The Perfect Combinatorics Problem

In this post, I’m going to extol the virtues of my favorite combinatorics problem.  You’ve probably heard it, or some version of it, before:

A pizza parlor offers ten different toppings on their pizza.  How many different types of pizza are possible to make, given that a pizza can have any number of toppings, or no toppings at all?

Just in case you aren’t familiar with this problem and want to work it out for yourself first, I’m putting most of this post after the jump.  First, a shout out: I remember doing this problem with Michigan State Professor Bruce Mitchell, who used to teach Saturday-morning math enrichment classes at my middle school, and whose enthusiasm and humor kept me coming back.  Second, some pizza:

http://www.neatorama.com

You may prefer to pretend you never saw that.

Let’s Ban the “Distance Formula”!

One of the things that can be hardest for kids learning algebra is to be able to understand the value of abstraction and of using symbols to help one analyze and think about a problem.  I myself remember learning algebra (way back in 1978)  from Dolciani/Wooton, a textbook that valued formal manipulation above all else; there was almost no motivation given for where the rules came from, just lots of practice in learning how to manipulate symbols correctly.  Indeed, for a number of years afterward I thought that formal manipulation was all there was to algebra.

The idea that there are ideas to be discovered in algebra was completely foreign to me.  I knew in 9th grade that perpendicular lines had slopes that were negative reciprocals, but if you had asked me why, I would never have known, or even thought I should have known.  Dolciani gives a proof, but I would bet a lot of money that very few students read it, as it is, to be frank, inappropriately abstract for a high schooler learning the subject for the first time. 

Our curriculum ( http://parkmath.org/curriculum/ ) approaches the topic by having students first draw a line on graph paper with a slope of 2, and then try to figure out experimentally what slope a line perpendicular to it should have.  Working out concrete examples using lines of different slopes can lead to a much deeper understanding for a 9^th grader than an algebraic proof that they have little chance of following, never mind retaining.  When students are older and have more experience with algebra, of course, a formal proof can make good pedagogical sense.  But for a freshman? 

Which is why I think we should ban(!) teaching the “Distance Formula”, at least for the large majority of 9th and 10th graders.  Why, you might ask, if I am trying to teach them the value of abstraction?

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Park School Math in the “Mathematics Teacher”

We’ve been fortunate enough to have an article published in the February 2012 issue of the Mathematics Teacher, “Geometry in Medias Res”.  We teach Geometry in a fairly unusual way, we think, so we decided to write about it and see what people thought about our approach.

One of our main ideas is that we want students to encounter interesting problems on the first day.  So we ask them non-trivial questions right away (e.g. can every triangle be circumscribed?), and in the process of them discussing/arguing with each other, we start to develop with the kids the necessity for a standard of proof other than “it really seems like it to me!”. 

There’s a lot more, but the idea is to have a more natural and intuitive introduction to the axiomatic nature of Geometry than one usually finds.  We are big believers that proof is completely accessible to all levels of students, but that it has to be introduced gradually, as a way of resolving questions students have, not as a forced superstructure like the way it was often taught in the past.

If you have a chance to take a look at the article, we’d love your feedback and thoughts.

Geometry Follow-Up: Proof in a Bag

The concept of proof-in-a-bag is simple.  Write out a two-column proof and then cut it up so that each statement or reason is by itself on a scrap of paper. Then put all the scraps in a bag (a small sandwich bag works well, though an opaque paper bag might have more of a dramatic effect) and have kids work on rearranging the scraps so that they form a coherent proof.  You can decide whether you want students to know ahead of time what it is they’re proving, or if you want them to figure it out by putting statements with “given:…”, “prove…” and a diagram in the bag as well.

Credit where credit is due: I got the idea for this from Laura Chihara while a student in her Algebraic coding class at the Carleton-St. Olaf Summer Math Program.

 It’s nice to have any activity where kids are physically doing something in a math class, of course, but I really like what kids get out of this activity.  It emphasizes the idea that you have to have enough evidence before you can conclude that triangles are congruent (otherwise, what are those “extra” statements doing in the bag?)  And it is very good for helping students understand what can be a statement vs. what can be a reason.  I often find that students want to use triangle congruence theorems like SAS when using properties of triangle congruence; the structure of this activity leads them to realize that they’ve already used SAS to justify the triangle congruence statement; they now need to use something else (CPCTC or the equivalent) to start using the congruence.

There are some times when I would definitely not use this activity.  If the proof is a particularly exciting one for kids to work out on their own, I wouldn’t rob them of the opportunity.  Proof-in-a-bag works best for simple, straightforward proofs, where the two-column proof format can be used without having to do a lot of extra explaining.  I generally use it for one day only, at a time when the class has had some practice writing proofs but has not yet reached a level of comfort with them.

Does anybody else have activities or techniques that they use to teach writing proofs?  I’d be especially interested in what people do who don’t insist on a strict two-column format all of the time.

On Algebra and Logic

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.

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”!