What learning math sometimes looks like: hard vs. not hard

I’ve said several times previously that the most difficult thing for me teaching my kids is understanding ahead of time what is going to be hard and what’s going to not be so hard for them. Today I had a fascinating example of this problem with both kids.

My older son and I started a section in our Introduction to Geometry book on translations. The problem we used for the movie was question about translating a hexagon. I didn’t know what approach he would take, but I expected the approach would be more “hand waving” than math since this was our first day in the section. Instead he gave a sensational geometric proof:

Since I expected a more hand waving approach to the problem, I planned to go to the kitchen table to do the translation after we talked about it on the board. This part of the talk was something that I thought would be easier and would help build on some thinking from the first part. What surprised me a little here was that he had a little trouble understanding how to label the points of the hexagons to match the labeling in the problem on the board. The trouble he had in this aspect of the problem makes me think that I should probably spend a bit more time with some “hands on” geometry.

My younger son and I continued in the basic statistics section of our Prealgebra book. I noticed a homework problem that essentially asked you to wrote down two lists of numbers – the first with mean 70 and median 50, and the second with mean 50 and median 70. Seemed like this would be a fun way to get him to talk through some of the ideas in the chapter.

He had a pretty clever way to think about the first list:

Given his nice way of thinking about the first list, I was surprised that coming up with the second list really gave him some difficulty. In fact, after nearly 6 minutes of talking in the video below, he’s still pretty confused about how to come up with the second list. The difficulty that he had here made me wish that I’d pressed for a few more explanations when he was writing down the first list.

In this last part of the talk my son finds the way to write down a list of numbers with median 70 and mean 50. After he makes his list, we talked through a couple of alternate approaches to the problem.

So, with my older son I was surprised to find a problem where the more theoretical part of the problem was not so difficult for him, but the “hold the objects in your hand” part gave him a bit of trouble. With my younger son I was surprised to find two versions of the pretty much same problem (just flipping two numbers) had really different outcomes when he tried to talk through them.

Days like this serve as an important reminder to me – learning math is never a straight line and often really hard to predict.

A challenging AMC 10 problem involving some basic statistics

I was traveling for work the last two days and asked my son to work on some old AMC 10 problems rather than working in his geometry book. When I got back home tonight I asked him to pick one of the problems that gave him trouble for us to work through together. He picked #19 from the 2005 AMC 10 b:

The 2005 AMC 10 b

Here’s the problem:

On a certain math exam, of the students got points, got points, got points, got points, and the rest got points. What is the difference between the mean and the median score on this exam?

This choice proved to be lucky since I’ve just started the section on basic statistics in Art of Problem Solving’s Prealgebra book with my younger son. It was a nice problem to work through with both of them.

We started by reading through the problem carefully and making sure that both boys understood what mean and median meant. The boys decided to solve the problem by assuming that 100 students took the exam. We solved for the median first:

Next we found the median by working through a long arithmetic calculation. Since the calculation itself isn’t really that interesting, I tried to focus more on building up number sense, and, in particular, on ways to make the calculation easier. My older son is definitely more comfortable working through calculations like this one, but I think my younger son was able to see a few good math ideas in action:

So, a nice problem giving some good practice in basic statistics as well as some good arithmetic review. It is also a nice illustration of why I like the problems from the old AMC 10 tests. This is #19 out of 25 problems on the AMC 10, meaning it is one of the more difficult problems. Not so difficult that the kids aren’t able to understand the solution, though. The AMC folks do a great job producing problems that strike that balance, which is something that I’d probably struggle mightily to do on my own.