Building the Foundation for Transfer
Few outcomes are more powerful than seeing students take a mathematical idea beyond the classroom, interpreting a news graph, applying proportional reasoning in science, or solving a real-world puzzle. This capacity to extend knowledge to new situations, known as “transfer,” is a hallmark of deep learning.
When learning is built on strong conceptual foundations, transfer becomes possible. Students don’t just recall procedures; they make sense of ideas, adapt them, and apply them with confidence.
Why Concepts Matter
Conceptual understanding is not an “extra” in mathematics; it is the foundation that gives skills meaning and longevity. While procedural fluency may yield short-term success, without underlying understanding, it often collapses when problems are presented in unfamiliar ways.
Frameworks illustrate this clearly:
- At Depth of Knowledge Levels 3 and 4, students reason and problem-solve beyond recall.
- At the relational and extended abstract levels of the SOLO taxonomy, students link and expand ideas, enabling transfer.
Current research reinforces this balance. The OECD’s PISA 2022 framework defines mathematical literacy as the ability to apply concepts and reasoning to explain and predict real-world phenomena. Similarly, the NCTM’s Catalyzing Change series (2018–2020) emphasizes that true proficiency requires both conceptual understanding and procedural fluency. Fluency without meaning is fragile; concepts without fluency lack power. When combined, they create durable and transferable learning.
Defining Conceptual Foundations
- Conceptual understanding: Knowing why a method works, not just how
- Foundations: Core principles such as place value, equivalence, and ratio that unlock future learning
These are not separate from skills, they strengthen and support them. Conceptual learning makes procedures meaningful rather than mechanical, ensuring that knowledge is both secure and adaptable.
Why This Matters for Equity
Conceptual teaching promotes lasting understanding for all learners. Students who only memorize steps may falter when problems change form, but students with strong conceptual grounding can reason through new challenges, invent strategies, and explain their thinking with clarity.
Classroom Example: Division of Fractions
Many adults recall the shortcut: Keep, change, flip. But ask why dividing by a fraction involves flipping and multiplying, and uncertainty often follows.
- Procedural only: Students memorize “invert and multiply.”
- Conceptual foundation: Students explore division as “how many groups?” With tape diagrams or number lines, they see that dividing by ½ means asking, How many halves fit into this amount?
In this approach, the procedure emerges naturally from the concept. One teacher reflected:
“Once my students saw fraction division on a number line, the ‘flip and multiply’ finally made sense, with one student suggesting that it wasn’t a trick anymore.”
Such moments transform mathematics into something coherent, purposeful, and empowering.
From Fragile to Flexible Knowledge
Surface-level knowledge is essential. It is at its most impactful when it leads to deeper-level learning. When students’ understanding remains at a “surface-level”, it can stay narrow and brittle. Students might treat decimals like whole numbers if place value has not been fully developed or hesitate with graphs if the slope’s rate of change is unclear.
By contrast, strong conceptual foundations foster flexible knowledge, promote questioning, probing, and problem-solving. Students adapt strategies to new problems, evaluate the reasonableness of answers, and articulate their reasoning with clarity. This flexibility equips them to approach unfamiliar challenges with both skill and confidence.
Supporting Concepts Through Dialogue
Purposeful dialogue is one of the most effective ways to deepen conceptual understanding. Teachers can invite reasoning and connections by asking:
- Why does this method work?
- Can you show it in another way?
- What links this model to the algorithm?
These prompts encourage students to justify their thinking, connect representations, and see mathematics as logical and interconnected. They also signal that what matters most is depth of understanding, not just speed.
The Role of Visuals and Representations
Concepts can become clearer when explored through multiple representations:
- Concrete: Manipulatives such as counters, base-ten blocks, or fraction strips.
- Pictorial: Diagrams, drawings, or graphs.
- Abstract: Symbols and equations.
This Concrete–Representational–Abstract (CRA) sequence (Bruner, 1966) provides a scaffold that bridges intuition and formal notation. For example, ratio tables and double number lines help students visualize proportional reasoning before moving to equations. Often, it’s a mixture of all three, enriched through regular classroom discussions and peer-to-peer interactions.
The Human Element: Teacher Expertise
Teachers are central to orchestrating the journey from procedures to concepts. They:
- Choose tasks that highlight relationships, not just rules.
- Model multiple strategies and representations.
- Encourage student reasoning, even when it takes longer.
As Hattie (2023) emphasizes, clarity and sequencing matter. Teachers who intentionally nurture conceptual foundations accelerate learning and empower students with tools that last.
Thinking Routines That Deepen Concepts
Simple, repeatable routines help make conceptual thinking visible:
- Same–Different: Present two problems or graphs. Ask: How are they alike? How are they different?
- Represent It Another Way: After solving, students show their thinking with a model, equation, or graph.
- Because–But–So: Offer a statement such as Fractions are numbers. Students complete it with “because … but … so …” to extend reasoning.
- A Simple Venn Diagram: Thinking about similarities and differences.
These routines create deliberate pauses for reflection and connection, supporting students to engage more deeply with the mathematics.
Join the Conversation
This blog is part of an ongoing conversation among educators. I’d love to hear from you:
- Share your reflections and stories using #MathSuccessJourney.
- Tag @CengageSchool, @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]
Looking Ahead
True fluency is not about recalling facts on demand; it’s an ongoing process of setting the right conditions for learning so that understanding grows. For students, procedural fluency is the ability to use procedures accurately, efficiently, and flexibly. How are you developing procedural fluency in your classroom?
Together, we can build a community where conceptual teaching empowers every learner to succeed.
Next blog: Strategy #4: Building Procedural Fluency.
References
- Biggs, J., & Collis, K. (1982). Evaluating the quality of learning: The SOLO taxonomy. Academic Press.
- Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
- Hattie, J. (2023). Visible Learning: The Sequel. Routledge.
- Hiebert, J., & Grouws, D. A. (2007). In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age Publishing.
- Nagle, C. (2024). High-quality instructional materials: Tools that build student success in mathematics. Big Ideas Learning.
- National Council of Teachers of Mathematics. (2018). Catalyzing change in high school mathematics: Initiating critical conversations. NCTM.
- National Council of Teachers of Mathematics. (2020). Catalyzing change in early childhood and elementary mathematics: Initiating critical conversations. NCTM.
- OECD. (2023). PISA 2022 results: Learning during and from disruption (Vol. I). OECD Publishing.
- Webb, N. L. (1997). Criteria for alignment of expectations and assessments in mathematics and science education. Council of Chief State School Officers.