A first post on geometry and proof

Euclidean geometry is for many students the first time they get a taste of what math is really about.  The problems don’t all fit the same pattern; it’s natural and expected that students will come up with their own arguments to prove something, rather than following a set of rules.  Ideally, geometry class also involves experimentation and conjecture.

I don’t think Park students are encountering these things for the first time in geometry.  Our students are used to investigating and asking their own questions.  And they are used to making careful arguments to support their claims.  Still, for Park students as much as students anywhere, geometry tends to be the first time that they are asked to write formal proofs.  Anyone who has taught geometry knows that writing proofs can feel to students like wearing a straightjacket.  For the first time, arguments that are correct but either vague or not axiomatic are inadmissible:

• Opposite sides of a parallelogram have to be congruent because lines with the same slope stay the same distance apart.
• Opposite sides of a parallelogram have to be congruent because there is no way to extend one of those sides without changing the angle of the side coming to meet it.
• The base angles of an isosceles triangle have to be congruent because the triangle is symmetric.

I’ve stopped telling students that these arguments are not convincing.  Anyone who understands the terms they’re using would be convinced.  And I’ve even stopped telling students that they are incorrect.  They’re not incorrect; they’re just not arguments from first principles.  They appeal to intuition and common sense, as most arguments we’d make in daily life do.

Acknowledging those things, we still need to make rigorous arguments that appeal to specific principles we’ve studied in class, such as theorems about parallel lines, and theorems about congruent triangles.  For this reason, I stick to the “statement/reason” model of proofs taught in most geometry classes.  I find that if students don’t write proofs this way it is too easy for them to fall into arguments that are merely intuitive.  It’s also easy for them to fool themselves into thinking that they have enough evidence to conclude that triangles are congruent when, say, they’ve really only found two pairs of congruent sides.

I don’t, however, insist on the degree of rigor that most geometry books do.  Students in my classes do not write proofs that contain the sequence, “If angles form a linear pair, then they are supplementary.  If angles are supplementary, then their measures add up to 180 degrees.”  They can go right from linear pair to adding up to 180 degrees.  I don’t think that this level of following tiny steps in a chain serves the purpose of helping students to build new theorems out of the knowledge they already have.

Generally, my rule is that if students are using congruent triangles to prove something, they need to

• Name the three pairs of sides/angles that they need to justify the congruence, providing a reason for each.
• Name the pair of congruent triangles and say which theorem (SSS, ASA, etc) they are using.
• Only after they’ve done all that, name the pair of sides or angles that they can now say are congruent.  To justify this, they will sometimes use the infamous “CPCTC,” or, since many students have trouble remembering what the acronym stands for, just say that they are using triangle congruence.

I think that, among Park faculty, I am one of the teachers who insists the most on some kind of standard template for proofs, even though I allow much more leeway in what can be used for a reason than most textbooks do.  I’d be interested in what other teachers ask of their students when writing geometry proofs.

But Who’s Counting? (Or: It’s Friday, Friday)

There are times when you can tell your class is a little fried and they need some short-term gratification rather than the usual drawn-out discussions.  Last period Friday, in particular, is a time when it is useful to have some math games ready to go.  So on this Friday I’ll describe a game that I stole from the wonderful Public Television show Square One.  (Remember that?)

As the “host”, I set up the digits of an arithmetic problem like _ _ _ + _ _.  (Actually, I make each digit a box rather than a dash, but that’s much easier to do on a whiteboard.)   Students copy this into their notebooks.  Then I use a calculator to randomly generate digits ranging from 0-9, repeats allowed.  As I read each digit, students place the number in one of the boxes, with the goal of making the largest sum possible.  The catch is that they are not allowed to move a digit once they place it.  The winner, at the end, is the student with the largest sum.  (You could also ask for the smallest sum, of course.)

Of course, this game is too easy for high school.  The strategy is clear to the students, and after that it’s all luck.  So after we play with the initial setup, to get the idea of the game, I begin to vary the game board, tailoring it to the level of the class.  _^_ + _ _ _ _ is an interesting one – is it best to use a nine, if it is chosen, as the base of the exponent, the exponent itself, or the first digit of the four-digit number?  Using fractions reinforces the concept that the bigger the denominator, the smaller the value of the fraction.  You can also involve “sin(_ _)” in your expression (specifying that you are using degrees), and students will need to think about what degree value will maximize the sine function.  For students thinking about limits, you can compare ratios of different types of functions so that students need to think about the shapes of their graphs and their end behavior.

I like this game for several reasons.  First of all, there’s an element of suspense when choosing the final few digits that always puts students on the edge of their seats.  Almost all students like this game and find it engaging.  Secondly, when the game is over (and they are allowed to talk about it), students want to argue about what the best possible result would be.  Because of this, they make all the mathematical arguments appropriate to the game board without my having to prompt them.  Third, it is extremely malleable to the needs of the class, and it can be anywhere from very easy to very hard.

One final note: I’ve been able to find only one web clip of this game, here.  They play a simple version in which the object is to make the largest five-digit number possible.  Watching it (if you can make it past the super-silly intro) might give you an idea of the principle of the game.

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?