AC 2007-2853: ENGINEERING STUDENTS' MATHEMATICAL THINKING: IN THE WILD AND WITH A LAB-BASED TASK

Although mathematics is considered to be a fundamental element of engineering education, little empirical research has been conducted to understand how engineering students actually use mathematics. This project takes a researchinformed approach towards understanding the role of mathematics in engineering design by combining two studies of engineering students’ use of mathematical thinking: a study of engineering students’ use of mathematics during an industrybased senior design project and a study of engineering students’ use of mathematics during a laboratory based design problem. The capstone study used a combination of qualitative methodologies to investigate engineering students’ use of mathematics during one of their first real-world design projects. For this study, a team of five industrial engineering students agreed to allow the investigator to observe their team meetings, individually interview each team member and analyze their work related to their capstone project. For the laboratory based study, eight industrial engineering seniors were asked to think aloud while completing a three-hour design problem. The findings from the capstone study guided the analysis of the data from the laboratory based study. Mathematical thinking behavior was investigated using Schoenfeld’s five fundamental aspects of mathematical thinking: knowledge base, problem solving strategies or heuristics, effective use of resources, beliefs and affects and mathematical practices. Additionally, Atman and Bursic’s design process coding scheme was used to investigate the engineering students’ design behavior, and identify relationships between mathematical thinking and engineering design behavior. In both contexts the engineering students engaged in mathematical thinking throughout their design processes. This paper presents: 1) a summary of the different mathematical thinking activities that the students engaged in during the capstone study, and 2) a summary of the mathematical thinking activities the students engaged in during the laboratory based study, and 3) some insights from the laboratory study into how the students engaged in mathematical thinking during specific design activities. The results of this study provide insights into how engineering students actually use mathematics, which can inform the way that mathematics is taught to engineering students as well as students at the pre-college level. Motivation from the literature Mathematics has been a central part of engineering throughout the history of the profession, as well as a typical part of an engineering curriculum. The reasons for its inclusion, however, and stakeholders’ perceptions on why it is included, are varied. Some students believe that they take mathematics courses simply because the mathematics courses act as a gatekeeper, ensuring that only the brightest students are able to take engineering courses, while other engineering students view mathematics as one of the many tools that are at an engineer’s disposal. P ge 12652.2 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), as “a medium of knowledge representation” and in developing the ability to reason (e.g. Underwood). Many educators continue to devote attention to how to structure classes to teach mathematics engineering students (e.g. Venable, McConnell and Stiller,Aroshas, Verner and Berman; McKenna, McMartin and Agogino). At the same time, however, there is a prevalent belief among practicing engineers that the mathematics they learned in college is not applicable to their daily work. Pearson estimates that of the thousands of engineers he knew, only three out of ten actually use calculus/differential equations and that “there were/are extremely competent engineers/scientists who never had a course in calculus” (p.8). He also found that many engineers get their mathematical needs met by specialists or mathematicians, and also often just look up the formulas in books. Pearson asks, “why do we continue to teach what the mathematics professors think the engineer/scientist needs, as contrasted with what is actually needed in industry and commerce?” (p. 8). In looking at engineering students’ use of mathematics, there is more to consider than just the mathematical content knowledge taught in mathematics courses. Indeed, mathematics courses may transform a students’ way of thinking and approaching problems; students have the opportunity to learn how to engage in mathematical thinking. Schoenfeld identifies five aspects that are part of the process of thinking mathematically: Knowledge Base, Heuristics (problem solving strategies), Monitoring and Control (cognitive mechanisms that the problem solver uses to monitor their progress and their use of cognitive resources), Beliefs and Affects (regarding mathematics) and Practices (context for teaching and learning mathematics). For the purposes of this research project, we consider an amended set of five aspects of mathematical thinking: knowledge base, problem solving strategies, use of resources, beliefs and affects, mathematical practices. These are defined in Table 1. Table 1: Aspects of Mathematical Thinking Aspect Definition/Description Knowledge Base Cognitive Resources: Mathematical Content Knowledge Problem Solving Strategies Global or local strategies learned from mathematics courses Use of Resources Social Resource: Peers, Experts Material Resources: textbooks, time, computers Use of Resources: metacognitive processes such as planning and monitoring Beliefs and Affects Beliefs about mathematics and one’s mathematical ability, Feelings towards mathematics, Emotions or feelings experienced Mathematical Practices Activities or actions that mathematicians engage in, or activities that involve mathematics. Because design is a practice that is both common to and integral to all branches of engineering, design is a prime context for considering how engineering students engage in mathematical thinking. Additionally, design problems and projects often give engineering students an opportunity to integrate and apply the content knowledge they have learned in their mathematics, science and engineering courses. While there are many different prescriptive models of design that engineering students may be exposed to through engineering textbooks or in their course P ge 12652.3 lectures, most of these models have many similarities. Atman and Bursic conducted a content analysis of the design process presented to students in seven engineering textbooks, and found that most models include ten activities: identifying a need, defining the problem, gathering information, generating ideas, modeling solutions, performing feasibility analysis, evaluating solutions, making decisions, communicating design decisions, and implementing a solution (see Table 2). In this synthesized prescriptive model, it is clear that mathematics is important for modeling solutions and evaluating solutions, and mathematics is also often associated with performing the feasibility analysis. It is unclear, however, from the prescriptive model, what the role of mathematics plays in the other seven design activities. Table 2: Design Activities 2 Code Activity Definition/Description N Identify Need Identify basic needs (purpose, reason for design) PD Problem Definition Define what the problem really is, identify the constraints, identify criteria, reread problem statement or information sheets, question the problem statement GATH Gather Information Search for and collect information GEN Generate Ideas Develop possible ideas for a solution, brainstorm, list different alternatives MOD Modelling Describe how to build an idea, measurements, dimensions, calculations FEAS Feasibility Analysis Determine workability, does it meet constraints, criteria, etc. EVAL Evaluation Compare alternatives, judge options, is one better, cheaper, more accurate DEC Decision Select one idea or solution among alternatives COM Communication Communicate the design to others, write down a solution or instructions IMP Implementation Produce or construct a physical device, product or system This paper addresses the question of how mathematics is used by engineering students throughout the design process—that is, in each and every design activity. One of the most prevalent methods used to study engineering design is verbal protocol analysis 13, . In this research, we use the results from the ethnographic study of engineering students working on their senior capstone design project to inform the analysis of verbal protocols collected from students thinking aloud while attending to a three-hour design task. Methods In this section we present the methodologies used in the two studies, the Capstone Study and the Lab-based Playground Study. A summary of the two studies is provided in Table 3. Study one: Capstone design project The capstone study used a combination of qualitative methodologies to investigate engineering students’ use of mathematics during one of their first real-world design projects. One team of five industrial engineering students allowed the investigator to observe their team meetings, individually interview each team member and analyze their work related to their capstone project. In addition, four engineering students (representing Aeronautics and Astronautics, Chemical Engineering and Materials Science Engineering) participated in interviews. In this section, we present a brief overview of the methodology used to collect and analyze the data from the Capstone Study. The findings from the interview portion of the study as well as a more detailed description of the Capstone Study methodology are presented elsewhere 15, . P ge 12652.4 Table 3: Comparison of the Two Studies Study One: Capstone Design Project Study Two: Lab-based Playground Task Method Observations of 22 team meetings Interviews w