Research Sampler
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The concept of function is central to undergraduate mathematics, foundational to modern mathematics, and essential in related areas of the sciences. A strong understanding of the function concept is also essential for any student hoping to understand calculus - a critical course for the development of future scientists, engineers, and mathematicians.
Since 1883, there have been repeated calls for school curricula to place greater emphasis on functions (College Entrance Examination Board, 1959; Hamley, 1934; Hedrick, 1922; Klein, 1883; National Council of Teachers of Mathematics, 1934, 1989, 2000). Despite these and other calls, students continue to emerge from high school and freshman college courses with a weak understanding of this important concept (Carlson, 1998; Carlson, Jacobs, Coe, Larsen & Hsu, 2002; Cooney & Wilson, 1996; Monk, 1992; Monk & Nemirovsky, 1994; Thompson, 1994a).
This article provides an overview of what is involved in knowing and learning the function concept. We have included discussions of the reasoning abilities involved in understanding and using functions, including the dynamic conceptualizations needed for understanding major concepts of calculus. Our discussion also provides information about the common conceptual obstacles that have been observed in students. We make frequent use of examples to illustrate the 'ways of thinking' and major understandings that research suggests are essential for students' effective use of functions during problem solving; and that are needed for students' continued mathematics learning.
Studies have revealed that learning the
function concept is complex, with many high
performing undergraduates (e.g., students
receiving course grades of A in calculus)
possessing weak function understandings
(Breidenbach, Dubinsky,
Hawks, & Nichols, 1992;
Carlson, 1998; Thompson,
1994a). We are beginning to understand
that the conceptions and reasoning patterns
needed for a strong and flexible
understanding of functions are more
complex than what is typically assumed
by designers of curriculum and
instruction (Breidenbach et
al., 1992; Carlson, 1998;
Thompson,
1994a). Students who think about
functions only in terms of symbolic
manipulations and procedural techniques
are unable to comprehend a more general
mapping of a set of input values to a set of
output values; they also lack the conceptual
structures for modeling function
relationships in which the function
value (output variable) changes
continuously in tandem with continuous
changes in the input variable
(Carlson, 1998;
Monk & Nemirovsky, 1994;
Thompson, 1994a).
These reasoning abilities have been shown
to be essential for representing and
interpreting the changing nature of a
wide array of function situations
(Carlson, Jacobs, Coe,
Larsen, & Hsu, 2002;
Thompson, 1994a); they
are also foundational for understanding
major concepts in advanced mathematics
(Carlson, Smith, & Persson,
2003; Cottrill, Dubinsky,
Nichols, Schwingendorf, Thomas, & Vidakovic,
1996; Kaput, 1992;
Rasmussen, 2000;
Thompson, 1994a;
Zandieh, 2000).
It is noteworthy that many of the reform
calculus texts of the early 90's, e.g.,
Ostabee-Zorn (1997),
Harvard Calculus (Hughes-Hallett
& Gleason, 1994), and C4L (
Dubinsky, Schwingendorf, & Mathews, 1994),
included a stronger conceptual orientation to
learning functions. Such past curriculum
development projects and the educational
research literature are pointing the way
for future curricular interventions to
assist students in developing a robust function
conception - a conception that includes a view
of function as an entity that accepts input and
produces output and enables reasoning about
dynamic mathematical content and scientific
contexts. Research suggests that the
predominant approach to calculus instruction
is not achieving the foundational understandings
and problem solving behaviors that are needed
for students' continued mathematical development
and course taking. It is our view that the
mathematics community is ready for a careful
rethinking of the precalculus and calculus
curriculum - one that is driven by past work of
mathematicians, as well as the broad body of
research on knowing and learning function and
major concepts of calculus. It is also our
view that if algebraic and procedural methods
were more connected to conceptual learning,
students would be better equipped to apply
their algebraic techniques appropriately in
solving novel problems and tasks.
As students move through their school and
undergraduate mathematics curricula, they are
frequently asked to manipulate algebraic
equations and compute answers to specific types
of questions. This strong procedural emphasis
has not been effective for building foundational
function conceptions - ones that allow for
meaningful interpretation and use of function
in various representational and novel settings.
Even understanding functions in terms of input
and output can be a major challenge for most
students. For example,
It is also common for developing students to
have difficulty distinguishing between an
algebraically defined function and an equation
(Carlson, 1998). This is
not surprising if one considers the various
uses of the equal sign and the fact that many
instructors refer to a formula as an equation.
For the student, this ambiguous use of the
word equation appears to cause difficulty for
them in distinguishing between the use of the
equal sign as a means of defining a
relationship between two varying quantities,
and a statement of equality of two expressions.
Our recent work has shown that students benefit
from explicit effort to help them distinguish
between functions and equations. We have
developed instructional interventions that
promote students' thinking about an equation
as a means of equating the output values of
two functions, and the act of solving an
equation as finding the input value(s) where
the output values of these functions are equal.
Many students also tend to believe
that all functions should be definable
by a single algebraic formula. This
focus often hinders flexible thinking
about function situations and can
lead to erroneous conclusions such as
thinking that all functions must
always behave "nicely" in some sense
(Breidenbach et al., 1992).
For example, many students tend to argue
that a piecewise defined function
like Why is the function concept so important?
Why is the function concept so difficult for
students to understand?
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Another common difficulty for students is distinguishing between visual attributes of a physical situation and similar attributes of the graph of a function that models the situation. Throughout the secondary school curriculum, we teach students to attend to increasingly subtle features of graphical representations of functions, ranging from extrema and monotonicity to concavity, inflection points, and curvature. When dealing with functions modeling concrete situations, there are often similar topographical structures within the real-world setting itself (e.g., the curves of a racetrack, the elevation of a road traveling across hilly terrain, or the shape of a container being filled with liquid). The considerable salience of these physical features often creates confusion, even for students with a strong understanding of function. Several types of errors can be traced to conflating the shape of a graph with visual attributes of the situation (Carlson, 1998; Monk, 1992; Monk & Nemirovsky, 1994). Consider the following problem:
| The following diagram is the side-view of an individual cycling up and over a hill. Draw a graph of speed vs. position along the path. |
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| Figure 1. Students often confuse physical features of the context with the graph of a function. |
In response to this problem, many students tend to directly copy features of the diagram into their graph ( Monk, 1992). Correct interpretation of the situation is a conceptually nontrivial task. A student must ignore the fact that the picture looks like a graph, think of how riding uphill (for example) affects the speed of the cyclist, then while ignoring the shape of the hill in the picture determine how to represent the result graphically.
When interpreting graphs such as the
ones in Figure 2, students often
confuse velocity for position (Monk, 1992) since the
curves are laid out spatially, and
position refers to a spatial
property. This confusion leads to
erroneous claims such as: the two
cars collide at
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| Figure 2. Students often confuse position and velocity. |
In both these examples, students are thinking of the graph of a function as a picture of a physical situation rather than as a mapping from a set of input values to a set of output values. Developing an understanding of function in such real-world situations that model dynamic change is an important bridge for success in advanced mathematics.
Students' procedural orientations to
functions have also been observed in
their inability to accurately
express function relationships using
function notation. When asked to
express s as a function of
t, many high performing
precalculus students did not know
that their objective was to write a
formula in the form of "
In our work to develop and validate the
Precalculus Concept Assessment
Instrument1
(Carlson, Oehrtman, &
Engelke, submitted), we found that
students' ability to respond correctly
to a diverse set of function-focused
tasks is tightly linked to two types of
dynamic reasoning abilities. First, as
mentioned above, students must develop
an understanding of functions as
general processes that accept input and
produce output. Second, they must be
able to attend to the changing value of
output and rate of change as the
independent variable is varied through
an interval in the domain.
Understanding limits and continuity
requires one to make judgments about the
value of a function on intervals of
infinitely many different (small) sizes.
Conceptualizations based on "holes,"
"poles," and "jumps" as gestalt
topographical features (corresponding to
removable discontinuity, vertical
asymptotes, and jump or one-sided
discontinuity, respectively) can lead to
misconceptions in more complex limiting
situations, such as the definitions of
the derivative and definite integral.
For example, students often develop some
intuitive understanding of the
Fundamental Theorem of Calculus with
which they can explain that the
derivative of the volume of a sphere,Dynamic conceptualizations
needed for precalculus and calculus
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To understand the relationship between average and instantaneous rates and the graphical analog between secant and tangent lines, a student must first conceive of an image as in Figure 3a, below (Monk, 1987). By employing covariational reasoning (e.g., coordinating an image of two varying quantities and attending to how they change in relation to each other), the student is able to transform the image and reason about values of various parameters as the configuration changes. Being able to answer questions that require such variation as "When point Q moves toward P, does the slope of S increase or decrease?" is significantly more difficult than being able to answer questions about the value of a function at a single point.
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| Figure 3. Foundational images for the definitions of a) the derivative and b) the definite integral |
Analyzing the changing nature of an instantaneous rate also requires the ability to conceive of functional situations dynamically. Consider the following question based on a classic related rates problem in calculus:
| From a vertical position against a wall, the bottom of a ladder is pulled away at a constant rate. Describe the speed of the top of the ladder as it slides down the wall. |
Reasoning about this situation conceptually is difficult for calculus students even when they are given a physical model and scaffolding questions (Monk, 1992) and is similarly challenging for beginning graduate students in mathematics (Carlson, 1999). The standard calculus curriculum presents accumulation in terms of methods of determining static quantities such as the area of an irregular region of the plane or the total distance traveled given a changing velocity (but as a completed motion). Equally important, however, is a dynamic view in which an accumulated total is changing through continual accruals (Kaput, 1994; Thompson, 1994). For example, in a typical "area so far" function as in Figure 3b, this involves being able to mentally imagine the point p moving to the right by adding slices of area at a rate proportional to the height of the graph. This requires students to engage in covariational reasoning (Carlson, Smith, & Persson, 2003) and is significantly more difficult for students than evaluating and even comparing areas at given points (Monk, 1987).
In interviews with over 40
precalculus level students, we
found that students who
consistently verbalized a view
of function as an entity that
accepts input and produces
output were able to reason
effectively through a variety
of function-related tasks. For
example, these students, when
asked to find the composition of
f with g, 
gg
According to several studies, calculus students are slow to develop an ability to interpret varying rates of change over intervals of a function's domain. (Carlson, 1998; Kaput, 1992; Monk, 1992; Monk & Nemirovsky, 1994; Nemirovsky, 1996; Tall, 1992; Thompson, 1994a). According to Thompson (1994a), once students are adept at imagining expressions being evaluated continually as they "run rapidly" over a continuum, the groundwork has been laid for them to reflect on a set of possible inputs in relation to the set of corresponding outputs (p. 27). Such a covariation view of function has also been found to be essential for understanding central concepts of calculus (Cottrill et al., 1996; Kaput, 1992; Thompson, 1994b; Zandieh, 2000) and for reasoning about average and instantaneous rates of change, concavity, inflection points, and their real-world interpretations (Carlson, 1998; Monk, 1992).
The following section provides additional elaboration of these essential process and covariational understandings of functions.
Developmental research has provided
insights about the reasoning patterns
essential for success in collegiate
mathematics. As we have previously
discussed, investigations of students'
function knowledge have consistently
revealed that students' underlying
conceptual view is important.
Researchers have formalized these
consistent observations by introducing
terms for referencing specific types of
conceptual views and their development.
Specifically, students must move from
what is called an action view of
functions to what is called a
process view of functions.
According to Dubinsky &
Harel (1992),The process view of functions - A more
formal examination
| An action conception of function would involve the ability to plug numbers into an algebraic expression and calculate. It is a static conception in that the subject will tend to think about it one step at a time (e.g., one evaluation of an expression). A student whose function conception is limited to actions might be able to form the composition of two functions, defined by algebraic expressions, by replacing each occurrence of the variable in one expression by the other expression and then simplifying; however, the students would probably be unable to compose two functions that are defined by tables or graphs. |
Students whose understanding is limited to an action view of function experience several difficulties. For example, an inability to interpret functions more broadly than by the computations involved in a specific formula results in misconceptions such as believing that a piecewise function is actually several distinct functions, or that different algorithms must produce different functions. More importantly, reasoning dynamically is difficult because it requires one to be able to disregard specific computations and to be able to imagine running through several input-output pairs simultaneously. This ability is not possible with an action view in which each individual computation must be explicitly performed or imagined. Furthermore, from an action view, input and output are not conceived except as a result of values considered one at a time, so the student cannot reason about a function acting on entire intervals. Thus, not only is the complex reasoning required for calculus out of reach for these students, but even simple tasks like conceiving of domain and range as entire sets of inputs and outputs is difficult.
Without a generalized view of inputs
and outputs, students cannot think of
a function as a process that may be
reversed (to obtain the inverse of a
function) but are limited to
understanding only the related
procedural tasks such as switching
x and y and solving for
y or reflecting the graph of
f across the line
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| Figure 4. Various conceptions of the inverse of a function. a) as an algebra problem, b) as a geometry problem, and c) as the reversal of a process. The first two of these are common among students but, in isolation, do not facilitate flexible and powerful reasoning about functional situations. |
Students who possess only the
procedural orientations of
Figures 4a and b, absent of
the understanding of why the
procedures work, will likely be
unable to recognize even simple
situations in which these
procedures should be applied.
Curriculum and instruction have
not been broadly effective in
building these connections in
students' understanding. A
recent study of over 2000
precalculus students at the end
of the semester (Carlson,
Oehrtman, & Engelke,
submitted) showed that only
In contrast to the conceptual
limitations of an action view,
Dubinsky and Harel
(1992) state that,
|
A process conception of
function involves a dynamic
transformation of quantities
according to some repeatable
means that, given the same
original quantity, will always
produce the same transformed
quantity. The subject is able to
think about the transformation
as a complete activity beginning
with objects of some kind, doing
something to these objects, and
obtaining new objects as a result
of what was done. When the
subject has a process conception,
he or she will be able, for
example, to combine it with other
processes, or even reverse it.
Notions such as |
With such a process view, students
are freed from having to imagine
each individual operation for an
algebraically defined function.
For example, given the function on
the real numbers defined by
Table 1. Action and process
views of functions
| Action View | Process View |
|---|---|
| A function is tied to a specific rule, formula, or computation and requires the completion of specific computations and/or steps. | A function is a generalized input-output process that defines a mapping of a set of input values to a set of output values. |
| A student must perform or imagine each action. | A student can imagine the entire process without having to perform each action. |
| The "answer" depends on the formula. | The process is independent of the formula. |
| A student can only imagine a single value at a time as input or output (e.g., x stands for a specific number). | A student can imagine all input at once or "run through" a continuum of inputs. A function is a transformation of entire spaces. |
| Composition is substituting a formula or expression for x. | Composition is a coordination of two input-output processes; input is processed by one function and its output is processed by a second function. |
| Inverse is about algebra (switch y
and x then solve) or geometry (reflect
across |
Inverse is the reversal of a process that defines a mapping from a set of output values to a set of input values. |
| Domain and range are conceived at most as an algebra problem (e.g., the denominator cannot be zero, and the radicand cannot be negative). | Domain and range are produced by operating and reflecting on the set of all possible inputs and outputs. |
| Functions are conceived as static. | Functions are conceived as dynamic. |
| A function's graph is a geometric figure | A function's graph defines a specific mapping of a set of input values to a set of output values. |
Understanding even the basic idea
of equality of two functions
requires a generalization of
the input-output process, the
ability to imagine the pairing
of inputs to unique outputs
without having to perform or even
consider the means by which this
is done. Students may then come
to understand that any means of
defining the same relation is the
same function. That is, a function
is not tied to specific
computations or rules that define
how to determine the output from
a given input. For example, the
rules
 |
vs. |
 |
Students with a process view are also better able to understand aspects of functions such as composition and inverses. They are consistently able to correctly answer conceptual and computational questions about composition in a variety of representations by coordinating output of one process as the input for a second process. Similarly, students conceiving of inverses as reversing a process so that the old outputs become the new inputs and vice-versa (Figure 4c), or by asking "What does one have to do to get back to the original values?" were able to correctly answer a wide variety of questions about inverse functions (Carlson et al., submitted).
A process view of function is crucial to understanding the main conceptual strands of calculus (Breidenbach et al., 1992; Monk, 1987; Thompson, 1994a ). For example, the ability to coordinate function inputs and outputs dynamically is an essential reasoning ability for limits, derivatives, and definite integrals. In order to understand the definition of a limit, a student must coordinate an entire interval of output values, imagine reversing the function process, and determine the corresponding region of input values. The action of a function on these values must be considered simultaneously since another process (one of reducing the size of the neighborhood in the range) must be applied while coordinating the results. Unfortunately, most pre-calculus students do not develop beyond an action view, and even strong calculus students have a poorly developed process view that often leads only to computational proficiency (Carlson, 1998). With intentional instruction, however, students can develop a more robust process view of function (Carlson et al., submitted; Dubinsky, 1991; Sfard, 1991).
Certainly not every aspect of an action view of functions is detrimental to students' understanding, just as the acquisition of a process view does not ensure success with all functional reasoning. However, a process view of functions is crucial to developing rich conceptual understandings of the content in an introductory calculus course. The promotion of the more general 'ways of thinking' that we have advocated should result in producing curricula that are more effective for promoting conceptual structures for students' continued mathematical development.
|
|
Figure 5. Which diagram
represents g |
Ask about the action of functions on
entire intervals in addition to single
points. Focusing on the image of a
function applied to an infinite set also
encourages students to think in terms of
a general process. Students should be
asked to coordinate such judgments with
basic compositions and inverses, asking,
for example, for the length of an
interval after being transformed by two
linear functions. Similarly, ask
students to find preimages of intervals
as in the definition of limit or
continuity and to reverse the process of
a function even if it is not invertible
(e.g., find the preimages of 1 under
Ask students to make and compare
judgments about functions across
multiple representations. Such
questions should include multiple
algebraic representations to
reinforce the independence from a
formula as well as the standard
representations of graphs, tables,
and verbal descriptions. Students
should make such determinations;
then compare the results for
consistency, justifying or
discovering why they are the same.
For example, asking how the various
techniques of inverting a function
are related reinforces seeing a
reflection across the line
| Mental Action | Description of Mental Action | Behaviors |
|---|---|---|
| Mental Action 1 |
Coordinating the dependence of one variable on another variable |
|
| Mental Action 2 |
Coordinating the direction of change of one variable with changes in the other variable |
|
| Mental Action 3 |
Coordinating the amount of change of one variable with changes in the other variable |
|
| Mental Action 4 |
Coordinating the average rate-of-change of the function with uniform increments of change in the input variable |
|
| Mental Action 5 |
Coordinating the instantaneous rate-of-change of the function with continuous changes in the independent variable for the entire domain of the function |
|
In our work to study and promote students' emerging covariational reasoning abilities, we have found that the ability to move flexibly between mental actions 3, 4 and 5 is not trivial for students. We have also observed that many precalculus level students only employ Mental Action 1 and Mental Action 2 when asked to construct the graph of a dynamic function situation.
When prompting students to construct the graph of the height as a function of the amount of water in a bottle (Figure 6), we found that many precalculus students appropriately labeled the axes (MA1) and then constructed an increasing straight line (MA2). When prompted to explain their reasoning, they frequently indicated that "as more water is put into the bottle, the height of the water rises (MA2)." These students were clearly not attending to the amount of change of the height of the water level or the rate at which the water was rising.
| Imagine this bottle filling with water. Sketch a graph of the height as a function of the amount of water that's in the bottle. |
 |
| Figure 6. The Bottle Problem. | |
We have observed that calculus students frequently provided a strictly concave up graph in response to this question (Carlson, 1998 ; Carlson et al., 2002). When probed to explain their reasoning, a common type of justification was, "as the water is poured in it gets higher and higher on the bottle (MA2)." In contrast, other students who were starting to be able to construct an appropriate graph began coordinating the magnitude of changes in the height with changes in the volume (MA3). This is exemplified in the strategy of imagining pouring in one cup of water at a time and coordinating the resulting change in height based on how "spread out" that layer of water is.
Other students have demonstrated the ability to speak about the average rate of change locally for a specific interval of a function's domain (MA4) but were unable to explain how the rate changes over the domain of the function. Even when calculus students produced a graph that was correct, they commonly had difficulty explaining what was conveyed by the inflection point and why the graph was "smooth" (i.e., C 1 rather than piecewise linear). Students frequently exhibited behaviors that gave the appearance of engaging in Mental Action 5 (e.g., construction of a smooth curve with the correct shape), however when prompted to explain their reasoning, they explained that they had relied on memorized facts to guide their constructions. They were relying on facts such as faster means steeper and slower means less steep, but they were unable to explain why this was true.
Generally, ask questions
associated with each of the mental
actions. For orientation to
any problem, MA1 skills and basic
function awareness can be
addressed by asking what values
are changing and what variable(s)
influence the quantity of interest
(i.e., the dependent variable).
Is there a single variable that
determines that quantity's values?
How are the variables related and
in what representations can this
relationship be expressed? For
MA2, ask whether a function
increases or decreases if the
independent variable is increased
(or decreased). Expect students
to make such judgments from
multiple representations. At an
MA3 level, ask students to make
judgments about amounts of change
in the function for constant
increments of the independent
variable. For a dynamic
situation, have students draw
diagrams representing changes from
one output variable to the other
for each of two nearby intervals
of the input variable, and
represent these changes
pictorially and algebraically.
Ask students to interpret these
representations in terms of rate
of change in the problem context.
To foster MA4 thinking, have
students compute several average
rates using various representations
and find various interpretations
for these values and explicitly
discuss the meaning of units such
as meters per second and even
non-temporal rates such as square
inches per inch or degrees Kelvin
per meter. For MA5, ask students
to anticipate second derivative
information based on the problem
context, e.g., whether the force
of gravity between two celestial
objects will increase at an
increasing rate or at a decreasing
rate with respect to a decreasing
distance between them. Ask
students to describe the rate of
change of a function event as the
independent variable continuously
and dynamically varies through the
domain. Ask where inflection
points are, what events they
correspond to in real-world
situations, and how these points
are interpreted in terms of
changing rate of change.
Ask for clarification of rate
of change information in various
contexts and representations.
Expect students to explain
statements about rates in
real-world contexts from algebraic
or graphical information, e.g.,
why does a steeper graph mean the
quantity represented by the
function is increasing faster?
Push beyond students' initial,
simplified statements such as
"the rate of change of position"
that ignore the role of time.
Require explication of both
variables involved and
relationships about changes in
both quantities. Finally, a student
may be able to make statements
indicative of a Mental Action 5 by
attending only to the geometry of
the curve and associated phrases
such as "increasing at a decreasing
rate." Ask them to unpack such
statements in terms of the
underlying mental actions, in this
case perhaps prompts that reveal
whether they understand what they
mean by the phrase "increasing at a
decreasing rate." Unpacking what
may be pseudo-conceptual
knowledge - knowledge that has been
memorized and is not based on an
underlying conceptual structure and
understanding, can be achieved by
posing pointed questions that
prompt students to reveal their
underlying conceptions (e.g., why
is the graph concave up or why is
the curve "smooth" rather than
piecewise linear?). Such questions
typically reveal whether the student
is merely spouting a memorized rule
or fact, or whether the statement is
supported by an understanding of why
the rule or statement is true.
Marilyn P. Carlson, Director, CRESMET
and
Michael Oehrtman
Engaging covariational reasoning
through analysis of function
situations
We offer the following suggestions
for strengthening students'
covariational reasoning abilities:
Concluding remarks
A mature function understanding
that is revealed by students' using
functions fluidly, flexibly, and
powerfully is typically associated
with strong conceptual
underpinnings. Promoting this
conceptual structure in students'
understanding may be achieved
through both curriculum and
instruction including tasks,
prompts, and projects that promote
and assess the development of
these "ways of thinking" in
students. We advocate for
greater emphasis on developing
students' ability to speak about
functions as entities that accept
input and produce output, a more
conceptual orientation to teaching
function inverse and composition,
the inclusion of tasks requiring
simultaneous judgments about
entire intervals of input or output
values, and the development of
students' ability to mentally run
through a continuum of input values
while imagining the changes in the
output values, etc. Our work
suggests that students would
benefit from explicit efforts to
promote their understanding of
function notation. We also call
for evaluations of students'
mathematical development and
readiness to include assessments
that measure the foundational
reasoning abilities needed for a
robust function conception.References
1The Precalculus Concept
Assessment Instrument is a 25-item
multiple choice instrument for assessing
students' understanding of the major
aspects of the function concept that are
foundational for success in beginning
calculus. The answer choices include the
correct answer and the common
misconceptions that have been expressed
by students in research studies (e.g.,
interviews that have probed students'
thinking when providing specific
responses to conceptually based tasks).
Center for Research on Education in Science,
Mathematics, Engineering and Technology
Arizona State University
Tempe, AZ 85287-1804
Email:
marilyn.carlson@asu.edu
Department of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287-1804
Email:
oehrtman@math.asu.edu
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