Functions: Looking ahead, beyond calculus

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Functions Looking ahead, beyond calculus

• Matthias Kawski
• Department of Mathematics Statistics
• Arizona State University
• Tempe, AZ U.S.A.kawski_at_asu.eduhttp//math.asu.
  edu/kawski

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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

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

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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??
  

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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 !

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Compositions 1
One of the most simple questions about
compositions success rate?
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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?

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Preserving structure 1

• What is the point of (fg)(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 ___________
  ?

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VCPreserving 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?
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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 ?
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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?
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Linear equation?? function!!

• Linear equation ??
• Linear function!!
• Linear differential operator (NOT equation
  )superposition principle
• Composition of differential operators(inverse of
  a linear function is ?)

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Quantifiers versus functions

• An equilibrium point xe of a differential
  equation is called asymptotically stable if
• An equilibrium point xe of a differential
  equation is called asymptotically stable if there
  exists a KL-function b such that for all tgt0 and
  all x0

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• Common strategy in analysisIn order to avoid
  excessive numbers of alternations of universal
  and existential quantifiers encapsulate these
  into functions
• How does this affect our teaching of functions?

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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.