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                                                                 6/6/'96
                Mathematical Modelling as a `Capstone Course'
                                        T.I. Seidman, UMBC (Math)
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Goals:
        The development of
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        an understanding of the roles of simplification, approximation, and
simulation in obtaining insight into complex phenomena,
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        an awareness of the wide range of situations to which mathematical
modelling and analysis can usefully apply,
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        modelling skills and familiarity with the modelling process,
especially the integration of mathematical formulations with scientific
reasoning, and
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        a deeper understanding of (previously learned) concepts of
Science and Mathematics by seeing them in action.


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Format:
        This is a one-semester 3-credit course, organized through
project-oriented group work and class discussion. Thus, students work in
teams (2--4 students) on their selections from a variety of proposed modelling
projects, each leading to a formal `Project Report'. Students will maintain
individual Journals (intended for comments on progress, reflection on
strategies, questions for consideration, reaction to the team interaction,
etc.)

        Over the semester, the class will move from some short, quite
structured assignments and introductory lectures and discussion of basic
modelling ideas to longer and more open-ended `major' projects.  Each student
will be expected to participate in at least two major projects (probably
involving different teams) and will provide a `reflective analysis' for each
project, commenting individually on the team Project Report.

        The class will meet three times per week week. Each Monday class hour
will be devoted to oral team activity reports to the class, describing
the team's accomplishments, results, difficulties, etc., for the week, with
questions from the class and the Instructor. Wednesdays will be devoted to a
mix of brief lectures (typically, 10--20 min.) and class discussion related to
basic modelling strategies.
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\footnote{  E.g., the use of conservation laws, the use of continuous models
for discrete phenomena (as population models or traffic flow), the `law of
mass action', etc., or the use of the `forward Euler' approximation for
handling differential equations.        [It might be noted that this last
has been understood and used successfully by 5-th grade students with
`some' programming skills.]
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Most team working sessions will necessarily be scheduled outside class time,
but Friday class meetings will be used for team discussions in which the
Instructor would be able to participate (moving from group to group) for
interactive mentoring. [The Instructor will also `always' be available by
e-mail as a consultant.]

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Assessment:
        The intent is to grade `developing insight' with respect to the goal
criteria so grading is inherently somewhat subjective. Grading will be based
on the students reflective analyses, especially a Course Summary, primarily
answering `What have you learned in this course?' This Course Summary will
be the main segment of a portfolio which will also include, as appendices,
the completed Project Report s (and related individual reflections) for all
projects in which that student participated and the cumulative Personal
Journal.  The Course Summary is to be analytic and reflective, utilizing the
Project Report s and Journal as documentation. Grades will not be based on the
team projects, but on the individual analyses of these and the insights shown
in the Journal, etc.; included in the portfolio would also be a self-
assessment justifying (in terms of the Course Summary) a proposed course
grade.

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Content:
        Fundamental concepts are: state, parameter, conservation, balance
and rates, dimensional analysis, linearity vs. nonlinearity as well as such
basic scientific ideas as density, flux, energy, etc. The mathematical
level will presume a good working knowledge of Calculus and will extend this
(differential equations, numerical approximation, etc.) as needed (note
Goal 4).  The scientific content will vary (note Goal 2) and will be strongly
dependent on the backgrounds and interests of the students in the class.

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Resources:
        It is not expected that hands-on (physical) experiments will be done
in class, so no apparatus will be needed. In order that the students see for
themselves how to `reason mathematically' about the world --- how to
construct/adapt and interpret models to gain insight --- rather than merely
hearing about how others have done this, there will be no text per se. Some
model Project Reports will be provided, both to exemplify `what might
reasonably expected' and to introduce fundamental modelling strategies.
Handouts will suggest a variety of project topics for student selection.
Outside faculty will be available as mentors, to provide subject expertise as
needed.  It is anticipated that enough of the students will (already) have
some computer skills that most teams will contain someone who can organize
simulations in connection with team projects. Students should have computer
access for e-mail interaction and for the use of WWW as a resource.

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Remarks:
        The course description here is an adaptation to an MCTP context of
modelling courses previously given at UMBC by the same Instructor (at a much
higher technical level in math). This course has already been run once,
essentially fitting this description, (with 15 students, one an MCTP Fellow)
at UMBC as sc MATH 385 in Fall, 1995; it is again scheduled for Fall, 1996.

        The Instructor was Prof. Thomas I. Seidman, a senior member of
the UMBC Math/Stat faculty (Ph.D. -- NYU, 1959) with a combination of active
research and publications in technical journals, modelling experience in
industrial employment and consulting, and some background in Educational
theory (MA -- Columbia TC, 1953). The particular Instructor's variety of prior
experience made the presentation somewhat less dependent on outside resources
than might otherwise have been the case.

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