Research Sampler |
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Advances in technology, together with an increased interest in dynamical systems, are influencing the nature of many first courses in ordinary differential equations (ODEs). In addition, there is an increased emphasis on nonlinear differential equations, systems of differential equations, and mathematical modeling, as well as on qualitative and numerical approaches that shed light on the behavior of solutions. Analytic techniques are still important, but they no longer tend to be the sole focus. As these new directions come into classrooms, research is beginning to illuminate aspects of learning and teaching ODEs that can inform ongoing curricular innovations.
Making Connections between Representations
The interplay between algebraic, graphic, and numeric
representations and the contextual situations that
particular equations are intended to model is a
common theme in reform efforts. Examples of problems
of these types can be found in a Special Issue of
the College Mathematics Journal [see West, 1994] and the Mathematical
Association of America Notes #50 [see
Kallaher, 1999]. Common amongst
these is a need for students to move flexibly between
algebraic, graphical, and numerical representations,
to make interpretations from the various representations
of situations being modeled, and to make warranted
predictions about the long-term behavior of
solutions. Are students successful on these types
of problems — why or why not? Research is beginning
to document students' accomplishments and
difficulties, as well as providing theories about
their possible cognitive and instructional origins.
In a French study aimed at exploring the teaching of qualitative solutions, Michèle Artigue conducted a three-year project with first-year students at the University of Lille I. Approximately 100 students per year received nearly 35 hours of instruction on first-order differential equations. Students attended common lecture sessions with smaller exercise sessions using computers. As evidenced by their lab reports and examinations, early on students were able to successfully complete tasks where information was given simultaneously in two settings and the problem to be solved required interpretation between the two settings. For example, one interpretation task asked students to find and justify the correct match between seven different differential equations and corresponding graphs of solution curves. With little or no intervention from the instructor, these students were successful because they were able to employ a variety of familiar criteria for determining and checking their answers. These criteria included connections between the sign of f (where dy/dx = f(x, y)) and properties of monotonicity for solution curves, zeros of f and horizontal slope, infinite limit of f and vertical slope, the value of f at a particular point and the slope of a solution curve at that point, and recognizing particular solutions associated with straight lines in the graphic setting and checking them in the algebraic setting. We would hope all students are familiar with these criteria from calculus, and thus, that such criteria can serve as a basis for further study of differential equations. [see Artigue (1992)]
One of the tasks required students to determine, with
reasons, the appropriate match between eight differential
equations and four slope fields (four of the differential
equations did not have a matching slope field). Although
all students were generally successful on this task
(using many of the same criteria outlined by Artigue),
Rasmussen found that behind students' correct answers
there often lay an incorrect conception of equilibrium
solution. In particular, three of the six students, at
various points in their solution processes, conceptualized
equilibrium solutions as existing whenever the
differential equation is zero. While this is true for
autonomous differential equations, it is not true in
general. For example, when dealing with the equations
dy/dt = y
Michelle Zandieh and Michael McDonald also studied
students' underlying understanding of solutions
and equilibrium solutions. They interviewed a
total of 23 students from two separate
reform-oriented differential equations classes, one
at a large state university in the southwest and one
at a small liberal arts college on the west coast.
In addition to asking students open-ended questions
such as, "What is a differential equation?" and "What
is a solution to a differential equation?", they
posed several tasks for students to solve. One of
the tasks was the same matching task used by
Rasmussen and another asked students to draw
representative solutions on a given slope field for
dy/dt = y + 1. Much like
previous research findings, 7 of the 23 students
overgeneralized the notion of equilibrium solution
to include all values for which dy/dt
is zero. When asked to draw representative solution
functions, 3 of the 23 students failed to sketch the
equilibrium solution y(t) =
In another task Rasmussen provided students with
the autonomous differential equation
dN/dt =
Why would students be able to do parts (a) and
(b), but fail to "see" the connection between their
sketches and the long-term behavior for various
initial populations? This question is especially
intriguing because these students had just created
for themselves what is from our perspective a
sketch like that in Figure 1 of various solution
functions. During the interviews, the answer to
this question became quite clear. Students did not
view the sketch they had just created as a plot of
the functions that solve the differential
equation. In the words of one student, his sketch
was "just a test for stability." These students had
learned a graphical approach for determining
stability where the graphs they created did not
carry the intended conceptual meaning. [see Rasmussen (2001)]
Research findings focusing on student understanding
in courses taking new directions in ODEs indicate
that graphical and qualitative approaches do not
automatically translate into conceptual
understanding. In a traditional course, a typical
complaint is that students often learn a series of
analytic techniques without understanding important
connections and conceptual meanings. Care must be
taken or else students are likely to supplement
mindless symbolic manipulation with mindless
graphical manipulation. Of course, how a student
thinks and reasons is as much a reflection of his
or her individual cognitive development as it is a
reflection of the mathematics classroom. For
example, if students are not routinely expected to
explain and mathematically defend their
conclusions, it is more likely that they will
learn to proceduralize various graphical and
qualitative approaches in ways that are
disconnected from other aspects of the problem.
The following is another intuitive or informal student
theory: Solutions that hint at converging continue
converging. Rasmussen arrived at this finding after
interviewing students on a task where they were given a
direction field for
dy/dx = y(x
Half of the six students incorrectly reasoned that
solutions starting off in the upper left-hand region
would tend to zero as x approaches infinity.
Despite evidence to the contrary, these students
demonstrated strong intuitive notions that asymptotic
behavior would prevail. For example, one student
figured out that all the slopes in the first quadrant
just above the x-axis would have a positive
slope, but rejected this as irrelevant to the
long-term behavior of solutions that initially
approached the positive x-axis. These students
are in good company. As James Gleick described in his
book, Chaos: The Making of a New Science, the
famous mathematician Stephen Smale once proposed that
practically all dynamical systems tend to settle
into behavior that is not too strange. For students
in a first course in ODEs, solutions that hint at
asymptotic behavior, but then change course, appear
too strange to be believable. [see
Rasmussen (2001)]
Since graphical predictions are playing an
increasingly prominent role in reform-oriented
approaches to ODEs (see for example, Blanchard, Devaney, and Hall, 2002;
Borrelli and Coleman, 1998; Diacu, 2000; Kostelich
and Armbruster, 1997), it makes sense to explore
the extent to which students are able to create
geometric proofs. Artigue's work specifically
examined this issue; she reported on students' work
on three types of tasks — prove that a solution
intersects a given curve; prove that it cannot
intersect a given curve; and prove that it has an
asymptote or rule out the possibility of such an
infinite branch. She found that students had great
difficulties generating these proofs. She
attributes this to two causes. First, students had
not been exposed to the delicate tools that they
needed to use in qualitative proofs in the graphical
setting. For example, the helpful ideas of
fence, funnel, and area had not
been introduced to students because, as Artigue
suggests, mathematics professors have been slow to
accept the graphical setting as a place for proof.
Second, many students have strong monotonic
conceptions that interfere with their proof efforts.
For example, students had an intuitive belief in the
following false statement: If f(x) has
a finite limit when x tends towards infinity,
its derivative f '(x) tends toward
zero. Yet another reason for students' difficulty
was that moving from predictions about how a
solution might look to actually proving these
statements requires the use of elementary analysis.
[see Artigue (1992)]
A different approach to proofs involves emphasizing
argumentation as a routine part of everyday classroom
discussions. In a multi-year project at a mid-sized
university in the Midwest, researchers2 are studying
student learning in a first course in ODEs as it
occurs in classrooms over the course of an entire
semester. An interesting example from this research
related to proof involves the arguments students
developed to justify that two solutions to a logistic
growth differential equation with different initial
conditions would never touch. Although these
students had not yet studied the uniqueness theorem,
they argued that since graphs of solutions to
autonomous different equations were shifts of each other
along the t-axis, there would never be a point
in time when the solutions intersected each other. For
another example of arguments involving short chains of
deductive reasoning, consider the following question
that students in this project asked and answered: Is it
possible for a graph of a solution to an autonomous
differential equation to oscillate? The typical
argument these students developed to reject this
possibility was to argue that since the slopes in a
slope field for an autonomous differential equation
would have to be the same "all the way across" the
slope field, a graph of a solution would not oscillate
because if it did, there would be a value for y
where the slope would be both positive and negative.
For students like those in this class who have little
to no experience in developing mathematical arguments
to support or refute claims, significant progress in
their ability to create and defend short deductive
chains of reasoning was observed. This progress was
due in large part to the explicit attention paid to
classroom norms pertaining to explanation and
justification. These social aspects of the mathematics
classroom are reviewed in the final section. [see
Stephan & Rasmussen (2002)]
In Rasmussen's study, students discussed a
previously completed Mathematica
assignment where they had generated and
interpreted graphs of the angular position (in
radians) versus time for the linear and
non-linear differential equations similar to
those shown in Figure 3. Each plot in Figure
3 shows a different set of initial conditions
for the solutions to the undamped linear model,
As might be expected, students experienced the most
difficulty interpreting the graphs in Plots C and D.
Students tended to interpret the graph as a literal
picture of the situation. For example, one student
said that the graph of the solution to the
nonlinear model in Plot C indicates that "it starts
increasing and remains at a constant distance from,
whatever, and then it would start increasing again
spontaneously, plateau again and then start
increasing." He also acknowledged that he had
never seen a pendulum do something like that, but
was unable to interpret the plot otherwise. In
Plot D, this same student explained that the graph
of the solution to the nonlinear model shows the
pendulum "increasing and increasing and this thing
wouldn't be able to hold it and it would just fly
off." [see Rasmussen (2001)]
The studies by Trigueros and Rasmussen suggest that
developers of both curriculum and instruction need
to be cautious about what is assumed will be
obvious to students when dealing with rich and
complex graphical representations. Perhaps
further and deeper classroom conversations
surrounding the interpretation of such
representations might help minimize the types of
student difficulties highlighted in these studies.
One study conducted by Thomas Klein concerned how
the use of a computer algebra system as a
demonstration tool affects achievement in solving
differential equations. Klein investigated four
sections of ODE classes consisting of 110
students at a large private university in the
southeast. In two classes, the instructor used
a computer algebra system (CAS) as an in-class
teacher-led demonstration tool and students were
given access to the same system out of class.
In the other two classes, no CAS was used. A
common posttest was given to all the classes,
along with questionnaires and interviews with
students in the CAS classes. In comparing the
classes, Klein found no significant difference
in students' ability to analytically solve ODEs.
This result is perhaps not surprising since the
power a CAS offers does not relate well to the
development of paper and pencil skills
necessary to analytically solve ODEs. However,
students in the CAS classes did show more
positive attitudes about the use of computers
in mathematics. [see Klein
(1993)]
In the study conducted by Rasmussen at the large
mid-Atlantic university, students worked on CAS
labs outside of class time and only rarely did
class discussion focus on interpretations or
analysis of their labwork. As documented by
interviews and surveys, students viewed these
CAS labs as unrelated to what they saw as the
main ideas of the course and they did not think
that the work they put into the labs furthered
their understandings of important ideas or
methods, which was contrary to the instructor's
goals of the course. However, when technology
is integrated into the course, there is
some evidence that this does help promote
better understandings of various graphical
representations. For example, in Habre's study
students used computer modules designed for
specific course goals of the course and
intended to introduce students to specific
concepts. Although this was not the focus of
his research, some of Habre's interview data
suggests that these modules might have been
helpful to students in their development of
mathematics in the graphical setting. For
example, when given a vector field for the
system of ODEs x'(t)
=
What this limited research does indicate is that
we need to be deliberate in how and why we decide
to implement technology in the classroom. It
shows that students' visual understanding of phase
portraits, slope fields, and solutions of
differential equations is an area where we might
consider integrating technology into students'
experiences in the classroom. Using a computer
algebra system as a separate lab component or
only as a demonstration tool seems less likely
to achieve the intended learning goals.
In two different semester long research studies
conducted at a midwestern university, Chris
Rasmussen, Erna Yackel, Michelle Stephan, and
Karen King investigated the feasibility of adapting
research-based approaches to instructional design
and teaching that have been effective at promoting
student learning at the school level (in
particular the theory of Realistic Mathematics
Education being developed at the Freudenthal
Institute in The Netherlands). The project
classes involved differential equations students
majoring in mathematics, science, or engineering.
All classes were video-taped and individual
interviews were conducted with a majority of the
students at various points throughout the semester.
The data consisted of classroom video-recordings,
field notes, copies of student work, and
video-recorded individual student interviews.
In this section, we highlight the work based on
these project classes. [see
Rasmussen & King (2000); Yackel,
Rasmussen, & King (2000);
Rasmussen, Yackel, & King (in press)]
Findings include the significant role of
explanation and justification as a normal part of
classroom discussion. The following classroom
features, critical to the success of the project
in terms of student learning, were initiated by
the instructor and sustained throughout the
semester: Students routinely explained their
thinking and reasoning (versus just providing
answers), listened to and tried to make sense of
other students' thinking, indicated agreement or
disagreement with other students' thinking, and
responded to other students' challenges and
questions. Such aspects of classroom social
interactions involving explanation and
justification that become routine are referred to
as social norms. The initiation and
maintenance of such norms was a challenge because
students in the project classes were used to and
expected traditional patterns of interaction
where the instructor talked and the students
listened.
Given that many undergraduate students are not
used to explaining their reasoning and making
sense of other students' thinking, a pervasive
and important question is: How can instructors (1)
initiate a shift in social norms, and (2) sustain
these norms over time? The studies conducted by
Rasmussen and colleagues offer useful responses.
For example, in the semester-long classroom studies
described, the instructor devoted explicit attention
to initiating the social norms described above.
During an approximately twenty-minute whole class
discussion on the second day of class, the
instructor led a whole discussion where he offered
no mathematical explanation himself. Rather,
he strove to initiate new social norms by inviting
students to discuss their thinking and reasoning
through remarks and questions such as:
Social norms are not rules set out in
advance on a syllabus. Although being explicit
about expectations can be useful, such explicit
statements are insufficient. Norms are
regularities in the ways individuals interact.
As such, an instructor alone cannot establish them.
They are constituted and sustained through
participation and interaction over time. As
students and the instructor act in ways that are
consistent with new expectations regarding
explanation and justification, they contribute to
their ongoing constitution.
Another point, which is illustrated in two case
studies at two different U.S. universities, is that
every class, from the most traditional to the most
reform-oriented, has social norms that are
operative for that particular class. It is not the
presence or absence of social norms that
differentiates one class from one another. Rather,
it is the nature of the norms that differ from
class to class. Of course the social norms
pertaining to explanation and justification might
apply to a history class or an English literature
class, as well as a mathematics class. The term
sociomathematical norm refers to the fact
that the subject being learned is mathematics. The
expectation that one is to give an
explanation is a social norm, but what is
considered to be an elegant solution, a different
solution, an efficient solution, or an acceptable
mathematical explanation are sociomathematical
norms. For example, when students develop
predictions and explanations about the future of
say, the population of fish in a lake, it is
imperative that these explanations move beyond
conclusions based solely on contextual reasons
(e.g., the fish are going to run out of food,
so their numbers are going to decrease) to
include reasons that rely on an
interpretation of the mathematical idea of rate
grounded in the differential equation. Fostering
a classroom learning environment that promotes
the types of explanations valued by the mathematics
community is a process that evolves over time as
students and instructor interact in the classroom
setting. If instructors are interested in
promoting a classroom environment where students
routinely give and evaluate mathematical arguments,
explicit attention to the processes by which norms
are constituted is a first step.
Finally, this research team documented how these
evolving norms fostered a shift in student beliefs
about their role as learners, about their
instructor's role, and about the general nature of
mathematical activity. These beliefs shifted from
seeing their role as passive absorbers of
information to active participants in knowledge
creation. When the classroom is viewed as a
dynamic system that includes the way in which
students participate in mathematical learning,
we can account not only for how student beliefs
evolve and develop, we can also promote student
beliefs about mathematics more compatible with the
discipline itself. [see Yackel &
Rasmussen (in press)]
2 The research in students' learning of
differential equations began in 1998 with Chris
Rasmussen, Erna Yackel, and Karen King. Since that
time the project team has expanded to include
Michelle Stephan, Karen Whitehead, Michael Keynes,
and Wei Ruan. At the time of this review, these
researchers were collaborating with Karen
Marrongelle, Oh Nam Kwon in South Korea, and Mark
Burtch.
Chris Rasmussen
What Lies Beneath Correct Answers?
Being able to correctly associate differential equations
with appropriate graphs of solution functions and/or
slope fields suggests that students might demonstrate
a certain level of coherence between representations.
However, we all have experienced situations where
students have arrived at correct conclusions, yet
underneath lay erroneous ideas and conceptual gaps.
Details about students' underlying difficulties can be
obtained by conducting in-depth one-on-one interviews in
which students are asked to think aloud as they solve a
variety of problems. In one study that took this
approach, Chris Rasmussen investigated the mathematical
development of six students in a reform-oriented
differential equations course of 16 students at a large
mid-Atlantic state university.
t and
dy/dt = t + 1, students tended to
reason that y = t and t = 1 were
equilibrium solutions, respectively, and they used this
to guide their work on the matching task. If we view such
errors as making sense to the student, how might we
account for this overgeneralization? One explanation
involves the difficulty of conceptualizing a solution as a
function that satisfies the differential equation. In
previous math courses, students were accustomed to
thinking of a solution as a number or numbers, but in
differential equations, solutions are functions. Thus the
letter y in a differential equation is meant to
represent an unknown function, as well as being a variable
in the differential equation itself. Moreover, students
often associate the derivative with the slope of the tangent
line at a point, which, when combined with our everyday use
of the term equilibrium as balance point, tends to result
in students considering equilibrium solutions as points
where the derivative is zero, rather than as constant
functions that satisfy the differential equation. [see
Rasmussen (2001)]
1.
Mathematically, we would expect students' notion of
equilibrium solution to be a subset of their notion
of solution, but for these students this did not
appear to be the case. Consistent with Rasmussen's
findings, these results underscore an important
conceptual difficulty that may lie beneath many
correct answers. [see Zandieh &
McDonald (1999)]
4N(1N/3)
(1N/6) and the corresponding graph of
dN/dt vs. N. He asked the
following three questions: (a) What are the
equilibrium solutions? (b) Which of the
equilibrium solutions are stable and which are
unstable? (c) What is the limiting population for
N(0) = 2, N(0) = 3, N(0) = 4, and
N(0) = 7? All six interview subjects figured
out the correct answers to parts (a) and (b) but
four of the six students were unable to address
part (c). This was particularly surprising because
the typical student approach to this problem was to
figure out the first two parts by creating a sketch
like that in Figure 1.
Numerical Approximations and Graphical Predictions
Similar to the way research is pointing to students'
intuitive or informal theories and notions regarding
equilibrium, research is also highlighting students'
informal or intuitive ideas regarding numerical
approximations and graphical predictions. Regarding
the latter, Artigue found evidence suggesting that
students' mental image of Euler's method is similar
to that of a semi-circle inscribed with a series of
line segments. In addition to finding further
evidence supporting this, Rasmussen found another
inappropriate image of numerical approximations,
namely, that numerical approximations "track" the
exact solution by using the slope of the exact
solution at the start of each new time step. Of
course, this notion is untenable since one only
knows the slope of the exact solution at the initial
condition, and not at, say t = 0.5, 1.0, 1.5.
y)
and asked to describe how the limiting behavior depends
on the initial point in the xy-plane.
y)Proofs of Graphical Predictions
The Cultural Status of Graphical Representations
Students typically enter a first course in
differential equations with a significant amount of
previous experience in mathematics courses where
answers more often than not involved numbers and
equations — objects that live in the analytic, or
symbolic, setting. This may well serve as a
stumbling block to using the graphical setting as
a way to understand solutions of differential
equations and to qualitatively understand families
of solutions. As a result, students may implicitly
believe that graphical solutions are less than
desirable. Samer Habre pursued this line of
inquiry in a study that investigated students' use
of visual representations of solutions to ODEs.
Students from a third semester four credit calculus
class at a large northeastern university where the
first half of the course covered multivariable
calculus and the second half was devoted to
differential equations were his subjects. Data
included classroom and lab observations, students'
exams and assignments, and a 45-minute interview
with nine of the students in the class. One of
the questions in the interview was, "What comes
to your mind when you are asked to solve an ODE?"
The initial response from all nine subjects
indicated that the students thought of an analytic
solution. Their dominant notion of what
constitutes a solution remained in the analytic
realm even though a significant amount of class
involved learning qualitative methods that relied
heavily on technology to look at vector fields
and other graphical representations. Habre's
research lends further support to the claim that
students' concepts about solutions as analytic
are resistant to change and that moving to the
graphical setting to understand ODEs is
extremely difficult. Students' reluctance to
value graphical solutions equally with analytic
solutions is likely a result of the
mathematical culture that they have experienced
in many of their previous mathematics
classrooms. [see Artigue (1992);
Habre (2000)]
Student Understanding of Systems and Second
Order Differential Equations
In addition to research on the learning and
teaching of first order ODEs, researchers are
beginning to examine students' understandings of
systems and second order differential equations.
In one study, Maria Trigeuros investigated
student learning of systems of differential
equations in two ODE classes at a small private
university in Mexico. Three individual
task-based interviews were conducted with nine
students in each class. Her analysis of the
interviews reveals that some students had
problems interpreting the meaning of
equilibrium solution (which was an issue for
students in single ODEs as well), interpreting
the meaning of a point in phase space, and
seeing the dependence of time in the phase
space. Students in her study also showed a
tendency to focus on just part of the
information provided by phase portraits.
Only a few students analyzed long-term
behavior of solutions in relation to
equilibrium solutions. [see Trigueros (2000)]
'' + = 0
and to the undamped nonlinear model,
'' + sin = 0.
Graphs of solutions to the nonlinear model are
indicated with NL.
Technology
Depending on institutional constraints, resources,
and instructor preferences, technology can be
utilized in many different ways, including as a
teacher-led demonstration tool, as a lab activity
done outside of class time, or as an integrated
part of daily class sessions. Research in the
learning and teaching of ODEs is beginning to
shed light on the advantages and disadvantages
of some of these.
x + 4y,
y'(t) = 3xy, all the
students he interviewed were able to draw an
appropriate trajectory in the xy-plane
and to draw reasonable x(t) and
y(t) graphs corresponding to this
trajectory. Some students, however, faced
difficulties in drawing the 3D-parametric curve.
Habre suggests that the role of such computer
modules in student learning warrants further study.
[see Rasmussen (1997); Habre (2000)]
The Differential Equations Classroom
As mentioned earlier, attention to explanations
and justifications in differential equations is
an emerging area of interest that is informing
both teaching and basic research into how
students can learn undergraduate mathematics
with understanding. Given that many university
instructors are increasingly interested in
developing their students' ability to
communicate their thinking and reasoning, this
current line of research is of both pragmatic
and theoretical interest.
To Sum Up
The teaching and learning of differential equations
is heading in new directions due in part to an
increasing interest in dynamical systems and a
broad interest in improving mathematics learning
and teaching across K-16. The use of technology
and the "rule of four" (concepts and methods should
be approached graphically, numerically, analytically,
and descriptively) are a springboard for looking at
new ways for students to understand central ideas
and methods in differential equations. In addition
to new pedagogical strategies, research into
students' thinking and reasoning is yielding new
insights into ways to create and sustain learning
environments where students can gain deep
understandings of mathematical concepts and
methods.
References
1 Support for this review was provided
by the National Science Foundation under Grant No.
9875388. Any opinions, findings, and conclusions or
recommendations expressed in this material are
those of the authors and do not necessarily reflect
the views of the National Science Foundation. We
would also like to thank Annie Selden for her
comments on an earlier draft of this paper.
Department of Mathematics, Computer Science & Statistics
2200 169th Street
Purdue University Calumet
Hammond, IN 46323
Email: raz@calumet.purdue.edu
Karen Whitehead
Department of Math and Computer Science
Valparaiso University
Valparaiso Indiana
Email: Karen.Whitehead@valpo.edu
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Last modified: Tue Jan 28 12:55:29 -0500 2003