The Mathematics of Fountain Design

Park teacher Marshall Gordon has an upcoming article in Teaching Mathematics and its Applications: An International Journal of the Institute of Mathematics and its Applications.  The article details the experience he had teaching a project-based unit on the mathematics of fountain design.  In the process of designing their fountains, students were naturally motivated to explore the different parameters affecting the trajectory of a parabola.

Click the link to read the article: The Mathematics of Fountain Design

Not My Best Moment

This happens more often than I’d like: we start by having a normal class discussion and end up trying to resolve something that doesn’t engage the students’ interest, or even their understanding.  It’s the exact opposite of the way I’d like my classroom to feel.

Last week in 9^th grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

Getting it right this time?

Today is the last day I can treat myself to the luxury of sitting in a coffeeshop on a weekday morning/afternoon.  I came here to think about what I wanted to do in the early days of my 9^th grade class.  In practice, this has translated into my spending most of the time solving and thinking about the “Tinker” problems.  This has worked remarkably well to help me set priorities.

I wanted to teach 9^th grade this year because I realized that I was not doing nearly as much as I could be to teach students how to be learners.  Assigning students nonroutine problems has its drawbacks: though we have great class discussions and kids learn to see math for the open book that it is, students also have the perfect excuse to say, “I just didn’t know how to do this homework problem”, and teachers have the perfect excuse to give large hints that don’t empower kids to feel that they could have solved the problems themselves. 9^th grade seems like a good opportunity to focus on changing some of my practices.  We’re all making a fresh start.

Here are the things I most want to work on with my students this year, things that we need to establish in the earliest days.

Postgame Analysis: the Towers of Hanoi

I recently gave my juniors the classic Towers of Hanoi puzzle to play with in small groups.  It went something like this:

You have three plates, and plate #1 has a stack of 5 pancakes, in order from the largest one on the bottom to the smallest on top.  The puzzle is to get the stack onto plate #2 using as few moves as possible.

Two rules: (i) you can only move the top pancake on a stack, and (ii) at no time can any larger pancake be on top of a smaller pancake.

They spent a couple minutes getting familiar with the mechanics of it, and then settled into working together, shifting pancakes and keeping a count of their moves.

The Problem That Never Fails

[This post also appears as a guest post over at Sam Shah’s blog.]

This is a post about a problem that never fails.  It’s the problem I used for my sample lesson when interviewing for jobs four years ago.  It’s the one I almost always use on the first day of class, and it’s also what I give to parents on back to school night.  Because… well…it. never. fails.  Seriously.

The perfect, ineffable jewel of a problem to which I refer is the classic Bridges of Konigsberg problem.   Here’s the story, in case you don’t know it:

(image from wikipedia) 

As shown in the image above, the town of Konigsberg once had seven bridges.  Back before some of these bridges were bombed during WWII, the residents of the town had a long-standing challenge: to walk through the town in such a way that you crossed each bridge exactly once—i.e., without missing any bridges, and without crossing any of them twice.

So, why is this problem so great for a high school classroom?  Well, first of all, whenever I tell this rather contrived tale to my students (or their parents, for that matter), they are inevitably scribbling on their scrap paper before I can even finish.  It’s a compelling puzzle, simple as that.

Before long, students have redrawn the thing enough times that they’re annoyed with all the extra time it takes to draw all the landmasses and bridges, and so they simplify it:

(image from wikipedia)

Voila: in a completely natural fashion, they have reduced the problem just like Euler did.  At this point, I usually bring them together for a moment to appreciate what’s going on here: reducing a problem to its essential components, finding the simplest way to represent the underlying structure of the situation.  (And I also mention to them that this is precisely the move that Euler made when he invented graph theory based on this initial problem.)  Even if they never went any further, this is already a nice lesson in problem solving.

[SPOILER ALERT: notes on the solution below the fold]

Learning to Read

Yes, this is math class, but especially in a problems-based curriculum we are teaching reading at the same time, right?

I recently gave two classes the following problem*: “How many triangles are there whose three vertices are points on this 3×3 square grid?”

Once they put their heads together, kids are able to make nice progress on this problem.  It’s intended to teach the habit of “taking things apart” (most kids tend to say, “break it down”), and comes with a suggestion to count the number of triangles you can make within a 2×2 grid, then a 3×2 grid, then the full 3×3 grid.  Not all kids take this suggestion (and good for them), but most wind up categorizing triangles by type somehow and then counting how many there are of each type of triangle.

Looking back, I think about the little things that tripped up the kids, all of which had to do with implicit assumptions about what the problem was asking.  Some kids wanted to know if the vertices of the triangles had to be on the dots in the grid, or if they could be in between.  Other kids assumed that the triangles had to be right triangles, or else asked if it was okay to use non-right triangles; these kids thought that there was real ambiguity in the question.  I want too much to be helpful, and for the students to be able to get on with their math, so in this case I answered their questions directly rather than doing what (I suppose) I should do: telling them to read the question again.  At times I have done a better job guiding students back toward the question.  I resolve to be so good again!

Other questions that the kids had about the wording of the problem struck me as more legitimate.  One question that came up was, “if two triangles are the same shape, should we count them as different triangles?”  Here I think the kids have a case that the question is ambiguous.  After all, we are just beginning to study graph theory, in which we’re about to call several pairs of isomorphic graphs that look nothing alike “the same.”  I’ve encountered many problems over the course of teaching that, while not ambiguous to someone “in the know,” can actually be read a few different ways by an attentive student.  I suppose that teaching students the conventions  of mathematical language is also “teaching them to read.”

*this problem appears in our textbook, inspired by a problem on a Math Counts competition.

A critical mass problem

One of the things I like best about the way we teach math at Park is that the problems themselves serve as intrinsic motivation. Sure, not every kid is perfect about doing their homework or working as hard possible, but we’re far away from the situation where it’s the impending test that motivates a kid to do their work. Most kids are interested in the conversation that happens in class and almost can’t help but give thought to the problems before them.

Every now and then I have a class that thinks that the material is too easy, despite my feeling that most students in the class are not giving the material the thought it deserves, and sometimes even despite the fact that I know there are basic skills that most students have not mastered yet. This could happen in a geometry class, where it’s easy to trick yourself into thinking that an informal argument appealing to symmetry, say, is sufficient, when actually a proof is needed. Or, if the topic is algebra, a “which of the two quantities is bigger” question: to which savvy students often know that the answer is almost always, “they’re the same size,” even if they can’t provide the algebraic justification.

Often, it’s very smart students who have this view – they’re able to intuit their way to an answer for some problems without needing to go through the thought process that the person who wrote the problem intended. It’s great if they can do that, of course, but they may be missing a chance to generalize their method to future problems. That is, they may be missing the core content of the class. Even more importantly, in their eagerness to get the problem done, they’re robbing themselves of the opportunity to be a mathematician. If a problem seemed dumb… what do you suppose you were supposed to get out of the problem? What is its larger significance?

In these situations, if I give a test that I feel is reasonable given my expectations of the students, they don’t do very well.

Because we only give tests once a month or so, it takes too long to give students the feedback that they don’t understand everything they think they understand. Part of me has the impulse, then, to give them quizzes to hammer home the point. But this is not really what I want to do. For one thing, I don’t want students in my classes to feel that they constantly have to be completely on top of the skills and content in the course. Too often, we are in discovery mode, where we are debating the appropriateness of the very skills I’d be quizzing on. It takes time for the dust to settle. And perhaps more importantly… is a test or a quiz the only way to give feedback to a student about how they’re doing? Shouldn’t there be a way to give that feedback more naturally? In most of my classes, when the majority of the students understand the spirit of the class and the exploration, students will let each other know if their arguments are too vague. In the type of class I’m describing, where there isn’t this critical mass, it’s harder.

The way I have dealt with this issue in the past is to collect homework more often, either for a small grade or just for written feedback. Still, I’d like a way to send a message to these students that even the easiest problem contains a world of follow-up questions, generalizations, and connections to other topics. A message other than “teacher says,” of course.

What does a good discussion look like?

One of my favorite moments in the classroom is when students are thinking about some really interesting problem… perhaps they’ve even posed an extension of a problem in their textbook… and they are excitedly discussing it.  They build on one another’s ideas, they inevitably argue, there is a back-and-forth that continues until they’ve really gotten somewhere.  Occasionally I will step in to resolve a dispute or get the students to think more carefully about some misconception they’ve been running with, but for the most part it is the material itself that drives the discussion.

There is a tension, though, between letting the discussion flow naturally and between creating a balance of voices heard in the classroom.  When things get exciting, it is much harder, and perhaps not even the right thing, to let the students speak in turn.  Because there is often one person who has had the crucial idea, the other students’ comments tend to be directed at that person, who may then be speaking every other comment.  Because the discussion is heated and the people who’ve just spoken want to respond right away, there is also less “space” in the discussion for people who are not as in the thick of it to jump in.  I worry in these cases about quieter students, students who take a bit longer than others to formulate their ideas, more tentative students, and students who’ve simply missed some of the framing of the discussion and aren’t quite sure what we’re talking about.

Here are two strategies I’ve sometimes used to make these conversations more friendly to every student.  1) Go to a strict hand-raising system, in which the two or three most ardent students have to wait to bring their ideas forward while we hear from other people who have more tentative and perhaps less-formed opinions.  2) Go to group work for five minutes and let each group the chance to discuss the material, then report back, at which point multiple groups might have definitively solved the problem, or, if not, at least we can begin the discussion again with more students “on the same page.”

While I do sometimes use those strategies, #1 especially feels strange, as if I’m killing the momentum of the discussion.  At a private school we have the luxury of small classes, but there is still something that seems artificial about having a discussion with more than, say, three people at once.  What does it look like to have an open-ended discussion in which most students are involved, that at the same time builds on an idea and approaches a conclusion?

How long should I let kids struggle with a problem?

I’ve been thinking a lot about that question the last few weeks. I teach math to three different “ability groupings” of kids, and yet in many ways they have similar sets of reactions in grappling with a difficult problem.

When they first hear the problem, there generally is interest and excitement, especially if I have chosen a good problem. (By the way, by “problem”, I don’t mean a routine exercise, but rather something that requires them to think, to use what they have previously learned but in a different way.) Students generally start talking among themselves about what they might do, or they raise their hands to ask clarifying questions and to propose a plan of attack. At this point, things have a really good feel– students are engaged, and they are anticipating solving a thorny problem if they just make a sincere effort. And indeed, that is often how things go– students spend a few minutes trying an approach or two, make a mini-breakthrough, and then solve the problem.

But often, after 5-10 minutes of thinking, students find themselves at an apparent dead end.