MATH 110

Elementary Mathematical Models

Section 0501

Spring 1995

Class Meetings:
MWF 1-1:50
MATH Building room 0201

J. Fey
Office in 3113 Mathematics Building
Phone: x53151 (voice mail)

Office Hours:
M 2-4, W 2-4, and other times by appointment
Room 3113 Mathematics Building

Course Goals

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 differences too.

Teaching/Learning Format

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 be tested.

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.

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 unfolds.

Developing and Demonstrating Learning with 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 following questions:
  1. What did you learn from a class, activity, discussion, or assignment?
  2. What questions do you have about some current class topic or activity?
  3. What discoveries are you making about the ideas or methods of mathematics or about your own mathematical development?
  4. What thought processes have you gone through in solving a particular problem or investigating a question?
  5. What challenges or confuses you? What do you like or dislike?
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.

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 graded.

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.