Since the days of Pythagoras, the history of mathematics is replete with mathematicians who studied the arts, languages, literature, religion, and philosophy.1 For example, consider Galileo and Newton who saw their work as a religious vocation, Durer’s treatises on perspective and proportions, Leibniz and Russell who attempted to (dis)prove the existence of God, and Galois who was a young radical during the days of the Second French Revolution.2, 3
Today, in the 21st century, I find it difficult to broach the multilayered and nuanced topics of the humanities in my high school math classroom. By default, I stick to the syllabus and make sure that students are prepared for the next steps — calculus, statistics, and higher-level math. Most students want to know what the answer is, to be prepared for the test, and then have me move out of the way. However, as their teacher, I recognize my responsibility to help them see how mathematics relates to other disciplines.
Figure 1: Pecos Mission and Kiva (a religious structure). © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
The Hungarian mathematician George Polya wrote:
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.4
My own interpretation of how to live out Polya’s charge to math teachers has been to make time for deep-diving projects as a way to link mathematics to other disciplines. In Mathematical Mindsets, Jo Boaler encourages teachers to instill a sense of numeracy, curiosity, and the spirit of tinkering in students.5 I want my students not only to learn about alternate solutions to a mathematical problem but to seek out problems for which there is no clear-cut solution. I also want students to make room for the “in between” or the gray that educators pray every humanities and liberal arts student will delve into and contemplate. No longer will I relegate my students’ philosophical musings to my colleagues from other departments.
This is where archaeology can serve as a bridge to make the humanities central to student understanding of mathematics. It is by crossing the math and humanities expanse that students can engage in discussions about anthropology, gender, history, and social justice and have a deeper appreciation for the role of math.
According to the Humanities Team at Presbyterian School (Texas):
There are many different ways to develop these higher-level thinking skills that are necessary to the 21st century student. Place-based learning is an invaluable opportunity for students to practice these skills in an authentic way. Cultural spaces provide diverse, varied, and previously unknown sites for exploration and analysis.6
Four years ago, I started working with the Robert S. Peabody Museum of Archaeology at my school, Phillips Academy in Andover, Massachusetts. The Peabody Museum is not a traditional museum of natural history; there are no exhibit halls, galleries, or rotating exhibits. Instead, there are classrooms, a permanent exhibit, and a museum staff of archaeologists, educators, and collection experts.
Figure 2: Diorama of Pecos Pueblo © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
Lindsay Randall is the Peabody’s museum educator. Her job is to work with teachers to expand on learning that takes place in the classroom. Over the years, Lindsay has partnered with teachers from the humanities to build engaging and rich curriculum. I remember being a bit nervous: When was the last time I put down my chalk, took my students outside of the math classroom, and talked about a real-world problem using cultural artifacts and materials as the basis of my lesson? I thought to myself, interdisciplinary “is something for humanities teachers.” The truth is that I had never facilitated a seminar-style discussion around a table in my years as a math teacher. Could I lead such a discussion? And would my students in math courses want to engage in discourse around archaeology, history, and politics, and would they understand the mathematical and statistical underpinnings of their study?
The museum’s main classroom is surrounded by a trove of artifacts, dioramas, and murals that depict scenes telling the story of Pecos Pueblo.
Beginning in the 13th and 14th centuries, people in the Pecos Valley region of New Mexico began to gradually leave their small communities and came together to form a larger community called Pecos Pueblo. When the Spanish first arrived in the Southwest (mainly in Northern New Mexico), Pecos Pueblo stood as one of the most powerful and formidable communities in the region. For centuries, the people of Pecos Pueblo dominated the area because they controlled trade between the other pueblos and tribes from the Plains. However, by 1838, the last surviving residents migrated to Jemez Pueblo, and the pueblo was abandoned.
Figure 3: Archaeologist Alfred V. Kidder (center) and colleagues. © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
In 1915, the archaeologist Dr. Alfred V. Kidder chose Pecos Pueblo as the area on which he would focus his research in the hopes of creating a more specific chronology of the Southwest through the ceramic remains that were abundant in the area. Kidder knew that pottery technology and decoration change over time; therefore, the bottom layer of an archaeological site is the oldest, and each layer above it is progressively younger. Kidder hoped that careful stratigraphic documentation would demonstrate the exact sequence, from older to younger, of the various ceramic sherds (pieces of pottery) being excavated. If successful, this “relative dating” would, for the first time, allow archaeologists to understand from broken pottery sherds when an archaeological site in the Southwest was occupied.
The museum’s collection came from Pecos and four smaller pueblos from surrounding areas: Forked Lightning, Rowe Ruin, Dick’s Ruin, and Loma Lothrop. Over time, because of hostile pressures, residents of the four smaller pueblos emigrated to Pecos to form the much larger and stronger pueblo. Through his work at Pecos Pueblo and at these smaller sites in the Pecos Valley, Kidder defined the first Pecos Classification system, which identified eight ceramic types that could serve as the markers for the chronology and dating of sites in the Pecos Valley. Since Kidder’s initial work, the Pecos Classification has been modified and refined by other scholars as more research has been conducted. However, Kidder’s original chronology of ceramics is still pivotal to our knowledge about the history of pueblos.
Figure 4: Pecos Merger. © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
After we decided that the museum’s collections from the Pecos Valley would be used, Lindsay contacted Dr. Linda Cordell, a preeminent Southwestern archaeologist and member of the National Academy of Sciences, to discuss the project and gain expert advice. It was Cordell’s suggestion that our students use the museum’s Southwestern ceramic collections to mirror the work she was beginning to test the validity of the original Pecos Classification system. Investigating the accuracy of the original chronology is important because if Kidder’s work was wrong, then archaeologists and scholars would need to reanalyze what is known about the prehistory of the Southwest. Sadly, Dr. Cordell passed away before our students began their investigation; however, the project went forward in discrete stages.
Figure 5: Students work with ceramic sherds. © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
Our students first separated utility wares (those used for cooking) from decorated serving vessels. The next step was to divide the serving vessels by their paint type: black-on-white, black-on-red, and polychrome (multicolored). Additionally, the students counted the number of rim and body sherds present in the samples. These were important steps for numerous reasons: One is that there is an expectation that there should be more utility wares than serving wares because, as in modern households, people have fewer fancy serving dishes than cooking and everyday ones. Also, there should be fewer rim pieces than body sherds (because the rim represents a smaller fraction of the vessel). Any deviations from these expectations might reveal something about a bias in our sample. The separation of the various styles of decorated serving wares is also an important step because Kidder asserted that black-on-white vessels were the first to be developed; black-on-red came next; and polychromes were the latest style to be produced.
During our lesson planning, Lindsay and I discussed how we would teach mathematics from a humanities point of view. To start, our students would need to appreciate the historic nature of Kidder’s work, how archaeological evidence was collected and analyzed in the early 20th century versus today’s modern methods, and the role played by women on Kidder’s team. We needed to understand these issues before addressing questions of sampling bias, our museum’s collection bias, and what, if any, hypotheses we could test. In looking through Kidder’s field journals and the artifact catalogs, we found that there was no mention of how Kidder’s team excavated the sites. My students knew that Kidder’s team could not have followed what today would be considered a sound statistical process for drawing conclusions (computer mapping and other advanced technological methods).
As my students sat around the seminar table, Lindsay walked them through a quick history of the Southwest, how ceramics are studied in archaeology, and the historic role our school played in the archaeology of the Southwest. Students also learned how one of Kidder’s researchers, Anna Shepard, studied the mineral composition of the artifacts that we would look at, a process called optical petrography. Shepard’s published laboratory findings seemed to contradict Kidder’s chronology hypothesis. However, her conjecture about trading and how shifting trade routes may have impacted the proportions of types of sherds at each site was ignored. The students wondered whether this was because of her gender.
From there, my students questioned whether there was statistical bias in how Kidder chose which pieces to collect, how they were entered in his logbook, and which pieces the museum retained. If so, this statistical bias could lead to conclusions about the data that were highly distorted. Since Kidder and his team’s sampling method was likely “this ceramic piece looks nice and intact,” as opposed to using a formal algorithm or modern accounting methods, my students chose to narrow their focus. Kidder is also known to have focused his energies on sites that had human remains.7 Rather than trying to generalize our results to draw a conclusion about the population of ceramics from Pecos and the other four sites, they would instead explore just the museum’s collection — a much smaller population from which to draw an inference. In my decade of teaching statistics, this was the first time my students were wrestling with difficult questions of bias (in multiple forms), variability, and a problem that was over a hundred years old. They were also being asked to interpret their answer, brainstorm questions about what working on an archaeological dig might be like in the 1920s, and offer up new hypotheses for us to investigate.
Students were divided into small groups to examine the museum’s collection of sherds from all five sites. Results from the groups were pooled to create an overall distribution for the sherds that were sampled from the museum’s collection. From these data, the following hypotheses were formed:
Null Hypothesis: The percentages of sherds of each type across all five sites are the same.
Alternative Hypothesis: The percentages of sherds of each type are different across all five sites.
Students then conducted a chi-square test for homogeneity, which compares the proportions of the different sherd types across the five pueblos. If there is a significant difference across the subpopulation strata, this would suggest that the populations have different compositions, whereas populations that have very little difference across strata suggest that they are similarly composed or homogeneous.
The formula for the chi-square test is
As part of carrying out the test, my students checked the two conditions that validate whether the chi-square test is appropriate. The first condition is that a random sampling method was used to obtain the sherds. My students presumed that Kidder did not randomly select the sherds in his sample. Rather, he likely favored sherds that contained a rim to use in chronology testing. In addition, the Peabody has given away clusters of sherds to other museums. Therefore, we cannot guarantee randomness in our samples.
The second condition to check is that the sample size is large enough to make the chi-square approximation valid. Using this “expected counts” formula
students found that the sample sizes were all above 5, which meant that the sample size from each population was large enough.
Figure 6: Graph of distribution of sherds across all five sites
The students’ statistical significance test yielded a chi-square value of 389.721 with 8 degrees of freedom and a p-value that was essentially zero. From this, they concluded that there was sufficient statistical evidence to show that the sherd distribution across all five sites was not the same. Put another way, the museum’s collection of sherds was not homogeneous. The data called into question Kidder’s ceramic chronology because there appears to be a lower proportion of black-on-red ceramics (assumed to be newer) at the newest site, Pecos Pueblo, but a comparatively higher proportion of black-on-red ceramics at the older sites.
For a number of reasons, the data collected in this project almost certainly do not yield a conclusive answer about Kidder’s ceramic chronology. First, Kidder did not properly use random sampling in his data collection, as shown by the greater number of rim pieces than body pieces. Second, the Peabody Museum has since undergone major reconstruction and regifting processes. Some of the collection has been given away, and some has been lost; little of this has been documented or recorded. Thus, our data cannot be said to be representative of the overall ceramic population.
Scholars looking to continue their study of Kidder’s ceramic chronology should seek to collaborate with other museums and other collections. By comparing the Peabody’s collection with those of other museums, we would be able to draw more meaningful conclusions that could support or reject Kidder’s chronology. It may also be worthwhile to examine the evidence supporting Anne Shepard’s theory about shifting trade routes.
Along these lines, my students started to grapple with the question of who the artifacts actually belonged to. Lindsay explained that, in 1990, the passage of the Native American Graves Protection and Repatriation Act (NAGPRA) gave grants to assist museums in returning cultural items that were deemed “sacred,” “funerary,” or “human remains.”
My students commented to me that this project gave them a real-world feel for how interdisciplinary statistics can be. Furthermore, statistics gave my students the language to analyze and debate archaeological evidence they had never encountered before. Students also appreciated using a statistical test to “confirm” what they were seeing from the data they collected: that Kidder’s chronology and the museum’s collection did not agree. The many issues presented in this project highlighted how the humanities could grow students’ appreciation and use of mathematics.
Figure 7: Descendants of Pecos Pueblo visit Andover to reclaim sacred objects. © Robert S. Peabody Museum of Archaeology, Phillips Academy, Andover, Massachusetts. All Rights Reserved.
We are grateful that we work at a school that values interdisciplinary education and experiential learning. I don’t know of many high schools that have an archaeology museum on campus or a group of educators to help teachers with lesson planning. Regardless of how well-resourced a school is, the key to exploring humanities and the connection to math is in investing the time for teachers to have cross-disciplinary conversations. After this project, Lindsay and I partnered again to look for more opportunities to introduce archaeology and the humanities to my Algebra I, trigonometry, and precalculus classes. We are happy to share these lesson plans electronically and be resources to other educators.
For those interested in learning more about the museum, check out https://www.andover.edu/learning/peabody
. To view the museum’s collection, visit http://peabody.pastperfectonline.com/
1. “Philolaus,” Stanford Encyclopedia of Philosophy
(Stanford, CA: Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University, 2003, rev. 2016); online at http://plato.stanford.edu/entries/philolaus/.
3. Paul Dupuy, “La vie d’Évariste Galois,” Annales Scientifique de l'École Normale Supérieure, 13 (1896): 197-266.
4. George Polya, How to Solve It: A New Aspect of Mathematical Method (Princeton, NJ: Princeton University Press, 1945).
5. Jo Boaler, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching (San Francisco: Jossey-Bass, 2015).