into detail on my sessions. I have just about memorized all my students' names, which makes it much more comfortable to start conversations with them, and they in turn have become much more comfortable with asking me for help with their classwork. I have given several talks to both classes by now, and have pulled out some lessons along the way which I'll share in upcoming posts. George and I have reached a good arrangement where when I talk, I take 15-20 minutes or so either at the beginning or end of the hour, with a primary goal of keeping the students engaged and inquisitive in some technical discussion, and a secondary goal of relating the discussion back to current material when possible.

Coming up next, in no particular order: my talks about primes, football & predicting the future, what we can learn from starlight, the snickers challenge, and more.

9/22/08: Lesson plans are tough

Last week I mentioned that George is about one day ahead of his lesson plans in Stats, since he's never taught the class before. While this isn't the most convenient thing for me in trying to plan what I want to talk about, after preparing my talks for today I have a whole new respect for just how hard it can be to make a lesson plan. Especially when you have not one class but two different classes each day. I mentioned in my last post that I wanted to try to do two talks, one for Stats and one for Algebra. Stats is about to do normal distributions (aka the bell curve, with the 68-95-99.7 rule), so I knew I could make a talk out of that, but algebra was actually much harder. How do you come up with something interesting for Algebra I?

Stats: The Normal Distribution

The normal distribution is created when you have a series of random events that add together. A simple example, and one we used in class today, was of flipping a coin ten times and counting how many heads you got. We should expect this to follow a normal distribution from 0 to 10 with a mean of 5. I brought a sack of pennies and had each student flip a coin ten times and call out their answers to me, and then I input them into an array in MATLAB and printed out a histogram really quickly. However, there are only about 15 people in each class, counting myself and George. That makes for a pretty poor normal distribution; it's not symmetric, the peaks aren't clear or are in the wrong places, etc. I asked the students what was wrong, why, for example, no one had gotten 7 heads. Is it impossible to get 7 heads out of 10? They knew that of course it's possible, and realized that if we had more coin-flippers we could fill in the data better. Planning for this, I wrote up a script in MATLAB last night (very simple, like 5 lines) to simulate a user-input number of coin flips over a user-input number of trials. In one stroke, I could show them what the distribution looked like with a hundred, or a thousand, or ten thousand coin flippers. Out popped the bell curve we all know and love, still bumpy at a hundred but nice at a thousand and perfect at ten thousand. The kids were pretty impressed with the rapid graphs, which was gratifying.

Then I showed another type of random event that generates a normal distribution, this time the random walk or the "drunkard's walk," as it was explained to me when I first learned about it. Basically, imagine a drunken sailor staggering along the dock. For every step forward, he is equally likely to stagger left or right. If you try to find how far to the left or right of his original point he is after some set number of steps, it follows a normal distribution. I got a lot of laughs demonstrating the stagger, Captain Jack Sparrow-style, down the center of the classroom. I wrote up a MATLAB script for this, too, simulating any number of sailors in a 50-step random walk and outputting the results to a histogram. Again we observed that with only 50 sailors, the distribution isn't clear. At 500 and 5000 it's better, and at 50000 it's almost perfect. For a wow factor I ran a simulation of 5 million random walkers last night, which took about half an hour on my laptop, and saved the resulting histogram. This paid off when one of the girls in class (I need to get better with their names! A personal copy of the seating charts is a must) asked what it looked like with a million tries.

After we did all this, I showed them the usual image of the bell curve with its 68-95-99.7 breakdown, and I think it helped to recognize the shape and what it meant since we'd just seen so many examples of it with our coin flips and random walks. I talked about things in nature that follow this distribution, like particle velocities in the air. I also talked about things we scale onto a bell curve, like test scores in classes and especially standardized tests, like the ACT. After seeing how random events build up a normal distribution, I got a lot of furrowed brows when I had them think about what that meant for using it as an intelligence distribution—trying to say how smart you are as if there were a series of points in your life where you had a random chance of either gaining a chunk of intelligence or not, like the flip of a coin. I finished by linking the topic to Brownian motion, so I explained what that is and showed a pretty sweet video I found online of a physics demo lab device that shows a bunch of tiny ball bearings hitting a hockey-puck sized disc and moving it erratically back and forth.

Algebra: Fun with x, Dividing by Zero and Ramanujan

About a week and a half ago I saw a play called A Disappearing Number at the Power Center, by a theater company called Complicite. The play was based on G.H. Hardy's A Mathematician's Apology, which was the story of his collaboration with the self-taught Indian savant Srinivasa Ramanujan. The play is heavily scattered with mathematical ideas, but the theater company does a remarkable job making the concepts understandable to a largely non-mathematical audience. I decided to use some of Complicite's tricks and some of my own to make my talk for Algebra.

I started with a riddle: Pick a number, and number. Now add five. Now double the number. Subtract four from what remains. Divide by two, and finally subtract your original number. The answer, assuming you did the math correctly, is always three. It's a verbal trip through an algebraic process, but to an untrained audience it is as good as magic. This brought gasps from the audience in the play, and it got some whoops and cries of "no way!" in the algebra class too. I went through the algebra on the overhead, and as we went I explained that this is the power of algebra. I can try these steps for each number individually, but there's always the question of whether there's some number I missed that doesn't give an answer of three. When I choose x as my number instead, it proves the process for any number I can think of, big or small.

With this example of the power of x in hand, we then went from another direction. I went through an old fallacy on the overhead, using x and dividing by zero under the guise of x to prove something nonsensical, that 1 = 2. No one was able to figure out where we went wrong, so I went back to a story from Ramanujan in school, a story of boys and fish that I picked up from the play. If I have ten boys and ten fish, says the teacher, how many fish does each boy get? One, of course. What about a hundred boys and a hundred fish? It's the same. In fact for any number of boys with the same number of fish, each boy always gets one fish – a number divided by itself is always one! Then the clever Ramanujan asks, "What if I have no boys and no fish? Does each boy still get a fish?" Or, is 0 / 0 = 1? This is a really deep question, and I went through some idea of how we could try plotting the function 1/x to get an idea of what happens with division by zero. Ultimately, I explained, mathematicians punted and called it "undefined." I tried to link this paradox for them with the limitations between math and the real world. I can always take things, like boys and fishes, and use mathematics to describe them, just as I can take those random numbers we picked and substitute in x to prove it for all cases. But the reverse, trying to apply something nonphysical, like zero boys each getting a fish, doesn't make sense, the same way that the things we do with x's don't always come out quite right, like in the case of dividing by x=0 to get 1 = 2.

This lecture was a stretch for a lot of the students. The concepts of infinity are really fuzzy at this stage for them, so thinking about plus and minus infinity is a trippy thing. I think I had a hair-blown moment when I told them to think about how, while there are an infinity of numbers 0,1,2,3,4, etc., and an infinity 0,-1,-2,-3,-4, there is also an infinity of numbers between any other two numbers, like between 2 and 3, or 2.2 and 2.3, or 2.22 and 2.23, and so ad infinitum. This had the side effect of leaving some of them pleasantly subdued when George took over. I lost many during the discussion though. This was perhaps an overly ambitious talk for a freshman class, but they seemed to enjoy it better than going over their homework.

9/15/08: First Day

(Writing from notes taken the first week)

We've settled on me coming in every Monday for hours 1-4, which is about 7 am – 12:30 pm. Mondays work for my schedule, and will never conflict with a test (weekends are anathema to short-term memory). Hopefully it will also give kids something to talk about to refer back to through the week. George (that's Mr. Lancaster to any student types who find this blog) teaches two classes of Statistics, an almost-AP course largely populated by seniors with a smattering of juniors, and two classes of Freshman Algebra, which is what the name suggests. As far as I've been able to deduce the math curriculum at YHS, they start freshman year with Freshman Algebra, then take Algebra II. Either simultaneously with Algebra II or subsequently, they can take Geometry. Most students don't double up like that, so I take that to mean that a typical junior is in Geometry, and a senior takes Pre-calculus or Statistics. I know there is an AP Calculus class, but I'm not sure how students can get to it by their senior year unless it's by doubling up as a sophomore. Moving on…

"Are you a teacher or are you fun?" That's one of the girls in first hour Stats this week, after I tell them who I am and start to tell them a little bit about myself. This is a very good, and very revealing, question. George and I have talked about this, and we have leaned toward me being fun, at least to start out. For now we don't plan to add homework problems around what I present, or anything of that nature. The focus is on me showing them something interesting in the science and engineering fields each week, which may or may not have a direct bearing on the current lesson plan, and trying to do it in under around twenty minutes so George can still go over homework. The state of the current lesson plan is worth noting – this is George's first year teaching Stats, so his lesson plan extends to approximately tomorrow at any given moment. Apparently the teacher's edition of his book makes it difficult to gauge how long different sections are going to take, so for the future I plan to loosely preview upcoming concepts I deem useful and try to work in a narrative around them. More on that further down.

Stats is a bunch of seniors, very engaged, not shy about asking me questions, laughing, joking around with me when I talk to them. Even the kids who aren't great students are still able to pay attention to me when I present. Algebra is a different story altogether. I had to tell kids to quit throwing wads of paper at each other, face forward in your seat, don't sleep in class, etc. In a word, they're freshmen. They were also a lot quieter in terms of responding to me while I talked, although they were a heck of a lot more likely to just keep talking to each other in spite of me. I ended up walking around the class a lot more as I talked, trying to shift the dynamic away from a me-lecturing-at-the-front to a me-right-in-the-middle arrangement, which helped some.

Since this is my first week, I decided to do an extended introduction of myself, about why I'm here at the school teaching a math class, and about what I do every day as a scientist and engineer. I read a blog from Megan DeFauw last year where she noted that students seemed to have trouble distinguishing between the Teaching Fellow position and a student teacher, so I tried to make very clear that distinction. Perhaps I hope I will carry more weight if I am perceived as belonging to "the real world;" certainly that's my recollection from when I was in high school. I talked about ion thrusters, which are my particular area of expertise. An ion thruster can be thought of as what those blue glows on the back of ships in Star Wars are pretending to be. They use electrical power to strip neutral atoms of their electrons and then accelerate the resulting ions at tremendous speeds, ~30-40 km/s in some cases. They are propulsion devices, so I got to tell the kids that I'm really a "rocket scientist." They got a kick out of that.

This week I gave the same talk to both classes, but I haven't decided if that's a good plan for the future yet or not. There's such a wide gulf between the Stats class and the Algebra class that it's no mean feat making a talk stretch the distance. This will be on my mind in the coming week.

Starting off

Introductions first: I’m a third-year graduate student in Applied Physics and Aerospace Engineering. I originally envisioned becoming a Teaching Fellow to help out in a physics class, a plan which persisted right up to the pairing meeting, when I first realized that physics was not an option this year and that I was in the “math” group. Interesting…

I remember not being terribly fond of math in high school or middle school. Algebra, geometry and statistics were all merged into a three-year integrated sequence in our high school (Plymouth Canton School District, quite nearby), and they were pretty easy, though how interesting they were was largely a function of the teacher’s charisma. I got lucky – in a three year sequence I got two good teachers. Precalculus was a dud, both because it was impossible to believe I would ever need most of that stuff (oh, how wrong…), and because the teacher couldn’t control a class so it was mainly a social hour. AP Calculus in my senior year was one of the first times I remember going “whoa” and really getting my hair blown back with something I wasn’t expecting.

Think back to that time; put yourself in those shoes. You’re pretty good with manipulating this ‘x’ character, you’ve got a handle on things like sines and cosines, though they seem to be taught all out of proportion to your opinion of the relative importance of triangles in the world, and if you had to you could muster some proofs of whether two triangles are congruent or merely similar based on a few axioms. Not bad.

Now we’re going to do two fundamentally new and different things. First, we’re going to take the old, and boring, formula slope = rise /run, and we’re going to give it a twist – if we go slowly enough, and keep track of exactly where our delta-x’s are, the same method of rise / run that worked for straight lines will work for curves. Meaning, even though it makes no sense to think of the slope of a curve (wait…curves have different slopes! they change! that’s not allowed!) if we’re careful and walk the line between not thinking about it and thinking too much, the math will work out and give us…something different. Something that we don’t yet fully understand, that doesn’t fit into our intuitive grasp of things. The slope of y=x^2 is 2x, because it’s variable and changes everywhere, but can still be described. This dovetailed nicely with taking algebra based physics my senior year, by the way. You could just tell that the mathematics of continuous change would eventually come in handy in physics where things are always moving. Second, we learned to add up an infinite number of infinitesimal pieces. I don’t think I realized it at the time, but this one is just as crucial as the other one, though there was no similar “eureka” moment to the formal definition of the derivative.

So, here I am, ambivalent about math for math’s sake, but as an engineer and a physicist, I use math all the time for real things. I probably know more math than 99% of the population, but it’s not like it’s my favorite thing to talk about at parties. The math only ever gets interesting when it describes something physical, something real.

What does all this mean? Why start a blog about my position for the next year with a post describing my mixed feelings about it? Because that’s what makes me exactly right for this job, ultimately. These kids are me writ small. For as long as I can bring something real to the table, something to convince them that someone, somewhere who is not teaching or taking freshman algebra or senior statistics still does stuff that will blow their hair back, I may get their attention. So let’s begin.