Making the thinking process "visible" so students can visualize concepts in their heads before moving to abstract symbols. Target Audience:
Searching for “Visible Thinking in Mathematics PDF” yields a wealth of structured routines, but the document alone is inert. The true transformation happens when a teacher prints a routine, projects it, and waits —allowing silence before asking, “What do you see?” The best visible thinking is not something you read; it is something you do. The PDF is merely the map. The journey is the classroom conversation where mathematical reasoning finally steps out of the shadows and onto the page.
What does it make you wonder? (Deep questions, like "Will the tenth step always be an even number?")
Unlocking Mathematical Understanding: A Comprehensive Guide to Visible Thinking in Mathematics
: Available for access and citation via the Semantic Scholar academic database. visible thinking in mathematics pdf
It builds observational skills and trains students to look for geometric and numerical patterns before diving into algebraic formulas. 3. Claim, Support, Question
Used when analyzing a mathematical strategy or a student's proposed solution. Claim: What is the student claiming about the answer? Support: What evidence supports this claim? Question: What questions remain? Before-During-After (BDA): Used to scaffold word problems. Before: What do I know? What am I looking for? During: What strategies am I trying? After: Did my plan work? How do I know?
Ready-to-use templates for routines like See-Think-Wonder and Circle of Viewpoints tailored for math.
: Connects initial thoughts on a topic to new learning after a lesson. Making the thinking process "visible" so students can
By following these recommendations and incorporating visible thinking strategies into their teaching practice, teachers can help students to develop a deeper understanding of mathematical concepts and relationships, and to become more confident and capable mathematicians.
"The Model Method in Singapore Primary Mathematics" Author: Ng Swee Fong Source: Mathematics Educator or similar educational journals focusing on Southeast Asian math pedagogy.
Use this routine when launching a new unit or a complex multi-step word problem.
What is going on in this problem? What mathematical operations or concepts do you think apply here? The PDF is merely the map
| 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.” |
Encourage students to share partial thinking and make mistakes, focusing on the reasoning rather than just the final answer.
What (e.g., fractions, algebra, geometry) are you planning next?