The setting is the crowded Mercat de la Boqueria in Barcelona, Spain. It is by reputation one of the world’s best fresh food markets. The history of the market goes back to the 13th century when street vendors, peasants, and nearby farmers would come to sell their wares from makeshift, open-air stalls in the city’s center. The name is believed to derive from the language of the region. In Catalan, boc means “goat”, so that a boqueria would be a place where goat meat is sold.
The setting is the crowded Mercat de la Boqueria in Barcelona, Spain. It is by reputation one of the world’s best fresh food markets (Figure 1). The history of the market goes back to the 13th century when street vendors, peasants, and nearby farmers would come to sell their wares from makeshift, open-air stalls in the city’s center. The name is believed to derive from the language of the region. In Catalan, boc means “goat”, so that a boqueria would be a place where goat meat is sold. The many shoppers and tourists who walk into La Boqueria are immersed in a world of delicate scents, vibrant colors, and rich flavors. In the central part of the market, they’ll find the rich seafood displays, and toward the back many of the butcher and delicatessen shops, offering aromatic meats, hams and sausages. Completing this mosaic are the displays of the scores of fruit vendors—each in just a few available square meters—with varieties of apples, pears, peaches, plums, apricots, cherries, grapes, and oranges. In competition with each other, they vie for the attention of locals and tourists who press ahead from stall to stall. One of several merchants sells Valencia oranges. His fruit is large, round, polished, and beautiful. Arranged in a large pyramid, they glow rich in color and light. A sign proclaims them to be “extra dulce” and a larger sign announces that they cost 0.99 euros per kilogram. See Figure 2. An excited boy reaches to pull an orange from the bottom of the stack … his mother, recognizing an impending disaster, stops him just in time. The boy’s interest in the oranges now turns to the question: “how many oranges are there in this pile, mama?” Mama shrugs, but tells him that this is a question for his older sister. The older sister, a mathematics student at the Universitat de Barcelona, reflects: “hmm, how many oranges might there be? Can their number be estimated?” After some back and forth with her brother, she assumes that the pyramid consists of oranges through and through and that there is no inner structure of wood or cardboard that supports the display. She regards the oranges at the very top of the stack to be rearranged in two horizontal layers, counts the oranges in the horizontal rows of each of the two rising sides of the pile, and lists the resulting numbers as
on a sheet of paper. Her brother counted also and objects that his sister’s count is not accurate. His sister reassures him, “we’re only looking for a good estimate, and you’ll see the method behind my choice of numbers in a moment.” In her approach to the question, she has replaced the stack in front of her by a streamlined pyramid that has a flat top, and consists of 12 horizontal layers of oranges, all arranged in squares. Starting at the bottom of the pyramid and going up, the sizes of these squares, in terms of the oranges that they contain, are
oranges. This is the older sister’s approximation of the number of oranges in the stack. Since it is hard to imagine that anyone would actually stack so many oranges, the two agree that the merchant’s pile must have an internal scaffolding structure that supports the oranges on display, and that with the exception of the top of the pile, the oranges that are visible are more or less all of them. Using the streamlined version of the stack once more, brother and sister compute the number of oranges in the first ten rows of each of the two slanting sides as
Since the oranges of the rising edge that the two slanting sides share are counted twice, the first ten rows of the two sides contain 110 + 110 − 10 = 210 oranges. Adding the 100 or so oranges at the top of the stack, they conclude that there are about 300 oranges in the display.
Our story about Barcelona’s Mercat de la Boqueria has illustrated some key features of a successful mathematics classroom. After all, at its best, a classroom is a marketplace of ideas. In the same way that the merchant emphasizes the quality of his oranges, the instructor of a mathematics course needs to make a strong case for the relevance of the subject. In the same way that the merchant displays his Valencias in an attractively organized way, the instructor needs to present the subject articulately and coherently. Just as a shopper examines the oranges, questions their quality, and asks for samples, the student needs to engage the relevant concepts, concentrate on the arguments that are made, examine difficult details, and study examples that offer tangible illustrations. Just as shoppers learn about the products being offered and their prices in conversations with each other, students should profit from carefully designed collaborative opportunities to explore concepts and facts, and to engage relevant problems.
By March and April of the year 2020, the coronavirus—having already wreaked havoc in China—had burst into Europe and the Americas, quickly infecting hundreds of thousands of people, putting medical facilities under siege, and killing thousands. In order to contain the spread of the virus, many governments put travel restrictions in place, ordered its citizens to wear masks, to keep several arm-lengths from each other when in public spaces, and to remain sequestered at home in non-emergency situations. The negative impact on the social, commercial, and economic life in the countries involved was predictable. In Spain, Barcelona and the surrounding province of Catalonia were hit especially hard.
How did the spreading virus affect the goings-on at Mercat de la Boqueria? A comparison of Figures 1 and 3 confirms that most of the market’s vibrant life had been snuffed out. A majority of its 250 food stalls were closed. Entire sections, normally a riot of colors and mouthwatering aromas, were locked behind gray metallic shutters. A cleaning team disinfected the spaces every day. In place of the 40,000 to 50,000 visitors that used to pass through the market each day, only a few dozen customers were allowed to enter at a time. Once inside, they needed to maintain the required “social distance” from one another. It was forbidden to touch any of the products on display. Payments with credit card or cell phone were allowed, but cash payments were not accepted. Not surprisingly, customers were no longer inclined to linger. Figure 3 shows a lone customer leaving an almost deserted market.
Barcelona struggled with the silence that the pandemic spread over the city. Commercial venues faced the difficult question of how to reopen and to restart their economic lives. This includes La Boqueria. The tourists who had been overrunning the market in recent years were now absent, and La Boqueria was beginning to reclaim its role as a neighborhood market. The proprietor of the fruiteria Vidal Pons comments, “we have to go back to our essence, that of being a neighborhood market,” and adds that “now with no tourism, we are focusing on online selling.” Accompanying the sharp decrease of the spread of the virus in Spain during May and June were calls for the easing of the restrictions that had been put into place. Not surprisingly, this included La Boqueria. A vendor at the fish stall Palmira made this explicit, when she urged a small group of customers to “spread the word, the markets are open, and the fish is fresh!”
However in July, 2020, a new reality began to set in. The numbers of infections in Spain—as the graph of Figure 4 indicates—began to rise again. This resurgence in coronavirus cases had a two-fold explanation. On the one hand, large numbers of people took advantage of the warmer weather to venture into the streets and to the beaches and ignored the mask-wearing and distance-keeping advisements. And on the other, the testing of the population for the virus, initially confined mostly to older patients arriving at hospitals, was expanded. Even though they displayed no symptoms or only mild symptoms, young people were also being infected. The age group from 20 to 40 years began to account for 40% of all cases in Spain. Around four million people in the Barcelona metropolitan area were instructed to remain indoors and to leave only for essential reasons. As a consequence, cinemas, theaters and nightclubs were closed, and restaurants and bars were limited to half capacity. Non-essential businesses had to interact on ‘appointment only basis’ with their clients. The health ministry of Catalonia announced that “we must go backwards so that we do not have to return to a total lockdown of the population in the coming weeks.”
In the meantime, in early March 2020 on the other side of the Atlantic in New York City, college students at Columbia University were huddled in libraries studying for their midterms. But on the evening before many of the first exams, an email from the university’s president confirmed what had previously been just a rumor: The entire university would be moving to online instruction as a result of the growing threat of the coronavirus in New York City. Figure 5 gives a sense of the rapid flow of infections through the populations of United States and Europe during the month of March. Once the World Health Organization declared the coronavirus a global pandemic, a follow-up recommendation was sent to Columbia’s students informing them not to come back to the university after spring break and that online instruction would extend to the remainder of the semester. Harvard, Princeton, as well as Notre Dame made similar announcements to its students, and thousands of colleges and universities across the country followed suit.
Online instruction meant that the interaction between instructor and students, traditionally in the classroom, person to person, and face to face, gave way to a computer screen to computer screen format conducted via video conferencing platforms such as Zoom. These tools can be used in simple and also sophisticated ways. A television-like transmission of a lecture is easy. However, to turn such a platform into an effective, multi-aspect teaching tool and learning environment represents a challenge. A combination of a high-functioning laptop or desktop with built-in camera and microphone, headphones, and an additional monitor allows the instructor to transmit the visual content of the lecture and to observe the students’ faces and the frowns, confusion, and distraction that they express. Basic functions of the instructor’s online setup, such as discussion boards, breakout rooms, and polling features, allow an instructor to pause for questions, respond with clarifying comments, quickly gauge students’ confidence with the material, divide students into small discussion groups, launch collaborative problem solving sessions, and facilitate one-on-one instructor-student check-ins and peer-to-peer mentoring. To succeed in doing all this is difficult enough in the real, face-to-face classroom. But in the online classroom, it requires a skillfully organized effort not dissimilar to that of a conductor presiding over a symphonic musical performance. Even then, it cannot replicate the three dimensional, live fabric of a dynamic, real classroom.
In this way, American universities were able to complete the spring semester of the academic year 2020. The fall semester would start differently. While many universities continued with the online approach, many others returned to traditional, in person instruction, even though—as Figure 5 indicates—there was a sharp rise in the number of corona infections in the month of July nationally. The University of Notre Dame was one of them.
A mathematics professor and an economics professor1 pursued the question as to whether a mid-size university could successfully proceed with in-person instruction during the pandemic. They developed a mathematical simulation that involved a highly transmissible hypothetical virus at a hypothetical university with 20,000 students and 2,500 instructors interacting daily for 100 days. It studied the spread of this hypothetical virus and evaluated the efficacy of various interventions. The study concluded that without serious restrictions, more than 2,000 people would be infected within 30 days of the first infection, and that in time over 20,000 people, or about 90% of the total campus population, would become infected. The model also studied the infection outcomes under a combination of strategies that included mask-wearing, daily randomized testing of 3% of the university community, quarantine and contact tracing, as well as the teaching of all classes with 30 or more students in an online-only mode. Under these assumptions, the cumulative infections were kept below 66 cases (out of 22,500) in more than 95% of the simulations. Conducting all classes with more than 30 students in an online format was determined to be the most effective measure for keeping infection rates within acceptable limits.
Only time will tell how well universities will be able to respond to the unforeseen and unimagined challenges that the coronavirus presents. While a number of universities have already been forced to return to all online instruction, Notre Dame held to its decision to bring all of its students to campus for an abbreviated fall semester of in-person classes. In the Fall of 2019, the university community consisted of about 12,600 students (of which 6,900 resided on campus, 5,100 off campus, and 600 off site), 1,350 faculty, and 6,100 secretaries, maintenance personnel, and food workers.
We’ll start our discussion of Notre Dame’s experience with a short digression. Unlike the earlier piece of fiction about the number of oranges in the display at the La Boqueria market, the following story is true. In early August, a philosophy professor at Notre Dame, took a break from the pandemically challenged preparations of his fall classes and turned his thoughts to an ongoing interest of his: the study of numbers. He noticed that small odd positive integers could be expressed as differences between two squares. He observed that
and thought that this pattern might continue. Could it be, he wondered, that any odd positive integer can be expressed as the difference between consecutive squares? When he did not see the path to a proof, our philosopher turned to a colleague in the mathematics department for assistance. After a brief consultation, he knew what to do. He let an arbitrary odd positive integer be given and expressed it in the form 2n + 1 for some integer n ≥ 0. His computation
verified that 2n + 1 is indeed equal to the difference between two consecutive squares. He had rediscovered a beautiful fact that Pythagoras had observed over 2500 years ago.
Our philosopher’s successful excursion into the field of elementary number theory has since found a parallel in Notre Dame’s return to in-person instruction on August 10th. Early on there was an alarming spike in the numbers of infected students. The suspicion was that off-campus partying had infected groups of students who then carried the infection to the campus. The university responded to this flareup with a precautionary two-week pause and a return to online classes. Administrators observed that students living off campus had a much higher coronavirus-infection rate than students living on campus. The pause kept those two groups of students apart. The contact tracing that followed, revealed that while off-campus students where the cause of the spiking infections, the source of the transmissions were not large parties, but small, indoor gatherings of students not wearing masks. Students thought erroneously that they were safe, as long as they didn’t gather in massive numbers. What was learned, moved Notre Dame to tighten its procedures for students, faculty, and staff. The two-week pause squashed the spike and was followed by a resumption of in-person instruction within a strictly enforced adherence to a thorough infection-controlling protocol. This included the following provisions:
Masks must be worn at all times and in all places—both outside and inside—except by students in their assigned residence hall rooms and by faculty and staff when alone in a private office. Physical distancing is required and any gathering is limited to 10 people or fewer (with some specified exceptions). This applies to on-campus as well as off-campus situations. Within the dorms up to two additional residents from the same section of the dorm may gather in another student’s room as long as the door is left open and all present wear masks and keep appropriate distances. Students need to eat their meals exclusively either outdoors or in their residence hall rooms. Students, faculty, and staff are required to complete an online Daily Health Check. Individuals who report corona related symptoms are issued a ‘red pass’ and directed to receive rapid tests, to self-isolate, and to get medical attention. Those who contract the virus are subject to quarantine. People that an infected person had been in contact with are traced and tested for the virus.
Faculty are expected to deliver their courses in “dual-mode” instruction that is simultaneously in-class and Zoom-assisted online. Faculty whose age or health situation falls within a high risk category or involves other special circumstances are permitted to teach exclusively online. Students scheduled for an “alternating attendance” course join the class in person or online on alternating class meetings. Those who receive a red pass on their Daily Health Check or test positively are moved into quarantine or isolation and are able to continue to participate in their classes online. All classes must be be recorded to allow students to review previously presented material and to facilitate a return to an exclusively online approach should this become necessary.
A snapshot or “dashboard” of the campus infection picture is published daily. Figure 6 depicts the dashboard for the 10th of September, 2020. The bar graph—the number of tests are shown in gray bars, the number of positive outcomes in brown, and the graph of the running 7 day average in green—confirms that the earlier flareup has ended and that the virus has been brought under control. In recent weeks, the number of students moving out of quarantine has been greater than the number moving into quarantine. All indications are that the measures and strategies that Notre Dame has adopted are dealing with the pandemic successfully. It seems probable—so far so good—that the university’s effort is sustainable.
The coronavirus stories for the University of Notre Dame and Barcelona’s Mercat de la Boqueria contrast sharply. Notre Dame is open and carrying out its educational mission. On the other hand—in response to the rising infection curve of Figure 4 and as Figure 3 illustrates—La Boqueria has only a small number of shoppers and its stalls are largely closed. The difference is easily explained by considering the two environments. Notre Dame is a community that is largely self-contained. It’s close to 20,000 individuals live and work in an environment that can be subjected to stringent protocols and careful monitoring. The market La Boqueria, on the other hand, is subject only to the already discussed weaker protocols and precautions of the city of Barcelona and the province of Catalonia. Notre Dame is a microcosm of a restricted number of people, La Boqueria on the other hand, is a market that is open to the millions of residents of the city.
While Notre Dame opted for a “primarily in person” instructional model, most colleges and universities have been more cautious. The pie chart of Figure 7 shows how the close to 3000 colleges and universities in the U.S. have approached the instruction of their students in response to the threat of the virus. In the same way that states, counties, and municipalities in the U.S. have needed to react to increases in the number of daily infections, U.S. colleges and universities have needed to adopt a fluid approach to control the infections on their campuses. Some, like Notre Dame, were able to respond to spikes successfully with temporary pauses in in-person instruction, but for others such pauses were not successful, and they were forced to turn to online instructional delivery.
We have had a glimpse at the worldwide turmoil that the coronavirus has caused. The challenge has been to keep the disruptions to the functioning of the social, commercial, and educational channels to a minimum while at the same time to safeguard peoples’ health and lives. This has brought about new patterns of behavior and new modi operandi. Time will tell what kind of “new normal” stable states will emerge.
Alexander Hahn is Mathematics Professor Emeritus at the University of Notre Dame. His most recent work, the book Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno, was published during the corona pandemic by Springer Publishing in April, 2020.