More About Mathematical Thinking
Yesterday I was reflecting on why I want my students to learn how to express their mathematical thinking and how it connects to both learning targets and state testing goals. We continued our work on this today and realised something very important about this skill: it is incredibly challenging!
As I walked around the room and listened to my students’ conversations, I realised that many, especially my more advanced students, know how to solve these problems, but they don’t know how to express it. Even when working with a simple one- or two-step problem, they can quickly do the computation, but they don’t actually know how they solved it. And, as I have told fourth graders time and time again for the past three years, it is not enough to just know the answer; students need to know how they got the answer and they have to know how to express it!
I remember a news article several months ago that had all sorts of folks upset about the new Common Core State Standards. The gist of the article was that a panel of teachers was answering questions about the standards and one had been asked about the math learning targets. Her response for the math goals was that the standards no longer focus on correct answers; they focus on mathematical thinking. Of course, what some took out of this comment was that it is totally okay for students to get the wrong answer. That is not accurate at all. What is intended with the new standards was that students cannot just find the right answer; they must be able to articulate the processes they used to solve it.
One of the single greatest frustrations I run into when checking students’ math work is when they provide an answer but don’t show any work. If the answer is wrong, I don’t know how they got it. If it is right, I still don’t know how they got it. Maybe they looked it up online. Maybe they asked a parent or a friend or an older sibling. Maybe they used a calculator. If there is no evidence of mathematical thinking, I simply don’t know how they got to the solution, and that’s what I am really looking for.
Do I want my students to know how to accurately compute and find the correct answers in math? Of course I do. But the correct answer isn’t enough. It is just the beginning. I want my students to be able to think about what they are thinking and how they are thinking. I want them to examine their processes and see if they can find a different method of solving the same problem. I want them to be able to explain their ideas to others. Even though they are just in fourth grade, my expectation is for my students to be able to think critically, not just about the world but also about themselves. As they increase their capacity for deeper thinking, I am confident they will be better prepared for any challenge they come across in the future. And yes, I really believe that solving problems about buying squirt guns, boxes of pencils, and bags of apples will help them in this process!