Students grasp the "why" behind formulas, leading to deeper, more durable learning.
When evaluating a mathematical assertion or a peer's solution, students must:
Implementing the Concrete-Representational-Abstract (CRA) Model
When proficient mathematical thinkers solve a problem, they engage in a silent dialogue: estimating, visualizing, testing hypotheses, and checking for reasonableness. Struggling students rarely see this process; they only see the peer who raises their hand first with the correct answer. Visible Thinking exposes the messy, iterative nature of problem-solving, showing students that confusion and revision are natural parts of doing math. 2. Formative Assessment in Real-Time visible thinking in mathematics pdf
A variation of Notice and Wonder, this routine pushes students toward deeper interpretation and hypothesis testing.
Move away from being the "sage on the stage" who validates answers. Instead, act as a facilitator. Use neutral, probing responses like, "How did you arrive at that conclusion?" or "Can anyone build on Sarah's thinking?" Use Non-Permanent Vertical Surfaces (NPVS)
Students manipulate physical objects to understand a concept. For instance, physically grouping plastic tiles into rows to discover multiplication. Students grasp the "why" behind formulas, leading to
Students learn that intelligence is not fixed and that struggle is part of the learning process. How to Implement Visible Thinking: Finding Resources (PDFs)
Ready-to-print layouts for routines like See-Think-Wonder or Claim-Support-Question formatted specifically with grid squares or workspaces for mathematical sketches.
When the focus is entirely on the right answer, mistakes feel like failures. When the focus shifts to the thinking process , errors become valuable data points. Students begin to understand that confusion is a natural, productive step in mastering complex math. Formative Assessment Mastery Visible Thinking exposes the messy, iterative nature of
: Originally from the arts, this routine is powerful for geometry or data analysis. Students observe a pattern or graph, state what they see, what they think is happening, and what they wonder about the next step. Representation & Structure
Making the Invisible Visible: Transforming Mathematics Education Through Visible Thinking
Students move around the paper writing their thoughts, drawing diagrams, solving a piece of the problem, or drawing arrows to connect their ideas to a classmate's comment.
Here are some best practices for teaching mathematics with visible thinking: