This post was written by Peter DeCraene, the 2020-21 Albert Einstein Distinguished Educator Fellow at the Library of Congress. This is one of several posts about using mathematics to develop and analyze data representations found in primary sources.
As the lone math teacher among humanities folks, I’ve been awash in the seemingly limitless sea of historical primary sources at the Library. The first islands of refuge I found were statistical atlases. I was amazed and delighted at the variety and beauty of how these atlases displayed the vast amounts of available data, and the possibilities for connecting math topics to American history. When I started to explore, the graphs on this page caught my eye. While I wasn’t overly excited about the topic of “Live Stock and Products,” the graphs offer a good place to begin a mathematical journey using primary sources.
First, ask students to look at the top half of the page and examine the bar charts and map and how they’re presented. After a minute or two of observing, direct students to gather in small groups to share with each other and reflect on what they’ve observed, then generate questions about the graphs.
When I use this strategy, I record student questions where everyone can see them while groups report, without offering any feedback other than “Thank you.” (Too often, we math teachers and students like to jump right to the solving part, instead of taking some time to appreciate the richness of information.)
Next, discuss which questions require additional information and which can be answered using a math strategy. For online instruction, use breakout rooms or a chat tool. The Observe-Reflect-Question strategy allows all students to participate, no matter their computational strength. Everyone can observe something and ask a question.
Here are possible questions about the data:
- “Which of the bar charts does the map illustrate? How might you create a map for the other bar chart? When I first saw the map on this page, I was surprised that states like Illinois and New York were so much darker than Texas; I thought Texas would be the “biggest cattle state.” It’s important to attend to graph titles!
- In a unit on statistics, you might ask, “What is the mean (average) of the data presented on the larger bar graph?” What would that average mean in this context? If we represent this data as a box-and-whisker plot (which would illustrate the median value and “spread” of the data), what else might we infer? Do we see any outliers in this representation?
- A unit on rates might include a discussion about how the smaller bar chart shows a rate. How might this data have been calculated? Or, could we find the area of each state from the information given here? Could you use the smaller bar graph to find the average number of cattle per square mile in the United States at that time? (This could be a way to start talking about weighted averages.)
You might ask questions about how and why different representations might be helpful:
- How might we represent the data with a pie chart? What information could we infer from a pie chart? When would this representation be helpful?
- Use the data on this page and the data from the top bar chart on the page about population to create a scatter plot showing “Population per Square Mile” on the horizontal axis and “Cattle per Square Mile” on the vertical axis. What information might we infer from this representation? A scatter plot of this kind can also be used to find a line of fit. What would such a line mean in this context? (This would be an opportunity to teach some spreadsheet skills as well.)”
- What other ways of representing the data might be useful?
Starting with primary sources and asking questions helps develop students’ proficiency with representing and interpreting data, and with reasoning, modeling, and communicating mathematical ideas.
What questions or models do your students come up with? Please take a moment to comment below.