Engineering students' mathematical problem solving strategies in capstone projects

Mathematics is generally considered to be a fundamental element of engineering education. However, there is little empirical evidence characterizing the role of mathematics in the engineering design process. The goal of this paper is to take a research informed approach towards understanding engineering students’ use of mathematical problem solving strategies while engaged in capstone design projects. This paper presents some initial evidence elicited through observation of a team of five industrial engineering seniors, interviews with these students as well as students from other engineering departments and document analysis. The data is analyzed using a framework based on the work of Alan Schoenfeld which consists of five aspects of mathematical thinking: the knowledge base (e.g., calculus), problem solving strategies or heuristics, effective use of one’s resources (or cognitive structures, such as memory), beliefs and attitudes (about mathematics), and engagement in mathematical practices. Results from this study provide insights for mathematics and engineering educators as they support engineering students’ integration of mathematics and mathematical thinking into their design practices. Introduction and Relevant Literature Mathematics has been a central part of engineering throughout the history of the profession and continues to be an important element of engineering education. Many members of the engineering education community are continuing to devote attention to how mathematics should be taught to engineering students—through the introduction of mathematics into design courses, the design of mathematics courses for engineering students and the integration of mathematics into engineering curricula. Most members of the engineering education community believe that mathematics is both important and helpful for students in designing and developing systems (e.g. Moussavi) and developing the ability to reason (e.g. Underwood). However, little work has been done to empirically investigate how engineering students use mathematics as they practice engineering design. In addition to supporting the community’s beliefs about the importance of mathematics, an empirical investigation of engineering students’ use of mathematics can inform future decisions about how to best teach mathematics to engineering students. One issue to address here is whether mathematics should be taught within an engineering-specific context or if general mathematics courses sufficiently prepare engineering students. Some educators suggest that mathematics should be taught by the engineering faculty because only engineering instructors are P ge 10559.1 “Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education” able to really convince engineering students of the utility of mathematics (e.g. Venable, McConnell and Stiller). Another fundamental issue here is a question of the similarities or dissimilarities between mathematics and engineering, particular in terms of problem solving. That is, are the mathematical problem solving process and the engineering design process similar enough to teach mathematics to engineering students from the mathematical perspective? Some members of the engineering design community consider problem solving to be very different from engineering design. For example, Holt, Radcliffe and Schoorl distinguish design as encompassing ill-defined problems, and define “problem solving” as solving well-defined problems. Part of the distinction is that well-structured problems can be reduced to an algorithm while ill-structured problems cannot. Additionally, design is a creative endeavor and a learning process . Within the mathematics education community, however, we find a similar distinction. Schoenfeld describes mathematical problem solving in terms of wrestling with the same types of ill-structured problems that Holt, Radcliffe and Schoorl associate with engineering design. His description of mathematical problem solving is part of his explanation of mathematical thinking—having a mathematical point of view and having competence with the “tools of the trade.” He includes problem solving strategies or heuristics as one of five fundamental aspects of mathematical thinking, along with: the knowledge base, effective use of one’s resources, mathematical beliefs and affects and engagement in mathematical practices. Schoenfeld’s problem solving strategies are based on the heuristics identified by Pólya who suggests that heuristics should be built on experience in solving problems. Other members of the mathematics education community also associate problem solving with illdefined problems. For example, Lesh and Harel have investigated the applicability of developmental psychology to problem solving. They focus on problem solving situations that simulate real life experiences where mathematical thinking is useful. In their studies, the problem solvers progress through a series of iterative modeling cycles, during which the problem solvers’ ways of thinking about the problem and given information need to be tested and revised, in order to create useful models of complex problem situations. McGinn and Boote also define mathematical problem solving as solving “open-ended” problems in their study of the experiences of problem-solvers. As they investigated both the socially situated and cognitive aspects of mathematical problem-solving, they distinguished the open-ended problems of problem solving from the well-structured end-of-the-chapter questions students are often assigned, and acknowledged the benefits of each type of assignment. Other members of the engineering education community agree with these members of the mathematics education community and likewise associate engineering design with problem solving. For example, Fadali, Velasquez-Bryant & Robinson report that most engineers define their main job description as being problem solvers. Wankat and Oreovicz advocate explicitly teaching problem solving strategies in the engineering curriculum. In their discussion of problem solving, Wankat and Oreovicz review problem solving basics, identify expert-novice differences, suggest a general strategy for solving problems and provide suggestions for teaching problem solving. Their general strategy for problem solving is based on Woods’ research-based, sixstage strategy for problem solving: engage, define-the-stated problem, explore, plan, do-it and P ge 10559.2 “Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education” look back. Between each of these stages, the problem solver makes a transition—which is mainly a monitoring step. Wankat and Oreovicz’s strategy differs from Woods’ in two ways: Wankat and Oreovicz term the first, motivational stage “I can” rather than “engage” and add “Generalize” as the final stage, during which the problem solver reflects on the problem and identifies lessons learned and opportunities to work more efficiently in the future. Woods and Wankat and Oreovicz also share a similar understanding of problem solving—Woods distinguishes “problem solving” from “exercise solving” and reserves the term “ill-defined” for problem solving and Wankat and Oreovicz acknowledge that problem solving is a complicated process and that problems can be classified as either well-defined or ill-structured. These definitions of problem solving—in particular the emphasis on the ill-structured nature of the problem—all resemble Holt Radcliffe and Schoorl’s definition of design. The literature suggests that there are many similarities between mathematical problem solving and engineering design. This study, which is part of a larger attempt to understand the role of mathematics in engineering design, further explores the similarities between mathematical thinking and design by investigating engineering students’ use of mathematical thinking during their capstone design project. The capstone project provides engineering students with one of their first opportunities to synthesize and apply the material they have learned throughout their undergraduate education to a “real-world” problem. In this paper, we focus our discussion on: How do engineering students use mathematical problem solving strategies while working on their capstone projects? In the next section we describe the research participants and data analysis methods we used as we present our data collection strategy and procedures. Next, we present an overview of how engineering students engaged in each of the five aspects of mathematical thinking included in Schoenfeld’s framework and then focus on the strategies that engineering students learned from their mathematics courses (as reported in interviews) and used during their capstone projects (as observed during their meetings or reported in interviews). Finally, we present some insights gained from our observations and some potential implications for engineering education. Data Collection Strategy and Procedures To investigate how engineering students use mathematical problem solving strategies while working on their capstone project, we chose to take a qualitative approach: a combination of interview and observation methods. Specifically, the first author interviewed students from four engineering disciplines about their mathematics courses and capstone design projects, observed one team of Industrial Engineering students to see how they engaged in mathematical thinking during their capstone projects and collected copies of the team’s work. The interview portion of the study offers insights into the engineering students’ experiences in their mathematics courses, including what the students believed they learned during their mathematics courses, as well as insights into how the engineering students believed that they used