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Critical Thinking: Does It Mean What You Think? On Conjugation of Cyclically-Generated Banach Spaces What’s So Critical About Thinking Anyway? Ocean’s Fourteen Auditions So What Exactly IS Critical Thinking? (And Can It Really Kill You?) Dave Sobecki Miami University* Hamilton * Miami University is not in Florida. When it was founded, Florida was a Spanish territory (1809). I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description ["hard-core pornography"]; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that. —Justice Potter Stewart, concurring opinion in Jacobellis v. Ohio 378 U.S. 184 (1964), regarding possible obscenity in The Lovers. Critical Thinking as Defined by the National Council for Excellence in Critical Thinking, 1987 A statement by Michael Scriven & Richard Paul {presented at the 8th Annual International Conference on Critical Thinking and Education Reform, Summer 1987}. Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action. In its exemplary form, it is based on universal intellectual values that transcend subject matter divisions: clarity, accuracy, precision, consistency, relevance, sound evidence, good reasons, depth, breadth, and fairness. HUH? “Dude, this class sucks.” “Dude, this textbook sucks.” “Math and science education is at a critical juncture in the United States.” • Analyze information provided in detail, and in a scholarly way • Evaluate a situation and judge what is useful in solving problems and drawing conclusions • A method of exercising one’s thinking ability in a way that is crucial to intellectual and personal development • A transition from childhood thinking to adult thinking • A process that leads to further intellectual development and encourages students to become lifelong learners. Definition By Contradiction Def: Critical thinking in math The OPPOSITE of memorization and mimicry. And now, the big question: The Obvious (and most common) Way: Critical thinking questions on HW/Tests But can we expect our students to excel at critical thinking questions if we don’t train them to, you know, THINK CRITICALLY? The More Important Way: TEACHING STYLE What do your students understand? What can they explain? ASK THEM! What is a variable? “It’s a letter, like x.” NO IT’S NOT Able to vary Able to vary What is an equation? “It’s a thing you solve for x.” NO IT’S NOT Statement Is 3y2 – 7y +12 true? What is solving an equation? “It’s getting x by itself on one side.” NO IT’S NOT Understanding vs. Doing What is solving an equation? “It’s getting x by itself on one side.” NO IT’S NOT Understanding vs. Doing What is a function? “It’s a formula, like f(x) = x2.” NO IT’S NOT Relationship “I solved the function.” What is a graph? “You plot points and connect the dots.” NO IT’S NOT Picturing Data Unemployment Rate by Year Year '92 '93 '94 '95 '96 '97 '98 '99 '00 '01 Rate (%) 7.5 6.9 6.1 5.6 5.4 4.9 4.5 4.2 4.0 4.7 Year '02 '03 '04 '05 '06 '07 '08 '09 '10 '11 Rate (%) 5.8 6.0 5.5 5.1 4.6 4.6 5.8 9.3 9.6 8.9 Assessment is important too! (A) Write a verbal description of the inequality x > –3, and a verbal description of the inequality x > 5. (B) The expression –3 < x > 5 is a combination of the two inequalities from part (A). Rewrite your two verbal descriptions from part (A) with the word “and” in between, then use the result to describe why the expression –3 < x > 5 is silly. Assessment is important too! ò 2 x +16 dx Assessment is important too! (A) Find the value of the expression (3 + 8)2. Don’t forget order of operations! (B) Find the value of 32 + 82. How does it compare to your answer from part (A)? (C) Rework parts (A) and (B) with any two nonzero numbers you choose. (D) What can you conclude about “distributing” an exponent? Assessment is important too! Explain what it means to solve an equation in your own words. But Dave, what about online homework? Can critical thinking play a role? Yes. Yes it can. Comments? Questions? Suggestions? Dirty jokes? [email protected] facebook.com/Teachingbackwards @teachbackwards