ISSUES IN EDUCATION

The value of open-ended questions, Part 2: Implementation

By Will Millhiser

In April’s column I proposed more open-ended problems in our classroom teaching, homework and quizzes/exams. To summarize, the mathematics education literature suggests that open-ended questions promote deeper and more conceptual learning, student confidence, communication skills and better assessment of what students know. This month’s column is dedicated to turning that theory into practice.

Education scholars identified dozens of ways that problems can be open-ended. The methods outlined below are due to Doug Clarke, Charles Lovitt [1], David Clarke [2], Thomas Cooney [3], Peter Sullivan, Pat Lilburn [4], John Van de Walle, LouAnn Lovin [5] and others.

To illustrate how one makes a problem more open-ended, consider a simple linear programming (LP) question [6] (Figure 1).

Figure 1: A simple linear programming question.

Figure 1: A simple linear programming question.

Alternatives to probe deeper into the student’s understanding and to promote learning are illustrated in Figure 2. The first category is based on a popular TV game show.
These examples are about improving one’s understanding of mathematical techniques. We might also imagine examples about interpreting results, performing sensitivity analysis and applying techniques in real settings.

Figure 2: Problems that probe deeper into student’s understanding and promote learning.

Figure 2: Problems that probe deeper into student’s understanding and promote learning.

Instructor Resources

Mathematics teachers have volumes of open-ended problems (e.g., see [4], [7]). Are similar resources available in OR/MS and Operations Management (OM)?

Case Studies. One might argue that the case study is to us what an open-ended problem is to a mathematician: concrete with room for multiple approaches. OR/MS professors who teach with cases say that compared to traditional approaches, students find cases more exciting, the subject matter more relevant, quantitative and analytical subjects more interesting and the tools easier to apply in unfamiliar situations.

Where can one find cases? In addition to the usual publishers, there is the INFORM-ED case competition (see informs.org/Community/INFORM-ED/Teaching-Materials) and the journal INFORMS Transactions on Education (ITE). According Jim Cochran, ITE “is rapidly building a collection of [refereed] cases that are freely available to instructors and students” (see informs.org/Pubs/ITE). Some may find OR/MS Today’s “PuzzlOR” column useful (see informs.org/ORMS-Today/Past-Issues starting Feb. 2008), though these tend not to be open-ended. Finally, my OM colleagues may know that searching one’s favorite bookseller for “cases in operations management” reveals several collections.

Alleviating the Grading Burden. A concern is that introducing open-ended problems increases one’s grading load. Since no two responses are identical, “Open-ended questions require teachers or evaluators to interpret and use multiple criteria in evaluating responses” [4]. To reduce grading subjectivity, instructors create and communicate grading rubrics. (For a brief survey on rubrics, see [8].)

Some universities are finding solutions to this time cost. Professors can find grants to improve the quality of instruction. This author was part of a Baruch College task force that recommended the provost give academic departments funds to cover a mix of release time and homework graders to make it possible for faculty to give richer assignments that encourage deeper learning.

Your Homework Assignment

Let me leave you with questions. Should the use of open-ended questions change the way OR/MS and OM is taught and our textbooks written? Also, does adoption of Web-based homework systems occur at the expense of open-ended problem solving? What’s the right amount of open-ended problem solving in exams, homework and classroom teaching, anyway? Must open-ended questions have unstructured answers?

Will Millhiser (william.millhiser@baruch.cuny.edu) is an assistant professor in the Operations Management Group at Baruch College, City University of New York. He has taught high school mathematics and studied mathematics teaching methods.

Acknowledgments

The author is grateful to Professor Emeritus of Mathematics Richard Evans for his mentorship. The four categories of open-ended problems and references are revised from his lecture notes, with kind permission. Thanks also to Fredrik Ødegaard, Peter Bell, Stephen Powell, James Cochran and James Grayshaw for useful discussions. Special thanks go to Armann Ingolfsson and others at INFORMS Transactions on Education for comments that influenced several aspects including the “homework assignment.”

References

  1. Clarke, D., C. Lovitt, 1987, “Assessment Alternatives in Mathematics,” Australian Math. Teacher, Vol. 43, No. 3, pp. 11-12.
  2. Clarke, D.J., 1992, “Activating Assessment Alternatives in Mathematics,” Arithmetic Teacher, Vol. 39, No. 6, pp. 24-29.
  3. Cooney, T.J., 1996, “Thinking About Being a Mathematics Teacher,” in T.J. Cooney, S.I. Brown, J.A. Dossey, G. Schrage, E.C. Wittmann (eds.), “Mathematics, Pedagogy, and Secondary Teacher Education,” Heinemann.
  4. Sullivan, P., P. Lilburn, 2002, “Good Questions for Math Teaching,” Math Solution Publications.
  5. Van de Walle, J.A., L.A.H. Lovin, 2005, “Teaching Student-Centered Mathematics: Grades 5-8,” Allyn & Bacon.
  6. Example taken from Mathur, K, D. Solow, 1994, “Management Science: The Art of Decision Making,” Prentice Hall.
  7. Cooney, T.J., W.B. Sanchez, K. Leatham, D.S. Mewborn, 2004, “Open-Ended Assessment in Math: A Searchable Collection of 450+ Questions,” Heinemann (books.heinemann.com/math).
  8. Robertson, D., C. Lehane, 2009 (eds.), “Do-It-Yourself Rubrics,” The Advocate, Vol. 26, No. 6, pp. 5-8 (nea.org/home/33605.htm).