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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Functions: Looking ahead, beyond calculus Matthias Kawski Department of Mathematics & Statistics Arizona State University Tempe, AZ U.S.A.

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Background Largest US university campus (52,000+ students) at public research university (14,000 stud math/sem) continuing push twds smaller classes (19max stud/class) dual system: research faculty – “1 st year math” instructors “unhappiness” w/ students’ understanding of the concept of functions upon entering post-calculus courses prominent math education research claiming to study learning of “functions” – mismatch w/ fcn beyond calc personal interactions w/ middle/hi school math teachers

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Mathematics education research Personal concern about this “authoritative article” about what matters about functions: 16 pages consider only real [?]-valued functions defined on (unions of) intervals A R [?] CERTAINLY, NOT everyone in math education, but a prominent large group (eg recent ARUME program)

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Textbooks versus what do the teachers and students see, what do they skip? Definitions from standard calculus textbook by Stewart (5 th edition) The teacher’s decision: ignore, or how much to explore other than the “usual” (in this class) examples of functions (“are they on the exam”?)

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS An “awesome” text [?]: The teacher finds what (s)he is looking 4, while the student can safely ignore these “decorations” Textbooks define “functions”, but… Definitions from standard calculus textbook by Stewart (5 th edition) The teacher’s CHOICE: Ignore, or how much to emphasize that these are just more “examples” of functions. DECIDE whether to discuss their properties in this specific context or merely as other “instantiations of universal properties of functions” (“what will be on the exam”?) An “awesome” text [?]: The teacher finds what (s)he is looking 4, while the student can safely ignore these “decorations” which are there only 4 the teacher, not in the exercises and will not on the exams…

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Calculus Linear Algebra Algebra Geometry AdvCalc / IntroAnalysisDiff Equations Vector Calculus ComplexAnal, PDEs, … Abstract Algebra Transition, abstraction, authoring proofs Everyone teaches functions High school teachers, instructors Research faculty ensuring the continuity of an EVOLVING concept what other classes do the teachers teach? small classes large lectures at some placessmall classes mostly equationscontinuous evolution of functions all the way to functional analysis, categories The picture at ASU and alike, and their “feeders” College algebra Precalculus Definite need for bridge courses

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Everyone teaches functions ensuring the continuity of an evolving concept what other classes do the teachers teach? Research faculty small classes functions in view of preparing for calculus Functions: no continuity, need to first wipe the slate clean. Start over. High school teachers, instructors small classes Linear Algebra Algebra College Algebra Geometry Precalculus Calculus AdvCalc / IntroAnalysisDiff Equations Vec tor Calculus ComplexAnal, PDEs, … Abstract Algebra Transition, abstraction, authoring proofs The picture at ASU and alike 2004: 175,000 (50,000) students take AB (BC) AP-calculus tests, many more take hi-school calc classes`

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Selected typical questions (Low pressure) 1 st day of class diagnostic tests –amazing insights into students preparation –interesting correlation students’ preparation - success Examples of simple functions post-calculus

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Domain Find the derivative of of y = log (log ( sin(x)) and overlay the graphs of y and y’. The domain of y is empty – yet most everyone finds a function y’ with nonempty domain??

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Mapping – computer algebra Many students consider to be hard But the detour via complicated functions works “You mean a function is, -- is, just like / the same as a subroutine/procedure?” Take advantage of the students’ programming classes !

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Compositions 1 One of the most simple questions about compositions… success rate?

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Simplify If g = f -1, then the inverse of x g(x-1)= …..? Solve for x IN ONE STEP what is this important for? Compositions 2

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Preserving structure 1 What is the point of (f+g)(x)=f(x)+g(x) ? Does it matter? What for? Who cares? What structures does Y X inherit from X? from Y? If f and g are decreasing (order reversing), then f -1 is __________ and (f ◦ g) is ___________ ?

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS When teaching “linear functions”, what are the key points? What are we looking at as the long term goal? What definition of linearity for whom? Vector fields are functions. Which is / are linear? VC:Preserving structure 2, linearity

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS LA: Multiplying tables Why multiply matrices the way we multiply matrices ? Multiplication by a matrix is a function, just like “times 3” is a function. Do the teachers teach and the students learn about functions like *3 ? Where is the function? Where are the functions? Associativity ?

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS From equations to functions Sketch the graph of How big a step is it to ? Think how it helps in Are we thinking ahead – preparing for the next incidence of the same step, or will the students have to do everything again from scratch?

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Linear equation?? function!! Linear equation ?? Linear function!! Linear differential operator (NOT: equation ) “superposition principle” Composition of differential operators (inverse of a linear function is ……………?)

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Quantifiers versus functions An equilibrium point x e of a differential equation is called asymptotically stable if An equilibrium point x e of a differential equation is called asymptotically stable if there exists a KL-function such that for all t>0 and all x 0

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Common strategy in analysis: In order to avoid excessive numbers of alternations of universal and existential quantifiers encapsulate these into functions How does this affect our teaching of functions?

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Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS Summary and conclusion Maybe my worries are unfounded, or my home institution is highly unusual…. would be great news. (My daughters are in grades 7 and pretty scary [only ??] in the US.) In any case, we all want teachers to know / look ahead significantly beyond the class they teach (compare Liping Ma, grades K-4), so that they can make well-informed decisions (depending on their specific environs) what to emphasize, what to barely discuss at all. It is us mathematicians / math-education researchers are responsible for the curriculum of current in-service and future teachers. Personal worry: Next spring I’ll teach point set topology and applied complex analysis, each for 2 nd time in 10 or 15 years. Do I know enough about functions ?

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