Many students believe that solving a physics problem requires them to know the right equation(s) for that particular problem. So part of their plug-and-chug and pattern-matching techniques is the memorization of all the equations in each chapter of their textbook. These equations are of equal importance to students.
They do not distinguish the mathematical formulations of fundamental principles from equations that are the consequence of those fundamental principles to specific circumstances or for specific types of interactions. Worse yet, the equations memorized for one chapter are promptly forgotten while memorizing the formulas for the next chapter.
So how should the Third Law of Education be applied here?
Make it easier for students to do what you want them to do and more difficult to do what you don’t want.
What can you do to promote the development of a coherent, connected network of knowledge organized around fundamental concepts, while discouraging the memorization of disconnected formulas?
At first glance the answer may seem simple: allow students to write equations on a single file card or sheet of paper and bring their equations to your exams and examinations. While this solution discourages memorization, it does not address the difficulty students have in developing a coherent knowledge base. Those who have tried this technique know too well that students will write as small as possible to cram everything onto a single piece of paper. They then complain
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Part 1:Teaching Physics Through Problem SolvingFigure 3.4. Example information sheet for exams: algebra-based course
This is a closed book, closed notes quiz. Calculators are permitted. The only formulas that may be used are those given below. Credit is given only for logical and complete solutions that are clearly communicated. In the context of a complete solution, partial credit will be given for a well-communicated solution based on correct physics.
5 points: A useful picture, defining the question, and giving your approach.
7 points: A complete physics diagram defining the relevant quantities,
identifying the target quantity, and specifying the relevant equations.
6 points: Planning the solution by constructing specific equations and checking for
sufficiency
5 points: Executing the plan to get algebraic and numerical answer 2 points: Evaluating the validity of the answer.
Useful Mathematical Relationships:
Chapter 3: Combating Problem Solving That Avoids Physics
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that they would have done better on the exam if they had just had the foresight to write down one more equation.
We have found it useful to supply students with only the information they need to solve exam problems from the basic ideas emphasized by the course. An example of such an information sheet used in an algebra-based course is shown in Figure 3.4. There are three features of the information sheet that are
designed to promote student development of a coherent, integrated network of knowledge and reinforce logical analysis of a problem using fundamental concepts.
Supply a limited number of equations that students may use.
Supply only the equations that state the fundamental physics principles and concepts that are stressed in the course. Students are not allowed to use any other equations to solve a problem. A distinction is made between physics that underlies everything and the important physics that depends on a specific but reasonably general situation. The symbols are not defined to encourage students to know the meaning of the equations.
The choice of equations depends on the course and the emphasis of the instructor. For example, the kinematics equations in Figure 3.4 would be given to the students in the algebra based physics course. In the calculus-based course, we replace the equation for the special case of constant acceleration
2 v vx vxi xf
that builds on students intuitive understanding of an average with o
an equation directly connected by calculus to the solution of the equation defining acceleration. [See Chapter 11, page 141 for an example of an information sheet for a calculus-based course.
The information sheet grows with time.
Nothing is ever taken off the sheet, but new equations and needed constants are added. The “old” equations are often used in problems for new topics to emphasize that the underlying physics does not depend on the context. For example, the information that is not shaded was supplied for the second exam in the course, the information in light-gray was added by the end of 8 weeks, and the information in darker gray by the end of about 12 weeks in the course.
Allow students to use calculators.
Students love their calculators and feel
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Part 1:Teaching Physics Through Problem Solvingpersecuted if they are not allowed to use them on exams. Unfortunately modern calculators can hold an enormous amount of information. Since all students have an information sheet with all the equations that they are allowed to use, there is no advantage in using the calculator memory to store equations for exams. The same holds true for any temptation to bring written crib sheets to exams. If students are also required to communicate their solution in a logical and complete manner, the ability and temptation to cheat is almost completely eliminated. For these reasons, we see no benefit of restricting the use of calculators in CPS.
Endnotes
1 The definition of a problem and problem solving has not changed
substantially in the last forty years. For example, see Newell, A., & Simon, H.A. (1972), Human problem solving. Englewood Cliffs, NJ, Prentice-Hall, Inc;
Hayes, J.R. (1989), The complete problem solver (2nd ed.), Hillsdale NJ; Lawrence;
and Martinez, M. (1998), What is problem solving? Phi Delta Kappan, 79, 605-609.
2 A number of researchers have concluded that the typical problems assigned in physics courses are actually counterproductive to learning physics. Other types of “nonspecific goal problems” and “ill-structured problems” have been found to move students towards more expert-like problem solving. See, for example, Sweller, J., Mawer, R. & Ward, M. (1982), Consequences of history: Cued and means-end strategies in problem solving, American Journal of Psychology, 95(3), 455-483; and Shekoyan, V., and Etkina, E. (2008),
Introducing ill-structured problems in introductory physics recitations, Proceedings of the 2007 Physics Education Research Conference, PERC Publishing, Rochester, NY, 951, 192-195. In our own research, we also found that standard problems seemed to promote the use of the novice, plug-and-chug or pattern-matching strategies rather than the use of a more logical and organized strategy. See Heller, P. & Hollabaugh, M. (1992), Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups, American Journal of Physics, 60(7), 637-644.
3 For reviews of student difficulties solving problems, see Maloney, D.P. (1994), Research on problem solving in physics, in D.L. Gabel (Ed.), Handbook of Research in Science Teaching and Learning, (pp. 327-354), NY, Macmillan; and Woods, D.
(1989), Problem solving in practice, in D.L. Gabel (ed.), What Research Says to the Science Teacher Vol. 5, National Science Teachers Association. See also, Hsu, L., Brewe, E., Foster, T. M., & Harper, K. A. (2004), Resource letter RPS-1:
Research in problem solving. American Journal of Physics, 72(9), 1147-1156.