In a Journal of Engineering Education article (referenced below), Richard Felder and Rebecca Brent propose an instructional model that promotes the intellectual development of science and engineering students.
Among a number of conditions they identify as being relevant to intellectual development, they suggest particular kinds of problems for students to solve. Their list (summarized below) offers ideas relevant in any course where students solve problems.
Predicting outcomes—“Describe physical demonstrations or experiments and have students predict the outcomes and then describe (or if possible, carry out) the demonstrations or experiments and show the actual outcomes.” (p. 281) It is beneficial when students make incorrect predictions. If they are directly confronted with wrong mental pictures, they will be very motivated to make corrections and learn the right perspective on the problem.
Interpreting and modeling physical phenomena—In these problems, students are provided with data from a real or a hypothetical experiment, and then they are asked to use course concepts to explain the results.
Generating ideas and brainstorming—The idea here is to use open-ended exercises to disconnect students from their belief that every problem has one right answer. For example, a teacher might present students with a product design and have them brainstorm as many possible flaws and failures as they can think of. No answer is considered wrong during the brainstorming process.
Identifying problems and troubleshooting—Describe a device (such as a process or system) that is not working effectively, and ask students to speculate on the possible causes of the problem. They might also be asked to devise experimental tests that would confirm or refute their suppositions.
Formulating procedures for solving complex problems—In this situation, students are given “incompletely specified problems.” (p. 282) They start by itemizing what they know. Next they list what they need to know, and finally students explore how they will determine those unknowns. For example, would they look up the unknowns? Calculate them? Measure them? Estimate them from empirical correlations? Use rules of thumb?
Formulating problems—Rather than always giving students the problems, turn the tables. Have students look at previous course content from a designated time period (for example, one week, three weeks) and make up the problems that they then also solve. Challenge students (maybe by giving more credit) to come up with problems that require complex analysis, critical examination, or creative thinking.
Making judgments and decisions and justifying them— “Call on students to make and support judgments on ambiguous or controversial matters.” (p. 282) The point here is not the conclusion per se but the quality of the evidence and reasoning mustered to support their position. In order to do this, students must be taught to evaluate evidence in terms of its reliability and validity.
“Including a variety of problem types in assignments serves an important purpose besides promoting intellectual growth and adoption of a deep approach to learning. Some students are gifted in ways that may not show up on straightforward homework problems. When they are assigned problems that call for different skills, they sometimes discover talents they may not have known they possessed. The effect of this discovery on their self-confidence and subsequent performance levels—even on more conventional problems—can be quite dramatic.” (p. 282)
Reference: Felder, R. M. and Brent, R. (2004). The intellectual development of science and engineering students. Part 2: Teaching to promote growth. Journal of Engineering Education, October, 279–291.