Student interpretations of the terms in first-order ordinary
differential equations in modelling contexts
DAVID R. ROWLAND
Student Support Services, The University of Queensland,
Brisbane, QLD 4072, Australia e-mail:
d.rowland@courses.uq.edu.au
ZLATKO JOVANOSKI
School of Physical, Environmental and Mathematical Sciences,
The University of New South Wales at The Australian Defence
Force Academy, Canberra, ACT 2600, Australia e-mail: z.jovanoski@adfa.edu.au
(Received
2 June 2003)
A study of
first-year undergraduate students’ interpretational difficulties with
first-order ordinary differential equations (ODEs) in modelling contexts was
conducted using a diagnostic quiz, exam questions and follow-up interviews.
These investigations indicate that when thinking about such ODEs, many students
muddle thinking about the function that gives the quantity to be determined and
the equation for the quantity’s rate of change, and at least some seem unaware
of the need for unit consistency in the terms of an ODE. It appears that
shifting from amount-type thinking to rates-of-change-type thinking is
difficult for many students. Suggestions for pedagogical change based on our
results are made.
1. Introduction
Reform in the teaching of differential equations at the
tertiary level has been driven in part by the same sorts of disappointing
observations on student learning outcomes that have driven calculus reform [1,
2]. In addition, the ready availability of programmable graphics calculators
and symbolic algebra packages such as Mathematica and Maple have raised the
question of what it is important for students to be able to do themselves and
what can be left for the technology to calculate. Furthermore, these packages
have opened up opportunities for more sophisticated analyses and investigations
of ordinary differential equations ( ODEs ) than was previously possible.
Consequently, as Boyce [2] describes, in reform efforts there has been a move
away from ‘mere manipulative skills’ teaching, to a greater emphasis on conceptual
understanding, exploration and higher-level problem solving. Some of this
reform work has been documented in a special issue of The College Mathematics
Journal (Issue 5, 1994) and examples of the uses of graphics calculators and
Mathematica are given in [3] and [4] respectively.
Aside from courses in
differential equations, some authors as part of their reform calculus efforts
have also introduced, early on in the curriculum, differential
International
Journal of Mathematical Education in Science and Technology
ISSN 0020–739X print/ISSN 1464–5211 online # 2004 Taylor &
Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/00207390410001686607
equations via physical models [3,
5]. The reasons given for doing so are partly motivational (the physical models
illustrate the applicability and usefulness of the mathematics; the solution of
differential equations provide a natural motivation for the learning of
techniques of integration; also, introducing differential equations makes the
subject fresh and therefore more interesting for students who have already
studied calculus at the secondary level), though on the pedagogical side, it is
believed that the physical applications will help deepen student understandings
of the mathematics they are learning [3].
Despite these considerable reform efforts involving ODEs,
not much research into students’ understandings of ODEs appears to have been
done, however. Rasmussen [6] has investigated student understandings of various
aspects of solutions to ODEs, including graphical and numerical solutions. Two
things of particular note for this research come from Rasmussen’s work. First,
Rasmussen posited that the switch from conceptualizing solutions as numbers (as
is the case when solving algebraic equations) to conceptualizing solutions as
functions (as is the case when solving ODEs) is akin to a paradigm shift and is
non-trivial for students. Secondly, Rasmussennoted that some of the
difficultiesstudents had with graphical approaches stemmed from either thinking
with an inappropriate quantity and/or losing focus of the intended underlying
quantity. (This observation may be related to the height–slope confusion
previously identified in the calculus literature [7, 8]).
Another
piece of research in this field is by Habre [9], who explored students’
strategies for solving ODEs in a reform setting. Of interest from this research
is that despite an emphasis on qualitative (graphical) solution methods in the
course, the majority of students interviewed still favoured algebraic
approaches over graphical approaches at the end of the course, possibly
reflecting the heavy algebraic focus of previous mathematical experiences. The
research also suggested that students find it difficult to think in different
modes (i.e. algebraic and graphical) simultaneously, which might also help
explain why students typically don’t use multiple modes to
tackleproblems.(Asreportedin[10],typicallyonly20%ofengineeringstudentsuse
diagrams to aid their physics problem-solving in exams, and it has been found
that even top students used graphs in only one quarter of their solution
attempts in a test with non-routine calculus problems [11]).
An aspect of conceptual understanding not addressed by the
above research is students’ ability in modelling contexts to both interpret in
physical terms the various terms of an ODE and to translate from a physical
description into a mathematical description. These two abilities are the focus
of the present research and are of course flip sides of the same coin. These
abilities are important as they are needed if students are to reason
appropriately about solutions and ultimately if they are to develop skills in
modelling themselves.
Although these aspects of student understanding of ODEs do
not seem to have been previously investigated, similar things have been
investigated in the contexts of algebraic word problems and various aspects of
calculus problems. Thus, for example, it has been found that in algebraic word
problem translations, common problems were word order matching/syntactic
translation and static comparison methods [12]. Similarly, student difficulties
with correctly distinguishing between constants and variables, and between
dependent and independent variables in rates of change contexts has also been
identified [13–15]. In addition, research on student understanding of
kinematics graphs [7] and velocity and acceleration [16, 17], reveals that many
do not clearly distinguish between distance, velocity and acceleration. It is
also known that prior to their development of the concept of speed as an
ordered ratio, children typically progress through a stage where they think of
speed as a distance (the distance travelled in a unit of time) [18].
The goal of the research reported in this paper, which is to
investigate student difficulties with translating between words (physical
descriptions) and mathematics, is motivated by the observation that many
student ‘errors’ in science and mathematics are not just careless slips, but
are in fact systematic and furthermore common to significant numbers of
students across a wide range of contexts. Furthermore, these systematic
‘errors’ have been found to be resistant to change by traditional instruction,
and there is a general consensus that teaching needs to be cognizant of these
systematic ‘errors’ if they are to be effectively addressed [19–22].
Why precisely the above should be the case is still a matter
of debate and research [19], although there seems to be general agreement that
fundamentally it is because students are not blank slates for their teachers to
‘write upon’, but they come to class with ideas and conceptions which may both
aid or hinder further learning and also because students do not unproblematically
absorb new teachings, but what they learn (or fail to learn) is affected by
both their beliefs about learning and by how they attempt to make sense of what
they are taught. (This is the
‘constructivist
principle’ [23, 24].)
Beyond this general agreement though, there are a range of
different ways of conceiving of student ‘errors’ (and hence the scare quotes
around ‘errors’, as sometime these are conceived of as primitive understandings
that are to some extent ‘correct’, and need to be developed and refined more
than ‘corrected’), some of which seem to be competing conceptions, while others
appear to be describing different phenomena. Confrey [19] and Hammer [25]
provide reviews of the various conceptions of student ‘errors’ and a discussion
of their implications for pedagogy are provided in these reviews, while the
ongoing nature of research in this field is illustrated in [23, 26].
One conception that seems particularly relevant to the current
study, however, is that of a ‘paradigm shift’ or ‘knowledge in transition’. As
posited by Rasmussen [6], moving from the context of algebraic equations where
solutions are constants, to that of ODEs where solutions are functions,
represents a paradigm shift for students. Consequently, as with the scientific
paradigm shifts as discussed by Kuhn, it can be expected that some students
will have difficulties moving from the old ways of thinking to the new. Related
to this view is the phenomenon of ‘binary reversion’ [20] where older and more
familiar knowledge is inappropriately triggered by the context because it is
more readily cognitively accessible than the newer, less familiar but relevant
knowledge. These ideas are relevant because we will argue later that the ‘rates
of change’ contexts of first-order ODEs can trigger both thinking about the
function that describes a quantity rather than the function that describes the
quantity’s rate of change and constant rate of change concepts, which are
inappropriate in a variable rate of change context.
To help understand why the
above-identified problems should be so common (and why standard instruction
almost encourages things to go wrong), and to help predict the sorts of
problems one might expect in any area of learning, we use Perkins’ theory [27]
of default modes of thinking as a cognitive framework. According to this
theory, the pattern-driven nature of human cognition leads to four default
modes of thinking which, while they serve us well most of the time, can cause
problems in novel situations or familiar situations that have been subtly
changed (i.e. the typical sort of situations any student faces). These default
modes are (giving only the negative side of the mode):
Fuzzy thinking: exemplified by a failure to clearly
discriminate between closely related terms [20, 21, 28–30]; and
overgeneralizing or having deficient applicability conditions [20, 26, 29].
Hasty thinking: exemplified by too rapidly deciding on a
solution strategy or solution on the basis of a superficial examination of the
most obvious features of a problem, rather than on deep processing (i.e. trying
to pattern match the problem to one seen before) [30–32]. This thus represents
a weakness in problem-solving approach, that is, it is a metacognitive
weakness.
Narrow thinking: related to hasty thinking, narrow thinking
also represents a metacognitive weakness in that it is exemplified by a failure
to consider alternative perspectives or solution strategies.
Sprawling thinking: may be
useful when one is brainstorming, but is a problem when it leads one to lose
track of what one is doing or ‘to change horses midstream’ [20]. This also
represents a failure to develop effective problem-solving control and
monitoring strategies.
2.
Method
2.1.
Participants
Participants
The participants were 59
first-year BSc students enrolled in a two-semester sequence of calculus and
linear algebra. These students had all studied some calculus at secondary school,
and had achieved reasonable results in these studies. During their semester of
university study, the students had worked on a variety of physical systems
which could be modelled by first-order ODEs, including unconstrained and
logistic population growth, radioactive decay, the mixing of solutes in a tank,
and Newton’s law of cooling. Apart from solving the resultant ODE, during the
course students were also expected to be able to interpret the physical meaning
of the terms in an ODE given a description of the physical problem, and, given
the description of a physical system covered in the course, to determine the
governing ODE of that physical system.
2.2.
Procedure
Probing student conceptual understanding is neither easy nor
straightforward, and various methods for doing this each have their strengths
and weaknesses as discussed below. Consequently, we used three different
methods––a multiple choice diagnostic quiz, a short answer exam question, and
one-on-one interviews–– in order to triangulate our observations. We did not
investigate students’ ability to model a physical problem completely by
themselves for two reasons. First, as part of the course aims, students were
only expected to be able to model a problem that was a variation on one covered
by the course. Thus their performance on such a question may simply reflect
what they had memorized, not what they really understood. Second, it was
envisaged that modelling a new problem would bring too many factors into play
to allow easy interpretability of student answers. Consequently, all of the
questions asked only focus on aspects of the whole problem of modelling.
2.2.1. Phase 1: Diagnostic quiz
In the first phase of the investigations, the participants
were given a multiple choice diagnostic quiz in the last week of classes of
their first semester of university study (48 out of the 59 were present at
class on that day). The students were not given warning of the quiz, and so did
not make any specific preparation for it (it was assumed that the quiz questions
were either sufficiently basic and/or conceptual that this lack of preparation
should not have been a problem for them). The timing of the quiz was such that
the students had at that time spent some time studying first-order ODEs,
methods for solving them, and various physical systems that can be modelled by
these differential equations.
The diagnostic quiz included two questions chosen by the
authors to be relatively simple models, similar to but not identical to models
the students had seen either in class or on tutorial sheets (so that students
could not simply ‘remember’ the correct answer), with the responses being
chosen to represent hypothesized ‘error types’ (these hypotheses were based on
a combination of teaching experience, results in the literature on related
calculus questions, and an application of Perkins’ [27] default thinking modes
theory). (Normally, one would construct such a quiz after first doing some
qualitative research on student thinking with a number of students. However, in
this case, the choice of distracters was validated post hoc by the students’
open responses to the exam question and by the one-on-one interviews.)
The two
modelling questions on the quiz were:
Quiz Question 1. As light
passes through a liquid or a solute it is absorbed ( i.e. its intensity I,
decreases) because it interacts with the molecules of the liquid or solute. For
a solution of anthracene dissolved in dioxane, the absorption rate is proportional
to its intensity I at that point (the proportionality constant is 0.0693 per
mm). The intensity of light I, as a function of distance travelled x (in mm),
into this solute is therefore described by the differential equation:
dI dI dI dI
ðaÞ ¼ 0:0693 ðbÞ ¼
0:0693 ðcÞ ¼ 0:0693I ðdÞ ¼
0:0693I dx dx dx dx dI dI dI dI
ðeÞ ¼
0:0693x ðfÞ ¼ 0:0693x ðgÞ ¼
0:0693Ix ðhÞ ¼ 0:0693Ix dx dx dx dx
Quiz Question 2. The population
P of fish in a pond at a fish farm as a function of time t will grow at a rate
proportional to the population if left undisturbed and if there is plenty of
food. In addition to this undisturbed growth rate, 5000 fish per year are also
removed from the pond for sale. The differential equation which describes the
growth of the fish population with time (in years) is given by:
(a) dP=dt
¼
term describing undisturbed growth 5000ðdP=dtÞ
(b) dP=dt
¼
term describing undisturbed growth5000
(c) dP=dt
¼
term describing undisturbed growth 5000t
(d) dP=dt
¼
term describing undisturbed growth 5000P
(e)
dP=dt ¼ term describing undisturbed growth
5000Pt
For question 1, the responses are in four pairs, with each
pair differing only in sign to check for student awareness of sign issues.
Responses (a) and (b) are of the simple form, dI/dx¼rate constant; (c) and (d)
are of the form, dI/dx¼rate constantdependent variable, a form familiar to
students from unconstrained population growth, and with (d) being the correct
answer; (e) and (f) are of the form, dI/dx¼rate constantindependent variable, a
form which represents thinking in terms of ‘amounts’ rather than ‘rates of
change of amounts’ in the familiar context of constant rate problems; and
finally, (g) and (h) are as per (e) and (f) but with a factor of I included
because the question states that rate of absorption is proportional to I. Apart
from dropping the sign variation, the responses to question 2 were constructed
along similar lines.
As can be seen, these two questions assessed students’
ability to translate from a word problem to a mathematical equation. As translation
is very difficult for most students and can involve a host of difficulties, to
simplify the interpretation of the results it was decided to use the multiple
choice format where students had to match the correct equation to the physical
problem, a presumably much simpler task than developing the whole equation
oneself.
The strengths of such quizzes,
if the alternative responses are in fact distracters which match student
thinking (see [33] for a discussion on the construction of useful multiple
choice diagnostic questions), are that they indicate the prevalence of certain
types of thinking [34] and if used as pre-tests and post-tests, can be used to
assess how effective a course of instruction has been in helping students think
in the accepted ways [35]. Their weakness, however, is that they don’t
necessarily reveal why students pick certain responses, and being forced
response type questions, may force students to respond in different ways than
they would if left to their own devices. Also, the results need to be
interpreted with some caution, as it has been shown that students are not
necessarily consistent in the way they answer questions which are presumably
testing the same concepts [36, 37]. (This inconsistency presumably reflects the
fact that the knowledge of many students is fragmented and that the way they
think about problems can be context dependent [24, 29].) Despite these
shortcomings however, we argue that they still provide useful information when
correlated with other sources of information.
2.2.2.
Phase 2: End-of-semester exam
The end-of-semester exam
consisted of a variety of calculus questions, and included two questions on
first-order ODEs. The part of the question which is relevant to this study is
as follows:
Exam Question 3(c). A new type of drug is administered to a
patient in a hospital by continuous intravenous drip. The differential equation
describing the amount D (mg) of the drug in the patient at time t hours after
it was first administered is dD=dt ¼ 100
0:01D2. Give the physical meaning of each of the three terms
dD/dt, 100 and 0.01D2 in the differential equation.
Thus, this question assesses students’ ability to relate the
terms in an ODE to processes in the physical model. Since this is a free
response question, it overcomes some of the limitations of the forced response
multiple choice questions in the diagnostic quiz. However, it too suffers from
the limitation that because student responses were generally not justified,
these responses only indirectly tell how students are thinking about the
question. There is also the problem that some students may not choose their
words very carefully and so what they write may not be precisely what they mean
to say (this is another reason why multiple sources of information on student
thinking are required). Nevertheless, in combination with other sources, this
question can provide insights into student thinking. Furthermore, as argued in
the Introduction in relation to ‘fuzzy thinking’, this lack of precision is
undoubtedly one of the causes of errors in student thinking.
To aid the analysis of the
results, the students’ answers to this question were tabulated in order to ease
comparisons both within and between students. After tabulation, each
investigator read through the responses several times in order to get a feel
for the sorts of categories answers could be sorted into. Each investigator
then independently sorted responses into the agreed categories and compared results.
There was high agreement between the investigators and the few disagreements
were then discussed and a final categorization was decided upon. In addition to
this, consistency of types of responses within this question and with the other
questions was also determined.
2.2.3.
Phase 3: Follow-up interviews
To check our interpretations of student responses to the
first two phases of the study, during second semester, one-on-one follow-up
interviews were conducted with eight students who had exhibited commonly
occurring misconceptions on the diagnostic quiz. While the students targeted
for these interviews were ones exhibiting common errors, participation was
voluntary. The students interviewed scored from the 14th to the 71st percentile
on the final exam, with five of the eight scoring near or above the class
average. These facts give confidence that most if not all of the students
interviewed had made a reasonably serious attempt at the course and that their
(mis)conceptions are likely to be representative of the (mis)conceptions of the
bulk of students who had problems with the questions investigated. The protocol
for the interviews was as follows.
Each student was first reminded of their answer for
questions 1 and 2 of the diagnostic quiz and was then asked to explain why they
chose that answer. If their response seemed incomplete or unclear, they were
prompted to elaborate (sometimes this involved asking the student to explain why
they didn’t choose some of the other answers).
Following this, each student was asked to more fully explain
their answers to Exam Question 3(c). To flesh out their thinking more, among
other things they were also asked what they thought the units of each term in
the ODE were.
To investigate whether student
misinterpretations of the constant ‘100’ term in exam question 3(c) was due to
unfamiliarity with the operation of a continuous intravenous drip (though this
was discussed in the course) or whether it represented something more general,
the following more familiar physical situation was presented to the interviewed
students.
Interview Question. A car
initially travelling at 100km/h suddenly loses engine power and so starts to
decelerate at a rate proportional to its velocity squared (the proportionality
constant is k) due to wind resistance. The car’s velocity v as a function of
time t since the loss of power is thus described by the differential equation:
(a) dv/dt¼100kv2
(b) dv/dt¼kv2
During each interview, the
interviewer made a written record of student responses and at the end of the
interview, also made a few additional notes regarding how the student had
responded (such as recording that a particular line of questioning had been
curtailed as the student had appeared to be getting quite frustrated by their
inability to sort things out in their mind). After the interviews, student
responses to the various parts of the follow-up interviews were tabulated to
allow for easy comparison of student responses in order to identify
similarities and differences in responses. This table also allowed for easy
comparison of these students’ interview responses with their answers to other
phases of the investigation.
3. Results
Although not a part of this study, we’d like to note that
the students investigated developed reasonable manipulative ability with ODEs,
as evidenced by the fact that on the final exam question 2(a), which asked
students to ‘find the explicit general solution to dy/dx¼yln(x), x>0’, the
average mark was 5.25/7 , with 42% of the class getting full marks for the
question. (The main reasons for losing marks were algebraic errors, or errors
in performing the integration.) This relatively good performance on a
‘traditional question’ in comparison to the relatively poor performance on the
conceptual questions, as will be shown below, reinforces previous observations
in the literature that performance on ‘traditional (manipulative or
algorithmic) questions’ does not necessarily give an instructor a clear idea of
how much students have learned conceptually.
In
presenting the results of our investigation, it is most informative to start
with Phase 2, student interpretations of the terms in the ODE of Exam Question
3. As shown in table 1, and in contrast to their performance on Exam Question
2( a), many students performed rather poorly on this question.
Apart from this general
observation, a detailed analysis of student responses revealed various patterns
of responses. Firstly, as one would expect, the majority of students could
interpret the dD/dt term adequately, but even here almost a quarter of students
made incorrect interpretations. Some of these students mixed amount and rate
terminology in their answers, for example, ‘...is the amount of drug (D) in the
body with relation to time, or the rate of change of concentration of the
drug’, while some others just talked in terms of amounts, for example,
‘...represents how much the amount of drug changes due to time’. One might
think, particularly in relation to the first answer, that these students are
just being imprecise in their language use, and that it is likely they have
correct conceptions. This interpretation cannot be ruled out for some students,
as two students also interpreted the constant ‘100’ in terms of ‘amounts
in/administered’, and yet got
|
Student Interpretation
|
|
Term
|
|
|
dD/dt
|
100
|
0.01D2
|
|
|
Correct
|
42
|
10
|
21
|
|
No attempt
|
3
|
6
|
6
|
|
Other incorrect
|
14
|
43
|
32
|
Table 1. Student
results on Exam Question 3( c ).
both the diagnostic questions
correct, suggesting that they had some level of understanding. However, as will
be shown below, many student answers clearly indicate that the issue is not
merely one of imprecise language usage, but of a confusion between related
‘amounts’ and ‘rates of change of amounts’.
Consider now student interpretations of the constant ‘100’
term. These answers could be categorized as follows: there were six ‘no
attempts’, ten answers were deemed ‘correct’ and five were categorized as
miscellaneous. Seven answers were in terms of ‘amounts’, and as mentioned
above, some of these may have been just imprecise language use. Examples
include: ‘...represents the amount of drug going into the patient’s body’, and,
‘...is the amount of drug D going in’.
The most common response, though, was in terms of ‘an
initial condition or arbitrary constant’ (16 students), for example: ‘...is the
initial amount of the drug in the body’, ‘...is a constant that represents the
initial level of drug administered’, and ‘...arbitrary constant’. Note that
these answers are clearly in terms of ‘amounts’ and cannot simply be imprecise
language use.
One possible reason as to why so many students interpreted
the ‘100’ term in this way is given by one of the interview responses where the
student said that the ‘100 is a constant’ and so must be ‘the initial amount
administered’. Furthermore, this student argued that ‘you need a variable for
it to be a rate of change’. This is reminiscent of Elby’s [24] what-you-see-is-what-you-get
(WYSIWYG) conjecture, in that because ‘100’ was seen as ‘constant’, it was not
seen as a ‘rate of change’, and the ‘obvious’ link to make (when making a
‘fuzzy’ overgeneralization) is that constants in ODE problems are ‘initial conditions’.
Another, though very similar
possibility revealed in the interviews, came from four students’ answers to the
decelerating car question. These students all argued that the ODE modelling the
situation was ‘dv=dt ¼ 100 kv2’
because the car started at 100km/h and the kv2 showed how it slowed
down. For example, various students argued:
‘...car
traveling at 100 km/hrate at which slowing’;
‘...initial velocity ¼ 100 km/hsomething because you will slow
down’; ‘...100 is initial velocity and velocity decreases from something’ and
‘...initially traveling at 100 and can’t decelerate from zero’.
Clearly these students are all
thinking about an equation for velocity rather than for its rate of change as
required, and because of this, they have put the initial condition into the
ODE. (This is reminiscent of the student descriptions of dD/dt as being for
‘the amount of drug D in the body in relation to time’.)
Moving back to the exam question, the other common
interpretation (15 students) of the ‘100’ term was that it represented an
‘equilibrium amount’ or a ‘maximum amount’, for example: ‘...is the equilibrium
constant’ and ‘...is the limiting/max amount. It is the equilibrium solution’.
Again it is unlikely that these students are simply being imprecise in their
use of language, but have again started thinking in terms of ‘amounts’ rather
than in terms of ‘rates of change of amounts’.
This last set of responses clearly shows the power of
context on student thinking, as the most likely explanation for why they
interpreted the ‘100’ term as an ‘equilibrium or maximum amount’ is because
Question 3(a) asked the students to ‘find the equilibrium solution of the
differential equation’, and for this ODE, this happens to numerically evaluate
to 100. Contextual influences were also evident in student interpretations of
the 0.01D2 term as well, as eight students who had interpreted the
‘100’ term as either an ‘initial condition/amount’ or as a ‘maximum or
equilibrium amount’, then went on to interpret the 0.01D2 term as a
‘rate or amount administered’ in order to account for their knowledge that a
drip continuously delivers drug to a patient. For example: ‘...is the rate at
which the drug is being administered’ and ‘...is the amount of drug that the
continuous intravenous drip gives the patient’.
Another prominent category of interpretation for the 0.01D2
term was in terms of amounts, with 13 (22%) having answers like, ‘... is
the amount of drug being used up by the patient’s body’. In comparison, these
students’ interpretations of the ‘100’ term were fairly evenly spread between
‘amount in’, ‘amount initially administered’ and ‘equilibrium or maximum
amount’ type answers. It is possible that those with the combination ‘amount
in/amount used up’ for the ‘100, 0.01D2’ terms may have just been
sloppy with their use of words, but the others are clearly on the wrong track
and are thinking in terms of ‘amounts’ rather than in terms of ‘rates of change
of amounts’.
Further evidence of the above interpretation that many
students are thinking in terms of amounts rather than rates of change of
amounts comes from the interviews where five of the eight students gave the
units of dD/dt as being ‘mg/h’ while giving the units of the ‘100’ term as
being ‘mg’. Two of these five students realized that this was a problem (thus
indicating that they hadn’t thought to check for unit consistency originally)
but could not resolve it, while another two explicitly stated their belief that
the units did not have to be consistent! Either way, the idea that one can use
unit consistency to check one’s thinking was not a part of any of these
students’ problem-solving strategies.
Moving
now to the diagnostic quiz, the above results indicate that we were correct in
our anticipation that amount type thinking would be the most likely error in
student thinking. Table 3 in the appendix gives the full breakdown of student
answers to the two diagnostic quiz questions, while table 2 combines responses
into broader categories to ease interpretation.
There are three striking things
about table 2. First, only eight students got both questions correct. Secondly,
consistent with observations made above, 20 students (45%) used amount-type
thinking for their answers to both questions. Finally, some 14 students (32%)
appeared to give inconsistent answers (i.e. answers which appear to involve
different types of reasoning) for the two questions. As mentioned above, this
apparent inconsistency might reflect the fact that many students’ knowledge is
highly fragmented and consequently is highly context-dependent [24, 29]. Note
that while this inconsistency does not bode well for the rationale of this
research, the fact that 45% nevertheless did seem to be consistent in the ( not
|
|
RC (2b)
|
DV (2d)
|
IV (2 c,e )
|
|
RC (1a,b)
|
1
|
–
|
1*
|
|
DV (1c,d)
|
8**
|
1
|
6*
|
|
IV (1e–h)
|
7*
|
–
|
20
|
Table 2.
Cross-tabulation of student responses to diagnostic quiz Questions 1 and 2.
Here therowandcolumnheadersprovidedescriptorsoftheright-handsideoftheODEs,with
RC¼rate
constant, DV¼dependent variable (i.e. I or P), IV¼independent
variable (i.e. x or t), and * indicates apparently inconsistent answers, while
** indicates both quiz questions correct. Total number of participants¼44.
quite correct) ways they thought
about these problems, supports the value of this kind of research to find out
patterns in the errors students make.
The follow-up interviews confirmed that students were
picking answers 1(e)–(h) and 2(c), (e) on the quiz because they were thinking
in terms of amounts, and furthermore that there were two basic types of
amount-type thinking going on. First, for the fish removal question, six of the
eight students argued that the answer was 2(c) because 5000t gives the number
of fish removed after t years. This result suggests that the issue is one of
‘knowledge in transition’ or of a ‘paradigm shift’ as the statement that ‘5000t
gives the number of fish removed after t years’ is actually a correct statement
(and presumably these students intuitively drew on their experience with
constant rate problems to quickly come to this conclusion), it is just
inappropriate for the current context.
Student
reasoning on the light absorption question was somewhat different however,
presumably because this question was sufficiently different from previously
seen constant rate problems so as not to automatically trigger that schema. For
this question, nearly all of the students interviewed basically argued that the
ODE had to have an x on the right-hand side because I was a function of x. (
Again note that these students are thinking in terms of I(x) instead of its
rate of change.) This suggests not only amount-type thinking (one student said
that ‘the x accounts for more light being absorbed with distance travelled’),
but also a lack of insight into implicit functions (three students explicitly
ruled out dI/dx¼0.0693I as a solution because it was not explicitly a function
of x as they expected). A further lack of understanding of implicit functions
was revealed by one student’s comment that the P in diagnostic quiz Question 2
was the maximum amount of fish, rather than P(t) (an example of the
constant/variable confusion found by other researchers [13–15]). It is unknown
whether similar interpretations were made by other students of the I in
diagnostic quiz Question 1, thus causing them to rule out solutions with an I
in them.
4. Discussion
Although we found some context dependence in student
answers, we also found that a significant proportion of students muddled
‘amount’ and ‘rate of change of amount’ type thinking when thinking about
first-order ODEs used to model physical processes. In some instances, this may
be simply imprecise language use by the students involved, but in many others
it is clearly a lack of discrimination between these closely related terms. In
fact, one student who was interviewed even went so far as to say, ‘rate and
amount of change mean the same to me’. This lack of discrimination between
closely related concepts is often found in novices [20, 21, 28–30], and may
simply reflect the human tendency for ‘fuzzy’ thinking [27], but its similarity
with a stage that children typically go through in their thinking about speed
(namely considering speed as a distance––the distance travelled per unit time
[18]) suggests also that this may be an instance of ‘knowledge in transition’
or of difficulties making a ‘paradigm shift’ (cf. [6]) from thinking about the
function which describes how a quantity varies to thinking about the equation
which describes how a quantity’s rate of change varies.
We found several consequences of this muddling of ‘amounts’ and
‘rates of change of amounts’. First, a constant term in an ODE was interpreted
by many either as an initial condition or as an equilibrium/maximum amount
rather than as a constant rate of change. This may reflect Elby’s [24] idea of
WYSIWYG, namely that a ‘constant’ is a ‘constant’, that is something that is
not changing and hence not a ‘rate of change’, with ‘sprawling thinking’ (i.e.
losing track of what one is after [6, 20, 27]) also possibly contributing to
the error. Similarly, several of the students interviewed
put an initial condition into the ODE modelling a decelerating car. In this
case, the students were clearly thinking in terms of the car’s velocity rather
than its rate of change. Interestingly, one student who from their knowledge of
physics recognized dv/dt as representing acceleration, did not make this
mistake as they realized that they were looking for an equation for
acceleration rather than velocity.
Thisraisestheintriguingpossibilitythatrephrasingquestionssothattheyareabout
finding, for example, the equation for ‘acceleration’ rather than finding the
equation for ‘the rate of change of velocity’, may help students stay focused
on ‘acceleration’ insteadof thinking about ‘velocity’ (cf. [38]). In all these
cases, ‘hasty’thinking [27] , namely pattern matching based on superficial
similarities with similar problems seen before, also undoubtedly contribute to
the muddling.
The second major example of muddling of ‘amount’ and ‘rate
of change amount’ type thinking was the common tendency for students to use
ideas from constant rate problems where a change in a quantity is given by the
rate constantthe independent variable. Thus for example, if P is the number of
fish in a fishery and 5000 are removed per year, then ‘dP/dt¼5000t’. A contributing
factor to this confusion for at least some students was their thinking that the
dependence on the independent variable had to be explicit, thus failing to
recognize that a dependence on the dependent variable is also implicitly a
dependence on the independent variable. Again there is evidence of superficial
pattern-matching to familiar problems and losing track of what one is after.
Another
area where a lack of sufficient discrimination was evident was the lack of
awareness, even appreciation, by some students, that all the terms in a
physical equation must have the same units. This may reflect the possibility
that they have not observed or realized (or not been required to realize) that while
each individual factor in a physical equation or ODE may have differing units,
each term as a whole must in fact have the same units. (Additionally, some
students did not realize that proportionality factors in ODEs often have units
too.) Since this problem was only revealed in the interviews (though student
answers to other questions showed that few, if any, took units into
consideration when thinking about the ODEs), it is unknown how common this
problem is and thus is an area for further research.
5. Suggestions
for pedagogical improvement
As a broad claim (and one oft made in both the calculus
reform and physics reform literature, and by [1] in relation to reform in the
teaching of differential equations), the inclusion of more qualitative or conceptual
type questions into the curriculum is likely to be of some benefit, as these
will force students to move away from a purely manipulation focus to more of a
focus on understanding. More specifically, based on the experience of physics
reform efforts, an effective way of dealing with the student misconceptions
brought to light by questions 1 and 2 of the diagnostic quiz is likely to be
Mazur’s [39] Peer Instruction method ( see also
http://mazur-www.harvard.edu/education/pi.html). In this method, students would
first be given in class a question like 1 or 2 from the quiz to think about
individually and to choose an answer. A vote on the answer would reveal a
diversity of opinion and consequently provide the cognitive dissonance required
to motivate the students to discuss the question in small groups. (Having to
defend their choice and argue against others is likely to lead to more learning
than if the instructor were simply to resolve the dispute.) Mazur found that
these group discussions usually zone in on the correct answer (provided that at
least 30% of the students have a correct conceptual understanding), and the
instructor can check that this has in fact happened by a second vote. At this
point, the instructor can also confirm that the students have discovered the
important checking strategy of confirming that the units of both sides of the
differential equation agree by asking if the students know how they could
positively prove an answer is incorrect. (Proving that an answer is in fact
correct is of course considerably more difficult.) A possible drawback of this
method, though, is that students may need to experience it regularly for it to
work well.
Another technique that has been used in the past is to give
students questions of the sort: ‘A student has provided the following answer to
this problem. What is the error in their reasoning?’ If the error has been
shown to be common by research, then many students will consequently become
aware of their error through questions of this sort, and, by having to figure
out where they’ve gone wrong themselves, are likely to learn a lot.
The
above two suggestions work on the idea that misconceptions need to be
confronted if they are to be overcome. Another approach is to recognize that it
is common for humans to only discriminate between terms as much as appears to
be necessary [20], and that the most common error identified above is most
likely due to a failure to clearly discriminate between amounts, changes in an
amount and the rate of change of that amount. Consequently, questions which
force students to clearly discriminate between these terms may help reduce the
sorts of errors we have identified.
Acknowledgment
A
preliminary version of this paper was presented at the 5th Biennial Engineering
Mathematics and Applications Conference, 29 September–2 October, 2002.
Appendix: Student
results on the diagnostic quiz questions
|
|
2a
|
2b*
|
2c
|
2d
|
2e
|
blank
|
Total
|
|
1a
|
|
|
|
|
|
|
0
|
|
1b
|
|
1
|
1
|
|
|
|
2
|
|
1c
|
|
|
2
|
|
1
|
|
3
|
|
1d*
|
|
8
|
2
|
1
|
1
|
|
12
|
|
1e
|
|
1
|
5
|
|
|
|
6
|
|
1f
|
1
|
5
|
8
|
|
1
|
|
15
|
|
1g
|
1
|
1
|
6
|
|
|
|
8
|
|
1h
|
|
|
|
|
|
|
0
|
|
blank
|
|
|
|
|
1
|
1
|
2
|
|
Total
|
2
|
16
|
24
|
1
|
4
|
1
|
48
|
*The correct responses.
Table 3. Complete
cross-tabulation of student responses to questions 1 and 2 of the diagnostic
quiz.
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