Using Real-World Data

in the Teaching and Learning of Mathematics

Florence S. Gordon and Sheldon P. Gordon

New York Institute of Technology SUNY at Farmingdale

fgordon@faculty.nyit.edu gordonsp@farmingdale.edu

The use of real-world data to motivate the teaching and learning of mathematics is one of the strong threads that runs through much of the reform movement in mathematics, but has received relatively little attention compared to themes such as the use of technology or new learning paradigms. Those who advocate the use of real-world data find that it simulta
neously motivates both the mathematical development and the students' appreciation for the value of mathematics. This use of data is beginning to permeate all levels of the beginning undergraduate curriculum, in courses ranging from introductory algebra [1] through college algebra and precalculus [2] and on to calculus [3,4] and differential equations [5]. It is also a major theme in the precollegiate mathematics curriculum, as strongly advocated by the NCTM Standards. Most textbooks today introduce some real-world data as motivation for the mathematical topic about to be developed. Many of the "reform" projects go further to use the real-world data as a source of problems to study, anticipating what most people in the quantitative disciplines actually do.

Unfortunately, many members of the mathematics community are not familiar with these ideas and often see them primarily as statistical tools. Consequently, even though such ideas are beginning to appear in new editions of standard texts and not just in innovative project materials, many faculty choose to avoid data analysis ideas and problems as unnecessary or optional topics. We hope to demonstrate that these ideas can and should assume a far more central role in the curriculum, one that drives the mathematics and convinces students that the mathematics is connected to the world around them.

At the developmental level, data taken from the daily newspaper can provide sources for interesting and motivating problems that make the mathematics come alive. For example: During the 1998 regular season, the Yankees won 114 of 162 games. What was their winning percentage? During the playoffs, the Yankees won 11 of 13 games. Was their winning percentage in the postseason higher or lower than that during the regular season? What was the Yankees' winning percentage over all games, both regular and postseason that year?

Alternatively, one can introduce in-class activities to collect data for analysis. For instance, students can be asked to perform a variety of measurements on their physical characteristics, such as height, arm span, neck size, length of foot, and so forth. Students can then be asked to examine the data and look for relationships between the different quantities at a level prior even to the introduction of formal variables and equations. In particular, it turns out that arm span is typically about two inches less than height. Once students begin to see that such relationships between quantities are all about them, it is easier and more effective to introduce the mathematical notations that allow them to work with the relationships.

In traditional courses at the college algebra or precalculus level, we repeatedly ask students routine questions such as: Find the equation of the line through P(1,3) and Q(4,12). As a far more realistic application of linear functions, we could ask instead: In 1986, 12,099 men and 1048 women died of AIDS. In 1992, 39,160 men and 6312 women died of AIDS. Assuming (incorrectly) that the growth in AIDS for both groups is linear, (a) Find the equations of the linear functions for the number of deaths from AIDS among men and women as functions of time. (b) What is the practical significance of the slope in each case? (c) Since the slope of the function for women is larger than that for men, determine when the same number of men and women will die from AIDS if the trends were linear and continued indefinitely. (d) Use the fact that, in 1990, 36,770 men and 4883 women died from AIDS to explain why it is incorrect to assume that the growth trends are linear. This is certainly harder than the standard problem to find the line through P(1,3) and Q(4,12), but is more interesting, more realistic, more motivating, and more likely to get the students to focus on doing it rather than ignoring the long list of indistinguishable problems we traditionally give. Also, once we have such a set of data, we can return to this context when studying exponential functions and power functions to ask the comparable questions regarding those mathematical models.

In their other courses and on the job, what students actually encounter are sets of data that fall into a roughly linear (or some other) pattern and they have to find the equation of the line (or other function) that comes closest to all of the points, usually in the least-squares sense. The resulting problems are far more challenging than the simple abstract one we cited; the slope and intercept of a regression line certainly are not one-digit positive integers when you work with real-life data. Yet the fact that the mathematics arises in practical contexts that the students see as something they will do (and likely have already done in other courses) motivates them to rise to the occasion. This is all the more true if the students are encouraged to obtain their own data on topics of interest to them, to create the best-fit line, to raise questions of a predictive nature regarding the situation, and to answer those questions using the linear model they have created.

Further, examples such as the growth of populations, the growth in the stock market, the progression of natural phenomena over the course of a year, or other nonlinear processes provide the ability to motivate, compare and contrast different families of monotonic functions such as linear, exponential, power, polynomial, or trig functions. They also present the opportunity to raise predictive questions in the context of the situation being studied that can be answered by applying the functional models that have been created to fit the data. More importantly, they necessitate the development of judgment on the part of the students -- they cannot simply stick numbers into formulas without regard to the significance of the answer.

The shape of the Gateway Arch in St. Louis presents a lovely illustration of the difference in behavior between quadratic and higher degree polynomials. At first, most students think that the arch is a parabola, but the best-fitting quadratic (see Figure 1)does not truly capture the pattern because the arch is flatter near the top. A quartic polynomial is an exceptionally good match to the arch because the quartic is flatter about its vertex (see Figure 2), a dramatic lesson that connects solidly with the students. (Incidentally, the arch is actually an inverted catenary.)

Similarly, the number of hours of daylight as a function of the day of the year and historical temperature readings over the course of a year are highly effective examples to motivate and emphasize the significance of the trig functions as providing the mathematical models for periodic phenomena. One must deal with functions that are considerably more complicated than the usual; the function that models the number of hours of daylight in New York is H(t) = 12 + 3.1 sin(2p(t - 80)/365). (Data from which such a function can be constructed is in the

Farmer's Almanac.) In context, the concepts of amplitude, period, frequency, vertical shift, and phase shift make far more sense to students in such a function than they do in y = 2 sin 3x.

When studying polynomials, consider the set of data on the spread of AIDS:

Year	1983	1984	1985	1986	1987	1988	1989	1990
Number of cases	4,589	10,750	22,399	41,256	69,592	104,644	146,574	193,878
1991	1992	1993	1994
251,638	326,648	399,613	457,280

From the intuition that has been developed prior to this point, or from an examination of the scatterplot of these data, most students suspect that the best mathematical model would be an exponential function. As we see in Figure 3, however, this is not a particularly good fit; if anything, the growth of AIDS is considerably slower than the exponential model predicts. On the other hand, the best cubic function to fit this data is shown in Figure 4, where we see that it appears to be an almost perfect fit. Of course, this leads to a number of solid mathematical questions; for instance, knowing the growth pattern for a cubic, what portion of the curve are we seeing? One can answer this by zooming out on the graph or examining the sign of the leading coefficient of the cubic, A(t) = -68.8t3 + 4781.7t2 - 2666.2t + 7094.5. Also, this gives another opportunity to discuss the limitations of extrapolation.

In a calculus setting, one can then ask questions regarding the behavior of this function. For instance, where do the maximum and minimum occur, and what significance, if any, do they have? Or, more significantly, where does the cubic have its point of inflection where the number of cases of AIDS is growing most rapidly.

Or, in a calculus course where models based on differential equations is a major theme, the use of real data can be an essential component in terms of estimating parameters for a solution. For instance, the population of the U.S. can serve as the basis for a lovely student exercise to determine values for the coefficients in a logistic model. As the students are learning the ideas of the calculus, they are simultaneously replicating the kind of work that a professional scientist would do to analyze the data.

In a differential equations course, either the instructor (on a classroom demonstration basis) or the students (in a lab setting) can conduct some simple data collection experiments using tools such as the CBL. For instance, the following data was collected in an experiment on the rate at which the temperature (in °C) drops as a function of time (in seconds) when a heated object is immersed in cold water.

Time	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Temp	42.3	36.03	30.85	26.77	23.58	20.93	18.79	17.08	15.82	14.77	13.82	13.11	12.51	11.91	11.54	11.17

17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
10.67	10.42	10.17	9.9	9.8	9.67	9.54	9.42	9.29	9.16	9.16	9.04	8.91	8.83	8.78	8.78	8.78	8.78	8.66	8.66

At the precalculus level, we could simply attempt to fit a decaying exponential function to the data, after first transforming the data with a vertical shift to reflect the fact that the temperature decays to the temperature of the water, not 0. At a more sophisticated level, we can look at a plot of the differences in temperature, T (as an approximation to the derivative T'), versus the temperature readings, as shown in Figure 5. We immediately observe that the data falls into a remarkably linear pattern. The equa tion of best-fit line is

T = -0.1851 T + 1.6740.

The corresponding correlation coefficient is r = -0.9976, which indicates an almost perfect linear fit. If we now factor out the coefficient -0.1851, we get

T = -0.1851 (T - 9.04).

Notice that the value 9.04 is actually fairly close to the temperature of the cold water, so that this suggests that the change in temperature is proportional to the difference between the temperature of the object and the temperature of the medium. This, of course, is just a statement of the discrete form of Newton's Law of Cooling. In the limit as the length of the time interval goes to zero, we obtain the continuous formulation, which is one of the standard models studied in differential equations courses. But, by looking at the actual data, we can deduce the principal rather than stating it as the starting point for solving a differential equation. Certainly, that far better mirrors the scientific method than a revelatory approach does.

Similarly, at the differential equations level, we can introduce a set of data such as the population of the United States over time, which can be modeled by a logistic function. Working with and analyzing such a set of data, however, provides a lovely context in which to introduce some fairly sophisticated methods for estimating the coefficients of the logistic differential equation. At the same time, it provides the motivation to see why it is important to study these methods.

At any curriculum level, there are several additional advantages and opportunities for emphasizing real-world data. One of the most significant is that it allows us to broaden our assessment techniques while simultaneously letting the students make the mathematics their own. This is accomplished by having each student (or small group of students) conduct one or more individual studies in which each student collects or finds a set of data on some topic of interest to himself or herself. The data might come from another textbook, from an experiment conducted in a lab course, off the Web, or from some general reference source such as the Statistical Abstracts of the U.S. At an algebra level, the student would have to plot the points, estimate the best-fit line by eye, and use it to answer questions that the student raises about the process being studied. The results would then be submitted in a formal project report. At a more sophisticated level, when different families of functions are being studied, students can be asked to conduct comparable studies where they find the best function from each family and answer the questions they raise using each of the models they created. Similarly, at the calculus level, comparable studies can be performed to determine the behavior of the functions created. This is especially appropriate in applied calculus courses for business and other non-science majors.

These types of projects are key to getting the students to realize that the mathematics they are learning is their own, not ours. It is not the applications that we as faculty are interested in, but rather the topics that the students find interesting that they are studying; as a result, they see that the mathematics they are learning is definitely something that they can and will apply in their personal and professional lives.

In each of these activities, the students are learning more mathematics at a more sophisticated level, and simultaneously gaining a deeper appreciation for the value of what they are learning. As mathematics educators, what more can we ask for?

References

1. Patrick, A. Richard, Arlene Kleinstein, et al, (The Consortium for Foundation Mathemat
ics), Mathematics in Action, Addison-Wesley, 1998.

2. Gordon, Sheldon P., Florence S. Gordon, et al, Functioning in the Real World: A PreCalculus Experience, Addison-Wesley, 1996.

3. Hughes-Hallett, Deborah, Andrew Gleason, Patti Fraser Lock, et al, Applied Calculus, John Wiley & Sons, 1999.

4. Smith, David A. and Lang C. Moore, Calculus: Modeling and Application, Houghton Mifflin, 1996.

5. Borrelli, Robert and Courtney Coleman, Differential Equations: A Modeling Approach, John Wiley & Sons, 1998.