I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers.

People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

We continue with our counting lessons — and once again, Kitten proves that she doesn’t think the same way I do. In fact, her solution is so elegant that I think she could have a future as a mathematician. After all, every aspiring novelist needs a day job, right?

If only I could get her to give up the idea that she hates math…

Permutations with Complications

How many of the possible distinct arrangements of 1-6 have 1 to the left of 2?

In a lazy, I-don’t-want-to-do-school mood, Princess Kitten was ready to stop after three math problems. We had gotten two of them correct, but the last one was counting the ways to paint a cube in black and white, and we forgot to count the solid-color options.

For my perfectionist daughter, one mistake was excuse enough to quit. She leaned her head against me as we sat together on the couch and said, “We’re done. Done, done, done.” If she could, she would have started purring — one of the most manipulative noises known to humankind. I’m a soft touch. Who can work on math when there’s a kitten to cuddle?

Still, I managed to squeeze in one more puzzle. I picked up my whiteboard marker and started writing:

Kitten complained that some math programs keep repeating the same kind of problems over and over, with bigger numbers: “They don’t get any harder, they just get longer. It’s boring!”

So we pulled out the Counting lessons in Competition Math for Middle School. [Highly recommended book!] Kitten doesn’t like to compete, but she enjoys learning new ideas, and Batterson’s book gives her plenty of those, well organized and clearly explained.

Pizzas at Mario’s come in three sizes, and you have your choice of 10 toppings to add to the pizza. You may order a pizza with any number of toppings (up to 10), including zero. How many choices of pizza are there at Mario’s?

[The book said 9 toppings, but I was skimming/paraphrasing aloud and misread.]

I know other teachers have done mathquilts, but I’ve never gotten around to trying it in any of my classes. Still, this image caught my eye and practically begged me to make it into a math lesson for my elementary math club.

I thought of at least two ways I could go with this, but I bet that if we put our brains together, we can come up with even more creative ideas. So here’s the question, ala Dan Meyer:

What can you do with this?

How could you use this image as a springboard to doing math? What questions would you ask? What concepts would you try to get across? What would you follow it with? Please comment!

The Art of Problem Solving people recently announced their new Alcumus program, which provides online lessons on assorted math topics, including probability and combinatorics, which most math textbooks do not cover well, if at all.

Update October 2011:

Alcumus currently complements our Introduction to Algebra, Introduction to Counting & Probability, Introduction to Number Theory, and Prealgebra textbooks, as well as our Algebra 1, Algebra 2, Introduction to Counting & Probability, Introduction to Number Theory, and Prealgebra 1 online courses. We expect to continue to expand topics in Alcumus.

The last couple of weeks, in Math Club, we’ve been learning to count. My new set of MathCounts students have never heard of combinatorics, so we started at the very beginning:

Most of our math contest preparation consists of working lots and lots of old test problems. Occasionally, however, I put together a tip sheet summarizing a topic that my students have trouble remembering.

In the first section of George Lenchner’s Creative Problem Solving in School Mathematics, right after his obligatory obeisance to George Polya (see the third quote here), Lechner poses this problem. If you have seen it before, be patient — his point was much more than simply counting blocks.

A wooden cube that measures 3 cm along each edge is painted red. The painted cube is then cut into 1-cm cubes as shown above. How many of the 1-cm cubes do not have red paint on any face?

And then he challenges us as teachers:

Do you have any ideas for extending the problem?
If so, then jot them down.

This is strategically placed at the end of a right-hand page, and I was able to resist turning to read on. I came up with a list of 15 other questions that could have been asked — some of which will be used in future Alexandria Jones stories. Lechner wrote only seven elementary-level problems, and yet his list had at least two questions that I had not considered. How many can you come up with?