An Assessment Of Problem Solving Processes In Undergraduate Statics

Four well-articulated models that offer structured approaches to problem solving were identified in the engineering research literature. These models provided a conceptual base for the study reported here. Four undergraduates enrolled in statics and two engineering faculty members provided think-aloud data as they solved two statics problems. The data were used to develop a coding system for characterizing engineering students’ behavioral and cognitive processes. These codes were used to analyze students’ problem solving procedures in a detailed manner, particularly differences between goodand not-so-good problem solvers. The analyses provide a picture of how students and faculty solve problems at a cognitive level, and indicate that published problem-solving models are incomplete in describing actual problem-solving processes. Wankat and Oreovicz asserted that “engineering education focuses heavily on problem solving. ” This assertion would find significant agreement among engineering educators. The high proportion of time spent solving textbook problems outside of class by engineering undergraduates has been documented in the engineering research literature. The central place of problem solving in engineering has led some scholars to inquire about the nature of effective problem solving, asking about the processes that underlie good problem solving procedures. Engineering educators have also developed didactic models meant to guide classroom practices. The research presented here is based on four well-articulated models that offer structured approaches to problem solving. The models have been developed as a response to students’ use of a “hodgepodge of tricks” to solve statics, dynamics, and thermodynamics problems, and they were regarded by their authors as useful to students in developing good problem solving skills. Therefore the models were considered appropriate for an empirical study of problem solving by undergraduates. The goals of this study were to develop a descriptive language for characterizing engineering students’ behavioral and cognitive processes related to problem solving. This descriptive language was developed as a coding system that was used to analyze students’ problem solving procedures in a detailed manner. These codes were used to evaluate the extent to which the four underlying models captured students’ problem solving processes. The codes were also used to characterize processing differences between good and not-so-good problem solvers. In summary, the goals were: • To develop a coding system for describing problem solving processes • To test the adequacy of four models for describing problem solving processes • To use the coding system to examine differences between good and not-so-good problem solvers. The central method for addressing the questions in this study was the collection and analysis of verbal protocol (“think-aloud”) data. Verbal protocols are open-ended thinkP ge 13175.2 aloud reports, through which participants are asked to verbalize what they are thinking as they work through a task. Four Problem Solving Models During the past several decades, extensive efforts have been directed at developing an ideal problem solving model. These efforts express the complexity of problem solving by incorporating cognitive, metacognitive, and attitudinal elements into problem-solving models. The models are expressed in specific terms, with the goal of making the processes of problem solving explicit, and thereby allowing educators to reflect on and incorporate the detailed processes of the models into effective instructional practices. The four models presented here are fleshed out in a manner that strives to present their elements in a uniform terminology and at a comparable level of expression. Formulating this level of descriptive consistency across the four models was a necessary step in developing a coding table that would allow a consideration of the adequacy of the models and meaningful comparisons of the models to students’ problem solving behaviors, which reflect the goals of this study summarized above. The problem solving processes for each model were derived from a combination of authors’ descriptions of problem solving in the text of their articles, as well as from the tables and figures in the respective papers. The Wankat and Oreovicz Model (W). In considering problem solving processes, Wankat and Oreovicz note that novices tend to be anxious, have information organized into small pieces, do not know what information is relevant in the problem, reason from superficial problem details, jump to conclusions about what the problem is asking, do not analyze the problem into parts, often do not sketch the problem, use a trial and error strategy, do not check their solutions, and ignore corrective feedback. Experts, on the other hand, are typcially confident, organize information into “chunks,” know what information is relevant in the problem, reason from fundamental principles, take time to define and redefine the problem to themselves, analyze the problem into parts, look for familiar patterns in the problem, spend considerable time sketching the problem, apply well-developed strategies, check their solutions, and learn from errors. The essential elements of problem solving in this model are summarized in Table 1a. Table 1a. Problem-Solving Processes Based on the Wankat & Oreovicz (1993, pp. 7172) Problem Solving Model (A Prestep and Six Operational Steps) I Can 1 – Expresses anxiety or uncertainty 2 – Expresses confidence Define 1 – Lists knowns and unknowns 2 – Draws figure 3 – Identifies constraints on the solution 4 – Identifies criteria for solution Explore 1 – General explorative questions about the problem 2 – Determines if there are required data that are unavailable in the problem 3 – Notes if entire problem is routine 4 – Breaks the problem into parts P ge 13175.3 5 – Identifies parts/segments that are routine 6 – Considers alternative solution methods 7 – Considers the most convenient basis of representation 8 – Considers whether there is a more important underlying problem 9 – Calculates limits on the solution Plan 1 – Develops logical structure of how to solve the problem 2 – Sets up steps to solve the problem 3 – Works through equations without numbers Do It 1 – Inserts values into equations and calculates Check 1 – Checks calculations 2 – Compares answer to problem requirements in Define and Explore 3 – Compares answer to common sense (“doesn’t look right”) Generalize 1 – Indicates what has been learned 2 – Indicates how they should have solved the problem to eliminate errors 3 – Considers how to solve the problem more efficiently in the future Other Processes Paraphrases and looks at different ways to interpret the problem Employs deep processing Generalizes the problem, in order to understand it Substitutes in numbers, in order to understand the problem Simplifies problem (especially if stuck) Relates the current problem to one he knows how to solve (especially if stuck) Searches for patterns (interconnected knowledge) instead of single facts or elements Changes the way the problem is being represented (especially if reaches obstacle in solution) Retrieves memorized equations Uses fundamental relations to generate equations Considers whether solution plan is reasonable Guesses the solution and then checks the answer Monitors solution progress If stuck, uses heuristics, perseveres, brainstorms If stuck, guesses, quits Uses broad experience to evaluate results The Gray, Costanzo, & Plesha Model (G). Gray et al. present a structured approach to problem solving. They regard this approach as a useful one to students throughout their careers. They developed this approach in response to students’ use of a “hodgepodge of tricks” to solve statics and dynamics problems, but regard the method as able to guide students to the solution of any problem they encounter in mechanics. They present their method as “universally applicable” and appropriate for students as early as sophomorelevel mechanics. P ge 13175.4 Gray et al. note that in solving homework and exam problems, students engage in “pattern matching” of the problem to equations they know “coming up with any n equations in n unknowns. ” In their structured approach they provide a set of basic equations from which students can derive the equations they need for a specific problem. The essential elements of problem solving in this model are summarized in Table 1b. Table 1b. Problem-Solving Processes Based on the Gray et al. (2005) Structured Approach (from “Our Five Steps of Problem Solving”) Road Map 1 – Identifies given information 2 – Determines what needs to be solved for 3 – Outlines an overall solution strategy Modeling 1 – Notes assumptions or idealizations in problem 2 – Constructs free body diagram (model of problem) Governing Equations 1 – Writes equations for solution 2 – Organizes equations using key relations (equations) 3 – Checks number of unknowns against number of equations Computation 1 – Manipulates and solves equations Discuss & Verify 1 – Verifies that solution is correct 2 – Considers what the solution physically means 3 – Considers roles of assumptions in solution Other Processes Chooses a coordinate system (e.g., Cartesian) Assesses the adequacy of the problem model Assesses the accuracy of the problem model Checks signs on equations Compares reasonableness of solution (common sense) The Litzinger, Van Meter, Wright, & Kulikowich Model (L). Litzenger et al. were interested in problem analysis as a critical element in problem solving. Based on a review of the literature, they identified three factors that were closely related to analytical skills: content knowledge in the domain of the problem, knowledge of and ability to implement problem solving processes, and the ability to translate between representational systems, particularly translating between a verbal problem description and a diagr