Elementary Mathematical Models
MATH Building room 0201
Office in 3113 Mathematics Building
Phone: x53151 (voice mail)
M 2-4, W 2-4, and other times by appointment
Room 3113 Mathematics Building
The primary goal of this course is to explore some of the fascinating and powerful ways
that mathematical ideas and methods help us to understand the world of our experiences. In
contemporary mathematics the central unifying way of looking at connections between concepts
of mathematics and patterns observed in the world of our senses is to think of mathematical
systems as mental models of the quantities, shapes, and patterns of change that we see, hear, feel,
and measure. Those models help us to describe our ideas to others, to develop theories that
explain phenomena in the biological, physical, social, and management sciences, and to use those
theories to make predictions.
The broad purpose of MATH 110 is to survey some of the most common ways that
mathematical ideas are used as models and to develop some skill in application of those models
to important quantitative problems. This special section of MATH 110 (0501) shares that
common overall purpose with all other sections of the course this Spring. However, the special
section will explore several alternative approaches to that goal.
The content covered in this special section of MATH 110 will focus on units based on
four kinds of models for applying mathematics in solving important real-world problems:
Many topics in these units will also appear in regular sections of the course, but there will be
- Geometric models of size, shape, location, and motion
- Algebraic models of change
- Network optimization models
- Probability models of randomness
The most common instructional format in regular sections of MATH 110 emphasizes
lecture/demonstration time in which the teacher answers questions from previous homework
assignments and explains concepts and problem-solving techniques for new topics. In the
experimental section we plan to use class time quite differently. The instructor will spend some
time at the beginning of each class period to set the stage for investigation of a significant
mathematical question or problem. But substantial portions of each class will be devoted to
students working collaboratively in smaller groups to investigate the questions and solve the
problems that have been posed. Results of those small-group investigations will then be shared
with and discussed by the whole class.
The goal of a collaborative, active class format is to engage every student in constructing
a personal understanding of the key mathematical ideas - not memorizing in a rote fashion the
mathematical techniques demonstrated by the instructor. The mathematical investigations will
often include collecting and analyzing data from simple experiments, finding mathematical
models that represent patterns in that data, and using models to make predictions which can then
Assessment of Student Learning
In keeping with the commitment to collaborative learning, assignment of semester grades
will not be based on competitive ranking of scores on the common MATH 110 hour exams and
quizzes. Assessment in the special section will be based on two general principles. First,
everyone is striving to master mathematical ideas and methods - not to compete against other
students for a limited number of A's, B's, or C's. If everyone develops excellent command of
the material covered in the course, everyone will get an A!
Second, it is now fairly widely accepted that there are many different ways that one can
demonstrate learning - not only by scores on timed examinations that emphasize computational
problem solving. Therefore, each student will have some options (within certain limitations) in
how their semester grade is determined.
The exact combination of evidence you choose to offer for your own grade
will be negotiated and agreed upon no later than mid-semester.
- At least 50% of the course grade will be based on the two hour exams and the
final exam (weighting those scores in the ratio 2:2:3).
- Weekly assignments - varying from individual and group written reports of
mathematical investigations, problems for homework, quizzes, and other learning
checkpoint activities - will constitute 15 - 20% of the semester grade.
- Class participation - including regular attendance and active participation in the
cooperative problem solving activities of the class and regular writing in a journal
of reflections on content and learning throughout the course - will constitute
15 - 20% of the semester grade.
- At your option, you can choose to have 15 - 20% of your semester grade based on
a portfolio of work illustrating the topics and results of the course and your best
work on those topics.
Text and Materials
While we will cover many topics that appear in the textbook for regular sections of
MATH 110, we will not be following the presentation of that book. The instructor will provide
text materials as the course unfolds - generally introductions to the problem contexts and
questions to be addressed by collaborative learning in class or independent work outside of class.
Your results from those investigations, when written up carefully, will comprise the balance of
the course "text".
Because the class will involve frequent investigations that involve data collection,
sketching of ideas, and written communication of results, it will make sense to have a notebook
in which copies of individual papers can be entered neatly. Graph paper will be helpful. Some
experiments will require rulers and angle measuring devices. For major sections of the course a
graphing calculator will be an essential tool. While the regular sections of MATH 110 require
only a TI-81 graphing calculator, many things we will do are aided by features of the somewhat
newer TI-82 model that will cost only slightly more than the TI-81.
The University now provides excellent computing resources on campus and by remote
log-in from off campus. In the WAM labs you can do mathematical calculations using powerful
software and word processing for writing reports of those calculations. You can also do
electronic mail with others in the class and with the instructor. You are encouraged to get a
WAM account and use it. This service is free and you can get your account set up at the
Computer Science Center. We intend to provide some help in getting started as the course
Developing and Demonstrating Learning
Journals and Portfolios
In this special section of MATH 110 there will be two mid-term exams and a final exam.
However, you are urged to take advantage of two other strategies for developing and
demonstrating your understanding of the mathematical ideas in the course.
Journal Writing - Many people find that their thinking about difficult concepts and
problems can be stimulated and clarified by attempts to express ideas in spoken or written form.
In a mathematics class, periodic writing about progress and problems encountered can also be a
powerful strategy for communicating what you have learned and what you are puzzled about.
For these reasons, part of the class participation requirement involves keeping a mathematical
journal with a signficant entry at least once each week of the course.
The focus of your journal entries should be your learning of mathematics - what you do,
feel, discover, and wonder about. The journal entries will then comprise a reflective record of
the questions and insights that help you make sense of the material covered in the course. In
your writing you may choose to focus on any aspect of the course and its connections to your
other mathematical, academic, and out-of-school experiences, as long as you are willing to let
the instructor see what you have to say . There are no "right" or "wrong" journal entries.
As general guidelines to consider in crafting journal writing, you might consider the
The journal will be read and reactions given several times during the semester. For
purposes of contributing to a course grade score, journal entries will be rated on the frequency,
breadth, and depth of commentary. We are basically looking for indicators of probing thought
and growth of insight into mathematics and your own thinking about mathematics.
- What did you learn from a class, activity, discussion, or assignment?
- What questions do you have about some current class topic or activity?
- What discoveries are you making about the ideas or methods of mathematics
or about your own mathematical development?
- What thought processes have you gone through in solving a particular
problem or investigating a question?
- What challenges or confuses you? What do you like or dislike?
Portfolio Construction - For many years artists and architects have used portfolios of
their work to showcase their talents and interests. That sort of vehicle for demonstrating
capability is now becoming more widely accepted in other fields as well, as an alternative to
examinations in which questions are answered under strict time and resource constraints.
Since this special section of MATH 110 will emphasize collaborative work on analysis of
complex problems, with written and oral reports shared in full-class discussions, your work in
the course will naturally lead to a portfolio of findings and reports. From time to time you will
be asked to submit those reports for review, but you might also consider making a more complete
portfolio of your work a significant component of the material on which your course progress is
To construct a portfolio of your work, begin early to collect the problems posed and
worked on in class and assigned as homework and your own write-ups of findings in work on
those problems. As you gain new insights and see ways to solve previously unsolved problems,
rework earlier results, crafting them into the sort of polished form you'd be pleased to present as
reports to a supervisor at work. By the end of the course you should have assembled a collection
of at least 15-20 such pieces of work, selected to illustrate the major ideas of the course. That is,
your final submitted portfolio should include problems and results that would convey to someone
else the main ideas of the course and your insights into those ideas. A brief commentary on each
major section of problems and results should set the context and guide the reader.
To help you feel comfortable about the direction of your portfolio work, there will be two
mid-term reviews of what you have assembled. You are, of course, welcome to discuss your
progress with the instructor at any time. Grading criteria for portfolios will include breadth of
the selection as indicators of course content, completeness and accuracy of your written work,
and clarity of your presentation.
HTML coding by Tom O'Haver, Feb. 24, 1995.