The forumla for combinations and thinking in larger chunks

I hate the formula for combinations. Here it is:

For n objects, the number of different ways to choose k of them is: n! / (n-k)! k!

For example, if I have 25 students and need to choose a committee of 5 of them, there are:
25! / 20! 5! possible committees.

The reason I hate the formula is because it gives virtually no clue about where it comes from or how it makes sense. Students forget it and misapply it because it doesn't fundamentally make any sense.

I always think of this kind of problem in this way: 25*24*23*22*21 / 5!

The numerator connects very straightforwardly to other counting problems. The 25*24*23 etc. represent how many students I could choose as the 1st student for my committee, the 2nd student for my committee, the 3rd student, etc. I have to divide because I'm over-counting the number of committees that are really different. For example, the committee with students {A, B, C, D, E} is really the same as the committee with students {B, C, A, D, E}--the order that I choose the students in doesn't matter. How much to divide by? Each committee I might choose is the same as 5! -1 other committees, because that's how many ways to re-order the same 5 people.

This is probably too quick an explanation to feel comfortable unless you've worked with other combinatorics problems recently--but my students have taken to it quite well after understanding the more fundamental types of counting it relies on.

This seemed to me a great leap forward in student understanding. I didn't tell them that there was a "formula for combinations" because I feared that they would be unwilling to just think about how the problem made sense. As I said, this worked out very well until I had to give them more elaborate problems:

How many 5-card poker hands have (exactly) 2 aces and 2 kings? This is a trickier problem to think through from first principals. It's a fairly long process with many subtle places to go wrong. It would be much easier to think about it in terms of a few independent choices:

(# ways to choose 2 aces)*(# ways to choose 2 kings)*(# of ways to choose another card), or

(4 choose 2)*(4 choose2)*(43 choose 1)

This is where the abstraction of (n choose k) would be handy. It would let students chunk the problem more easily.

This reminded me of something I read somewhere about quick thinkers. The author compared the mind to a pipe with water (thoughts) flowing through it. A lot of people think that fast thinkers are like pipes with faster running water; the thoughts come sequentially at a faster rate. This might happen sometimes, but far more common is for a quick thinker to have a wider pipe--they use abstractions so their thoughts simply contain more, even if they happen at the same speed as everyone elses.

The use of combinations and permutations seems to me a good example of this. A student /could/ reason their way, step by step to the correct solution, but that's many sequential steps. If they can think of the problem in terms of combinations and permuatations, each of their fundamental operations encapsulates several of the old sequential steps at once. This lets their minds arrive at the same place faster, and with less chance of error (or wandering, or boredom).

In my class, students who have an ability to focus their attention for longer periods didn't have trouble with the longer problems. It was students who "got lost" in the steps due to lapses of attention, or an inability to conceptualize the process as a whole, that had trouble.

Next time I'm going to try and achieve a best-of-both-worlds. I'll teach permuations and combinations my way, but then make sure they recognize each as a fundamental type of situation; and help them make the gestalt switch to breaking larger problems down into them, instead of building them from the ground up, as they're used to.

Mental models

We had a department meeting last week in which we briefly discussed one of my list-item goals: to explicitly teach mental models instead of just procedures.

Some teachers wanted clarification about how I thought of a mental model. The short answers was: a mental model is what a concept means, instead of a procedure to get a certain kind of answer. This can be tricky because often students are looking for "the steps to get the answer"--they're not attuned to what something means. It's also common to accidentally "proceduralize" a question--that is, take a question that required applying one's understanding of the meaning of a concept, and instead turn it into a question that students can solve by following a procedure they've memorized. Typically this happens by giving the same type of problem repeatedly, allowing the students to notice the steps you follow to solve it. Then they can follow those same steps without really understanding where they came from or why they make sense.

Here's an example of this that occurred to me while writing questions for my final.

The Topic: What does the "end behavior" of a function refer to? Specifically, what does it mean to say (for example): as x->infinity, y-> 0; as x-> -infinity, y->0 ?

The original question: The target sort of question is to give students an equation, and have them describe its end-behavior. For example: Describe the end-behavior of f(x) = 1/x

I discovered that I had inadvertantly proceduralized this kind of question for students. The procedure most of them know at this point is:

1). Plug in a very large x-value (or a few).
2). If it's large then write: "As x->infinity, y-> infinity". If it's close to 0 write: "As x->infinity, y->0", etc.
3). Plug in a large negative value for x, etc.

This process usually yields correct answers to questions like the target.

Unfortunately, many of those same students were stumped when I asked the following:

Say that the function f(x) has the following end-behavior: As x->infinity, y -> 0. For each point, explain whether or not you think it's likely to be on the graph of f(x). [if you don't have enough information to know for sure, clearly explain why]

a). (0, 2) b). (99999, 0.9999) c). (9999, 9999) d). (9999, -0.00001)

It wasn't that students gave wrong answers (although many did). Many students didn't even know where to start. This is a clear indication that they don't really understand the /idea/ of end-behavior.

A similar phenomenon occured when I asked them the following:

Say the function f(x) has a horizontal asymptote at y = 3. What is it's end-behavior?

Many students didn't know where to start, because they had memorized procedures to /find/ the horizontal asymptote, without ever really thinking about what it means to be a horizontal asymptote, and how that idea is related to the idea of end-behavior.

So, to get back to mental models, what would the mental model(s) be for end-behavior? I think it's hard to capture what someone's mental model is exactly; it's whatever they use to think about what a question means.

When I see: "as x -> infinity, y -> 0". I always say out loud in my head "as the x-values get bigger and bigger, the y values get closer and closer to 0"

I also always picture a graph in my head something like this...

As the x-values of my points go to the right, the y-values of my points will get closer to 0 (the x-axis). In other words, as I keep plotting points, their height (the red dotted lines) will get shorter and shorter. (Actually I imagine an animation with the dots appearing from left to right, as the x-values get bigger)

Since the idea of end-behavior is a slight formalization of a very natural question to ask about a graph, I think a good way to introduce it would be to have students do their best to answer the natural question as best they can.

It could be a "describe the graph over the telephone"-type activity. Students could have a series of graphs, and their goal is to write a short (i.e. 1-2 sentence) statement explaining what the the trend at the "ends" of the graph are, so that someone else could re-create it only from the description. After discussing student descriptions, you could unveil the sort of description mathematicians have agreed upon in a context that makes sense.

This would also be a good lesson for a party game I like. Take a vertical strip of paper and fold it into (say) 4 sections. The first person in the group draws a graph. The second person in the group looks at the graph and describes its end-behavior. The paper is folded so the 3rd group member can only see the 2nd member's writing. The 3rd member has to draw a graph that matches the end-behavior described by the 2nd member, and so on. It's like "telephone" using paper, and switching back and forth between representations. At the end, the group checks to see if they ended with the same kind of end-behavior they started with.

To emphasize to students that they should be learning what the concept means, it would be good to give them assignments that require them to express this understanding directly--not use it to get an answer to another sort of problem.

Since the concept of end-behavior has a natural interpretation in terms of a progression, I think a nice assignment would be to have students create a short multi-panel comic strip explaining or illustrating the idea of end-behavior for someone in a younger grade (like 9th).

In my experience, some students won't produce anything useful, but many students will produce clearer explanations than anything I was likely to give; so re-distributing some of the better comic strips as "notes" would be a good follow-up.