Dividing a Circle

I came across this item (https://www.loc.gov/resource/g3200m.gcw0013960/?sp=9) from an online copy of the 1862 Johnson's new illustrated family atlas.  The picture intrigued me for lots of reasons, but the one that stuck in my head was the fact that it shows a circle divided into nineteen equal sectors.  That's pretty remarkable, since the 360 degrees in a circle are not nicely divided into 19 equal pieces, and I did not think 19 pieces was one of the divisions possible using compass and straightedge constructions.  (I checked; it isn't.)

I wondered how the draftsperson who created the image divided the circle?  Protractors have been around for centuries, so it is possible that they simply measured the necessary angle with a protractor.  I wasn't satisfied with that, because it seems not quite precise enough.  One would need a really carefully scaled protractor to measure an angle of just under 18.95 degrees.  Maybe the draftsperson just used 19 degrees?  After all, 19 sectors at 19 degrees each would be 361 degrees, which at the scale of the drawing might have been accurate enough.  So maybe they did use a protractor.

But I wanted something precise and elegant.  Something that could be done simply, and would provide an accurate division of the circle, without losing even a fraction of a degree.  And if the process were scalable to divide the circle into any number of sectors, that would be the icing on the delicious mathematical cake.  I had not seen such a process or tool, but its existence seemed possible and even reasonable, even if not with a compass and straightedge.

After some searching, I found an amazing device called ... get ready for it ... the Circle-Divider! (https://babel.hathitrust.org/cgi/pt?id=uiug.30112037739783&view=1up&seq=220&skin=2021).  The article in an 1885 issue of Scientific American Supplement even used 19 divisions as an example. 

The basic idea uses a small wheel with radius of one unit attached to the end of an adjustable arm, so that it could roll around the perimeter of a circle with radius n units.  (It doesn't matter what units we use, as long as they are the same for the wheel and the rotating arm.)  A mark would be positioned at the bottom of the wheel, and the arm would be rotated around the center of the radius n circle, with the wheel rolling along the perimeter.  Each time the mark on the wheel reaches the lowest point, you can mark that position on the circle, and after one rotation, the circle is divided into n sectors!  (And I love that the illustration shows what appears to be a woman's hand using the device.)

This was beautiful and simple!  All it uses is the formula for circumference, which middle school students typically know.  Since the circumference of the circle on the paper is 2pi times its radius, n, and the wheel has circumference 2pi, the wheel will rotate exactly n times as it rolls around the perimeter of the circle.  (And if your circle divider draws a 19-inch circle with 19 sectors, but you want a five-inch circle with 19 sectors, just make your smaller circle concentric with the larger one, and the sectors you want will match with the sectors you have.)

It's not a traditional compass and straightedge construction, but awfully close!  No need for a ruler (since you can construct a segment n units long, given the length of one unit).  The result is theoretically exact.  And the process is scalable to any size circle with any number of sectors!  This is what I consider a precise and elegant solution to the problem.

Here's the difficulty ... I have not been able to find this tool referenced anywhere but in this short Scientific American article about it.  And the only name I have is "circle-divider" from that article.  It's not part of a typical drafting toolkit, either modern or 19th century as far as I can tell.  A librarian from the Science, Technology, and Business Reading Room at the Library of Congress is helping me track it down, but neither of us has found another reference so far.

I'm not sure if I'm hoping to be able to find an actual circle-divider (I love old tools), or if I'm more excited to actually build one (I've got plenty of cardboard and other scraps around).  Either way, the circle-divider will certainly be making an appearance in my Trig/PreCalc class next year!

Basic arithmetic is not so basic

 I found a book on "jettons" a few months ago, as I was working on a project with a Business Librarian at the Library of Congress.  It sat on my desk, mostly untouched until recently, as it was not directly related to my other work.  What intrigued me about the book initially was a diagram like the one below.  It reminded me of a musical staff or an abacus, and reading the caption, I saw that it represented an addition problem: 8342 + 2659.

It's actually a diagram of a counting board that had been used in the 15th and 16th centuries as an aid to calculation.  To add the numbers, start at the bottom.  There are a total of six counters, or jettons, on the bottom line, so five are removed, and one is placed in the space directly above, representing five, and leaving one counter on the bottom line.  Next, there are two counters in the 5 space (the original one shown on the right and the one we just placed), so those are removed, and a new counter is placed on the tens line.  Again, 5 tens make fifty, so all five of the tens counters are removed, and a new counter is placed on the 50 space.  And so on up the board.  You can find the complete process on page 257 of Francis Pierrepont Barnard's The Casting Counter and the Counting Board, published in 1916.*

The process reminds me of the "Exploding Dots" lessons developed by James Tanton, which I had a chance to teach to 7th and 8th graders online last year.  (Shout out to Ms. Anna (@ampacura) for sharing her students with me!)  Exploding Dots can take you from basic counting and arithmetic, through any number base you want, and into polynomials.  (Do our "standard" algorithms work well for all that?)

It's also interesting that the counting board is a combination base-5 and base-10 system.  As I study math history and culture, I'm realizing that base-10 by itself is not the "natural" way to count for everyone.  If you use your thumb as a pointer and the three bones in each finger as one unit, you can count to twelve on one hand, or some cultures historically pointed to not just their fingers, but also to locations on their arms and head to represent numbers up to 27.  I thought I might have the title of the book where I found this information, but I can't find it in my notes right now.  Of course, I can picture the location on the shelves ...

All this to say that there are lots of different ways of representing and calculating with numbers.  And our "standard" algorithms for doing basic operations by hand were developed as a way to save ink and space on the paper.  (Perhaps they also arose as a way of notating how the counting boards were used?)  The more standard algorithm in the 19th century, as far as I can tell was to add numbers from left to right, no carrying needed, but more space required.  Or, you could just do the work in your head, using one of the many methods published in a multitude of pamphlets at the time.  How we do basic arithmetic is definitely not set in stone!  (Unless you're an ancient Babylonian using cuneiform.)

*It's really cool to have old books sitting on my desk.  I have a few from the early 1800s piled there as well.  I am terribly grateful to have the time to actually work inside the Library.

Stop Thinking About Math Education Politically

A colleague passed along an article from The Economist, "America's Maths Wars" (https://www.economist.com/united-states/2021/11/06/americas-maths-wars) that discussed how teaching math in the U.S. has become a political issue: "Conservatives typically campaign for classical maths: a focus on algorithms (a set of rules to be followed), memorising (of times tables and algorithmic processes) and teacher-led instruction. ... Progressives typically favour a conceptual approach to maths based on problem-solving and gaining number-sense, with less emphasis on algorithms and memorizing."

The more I read and think about how we learn math, the more annoyed I get by articles like this (and there have been many!), and those that espouse the "best practices" of one side or the other.  The argument that learning math is about the basics versus about conceptual thinking is an old and dusty one, and at the moment, I believe neither side is valid.

People argue that math, like reading, is fundamental, so we need to make sure every student learns it. I don’t disagree with this, but how we teach reading is also hotly debated (https://www.washingtonpost.com/education/2019/03/27/case-why-both-sides-reading-wars-debate-are-wrong-proposed-solution/). I wonder if the perceived importance of math and reading inclines people to think of them as part of the zero-sum "pool of power" and less thoughtful about how to teach them?

I think that math (and perhaps reading?) should be taught more like we teach art, music, or creative writing. In these subjects, we encourage students to get messy and draw/play/write what they know, provide tips and critique on their work, teach basic principles of structure, suggest edits based on historical/cultural norms (hopefully a variety of them), and how to break those norms effectively. And many students find joy in these subjects, and many students who don’t pursue them as careers still continue to draw, play, or write into adulthood.

Teaching math as SOMETHING YOU MUST KNOW AND HERE’S HOW TO LEARN IT misses the point. It loses sight of the fact that there is joy in math, and helping students to get messy and use what they know, providing feedback, and teaching the history and basic structures of the subject are all important parts of the learning process.

The “conservative” idea about “teach the basics” is good: learning a basic addition algorithm is good, but needs to be connected to students’ experiences instead of valued as an isolated skill. If we only focus on the “basics” we suck the joy out of and ignore the students’ connections to the subject.

Likewise, the “progressive” idea about “teach concepts” is good: seeing how the basic addition algorithm and all sorts of methods work are good, but need to be connected to students’ experiences and thinking instead of scatter-shotted without providing focus. If we only focus on the “concepts” we ignore any procedural fluency.

Art, music, and creative writing teachers know how to teach the basics of their fields and provide feedback that values the student’s contribution. They teach some basic techniques and encourage their students to practice those and to indulge their curiosity and imagination. They critique work (including their own), identifying good and interesting ideas and suggesting changes to make the work better. They recognize that not all students will pursue careers in these fields, but help students appreciate and find joy in the work.

The back-and-forth arguments about "best practices" for math education miss the mark, in that they both make assumptions about how ALL students learn math. I learned many years ago that successful teachers build up a repertoire of strategies that they can match to their students, content, and context.* It's not about one way being right and the other wrong. Teaching math is about figuring out when to teach the algorithm and when to let students explore; who needs clear and actionable feedback and who needs to just know if the work is right or wrong; and how to demonstrate perseverance and express joy in the messy, creative, and sometimes straightforward work of doing math.

I agree that math is something you should know, but ignoring the students’ and our own ideas and experiences in order to adhere to a “conservative” or “progressive” curriculum is not the way to help students know it. Do we need to teach basic techniques and have students practice them? Of course. Do we need to encourage students to indulge their curiosity and imagination? Certainly. Let’s also learn to critique their work (and our own) effectively, to identify the good and interesting ideas, and to suggest ways to make improvements. We need to help students appreciate and find joy in math, even if it’s not part of their career plans.


(*I was lucky enough to learn from Jon Saphier and his book The Successful Teacher about repertoire and matching many years ago. I still refer to the book for guidance when I start to feel frustrated about some aspect of my teaching.)

It's Been Another Rough Week

What's a math teacher to do?

The headlines this week have been about anti-vaxxers, climate deniers, partisan fighting, and a missing white woman.  So many of the problems stem from a lack of critical thinking (about information, about race, about consequences ...) and critical thinking is what I'm supposed to be teaching!  Except I'm not in the classroom this year, and the stuff I have typically taught has been so far removed from the lived experiences of my students, that I've been finding it difficult to even think about how I can have any impact.  And while I've felt disconnected in the past, being a middle-aged, cis-gendered, white male, I really don't like this feeling of powerlessness.

I have been trying to keep up with reading articles and books and attending webinars about issues and practices around diversity and inclusion and about promoting student agency through instruction and assessment. At the moment, I'm feeling really overwhelmed by the depth and breadth of the information available and the apparent lack of effectiveness my learning is having on the world around me.  

With so much information out there, why is it that states and school boards still think it's okay to bury uncomfortable truths?  How come law enforcement can rally to find a missing white woman in a state where hundreds of indigenous women are unaccounted for?  And what can be the motivation to avoid providing vaccines and other aid to communities and even countries who are struggling (often because of policies and actions of more powerful groups)?

Even as a middle-aged, cis-gendered, white male, brought up in a religious household, I know the answer to those questions.  Too many people who look like me are too comfortable with their own power to look beyond themselves.  And aside from trying to be aware of the impact I have on others and learning from my mistakes, I'm not sure at the moment what I can do.

My response today is to write about this.  And try to figure out what else I can do.

Games and Education

A couple weeks ago, I attended almost every session at the GENeration Analog tabletop games and education conference, sponsored by Analog Game Studies.  The first day focused on board games and the second day on role-playing games (the tabletop kind, not the video game kind).

So many good speakers and interesting topics, and I ended up doubling my articles/website reading list, and following more folks on Twitter.  I want to capture some of my thoughts about the conference before other shiny objects distract my attention more than they already have.

Here are some take-aways I want to remember:

• Jorge Moya-Higueras presented some research on gamification versus playing games that indicated a lower engagement rate for gamification, possibly because of the extrinsic rewards of earning badges and points, versus the intrinsic reward of enjoyment that simply playing games produces.
• I want to pick up a copy of Critical Play by Mary Flanagan.  Her talk on "Values and 'Enculturation' in Tabletop Game" was really intriguing. 
• Steven Dashiell's talk on "Gamer Stores and Gilded Doors: Narrative analysis of minority gamers' experiences at analog game spaces" included a couple ideas I need to think more about.  In particular, he pointed out that understanding the rules is not the same thing as understanding the culture, and that immersion requires socialization.  It makes me think about how I "find my tribe" in role-playing and board games, because of the shared language I typically experience outside of the game, like Monty Python jokes and references to books like Lord of the Rings or TV shows like Firefly.  Dashiell talked about his experiences as a kid enjoying RPGs at his local game store, but not feeling part of the crowd because he did not understand that shared language.  This reminds me that I need to be cognizant in all spaces about how I help or hinder others from being welcome; signaling my own geek culture is okay, but do I also value other cultures present around me?  Also, just as Dashiell spent time on his own as a kid going back to watch and read what the others were talking about, it is up to me to do the same when I am in spaces where I am an outsider.  It also reminds me of the importance of spaces created specifically for marginalized populations, and makes me appreciate people who are willing to share those spaces with wider audiences so that me, as a white, straight, cis-gendered, mostly neuro-typical male can get a glimpse inside the thinking, emotions, and relationships of people who are different from me.  I am grateful for people like Tanya DePass, who hosts Into the Motherlands, an actual-play game, on her YouTube channel for opportunities like this.
• The keynote presentation from B. Dave Walters on "Diversity and Inclusion" was outstanding!  I loved his quote "Storytelling is sacred.  It makes us human."  And I appreciated his advice to "tell the stories you want to see in the world."  These are definitely ideas I want to remember when I return to the classroom and tell the story of math, and when I write stories and adventures.

There were other talks that included information on specific games or research about using games in classrooms.  I especially enjoyed the presentations by Catherine Croft of Catlilli Games, Scott Nicholson on game design, and Rebecca Y. Bayeck on Historical African Board Games.  I definitely plan to attend GENeration Analog next year, hopefully in person!

Test Questions

 There's been lots of online discussion and articles lately about holding students accountable, while still showing grace and compassion, especially during the pandemic and as we move out of it.  My school has chosen to eliminate semester exams entirely, even after we are back in school full time next year.  We're also moving to block scheduling.

I've been thinking about alternatives to testing, because what I really want my students to learn is much more than some math facts and procedures they can quickly produce on a timed test.  So I'm wondering if portfolios might be the way to go, for at least part of the grade.  But what do I want my kids to learn, and what goes into a portfolio?

First, I do want them to know some math when they leave my class.  It would be horribly unfair of me to send them on to the next level without making sure they can perform some of those procedures.  (And I'm also thinking about how I can put some of those into a historical or cultural context; students will be far more likely to buy into procedural stuff if they understand its context.  I'm not talking about "making math useful" - I find the useful math becomes really boring really quickly, especially since we can always whip out our pocket computing devices and find answers.)  The other things I want my students to be comfortable and confident with are perseverance in the face of uncertainty and imagination in using their backgrounds and skills.  These last two would be really hard to show or evaluate on a timed test.

Here's what I would want to see in a portfolio problem:

• A significant problem.  Not just an exercise of a math procedure, but something the student had struggled with, revised, and thought about.
• Evidence of their problem solving: persistence, creativity, use of prior knowledge, verifying their results, finding further questions.
• Evidence of metacognition: What were their main struggles?  What are the key learnings they take from this problem?
• Evidence that they understand some of the important math of the class.  For this, I'm thinking that each portfolio problem must address a different topic, and the work must demonstrate an understanding of that topic.

I would want to have each student solve and write about maybe three portfolio problems per quarter?  I want the students to have opportunities to get feedback and make revisions.  But how do I describe the bullet points above on a rubric that the students and I (and their parents) understand?  What's the criteria for success?  And is it even possible for me to provide quality feedback on so many portfolios in a timely manner?

And more questions ... By using portfolios, am I actually grading a student's writing abilities?  What about students who have trouble keeping their work organized?  Might there be an opportunity for an "interview" version where I have the students talk to me about their work?  I'll need to give some sort of homework or some way for the students to practice the skills and procedures; should I assign points to that as well?  And should there still be small quizzes to check progress and give the students feedback on that aspect of the work?

I'm glad I have extra time to think about this, rather than turning it in at the end of the period.

Thank You, and Things to Think About

Last week, I finished teaching several Exploding Dots lessons to a group of seventh and eighth graders.  I was remote, and their classes were hybrid; their teacher, Ms. Anna, and a few of the students were in the classroom with masks, and the rest of the students at home.  Most of the kids did not have their cameras on, and several were active in the chat.  When I asked them to think about a problem, I could hear nothing because everyone was muted, but I could see the students in the classroom having some animated discussions.  I was really glad to be talking to students again, and recognized that while my experience this year is different (and much less difficult) than Anna's or other teachers, I still learned a few things.

First, the students were amazing!  I'm not surprised by this, but I think I was starting to forget how exciting it is to be with kids.  I was new to their class and only coming once a week, so the novelty of the experience probably had something to do with their engagement.  Even so, learning their names and a few of their faces, and getting to hear some of their thinking was the highlight of my weeks with them.

I was also reminded how difficult teaching remotely really is.  I missed seeing the students who were not on camera, and I wasn't able to see what was on anyone's papers and whiteboards, or hearing their discussions.  Some colleagues have shared stories about their remote and hybrid experiences, and I know my few times in this classroom are nothing compared to what they've been going through.  Many kudos to Anna, Quinton, Sara, and Matt, along with all the teachers this year; you are all incredible professionals making a very difficult situation work.

Building relationships with students takes practice, patience, and perseverance, and is much more necessary and difficult in a remote situation.  So thank you to James Tanton, who trained a group of us on Exploding Dots, for demonstrating how building relationships can look, even when you can't see or hear some of the students.  I was gratified that one of the students would say "Hello, Mr. Peter!" the minute he joined the zoom call, another would make sure to appear on camera at the beginning to say "Hi!" and others would share their favorite movies, or how they were feeling when we did a quick ice-breaker question at the beginning of class.

It is likely that because I wasn't grading the students and I was teaching them something outside the ordinary curriculum, they showed a great willingness to participate.  Which has me thinking again about how I use grades and feedback in my own classroom, and what ideas I choose to teach and focus on.  The first few lessons of Exploding Dots are more about seeing something familiar in a new light and learning how mathematics works, than about one more procedure or fact.  Along with that, it brings some humanness and history to math, which I supplemented with things I've found at the Library.  I think the most important bit is that it also encourages students to play with math ideas and connect some things they already know in new ways.

During one lesson, we played the game NIM, and the students quickly figured out that there must be some sort of trick that allowed me to win the game.  In their breakout rooms, they played the game against each other and discussed strategy.  One student, thinking about her next move, said, "Wait while I look into the future."  By the end of the lesson, some students had figured out some strategy, and they were excited to show me what else they learned about the game the next time we met.  So, how do I incorporate games in my classes on a regular basis?

In the last lessons, I showed the students how Exploding Dots related to multiplying polynomials, which they were working on during the days I wasn't with them.  I took a poll in one class about how well they were understanding what I was talking about, and found very mixed results.  So I put three different problems, of varying difficulty on screen, and asked the students to solve whichever one looked challenging but doable.  On camera, the students in the classroom raced for their whiteboards, and after a few minutes of working, I asked the students if they were ready to discuss their thinking.  Almost everyone, including those off camera gave a "wait" sign, and I realized that the students who started with the problem that looked the easiest were trying out the other problems as well.  By the end, everyone felt comfortable with the middle level problem, and a bunch more were satisfied with the harder version.  How will I build this kind of differentiated activity into my lessons when I return to the classroom?

One of the best moments for me was when a student who was mostly quiet and off camera responded in the chat with "I'm not sure about this, but I think ...".  His thinking was excellent, even though he was unsure where to go with it.  I asked him if he would mind turning on his mic while I asked him some questions.  He agreed, and as he explained and clarified his own conceptions, other students started typing ideas in the chat.  For me, it was an amazing example of trust and flexibility, and both Anna and I praised his participation, pointing to all the ideas he expressed and helped other students generate.  And this happened with lots of students as we went along.  So, what was it that Anna and I were doing that helped the students be willing to publicly wrestle with new ideas?

I want that kind of productive struggle and student engagement in my classroom all the time, and I need to think about how to bring that to life even more.  In the meantime, a big "Thank You!" to Ms. Anna and all the students in 701, 801, and 802!  You were wonderful to work with, and have given me lots to think about!