Matthias Kawski Functions Looking ahead beyond calculus Kingfisher

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Functions: Looking ahead, beyond calculus Matthias Kawski Department of Mathematics & Statistics Arizona State University Tempe, AZ U. S. A. kawski@asu. edu http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 (19 max 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 http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Mathematics education research CERTAINLY, NOT everyone in math education, but a prominent large group (eg recent ARUME program) 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 [? ] http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Textbooks versus what do the teachers and students see, what do they skip? 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”? ) Definitions from standard calculus textbook by Stewart (5 th edition) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Textbooks define “functions”, but… 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… 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”? ) Definitions from standard calculus textbook by Stewart (5 th edition) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Everyone teaches functions • • ensuring the continuity of an EVOLVING concept what other classes do the teachers teach? The 1985 -1995 picture at ASU and alike, and their “feeders” Definite need for bridge courses Algebra Geometry High school teachers, instructors small classes mostly equations Transition, abstraction, authoring proofs College algebra Precalculus Diff Equations Calculus Adv. Calc / Intro. Analysis Linear Algebra Vector Calculus Abstract Algebra Complex. Anal, PDEs, … Research faculty large lectures at some places small classes continuous evolution of functions all the way to functional analysis, categories http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Everyone teaches functions • • ensuring the continuity of an evolving concept what other classes do the teachers teach? The 2000 -2005 picture at ASU and alike Transition, abstraction, authoring proofs Algebra College Algebra Geometry Precalculus Calculus Diff Equations Adv. Calc / Intro. Analysi Linear Algebra Abstract Algebra Complex. Anal, PDEs, … Vec tor Calculus High school teachers, instructors small classes functions in view of preparing for calculus Research faculty small classes Functions: no continuity, need to first wipe the slate clean. Start over. 2004: 175, 000 (50, 000) students take AB (BC) AP-calculus tests, many more take hi-school calc classes` http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Domain • Find the derivative of of y = 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? ? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 ! http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS Compositions 1 One of the most simple questions about compositions… success rate? http: //math. asu. edu/~kawski@asu. edu

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

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 YX inherit from X? from Y? • If f and g are decreasing (order reversing), then f-1 is _____ and (f ◦ g) is ______ ? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS VC: Preserving structure 2, linearity 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? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 05, Fraser Island AUS LA: Multiplying tables • Where is the function? Where are the functions? • Why multiply matrices the way we multiply matrices ? • Associativity ? 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 ? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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? http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 ……………? ) http: //math. asu. edu/~kawski@asu. edu

Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 8 --- 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 ? http: //math. asu. edu/~kawski@asu. edu