"Interactive Projects for Mathematical Modeling for a General Audience"
A Proposal for Spring 2002
Abstract
I will develop technological modules during the Spring 2002 semester for use in a low-level mathematical modeling course, designed to appeal to a broad audience. Technological supports will primarily be Excel and the computer algebra system Mathematica. The technological elements are designed to help students internalize course content and build intuition.
Detailed Description of the Project
BACKGROUND Rationale for Overall Project
The rationale for this proposal is described well in the NCTM (National Council for Teachers of Mathematics) Standards: http://standards-e.nctm.org
Imagine [an environment] where all students have access to high-quality, engaging mathematics instruction. ... Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals. The curriculum is mathematically rich, offering students opportunities to learn important mathematical concepts and procedures with understanding. Technology is an essential component of the environment. Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from a wide variety of mathematical topics.
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In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. Mathematical competence opens doors to productive futures. A lack of mathematical competence keeps those doors closed. NCTM challenges the assumption that mathematics is only for the select few. On the contrary, everyone needs to understand mathematics. All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding. There is no conflict between equity and excellence.
For several years, I have taught two courses in mathematical modeling at Southwestern University, one serving as the senior capstone and the other with a pre-requisite of Calculus II. I am revising the mid-level course to have either no pre-requisites at all to serve a much broader audience. Technology will be a key element to this accessibility. The revisions will enrich the portion of the curriculum targeting the nonmajor.
Some of the topics overlap and complement those in other ACS funded projects, notably the following: 1) "There's more to Math than Calculus: A proposal to develop the technology-based portion of a survey course in mathematics", Johnston & McAllister of Center College (Spring 2001); (http://web.centre.edu/mat/acs01/) and 2) "Mathematica-based, Text-Independent, Interactive Instructional Materials for Introductory Calculus and Finite Math", Craft of University of the South (Summer 2000) (proposal). I will try to use portions of these projects and will create new modules to augment this collection. I will also learn more about the NCTM standards.
TECHNOLOGY Technical Requirements for the Project
I have the hardware, software, and skills needed to successfully complete the project: Mathematica, Excel, and various software needed for web publishing. I also have the resources of Southwestern's ITS staff and facilities as well as access to the ACS Technology Lab.
OTHER SUPPORT Institutional and/or Outside Support for Project
The Department of Mathematics and Computer Science has a solid history of incorporating technology into the mathematics curriculum when that technology will enhance the educational experience of the student. Southwestern has already committed considerable resources to the technological enhancement of education. I have received support from the Institution repeatedly for work consistent with that in this proposal. More specifically, I am working under a half load of teaching for the 2001-2002 academic year, teaching one course Fall 2001 and two in Spring 2002 rather than six for the year.
TIMELINE Deliverables/Milestones for ACS Funded Part of Project
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DESCRIPTION Part of the Project to Be Done Under ACS Funding
I will have adequate time in the Fall 2001 semester to create some modules (without ACS funding) prior to the beginning of the class. The release time during the Spring 2002 semester will provide the time needed while I am teaching the to create more modules.
I will develop a brief instruction guide regarding its use in the classroom for each module. The web site created for this project will include an annotated bibliography. Rather than trying to re-invent the wheel, I will implement other ideas from existing sources, too.
CURRICULUM How the Project Will Be Integrated into the Curriculum
I will teach the course in the Spring of 2002, at which time the modules will be incorporated into the classroom. I will teach the first 6 or 7 chapters of "A First Course in Mathematical Modeling" by Giordano, Weir and Fox. Topics and example models: graphs of functions as models (arms race, effects of the strike on baseball); modeling process (vehicular stopping distance, gas mileage, morning rush hour); discrete dynamical systems (home mortgage, growth of yeast, spotted owl populations, cooling of a heated object and many more); modeling using proportionality (geometric similarity, raindrops from a motionless cloud); least squares fitting; experimental modeling (includes cubic splines), simulation modeling ( generating random numbers, a simple queue). Note that these complement the topics in the two ACS projects mentioned in "BACKGROUND".
LEARNING OUTCOMES How the Project Will Enhance Teaching/Learning
Typically at Southwestern, if a student does not plan to take multiple math courses, he or she takes Elementary Functions (a pre-calculus course), Math Concepts (a survey course which also serves those training to be pre-college teachers), or Statistics. One advantage of the revised modeling course is the appeal of something very different to those with difficulties in math. Furthermore, the revised course would help prepare pre-college teachers for the recent revisions of the NCTM standards, which place a greater emphasis on modeling, probability, and statistics.
Consider the following example of a topic to be covered. Difference equations can be used to model a wide variety of phenomena: financial, biological, etc. A portion of this rich topic is accessible without calculus. Excel and Mathematica will be used to generate tables of values and graphs to illustrate convergence or divergence. These topics are at the core of mathematical thinking and, aided by technology, can be taught at this level. Hopefully, it will motivate some students to take calculus, which will enable further analysis.
The value of visualization tools can best be appreciated by a visit to a classroom but has also been documented (see "Sources" below). For instance, I used an innovative TI-86 graphing calculator animation for parametric plots in Calculus III Fall 2000 with great success. The reactions of the students made its success so obvious: heads were nodding, understanding was visible on their faces. One student declared, "This is SO COOL!", which was very gratifying espeically since this was the poorest student in the class. I want more students to have this "aha!" experience, including the nonmajor. Technology levels the playing field for students by accommodating a wider range of learning styles. Technology is empowering.
For other outcomes, please re-read the portions of the NCTM Standards quoted at the beginning of the proposal.
ASSESSMENT How the Project Will Be Evaluated
I will administer a pre-course technology assessment survey and a diagnostic test in background material at the beginning of the course. The course will end with a similar post-course technology assessment as well as a survey about the effectiveness of materials. These will provide a comparison to help assess the summative value of the technological modules. Hopefully, I will also get feedback from other ACS faculty members who use the modules. Formative evaluations of student understanding (quizzes, in-class discussions, etc.) will continue be given throughout the semester and used to modify the teaching/learning environment when appropriate.
I will evaluate these assessment tools to improve the modules. The webpage will include a discussion of strengths and weaknesses as perceived by the students, by me, and by any faculty who provide input.
DISSEMINATION How the Project Will Be Shared with ACS Colleagues
The project results will be made available to Southwestern and other ACS faculty through (1) my webpage http://www.southwestern.edu/~shelton, (2) presentation at a professional meeting, and (3) the ACS On-line Course Materials Database. I will (4) notify ACS math department chairs of the results. I will (5) add links on my site to the Johnston & McAllister and Craft ACS sites and will suggest that they include links as well.
Possibilities for the professional conference presentation include the Texas Section Meeting of the MAA (Mathematical Association of America) Spring 2002, MathFest Summer 2002 (a national meeting of the MAA), and the International Conference for Technology in Collegiate Mathematics (ICTCM) Nov 2002.
Shelton's webpage will provide electronic access to the files produced. File formats will include at least html, Excel files, and Mathematica notebooks. The resulting Mathematica notebooks will be most readily available to those with Mathematica. Those without Mathematica will be able to view and print the notebooks after downloading the freeware MathReader from the http://www.Wolfram.com website. This will be most useful for those with another CAS such as Maple. TI files will be available for download for those with the TI-GraphLink software and cables. If time permits, pdf format will also be included, which would be a great improvement over the formats provided by the two other ACS projects mentioned in "BACKGROUND", as it would be more accessible to a variety of CAS products.
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