Goal-Free Problems

Asking a student to find the solution to a problem creates high cognitive load that she may end up making a lot of mistakes. This is especially true when the problem requires many sub problems to be solved, and students tend to make many more errors in these sub goal stages than at the final goal stage. This effect is called the stage effect.

An alternative to help students learn problem solving is to ask them to find the value of as many unknowns as possible, rather than finding a value for a specific goal. As an example, given a programming assignment, students are asked what are the unknowns rather than asking them to create a final program. This can be open ended and the students may go off on a tangent if not properly guided. Of course, no one would hire these students if all they can do is to come up with unknowns(!), but this strategy can be used as a scaffolding device to help students connect what they already know and what we want them to know. New knowledge can be built up as the unknowns are identified, and how these unknowns are related to what has already been learned.

Reference:

Ayres, P. (1993). Why Goal-Free Problems Can Facilitate Learning. Contemporary Educational Psychology. 18, pp 376 - 381.

Good Problems and Effective Structures for Groups

Context-rich group problems help students to focus on the concepts and principles that are needed to solve them. They have the following general characteristics:

• Problem statement does not always specify the unknown to be computed.
• More information may be available than is needed to solve the problem.
• Some of the information needed to solve the problem may be missing from the question. Students need to determine what the missing information is and how to come up with it.
• Reasonable assumptions may need to be made to simplify the problem and allow for a meaningful solution.

Groups of three and four members are found to generate better plans for solving problems and a solution with fewer conceptual mistakes than pairs. Pairs usually have no mechanism for deciding between two strongly held viewpoints. In groups of four, one student was invariably left out of the problem-solving process. That person is usually the most timid or the most knowledgeable.

Homogeneous gender groups and mixed gender groups of two females and one male performed better than groups with two males and one female.

Groups with mixed ability performed as well as groups consisting of only high-ability students (who tend to make problems more complicated than necessary or overlook the obvious), and better than groups with students of only low or medium ability. Low ability students contribute by keeping the groups on track by pointing out the obvious and simple ideas, and requesting for clarification of the concepts and procedures that are needed to solve the problems (which the higher ability students sometimes realize their wrong assumptions and mistakes when they justify their solutions to them).

To avoid dominance of any student in a group, or to avoid a group from jumping at the first possible solution to avoid conflict in the group, two strategies can be used:

1. have students take on special roles. In a three member group, the roles of Manager (who designs plans for action and suggests solutions), Skeptic (who questions premises and plans), and Checker / Recorder (who organizes and keeps track of the discussions) can be assigned.
2. have the students reflect on how well their groups have worked and suggest ways of improvement at the end of each activity.
   

Reference:

Heller, P., Hollabaugh, M. (July 1992). Teaching Problem Solving Through Cooperative Group. Part 2. Designing Problems and Structuring Groups. American Association of Physics Teachers. 60(7). pp 637 - 644.

Is Collaborative Group Learning Useful?

A study on the effectiveness of group problem solving was conducted by Heller et al. (1992). The instructional approach was as follows:

• students were taught general problem-solving strategies
• a set of context-rich practice and test problems were given to help students focus their attention on the need to use conceptual knowledge to analyze a problem
• students worked in carefully managed groups to practice solving context-rich problems

Students' work were judged based on "expert" level of problem solving which is characterized by the following:

• evidence of conceptual understanding
• usefulness of information identified to solve the problems
• match of equations with information identified
• reasonable plan
• logical progression
• appropriate mathematics

Results: Group problem solutions were significantly better than those produced by the best problem solvers from each group on matched problems. Individual problem solving performance also improved over time. The key seems to be explicit problem solving strategy instruction and having the students practice using the strategy in groups.

Reference:

Heller, P., Keith, R., Anderson, S. (July 1992). Teaching Problem Solving Through Cooperative Grouping. Part 1: Group versus Individual Problem Solving. American Association of Physics Teachers. 60(7). pp 627 - 636.

Problem Solving

Definition: Problem solving is cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver. (Meyer, 1992)

Proposition #1: Problem solving abilities do not transfer between disciplines. (Maloney, 1993)

Proposition #2: A student's strengths and weaknesses in problem solving are the same regardless of the environment. As an example, a student's strengths and weaknesses in solving a complicated trip planning problem are the same in solving a physics problem or performing in a work place. (Adams and Wieman, 2007)

Implications: if #1 is true, the argument that math and logic help students in Computer Science is no longer valid?

If #2 is true, all Computer Science students should play a lot more video games?

References:

Adams, W. and Wieman, C. (2007). Problem Solving Skill Evaluation Instrument - Validation Studies. Retrieved on October 22, 2009 from here.

Maloney, D.P. (1993). Research on Problem Solving: Physics, in Handbook of Research on Science Teaching and Learning edited by D.L. Gabel. Toronto: Macmillan. pp 327 - 354.

Meyer, R.E. (1992). Thinking, problem solving, cognition (2nd ed). New York: Freeman.