Whose Problems?

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:

  1. The point of solving problems is to understand mathematics better.
  2. 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.

The Canuto 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.


Loop-de-Loops! And a contest!

As some of you might know, I recently wrote a book!

Cover image
Credit: Kane Miller Publishing

It’s a math and art activity book, containing all of the amazing MArTH activities developed by me and the Saint Ann’s School math department over the five years that I taught there. There are more than 30 different activities in the book, ranging from favorites like Pascal’s Triangle coloring to things you may not have seen before, like the Stomachion puzzle. I had SO MUCH FUN writing it– and I hope that someone out there has as much fun using it and I did putting it together.

(If you’d like to browse the book and maybe even buy one, click here.)

I’m posting this week not just to make this announcement, but because of an exciting opportunity! Dan Meyer had the amazing idea to run a contest for teachers and their students on his blog featuring one of my favorite activities from the book– Loop-de-Loops.

Credit: Kane Miller Publishing

Visit Dan’s blog to find out more about the contest! In a nutshell, it involves making Loop-de-Loops with your math class, sending them to Dan, and some math art friends of Dan’s choosing their favorite loop. You and your students could win a class set of This Is Not a Math Books!

Now, say you’re a teacher thinking of making Loop-de-Loops with your students, hopefully to enter Dan’s contest. Say you’re also wondering, what else can I do with Loop-de-Loops other than just have my students draw some? That’s a fun, mathematically interesting activity on its own– and I’m sure your students will have a blast if you stick to that. But say you wanted more…

So, for the next few posts, I’ll be sharing some more mathematically rich things you and your students can do with Loop-de-Loops. The first is pretty simple. I call it “Detective Loop-de-Loop.”

It’s one thing to come up with your own number sequence (like 2, 3, 4) and make a Loop-de-Loop out of it. It’s another thing entirely to look at a Loop-de-Loop and figure out what number sequence was used to create it!

1 3 6 LoopFor instance, what number sequence made this Loop-de-Loop? (Hint: It had three digits.) This one is easy compared to…

2 3 4 5 6 Loop(Hint: This one uses five digits!) Figuring out how a Loop-de-Loop was made requires looking for patterns, thinking backwards, and checking your hunches with a diagram– all math habits of mind that are important for elementary and middle school students.

“Detective Loop-de-Loop” doesn’t have to stop with deciphering and trying to recreate tricky Loop-de-Loops. Lots of interesting questions come up. Like…

Dual 2 3 4 and 2 4 3 LoopsThese two Loop-de-Loops look pretty similar– the only difference is that they seem to be looping in different directions, one clockwise and the other counter-clockwise. How were they made? What difference between their number sequences caused them to have different “chirality,” which is the technical math and science term for the direction something rotates?

Once you start making Loop-de-Loops and playing Detective Loop-de-Loop, you’ll notice that there are many interesting questions to ask about how different loops were made– and why small changes in the number sequence behind a loop make such big changes in the way the loop looks.

3 2 5 Loop

Or doesn’t… Here’s a 3-2-5 Loop-de-Loop. What would a 2-5-3 Loop-de-Loop look like? How would it be different, if at all? Can we figure out how to make a Loop-de-Loop that looks exactly like the 3-2-5, but turns in the opposite direction?

I think Loop-de-Loops are rich mathematical objects– meaning that playing with them can lead to an enormous number of interesting questions and investigations that take you and your students into the worlds of geometry, number theory, patterns, functions and more. Even just making up your own Loop-de-Loops and playing Detective Loop-de-Loop presents many opportunities for students to wonder about math and use their wondering to create something beautiful.

You can print these Loop-de-Loops for you and your students to play Detective Loop-de-Loop with. But you should also make more of your own– it’s fun, so why not?

To wrap up, I’ll give you a starters’ list of interesting questions to investigate while you and your students make and play detective with Loop-de-Loops. Of course, you might not use any of them– you might have so much fun drawing, coloring, and decorating your loops for Dan’s contest that you forget all about questions. And that’s great, too! Math class needs more time for free creativity, in which kids can feel proud of what they’ve made. At least, that’s what I think.

2 5 3 6 2 LoopInteresting questions:

  • What different number sequences make identical Loop-de-Loops? What do these sequences have in common?
  • What different number sequences make Loop-de-Loops with opposite chirality? What patterns can you find in those sequences?
  • 2 4 3 3 5 LoopWhat kinds of number sequences make Loop-de-Loops with long arms, like the first one in this post, or that are tightly packed, like the loop to the right? What about the sequences causes this difference?
  • Have you made a Loop-de-Loop that surprised you? What about it was surprising?

Post your loop drawings and classroom activity ideas in the comments section! Don’t forget to enter Dan’s contest, and have a lovely loopy day!


Assessing the Un-Testable

I think I mentioned the Talking Right Past Each Other Paradox in my last post. It’s the problem that some education researchers see where they want to do research that helps teachers, but they feel like teachers aren’t reading their work. At the same time, teachers have problems that they could use some help with—and it’s nice to have someone with lots of time to think about your problem give suggestions. But the suggestions don’t always come. Hence, Talking Right Past Each Other.

I had an encounter with said Paradox the other day in one of my classes. We were reading an article that another grad student (Kyle) picked out about “epistemic cognition.” If you feel like spending a half hour with your dictionary, by all means skip my so-so definition of epistemic cognition. If you’d rather not, here’s what I think it is: what people think about knowledge and learning.

The article was about the problems researchers have with figuring out exactly what people think about knowledge and learning. Researchers want to figure this out because they’d like to have a clear trajectory of the development of people’s thoughts about knowledge and learning, tracked as people get older and learn more about particular subjects. Researchers would also just like to be able to assess what people are thinking so that they can help them think better things. Thoughts about knowledge and learning have a lot to do with how you actually learn, so it would be nice to know what you’re thinking about what you’re learning to help you learn it better.

All of this makes sense. From the article, I could also clearly see how the researchers were having problems with their ways of assessing epistemic cognition. I won’t go into that (by all means look it up, it was interesting: by Greene and Yu, called, “Modeling and measuring epistemic cognition: A qualitative re-investigation”). But this was about as far as my thinking went. First, I agree that it’s useful to know about what people think about knowledge and learning. Second, I see how those researchers could be having a hard time.

We were wrapping up the class when Kyle made a comment that jogged my brain. To close the discussion, he said something about how he hopes that researchers figure out how to measure epistemic cognition. I hadn’t had that thought. So I asked him why—why did he hope that researchers figured out how to measure epistemic cognition? His reply was so that he could do a better job of teaching. He wants his students to have nuanced and well-rounded ideas about knowledge and learning, and he felt like have a research-proven, efficient test for epistemic cognition would be helpful.

Somehow this didn’t make sense to me. I mean, it does make sense—I suppose a test for epistemic cognition would be awesome. But I was having two problems. First off, I couldn’t picture what a closed test of epistemic cognition—one that you could administer within one day—would be like. I really don’t know enough about designing epistemic cognition assessments to know, though. The second problem was the bigger one. Even if someone did come up with a test that measures epistemic cognition, I couldn’t see myself using one in a classroom in a natural way.

It’s not like I didn’t keep track of what my students were thinking about knowledge and learning while I was teaching. In fact, helping my students develop a particular set of thoughts about and attitudes towards math knowledge and learning was a major focus of my teaching. And I totally assessed it. I just didn’t give a one-off test for it.

I did raise these complaints in class. (Case-in-point of me being a terrible student—being argumentative right at the end of class.) We talked about them a bit, and then the professor asked Kyle whether he assessed epistemic cognition when he was teaching. Kyle said he did. How did he do it? Kyle thought for a moment and answered, by seeing what kinds of questions his students asked.

Yeah, I thought, that’s partly how I did it, too. I think that works really well. When kids ask probing, authority-challenging, curiosity-sparking questions, that shows they have top-notch epistemic cognition. (Not sure if “top-notch” is a correct measure of epistemic cognition, but oh well.) Kyle seemed to think that this way of measuring epistemic cognition worked really well, too.

But kids don’t ask their awesome, top-notch-epistemic-cognition questions all at the same time. They don’t even do it all the time, even if they do have that cream-of-the-crop epistemic cognition. That’s just not how it works. Sure, the fewer awesome questions they ask, the more likely it is that they think that knowledge is something you pour into brains. (Or that the teacher’s class isn’t open enough to students’ ideas.) But there’s also a personality thing. And the fact that asking awesome questions isn’t easy, even if you do have glorious epistemic cognition. And the fact that kids strut their epistemic cognition stuff in the context of class itself, not on a test that’s just about epistemic cognition.

So can you—should you—turn Kyle’s question-asking way of measuring epistemic cognition into a closed assessment? Would it still work in a real-live classroom full of real-live kids? If I’m trying to teaching my students robust ways of thinking about knowledge and learning, is closed, fast assessment the part I really want help with? I don’t know! But here we go again, Talking Right Past Each Other.

I’m not saying that there’s anything wrong with researching a tool for measuring epistemic cognition. It’s an interesting problem. But I don’t know if it’s the most interesting question about epistemic cognition from a teaching perspective. I think the discussion we had, as thoughtful folks straddling the divide between teaching and research, demonstrates that very well.

There Are No Kids Here

I took a bit of a break from writing because I’m actually not teaching at Saint Ann’s anymore! I started a Ph.D. program in math education at UC Berkeley in September, so the past six months have been consumed with homework, readings, papers—that sort of thing. The first thing I’ll say about that is that while teaching, I have definitely become I worse student. I hear this is common. If anyone would like to do some research on the terrible students that former teachers become, I volunteer myself as a case-study.

I’m writing again for two reasons. First of all, I re-found a great education blog by someone who is also on the other side—Ilana Horn! I was in the middle of a small bout of despair about education writing when I re-found her blog. I have small bouts of despair about things related to education research every now and then, and this one was sparked by what I’ll call the Talking Right Past Each Other Paradox: Education researchers say their main goal is to help teachers, but they feel like their work doesn’t often get read by teachers. My first thought when I heard this was, “Because the researchers don’t blog about it!” And then I remembered that one of them does blog about it! And I realized that it is possible to write something about education research in a human, intriguing, and useful way. And I wanted in.

I’m obviously not ready to do this on even 1% of the same level as Ilana—so that’s not really why I’m writing again. The second, and primary, reason why I’ve jumped back in is basically the same as why I started blogging in the first place—because I’m having teaching problems.

Just like I was totally unprepared to teach math to kiddos when I first started at Saint Ann’s (not the fault of a teacher prep program—I didn’t go to one), I am totally unprepared to teach teaching to college students. This also isn’t the fault of a teacher-teacher prep program, because there isn’t one. I’m really not sure why. It’s not like college students aren’t kids, too, who need care and personal attention. It’s also not like teacher educators have got the whole teaching thing figured out, either.

Anyway, a little back-story to my current teaching problem—I’ve been “TA-ing” and “researching” in an education class for undergrads who are in one of Berkeley’s teacher prep programs. But the other day, I was launched from my cozy position as TA to the much scarier position of actual teacher. The instructors didn’t have anything planned for an hour chunk of class and one of them was going to be late. So they pulled out their secret weapon, the TA, to fill in the gap.

What should I do with 40 pre-service teachers for an hour? I first approached the task like I would if they were 40 high school students with whom I was asked to share something cool and mathy. They’d just read some articles about inquiry learning, so I thought we could do a little math inquiry together. But then I realized that my lesson was missing something essential. I had a great math problem picked out. But they weren’t going to do any teaching. I didn’t have a “teaching problem.”

And this is when the sheer challenge of the task hit me. I needed a teaching problem—but there are no kids here.

If the material of math problems is math, then the material of teaching problems is kids, right? You use math to do math, and you use kids to do teaching. Sure, plenty of math problems you find are missing the “real math”—but if you dig around enough and know what you’re looking for, you’re sure to find something. But, search all you like, you will not find kids in this class.

You may now be wondering whether I’ve been paying attention at all during the last six months of grad school. Yes, I knew before yesterday that there are no kids in education grad school. And, yes, I have been reading my Pam Grossman and I know a bit about “approximations of practice” and that sort of thing. I knew that folks had already identified the lack of kids as a problem in courses about teaching. I guess it didn’t hit me how much of a challenge this really is for developing good activities for pre-service teachers until I had to develop one of my own.

I did come up with something to do with the pre-service teachers. It did not involve the miraculous appearance of kids. Like most spur-of-the-moment, first-time activities run by new teachers (because I’m definitely a new teacher now, as new as I’d be if I switched to teaching English), it didn’t go as planned. I want to try it again, with Round 1 under my belt, and then I’ll write about it. I’m not sure if it was any good.

Until then, though, I just wanted to reflect about this problem for myself and anyone who is interested. I also wanted to ask for help. Does anyone have any good teaching problems? I’m calling them that because it makes a parallel with math problems that’s helpful for me. I don’t want teaching exercises, I want teaching problems, ones that really make people think and engage with genuine teaching. If I tried my hardest to not give kids math exercises when I was a math teacher, I want to try my hardest not to give kids teaching exercises now that I’m a teaching teacher.

It seems like the task might be more difficult, though, because like I said, there are no kids here. That seems like a real problem to me.