Enhancing Conceptual Transfer Of Mathematical Concepts And Motivating Students Using Engineering Performance Tasks

This paper presents a discussion of the usage of engineering performance tasks in two sections of an innovative Pre-Calculus course to enhance conceptual transfer while motivating engineering students. A description of the entire process from concept to implementation is given including: 1) the role of performance tasks in general; 2) the faculty development that occurred; 3) engineering performance tasks and their development; 4) performance tasks implementation; and 5) assessment and evaluation. Specific reasoning processes and mental habits associated with the Dimensions of Learning pedagogy are designed in the performance tasks, and are presented. Aligning the performance tasks with the topics covered in the lecture is the approach taken, and details of this process are provided. Results are positive. Approximately ninety percent (90%) of the students were proficient at solving the engineering performance tasks. Formative assessment results of the two-minute paper indicated that students understood and could explain the specific procedural and declarative knowledge addressed. In addition, the instructional framework increased students’ Pre-Calculus knowledge, and confidence for solving Pre-Calculus and engineering problems. These results were validated by the use of Pair Sample T-tests. The results of this study have implications for improving conceptual transfer of mathematical concepts and motivating engineering students. Introduction Institutions have developed various strategies for motivating students and exposing them to engineering, including the use of freshmen engineering design courses. On the other hand, some institutions have used a “just in time” approach in teaching the necessary mathematics for engineering in order to spark students’ interest and because of the difficulty students have in negotiating mathematics courses, particularly early in their program 1 . Moreover, when students initially take engineering courses, they often have difficulty applying mathematical concepts and knowledge to solve engineering problems. This paper presents a performance assessment approach to improve motivation and to enhance learning and conceptual transfer of mathematics concepts, using performance tasks and rubrics in an innovative Pre-Calculus course. A multidisciplinary team of faculty from the Schools of Engineering, Science, and Education collaborated in order to develop a series of performance tasks in the disciplines of civil, electrical and industrial engineering. The performance tasks incorporate the Dimensions of Learning (DOL) pedagogical framework, which is a powerful, effective and comprehensive model of five dimensions that uses what researchers and theorists know about learning to define the learning process. 2 P ge 10567.1 “Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education” The innovative Pre-Calculus course, developed and taught using the pedagogical framework previously mentioned, was initially offered in the Fall 2003 semester. The fourcredit course consisted of a lecture component that met four days per week, fifty minutes each day. In the Fall of 2004, a well-developed set of performance tasks were added to the course which necessitated a format change. A mathematics faculty continued to teach the lecture part of the course, and an engineering faculty taught the added laboratory component, where the performance tasks were implemented. No additional course credit was given for the added twohour laboratory component, which met once a week. The fifty-two engineering freshmen enrolled in the course were selected based on their placement exam results. Basic Algebra, PreCalculus I, Comprehensive Pre-Calculus and Calculus I are the possible entry courses for new freshmen. The innovative Pre-Calculus course content was most closely aligned with the University’s Comprehensive Pre-Calculus course. Students who were Comprehensive PreCalculus or Pre-Calculus I ready, based on their placement tests scores, were placed in the special course. Some who placed in Basic Algebra were placed in the course as well, considering their high school Pre-Calculus or Calculus exposure and course grades. The mathematics instructor identified the non-negotiable Pre-Calculus concepts and a questionnaire for the participants was developed based on this. A pre-test and post-test design was used to determine the effects of the instructional framework on outcomes. The tests were scored using a rubric. Role of Performance Tasks A performance task is a performance assessment mechanism. It is a carefully constructed activity or project designed to achieve a viable goal or outcome, and to enhance the knowledge, skills and dispositions of all students. In other words, performance tasks are assignments and projects used to provide students with practice necessary to determine competency in the skills and knowledge deemed important. A carefully constructed activity is one where assurance is made that “what students should know and be able to do” has been identified, and the tasks are developed to accomplish this. According to Danielson and Marquez 3 , they may be used for assessment and instructional purposes. Good performance tasks are engaging; are realistic, elicit knowledge and skill, enable assessment of individuals, and contain clear directions to the students. Because of its role, the performance task was selected as one of the assessment mechanisms to be used in the innovative Pre-Calculus course. Dimensions of Learning (DOL) Pedagogical Framework The pedagogy used in the course is based on the Dimensions of Learning (DOL) framework. A detailed discussion of the initial work performed using the DOL pedagogical framework during the Fall of 2003 has been previously reported 4 . The performance tasks were developed within this framework as well. Robert Marzano 2 lead the development of this framework which has a premise that five types of thinking, or five “dimensions of learning,” are essential to successful learning. The framework grew out of many years of research into how we learn most effectively, and is designed to translate research into a practical classroom application to improve teaching and learning in any content area 2,5 . The five dimensions of learning include 1) Positive Attitudes and Perceptions about Learning; 2) Thinking Involved in Acquiring and Integrating Knowledge; 3) Thinking Involved in Extending and Refining Knowledge; 4) Thinking Involved in Using Knowledge Meaningfully; and 5) Productive Habits of the Mind. P ge 10567.2 “Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education” From an engineering perspective, DOL3 and DOL4 were thought to be most applicable to conceptual understanding and being able to transfer the knowledge from mathematics to engineering. Dimension 3 indicates that once knowledge is acquired, it can then be extended and refined by applying reasoning processes of comparing, classifying and abstracting. Learners rigorously analyze what they learn by applying these reasoning processes that will help them extend and refine information. Once this has occurred, the knowledge can be used meaningfully, as denoted by Dimension 4. In this dimension, performance tasks can be constructed around the reasoning processes of decision-making; problem solving; invention; experimental inquiry; investigation; and system analysis. It was a requirement that these two dimensions be included in performance tasks development. For each of these two dimensions, performance tasks were created with particular reasoning processes in mind. Since the performance tasks are based on declarative knowledge (what students should know) and procedural knowledge (what students should be able to do), DOL2 is naturally a part of the performance tasks. Faculty Development and Training It was planned that engineering students from the three departments of civil engineering (CE); electrical and computer engineering (ECE) ; and industrial, manufacturing; and information engineering (IMIE) would be enrolled in the innovative Pre-Calculus course. In order to ensure that the performance tasks were interesting to all student students from the three departments, a diverse set of performance tasks were developed by involving faculty from each of the three disciplines. This included two from ECE, one from CE and one from IMIE. In addition, the mathematics and engineering instructors, the project director and two other mathematics faculty all received training. Faculty received training in three areas including 1) the Discourse software; 2) the Dimensions of Learning pedagogy; 3) performance assessment (including performance tasks); and 4) rubric development. Although not all faculty was going to use all of these components as it related to the course, it was desirous that everyone had a good understanding of all aspects of the project underway. In addition, this armed faculty with new tools and education that could be incorporated in the engineering and mathematics courses that they normally teach. During the summer of 2004, ETS personnel provided a 2-day workshop on Discourse, and the education faculty member conducted assessment and pedagogy workshops, all which occurred over a two-week period. The remainder of the summer was used for performance tasks development, and the education faculty continued to provide guidance. Engineering Performance Tasks and their Development Prior to the creation of the performance tasks, a reexamination of the desired course content was performed. Since a multidisciplinary team of engineering faculty was involved in the project, a review ensured that each Department had input on the mathematics topics deemed most important. The learning outcomes (the declarative and procedural knowledg