David B. Sher
Nassau Community College
Garden City NY 11530
I intend to present a variety of interactive web pages designed to enhance my precalculus course. The material I will present was developed under NSF support from grant # DUE-955-4949. There are three tested web pages:
These web pages can be accessed through http://www.polar.sunynassau.edu/~sherd/sware/sware.htm . A fourth web page is under development that will generate precalculus problems from user input.
Our project combines two approaches for teaching the precalculus and calculus sequence.
The reform approach to teaching calculus and precalculus was developed at Harvard and is increasingly popular across the country. At Nassau Community College we use the precalculus and calculus materials developed for this project. There are two themes to this approach:
a) Principle of Archimedes
Formal definitions and procedures evolve from the investigation of practical problems. Thus we use real data in many of our problems.
b) Rule of 4
Every topic should be presented geometrically, numerically, algebraically and descriptively. Another way to state this is: mathematics should be taught using multimodal techniques.
The Wilkinson and Sher developed a reform calculus by applying Treisman's work to the community college environment. Their approach encourages students to work together by forcing them to share computers in a computerized lab class. The students use the computers to develop projects using mathematical software. These projects are collected in a portfolio that they can use as a record of their accomplishment. A tangible record of accomplishment and the positive experience of working as a group encourages enrollment in higher level mathematics courses. Their approach resulted in a sixfold increase in enrollment in higher level mathematics courses.
Providing students with real world data shows them that mathematical modeling applies to the real world. They intuitively understand the importance of predicting income, weather or population. The students in Dr. Sher's precalculus section in the 1995 and 1996 semesters produced models for datasets on income, weather and population.
Students work on the project together in groups of three, a technique borrowed from the Wilkinson-Sher approach. From an immigration database, retrieved from the internet, we selected a different data set for each group of students. Thus, weaker groups can not copy the results from the stronger ones, though they can mimic technique.
A web page helps the students model and present this data. This web page has the algebraic models atop, followed by numerical tables and a graph of the models on the bottom. The groups see graphical, algebraic, numerical, and textual representations of their work combined in their project reports.
We introduce the web page to the students with a competition. While each group has its data and models password protected, there is a generic set of data available to anyone. This page is http://www.polar.sunynassau.edu/~sherd/mat111/projects/ . We break the students into three groups and have them compete to find the best linear model for this generic data set. The attendees participated in a demonstration of such a competition.
The internet facilitates the competition by showing each participant the models proposed by the other groups. Each time a group updates their model the latest models of all three groups are shown. The same feature also allows several students in the same group to see what the other students are doing with their models. Thus, students working at different sites can still cooperate.
The graphing calculator web page allows users to enter up to five functions. It presents the data as a graph and table simultaneously. The user selects one of these five colors for each function: red, green, blue, magenta, and cyan. The red function will be represented by a red curve on the graph and by red numbers on the table. This makes it easy to get the correspondence between the graph and the table. The calculator can be accessed at http://www.polar.sunynassau.edu/~sherd/sware/graph_calculator.htm .
Some immediate advantages of using the web page rather than graphing calculators are:
|Using a web page calculator promotes facility using the world
||The color presentations allow us to visually connect the graph
|| Students do not need to purchase graphing calculators.
||The large screens allow many students to observe the results
Another advantage is the ability to create links to specialized calculators in other web pages. http://www.polar.sunynassau.edu/~sherd/sware/harvprob.htm is a lecture notes page which consisted of references to book problems and additional problems. Each topic has a link to a graphing calculator. The sections on trigonometry have links to a specialized version of the calculator for trigonometry. This version had the units for the dependent variable (x) in terms of ¼.
Another way such a page could be used is to produce graphs for class materials. Dr. Sher's final exam last semester contained a graph produced on the calculator. Creating such a graph is a simple matter of saving the graph file using the right button on the web page viewer. Then you need only insert the graph into your word processor document.
There are two especially interactive versions of the calculator, a zooming version and a pointing version. The zooming version allows one to zoom in on a point on a graph just by clicking on it. Intersection positions are approximated by repeated clicks. The pointing version puts dots on the graph by clicking. When the graph is rescaled the points are moved as appropriate.
Dr. Sher discussed several in class group activities done with the calculator such as finding several lines through the point (3,4) or finding several quadratics through the point (3,4).
Forcing the students to find practical applications for precalculus topics can help convince them of the worth of that material. A typical essay would require the students to discuss how their knowledge of linear functions can assist them when moving. They discuss how the price is a linear function of the distance they are moving. The time is also a linear function of the distance, etc. Writing such an essay forces them to think about precalculus topics and makes them realize that the topics we teach them have applications in the real world.
http://www.polar.sunynassau.edu/~sherd/essay-tool.htm helps the students write such essays. The typical student has trouble remembering to make the connection between the precalculus topic and the real world problem explicit. For example, they would mention the time to load the truck but fail to show how it is a linear function. Or they could discuss the fact that the intercept represents the fixed part of the function but fail to mention its relevance to moving. The tool asks them to discuss the relevance of each idea to both topics (in the example: linear functions and moving). It also formats, spell checks, and emails for the students.
After perfecting the tools already discussed and making them easier to use, a new set of web pages will be constructed. These web pages will construct precalculus problems using graphical input.
A professor or student would click on a point. The page would respond by showing which points are an integral slope away. When the user clicked on one of these points the formula for the line specified by the two points would be presented and also the coordinates of the two points. If the user specified another line by clicking on two points, the formula for the second line and the coordinates of the intersection point would be presented to the user. The tool would present a variety of problems that use the points the user clicked. The answers to these problems would be available for checking student work.
Summary for Program Booklet
Dr. Sher will present interactive web pages designed to enhance a precalculus course. This material was developed under N.S.F. grant # DUE-955-4949. There are three tested web pages:
. A modeling project page.
. A graphing calculator simulator.
. An essay writing assistant.
These web pages can be accessed through http://www.polar.sunynassau.edu/~sherd/sware/sware.htm .
Dr. Sher has been teaching mathematics and computer science at the college level for 10 years. He has been the principal investigator on 3 grants that focused on teaching precalculus, two of them from the National Science Foundation. He has published journal papers on mathematics education, decision theory, and image analysis.
If a computerized venue with web access can be arranged I would like the audience to participate by browsing the web pages and working together to solve problems with them. If such is available a 2 hour session may be better since participation can take more time. Depending on what technology is available I intend to try to maximize audience participation and group effort on their part.
The best venue would be a computer lab, where workstations with access to the world wide web (Netscape or Microsoft Internet Explorer) were available. However a single such workstation with a projector can also be used for a presentation on this topic. Even if both of these prove impossible, I will have transparencies with which I can make a presentation. Note that for religious reasons I can not make a presentation after 2 PM on Friday Nov. 14.