I have changed much of my teaching in the last year. One big change has been the way I approach teaching problem solving, and it has to do with a larger shift in my thinking about the difference between means and ends in education.
Here are some fun problems:
#1: You have two strings that take an hour to burn from end to end, although they do not burn at constant rates (maybe the first half of a string burns in 5 minutes, and the second half burns in 55).You also have a lighter and a pair of scissors. How can you time 15 minutes?
#2: 100 prisoners are slated for execution. They will be lined up, and each prisoner given either a blue or a red hat. He can’t see his own hat, but he can see the hat of every man in front of him. Starting at the back of the line, the executioner will ask each prisoner what color his hat is. If he is right, he is spared. The men can make a plan together before the execution. How many men can be saved?
I’ve enjoyed working on both of these problems. I would argue that someone who possesses strong problem solving skills is likely to do well with these problems, and that those skills are an important goal of math education.
But these problems are the ends of a math education. In the past, I have “taught problem solving” by giving students problems like these and letting them struggle. But I think I was confusing means and ends — while these problems, and problems like them, are useful to gauge problem solving skills, I don’t think giving them to students is a particularly effective way to teach problem solving. I was confusing the ends of a quality math education with the means of getting there.
One approach to lesson planning floated at my school this year that has caught on with several teachers is the idea that, in each lesson, students should read, write, speak, reflect, and move. This strikes me as a useful goal, but is not an end in and of itself. The end is student learning and engagement. This approach is one means of getting there — but it isn’t the only one, and it is far from perfect. When I teach Des-Man, one of my favorite lessons, students are typically sitting, working, largely silently, for a long period of time. And it is one of the most engaging and, I think, one of the most successful lessons I teach. To read, write, speak, reflect, and move in a lesson is a useful means to move more consistently toward high student engagement, but it is not an end in and of itself.
I’ve had a similar thought about number talks recently. I don’t use number talks as often as I did when I taught middle school, but in looking back on those lessons, I think my number talks may have focused more on trying to get kids to produce expert mathematical thinking — focusing on the ends — rather than thinking about the means of building that thinking component by component. Putting kids in a situation where they might excel if they have the knowledge, no matter how clear and valuable my vision, is not a substitute for thinking carefully about the means of getting there.
Vertical Non-Permanent Surfaces
Vertical non-permanent surfaces are one of my favorite tools I’ve learned about in the last few years, but putting students in front of a VNPS with an Expo marker does not create learning. The purpose of a vertical non-permanent surface is to increase knowledge mobility throughout the room and decrease barriers to trying new ideas and making mistakes. If they aren’t serving those functions, then I’m focusing on the means at the expense of the ends, and shorting my students.
Means & Ends
I could give many more examples. I’m writing this post as a reminder to myself that the goals of my class are for students to learn math, to be able to transfer some of that knowledge outside of the math classroom, and to enjoy the whole process. Those are the ends, and there are lots of little pieces that bring my students in that direction. It’s easy to get caught up in cool-sounding trends and new clever ways of doing things that distract from my larger goals. It’s also easy to just focus on where I want students to end up without thinking critically about the little steps along the way that they need to get there. I hope this perspective questioning my view of means and ends can help me look at my pedagogical decisions more critically, and teach a little better for my students.