As a mathematician who later moved to education, I’ve noticed a strange contradiction: educational discourse often suffers from both ambiguity and certainty. In mathematics, precision is everything. Terms are defined with rigour, conditions are outlined meticulously, and assumptions are explicitly stated. This discipline demands clarity, as even a small misstep in language can unravel a proof or distort a concept. Yet, in educational discourse, clarity often seems to be the exception rather than the rule.
Education is a field of paradoxes. On one hand, vagaries around concepts obscure meaning and create confusion. On the other, certainty around educational practices is pervasive, with rigid ‘evidence-based’ methods often held up as universally applicable. Together, these twin traps—ambiguity and certainty—have real consequences for how we teach, how students learn, and how educational policies are shaped.
Terms that seem straightforward in education often prove slippery. Educators frequently use ambiguous terms like problem solving—a phrase that can mean anything from solving simple, routine exercises to tackling complex, open-ended problems—leading to confusion and misinterpretation. This phenomenon, known as the jingle-jangle fallacy, occurs when two concepts are mistakenly considered the same because they share a name (jingle) or where identical concepts are treated as different because they are labelled with different terms (jangle). Other common jingles include engagement, which can refer to anything from participation to interest, and feedback, used interchangeably to describe both formative and summative purposes. Likewise, metacognition and self-regulated learning are examples of jangles—terms with significant overlap that are often treated as distinct concepts.
In mathematics, precision extends beyond terminology: mathematicians are also careful to specify the conditions under which their results apply. Dynamical systems, such as the Lorenz system or the double pendulum, demonstrate that even a slight adjustment in initial values can result in a drastically different result—despite using the same system of equations.
The same principle applies in education: even with identical teaching methods, different inputs—such as students’ interest, readiness, resources, and classroom environment—can produce vastly different outcomes. Crucially, the outcome we aim for, whether understanding a concept, developing a skill, or fostering an attitude, also influences the choice and success of a teaching method. Certainty in applying a method without accounting for both inputs and desired outcomes can result in practices that fail to achieve their intended impact.
With this in mind, it becomes critical to evaluate educational claims with the same rigour mathematicians use. Mathematicians routinely ask two fundamental questions when faced with a claim: Is it true? and When is it true? This reminds me of the Always, Sometimes, Never1 routine, which encourages students to consider whether a mathematical statement is universally true, true in certain situations, or never true.
Educational claims and results should be approached with the same rigour. As Guy Claxton2 points out, “research can’t tell us ‘what works’ in teaching without specifying for what purpose, in what context, and with whom.” This is precisely where I believe pedagogical puritans—those who rigidly advocate for one method regardless of context—fall short. They advocate for their methods with absolute confidence but rarely appear to ask the fundamental questions: Is it true? When is it true? How often do they clearly outline the conditions under which their approach is effective? How often do they provide a clear picture of the specific outcomes they aim to achieve? Most importantly, how often do they ask you how their method fits into your context or aligns with your educational goals? I’d venture to say almost never.
We must hold all educational advocates accountable for answering these critical questions. And if they aren’t, it falls on us to ask. When presented with sweeping educational claims or a method that promises to be best practice, we must ask: Under what conditions does this work? What assumptions are being made? What is the specific purpose, and who does it truly serve?
Ambiguity and certainty may seem like opposites, but both hinder effective teaching. Imprecision creates obstacles, and rigidity reinforces them. When these forces dominate, we risk misapplying practices, misunderstanding outcomes, and ultimately failing our students.
To move forward, educators must avoid these traps by being intentional about language, approaching teaching with flexibility, and—crucially—fostering a mindset of critical thinking, regularly asking questions like: Under what conditions does this work? How does this fit into my context? and Is this the outcome I’m looking for? These questions are essential for evaluating the methods we use, the outcomes we expect, and the contexts in which our practices succeed or fail. Exercising our professional judgement in this way ensures that each teaching choice is both purposeful and responsive to the unique needs of our students.
- nrich has a good explanation of the routine and Tracy Zager has curated a great collection of statements. ↩︎
- Claxton, G. (2022, August 17). The Future of Teaching, and the Myths That Hold It Back (blog post). ↩︎
Wonder in Mathematics 