Problem Solving In Engineering, Mathematics, And Physics – Part 2

This is a work in progress dealing with problem solving across disciplines in an attempt to make engineering students better problem solvers. The purpose is to enable students to identify common types of problems in a variety of subject areas and to help them learn appropriate strategies suggested by each problem type. A previous investigation reported on a survey of math, physics and engineering faculty with respect to the types of problems they employed in their instruction. A major result of this study was that little common vocabulary is used to describe problems and problem solving. Therefore, the additional result that the disciplines do not share a common approach to categorizing problem types and appropriate solution techniques is not surprising. In order for interdisciplinary efforts to make further progress, it appeared that a common language and framework were needed. The current investigation deals with developing a problem-solving vocabulary and then a method of problem categorization that could be agreed upon by STEM disciplines. Starting with problem-solving words that appeared in transcripts of the faculty interviews, a vocabulary list was developed by consulting dictionaries, faculty, and national problem solving experts. With this in hand, a matrix was developed to categorize problems. This framework shows some promise as a means for promoting useful problem-solving conversations among faculty, and may have explicit applications in the classroom, as well. Work in the immediate future will focus on sharing, testing, and improving the matrix. From there, it will be employed as a tool for curriculum design. Ultimately, studies will investigate if students in courses affected by this categorization scheme are more efficient and effective problem solvers, and if they more readily transfer problem-solving skills from one course to another. Background The Fundamentals of Engineering for Honors (FEH) program at Ohio State has included some coordination of topics in physics, engineering, and mathematics since 1997 in an effort to 1) help students have appropriate background for each course and 2) assist students in making connections between the different subject areas. One element that is particularly common in all three disciplines is problem solving, but until recently there had not been much discussion of this prevalent aspect of STEM education in the coordination efforts. Some of the literature indicates that typical college experiences do not lead to much improvement in student problem solving skills and that the problem solving skills that may develop in one discipline are not readily transferred to another content domain. It was postulated that the FEH program with its interdisciplinary nature might serve as a useful setting for a successful attempt to impact these issues. In initial conversations during FEH meetings, it was observed that the physicists had names for some of the sorts of problems they assigned, but it did not appear that the engineers or mathematicians did. Did this mean that different disciplines were assigning different kinds of problems or just that the physicists had developed terminology of which the other disciplines were unaware? If the instructional team was assigning common types of problems, it would be a useful thing for the members of the team to know. Further, given that novices have a difficult time seeing commonalities within one discipline area, let alone across disciplines, it would be good to make any problem solving links between courses clear to the students. The thought was tha t if common problem types could be identified across the disciplines and described in a way that the instructors basically agreed upon, and that if the instructors referred to these common types of problems in their courses, that students might begin to see some connections between their courses. Further, if the instructors would help students see that certain strategies tend to be successful in approaching certain types of problems, the students might become more adept at interdisciplinary problem solving. If the students could match cues about the nature of a problem or the nature of its solution with a set of often useful skills, their approaches should be more effective than the random trial-and-error approach so often seen. With these goals in mind, a number of faculty from each of the three disciplines were interviewed about the kinds of problems they utilized in their teaching; all were involved in teaching first-year engineering students. The results of these interviews were reported previously. Two of them have a particular bearing on the work described here: 1) Some faculty cannot clearly articulate the types of problems they assign to their students, apart from detailing the content and/or context of the problem itself. 2) There were very few commonalities in the language the faculty used to describe the problem types they used. It became clear that before meaningful conversations could occur between the faculty, a common vocabulary would need to be developed and agreed upon. This vocabulary would be increasingly valuable when instructors would talk to students about common problem types across the disciplines, as well as about useful strategies for approaching them. Development of the Vocabulary The transcripts of the interviews from the previous phase of the study were carefully analyzed to identify terms faculty used to describe problems and problem solving skills. The research team added further terms that came up in their discussions of the interviews. Next the team drafted definitions for these terms as they specifically applied to problem solving, utilizing the interview transcripts, the relevant literature, and several dictionaries. After several iterations, the draft of the vocabulary was shared with the interview subjects for their feedback. This was an important step, because it verified that the definitions that had been developed were in accordance with the way the terms had originally been used by the faculty. Additionally, feedback was solicited from problem solving experts throughout the country. Moderate modifications were made based upon this input. The resulting list of forty-two terms is shown in Appendix A. A caveat given by one of the experts that is important to keep in mind is that it is unlikely that all people will agree on every aspect of every one of these terms. However, this list appears to be fairly acceptable to those who have been consulted, both problem solving experts and STEM faculty. As an example of how this process worked, consider Figure 1, which shows the evolution of the definition for “qualitative.” Iterations Sources Involved adj 1: involving distinctions based on qualities; "qualitative change"; "qualitative data"; 2: relating to or involving comparisons based on qualities; 3: not mathematical or numeric, not expressible as a quantity Dictionaries Relating to or involving comparisons based on qualities but not mathematical or numeric; a feature or characteristic not expressible as a quantity; not mathematical or numeric, not expressible as a quantity Research Team Relating to or involving comparisons that are non-numeric Research Team Relating to or involving characteristics, relations, or concepts that are non-numeric OSU Faculty, National Experts Figure 1, The Evolution of “Qualitative” Now that a common vocabulary was developed, the next step was to utilize this language to categorize and describe different types of problems. There are certainly a number of different ways this might be attempted. The team decided to begin with a two-dimensional matrix, where one axis indicates the nature of the solution (no possible solution , exactly one correct solution, or multiple correct solutions) and the other describes the nature of the given information (insufficient information, exactly sufficient information, or excess information). As an example of how this matrix works, consider Figure 2. Each cell contains a list of skills that might be appropriate to employ when solving a problem of this nature. None One Two or more Analyze Analyze Analyze Approximate Approximate Approximate Assess/Evaluate Assess/Evaluate Assess/Evaluate Assume Assume Assume Estimate Estimate Estimate Model Model Model Research Research Optimize Verify Verify Research Verify Analyze Analyze Analyze Assess/Evaluate Assess/Evaluate Assess/Evaluate Model Model Model Verify Verify Optimize Verify Analyze Analyze Analyze Assess/Evaluate Assess/Evaluate Assess/Evaluate Discriminate Discriminate Discriminate Filter Filter Filter Model Model Model Sort Sort Optimize Verify Verify Sort Verify Nature of the Answer N at ur e of P ro vi de d In fo rm at io n E xa ct ly S uf fi ci en t E xc es s In su ff ic ie nt Figure 2, General Problem Categorization Matrix with Associated Skills As an example of how problems that are similar both in terms of basic content area and presentation fit into different blocks of the matrix, consider Figure 3, which shows a set of 9 similar yet different statics problems, each in the appropriate block. The point of this is that problems which share a number of common characteristics can be quite different in the manner in which they are successfully approached. At this time, it appears that most problems typically encountered in introductory courses will fit into one of these matrix blocks. The point is not that any problem situation can be modified to fit in a different block of the matrix. In fact, the team came up with several problem situations that were not easily modifiable to fit all the blocks of the matrix, but that would fit in 6 blocks rather easily. Recall that the purpose of this matrix, as well as the vocabulary list, is to facilitate discussions among interdisciplinary faculty concerning problem solving. In current plans, it is highly unlikely that students will see the exact matrices presented here, but they may see some modi