This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.
Before I begin, I’d like to acknowledge that it’s been three years since I wrote a blog post. I’m back at it because I’m teaching this year!! Thank you to Michael Pershan and Anna Blinstein for inviting me to participate in their week of the Virtual Conference on Mathematical Flavors.
Michael and Anna gave us an essay to respond to when they invited us to write for this week of the conference. The essay was “The Two Cultures of Mathematics,” by mathematician W. T. Gowers. In a nutshell, the essay discusses the role of problems in the culture of mathematics. Gowers argues that there are two cultures of mathematics, one of “theory-builders” and another of “problem-solvers.” Those cultures are characterized by one of the following claims:
- The point of solving problems is to understand mathematics better.
- The point of understanding mathematics is to become better able to solve problems.
Gowers spends most of the essay defending the value of the second culture. He claims that mathematicians tend to value “theory-builder” culture more than “problem-solver” culture because coming up with abstract theories has more status than solving individual problems. But Gowers disagrees with this ranking. Solving individual problems can give mathematicians important insights that also develop the field. He argues that if mathematicians can ascribe status to both cultures, the field will be more inclusive for more mathematical learners. (He’s talking about PhD students, but his arguments probably apply to kids, too.)
His argument is interesting and worth reading– even if it does get bogged down with lots of mathematical details that I didn’t understand. It’s message about how the culture of mathematics excludes or demeans some who actually do valuable work in it for no good reason. It makes an argument for learning more about different cultures– in this case the “problem-solving” culture– because when mathematicians blindly exclude people, they hurt the field.
For this post for the virtual conference, I’d like to flip his question around. Instead of thinking about the role of problem-solving in mathematical culture, I’d like to explore the role of mathematical culture in problem-solving. I’d like to argue that how mathematicians think about problem solving excludes ideas and ways of working that are not only valuable to the field of mathematics (and students in our math classrooms), but that are essential when math is applied outside of pure math contexts.
I’ve been thinking about this a lot lately. In fact, I’ve been writing something about it– something that hopefully one day you’ll all get to read! In the meanwhile, though, I’ll share a story that I think highlights some gnarly issues in the culture of math problems. These issues are common to both of the cultures Gowers identifies– to solving them and posing them, to building mathematical theories from them and using mathematical theories to approach them.
This story begins in the time of the Incan Empire,. The Incan Empire didn’t exist for very long– only around one-hundred years– but in that time, it stretched across modern-day Bolivia, Peru, Ecuador, and parts of Colombia, Argentina, and Chile. To sustain such an empire, the Inca needed a way of communicating across long distances and storing information over time. In European culture and its derivatives, writing does this job. But the Inca did not write. So how did they communicate?
Unfortunately, even up to the present day no one knows the answer to this question. Really and truly no one– not even the descendants of the Inca, although they have their theories. This is because when the Spanish Conquistadors invaded Incan territory in the 1500s, they so thoroughly wiped out the Incan people and their culture that very little remained of that vast, rich, and complex society. A Spanish soldier who travelled through the ruins of the Incan empire not long after the first Conquistadors wrote, “it is as though a fire had gone, destroying everything it passed.” The Inca who survived the Spanish inferno struggled to save their cultural records, but the records and memories of how to interpret them alike were largely wiped out.
“How did the Inca communicate in writing?” is a problem that many people have tried to solve. Most relevant to the story I’m telling, mathematicians are among those who tried to solve this problem. Unfortunately, they bungled it.
Several hundred years after the Spanish conquest, European and U.S. anthropologists tried to salvage what they could of Incan culture. They dug up ruins, sifted through wreckage, and carted what they found off to museums and university research labs. Much of what they found was familiar– bowls and pots, shelves and oven, relics of daily life common throughout the world. But one particularly common artifact baffled them: the tangle of knotted, colored string that we now call khipu.
If you’re a fan of math history and ethnomathematics– the field devoted to the math of non-Western cultures– you may have heard of khipu. You probably heard that khipu are math. Khipu, as the story most often told by math historians goes, are records of numbers. They were probably used to record transactions and inventories. Khipu conveniently record numbers in base ten, the number system with which we are most familiar. Different knots positioned in different ways stand for digits with place value.
The first person to conclude that khipu were math was a mathematician named L. Leland Locke. Locke worked in the early 1900s. He got his hands on some khipu that were being stored in a U.S. museum and set to work deciphering the code. He found some patterns in the knots that he connected to numbers and counting. He found other patterns that he could not connect to numbers and counting, and these he tossed aside as possible forgeries or stylistic flourishes.
Locke wrote articles and books in which he proclaimed that he had solved the problem of how the Inca communicated across distance and time. That problem was a math problem, Locke declared, and the answer was, “In base 10! With knots!”
Locke defined the question, “How did the Inca communicate?” as a math problem. The culture of mathematics (and history, archaeology, and anthropology) in which he lived and worked supported him to do this. Dot-like knot patterns look a lot like math to Western mathematicians. Base ten is a very familiar number system. It was almost too good to be true. Locke’s definition of and solution to the problem of Incan communication has been cannon for nearly one hundred years.
But it missed a big piece of the story.
For instance, some Spanish records from the time of the conquest document Incans describing khipu as narratives. Once such record tells of an Incan woman who brought a khipu to Catholic confession because it contained her life story. Other modern communities of Incan descendants cherish khipu that they miraculously saved from the Spanish, khipu that they think tell the history of their people.
Are these histories written exclusively in base ten numerals? Unlikely. But– and again, I’d like to stress how remarkable this is, given the extent of the Incan empire at its height only five hundred years ago– no one alive today knows how to read them. And L. Leland Locke, charter member of the Mathematical Association of America, said that khipu are math. The khipu problem is a math problem. Narrative history khipu cannot possibly exist. They would be inconsistent in the mathematical world that Locke defined.
But such inconsistent khipu do exist. Anthropologist Sabine Hyland and her research partners in the Incan-descendant community of San Juan de Collata have them. And they’ve translated two of the first khipu words. Not numbers. Words— words that violate Locke’s mathematical pattern and turn his math problem on its head.
San Juan de Collata is perched high in the Andes mountains. It’s an isolated community– and its residents like it that way. Ancestors of modern Collata residents participated in a revolt against the Spanish in the 1700s, and modern-day Collata natives think that the khipu in their possession tell the story of that revolt. But they did not know how to read it. Until they met Hyland, they did not trust any outsiders to help them translate their cherished text. Hyland managed to gain their trust, however, because she did something that almost no khipu researcher before her had done: She believed the stories that Incan descendants told her about their documents and asked them to help her translate them.
Translating those two words at the end of the Collata khipu was hard work. If you want to learn more about how Hyland and her Collata colleagues translated the khipu, you should read her article. The Collata khipu are of the variety that Locke would have trashed. So Hyland and the Collata residents who worked with her had little prior research to go on. But they did have the expertise in Incan history and culture held by the Incan descendants– which Locke did not have.
Locke did not have the expertise that Hyland and her Collata colleagues leveraged in part because the Spanish torched Incan cultural memory. But he also did not have it because he did not seek it out. The mathematical culture in which he worked allowed him to define the khipu problem in a way that silenced the faint but still living voices of the Inca. The Inca had the most at stake in defining and solving the khipu problem. And yet Locke and mathematicians who supported and listened to him defined the problem in a way that excluded them.
It probably was easy to exclude the Inca from the problem-posing and solving process. The Inca had long been marginalized and were rightfully nervous of making connections with academics from Europe and the U.S. But history does not exculpate Locke and mathematicians like him from allowing mathematical culture to blind them from the role that Incan descendants should have played in solving the khipu problem. The khipu problem was and is not just a math problem. And it certainly does not belong to mathematicians alone.
Gowers in his essay defends the value of the mathematical culture of problem-solving. Problem-solving builds theory, he argues. Problem-solving has wide-ranging applications beyond mathematics, he claims. Not all math problems turn out to just be math problems. Sometimes they are also computer problems, science problems, and– in the case of khipu– historical and cultural problems. Mathematicians who solve problems contribute to the world as much as mathematicians who build the theories on which the field of math rests.
At its core, Gowers’s essay is about inclusion and exclusion. Gowers comes down on the side of inclusion. For this he should be celebrated. But he fails to address questions about the mathematical culture of problem-solving that are key if we want to include, not exclude. Whose problems are mathematicians solving? Who is solving those problems? Who defines the problems, and who decides when they’ve been solved? The story of khipu is a story of mathematicians defining and solving a problem without including the most essential stakeholders– and getting the problem wrong.
In this Virtual Conference, we are encouraged to tie our posts back to the classroom. My story does not directly relate to the classroom– although it does implicate what’s been taught about khipu for almost a hundred years. It is relevant to teaching, however, in that when we have kids solve problems in class, those problems rarely belong to the kids. Whose problems are they? They are the teacher’s problems. If not the teacher’s, then they are the textbook’s problems. Or they are the problems posed (and typically solved) by long dead, white, male mathematicians of European or U.S. descent. A very insular and exclusive culture of mathematicians defines the problems, and our students solving them in turn perpetuates that culture– but without fully becoming a part of it.
Problem-solving is a deeply cultural practice. When our students solve math problems but do not pose them or decide on how solutions will be evaluated, they only partially participate in that culture.
Whose problems? We must consider this question when thinking about mathematical culture, in particular the mathematical culture of problem-solving.