Mathematical understanding is built through problem-solving, not as a result of direct instruction alone. Research supports the idea that students develop deeper mathematical reasoning when they solve problems rather than being taught isolated skills first.
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Hiebert et al. (1996) emphasize that students should encounter mathematical ideas as part of the problem-solving experience rather than learning procedures in advance.
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The Standards for Mathematical Practice (SMPs) reinforce that students must make sense of problems, reason abstractly, construct viable arguments, and model with mathematics as they engage in problem-solving.
Well-designed tasks provide students with opportunities to explore, test ideas, and build connections between concepts and procedures. When teachers structure lessons to support this, students develop confidence in their ability to think mathematically.
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Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development (Stein, Smith, Henningsen, & Silver) provides strategies for shifting instruction to center student reasoning and problem-solving rather than direct demonstration of procedures.
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The Coherence Map (Achieve the Core) helps identify how concepts build over time, ensuring that students experience math as a logical, connected progression rather than disconnected skills.