Plotting and Analyzing: Graphing Calculators for Social Inquiry

 

Joe Garofalo, Clifford Bennett, and Cheryl Mason

The crux of social studies education has long been to promote civic competence. An effective citizen in a democratic society must possess a broad range of knowledge and an array of skills to engage in social inquiry. Indeed, as Martorella emphasizes, social inquiry involves the intertwining of a well-grounded body of knowledge with a repertoire of skills:

[Social inquiry is] a way to better organize and interrelate existing knowledge, as well as acquire new information. Students do not first learn facts, then engage in problem solving. They use knowledge already acquired and add new elements, such as facts, concepts, and generalizations, as they engage in [social inquiry].1

National Council for the Social Studies (NCSS) has identified ten thematic strands of knowledge in social studies, and three categories of skills essential for civic participation: acquiring new information, organizing and using information, and developing interpersonal relationships and social participation.2 Related organizations, such as the National Council on Economic Education (NCEE), the Center for Civic Education (CCE), and the Geography Education Standards Project (GESP) have also developed more specific knowledge and skills standards for social studies education.

Graphing calculators, most often used in mathematics classrooms to enhance students’ learning, can also be used in social studies classrooms to help students broaden their knowledge base and develop the skills needed for social inquiry. Graphing calculators have large screens (approximately 1.5 by 2.5 inches) and can plot data, graph functions, perform statistical tests, and make complex calculations quickly (see Figure 1). These features allow students to display and analyze data, make inferences, and simulate phenomena. Using examples from economics, geography, and civics, this article illustrates how graphing calculators can be used to address knowledge and skills promoted by NCSS and others.

 

Plotting Data

One of the features of graphing calculators useful in social studies education is plotting capability. What follows are two examples of how graphing calculators can be used to display data and facilitate the observation of patterns.

 

Data Plotting: Evaluating Multiple Perspectives

The scale of a graph can be manipulated to expose or conceal certain trends in data. Often this is done to persuade or mislead readers and influence public opinion and policy. Figure 2 contains calculator screenshots of three graphs of the number of new AIDS cases reported from 1981-1996, shown in three different viewing windows. The first graph (y ranging from 18,000 to 81,000) shows the growth and decline in the number of new AIDS cases, the second graph (y ranging from 250,000 to 1,000,000) hides the changes over the time period, and the third graph accentuates the changes (y ranging from 18,000 to 81,000 and x ranging from -40 to 200).

Students can easily enter real data in their graphing calculators and construct scatterplots. They can quickly change viewing windows for the plots, and examine what effects such manipulation has on graphical displays. Class discussions can address the various interpretations of the data supported by each of the displays, and the reasons why one would choose to display the data in each of these windows. Such experiences will make students aware that they must look carefully at the scales of graphs before judging the appropriateness of any argument being supported or refuted by graphically displayed data.

Activities like this actively engage students in developing two of NCSS’s essential skills—interpreting graphs and detecting bias in visual materials—and can be expanded to incorporate other essential skills, such as evaluating sources of information. Similar activities can be designed to address knowledge falling within NCSS’s Thematic Strand 2 Time, Continuity, and Change (e.g., patterns of historical change).

Furthermore, data graphing activities can be used to address knowledge and skills listed under CCE’s Grade 9-12 Standard III, which deals with principles of American democracy (e.g., political communication) and Standard V, which deals with the role of the citizen in American democracy (e.g., forms of political participation).3

 

Data Plotting: Analyzing Climate Patterns and Locations

The plotting capabilities of graphing calculators can also be used to help students examine data for patterns, and to relate their observations to social studies concepts.
For example, for many students it is easier to see patterns in the numerical data presented in Table 1 when it is displayed graphically.

These data can quickly be entered into a graphing calculator and plotted. The screenshots in Figure 3 show the temperature data plots for Washington, D.C., alone, Washington and Verhoyansk, and Washington and Buenos Aires, respectively. From these graphs, students can see how the average monthly temperatures for all three cities oscillate within one year. Students should be able to see that it is colder in Verhoyansk than in Washington, and that winter and summer in Washington and Buenos Aires are “out of phase.”

Students can be asked to relate these graphs to the locations and other geographical features of the cities involved (e.g., latitude and proximity to the ocean). Connecting data and geographical features can then be used as a springboard for discussions about relationships between the physical environment and a population’s culture (e.g., forms of occupation and recreation).

Activities such as these help students to practice several of NCSS’s essential skills, in that they are called upon to present visually information extracted from print, to synthesize information, and to draw inferences from visual material. Similar plotting and pattern exploration activities can be developed to address other knowledge included under NCSS’ Thematic Strand 3 People, Places and Environments (e.g., physical patterns), and Thematic Strand 9 Global Connections (e.g., environmental quality).

Such activities can also support aspects of GESP’s Grades 9-12 Standards 1 and 5, pertaining to the world in spatial terms (e.g., geographic representation), and the complexities of places and regions (e.g., climate).4

 

Analyzing Data

Besides plotting data, graphing calculators can also facilitate various types of data and information analysis. The following are three examples of how graphing calculators can be used to investigate issues important in social studies using statistical analysis, simulation, and recursive calculations, respectively.

 

Statistical Analysis: Interpreting Political Polls

Last year, the Charlottesville Daily Progress reported the results of a Mason-Dixon Political/Media Research telephone poll of 813 registered voters during the Virginia gubernatorial race. The pollsters reported “44% of those surveyed favoring Gilmore, 43% favoring candidate Beyer, and 13% undecided.” The newspaper stated that the results have “a margin of error no more than plus or minus 3.5 percentage points.”

Students using graphing calculators with statistical features can verify such statements. The first screenshot in Figure 4 shows the input for calculating a 95% confidence interval for the true population proportion when 813 voters are polled and 358 express a preference for Gilmore using a one proportion z-test. The second screenshot displays the results. Notice that the estimated population proportion favoring Gilmore is 44%, and that there is a 95% chance that the true proportion is within the interval 40.6% and 47.4%, or 44±3.4.

Students can also use graphing calculators to explore the effect that sample size has on the confidence intervals, and to determine reasonable sample sizes for surveys. Table 2 shows calculated confidence intervals associated with different sample sizes. Students can see that there is a point of diminishing returns. Furthermore, they can discuss whether or not ±3 is an acceptable margin of error for their purposes, and also whether or not decreasing the margin from ±3 to ±1 is worth the time and expense of increasing the sample size from 1,000 to 10,000.

This activity allows students to practice at least two of NCSS’s essential skills: predict likely outcomes based on factual information and interpret what is read by drawing inferences. Related activities can be developed to include knowledge within NCSS’s Thematic Strand 6 Power, Authority, and Governance (e.g., political structures and political processes).

The content of this activity directly addresses CCE’s Standard III, which calls for students to be able to understand the roles of political parties, campaigns, and elections. Related activities can meet CCE’s Grade 9-12 Standard II, which is concerned with the American political system (e.g., constitutional government), and Standard V, which addresses the role of the citizen in American democracy (e.g., civic responsibilities).5

 

Simulation: Exploring Interest Rates and Investments

Graphing calculators can also be used to simulate various economic phenomena relevant to social studies. Students studying interest rates and investment can be presented with situations such as the following:

Suppose you have $1,000 to invest for a period of 20 years and are trying to decide whether to put your money in market, bond, or stock mutual funds. Over the past 10 years, money market funds have had average returns of 5% per year, bond funds have had average returns around 8% per year, and stock funds have had average returns of 16% per year. Estimate and calculate the return for each of these investments, assuming similar growth rates over the next 20 years.

Students can use their graphing calculators to simulate graphically the growth of the $1,000 at each rate of interest simultaneously. The screenshots in Figure 5 show the growth after 240 months. We have found that most students, and teachers as well, are surprised by these dramatic differences after 20 years. Very few come close to estimating the amounts $2,700, $4,926, and $24,019.

After calculating these amounts, students can see that there is a big potential payoff with stock funds. They are better able to discuss the levels of volatility and risk associated with each type of investment, and to compare risk versus return. (This, of course, oversimplifies investment strategy, but can be very instructive nonetheless).

This activity involves a variety of NCSS’ essential thinking skills, such as securing factual information relevant to making a decision, and arranging information. Supplementary activities can be designed to develop other content within NCSS’ Thematic Strand 7 Production, Distribution, and Consumption (e.g., applying economic concepts).

The activity directly helps students develop an understanding of interest rates and investments as called for by NCEE. It can also be extended to support NCEE’s Standard 11 pertaining to the value of goods and services, and CCE’s Grade 9-12 Standard V, which defines the role of the citizen in American democracy (e.g., economic rights).6

 

Recursive Calculations: Adjusting the Consumer Price Index (CPI)

Graphing calculators can be used to carry out complex calculations that cannot be done with less sophisticated calculators. The following is an example of a calculation that can help students debate a contemporary social policy issue.

Many economists think that the current CPI overestimates inflation by 1.1% because of a number of biases, and many believe it should be adjusted downward to account for these biases. But adjusting the CPI downward is controversial because it is used to gauge inflation, and hence to determine cost of living increases for social security and other payments.

The newspapers reported recently that the cost of living increase for social security recipients would be 2.1%. Also reported was that the average monthly social security check this year is $749, and that the maximum check is $1,326.

Students can use their graphing calculators to calculate the total amount of payments retirees would receive over 20 years of retirement with the 2.1% increase and the total amount they would receive if the CPI was adjusted downward to 1.0%, using $8,988 (12 x $749) and $15,912 (12 x $1326) as starting amounts for average and maximum payments.

The screenshots in Figures 6 and 7 show that if monthly payments are increased by 2.1% a year over the next 20 years, a person receiving the average monthly payment today would receive a total of $234,192 over 20 years, compared to $208,873 with a 1% increase each year. A retiree getting the maximum amount today would receive $414,605 with 2.1% increases, and only $369,782 with 1% increases.

Students can use these figures to determine the effects of an adjustment of the CPI on the total federal budget, given that there are about 44 million persons receiving social security today. Since the CPI is also used to set food stamp and school lunch expenditures, extension activities can ask students to factor in these costs when determining how an adjustment will affect the federal budget. Classroom discussion can focus on the sociopolitical implications of these adjustments and how the CPI is biased (substitution bias, quality bias, outlet bias, and new product bias).

Activities such as this one would help students to develop several of NCSS’s essential skills, namely, secure needed information relevant to making a decision, form an opinion based on critical examine of relevant information, and keep informed on issues that affect society. Similar activities can be developed to address other knowledge falling within NCSS’s Thematic Strand 7 Production, Distribution, and Consumption (e.g., apply economic concepts), and Thematic Strand 6 Power, Authority, and Governance (e.g., individual rights in relation to general welfare).

This example directly addresses NCEE’s Standard 19, which incorporates a study of the CPI, and can be extended to address NCEE’s Standards 15 and 16 concerning investments and standard of living, abd the government’s role in market economy, respectively.7

 

Discussion

These are just a few examples of how graphing calculators can be used to help students engage in social inquiry. Students participating in activities like these can develop skills necessary to interpret graphs and statistical data and to derive the information needed to draw inferences and make decisions. These activities promote authentic learning in that students are manipulating, determining, interpreting, and analyzing information relevant to real world situations.

Exploring the numerical and graphical aspects of social situations can help students to develop richer understandings of social studies concepts. Effective social inquiry requires students to investigate all dimensions of a situation, and this often involves the gathering of multiple types of information as the bases for decision making. The level of rigor involved in such activities will enhance students’ knowledge of social studies concepts and thus better enable them to become informed and responsible democratic citizens.

Social studies teachers interested in using graphing calculators will need to learn about their various features, and some of the basic mathematics that underlie social phenomena. We suggest that social studies teachers collaborate with their colleagues in mathematics to meet these needs. Such collaboration can also serve as a springboard for broader connections between social studies and mathematics curricula.

 

Notes

1. Peter Martorella, Teaching Social Studies in Middle and Secondary Schools (Columbus: Prentice Hall, 1996), 186.

2. National Council for the Social Studies, Expectations of Excellence: Curriculum Standards for Social Studies (Washington, D.C.: NCSS, 1994), 148.

3. Center for Civic Education, National Standards for Civics and Government (Calabass, Calif.: CCE, 1994), 110, 127.

4. Geography Education Standards Project, Geography for Life: National Geography Standards (Washington, D.C.: National Geographic Education Society, 1994), 184, 192.

5. Center for Civic Education, 99, 110.

6. National Council on Economic Education, “Standards” (Online 1998). Available on the Web at: www.economicsamerica.org/cs12.html

7. National Council on Economic Education, available on the Web at: www.economicsamerica.org/cs11.html and www.economicsamerica.org/cs19.html

 

Joe Garofalo is co-director of The Center for Technology and Teacher Education and associate professor at the Curry School of Education, University of Virginia, Charlottesville. Clifford Bennett is associate professor of social studies, and Cheryl Mason is assistant professor of social studies, at the Curry School of Education.

©1999 National Council for the Social Studies. All rights reserved.