A VIP ( Very Important Problem) ( for me anyway)

Consider a line of lockers, numbered consecutively, starting from 1 and continuing without bound.
(As you might have already noticed, this is not a real world problem)
Person 1 walks through and, starting with locker 1, makes sure all the lockers are closed.
Person 2 walks through, and, starting with locker 2, opens every other locker.
Person 3 walks through, starts at locker 3, and changes every third locker, opening the closed ones and closing the open ones.
Person 4 starts with locker 4 and changes the state of every fourth locker.
This continues in like manner, with person n, starting at locker n, and changing every nth locker.
(You might have noticed that this problem is not intended to practice necessary algebra skills)
This problem changed my life. I guess P.J. would call it a puzzle. I am sure most of the readers of this have heard the problem.
Oh, this is not exactly the puzzle that changed my life. That one also included a question. I think it is a better problem if the student has to decide what would be a good question. When I give this problem to students, they ask what the question is. My response is, "Oh no, did I forget to ask a question again? What do you think would be a good question? "
Someone will say something that is roughly equivalent to "Which lockers are open eventually?"
That is the question I was asked. However sometimes student's questions are better than the one I had in mind. Isn't it true that an important aspect of mathematics is deciding what questions to ask? Do we give our students a chance to do this?
But, as is my lifelong habit, I digress.
How did this problem change my life?
It was given to me in a course named "problem solving". I had been teaching for several years. I thought I was making progress but now know I had a lot to learn about how children learn. I now know I still do have a lot to learn about how children learn. I found the problem challenging and interesting, even though it has no basis in reality. I worked on it for a while, and an amazing thing happened. Once I had an answer, I became curious about why the answer ended up being so nice. I started thinking about it and without much effort I had a proof of a theorem about factors of integers, and I understood and therefore learned some interesting mathematics I did not know before.
I realized that what made this problem good was that I found it interesting, and that it created a situation where solving it helped me to understand something. It was not just an answer, but a revelation, that lead to a new understanding.
Was it important? Well, I have since used it to solve other problems, so I would say, yes.
But most importantly, I learned some significant mathematics by doing a problem. I was proud, and happy , and wanted to find other problems to work on. And I got this crazy idea that perhaps I could arrange for this sort of thing to happen in my classes for my students.
And so, for the next thirty five years, I tried to write problems that would lead my students to figuring out something for themselves.
It took a while. I tried and rejected the method of having them work a few examples in the hope they would notice the pattern I wanted them to discover. For one thing, it seemed phony but for another, they didn't learn any mathematics from working on the problem because they didn't really figure out any mathematics, they simply arrived at what I wanted them to arrive at.
I also tried worksheets. That was a total failure. For one thing, once give a worksheet, the objective of a student is to finish the worksheet. If another student has a question or observation, that interferes with the task at hand, which is to finish the worksheet. I also learned that students did not understand that they were supposed to learn something from finishing the worksheet. They were just supposed to finish it.
So, I eventually fell into a groove that I was happy with. One problem at a time. Everyone works on that problem. If you finish, compare your result with others. If you all agree, ask the next question. What if we changed the problem this way? Is there a generalization? Is there a special case? Meanwhile I walk around, rapidly, listening and looking at what is happening, waiting for the teachable moment, that moment when the entire class is ready to share their thoughts on the problem. I usually also share my thoughts at this point, especially with regard to connections and what might happen later.
Then I give them another problem to work on.
And learning happens, the students are actively engaged, and frequently I learn something I didn't know. I also learn a lot about how my students work.
Teaching by giving students a good problem is different than other ways of teaching. The teacher has to be ready for anything that comes up, and must have a plan if nothing comes up. The teacher must be willing to give students time to think of things , time to make up their mind, and time for ambiguity. The teacher must be prepared for long periods of time where everyone is wrong, and be willing to let that happen. It takes practice and a belief that eventually truth will win and incorrect reasoning will be exposed. It means giving up the role of the person who is the authority about truth. But in the end, it worked better for me than any alternative.

And, the best part is, I have a great time doing it.

And it all started with the famous locker problem, a puzzle that taught me how to become a better teacher.

Words on/Problems with Word Problems

Dan Meyer's and Christopher Danielson's excellent blogs have recently been weaving a thread about "word problems"; Danielson quotes one and writes

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers?

This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).

And while generally I agree with Danielson that many students find the most interesting problems to be ones drawn from their own real experience--not these contrived, puzzle-y types of problems--I think he's wrong in saying that problems like Vincent's have no place in the curriculum.

Why might they be important?

Consider the following situation we considered in geometry yesterday:

 

Students quickly identified that all three triangles are similar to each other, but to justify the statement, they needed to "chase angles" around the figure.  So we labeled the measure of angle A as x.  What's the measure of angle ABF?

The algebra here is not hard, but it requires translating an idea--two angle measures add to 90 degrees--into an equation or expression.  There are lots of such situations in geometry:

 

To find the exterior angle sum of the pentagon, one of the easiest ways starts by writing:

      a' + a = b' + b = c' + c = d ' + d = e' + e = 180 degrees.

Or how about this early optimization problem: what's the largest area for a rectangle of a given perimeter, say 60?

So while I think it's true that many students find problems like Vincent's uninteresting, the underlying skill--and here, there's really no issue of transfer--is crucial to setting up and doing interesting math on other problems.  I learned this the hard way this week: we spent twenty minutes chasing angles I had thought we would find in just a few.

So what can we do?  Two ideas:

First, we can separate "problems involving translating words into symbols" from "application" and "real-life" problems.  And when we pose problems in the first category, we can be honest about the fact that we're trying to develop a skill students will use in attacking other math problems, not preparing them to do something most people do everyday.

But second, I think we can do a better job of honoring the category of puzzles.  My experience is that kids like puzzles when they know they're puzzles.  For example:

Here's a puzzle:  a 25-foot ladder is propped against a wall 7 feet away.  The bottom of the latter slips back 8 feet.  How far down does the top slip?

The Jain mathematician Mahāvīra poses the following problem:  There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers.  How many fruit were there altogether?

In England, there's a tradition of trying to ring all possible permutations (or "changes")  of a church's bells, without duplicating any permutations.  Unfortunately, because of each bell's mass and angular momentum, the only way to change the order is to switch two successive bells: so ABCDE can go to ABDCE or BACDE but not ACEBD.  Is it possible to ring all the changes on five bells without duplicating any?

Indeed, puzzles are an important mathematical tradition; as teachers of mathematics, it's up to us to help students appreciate this vital mathematical aesthetic.  (Great ideas about our role as teachers of mathematical aesthetics are in Natalie Sinclair's paper here.) We teach that aesthetic by including puzzles, riddles, and other mathematical games in our curriculum--not just what students "need", in some sense, "to use". [Not Danielson's quote or exact idea, to be fair!]

I'll go even further: I think that it would be great if we could develop our students' mathematical eyes so that they can notice and appreciate cool number patterns and symmetries just as much as we'd like them to notice and appreciate symmetries in geometric forms and patterns.  That is: our goal is not just that students can do math or that they know math, but that, at some level, they appreciate it.

A last note on puzzles and aesthetics.  Vincent's problem is truly a terrible problem.  But what makes it bad is not that it's contrived or abstract.  What makes it bad is that it presents a "puzzle" with nothing interesting in it to notice.  Who would possibly care about a sum of 2479?  There's no symmetry or pattern.  It's like a painting that is composed of two discrete forms, but with no balance, interesting details.  Even a tiny tweak makes it better--like using the year, or an interesting number, like this:

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2011. What are the two numbers?

or this

Don and Leo are going to visit their friend Vincent, who lives in an apartment building where units are numbered consecutively on each floor.  As a puzzle, Don writes down the sum of Vincent's apartment number and his neighbor's, using a scrap of paper he finds in his pocket.  When Leo gets the paper, he asks Don "Which way should I read this?" but then says "Wait, it doesn't matter!" If Don's number has four digits, what's the smallest possible apartment number Vincent could have? 

 And then you can ask interesting, mathematical questions:

Assuming you don't run out of apartments, is every four-digit number a possible sum of adjacent apartment numbers?  [Is this puzzle possible every year?]

What is the sum of the numbers of the apartments on Vincent's and his neighbor's other sides?

A visitor notices that each apartment is directly across the hall from another apartment, and that each pair of across-the-hall apartments has the same sum.  How would you number the apartments so that this happens?

Challenge to readers: in the comments, add your own "apartment problems"!

Notes on Notes

“When I am talking, the students are either ignoring me or listening attentively and trying to take notes, but in neither case are they actually doing mathematics, which is the one activity that I can guarantee will produce learning gains.”

Point 3 on the previous blog (above) registered something I think is important enough for its own discussion. Several years ago I went to a presentation about emerging technology, specifically the tablet P.C., and how it might influence teaching. I thought I was going to a presentation on how the tablet improved learning, but that was not the case. I am a huge fan of the tablet as a way of helping students learn, but the emphasis of this presentation was how helpful and efficient the tablet is to “give notes.” The demonstration proceeded to show how the teacher could have all of the notes for the class prepared on power point slides and how easy it was to annotate them so the students could copy them into their notebooks. Huh?

I firmly believe that we should strive to use every minute of every class for learning useful things. However, as P.J. pointed out, attentive students who are taking notes are not working on problems. So what are they learning? They are probably not thinking about much other than: did I copy this correctly? It is clear to me: the process of copying notes is not a learning experience.

Perhaps there are other reasons why it is important to take notes?

I think there is a belief that if students have good notes, they will have something to refer to when they forget or when they need to study. I propose that if their textbook does not serve that purpose, it is not a good textbook and never should have been selected in the first place. Get rid of it as soon as possible! Replace it with one that does provide an organized summary of what students are studying, with examples and sample problems.

But perhaps you are stuck with a terrible text. Or perhaps you have topics that you think are important but are not in the text. Possibly you have an alternative approach to the topic that you think is better. (I hope both of the last two situations do occur frequently in your class.) I would suggest that it might even happen that a student has an alternative approach that is not in the book and that you had not thought of, and that you have become aware of this approach by walking around and paying attention to how students are solving the problem you gave them to work. If so, doesn’t it make sense to carefully write this up and either post it on your web page for all to read, or print copies and pass them out for all to have.

This approach will provide students with the much-needed supplement, and that supplement will be carefully written, error-free, and well-organized, attributes uncharacteristic of in-class notes. This approach will also encourage students to read mathematics, an important step towards helping them become independent learners. These real or virtual documents will be legible, something my hand-written notes rarely are. I think some would call these documents, “class notes.” I prefer to call them, “supplements to the text.”

There would not be many of them, as we want students to learn to work with their text. But most importantly, it eliminates all the time that is wasted when students think they should be writing down what the teacher writes.

I take notes at every meeting I attend. They are notes, not transcripts of what the speaker has just said, or written. They are thoughts that occur to me as I am learning, thoughts I don’t want to forget. They are nuggets that someone said that I want to remember. My notes from a class I took from Dan Teague include, “There are three kinds of mathematicians, those who can count and those who can not.” I laughed and realized that I wanted to remember it, so I wrote it down. We were working on an amazing problem, and I knew my mind would immediately forget his quip as it focused attention on the problem. I put it in my notes and have used it hundreds of times since.

I think we need to distinguish between notes and transcripts. I think many do not make that distinction and as a result, their students waste a lot of class time. I understand there are even teachers who waste their own time collecting and grading their student’s notes instead of thinking up interesting problems to ask their students to work on.