One of the major areas of emphasis in a session at the number lab is strengthening critical thinking skills. Teaching critical thinking skills is a complex process, but the mathematics classroom is fertile ground for helping students improve in this area and an excellent example occurs when a child is called to the board to explain his or her thinking to the rest of the group regarding a mathematics situation or problem.
Calling a student to the board to do a mathematics problem is certainly a practice one sees in many math classrooms, but at the number lab the message to the students is a departure from what you typically find. Students are brought to understand that when they are at the board, they are not up there to “perform.” Mathematics is often mistaken for a performance subject rather than the rich and fascinating discipline it is. Coming up to the board is not a performance task in which students just show off that they can answer questions correctly or demonstrate they know the math skill, as in many or most classrooms. Instead, at the number lab, kids are up at the board to learn and to help the rest of us learn.
When a student is at the board, the teachers are not asking him or her to tell the class the answer to the problem in question, or tell us how he or she did the particular problem. Instead, we reiterate to the kids that we want them to “show us how they thought about” the particular mathematics problem. It is their thinking we are interested in, as careful analysis leads both to clearer and deeper conceptual understanding of the mathematics and to stronger general critical thinking skills.
The student at the board and the rest of the class group are all improving their thinking skills during these moments.
First, the students who are not at the board are working to grow in their ability to listen attentively and persistently, to follow another person’s thinking, to analyze the clarity of thought of the person at the board, and to ask supportive questions that help the person at the board think more clearly. This is hard intellectual work, and especially difficult when the teachers, as we do, really push the students to actively do this work. Constant feedback, reminders, questions, and praise keeps everyone carrying the heavy weight of this intellectual work. You’ll often hear us saying things such as:
- “Think hard – it’s the work of mathematicians.”
- “I need to hear from someone new – someone that hasn’t commented on our colleague’s thinking.”
- “Just take a risk and put something out there for us to think about.”
- “When someone is at the board explaining it isn’t a time for you to stop listening. Mathematicians are very keen observers – you don’t want to miss a thing.”
- “Let’s hear from everyone else. Do you all agree or disagree with your colleague's statements about this problem?
And second, consider the child who is at the board articulating his or her thinking. The child is asked to very precisely walk step-by-step through his work, articulating it to the classroom community loudly enough so that everyone can hear, using mathematical vocabulary. He is pushed to not assume anything, and to explain each leap in his thinking. And he is held to high standards when explaining. We don’t guess what’s in his mind and assume we know what he “meant to say;” we ask him to do the bulk of the cognitive work and actually figure out the way to articulate each step of his thinking all by himself. Any steps that don’t make sense – either seem to be unclear or do not have a mathematical property governing them – are brought to the class community for discussion. The students agree, disagree, ask questions, and grapple with their peer’s work.
It is in these moments – when one of our students steps to the board to analyze a problem – that you see what it really looks like to grow critical thinking skills, expand kids’ ability to think in a disciplined manner, and empower children to understand the beauty and richness of mathematics.