STUDENTS’ CONCEPTIONS OF SOLUTION CURVES AND EQUILIBRIUM IN SYSTEMS OF DIFFERENTIAL EQUATIONS

María Trigueros G.

Instituto Tecnologico de Mexico (ITAM)

trigue@gauss.rhon.itam.mx

Abstract: University courses on differential equations are being reconsidered and reformed, building on the previous reform efforts in calculus. In this paper we present a detailed analysis of semi-structured interviews where 18 students faced problems related to the solution of systems of ordinary differential equations, presented in different settings. We classify students' strategies and show many instances where students' understanding of parametric functions and variation conflict and become an obstacle to their understanding of the meaning of phase space representation and the notions of solution and equilibrium. We also highlight some cognitive and curricular issues that may be taken into account when dealing with these types of problems.

Introduction and Theoretical Framework

The teaching of differential equations at the collegiate level has been questioned in the last years (Artigue,1992; Ramussen, 1998; Trigueros, 1993, 1994; Zandieh and McDonald, 1999). The use of technological support, the development of Dynamical Systems as a growing branch of mathematics and the movement of reform in the teaching of Calculus have raised new concerns about the topics that students should learn nowadays. The decisions taken in this direction would benefit a lot from research on students’ conceptions about differential equations, about the strategies they use when facing problems related to differential equations and about the way students deal with complex problems like those appearing in the contexts of systems of differential equations.

The problem of description of equilibrium and solution curves in systems of differential equations is a complex one. It involves the integration of knowledge from different parts of mathematics and the use of different representation tools. This complexity makes the analysis of students' conceptions and strategies a challenging task. Taking into account previous work on differential equations (Artigue,1992; Ramussen, 1998; Trigueros, 1993, 1994; Zandieh and McDonald, 1999), on functions and derivative (Breindenback, 1992; Monk, 1992, Trigueros y Cantoral, 1992, Zandieh, 1997) and on the integration of information when students deal with complex problems. (Baker, Cooley and Trigueros, In press), this study uses the notion of schema from APOS theory (Asiala et al. 1996) in order to understand students' conceptions of solution curves and equilibrium solutions in the context of systems of differential equations.

For that purpose the idea of a double-triad of schema development, as proposed by Baker et al. (In press), was introduced in the context of this problem. Experience from teaching and from previous students' work in different courses on Differential Equations suggested that two main schema might be used in understanding the way students deal with problems related to the systems of differential equations: a schema for the representation of functions in parametric form and a schema for systems of differential equations. A double-triad containing these two schema was thus developed and used in the design and analysis of students' interviews with the aim of understanding students' conceptions and strategies. It is important to remember that the nature of the stages involved in the development of a schema is functional and not structural.

The development of the parametric representation of functions can be described by means of the following stages or levels. At the Intra-parametric level the student interprets the parametric representation of functions as two isolated functions; the possibility of elimination of the parameter and the representation of a function in a two dimensional plane, where the parameter is not explicitly shown, causes confusion. At the Intra-parametric level, some relationships between the components of the function are found, but the difficulty of using and interpreting graphs for this function persist. At the Trans-parametric level students are able to describe the function and its different representations in terms of the parameter involved. Coherence of the schema is demonstrated by the student's ability to describe which parametric representations are possible for a given function and how they relate to different graphical representations.

The development of the schema for the solutions of systems of differential equations can be described by the following stages: At the Intra-solution level the student is able to solve a system but is unable to interpret it and to coordinate the system and its solution with its graphical representations. At the Inter-solution level, the student begins to relate some systems with their graphic representation and to interpret the meaning of the solution to some systems, but it is not clear to him or her when a particular representation is convenient or even possible for many systems of equations. At the Trans-solution level students are able to solve, interpret and describe graphically different systems of differential equations. Coherence of the schema is demonstrated by the student's ability to describe which systems can be solved using analytic methods and which graphical representations are possible depending on the system.

Methodology

Data were collected from two Differential Equations classes at a small private university. One of the courses was taught to 34 Applied Mathematics students and the other to 37 Economics students. Both groups were taught by the author. During the semester, three individual task centered semi-structured interviews were conducted with nine students of each group. Interviews covered only topics covered in class. The interviewer was another researcher not involved in this particular project. The interviews were audio-taped and transcribed, and all the work done by the students was collected. The analysis of the data was discussed with another mathematics educator who also taught courses on Differential Equations.

For this particular project we selected the following particular tasks to be analyzed: 1) Given a mathematical model that describes the growth of two populations by means of a predator-prey model, and given a a plot that shows two solution curves in a phase plane for different initial conditions, draw the plots that show how each of the populations grow in time and interpret the solution; 2) Given a different mathematical model given by two linear and autonomous equations, solve the system of differential equations, draw the phase plane representation and interpret the solutions found in terms of their representation in the phase plane; 3) Given the x vs. t and y vs. t plots of the behavior of the solution to a model represented by a system of differential equations with different initial conditions, represent the solution curves in a phase plane and interpret the meaning of different points on those curves in terms of the given model.

The focus of the analysis was students' conceptions of solution curves and equilibrium solutions on the one hand and their strategies on the other.

Some Results

The analysis of the interviews showed that the interaction between the schema mentioned earlier can help to understand most of the difficulties of the students. It was found that students have problems interpreting the meaning of equilibrium solution, some of them thought of it as a point in phase space but forgot about variation with time. Some students in both groups could not see the dependence of time in the phase space representation and even considered each differential equation to be a separate equation and the phase space representation as a means to compare the solution of both. Other students were able to recognize the system as such and even to solve it, but when faced with the interpretation of the solution curves in phase space and asked to draw the plots of the dependent variables versus the independent variable, showed confusion because the independent variable did not appear explicitly in the phase plane equations (intra-parametric, intra-solution level). Many students could solve the linear system and interpret correctly the curves in a phase plane, but had trouble justifying their explanations and explaining why they chose a particular method or strategy (inter-parametric, inter- solution level). Most of these students had trouble when the system of equations was not autonomous and/or when it was non linear, but their difficulties could be due to a lack of understanding of the geometric representation of parametric equations (intra-parametric, inter-solution level) or a difficulty in discerning when a particular representation was possible for the problem (trans-parametric, inter-solution level). The results regarding the meaning of equilibrium solutions were consistent with those of Ramussen (1998) and Zandieh and McDonald (1999) who worked with the meaning of a solution to a differential equation.

The meaning of a point in phase space proved to be a problem to most of the students in both groups. They were not able to see it as a representation of the state of an autonomous system at a particular time even when they were able to solve the system. However their difficulties could also be differentiated as related to their conceptions of parametric functions or to their conceptions of a solution to the system of equations. The construction of phase space, the interpretation of solution curves and the relationship between the graph of the solution in a phase plane and in a configuration plane proved to be a very difficult task for many students, even when they could handle tangent fields for ordinary differential equations.

Students showed some specific tendencies to use particular strategies to make sense of the proposed tasks. For example, many students would start by reading values from a curve represented in a phase plane to make a table of values, but most of them were not then able to draw the other plots or to interpret the solution because they could not see the dependence with the independent variable on the phase plane plot. Other students would first try to solve the equations and then use the solution to construct the graphical representation and to interpret the solutions; within this group of students some tried to use this strategy even if the system was not linear. A very frequent strategy for the task where the solutions in a phase plane were given was to start from the equations and reconstruct the given phase plane from interpretation of the signs of the derivatives in the system and the use of nullclines; but even when they were able to do it, the students showed difficulties in interpreting the solutions and using a different graphical representation for the task. In the construction of a phase plane, a very common difficulty was to relate a zero derivative for one of the variables with a vertical arrow on the phase plane. Two students used the idea of a car traveling along the curve in the phase plane or walking on the curve and centered the attention to the change of the variable shown in the horizontal axis as the moving object goes along the curve; this strategy helped them make sense of the independent variable for one of the dependent variables, but they were unable to use the same procedure for the variable shown in the vertical axis of the phase plane.

Conclusions

As a result of this research project it was found that the integration of different concepts and techniques is difficult for many students. The generalization of concepts and strategies used to deal with ordinary differential equations to systems is not direct and may need to be made explicit in class.

We were able to find different strategies that students use to make sense of solution curves. These strategies can be helpful to design new ways to work in class with students to help them develop a richer schema for the process of solution and interpretation of solutions of systems of differential equations.

References

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