The Dilemma: Too Much? Too Fast?
Have you ever reached the end of a math unit and thought: “We covered everything… but did my students really learn it?” If so, you’re not alone. Teachers everywhere face the tension between breadth and depth. The pressure to “cover it all” often leaves students with short-term recall but little ability to apply ideas in new contexts.
What Do We Mean by Focus and Coherence?
- Focus: Concentrating on the essential concepts students must carry forward
- Coherence: Ensuring today’s lesson builds on yesterday’s and sets the stage for tomorrow
Focus ensures depth over breadth. Coherence ensures that math makes sense across years of learning, linking fractions to ratios, or algebraic expressions to functions. Together, they transform a checklist of topics into a connected journey. As the National Council of Teachers of Mathematics reminds us: mathematics isn’t just a list of skills; it’s a web of ideas.
John Hattie (2023) identifies teacher clarity and curriculum sequencing as two of the strongest accelerators of student progress. When teachers make learning intentions explicit and structure content deliberately, students thrive. By contrast, when focus and coherence are missing, students often disengage, especially when math feels disconnected from their lives or is not seen as purposeful (Andrà et al., 2017; Fredricks et al., 2018; Grootenboer & Marshman, 2024; Lloyd, 2024). The pattern is clear: without focus and coherence, mathematics risks becoming abstract, fragmented, and irrelevant.
Why This Matters: Achievement and Equity
This isn’t just about test scores; it’s about equity.
Without focus, students may skim the surface and rely solely on working memory. Without coherence, math can feel random: fractions one day, probability the next. How can our students connect the dots if they are unsure about the dots or if the dots are not clear at all? By intentionally building focus and coherence, teachers ensure that all students experience mathematics as meaningful and connected. This supports both achievement and access, closing gaps and providing every learner with the opportunity to succeed.
From Procedure to Concept: A Classroom Example
Take a unit on fractions.
- A procedural approach might involve simplifying one day, comparing the next, and quickly shifting to decimals.
- A focus-and-coherence approach emphasizes underlying ideas:
- What fractions represent
- How do they connect to ratios
- How both reappear later in slope, probability, and proportional reasoning
Using visuals, story problems, or real-world contexts (recipes, migration data, climate statistics), fractions become part of a coherent narrative. Students don’t just manipulate numbers in isolation, they see patterns that stretch across years of learning. When math feels relevant, students are far more likely to describe it as engaging and worthwhile, rather than just steps to memorize.
Supporting Focus and Coherence with Dialogue
Classroom dialogue is key to deepening focus and coherence. Probing questions stretch student thinking and encourage reasoning:
- “What happens if we double the numerator and denominator of this fraction?”
- "How is this ratio table like a line graph?”
The use of Generative AI can support teachers to generate question prompts and discussion starters within a lesson as it is taking place. Noting that, AI doesn’t replace teacher expertise; it can amplify it. Try generating mathematical question prompts to plan and guide thinking, allowing more time for engaging in rich conversations with your students. How are you using AI in your classroom to enhance focus and coherence? I would love to hear from you and share this with our math community.
The Human Element: Teacher Expertise
At the heart of focus and coherence is teacher expertise. Teachers can:
- Select which concepts to emphasize
- Connect ideas across lessons and years
- Foster dialogue that helps students reason and justify
As Hattie (2023) reminds us, when teachers are deliberate and clear, student progress accelerates. Technology can assist, but teachers orchestrate the learning.
Thinking Routines That Build Coherence
Here are three easy-to-use routines to highlight connections and slow down thinking:
- See–Think–Wonder: Use visuals (graphs, ratio tables, number lines). Ask: What do you see? What do you think is happening? What do you wonder?
- Connect–Extend–Challenge: After a lesson, prompt reflection: How does this connect to what we know? How does it extend our understanding?
- I Used to Think . . . Now I Think . . . After exploring a concept (fractions → ratios → slope), invite students to articulate how their thinking has grown.
These routines highlight patterns, strengthen coherence, and give every student a voice in the conversation.
Join the Conversation
This blog series isn’t meant to be read in isolation. It’s an ongoing conversation among educators. I’d love to hear from you, hear what works for you, and your context.
- Share your reflections and stories using #MathSuccessJourney.
- Tag @NationalGeographicLearning, @BigIdeasLearning, and @SophieSpecjal
- Join the mid-series webinar for interactive Q&A and teacher spotlights.
- Contribute classroom examples for future spotlights by contacting me at [email protected]
Your experiences bring these strategies to life. Together, we can show how focus and coherence transform classrooms everywhere.
Looking Ahead
Mathematics education has never had a stronger research base or a more dedicated community of teachers. The future is bright, and it begins with you—the educators who spark curiosity, build confidence, and prepare students for a lifetime of mathematical thinking.
Next blog: Building Success in Mathematics: Teaching with HQIM to Support Balance and Rigor.
References
Andrà, C., Di Martino, P., & Sabena, C. (2017). Perceptions of mathematics. Educational Studies in Mathematics, 94 (3), 307–324.
Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2018). Student engagement in mathematics: Findings from middle and high school students.
Journal of Educational Psychology, 110 (2), 295–312.
Grootenboer, P., & Marshman, M. (2024). Young learners’ attitudes towards mathematics in New Zealand. Springer.
Hattie, J. (2023). Visible Learning: The Sequel. Routledge.
Lloyd, J. (2024). Perceptions of mathematics among primary students. British Journal of Educational Studies, 72 (1), 55–71.
Lloyd, M. E. R. (2013). Transfer of practices and conceptions of teaching and learning mathematics. Action in Teacher Education, 35 (2), 103–124.
Nagle, C. (2024). High-quality instructional materials: Tools that build student success in mathematics. Big Ideas Learning.
OECD. (2023). PISA 2022 results: Learning during and from disruption. OECD Publishing.