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A Course Emphasizing Logic and Reasoning: The Language of Mathematics Warren Esty Professor of Mathematics Montana State University [email protected] Slide 2 The world is changing rapidly. Is teaching keeping up? What do we teach? Slide 3 Information Skills Slide 4 Impart information, computational skills Information is cheap Facts are on the web Computational skills are cheap If anyone can compute it, calculators and computers can do it accurately and faster Expert systems are replacing people Slide 5 Theorem: The value of traditional mathematical skills has gone way down. Corollary: We should refocus our teaching toward skills that add value. Slide 6 What adds value? Knowing how to use calculators and computers Knowing when to use various math algorithms Experience with problems Also, of course, many traditional activities (especially developing concepts) Slide 7 Add value by Preparing for lifelong learning Learn to read! Learning to make (the right) things come to mind Learning to reason logically Slide 8 The problem with searches I can just look it up. However, it must come to mind And you must be able grasp it when you find it Slide 9 Slide 10 We found that kids hit a wall, and that wall is called fourth grade. At that moment, a kid shifts from learning to read to having to read in order to learn. -- David Britt, Childrens Television Workshop President, in 1992 Slide 11 Half of all kids never make that transition. -- Colette Daiute, Harvard Professor of Human Development, in 1992 Slide 12 Imagine how much worse the statistics would be if they were about the fraction of kids who can read mathematics to learn mathematics. Slide 13 Why is reading important? Slide 14 Do you really think students can read math? Why should they be able to? Who ever taught them to? Who ever required them to? Slide 15 Math is difficult to read It is concise and precise in a non-concise and non-precise age [0, 1] (0, 1) {0, 1} are different in important ways that are alien to our students Slide 16 43 College calc profs asked their students 8% read most of the chapters 17% read sections they didn't understand from lectures 69% typically started by working homework and turned to examples if they had trouble 3% said they never opened the book. Slide 17 Is Math a Language? Communication By symbols Non-instinctive Conventional, learned meanings Shared by a community Slide 18 Algebraic Language has Vocabulary nouns, pronouns, verbs, expression, factor, Grammar Syntax2x 2 is not (2x) 2 Pronunciation {x | x 2 > 25} Synonyms If x > 5, then x 2 > 25, For all x > 5, x 2 > 25. Negations negate: If x 2 > 25, then x > 5. Conventions 3x 2 Abbreviations Sentence and paragraph structure Slide 19 Placeholders 3(x + 4) = 18 3(x + 4) = 3x + 12 3(c + 4) = 3c + 12 Let f(x) = x 2. f is not a number, f(x) is. Find f(x+h) = Slide 20 How do you add fractions? Explain this in English Slide 21 Explain this in Mathematics Slide 22 How do you solve these? x + 4 = 13 or x 2 + 12 = 100 x/3 + 7 = 42 Generally, for the first step: x + a = b iff x = b - a Slide 23 Pattern recognition x + a = b iff x = b a problem-pattern solution-pattern Slide 24 How do you factor this? x 2 + bx + c x 2 + 10x + 16 x 2 + bx + c = (x + d)(x + k) iff b = d+k and c = dk. The left-side pattern factors into the right-side pattern under certain conditions. Slide 25 What is the best way to learn a language? Spanish? German? Start very young Interact with others who use it Slide 26 Our children cant Start young Many elementary-school teachers dont know the language And avoid it And the curriculum lets them We often start the language in 8 th or 9 th grade (late!) Slide 27 Few El-Ed students choose extra math They dont have time in their curriculum They are not expected to be responsible for algebra A math-as-a-language course is not traditional Few colleges have one Slide 28 Linguists assert: It is difficult to have and retain thoughts without the proper language in which to categorize and express them. Musical notation Symbolic mathematics Slide 29 Learn to read Theorem: 1+2+3++n = n(n+1)/2. Find 1+2+3+4++70 Find 1+2+3++n++(n+5) Slide 30 The Quadratic Theorem If ax 2 + bx + c = 0 and a is not 0, Then x = Find x when 2x 2 kx = 12 Find y when x 2 + 3x + 5y 2 12y = 100 Find b when c 2 = a 2 + b 2 2ab cos(C) Find x when sin x + (sin x) 2 = 0.82 Slide 31 The Language of Mathematics 1. Algebra as a language Abstraction, Patterns, Order, Reading, Arithmetic methods expressed 2. Sets, functions, algebra Notation, Methods expressed 3. Logic for Mathematics (logical equivalences) 4. Sentences,Variables, Generalizations, Existence Statements, Negations 5. Proofs (paragraphs in the language) Slide 32 New courses Who will go to bat for one? It is not a traditional course Previous teachers, administrators, parents, didnt take this course (It didnt exist) Math profs usually dont care much about elementary ed, or have much influence over it Not everyone realizes the language aspects of mathematical symbolism shouldnt students just get that by osmosis in their math classes? Colleges readily accept new courses if students will take them, but Who will take it, if it is not required? Slide 33 The world has changed. Conclusion Information is incredibly cheap Calculations are incredibly cheap Theorem: Much of the math we have been teaching is not worth much. Learning to read is not easy Learning to read is worth a lot We must enable our teachers to help students learn to read Mathematics. Slide 34 References http://augustusmath.hypermart.net/ Language Concepts of Mathematics, by Warren Esty, Focus on Learning Problems in Mathematics, 1992, Volume 14, number 4, pages 31-54 A General Education Course emphasizing Mathematical Language and Reasoning, by Warren Esty and Anne Teppo, Focus on Learning Problems in Mathematics, 1994, Volume 16, number 1, pages 13-35. The Assessment of Mathematical Logic: Abstract Patterns and Familiar Contexts (joint with Anne Teppo and Kay Kirkpatrick), Psychology of Mathematics Education (Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education), 2003, 283-290. Slide 35 Users Profs. Robert Fisher and Cathy Kriloff, Idaho State University. [email protected] [email protected]@isu.edu Prof. Genevieve Knight, Coppin State University, Baltimore. [email protected]@coppin.edu Prof. Mircea Martin, Baker University, Baldwin City, Kansas. [email protected]

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