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2.3 ConceptsStudents should discuss their reasoning on a regular basis with the teacher and with one another, explaining the basis for their conjectures and the rational for their mathematical assertions. Through these experiences, students should become more proficient in using inductive and deductive reasoning appropriately. NCTM Standards 2000, page 262A conditional statement of the form, if an element is in K, then it is in M, can be illustrated using Venn diagrams and placing a circle representing all the elements in K inside the circle representing all the elements of M. Write the converse and the contrapositive of this conditional statement and sketch the Venn diagram for each of these new statements. Assuming that the original conditional statement is true, write an explanation as to whether or not the converse or contrapositive statements are true by referring to the visual diagrams in your explanation. 2.3 Teaching One of the 10 major topics recommended in Standards 2000 for all grade levels from preschool through grade 12 is Reasoning and Proof. Discuss the difference between inductive and deductive reasoning and give examples of both. Are these types of reasoning appropriate to include in the elementary or middle school curriculum? Support your position with examples and reasons. 2.3 Concepts Students need to explain and justify their thinking and learn how to detect fallacies and critique others’ thinking. They need to have ample opportunity to apply their reasoning skills and justify their thinking in mathematics discussions. NCTM Standards 2000, page188From given statements, it is possible to derive a new statement which is false using valid reasoning. It is also possible to derive a true statement using invalid reasoning. Give examples of both of these cases and discuss both situations. What conditions are necessary for the given statement in order for valid reasoning to produce a true statement or for valid reasoning to produce false statement? |