Students’ covariational
reasoning abilities: A
literature review
Marggie D. Gonzalez
November 28, 2011
Importance of the Topic
 Function is an important and unifying concept in modern
mathematics, central to many different branches of mathematics,
and essential to related areas of the sciences (Carlson, 1998).
 It is well documented that high performing pre-calculus and
calculus students have weak understandings of the function
concept and difficulty modeling function relationships involving the
rate of change of one variable as it continuously varies with another
variable (Carlson, 1998; Monk & Nemirovsky, 1994; Oehrtman,
Carlson, & Thompson, 2008; Thompson, 1994).
 This ability to reason covariationally has been shown to be critical
in the understanding of functions, central concepts of calculus
(Carlson, 1998; Carlson, Jacobs, Coe, Larsen, & Hsu, 2002;
Thompson, 1994).
Importance of the Topic
 As future educators of students that are looking into taking
calculus in their near future, it is imperative for them to have
a strong conceptual understanding in the areas of functions
and rate of change.
 Little research has been done to describe prospective
secondary mathematics teachers’ abilities to reason in
dynamic situations.
NCTM (2000)
 Algebra Standard for Grades 9-12
 analyze functions of one variable by investigating rates of
change, intercepts, zeros, asymptotes, and local and global
behavior
 “In high school, students should analyze situations in which
quantities change in much more complex ways and in which the
relationships between quantities and their rates of change are
more subtle… Working problems of this type builds on the
understandings of change developed in the middle grades and
lays groundwork for the study of calculus”
Example:
The given graph represents velocity vs. time for two cars. Assume the cars
starts from the same position and are traveling in the same direction.
Carlson (1998)
Common Core Standards
 Grade 8 - “Students grasp the concept of a function as a rule
that assigns to each input exactly one output” (CCSSI, 2010,
p. 52).
 High School – “Because functions describe relationships
between quantities, they are frequently used in modeling
(CCSSI, 2010, p. 67).
 “For a function that models a relationship between two
quantities, interpret key features of graphs and tables in
terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship”
(CCSSI, 2010, p. 69)
Functions:
 According to the literature, functions can be seen as static or
dynamic:
Static
Dynamic
Point-Wise
Across-Time
Correspondence Approach
Covariational Approach
Action View
Process View
 A dynamic conceptualization of functions is essential to be
able to coordinate changes in two variables simultaneously
(Confrey & Smith, 2005; Monk, 1994; Oehrtman, Carslon &
Thompson, 2008)
Covariational Reasoning
 Carlson (1998)
 Participants: college algebra (n=30), second semester calculus
(n=16), and beginning graduate study in mathematics (n=14)
 Data collection: pre-test and clinical interviews (n=15)
 From results a Covariational Framework was developed.
 What is covariational reasoning? Are the "cognitive activities
involved in coordinating two varying quantities while attending to
the ways in which they change in relation to each other" (Carlson et
al, 2002)
 Change refer to both: how things change and at what rate.
 Direction of change (increasing, decreasing)
 Amount of change (average, instantaneous)
Task:
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.
The Bottle Problem:
Carlson (1998)
Carlson, Jacobs, Coe, Larsen, and Hsu (2002)
Covariational Framework
Mental action
Description of mental action
Behavior
Mental Action 1
(MA1)
Coordinating the value of one variable with
changes in the other.

Labeling the axes with verbal indications of coordinating the
two variables (e.g., y changes with changes in x).
Mental Action 2
(MA2)
Coordinating the direction of change of one
variable with changes in the other variable.


Constructing an increasing straight line
Verbalizing an awareness of the direction of change of the
output while considering changes in the input
Mental Action 3
(MA3)
Coordinating the amount of change of one
variable with changes in the other variable.


Plotting points/constructing secant lines
Verbalizing an awareness of the amount of change of the
output while considering changes in the input
Mental Action 4
(MA4)
Coordinating the average rate-of-change of
the function with uniform increments of
change in the input variable.


Constructing contiguous secant lines for the domain
Verbalizing an awareness of the rate of change of the output
(with respect to the input) while considering uniform
increments of the input
Mental Action 5
(MA5)
Coordinating the instantaneous rate of
change of the function with continuous
changes in the independent variable for the
entire domain of the function.

Constructing a smooth curve with clear indications of
concavity changes
Verbalizing an awareness of the instantaneous changes in the
rate of change for the entire domain of the function (direction
of concavities and inflection points are correct)

Carlson, Jacobs, Coe, Larsen, and Hsu (2002)
The Car Problem:
 A car is traveling down the road between cities A and B.
Sketch a graph of his journey where his distance from A is
noted on the horizontal axis and his distance from B on the
vertical axis.
Examples from students:
Future Research!
 What needs to be done in middle/high school so that
students in college level enter calculus with a stronger
understanding of the function concept?
 We need more studies done with middle/high school students
(Only 5 studies focused on out of 21)
 What about prospective secondary mathematics teachers?
 From 16 studies done with undergraduate students, only 4 were
focused on prospective teachers.
Future Research!
 Need to be able to measure growth in students covariational
reasoning abilities.
 Kevin Moore, Marilyn Carlson and Michael Oehrtman, from
ASU, designed a course ”informed by theory on the process of
covariational reasoning and selected literature about
mathematical discourse and problem-solving” (Moore, Carlson,
and Oehrtman, 2009, p. 7).
 Most of the research that has been done has reported results
based on clinical interviews.
 What about the use of a pre and a post instrument?
Future Research
 What about the creation of a common instrument? Is that
possible??
 Use of the same tasks so that qualitative results are parallel.
 Use of a quantitative instrument to be use as a pre-test and as a
post-test that is common among researchers trying to
understand students covariational reasoning abilities.
Questions?
Function as a static situation
 Point-wise (Monk, 1994)
 Students tend to calculate value of a function for every input values by
making precise measurements
 Correspondence Approach (Confrey & Smith, 2005)
 “one initially builds a rule that allows one to determine a unique y-value
from any given x-value… build correspondence between x and y” (p. 33)
 Action view (Dubinsky & Harel, 1992; Oehrtman, Carslon & Thompson,
2008)
 Involve the ability to plug numbers into an algebraic expression and
calculate.
 Subject tend to think about it one step at a time.
 Able to form the composition of two functions defined by algebraic
expressions, but not when the two functions are defined by tables or
graphs.
Function as a dynamic event
 Across-time (Monk, 1994)
 “How does change in one variable lead to change in others? How is
the behavior of the output variable influenced by variation in the
input variable?” (p. 21)
 Covariational Approach (Confrey & Smith, 2005)
 “Entails being able to move operationally from ym to
ym+1
coordinating with movement from xm to xm+1“ (p. 33).
 Process view (Dubinsky & Harel, 1992; Oehrtman, Carslon &
Thompson, 2008)
 “it requires one to be able to disregard specific computations and to
be able to imagine running through several input-output pairs
simultaneously” (p. 5).