«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
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390 15TH Annual Conference on Research in Undergraduate Mathematics Education Mathematical Modeling and Engineering Majors Jennifer A. Czocher The Ohio State University Abstract: A ﬁrst course in diﬀerential equations for engineers and scientists is intended to introduce the students to key principles and techniques involved in using mathematics as a modeling tool. However, a great many students emerge with only a limited number of analytic techniques that are applicable only to a narrow selection of equations, despite the inclusion of word problems in the curriculum. Previous research into mathematical modeling competencies indicates that the students’ diﬃculties can be traced to coordinating mathematical with physical reasoning. The purpose of this research is to develop tasks for a data collection instrument that will allow for the development of a cognition based model of how such skills grow.
Keywords: diﬀerential equations, mathematical modeling, design experiment Background Diﬀerential equations, as a mathematical domain, arose from the study of change in physical systems over time. Many mathematics faculty and even some engineering faculty (Standler, 1990) insist that linking a diﬀerential equation with the particular applied problem it embodies should receive little attention in mathematics courses because such is the purview of engineering courses. In contrast, other educators argue that modeling is an interdisciplinary enterprise (English, 2010) and so coordinating the mathematical model and the situation it represents is critical (Shternberg & Yerushalmy, 2003).
According to Blum (2011), modeling is cognitively diﬃcult for students because of the dialectic nature of modeling tasks: they require Grundvorstellungen (appropriate fundamental mathematical ideas), real-world knowledge, and the ability to translate back and forth between the two. Niss, Blum, and Galbraith (2007) identify two reasons to teach with applications in mathematics: (1) to use mathematical modeling (MM) and applications for the learning of mathematics and (2) to learn mathematics in order to develop competency in applying mathematics and building mathematical models. The latter is the primary motivation for including diﬀerential equations in science and engineering programs. Engineering majors struggle in applying mathematics to build mathematical models or to manipulate them.
The two goals of the broader research project are (1) to study the development of students’ “ability to construct and to use mathematical models by carrying out those various modeling steps appropriately as well as to analyse or to compare given models” (Blum, 2011, p. 18), and its components, from a cognitive perspective and (2) to create tasks that could be adopted by mathematics faculty as instructional aids. The focus of the present research is instrument design. The intention is to present a set of modeling tasks appropriate to engineering students and diﬀerential equations and to also provide some preliminary data and analyses arising from ﬁeld testing the tasks with engineering students.
Mathematical Modeling In the mathematics, mathematics education, and engineering education literature bases, MM is presented as a process that bridges two worlds: the real and the mathematical. As in problem-solving, there are four overarching stages: identify a real world problem, idealize and express the phenomenon mathematically, analyze the mathematical model, and interpret the solution in real world terms. Kehle and Lester (2003) 15TH Annual Conference on Research in Undergraduate Mathematics Education 391 presented four processes that link these stages together: the modeler begins with a realistic problem which he simpliﬁes into a realistic (idealized) model. The idealized model is then abstracted into a mathematical model and calculations lead to mathematical results which must be interpreted. The process students have the least success with is mathematiziation.
The term “mathematizing” is used to encompass activities like symbolizing, algorithmatizing, and deﬁning (Rasmussen, Zandieh, King, & Teppo, 2005) or the the application of mathematical tools such as creating standard representations (Kwon, Allen, & Rasmussen, 2005; Rasmussen & Blumenfeld, 2007). In the MM literature (see, for example, Blum, 2011;
Niss et al., 2007; Lesh, Doerr, Carmona, & Hjalmarson, 2003), the term refers to the arc of cognitive activities that lead from the description of a life-like problem to rendering that problem in mathematical terms so that well-known tools (e.g., equations) can be identiﬁed or expressed. Lesh and Yoon (2007) distinguish between “mathematizing reality” and “realizing mathematics,” where the latter refers to dressing up mathematical problems with language of lifelike situations. Mathematizing reality involves simpliﬁcation and abstraction (Kehle & Lester, 2003), specifying assumptions and making mathematical observations (Zbiek & Conner, 2006, see Figure 1), and distilling life-like problems into an idealized “situation model” (Haines & Crouch, 2007). A series of processes inverse to mathematizing, but less well-theorized, is carried out after the mathematical analyses take place. In this phase, the modeler examines the results of the mathematical analysis in light of the purpose for building the model. Finally, the model must be validated and reﬁned.
There may be a great deal of oscillation among portions of the modeling process before a stable idea or representation is reached (let alone a viable model). Most diagrams represent the modeling process iteratively. Zbiek and Conner’s (2006) schematic details critical sites within the overarching cycle where deliberations may occur as well as which and how they recruit cognitive processes. Kehle and Lester (2003) explained these cognitive transitions between the two worlds as diﬀerent modes of inference. Abduction bridges experience to a sign system, deduction is the drawing of conclusions based on the manipulation of those signs according to rules, and induction applies a sign system to an experience that is thought to correspond to the structure of that system. Induction and abduction work together to help interpret experience. Students are best prepared in their mathematics classes for the analysis portion of the cycle (Gainsburg, 2006) and they need experience in connecting real world to mathematical world connections in order to develop modeling competency (Crouch & Haines, 2004).
The questions guiding this project are: How do engineering students carry out mathematization? How do they validate their models? Are their techniques stable or do they change over time? How do the students “keep track” (Gainsburg, 2006) of the transitions among realistic situations, idealized situations, and mathematical models? What features of life-like situations to students attend to? What criteria do they use to analyze and evaluate models? What mathematical, and in particular diﬀerential equations, competencies are modeling tasks most suited to enhance (Niss et al., 2007)? What elements, behaviors, or cognitive activities of the modeling process might be unique to diﬀerential equations?
Methodology Given the two objectives for the larger research context, a design experiment methodology was selected (Cobb, Confrey, DiSessa, Lehrer, & Schauble, 2003; Kelly, Baek, Lesh, & Bannan-Ritland, 2008). Task development has focused on modeling competencies 392 15TH Annual Conference on Research in Undergraduate Mathematics Education and has proceded iteratively. Modeling competencies include “the ability to identify relevant questions, variables, relations or assumptions in a given real world situation, to translate these into mathematics and to interpret and validate the solution of the resulting mathematical problem in relation to the given situation, as well as the ability to analyse or compare given models by investigating the assumptions being made, checking properties and scope of a given model” (Niss et al., 2007, p. 12). Selection of appropriate tools, whether mathematical or cognitive, depends on recognizing the underlying structure of a problem (English, 2010). Since many application and modeling problems emphasize the analysis of an already mathematized situation, Lesh, Hoover, Hole, Kelly, and Post (2000) developed model eliciting activities (MEAs) to serve the dual role of revealing students thought processes as they solved signiﬁcant mathematics problems while simultaneously providing learning experiences for the students. However, one of the primary challenges in using MEAs in undergraduate engineering courses is to discover ways to blend them with other pedagogies (Hamilton, Lesh, Lester, & Brilleslyper, 2008), particularly those often used in undergraduate mathematics classrooms.
These ideas guided the initial creation of the modeling tasks, which highlight diﬀerent stages of the modeling process and a variety of modeling competencies. Thus, both whole modeling tasks and competency-speciﬁc tasks were developed drawing on multiple mathematical domains. A series of one-on-one task-based clinical interviews will be conducted with engineering students enrolled in a diﬀerential equations course, in accordance with the design experiment methodology, in order to assess and modify the tasks relative to students’ knowledge and development.
Results At the time of this submission, instrument construction has proceeded iteratively with content-validity checks. Concurrent validity will also be assessed. Relevant literature has indicated various phases of the modeling process that students should encounter as they solve the problems and these phases will be used to frame the students’ activities while addressing the tasks. Through analysis of the protocols and students’ written work, I expect to assess the feasibility of using the instrument to identify and map cognitive activities crucial to the development of modeling competencies. My goal for this presentation is to generate feedback from other researchers about how to best improve the data collection instrument.
Questions Based on the preliminary data collected, I would request feedback to improve
• Are these tasks representative of the diﬀerent stages of the modeling process? Are there aspects of the modeling cycle that are being neglected? Aspects that could be better assessed?
• How can tasks be modiﬁed, extended, or added to include more student reﬂection?
• What additional paradigms that could be used to explore students’ development of modeling skills?
• What other literature, frameworks, theories, or considerations might be essential to this work?
• Does authenticity of the tasks matter? To what extent?
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ReferencesBlum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. In G. Kaiser, W. Blum, R. Borromeo Ferri & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (Vol. 1, pp. 15–30). International
Perspectives on the Teaching and Learning of Mathematical Modelling. Dordrecht:
Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003, January). Design experiments in educational research. Educational Researcher, 32 (1), 9–13.
Crouch, R., & Haines, C. (2004). Mathematical modelling: Transitions between the real world and the mathematical model. International Journal of Mathematical Education in Science and Technology, 35 (2), 197–206.
English, L. D. (2010). Preface to part iii, In Theories of mathematics education: seeking new frontiers (pp. 65–66). Springer.
Gainsburg, J. (2006). The mathematical modeling of structural engineers. Mathematical Thinking and Learning, 8 (1), 3–36.
Haines, C. R., & Crouch, R. (2007). Remarks on a modeling cycle and interpreting behaviors.
In R. Lesh, P. L. Galbraith, A. Hurford & C. R. Haines (Eds.), Modelling students’ mathematical modelling competencies (pp. 145–154). New York, NY: Springer.
Hamilton, E., Lesh, R., Lester, F., & Brilleslyper, M. (2008). Model-eliciting activities (meas) as a bridge between engineering education research and mathematics education research. Advances in Engineering Education, 1 (2), 1–25.
Kehle, P. E., & Lester, F. K. (2003). A semiotic look at modeling behavior. In R. Lesh & H.
M. Doerr (Eds.), Beyond constructivism: a models and modelling perspective (pp. 97– 122). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Kelly, A. E., Baek, J. Y., Lesh, R. A., & Bannan-Ritland, B. (2008). Enabling innovations in education and systematizing their impact. In A. E. Kelly, R. A, Lesh & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 3–18).
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