Using Technology and Real World Connections to Teach Secondary Mathematics Concepts

by Hollylynne Stohl Drier, Kara M. Dawson, and Joe Garofalo

The National Council of Teachers of Mathematics (NCTM, 1989) advocates that problem solving, reasoning, communication, and connections be woven throughout K-12 mathematics instruction. However, real world connections are often missing from mathematics teaching. Technology facilitates applications of school mathematics to real world situations by providing access to worthwhile data and tools that alleviate the computational constraints often involved in the analysis of real world "messy" data. In this article we illustrate a few activities that help students develop conceptual understanding through the use of real world data and electronic simulations of real world events.

Using real data to teach mathematical concepts

Technology makes a vast amount of information readily available to students and teachers. Data found on the Internet can be quickly transported to a spreadsheet, and subsequently downloaded to a graphing calculator. Simplifying data gathering allows more time for emphasis on data analysis, interpretation, and conceptual development. For example, students can use the Internet to gather information from the National Center for Health Statistics (http://www.cdc.gov/nchswww/) on the number of live births in the United States, numerically and graphically analyze the data in a spreadsheet, and connect their analysis to events in US history. Figure 1 shows a spreadsheet containing birth rate data, the absolute and relative yearly increases and decreases, a graph of the data, as well as interdisciplinary questions related to US history. This data can help students understand "percent increase" and "percent decrease" by giving the numerators and denominators contextual meaning and connecting the calculations with historical interpretations of the data. In addition, algebra students can use the absolute differences from 1951 through 1955 to find a local slope for these data points. The typical definition for slope as "change in y over change in x" now has contextual meaning as "birth rates increase approximately 7 per 1000 women in each year."

Figure 1: Birth rate data from

Other current and archived data is readily available on the Internet at sites such as the National Oceanic and Atmospheric Administration (http://www.noaa.gov) and the National Geophysical Data Center (http://www.ngdc.noaa.gov). Much of the data at these two sites can facilitate understanding of mathematical concepts. Real world phenomenon such as the tides, planetary orbits, and average monthly temperatures provide contextual situations for studying oscillating functions. Figure 2 shows temperature data for Washington DC, Verhoyansk, Russia, and Buenos Aires in both tabular and graphical form. From the graph students can see that the average monthly temperatures for all three cities oscillate with a period of one year. They should also notice that it is colder in Verhoyansk than in Washington, and that the temperatures in Washington and Buenos Aires are "out of phase."

Figure 2: Average monthly temperatures for Washington DC, Verhoyansk, and Buenos Aires

Trigonometry students can incrementally fit a sine curve to each city's data by estimating the amplitude, vertical shift, period, and horizontal shift (figure 3). Students and teachers can then relate the coefficients in the general form of the sine equation (y = AsinB(x+C)+D) to the geographical locations of the cities. If this data is entered into a graphing calculator, students can also calculate a sine regression and compare the equation generated by the calculator (figure 3f) with the one determined incrementally from the data and graphs.

3a: original data	3b: data and graph of y = 21.5sinx
3c: data and graph of y = 21.5sinx + 55.5	3d: data and graph of y = 21.5sin(2pi/12)x + 55.5
3e: data and graph of y = 21.5sin(2pi/12)(x-1) + 55.5	3f: sine regression

Figure 3: Incrementally curve fitting the Washington DC data

Using simulations to develop conceptual understanding

Another very powerful feature of technology is the ability to electronically simulate real world events. Graphing calculators, spreadsheets, and the Internet offer students and teachers the capability to create and use simulations. Such simulations allow students to visualize and explore important mathematical concepts as well as real world and interdisciplinary connections. One such simulation on a graphing calculator can be used to explore exponential functions by applying them to investment strategies and the study of economics. Students can explore the effects of investing $1,000 over the next 20 years in money market, bond, or stock mutual funds. There are varying degrees of risk connected with each of these types of funds. In order to weigh risk versus return, students can estimate the average return for each of these investments over the next 20 years, assuming monthly compounding and growth rates of 5%, 8% and 16% respectively (Figure 4).

Figure 4: Growth of $1000 over 20 years at 5%, 8%, and 16%

Although this is an oversimplification of actual investment returns, it is instructive nevertheless. Students should notice that after 20 years the difference in accumulations is dramatic at $2,700 (money market), $4,926 (bond) and $24,019 (stock). By using the parametric mode in a graphing calculator, the exponential functions defining the growth of money are no longer static since parametric functions allow students to watch their money "grow" as the number of years increase.

Physical events can also be simulated through technology. For example, the study of parametric equations and trigonometry can be connected with the study of projectile motion in physics. The spreadsheet in Figure 5 was created in Microsoft Excel to dynamically simulate projectile motion. Using the sliders, students can manipulate the initial velocity, angle of projection, and height from the ground to observe the effects on the subsequent path of an object.

Figure 5: Excel spreadsheet created to analyze projectile motion.
Available from www.teacherlink.org/content/math/interactive/

With the ability to manipulate time, students can animate the motion of the object and explore how long it takes for the object to reach its maximum altitude and when the object will hit the ground. By varying the other parameters, the students can explore how each effect the path of the object and connect these parameters with the coefficients in the sine and cosine equations. Other investigations could include maximizing or minimizing altitude and horizontal distance. Manipulating different parameters and visualizing the path of the object can make the mathematical equations used to describe projectile motion more relevant and meaningful. In this way, creating and using such a simulation allows teachers and students to explore mathematical and physical concepts in an open-ended environment and use higher order thinking skills to analyze relationships.

Summary

In the examples above, technology is not the focus of learning. Rather, it empowers teachers and students to explore mathematical concepts through the use of real world data and simulations of real world events. When technology is used in this way, interdisciplinary and real world connections become a natural and powerful way for students to make sense of mathematics (Drier, Dawson, and Garofalo, 1999).

The Curry Center for Technology and Teacher Education at the University of Virginia is currently funded to develop materials to help preservice and inservice secondary mathematics teachers incorporate appropriate uses of technology into their teaching. The examples presented in this article are excerpts from these materials that have been developed using the following guiding principles:

• Introduce technology in context of mathematical investigations
• Address worthwhile mathematics with appropriate pedagogy
• Take advantage of technology to go beyond what is capable with paper and pencil
• Apply mathematics topics to real world and interdisciplinary contexts
• Incorporate multiple representations of mathematical concepts.

Activities available at the Center's Mathematics Education web page (http://curry.edschool.virginia.edu/teacherlink/math) utilize a variety of technology tools including spreadsheets, graphing calculators, The Geometer's Sketchpad, MicroWorlds and a host of other mathematics software. We encourage you to visit our website and test our activities in your classroom. We also want to know how these activities worked with your students and what modifications you may have made. For more information about the Curry Center for Technology and Teacher Education, visit the Center's website http://curry.edschool.virginia.edu/teacherlink.

References

Drier, H. S., Dawson, K. M., & Garofalo, J. (1999). Not your typical math class. Educational Leadership 56(5), 21-25.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: author.

Hollylynne Stohl Drier is a doctoral student in mathematics education at the University of Virginia and a graduate fellow for the Curry Center for Technology and Teacher Education.

Kara Dawson is an assistant professor of instructional technology at the University of Florida in the department of Instruction and Curriculum.

Joe Garofalo is an associate professor of mathematics education at the University of Virginia and co-director of the Curry Center for Technology and Teacher Education.