Download - 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
http://math.asu.edu/~kawski [email protected]
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 – “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/~kawski [email protected]
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)
http://math.asu.edu/~kawski [email protected]
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 (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”?)
http://math.asu.edu/~kawski [email protected]
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 (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…
http://math.asu.edu/~kawski [email protected]
Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS
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
http://math.asu.edu/~kawski [email protected]
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 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`
http://math.asu.edu/~kawski [email protected]
Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher elta 05, Fraser Island AUS
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/~kawski [email protected]
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??
http://math.asu.edu/~kawski [email protected]
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 !
http://math.asu.edu/~kawski [email protected]
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
http://math.asu.edu/~kawski [email protected]
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 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/~kawski [email protected]
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
http://math.asu.edu/~kawski [email protected]
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 ?
http://math.asu.edu/~kawski [email protected]
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?
http://math.asu.edu/~kawski [email protected]
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 ……………?)
http://math.asu.edu/~kawski [email protected]
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 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 ?