Prove Your Mathematical Thinking

Good mathematicians explain their solutions and can prove their reasoning.
It’s not enough for a student to get the correct answer. We learn through discussion, through sharing our ideas with others. When teachers ask, “How do you know?” How can you prove what you know?” It compels students to think more deeply and to justify mathematical concepts through informal deduction. Stage three along Van Hiele’s scale of Geometric thinking encourages us to look at the properties of things. Stage one: visualize. Stage two: analyze. Stage three: informal deduction. Stage 4: deduction. Stage five: Rigor. This final stage address non-Euclidian (Spherical) Geometry where all lines meet at the poles.

What do I have questions about?
Questions I have are how can we encourage quiet students to engage in dialog in sharing their solutions? In a classroom where proving solutions is the expected norm, what does a forum for ongoing dialogue look like? How can we record, track, post, and reflect on student work in a more public manner. Is there a Math wall dedicated to the days thinking and the “whole class” ideation? How can we elevate the status of low performing students and give them an opportunity to succeed in a more dynamic conversational environment?

What are the implications for classroom practice?
Asking higher-level questions and not settling for correct answers, by asking kids to prove their ideas, is a way to invite discussion and reflection into a lesson. We can ask why? How did you solve that? What does another student think of what you said? Can you Prove it? A second implication for classroom practice is collaborative work – SOLVING PROBLEMS IN TEAMS -  I believe this is an essential strategy for teaching math to kids in the classroom. It’s easy to over think group work. A way to simplify this in my mind, is to imagine playing games in teams. 2-on-2. For instance, to extend a manipulative lesson, ask kids to solve their problem in pairs. The first one with a justifiable solution earns 2 points and moves 2 places around a board game, a bonus round could include proving how they know this is true (deconstructing the problem), before moving to the next shape. And so on.