When you watch a typical math class, you see a lot of rapid fire questioning going on.
Teacher: Number 1?
Student: Five and ¾
Teacher: Number 2?
Student: 3 and ¼
Teacher: Number 3?
Student: 2 and ½
Teacher: Anybody get a different answer?
Student: 2 and ¼ .
Teacher: Good. Next problem?
Now in some classrooms, a teacher will ask a student to come up to the board and show their work. Sometimes a student will even be asked to explain their work. However, that often sounds like this: “First, I looked at the denominators. They weren’t the same so I made them the same by multiplying. Then I added the numerators but left the denominators the same.”
This explanation is a summary of what the student did, but it doesn’t give any insight into how the student thought about the problem. Why did the student make any of these decisions along the process? How did he know why to change the denominators and not the numerators?
For the students who didn’t get the correct answer, this explanation doesn’t help. They just heard a list of steps that they need to memorize and remember correctly next time. The process of why these steps work is a mystery. It’s a black box.
For the students who can explain why it works, they are often not given the opportunity to explain. They are never asked to make their thinking explicit. What if they were given opportunities to try to explain their thinking so that it becomes an explicit process instead of a black box? Would all the students in the class benefit from it? I think they would.
At first this process can be uncomfortable. Students aren’t used to talking about their thinking. It seems private, something that no one asks you about.
But slowly, if we ask questions like:
• Why did you do that?
• How do you know that is the correct answer?
They will begin to be comfortable talking about their thinking.
Now if the students only know because they memorized a set of steps, that is all they will be able to explain. But if they are given tools to start thinking about why things work, asking them how they know can change the way a classroom operates.
If the goal in math is to figure out how rules work or why they always work, the questions in math class change, the activities change and everyone is engaged in trying to discover what is in the black box.
I’d love to hear what happens when you ask students, “How do you know?”