Students need to opportunity to work non-routine problems, but they also need to be taught what it sounds like for a mathematician to work through a tough problem.
Consider this item from the 2013 Texas 4th Grade STAAR test:
This item was coded as a perimeter problem, but because it has process skills embedded, a child could totally know how to find the perimeter of a figure and still miss the problem. Stop for a minute and consider the skills needed to solve this problem.
Here’s where modeling comes in. Students need to hear your thinking about how to work this problem. That’s the way to develop true problem solving/critical thinking skills. It might sound something like this…
[teacher reads the problem, which is projected on an overhead, document camera, or interactive white board]
Wow! That’s a pretty complicated problem! I heard several math vocabulary words as I read, so I’d better think carefully about the meaning of each word. Let me break this down and see if I understand each part.
[re-reading] Use the ruler provided to measure the side lengths of the figures below to the nearest centimeter. Okay, I need to measure the sides, and I’m using centimeters, not inches. Let me make sure I’m using the correct side of my ruler. [measuring the first side] I’d better be sure I line my ruler up carefully. I know that’s important when I’m measuring length. As I measure, I should write my measurements on each side. That way, I’ll have the information I need to work with. [measures all sides and record the measurements] Hmmm, I notice that all the sides on this hexagon, because it has six sides, are congruent. They’re all the same length.
I’m done with the measuring. I’ll read more and see if I can figure out what to do next. [re-reading] What is the difference between the perimeters of these figures? Okay, it mentions the perimeters of the figures. I know that perimeter is the distance around a figure. I’ll find the perimeter of each figure by adding up the side lengths. Oh! The sides on the hexagon are equal, so I can multiply those. [finds perimeter of each figure, recording the calculations].
Hmmm, now I’ve got the perimeters, but is that my answer? I’d better re-read the question again. [re-reading] What is the difference between the perimeters of these figures? Oh!! I need to find the difference between the perimeters. I remember that difference is like subtraction. I’m comparing the perimeters of the two shapes, so I’ll subtract the smaller perimeter from the larger perimeter. Oh! I already know that 29 can’t be the answer, because that number is too big. I’ll bet that’s adding the perimeters. The perimeter of the triangle is 17 cm and the perimeter of the hexagon is only 12 cm, so 17 – 12 equals 5.
Wow! That wasn’t really so hard after all. It had a lot of steps, but each of the steps was pretty easy once I read it and broke it down.
Notice a couple of things:
- The teacher does all the talking. This is not the time to have students help you solve the problem–that can come later.
- It really doesn’t take all that long! Model one problem like this a day, and you’re sure to see your students’ problem solving abilities grow.
- Pack as much math into the problem as possible. This was not a geometry problem, but I took the opportunity to use the names of the geometric figures as well as the word congruent.
- Be sure to model the habits you want your students to develop. By recording all of your work when you model the problem, and commenting on why it’s important to do so, you are showing the students that mathematicians communicate their thinking using words, numbers, and pictures. That’s better than just telling your students, don’t forget to show your work.
Bottom line, our students don’t come to us knowing how to think critically. It’s part of our job to help students develop analytical skills right along with the math content.