Don’t you love when you learn something that makes your life better? Especially if that something is something you wanted to learn, but thought would be impossible to learn.
This happened to me in Palm Springs, at the CMC-South Conference. Dan Meyer was talking about math problems that had “open middles”. Open Middle problems had a somewhat traditional problem background, without the truly traditionally predictable solution path. Traditional problems (i.e. factor the expression x^2 + 6x + 8) had closed beginnings, middles and ends because the way in which a student would arrive at the solution was clearly paved by the teacher…so all a student would have to do is mimic the teacher’s steps. Open middle problems allowed students to discover the best path (for them) that would help them get to the answer.
This may seem a little unclear so here is the first example I came up with.
So, here’s what I like.
1. There are many paths to the answers. I presented this to a group of Algebra 1 teachers, with hopes of helping them find problems that would help transition to the CCSS and incorporate most of the SMP. They used guess and check, “the box” (some call this the array or area model) and algebra tiles.
2. With only a couple of problems, they were able to “prove” their proficiency at factoring. Ideally, students would find a number and check to see if the quadratic expression was factorable. In a sense, they are still factoring the traditional type problem.
3. Math, in general, tends to teach a concept, then, teaches the “backwards” of that concept. These types of problems do not go forwards, nor backwards, but almost inside-out because a student has to truly understand factoring in order to find the answer.
4. There are multiple entry points for these problems for students who only have a superficial understanding (they can guess and check to see if the quadratic was factorable) to students who are proficient at factoring and can use a variety of ways to find the missing numbers.
5. It encouraged collaboration and mathematical discourse. Because the solution path was not clearly laid out for the students, they had to rely on each other to think things through. I was simply amazed at how these problems naturally promoted collaboration.
6. There are a couple of answers for number 1 (4, -4, 5, -5). Most found the positive answers. After they read the directions carefully (“integers”), they were able to find more of the answers. Click here to see the rest of the answers.
So, to make a long story short. These open middle problems are my new quest. I want to find more problems like these. Robert Kaplinsky and I are collecting/making problems like this at OpenMiddle.