Transcript Functions: Looking ahead, beyond calculus

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.
[email protected]
http://math.asu.edu/~kawski
http://math.asu.edu/~kawski
[email protected]
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 (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 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
[email protected]
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 (5th edition)
http://math.asu.edu/~kawski
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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 (5th edition)
http://math.asu.edu/~kawski
[email protected]
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
Transition, abstraction,
authoring proofs
College algebra
Precalculus
Algebra
Geometry
Diff Equations
Calculus
Linear Algebra
Vector Calculus
High school teachers, instructors
small classes
mostly equations
AdvCalc / IntroAnalysis
Abstract Algebra
ComplexAnal, 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
[email protected]
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
AdvCalc / IntroAnalysi
Linear Algebra
Abstract Algebra
ComplexAnal, PDEs, …
Vec tor Calculus
High school teachers, instructors
small classes
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
[email protected]
Matthias Kawski “Functions: Looking ahead beyond calculus” Kingfisher Delta 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 Delta 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 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
[email protected]
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
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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
[email protected]
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-1is __________ and (f ◦ g) is ___________ ?
http://math.asu.edu/~kawski
[email protected]
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
[email protected]
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
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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
[email protected]
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
[email protected]
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 2nd time in 10 or 15 years. Do I know enough about functions
?
http://math.asu.edu/~kawski
[email protected]