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Deborah Hughes Hallett
Math Department, University of Arizona
Tucson, AZ 85721
520-621-6886
HYPERLINK mailto:dhh@math.arizona.edu dhh@math.arizona.edu
Eric Connally
Science Center 224, Wellesley College
Wellesley, MA 02481
781-283-3081
HYPERLINK mailto:econnally@wellesley.edu econnally@wellesley.edu
Overview and Philosophy
Information, data, and decisions (IDD) is an Internet-based course designed to enable students to tap into the enormous quantity of data on the web, and to learn the mathematics necessary to handle that data critically. The course has been used for two very different audiences: incoming Wellesley undergraduates and mid-career public servants from developing countries at Harvards Kennedy School of Government. However, the philosophy and structure of both courses were the same. After introducing the tools - a browser and a spreadsheet - we gave the students access to interesting data and asked them to make judgements or decisions based on the data. The mathematics was introduced in the context of analyzing the data. This had the effect of engaging students in the mathematics in a way that they would not have been otherwise. Since the data were interesting in their own right, the students became interested in what the mathematics could tell them.
Description of Implementation of the Course
In the summer of 1998, about 65 mid-career students at Harvard and about 45 Wellesley students took two different versions of the course.
The outline of the Harvard course is as follows:
Introduction to Netscape: Students learn how to move around on the web and how to perform a search. The assignment is a treasure hunt which asks students to find several pieces of useful information.
Using a Pre-Fabricated Spreadsheet on the Web: A statistician in Vienna has loaded the data from the previous Austrian census into a spreadsheet. This allows students to analyze the effects of various policies (such as the retirement age or the immigration laws) on the solvency of the Austrian pension system.
Constructing Your Own Spreadsheet: Students build a spreadsheet to illustrate the gloomy predictions of Malthus: that exponential population growth eventually outstrips the linear growth of food production. Students then analyze the effect of varying the parameters in the model.
Downloading Data from the Web into a Spreadsheet: Census, CIA, United Nations, Worldbank
Projects and Presentations by Students
The Wellesley version of the course is taught from the same materials as the Harvard course but with much greater emphasis on scientific applications. The outline of the course is similar, with the exception that the unit on the Austrian pension system has been replaced by a unit on monitoring water flow in the Charles River, using data downloaded from the U. S. Geological Survey.
Both the Harvard and Wellesley programs have students with a wide range of mathematical skills. Most of the public servants in the Harvard program do not have a strong math background, although a few were trained as engineers. Many students in Wellesleys program lack math skills, but not all. Consequently, both programs need materials that are accessible to weak students, yet challenging for strong students. The web and spreadsheet materials we used are very satisfactory in this regard. The materials involve only basic mathematics (percentages, linear growth, exponential growth) yet some of the questions asked are quite deep. Thus, weak students can review a variety of skills (arithmetic, graphing, basic algebra) at the same time as strong students explore mathematical modeling. The assignments help students gain a deeper understanding of topics such as units and dimension, linear growth, exponential growth, logistic growth, rate of change and accumulated change. The exercises require students to construct models to describe realistic situations and, in so doing, to think about the mathematical relations exhibited by various real-world phenomena.
Success of the Program: The Mathematics Learned
The web gave the students immediate access to authentic data and a great sense of power. Working with spreadsheets forces students to make the connections between different quantities. It is impossible to get the cell references right without understanding the relationships between different quantities and impossible to enter the formulas correctly without knowing algebra. To make charts correctly, the student must understand the relationship between numerical data and its graphical representation. Thus, using spreadsheets provides a fresh way of reviewing algebra, percentages, and graphs. The student reports show that this review was well received. In addition, slower students were able to concentrate on the arithmetic and algebra without hindering the faster students, who were free to explore the spreadsheet models at a deeper level.
In addition to strengthening the basics, IDD enabled students to explore some significant mathematics. As an illustration, two examples are described below.
In the lab on the Charles River water flow, students investigated rates of change and accumulated change thereby receiving an introduction to calculus. This lab was also useful in another important, though less deep, way. It ensured students thought about how to convert units both linear and cubic units, as well as the units of rate of change.
In making their presentations, some students went to the UN site and got data on the Human Development Index (HDI) of several countries. The HDI is a weighted average of several other quantities including per capita income, GNP, life expectancy, and adult literacy (all indexed so that they are percentages). One groups presentation, which showed the regression of HDI against GNP, set off a firestorm of argument in the class. Students felt that it was not legitimate to fit a regression line to this data since HDI contains GNP as one of its components. This was an extraordinarily interesting and thought-provoking discussion, which caused students to do a great deal of independent mathematical investigation.
Success of the Program: Student Responses
At both Harvard and Wellesley, student response to IDD was extremely positive. Marian Stas, a major in the Romanian Army who taught for 14 years at the Technical Academy in Bucharest, came to Harvard particularly interested in seeing how mathematics related to management and technology and reported that the summer program at [the Kennedy School] fulfilled my expectations in this respect. He went on to elaborate:
Of a particular relevance to myself was the creative approach to the course, undertaken by the course coordinator, Deborah Hughes Hallett which was possible . . . due to the way in which this course was designed. As members of the advanced group in the Summer School Program, some students had the opportunity to literally be part of the teaching process, presenting specific aspects of the Excel software package, as well as topics in mathematics . . . As a direct result of this approach, the level of confidence and participation of all students significantly increased throughout the whole course.
I am pleased to strongly recommend the IDD-100 Course: Information, Data and Decisions taught at John F. Kennedy School of Government, for recognition as a successful innovative program in using technology in the teaching of Mathematics. I also believe that such recognition will enhance further development of this very rich teaching area in other various programs at Harvard University.
Three weeks into the fall semester, many of the Wellesley students report that they have already used what they learned in the summer program in both biology and economics courses. One describes what the course taught her:
Using spreadsheets made me look more closely at the process involved in an equation. I had to pay attention to the meaning/role of each variable and to decide how the values should be organized on the spreadsheet. Ive already created a spreadsheet document to keep track of the distribution requirements and, soon major requirements, which I have or still need to acquire. I also plan to keep a record of my finances using Excel.
Another Wellesley student wrote:
The whole [summer program] experience was wonderful and I did not even know how to use Microsoft Excel or how to do spreadsheets before the program. It helped me tremendously. I will now be using spreadsheets in the research I am doing in the Minority Mentorship Program.
Significance, Transferability and Presentations of IDD
Janice Rand-Vaughn, Director of the Kennedy School Summer Program, concludes: Evaluations of the course were very positive. Not only did those students with a weak foundation gain confidence and skills, those with more experience created an extremely cooperative learning environment where students could learn a great deal from each other. This was a big innovation for us. We were teaching a web-based math curriculum in the classroom. We were teaching math invisibly. Those students (and there are many) who had avoided math for most of their life, were introduced to quantitative skills though a fun and non-math focussed medium: the Internet. And they learned.
The web and spreadsheets are tools of great importance to science and industry. It is urgent that we learn how to teach mathematics in a way that uses these tools.
Acknowledgements
The IDD materials were developed by Eric Connally with funding from the Kennedy School of Government in response to a proposal from Deb Hughes Hallett. The materials were based on ideas generated by Deb Hughes Hallett, Eric Connally, Ralph Teixeira, Rajini Jesudason, and Graeme Bird.
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