Using Mathematics to Understand Our World

Using Mathematics to Understand our World

Spring 2007

COURSE OVERVIEW

Graduate Credit Hours: 3

Usual Delivery Format: This course follows the on-line distance course design with initial 2-day, on-site component, completed during an academic semester (see Face to Face and On-Line Components below).

Principal Developer: This course is designed and offered as part of the Math in the Middle Institute Partnership at the University of Nebraska – Lincoln by Gwendolen Hines, Professor of Mathematics at the University of Nebraska – Lincoln.

Course description: Students in this course will examine the mathematics underlying several socially-relevant questions from a variety of academic disciplines. Students will construct mathematical models of the problems and study them using concepts and skills developed in this or previous mathematics courses (such algebra, linear and exponential functions, statistics and probability). Sources will include original documentation whenever possible (such as government data, reports and research papers) in order to provide a sense of the very real role that mathematics plays in society,both past and present.

Course Goals: In this course the students' mathematical perspectives will be broadened and strengthened as they are exposed to a variety of interdisciplinary settings to which mathematical topics can be applied. The mathematical topics include exponential growth and decay, logarithmic functions, Newton’s Law of Cooling, simulations, graphing data, making predictions, analysis of the effects of error, probability and quality control. The disciplines to which the mathematics will be applied include biology, medicine, natural science, forensics, and industry. In addition to broadening the participants’ mathematical perspectives, these experiences will provide the teachers taking this course with concrete examples of how they might collaborate with colleagues in other disciplines in their own classrooms and schools.

The course goals are threefold:

Mathematical Modeling and Problem Solving: Students in this course will learn to identify the mathematics found in questions stemming from a variety of real-world settings and academic disciplines. Complex problems in which the mathematics is not readily apparent will be modeled and analyzed from a mathematical perspective, providing a very real application for concepts and skills which will be developed in the course or were acquired in previous mathematics courses. As is the case with mathematicians who apply their expertise to a variety of scientific and industrial settings, students will be required to synthesize considerable background information in order to make sense of and identify the mathematics in the project problems.

Reading ScientificResearch: Students will be able to read professional mathematical reports, such as reports from the Center for Disease Control or elementary research articles. Students will prepare for this by reading background information provided to them by the instructor and by analyzing problems which are similar too, but simpler than the problem discussed in the professional report. Students will gain an appreciation for the way in which mathematics is used to solve these very real problems and at the same time recognize how an understanding of a simplified problem leads to an enhanced understanding of a more complicated one.

Communication: Students will strengthen their communication skills in mathematics by working collaboratively, sharing ideas on discussion boards, and submitting written descriptions and justifications of their mathematical models and solutions. Participant’s written reports will incorporate mathematics into language intended for non-mathematical audiences, thereby developing students’ skills in articulating the connections between a mathematical study and its concrete applications. The emphasis on articulating mathematical explanations in both verbal and written forms will address the NCTM process standard of Communicating Mathematics.

Course Outline

Workshop: (face-to-face session) linear functions (review), exponential functions, logarithms, exponential growth and decay, modeling with linear and exponential functions, problem solving

Project I: Measuring Temperature & Newton’s Law of Cooling

Project II: Using Body Temperature to Estimate Time Since Death

Description: In these projects students will examine methods for the derivation of number scales for measuring temperature, construct their own thermometer and determine its temperature scale. Students will learn how Isaac Newton first measured very high temperatures (temperatures which exceeded the boiling temperature of his thermometer) and how Newton's Law of Cooling came out of this study. Participants will see Newton's original paper on cooling (in Latin and in English), use his ideas to predict the temperatures of simple cooling objects, and study methods for adapting Newton’s Law to more complicated objects. In particular, students will examine how these mathematical ideas are used to estimate time of death of a human body and use standard coroners’ algorithms to solve a murder mystery.

Mathematical Tools: exponential functions, logarithms, modeling, graphing, linear functions, proportional reasoning, conversions between scales, basic statistics
Project III: Containing Infectious Diseases
Description: Due to concerns about bioterrorism that have resulted from the September 11 attack, the U.S. government commissioned the CDC (Center for Disease Control) to come up with recommendations for how to react in the event of a smallpox attack. With the goal of containing the outbreak to prevent an epidemic, the CDC came up with a plan that involves quarantining infected individuals and vaccinating a percentage of healthy individuals (since the vaccination itself is dangerous it is not recommended to vaccinate everyone). The CDC used mathematics to solve this problem and the goal of this project is to understand it. Students will begin by analyzing the simpler (fictitious) disease, schoolpox, and various strategies for containing schoolpox outbreaks.After the simplified problem is understood, students will read the actual CDC smallpox report. During this project, students will study strategies (based on mathematics) employed to contain cholera outbreaks in the 1800’s, strategies which virtually eliminated cholera epidemics and saved literally tens of thousands of lives.
Mathematical Tools: modeling and simulation, iterative processes, basic statistics, interpretation of graphs, linear functions

Project IV: Can You Save Tom's Marriage?

Description: Tom wants to maximize the amount of time his VCR records a movie in high quality mode versus low quality mode. In high quality mode, he can only record a two hour movie on one tape, but most movies are longer than that, so he tries to program his VCR to record first in high quality mode and then, at some point, switch to low quality mode. But he always messes up and, as a result, the tape finishes before the movie does. His wife is very frustrated with him!

Mathematical Tools: linear functions, trade-off problems

Project V: Building Up Savings and Debt

Description: In this project, we will build on what we learned about compound interest during the January Workshop. We will learn how to compute the value of an annuity when we make regular deposits into the annuity. We'll see how surprisingly quickly an annuity can grow and we'll see the advantages (mathematically) in starting an annuity early. We will also apply these ideas to credit card debt to see how debt grows astronomically if part of the principal is carried over from month to month. And we will apply this mathematics to the national debt.

Mathematical Tools: exponential functions, geometric series

Project VI: Childhood Growth Charts
Description: In this project students will study the development of growth charts used by pediatricians and how they are utilized to predict health risks in children. The charts produced by the CDC) and the report containing the data which led to the development of the charts will be studied. Students will use data from the CDC to construct their own growth charts, analyze data from children in their own classrooms, and learn how to use their charts to determine the risk of obesity for real children.
Mathematical Tools: basic arithmetic, metric conversions,percentages, statistics,graphing and interpreting graphs, error analysis

Course Structure: The course will be structured around a series of projects problems. For each project, participants will complete a pattern of activities that is similar to what follows:

Read the problem and essential background information (using original documentation whenever possible).
Identify the mathematical aspects of the problem in order to develop an appropriate mathematical model.
Analyze the mathematical model and connect the results to the problem at hand.
Read further documentation of the problem, using the model and analysis to synthesize information in a research document describing a more complicated situation (again, using original documentation whenever possible).
Provide written reports which include mathematical explanations and justifications for solutions and recommendations for the problems.

Schedule:

This course, offered as a “blended” distance education course, will have two components.

Face-to-Face Component:

Participants will meet on campus for a 2-day intensive weekend in which they have 16 hours of class time. The course will begin with a study of exponential and logarithmic functions to provide the necessary background for the first project problem.

During the face-to-face component of the course, instruction will be offered by an instructional team consisting of two faculty, two graduate students and one master teacher. Each member of the instructional team will serve as the “advisor-coach” for approximately 6-7 participants. The advisor will check in with their participants regularly and offer comments about the homework solutions. In addition, participants will be able to meet individually with their advisor at various times during the day to discuss assignments, review material that has been presented and to discuss the successes or difficulties the participant is having in the course. In addition, advisors will meet with participants in the evenings to offer help with assignments.

On-Line Component: The remainder of the material during the semester will be communicated using Blackboard. One faculty member and one graduate student will continue to oversee the course during this time. Homework will be collected at the end of each project and given feedback. Participants and instructors will actively communicate via email, Blackboard, telephone, face-to-face meetings, and Breeze (an internet conferencing system) meetings with one another about the course content, assigned projects and written reports.

End of Course Requirements: At the conclusion of the course, participants will be required to complete the following

Teaching and Learning Project. A description of a project in which participants are asked to share what they have learned in their own classrooms and reflect on the experience.

Reflection Assignment. A description of an assignment through which participants reflect on the nature of the learning that has taken place.

Final Project. A description of the expectations of the final project; a revision and extension of a project previously submitted.

Grading: Grades will be based on course projects submitted throughout the semester, class participation, the Teaching & Learning project, the Reflection, and the revised Final Project.

A sample rubric for grading the homework and end-of-course problem set:

A project submitted with no mathematics or incorrect mathematics or little to do with problem. F
Project contains essentially correct mathematics but with little explanation of reasoning. B-/C+
Mathematics in the project is essentially correct with some logical explanations. B/B-
Project contains correct mathematics with coherent logical explanations. A/A-/B+
Project contains correct mathematics with well-written logical explanations and motivation and proposed alternate solutions when appropriate. A+/A/A-