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Multiple Solutions: Defined and Explored

Multiple Solutions: Defined and Explored

Generating more than one option to meet a given set of criteria.

Satisficing vs. optimizing

"Satisficing" describes the act of settling for (being satisfied with) something that will suffice, but may not be the best possible solution. "In the simplest context, one has a set of alternatives clearly ranked in terms of their utility as means to meet one's ends. If one is an optimizer, one chooses an alternative that ranks at least as high as any other. In contrast, if one is a satisficer, one settles for any alternative one considers satisfactory " (Schmidz, 2004, p.31). The two forms of problem solving use different rules for deciding when to stop generating solutions, and research on various forms of bounded rationality shows that people tend to be satisficers (Simon, 1982). As with any form of bias, this habit is worth addressing in our classrooms because we do not just want our students to be problem-solvers, we want them to be good problem solvers.

The anatomy and sequence of problem solving

Halpern's (2003) summary of problem solving literature begins with Newell and Simon's (1972) notion that all problems boil down to the same basic anatomy: an initial state, a goal state, a problem space between the two states, and solution pathways across the problem space.

Anatomy of a Problem

Solving a problem then consists of generating at least one solution path, and ideally several so that one can choose the best among alternatives. As described by Treffinger (1985), this process unfolds across three phases, each of which have strategies one can use to get the most out of them:

(1) Understanding the problem involves identifying the features of the initial and goal states, to get clarity on the problem space. In addition to emphasizing the importance of deeply understanding a problem before rushing to a solution, three strategies can help one incrementally understand the problem:

(2) Idea finding involves generating many options, of great variety, including novel or unusual options, and may include very refined options. These tasks tap fluent thinking, flexible thinking, original thinking, and elaborative thinking, respectively.  Brainstorming and hierarchical solution finding are two specific methods one can use structure this experience.  (Importantly, Dillon, Graham, and Aidells (1972) found that practicing brainstorming in groups followed by individual brainstorming was an effective method for generating multiple solutions.)

(3) Planning for action involves assessing the solution paths generated in order to select among them.

These strategies are helpful, but it seems that not enough students use them in formal way. Specifically, Garst et al. (2002) found that when generating hypotheses in a simulated scientific experiment, satisficing was the predominant mode of solution generation.

Helping students learn to generate multiple solutions

Many teachers have described their first year students arriving with some form of a dualistic perspective on knowledge (as described by Perry, 1998), in which every problem is understood to have a single correct answer. It is therefore important for instructors to put effort into helping their students develop the perspective that "there is more than one 'correct' way of solving most problems, and acknowledge creative alternative solutions" (Greenfield, p. 19, 1987). Rubenstein and Firstenberg (1987) suggest students must be able to tolerate uncertainty and work within ambiguity in order to successfully internalize a more flexible approach to thinking and willingness to adapt one’s approach that are required for finding multiple solutions to problems.

Finally, Ray (1966) found that simply giving students practice with multiple solution problems increased their ability to generate more solution pathways across a problem space. In addition to motivating students to acquire this practice by pursuing their own problems , many teachers use in-class problem-based learning activities.


Butler, D.L., & Kline, M.A. (1998). Good versus creative solutions: A comparison of brainstorming, hierarchical, and perspective-changing heuristics. Creativity Research Journal, 11(4), 325-331.

Dillon, P., Graham, W., & Aidells, A. (1972, December). Brainstorming on a 'hot' problem: Effects of training and practice on individual and group performance. Journal of Applied Psychology, 56(6), 487-490.

Garst, J., Kerr, N., Harris, S., & Sheppard, L. (2002). Satisficing in hypothesis generation. American Journal of Psychology, 115(4), 475-500.

Greenfield, L.B. (1987). Teaching thinking through problem solving. In J.E. Stice (Ed.), Developing critical thinking and problem-solving abilities (pp. 5-22). San Francisco: Jossey-Bass.

Halpern, D.F. (2003). Thought and knowledge: An introduction to critical thinking (4th Edition). Mahwah, NJ: Lawrence Earlbaum.

Newell, A., & Simon, H.A. (1972) Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.

Perry, W. G. (1998). Forms of intellectual and ethical development in the college years: A scheme. Jossey-Bass.

Ray, W. (1966). Originality in problem solving as affected by single-versus multiple-solution training problems. Journal of Psychology: Interdisciplinary and Applied, 64(1), 107-112.

Runibinstein, M.F., & Firstenberg, I.R. (1987). Tools for Thinking. In J.E. Stice (Ed.). Developing critical thinking and problem-solving abilities (pp. 23-36). San Francisco: Jossey-Bass.

Schmidz, D. (2004). Satisficing as a humanly rational strategy. In Byron, M. (Ed.). Satisficing and optimizing: Moral theorists on practical reason. New York, NY: Cabridge University Press.

Simon, H.A. (1982). Models of bounded rationality. Cambridge, MA: MIT Press.

Treffinger, D.J. (1995). Creative problem solving: Overview and educational implications. Educational Psychology Review, 7(3), 301-312.