| Routine | Purpose | Math Prompt Example | |---------|---------|----------------------| | | Initial exploration of a problem, graph, or pattern | See : three blue shapes, Think : maybe it’s a pattern of +2 sides, Wonder : what comes after 9 sides? | | What makes you say that? | Justifying reasoning | “I think 17 is prime.” — “What makes you say that?” | | Claim-Support-Question | Building arguments | Claim: “The sum of two odds is even.” Support: “odd+odd = (2m+1)+(2n+1)=2(m+n+1).” Question: “Does this work for negative odds?” | | Connect-Extend-Challenge | Linking new math ideas to prior knowledge | After learning integer division: Connect to sharing cookies; Extend to zero; Challenge: what does ÷ by a negative mean? | | I used to think… Now I think… | Metacognitive change | “I used to think commutative works for subtraction; now I think it doesn’t because 5–3 ≠ 3–5.” |
To make mathematical thinking truly visible, instruction should mirror the instructional sequence. This sequence provides the physical and visual scaffolding required for deep thinking. Description How It Makes Thinking Visible 1. Concrete
Thinking routines are simple, repeatable structures that become part of the classroom culture. Common examples include: Visible Thinking in Mathematics 2A | PDF | Thought - Scribd
What literal mathematical attributes are present? (e.g., "I see three blue triangles.") visible thinking in mathematics pdf
Teachers can share annotated PDFs, allowing students to exchange summaries and notes while keeping the original routines intact.
: Thinking is made visible when teachers and students explain their reasoning out loud, record it in journals, or use manipulatives and technology to demonstrate their process.
This involves overt, conscious, and deliberate efforts by both students and teachers to articulate, share, and reflect on their thought processes. Research indicates that in a math classroom, making thinking visible is a key to unlocking mathematics learning and success, as it moves students beyond simply getting the right answer to truly understanding the "why" and "how" of problem-solving. Visible thinking moves the focus from procedural performance to the development of deep, conceptual understanding. | Routine | Purpose | Math Prompt Example
The core idea is simple yet profound: This involves using tools like diagrams, drawings, and discussions to externalize the thought process. In mathematics, this means transforming problem-solving from a private, often internal monologue into a tangible, shareable artifact.
: It encourages mathematical discourse, where students learn to argue claims and provide evidence.
Students voice questions or uncertainties. ("I wonder what happens if we add another row? I wonder if the blue dots change the total value?") Routine 2: Think-Puzzle-Explore | | I used to think… Now I
Thinking must be captured to be visible. This involves using vertical non-permanent surfaces (whiteboards), math journals, digital annotation tools, and screencasting apps where students record their voices as they solve problems. 3. A Metacognitive Culture
Visible Thinking is an intentional instructional approach that uses practices, routines, and artifacts to externalize students' internal cognitive processes. In a traditional math classroom, thinking is invisible; teachers only see the final product (the answer) or the steps of an algorithm. If a student gets an answer wrong, it is difficult to diagnose where the misconception occurred.