Science, mathematics, and geography are just a few subjects that require students to analyze data on standardized tests yet graphical analysis is a process skill that students continue to struggle with. In, “Raising the Bar”, students practice graphing and graphical analysis for the purpose of discovering the characteristics of developed and developing countries. Exploration of these characteristics provides students with a purpose to graph data and to analyze it.

# Narrative

Few would argue that is acceptable to be illiterate. In fact, as the lesson “Raising the Bar” discusses, literacy is a significant characteristic of developed countries. Yet, many of us are satisfied with being innumerate—meaning we are satisfied with not having the capacity to interpret or manipulate numbers in a meaningful way. So often passed as a mild disinterest and, in some cases, an extreme hatred, our opinions of mathematics very rarely fall to praise. John Allan Paulos, in his book *Innumeracy* (1988), first introduced me to this severe dysfunction of the American attitude. His argument is succinct; as a culture, we abhor illiteracy and readily embrace innumeracy.

As a math teacher, my days are filled with students, teachers, and parents exclaiming their distaste for math and defending their position by giving account of bad teachers, lack of application, and, if you can believe it, genetics. Yes, the most common excuse I hear is to the effect of, “My parent(s) weren’t good at math, so they don’t expect me to be good either.” This persistent and widespread complacency has no place in the 21^{st} century.

Alicia Crowe (2010) presents an argument for the inclusion (or integration) of mathematics into the social studies. This argument for incorporation is not completely new. Bowen and Roth (2003) argue that, “Graphs are a central aspect of science” (p. 500). Although their primary research focuses on the development of new science teachers, they stress the necessity of graphs for all teachers and subject areas. “Graphs assume this important role because they efficiently summarize trends over time… [and] compellingly depict theoretical models of relations between variables…” (p. 500). As science teachers become more aware of the need to include graphing, Crowe’s argument is quite timely. She presents four types of numeracy that are “fundamental for citizens to know to be thoughtful, informed, active members of both a democratic republic and an ever-growing global society” (p. 106). These are 1) *raw numeric data*, 2) *percentages*, 3) *averages*, and 4) *interpreting and questioning graphs and charts* (p. 106). The lesson, *Raising the Bar: Graphical Analysis of Developing and Developed Countries*, is a perfect example of the fourth type of numeracy, which must become standard in the social studies classroom.

In the lesson, students create graphs to represent data comparing infant mortality, life expectancy, literacy rates, and urbanization of various countries. This investigation places a significant amount of data in the hands of the students and asks them not just to interpret the *raw numeric data*, but also to create a *visual/graphical representation* of that data. I like to think of the data as paint. Very few students (for that matter, adults) are able to look at a blank canvas and tubs of paint and imagine how the texture, value, etc. will come together as a beautiful painting. Bar graphs, if we continue this metaphor, are kind of like finger paintings; with a couple strokes, the trend of the data emerges.

As students create their graphs, they become comfortable with this type of representation. The lesson continues, challenging students to evaluate, extrapolate, and interpret their graphs. Perhaps the most interesting aspect of the lesson is the process by which the y-axes are numbered. The intent of the lesson is to narrow the focus of the students so that they are only concerned with the data they are given. This *default* process is the type of thing that very few students will debate. If the minimum data point is 43 and the maximum data point is 67, most students are perfectly content with numbering the y-axis from 40 to 70. For the purpose of the lesson, this range of values is sufficient. Ask students if it is necessary to include values from 0 to 60 and most would argue that that would just be silly. To some extent, they have a point. This is where the teacher shakes things up a bit.

It is a common understanding among mathematicians that the optimal range for the y-axis begins at zero. In fact, in an informal survey of fellow math teachers (N=13), 100% of the respondents agreed that the y-axis should always start at zero to ensure that skewing or misrepresentation of the data does not occur. For example, the graphs below represent a fictional set of data that shows 38 red items and 45 blue items. Despite various inaccuracies in the graphs, all of the participants in my informal survey concluded that Graph B is *more accurate*. Both A and B represent the same data, but Graph A, due to its narrow y-axis range, conveys a significant discrepancy between reds and blues. However, Graph B conveys a more accurate perspective in which there are more blues, but only by a few.

In whatever manner this part of the instruction occurs, the teacher must introduce the *more accurate* model to the students. The resulting debate is exactly the type of *interpreting and questioning *proposed by Crowe. “Raising the Bar” is a great lesson that challenges the perspectives of the teacher and students. It is the type of lesson that leaves students talking, making connections, and asking questions. In its own way, this lesson represents a change in attitude that will soon see numeracy added to the list of characteristics of developed countries.

# Snapshot

**1. Engage **

• How is the United States different from Afghanistan? – Students begin this lesson by discussing differences between these two countries.

• What countries have similar characteristics to Afghanistan and what countries are similar to the United States? This is the essential question students will be exploring in the lesson.

**2. Explore **

• Graphing the data– Students will be given a data set for a specific demographic characteristic for twenty countries. They will be asked to graph the data.

• Why do we use bar graphs?– Students will explore the purpose of bar graphs and when it is best to use them.

**3. Explain **

• Grouping countries – Students will group countries in two categories based on the characteristics displayed in student graphs. Students will discuss this in groups and each student will bring a different demographic graph to the discussion.

• Developed vs. Developing– Students are led through a discussion of characteristics associated with developed and developing countries.

• World Capita Map– The World Capita map will allow students to explore the per capita of each country from the lesson. Teachers can also use the map to assist students in exploring other developed and developing countries that were not discussed in the lesson.

**4. Extensions and Applications **

• “What If?” Activity: In this activity students are mentally transported from their lives

in the United States to life in a developing country. Students are asked to express how they would handle a variety of scenarios if they where living in the conditions of some of the poorest countries.

**5. Authentic Assessment: **

• A World Apart: In this assessment students are asked to complete a two-page article depicting the differences between developing and developed countries. The assessment places students in the role of a journalist on a mission to express to their community what life in a developing country might be like.

• Developing a Nation: Demographic Transition- This is a second K20alt lesson developed to help students explore how countries transition from developing to developed. This lesson focuses on two countries that have been experiencing significant transition in demographics in the past 50 years.

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