# Changing Historical Perspective with Mathematical Reasoning

This post was written by Peter DeCraene, the 2020-21 Albert Einstein Distinguished Educator Fellow, and Michael Apfeldorf, an Educational Resources Specialist, both at the Library of Congress.

At a recent National Council for History Education conference, we led a workshop that explored how mathematical reasoning might be used to offer fresh historical analysis.

Constituents of the Populations of the Great Cities-1890 from
Statistical atlas of the United States, based upon the results of the eleventh census, 1898

We started by presenting a portion of this chart from the 1890 Statistical Atlas, including only the city names and colored bars, and letting participants know that the chart was produced in the last half of the 19th century. Based on that limited information, participants observed and reflected on what they were seeing, and made conjectures about what the chart might be illustrating. Using a zooming out technique, we then showed the heading “Per cent” and the title of the chart, and participants again observed and reflected, then revised their conjectures. Finally, we showed the key describing the meaning of the different colors, and invited participants to share how and why they changed their conjectures at each step.

This kind of zooming out activity is useful in both math and humanities, because as we start solving a rich problem or researching an interesting topic, we often start with incomplete information, think about what additional information we need, then revise conjectures as we uncover new information. Furthermore, by revealing the chart slowly, participants were forced to grapple with the relationships presented and the intent of the representation. This process reinforces mathematical thinking skills and helps learners think more deeply about messages that are communicated to them in chart form.

As participants interrogated the chart further, questions arose about the data: Which city actually has the biggest native white population? (Further research is needed, since the chart does not show actual population numbers.) Why did the creator choose this order for the cities? Now sensitized to the chart’s construction, participants became aware that it was ordered specifically to rank cities in terms of percentage of native whites of native parents, perhaps laying bare the perspective of the chart’s creators.

Re imagining of the 1890 Populations of the Great Cities Chart, courtesy of Peter DeCraene

One way to change the perspective is to rearrange the information presented. We chose to reverse the order of the colors on the bars and reorder the cities by percentage of the “Colored” population, shown here. This raised more questions and conjectures that had not surfaced during the initial analysis. For example, some participants had barely noticed the gray sections on the right side of the original chart. Others wondered about immigration patterns for both the Black population and recent immigrants.

For teachers using charts as primary sources, ask students to represent the information in a different way to deepen their process of inquiry. Perhaps the students might change the color or order of the information, or use an entirely different type of chart. Then, ask the students what new conclusions they might reach about the information or creator’s intent.

Using a mathematical lens to look at different ways of representing data and finding patterns can change our perspective and lead to new and interesting questions and conclusions.

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