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1.1 ConceptsProblem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned. NCTM Standards 2000, page 182Write an explanation or definition of what it means for a question to involve “problem solving”. Select a topic and show and explain how “problem solving can provide a context in which a concept or skill in this topic can be learned”, as stated in the Standards. 1.1 Teaching Of the many descriptions of problem-solving strategies, some of the best known can be found in the work of Polya (1957). Frequently sited strategies include using diagrams, looking for patterns, listing all possibilities, trying special values or cases, working backward, guessing and checking, creating an equivalent problem, and creating a simpler problem. NCTM Standards 2000, page 53In years past it was a common practice for teachers to tell students not to draw pictures or sketches because “you can’t prove anything by drawings”. Today it is common for teachers to encourage students to form sketches to help solve problems. Discuss this change in approach to teaching mathematics. List some advantages and disadvantages to solving problems by drawings. Give an example to illustrate each disadvantage. 1.1 Teaching Doing mathematics involves discovery. Conjecture – that is, informed guessing – is a major pathway to discovery. Teachers and researchers agree that students can learn to make, refine, and test conjectures in elementary school. NCTM Standards 2000, page 57At one time teachers scolded students for guessing the answers to problems. In recent years mathematics educators have recommended that guessing and checking be taught to school students. Write a few sentences to discuss the advantages of teaching students to “guess and check”. |