In mathematics, metaphors can provide powerful insights into abstract concepts. In my last post, I explored mathematical proficiency, which is often depicted as a tightly woven rope. Each of its five strands—understanding, fluency, problem-solving, reasoning, and productive disposition—plays an essential, interconnected role in developing well-rounded mathematical learners.
Today, I’d like to approach mathematical proficiency through a different lens: exploring the interconnection of knowledge, skills, and attitudes using three different metaphors that illustrate how the elements work together to foster true mathematical proficiency.
Knowledge includes the acquisition and comprehension of factual, conceptual, and procedural understanding. But knowing mathematics is not enough on its own. We must also develop skills—competencies such as mathematical reasoning, problem-solving, metacognition, and collaboration, which allow students to apply their knowledge in meaningful ways. Finally, attitudes—the productive dispositions that guide how students approach learning—are just as essential. These include perseverance, curiosity, confidence, and openness to overcoming challenges.
Different educational ideologies may prioritise one domain over others. For instance, while the Scholar Efficiency1 perspective places primary importance on the various forms of knowledge, the Learner-centred approach often prioritises attitudes and personal growth. However, as Kilpatrick and others argue, true mathematical proficiency depends on integration of all three domains. Without knowledge, students lack the foundational understanding needed to engage mathematically. Without skills, they cannot effectively apply what they know to solve problems or think critically. And without positive attitudes, they may not have the resilience or motivation to persevere through challenges and setbacks.
We can imagine these three domains—knowledge, skills, and attitudes—as a Venn diagram, with each domain overlapping the others to create a unified whole. Two structures inherent in this visualisation highlight their interconnectedness. First, the Borromean rings—three interlinked rings that cannot be separated from each other, but that break apart into unlinked rings when one of the three is removed. This metaphor captures the idea that all components, whether these three domains or the five proficiency strands, must be present for successful learning.
Next is the Reuleaux triangle, a curved triangle of constant width that rolls smoothly. This metaphor, emerging from where all three domains overlap in the Venn diagram, represents the balance required for mathematical learning. (You might enjoy googling “Reuleaux triangle bikes” for some fascinating examples.) When knowledge, skills, and attitudes are all present, students’ learning moves smoothly, with each area supporting the others.
So, how do these three domains work together in practice? Guy Claxton’s metaphor of the river of learning 2 offers a useful perspective. On the surface, students acquire and comprehend knowledge; this is often the most visible part of learning. Beneath the surface, they develop the skills and expertise that allow them to apply and manipulate that knowledge. At the deepest level, along the riverbed, lie the attitudes and habits that shape how they approach learning. Just like a river, a classroom has different layers of learning happening at the same time. These layers—comprehension, competencies, and character—flow at different speeds and levels of visibility. However, as Claxton notes, sometimes the surface can look smooth, while the current below bubbles with activity. “If you judge the river only by looking at the surface, you’ll be misled.” The most significant learning often happens beneath the surface. All these layers are part of learning and of developing mathematical proficiency.
As educators, we must nurture all these layers. Are we offering students enough opportunities to build knowledge, develop skills, and cultivate the attitudes towards learning mathematics? Claxton reminds us that “in practice these three layers are blended together and constantly interact. Teachers don’t have to stop transmitting knowledge to do something different called ‘cultivating attitudes’. Nevertheless, we all have to be alert to what is going on at the deepest levels, lest we inadvertently teach in a way that keeps our students floating on the surface.“
Classroom climate plays a critical role. Whether or not we explicitly address students’ dispositions, we always influence them through the environments we create. Students pick up on the implicit messages we send — about mistakes, effort, persistence, and more—which can shape their beliefs about themselves as learners and about mathematics.
Metaphors like the Borromean rings, the Reuleaux triangle, and Claxton’s river of learning help us visualise the complex, interdependent nature of mathematical proficiency. By nurturing knowledge, skills, and attitudes together, we create learning environments where students can thrive. Just as each metaphor shows, these elements work together. As you reflect on your own teaching practice, consider how you develop knowledge, skills, and attitudes. How do you ensure that all three domains are being nurtured in your classroom?
By striking the right balance, we create a learning environment that fosters not just competence but a genuine appreciation of mathematics. When knowledge, skills, and attitudes come together, students become confident, capable, and curious—ready to engage with mathematics in the classroom and beyond.
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