## Literature Review: Student Understanding in Differential Equations

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Mel S. Sabella
E. F. Redish

Student Understanding of Topics in Differential Equations

Although differential equations is a critical course for the study of many scientific and engineering subjects there has been very little research on student understanding of the subject. The literature that presently exists mainly includes suggestions for a revised course in differential equations. There is some discussion of student understanding of the topic but it is mostly anecdotal. We have found three papers dealing with the teaching of differential equations using computers.

Both William Boyce and Paul Blanchard have employed the use of computers in a revised differential equations course. Boyce, who teaches at RPI, and Blanchard, who teaches at Boston University, have both adopted the revised course to give students more time to be engaged in what Boyce calls more valuable experiences such as conceptualization, exploration, and higher-level problem solving. A web page also exists dedicated to the Boston University Differential Equations Project located at http://math.bu.edu/odes/.

Students enrolled in the traditional differential equations class learn how to become proficient in symbolic calculations and show little understanding of the basic concepts involved, according to both Boyce and Blanchard. The course, in general, exclusively deals with the derivation of formulas for the solutions to different types of differential equations and the students become proficient at certain symbolic calculations but there is little understanding of basic concepts.

The introduction of the computer in such a course frees the student in many ways. A good deal of the traditional material, such as rote symbolic manipulation, could be dispensed with in favor of more valuable experiences. These valuable experiences are the processes of conceptualization, exploration, and higher-level problem solving (Boyce 364).

Conceptual skills can now be given more attention since the rote symbolic manipulations can be done on the computer. This allows students to devote time to develop skills in formulating problems and drawing conclusions from their solutions. Boyce states that the procedures and algorithms that are stressed in a more traditional course are quickly forgotten while the underlying concepts and ideas become part of the individuals mindset. Boyce suggests that mastering the underlying concepts is therefore beneficial to the student since the underlying concepts will be remembered. Blanchard, at Boston University, stresses the qualitative theory and requires the students to explain their solutions in everyday language to further strengthen a conceptual understanding.

Exploration is also easier when computers are used in the course. Students will have a much easier time, for instance, seeing how different parameters affect the solution. One example of this is how amplitude and period are related in the non-linear pendulum. This can be done without a complicated formula or even knowing that the solution exists. Numerical and graphical tools can be employed to carry out investigations and understand what the solution means and what type of information is present in the solution. The computer can also be used as a reference source for unfamiliar functions and their behavior. In this way the computer is used primarily as a tool for illustration and allows the students to actively participate in their learning.

Besides developing a better conceptual base, and allowing the students more avenues to explore, the computer allows students do obtain higher level problem solving abilities. Such abilities include modeling physical systems which both Boyce and Blanchard stress in their courses. Boyce states that students find modeling "to be notoriously difficult, partly because often it is indeed challenging, and partly because they have had relatively few opportunities to practice modeling--their prior courses have focused on algebraic skills (Boyce 367)." Due to the difficulty of modeling Blanchard uses discrete systems and difference equations which he feels are easier for the students to understand. This modeling of discrete systems is then used to introduce the modeling of continuous systems.

In the traditional course the differential equations that are solved are those for which a closed form of solution exists and we therefore only deal with a subset of differential equations. Blanchard notes that nonlinear systems are generally ignored in the traditional course. By using the computer more varied problems can be included in the differential equations course as well as problems which require students to obtain estimated solutions, when an exact solution does not exist.

Students also have the opportunity to consider alternative methods in presenting their results. Presentations of solutions are no longer just limited to formulas; it can now be done using formulas, plots, numbers, or as tables of numbers (Boyce 367). Students will have to decide on the best way to present their results. With the computer graphical analysis is easier so the student does not have to rely only on an analytical solution when interpreting results.

Using the computer, although making certain aspects of the course easier, makes the overall course more challenging and more complex for the student (Boyce 370). It also makes the course much more open ended by allowing approximate solutions, and many different ways of presenting solutions.

At present there no evidence that students improve their conceptual understanding when using computers in the differential equations course, although improvement was shown in calculus courses which use computers. Studies in the calculus courses also show that students do not lose their algorithmic skills when computers are used. Similar studies must be done in the differential equations course.

References

Barber, Fredrick, (1994). Technology, Cooperative learning, and assessment in the Teaching of Ordinary Differential Equations, Primus, 4 (4) 337-346.

Blanchard, Paul, (1994). Teaching Differential Equations with a Dynamical Systems Viewpoint, The College Mathematics Journal, 25 (5) 385-393.

Boyce, William, E., (1994). New Directions in Elementary Differential Equations, The College Mathematics Journal, 25 (5) 364-71.

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Comments and questions may be directed to E. F. Redish