What learning math sometimes looks like – counting edition

Ran across a nice example problem in Art of Problem Solving’s Introduction to Counting and Probability this morning – problem 5.5 in section 5.3:

In how many ways can a dog breeder separate his 10 puppies into a group of 4 and a group of 6 if he has to keep Biter and Nipper, to of the puppies, in separate groups?

The boys worked on the problem together for about 10 minutes and found two different ways to approach the problem. Unfortunately those two ways produced different answers. In the first movie I had them talk through the two approaches:


While explaining their initial thoughts about the problem my older son noticed a mistake in their approach which used complimentary counting. The boys worked through that mistake below:


Next we looked at the approach using case by case counting – could we find the mistake here?


I thought their work on this problem was a really nice example of what learning math can sometimes look like. They did a nice job working through the problem two different ways the first time around. The fact that they found two different answers with those two approaches meant that at least one of those approaches must have had a small error. Reviewing the two approaches helped us find the mistakes and hopefully helped the boys learn a little bit more about counting.

A Thank you to Max and Annie of the Math Forum

I don’t remember what brought it to my attention, but I saw the NCTM ignite talks in a link earlier in the week:

The 2015 NCTM Ignite Talks

The talks by Max Ray and Annie Fetter of the Math Forum really grabbed me. Here are direct links to their talks.

Max’s talk:


Annie’s talk:


I happened to be trying to think through some strange similarities between struggles that k-12 kids have understanding math and the struggles that some college chemistry majors have understanding chemistry when I heard these talks. A slightly unusual thing to be thinking through, I know, but some of the ideas in these two talks helped me gather my thoughts.

The talks had a much broader (and surely more important) influence, though, which I hadn’t noticed until reviewing the project that I did with my kids yesterday afternoon. While listening to the videos from that project on trip back to Boston from NYC, I struck me how much the ideas that were fresh in my mind from Max and Annie’s talks were influencing how I was talking to my kids about math.

That project is here:

Using Evelyn Lamb’s Infinite Earring article with kids

So, thanks to Max and Annie – 3+ months after you gave them each of your talks had a big influence on me. Glad they were filmed!