Functions: Looking ahead, beyond calculus

http://math.asu.edu/~kawski [email protected]

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.

[email protected]://math.asu.edu/~kawski



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



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)



Textbooks versus what do the teachers and students see, what do they skip?

Definitions from standard calculus textbook by Stewart (5th 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”?)



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 (5th 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…



Calculus Linear AlgebraAlgebraGeometry

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 places small classes

mostly equations continuous evolution of functions all the way to functional analysis, categories

The 1985-1995 picture at ASU and alike, and their “feeders”

College algebraPrecalculus

Definite need for

bridge courses



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 calculusFunctions:

no continuity, need to first wipe the slate clean. Start over.

High school teachers, instructors

small classes small classes

Linear AlgebraAlgebra College AlgebraGeometry Precalculus Calculus

AdvCalc / IntroAnalysisDiff Equations

Vec tor Calculus ComplexAnal, PDEs, …

Abstract Algebra

Transition, abstraction, authoring proofs

The 2000-2005 picture at ASU and alike

2004: 175,000 (50,000) students take AB (BC) AP-calculus tests, many more take hi-school calc classes`



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



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??



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 !



Compositions 1

One of the most simple questions about compositions… success rate?



• 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



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 ___________ ?



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



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 ?



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?



Linear equation?? function!!

• Linear equation ??

• Linear function!!

• Linear differential operator (NOT: equation )“superposition principle”

• Composition of differential operators(inverse of a linear function is ……………?)



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 2nd time in 10 or 15 years. Do I know enough about functions ?

Documents

Functions: Looking ahead, beyond calculus