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.

Puzzles for 11.11.11

Some questions we are asking our classes today:

Anand: How many times this millenium will the date consist of a single digit?

Bill: Today, the day, month, and year are the same. In how many days will this happen again?

Angela: How can you get an answer of 0.0909090909… using only one number, but as many times as you want, and basic arithmetic?

What are you all doing?

The Cruel Irony of Algebra

You would think it would have occurred to me sooner, but it wasn’t until a few years ago that it really hit me what I think is the biggest problem most students have with Algebra:  they don’t actually think the letters represent numbers.   Here’s the kind of question I’ve asked that illustrates what I mean:

Chuck says that (xy)(wz) is always equal to (xyw)(xyz).  Is he right?

Over the years, I have found that if a student is unsure of whether or not they are supposed to “distribute the xy”, they often just guess.  When asked why, they say that they were unsure of the rule they were supposed to use, so they just took their best shot.  For many years, I tried to show them which rule to use in various situations and the principles involved, hoping that over time they would catch on to the logic of algebra.

But invariably, for many of my students, even a slightly changed question presented what seemed like a freshly baffling challenge.  After all, does the question below really seem all that different (other than to a math teacher)?:

Chloe says that (xy)(w+z)  is always equal to (xyw) + (xyz).  Is she right?

So what to do?  While it may seem like taking two steps backwards in the march towards abstraction and generalization, these days I ask my students how they could possibly figure out for themselves if the two expressions are always equal, and the ensuing discussions leads us to the question of what the heck those x’s and y’s and z’s and w’s represent—numbers!  So why not try out these equations with numbers?  The cruel irony of algebra is that what is intended to make generalization easier actually becomes so abstract for many kids that numbers are the last thing on their minds. They end up seeing algebra as a bunch of arbitrary rules that are hard to predict.

Of course, just because two expressions are equal with a given set of numbers doesn’t mean they always will be—and we discuss that eventually as well.  But as an entry point into algebraic identities, and as a gut check to see if my students “get” what algebra is about, I find this works because students believe what they can test and see for themselves.  Having them practice distributing multiplication over addition (and also practicing NOT distributing multiplication over multiplication), while it has its place, I find isn’t a good substitute for the intuition that develops by playing with the raw numbers.

Next, semi-related entry:  The joys and sorrows of “flip and multiply”!