Analysis of proficient mathematicians’ problem-solving shows that their thinking is highly organised. It draws on a well-connected knowledge base of facts, methods and strategies that have been used to solve problems with a similar deep structure before. Successful problem-solving is therefore not just an activity but an outcome of successful learning of the facts and methods and their useful combinations as strategies. Conversely, if a problem-solver does not have conditional knowledge, they are more likely to be distracted by the surface features of problems. This has implications for how problem-solving as an activity is implemented in classrooms where teachers expect students to learn how to problem-solve by problem-solving.
Teachers could use a curricular approach that better engineers success in problem-solving by teaching:
Students need to be fluent with the relevant facts and methods before being expected to learn how to apply them to problem-solving conditions.
Teaching for Mastery has been at the heart of our lessons and in the forefront of our mind when teaching mathematics and planning our five-year curriculum for coherence. We have worked hard as a department, ensuring that key facts are mastered and effective representations expose the structure of the mathematics to enable conceptual understanding. We ensure fluency by carefully planning a wide range of questions and tasks in our lessons where students need to draw on prior knowledge. Our next target is to improve conditional knowledge by embedding variation tasks into our lessons to enable pupils to make connections and improve their reasoning skills. We are also embedding mathematical thinking strategies to enable pupils to improve their problem-solving skills.