October 26, 2010
Barbara Shipman teaches an analysis course at the University of Texas. She uses inquiry dialogues with her students to help them to hone their thinking and to guide them to proof. She used those methods with us to stretch our understandings of infinity. I think we felt infinitely stretched! We used perfect pairings to prove that the set of whole numbers can be mapped to the set of even whole numbers and explored several other pairings of infinite sets.
Much of what was discussed is well beyond what is required by AIMS since AIMS defines the minimum standard for all. Students need to be stretched further in their mathematical understandings, however. Some of the AIMS Standards that were addressed include:
HSMS1C1-02 Sort sets of numbers as finite or infinite, and justify the sort.
HSMS1C3-04 Estimate the location of the rational or irrational numbers on a number line.
HSMS5C2-01 Analyze a problem situation, determine the question(s) to be answered, organize given information, determine how to represent the problem, and identify implicit and explicit assumptions that have been made.
HSMS5C2-02 Solve problems by formulating one or more strategies, applying the strategies, verifying the solution(s), and communicating the reasoning used to obtain the solution(s).
MS5C2-03 Evaluate a solution for reasonableness and interpret the meaning of the solution in the context of the original problem.
MS5C2-04 Generalize a solution strategy for a single problem to a class of related problems; explain the role of generalizations in inductive and deductive reasoning.
MS5C2-05 Summarize and communicate mathematical ideas using formal and informal reasoning.
MS5C2-06 Synthesize mathematical information from multiple sources to draw a conclusion, make inferences based on mathematical information, evaluate the conclusions of others, analyze a mathematical argument, and recognize flaws or gaps in reasoning.
MS5C2-08 Use inductive reasoning to make conjectures, use deductive reasoning to analyze and prove a valid conjecture, and develop a counterexample to refute an invalid conjecture.
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| Barbara Shipman introduces her talk. | Silvia, John and Tatiana discuss the pairings of infinite sets. |
