Pedagogy: How Students Learn and How Teachers Respond
How students learn mathematics
Ontario frames instruction through the Mathematical Processes - problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating. Pedagogy items ask how to teach and assess, not just how to compute.
Key concepts: the three-part lesson (Minds On, Action, Consolidate); problem-based learning in which the task launches the mathematics; the concrete–representational–abstract progression; math talk and number talks; productive struggle; and fluency as flexible, efficient strategy use rather than speed. Assessment literacy matters: assessment for learning (diagnostic and formative, feeding next steps), as learning (self- and peer-assessment), and of learning (summative). Differentiation adjusts content, process, product, or environment using evidence - it is not fixed ability grouping. Universal Design for Learning and culturally responsive teaching, including First Nations, Métis and Inuit connections such as beadwork, birchbark biting, and star knowledge, are named expectations.
Traps
Equating fluency with timed drills; treating errors as failures rather than as evidence of a misconception; assuming manipulatives are only for struggling students (they support all learners and must connect back to symbols); confusing differentiation with streaming; choosing "restate the rule" over letting students construct understanding; and mislabelling a summative task as formative.
Worked example
A Grade 5 class keeps writing 1/2 + 1/3 = 2/5. The pedagogical move is to diagnose the misconception - adding numerators and denominators by analogy with whole numbers - then design a task with fraction strips or a number line so students see the sum must exceed 1/2. Assess formatively, then consolidate with a common-denominator representation.
Rule of thumb: when an option offers "explain the rule again" versus "elicit and confront the misconception with a representation," choose the second.