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Having students write problems shows the instructor where the gaps in their understanding are at the same time that it has students review for an upcoming test. Further, it can help students find the relevance of the course to their own interests.
Background and Purpose
Saint Joseph's University is a small, comprehensive institution with majors in the humanities, business, social sciences and physical and life sciences. Every student is required to complete two semesters of mathematics. The courses vary from one division to another.
In any course, we are challenged to construct tests that measure what students have learned. Introspectively, we often acquire a deeper understanding of a concept by teaching it, by explaining it to another, or by trying to construct problems for a test. The inadequacy of in-class tests for assessing the learning and understanding of my students led me to explore new methods of appraising their knowledge and reasoning. I wanted to ascertain student perception of which concepts are important and how each individual thinks about problems. Using student-created problems (and solutions) as an assessment procedure reveals both the thinking processes and interests of the students.
Student-created problems allow the instructor to determine the level of each student's comprehension of topics and her/his ability to apply the concepts to new problems. In the process, students hone communication skills, since problems and solutions are submitted in writing and often explained orally in class as well. With several days to create a problem, speed is not an issue; each student works at her/his own rate. This process fosters a commitment to personal achievement, and encourages the student to become more self-directed. Although some students submit only routine problems, each knows the grading scale in advance and so shares control in the teaching-learning environment. The difficulty level of the problems is entirely in the hands of the students, and hence they set their own personal goals for achievement.
The method of using student-created problems can be implemented in any course. I have used the technique successfully in freshmen level courses (with a maximum enrollment of 30) and in upper division courses with smaller enrollments. In this article, we demonstrate a use of the technique in a course for social science majors.
Method
In the courses for social science majors, the textbook For All Practical Purposes [1] is used. Broadly stated, the concepts include Euler and Hamiltonian circuits, scheduling, linear programming, voting strategies, population growth, bin packing, optimization, and statistics. The topics are current and relevant to today's world. During the course, each student is required to design a word problem and its solution for each of the major concepts. The instructor collects the problems, organizes them, and checks for accuracy. The corrected problems are used as a study guide and review aid for tests and the final exam.
As one would expect, the student-created problems run the gamut from mundane to exciting. Interesting problems are presented in class. A set of correct problems (without solutions) is given to the class as a review. Questions can be asked in a later class. Each student then becomes an authority on her/his problem and its solution, and can be called upon to explain or solve the problem in class. In practice, members of the class actively engage in questioning each other about the solution to a particular problem. Lively discussions take place. The grade on a problem becomes one part of the evaluation of a student's knowledge in the course. The instructor acts as an advisor in the development of problems and solutions of high quality.
In my classes, the collection of problems is given the weight of one test, which is 25% of the final grade. (There are two in-class tests, a collection of problems, and a final examination. I include student-constructed problems on the tests.) Problems are graded on the basis of correctness, relevance to the topic, and originality. If a student elects to correct a problem, the numerical value it receives is the same as if it had been submitted correctly initially.
Findings
Analysis of a problem allows me to determine if a student understands a concept. Possible errors include misusing the definition, confusion among the concepts, applying a concept to a problem for which the method is inappropriate, or in extreme cases, inability to apply the concept to a new situation. The interests of the students appear in the problems constructed and I use this information in designing examples for other topics in the course. Since my tests consist of routine exercises and problems drawn from the students' work, students' complaints about the test questions, when the test is returned, are virtually eliminated.
Since resubmission is encouraged for poorly posed problems or incorrect solutions, students tend to be more careful in designing the initial problem. Complimenting students on innovative problems, even if they are not completely accurate on the initial submission, improves students' self-esteem and self-confidence. They take more responsibility for the learning process, and develop pride in their work. Because students are actively engaged in the creative process, listening, writing, seeing and thinking are all involved. Furthermore, students learn to read the textbook carefully and to reread examples in order to develop problems.
Students are actively engaged in a cooperative activity and view me as a resource in developing problems rather than as the controller of their academic destiny. Each person has an opportunity to show the depth of her/his understanding without a time constraint. Since students may discuss ideas with each other and with the instructor before submitting a problem, more student-faculty contact, greater cooperation among students, and proactive learning result. As students internalize a concept, they can apply the concept to situations of personal interest. The instructor shows respect for the diverse interests, talents, and modes of learning among the students.
The outcomes of this process are well worth the effort. It is gratifying to observe the students as they find creative ways to use the concepts. There is a great variety in the problems, some of which are quite ingenious and truly enjoyable to read. An unanticipated benefit of the technique is that the instructor obtains a broader range of applications to discuss in the class. Below are a few examples of students' work.
A woman is campaigning for mayor. She needs to go door to door to gather votes. If she parks her car at 61st Street and Lancaster Avenue, design a path for her to use that minimizes deadheading. (Included with the problem is a map of a 3 block by 4 block section of the city.)
A chocolate chip cookie company ships cookies to a convenience store. The bags of cookies come in four different sizes: 1/2 lb., 1 lb., 2 lb., and 4 lb. The company packs these bags into cartons. Each carton holds at most ten pounds. The quantities of cookies that must be shipped are: ten 1/2-lb. bags, fifteen 1-lb. bags, ten 2-lb. bags, and two 4-lb. bags. Using the best-fit decreasing heuristic, place the 37 bags into as few cartons as possible.
In making a decision about which college to attend, the following tasks must be completed: (a) make a list of schools to consider (2 days), (b) send for information and applications (14 days), (c) research the schools (60 days), (d) apply to schools (30 days), (e) visit the schools (5 days), (f) receive acceptances/rejections (90 days), (g) make final decision about the school to attend (30 days). Construct an order requirement digraph and find the critical path.
Course evaluation forms from students indicate that the assignments are worthwhile, increase understanding, and (sometimes) are fun to do. A student's time commitment in designing and solving a problem exceeds that which a student usually spends solving exercises in the textbook. Course evaluations indicate that students spend more time and effort in this course than they usually devote to studying mathematics taught in a traditional style.
Use of Findings
This alternative to in-class testing is enlightening to me as well as beneficial to the student. I am more open to student ideas, and use a formative approach to the development of concepts as a result of this practice. Sharing control in the teaching-learning environment means that the topics discussed can be modified to reflect the interests of the students. When submitted problems have common errors or exhibit similar misconceptions about the application of a topic, I know I have not made the topic clear and delineated its parameters adequately. I need to find another approach or to give more complete examples. Usually, I respond by more carefully explaining a new example and examine how this problem fits the assumptions of the technique. In the following year, I know to be more thorough when explaining this topic. I also can tailor my examples to the interests of the students.
Success Factors
Using this strategy to determine a student's knowledge involves a considerable investment of time and effort on the part of the instructor, particularly in reading and commenting on the work submitted by the students. I estimate that it probably takes three times as long to read and return student-created problems as test problems. Grading the problems and checking the accuracy of the solutions can be a slow process. Solving each problem and checking for accuracy takes longer than checking routine problems. On an ordinary test, all students are solving the same problems. While grading a test, one becomes familiar with the common errors and easily sees them. With student-created problems, each one is different, so an instructor solves 20-30 problems rather than 8-12 problems. On the other hand, time is saved by not having to construct good test questions. Furthermore, class time is saved by eliminating at least one test, and perhaps another class used to return and review the test.
Depending on the availability of readers and/or typists, you may be performing the task of organizing and collating the problems alone. Compiling and editing can be completed more simply when the problems are composed on a computer and submitted electronically than when they are submitted typed on paper.
For other assistance in getting started and further references see Rash [2].
References
[1] COMAP. For All Practical Purposes, W. H. Freeman and Company, New York, 1991.
[2] Rash, A.M. "An Alternate Method of Assessment,
Using Student-Created Problems."
Primus, March, 1997, pp. 89-96.
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