Student misconceptions in using Euler’s method in ordinary differential equations and the importance of step size
Fox, William P
ABSTRACT: The purpose of this paper is to share a series of labs designed to improve student misconceptions in using Euler’s Numerical method to find solutions to ordinary differential equations. After close to twenty years of teaching differential equations and modeling with differential equations, it became apparent that something needed to be done to dispel these student misconceptions that were developed in calculus. These misconceptions have not been isolated phenomena at my current university; they also existed with students at my previous institution.
KEYWORDS: Differential equations, differential equations labs, Euler’s method, autonomous differential equations, numerical methods.
We all teach Euler’s method in an introductory differential equations course and we might even introduce this method during the differential equations section as part of calculus. In teaching ordinary differential equations, we try to educate our students using the rule of four. Students learn that mathematics is analytical, graphical, numerical, and has meaning relative to the context of the problem. We have students take a given scenario based on real world information and analyze the solution graphically, numerically, and analytically in terms of the scenario. Students require access to a computer for the graphical and numerical solutions. There is no requirement for any specialized ODE solvers (even though they are nice) -as we can do our analysis with the aid of a spreadsheet or a computer algebra system. Rather than use the black box approach within an ODE solver, we would rather the students obtain their solutions by programming Euler’s numerical method to approximately solve an ODE through writing macros (as in MathCad or Maple) or through a spreadsheet. Students learn that step size in Euler’s method influences the solution-sometimes significantly and sometimes counter intuitively. It is important for the users of numerical methods, whether students, instructors, or practitioners, to be aware of possible numerical errors and either have intuition about the nature of the solution or be able to produce a qualitative graph to compare to their numerical results.
We can use these estimates for the values of y(t) from 0 to t in order to get an approximate graph of the solution to the ODE. In the case of autonomous ODEs, we can also obtain a qualitative graph that allows us to directly compare the qualitative information to our numerically generated graphical solution.
This is important because numerical approximations can often produce error-ridden results. The user needs to be savvy enough to recognize, if possible, when the results of a numerical technique are erroneous. Our labs should provide some insights to students concerning Euler’s method.
The following is a short list of some prominent student misconceptions about numerical solutions that I have received from students during ODE classes:
* Decrease the step size because more steps are better.
* One numerical attempt, with any step-size, is sufficient and “models” the true result.
* The meaning of a small step size is relative to a student and their background (h=1 is considered small).
Many students believe that with any numerical method, the smaller step sizes always improve the solution. Many of these students recall the paradigm “more is better” from Riemann sums, Simpson’s rule, or the trapezoidal rule from calculus and are not ready to part with those concepts. This notion of “more is better” is prominent throughout calculus.
Students like to calculate using Euler’s Method only one time and believe that this result is the answer. If they use the computer or calculator to assist them then their confidence increases that they have found “the” solution. The difference between “analytical” and “approximate” is not well understood.
The meaning of step-size is a relative term. Most students prefer a step– size of 1 as it often simplifies calculations. Many then use the value of 1 as a relative benchmark. Smaller than 1 means we have decreased the step-size and greater than 1 we have increased the step-size.
We designed a set of labs using Euler’s method to dispel these misconceptions and force the students to think critically about the method, the form of the differential equation, and the numerical result.
Simply stated our goals for students completing these labs are:
* Improved skills in solving numerical related problems
* Improved understanding of Euler’s method
* Improved understanding of step-size and solutions
* Improved appreciation for the use of qualitative methods, where ap plicable.
* Improved student use of technology
The information will be presented in three parts for each lab. First, the lab is stated as it would be stated for the student. Next, the learning objectives for the lab are stated for the instructor’s reference. Then a sample solution with discussion is provided.
LAB 1: SIMPLE VAT MIXING APPLICATION
These experiences completing these labs have helped our students overcome their previous misconceptions in using Euler’s Method to approximate solutions to ordinary differential equations. This statement can be made in a comparative context. Prior to using these labs student responses to questions using numerical methods showed the misconceptions that were listed early in this paper. Since introducing these labs, students have done a much better job in trying to understand the numerical methods. In particular, the course projects have improved substantially
Our course in differential equations has three distinct blocks covering first order differential equations, second order differential equations, and systems of differential equations. Each block culminates in a project that stresses each of three aspects: qualitative solutions, analytical solutions, and numerical solutions. Students are required to compare their results and discuss the strengths and weaknesses in their different solution techniques. Project 1 is similar to Lab 2 in that it is a first order, non-linear ordinary differential equation modeling the spread of a rumor or a population growth model. Project 2 is currently either a chemical reaction or a Bungee Cord Jumping Project for the second order ordinary differential equations. Project 3 is a diffusion model for systems of differential equations. Students’ submissions to these projects, as compared to students before the labs, have clearly shown a distinct and marked improvement. Students routinely use multiple step sizes and compare their results to their qualitative solutions, analytical solutions, or both.
As a result of analyzing student projects and exam questions, it is clear that these current students (who have completed the labs) have a better understanding of the use of numerical methods. Furthermore, they are more experimental and willing to “test” ideas in numerical methods.
It is clear that these students compare and contrast step sizes while doing their projects. They no longer accept that a decrease in the step size is “good” because more steps are always better. They experiment and compare before stating their mathematical conclusions. Most students no longer perform only one numerical attempt, with any step-size, and stop. They try various step sizes and compare results without it being a direct requirement in the project. The last misconception, the meaning of a small step size, is relative to a student and their background (h = 1 is considered small), has been the hardest misconception and might still be somewhat present. Students try various step sizes but usually they are among h = 0.1, 0.5, and h = 1. They have not gone to smaller step sizes or larger without some additional suggestions. These values seem to be accepted as the student default step values to try even with technology readily available for any step size. Additionally, many of the students are also taking computational physics where step size is not currently emphasized.
Overall performance has indicated an improvement in student understanding Euler’s method and its basic concepts. This same performance indicates an improved awareness of the importance of step size that was not present with students before the introduction of these in-class labs.
1. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. 1997. Differential Equations. Pacific Grove CA: Brooks/Cole Publishers.
2. Fox, William P., Frank R. Giordano and Maury Weir. 2002. A First Course in Mathematical Modeling, 3rd Edition. Pacific Grove CA: Brooks Cole Publisher.
3. Giordano, Frank and Maury Weir. 1994. Differential Equations: A Modeling Approach. Reading MA: Addison-Wesley.
4. Zill, Dennis G. 1997. A First Course in Differential Equations, 6th Edition. Pacific Grove CA: Brooks Cole Publishers.
5. Cannon, Ray. 1999. “AP Calculus Professional Night Lecture.” Colorado State University. June.
William P. Fox
ADDRESS: Department of Mathematics, Francis Marion University, Florence SC 29501 USA. wfox@fmarion. edu.
Dr. William P. Fox is Professor and Chair of Mathematics at Francis Marion University in Florence, South Carolina. He is the Chair of the Calculus Strand for ICTCM, director of the High School Mathematical Contest in Modeling (HiMCM) and Associate Director of the Collegiate MCM. His interests are in mathematics modeling, optimization, and differential equations.
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