Functions: Looking ahead, beyond calculus

1. Functions: Looking ahead, beyond calculus Matthias Kawski Department of Mathematics & Statistics Arizona State University Tempe, AZ U.S.A.kawski@asu.eduhttp://math.asu.edu/~kawski http://math.asu.edu/~kawskikawski@asu.edu

2. 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 – “1st 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/~kawskikawski@asu.edu

3. 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/~kawskikawski@asu.edu

4. 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 (5th edition) http://math.asu.edu/~kawskikawski@asu.edu

5. Textbooks define “functions”, but… An “awesome” text [?]: The teacher finds what (s)he is looking 4, while the student can safely ignore these “decorations” 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 (5th edition) http://math.asu.edu/~kawskikawski@asu.edu

6. Definite need for bridge courses Transition, abstraction, authoring proofs College algebraPrecalculus AlgebraGeometry Calculus Diff Equations AdvCalc / IntroAnalysis Linear Algebra Abstract Algebra ComplexAnal, PDEs, … Vector Calculus 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” Research faculty High school teachers, instructors large lectures at some places small classes small classes mostly equations continuous evolution of functions all the way to functional analysis, categories http://math.asu.edu/~kawskikawski@asu.edu

7. Transition, abstraction, authoring proofs Algebra College AlgebraGeometry Precalculus Calculus Diff Equations AdvCalc / IntroAnalysis Linear Algebra Abstract Algebra ComplexAnal, PDEs, … Vec tor Calculus High school teachers, instructors Research faculty small classes small classes small classes 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 Functions: no continuity, need to first wipe the slate clean. Start over. functions in view of preparing for calculus 2004: 175,000 (50,000) students take AB (BC) AP-calculus tests, many more take hi-school calc classes` http://math.asu.edu/~kawskikawski@asu.edu

8. Selected typical questions • (Low pressure) 1st 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/~kawskikawski@asu.edu

9. 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?? http://math.asu.edu/~kawskikawski@asu.edu

10. 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/~kawskikawski@asu.edu

11. Compositions 1 One of the most simple questions about compositions… success rate? http://math.asu.edu/~kawskikawski@asu.edu

12. 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/~kawskikawski@asu.edu

13. 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-1is __________ and (f ◦ g) is ___________ ? http://math.asu.edu/~kawskikawski@asu.edu

14. 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/~kawskikawski@asu.edu

15. 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/~kawskikawski@asu.edu

16. 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/~kawskikawski@asu.edu

17. 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/~kawskikawski@asu.edu

18. Personal worry: Next spring I’ll teach point set topology and applied complex analysis, each for 2nd time in 10 or 15 years. Do I know enough about functions ? 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. http://math.asu.edu/~kawskikawski@asu.edu