I get by with a little help from my friends, and a good math problem.

 It's been a bit of a weird week.  I started out with all sorts of feelings about what's going on in the world, found out I will be co-teaching next year, went to a meeting that was very disappointing, fell behind on some writing I'm trying to finish, saw a really interesting puzzle on Twitter, then got together with some friends online last night to talk and play games.

I've already written about how Monday was going, so let's start with the co-teaching.  While I've met my co-teacher before, I don't know them very well, so we're going to need some time to get used to each other.  I've co-presented at workshops, conferences, and PD sessions lots of times, but having someone with me in the classroom on a daily basis is going to be a bit different.  I'm not sure how well I play with others over the long term.  Also, I really hope we have a common planning period.  That doesn't always happen between co-teachers, and it's going to be be important, for me at least, as I get used to co-teaching.  (I've had aides in my classes before, but this is not the same.)  Planning together will also mean that my planning time will be less flexible, and going back to the bell-scheduled day is going to be a big change from what I am currently doing.  On the other hand, working closely with someone on a daily basis is a great opportunity to learn from them -- about teaching, about the students we share, and about each other.

I'll skip over the disappointing meeting.  It's enough to say that there were lots of missed opportunities for learning, exploration, and collaboration.  As I think about it, I should probably let the meeting organizer know that an exit ticket with some critical remarks came from me.  I dislike providing difficult feedback without taking responsibility for it; I don't think negative feedback without the chance to follow-up is very helpful, and I don't like when it happens to me.

Most of the rest of the week felt unproductive.  My lack of focus on Monday carried through, and I was unable to finish some writing projects whose deadlines are coming up.  I'll have to push a little bit more this coming week.  I'm not disappointed or angry with myself about the delay, and I'm trying hard not to say "I should have done more" because it was probably important for me to process what I was thinking and feeling about current events.

As usual, James Tanton provided an interesting puzzle on Twitter, which led me to a little more research about "Langton's Ant".  It's an interesting situation that might be a good one for a non-curricular problem to use with my math students and an intro to automata for my computer science students.  I added it to my file of interesting math problems.

Finally, I met up online with some friends from the Chicago area whom I have not talked to in several weeks.  We had scheduled a D&D game, but spent the first 90 minutes catching up, and just enjoying each other's company.  We commiserated about current events, celebrated the end of the school year, and shared our hopes for the coming year.  The result is that I'm feeling more positive about the work I am doing, better able to cope with the ridiculousness in the world, and ready to meet the coming week.

The advice to "Check in with each other" is a really good thing, and just as good for the checker as the checkee.

How's your Monday going?

Today, Monday, has been rough.  I have too many thoughts and emotions to focus on much, which sucks because I have a lot I need to focus on.

So, here's me showing my work - the stuff that's going on in my brain right now.  And, yes, putting myself out there is work.  This is not easy.

First, a little background, as I keep this personal, local, and immediate:  I was born three months premature, back in the 1960s, and I spent the first three months of my life in the hospital, in an isolette (a clear box where I could get oxygen and be less exposed to infection).  I also went to Catholic schools from kindergarten through 12th grade.  So when people talk about "abortion rights" or "pro-choice" or "pro-life" or "right to life" it hits me personally.  I know that there have been people like me, who at six months gestation might have survived outside the womb, but unlike me, did not get that chance.  I know that life is sacred, a gift to be cherished, a blessing that requires nurturing, attention, and love.

I went to college and met people of other faiths and belief systems.  I met people whose social-economic situation was vastly different from mine, in both directions.  I met people whose race and culture and outlook on life were very different from what I was used to.  And, as I was taught, I believed that their lives were sacred, cherished, and blessed gifts as well.  Not all of them believed abortion was wrong, some had had abortions, and most believed the decision to end a pregnancy is never an easy one.  I realized that while my faith at the time prohibited abortion, I was not in a place to judge them or their beliefs - who among us has the right to cast the stone and all.

As I started my career in the math classroom, I took comfort in the fact that teaching math seemed objective, that I did not need to address difficult conversations around "right to life" or around any other hard topics for that matter.  Racism, sexism, classism, and all the other -isms, I thought, had no place in the math curriculum.

Then I met Hector who was terrified to show his father a failing grade, Ed who was about to be a father at the end of his junior year, Dulce who had to work every day after school to support her family, Marcus whose only meals were the ones he ate at school, Eliza whose anxiety caused her to miss many days of school, Jasmine whose anxiety kept landing her in the dean's office and labeled a behavior problem, and Jessyca who was homeless and sleeping on friends' couches.  And the list goes on, for thirty years.

And in that time, I realized that yes, Black children do have very different experiences than White children; girls do have very different experiences than boys; Hispanic, Asian, Middle Eastern, and most children have backgrounds I can never really know or understand.  But each of them is a life that is sacred, a gift to be cherished, and a blessing that requires nurturing, attention, and love.

Just as I learned, as the father of two sons, that loving and cherishing looks different for different people, so too, loving and cherishing my students looks different for each student, even in Math class, especially in Math class, a subject where the roots of classism and sexism and racism run deep.  So I try to meet students where they are, honor their backgrounds (in math and life), and see them as people, not just a brain to fill with math facts.  Besides, math facts without humanity are boring and devoid of the wonder, joy, and beauty that I see in the subject and in my students.

Which brings me to today.  I saw another tweet from someone stating that "two plus two always equals four" implying that math exists objectively, without human interaction.  I read stories about two more mass shootings over the weekend, one clearly racially motivated.  I heard teachers tell stories about being unable to teach current events in their history classes because it might make someone uncomfortable.*  And my son tells me he is participating in pro-choice protests.  

All of these things (and more) are rolling around in my head.  And I feel pride for my son taking a stand for what he believes in.  I feel guilt that I have not had the courage to do the same.  I feel anger at government officials for not clearly speaking out against racism and for embracing people who express xenophobic, misogynistic, ableist, or racist views**.  I feel disappointment that people still think we can teach math without thinking about our students.***  I am worried for my students who have to grow up in a society with people who claim to be "pro-life" but still refuse to do anything about gun violence, see violence against BIPOC and LGBTQ+ people as needing only "thoughts and prayers", or see less value in the life of a woman than in the life of someone yet to be born.  I feel helpless and frustrated that I don't know how to fix this.

I know my feelings, those of a cis-gendered, straight white male, are nowhere near as traumatic as what many other people are feeling today.
But, yeah, that's my Monday.
How's yours?

* If topics that make students uncomfortable have no place in school, all math teachers would be out of a job.

**And not just government officials.  I have family members who have "held their noses" about all the -isms a candidate espouses and voted for them just because they are "pro-life".  How do you tell someone you love that you think they are hypocrites?

***And for what it's worth, 2+2 is not always 4; sometimes it's 100, or 11, or even 10; sometimes it's |||| or IV or any number of symbols I can't get blogger to print (yet).  So stop using math to push your right-wing agenda; you're teaching your children to hate not only math, but also those who do it differently from you.

Two Hundred Years of Progress?

 I had the opportunity to look at some cyphering books from the 19th century this past week, and found a problem I thought interesting.  Ciphering books were books in which students copied math problems and solutions, often after first solving them on a slate and getting the approval of their teacher.  The books served as both math notebooks and reference books, and were often kept by students as they entered the business world, and sometimes passed from one member of a family to another.

Here's a page from a book composed by Christopher Render around 1800.  The book is part of the Ellerton-Clements Cyphering Book collection at the Library of Congress.  The collection has not been digitized, unfortunately, so is only available to those visiting the Manuscript Reading Room of the Library.

The problem at the top of the page reads: "Two men depart from one place suppose them to be James and Jerry.  James starts and travels 26 miles [per] day, seven days after Jerry starts and travels 37 miles [per] day.  I demand in how many days and in how many miles travel will Jerry overtake James?"

The first thing I thought about this was that the problem sounds very much like some of the word problems in modern text books.  Also, who travels 37 miles each day consistently until they catch up with someone else?!  (Apparently, I feeling a little salty about these types of problems.)  It turns out that many of the word problems posed to students in the 19th century came after the statement of a rule, possibly with explanation but often not, perhaps an example, and several (or many) practice problems without context.  The word problem itself provided information very much like the practice problems.  If a type of problem had a number of different variations, the rule would be broken up into cases, each with its own example, practice, and word problems.  After a few rules (and lots of practice problems) there would be a section called "Promiscuous Problems" or what we call in modern books, "Mixed Practice".

There's much more to look at on this page, but I'll write about Christopher's calculations later.  Right now, I just want to sit with the knowledge that many of the currently available textbooks and what students are often currently required to do looks very similar to what was happening over 200 years ago.  Have we really learned so little about how students learn math?!

Stretching with STEM Yoga!

A few weeks ago, I ran a short online session with the folks in my office.  It's part of a series my fellow Einstein Fellow and I dubbed STEM Yoga, to help stretch our thinking about using primary sources in STEM classrooms.  Most of the rest of the folks in the office have a background in the Humanities, so we've tried to tailor the series to be accessible to a wide audience.

I started out by showing this item (https://www.loc.gov/item/92518152/), and asking everyone to Notice and Wonder, a thinking strategy I've been using for a long time after seeing Annie Fetter give a talk about it at a conference, and later at a Metropolitan Math Club of Chicago dinner (Short video here: https://www.youtube.com/watch?v=a-Fth6sOaRA, and more on "Notice and Wonder" here: https://www.nctm.org/noticeandwonder/.  The Library of Congress uses a variation called "See, Think, Wonder" or "Observe-Reflect-Question" https://www.loc.gov/programs/teachers/getting-started-with-primary-sources/guides/).  

There were lots of items to notice in the picture:

• It is a woodcut.
• There are two men sitting at desks with an angel holding books in between.  (I pushed the thinking on this one, and asked "How do you know it's an angel?"  The answer was "She looks like she's floating and she has a halo.")
• There is a ribbon with words on it, possibly in Latin.
• One desk has math symbols on it, the other has an abacus.
• The man with the abacus has a pile of coins by his right hand.

There were some other things to notice as well, and then we went to questions:

• What do the Latin words mean?
• Is it really an abacus?
• Who made the drawing and why?
• Is this an allegory?
• Who are these people?
• What do the numbers mean?

We discussed which questions could be answered quickly, and which my take additional digging.  Quickly, we determined that the Latin writing was two names: Boethius and Pythagoras (on the ribbons near the two men) and the phrase: "Types of Arithmetic".  We also looked at the item record from the Library of Congress (scroll down on the linked page with the image) to find out the appeared in a book by Gregor Reisch in 1503, Margarita philosophica.  This led to all sorts of other questions about who these people were and what else was in the book.  Since that would require additional research, we moved on to the question about the "abacus".

I explained that it was probably not an abacus, as it has no frame, and is probably a medieval counting board.  This would have been made of lines on a table separating the space into regions representing ones, tens, hundreds, and thousands (and more decimal places as necessary).  The beads are actually counters called "jettons" (French for "token"), and the pile of coins under the man's hand were spare jettons*.

I asked if anyone knew how an abacus or counting board worked, and no one was really sure.  So to illustrate, I demonstrated how James Tanton's "Ten-One machine" from his "Exploding Dots" lessons worked.  I showed how the number 5 could be represented by five dots in the first (1s) box, and 10 by ten dots in the first box or one dot in the second (10s) box.  We talked about the meaning of the boxes, and then a bit about language:

• 12 could be represented by twelve in the one's box, or one 10 and two 1s, but then we might read that as "two-teen", just like 14, 16, 19, etc.
• Also, numbers like 42, 62, 92 are all read like "four-ty two" (four tens and two) or "six-ty two" (six tens and two), by 22 is not "two-ty two" because the English way to say numbers has some roots in base 20.
• "Eleventy" (110) was an actual word at one time (and not just from Tolkien)!
• We can read the number 1200 as "one thousand, two hundred" (one token in the 1000s space and two in the 100s place) or as "twelve hundred" (twelve tokens in the 100s space).

The word play got everyone excited (did I mention they are mostly Humanities folks?) and they were ready to try representing some numbers on their own.  I gave them a google jamboard with tokens and a counting board, and asked them to represent 357.  No problem.  Then I asked them to represent 265 just under that, and add the two numbers together using the tokens.  Here's a screenshot of what one of them did:

Others moved the jettons around:

And several folks explained their thinking, with several different methods.  Some translated to numbers, some worked left to right, and some right to left.  They asked each other a couple questions, and one person who had made a mistake originally talked about what she was thinking and what she learned.

We went back to the original picture, and folks said they could translate the numbers on the counting board, but wondered if the 1s was at the top or the bottom as they looked at the picture, and if the numbers meant anything.  We now had more questions to research.  And those Humanities folks (who often cringe about math) said they enjoyed and understood what we were doing!

As James Tanton would say, it was "brilliant"!

* You can read more about jettons in this book, by Francis Pierrepont Barnard, published in 1916: https://babel.hathitrust.org/cgi/pt?id=mdp.39015017345441&view=1up&seq=4&skin=2021, and this page has another example of addition: https://babel.hathitrust.org/cgi/pt?id=mdp.39015017345441&view=1up&seq=257&skin=2021.

 

Rollercoasters Are Scary!

The other day, I was giving my midyear presentation for the Einstein Fellowship, and talked about how I filter everything I do and learn through the lens of going back to the classroom.  I know that's where I belong, teaching math to students is what I do best, and watching them grow personally and mathematically is what I enjoy.

HOWEVER ...

As much as I am excited to go back to the classroom in August, I am just as afraid.  I won't have been in front of a class of students in over two years, thank you not at all, covid.  My experience as a Fellow has been so different from the experience of my colleagues, and even more so, as they've taught remotely, in a hybrid setting, and with masks on every day.  That's a situation that can bind people together, and I'm not there.  Additionally, my school has switched from a nine-period-a-day, 42-minute class period schedule to a block schedule with 85 minute periods.  The closest I've been to a block schedule is teaching summer school.  Also, I think two of the courses I have most recently taught have gone through some curriculum changes, and I think the third is in process.

The upshot is that at an age where I should probably start thinking seriously about retirement, I will most likely feel like a brand new teacher all over again.  That scares me; that's a discomfort I've not felt in a long time.  And it's a feeling I've been missing.  I applied for the Fellowship to shake me up a bit.  Well, I've been shaken, stirred, spun around, and turned upside down the last two years, and it's been great.  So while the roller coaster I'm going to be on next year will be familiar in some ways to the one I left two years ago, there will be new twists, unexpected drops, and exciting turns.  It's going to be scary.

And it's going to be fun.

Photo by Ittsky from pixabay.

Remember ...

Since I started teaching umpteenish years ago, I have only been out of the classroom for six years.  First, when I took off a year to start my Masters degree, then three years as Department Chair, and now two years working on a Fellowship with the Library of Congress.  Being outside the classroom this time has allowed me to really indulge my curiosity and flex my writing muscles in ways I never have before, and I'm always happy to share what I learn with others.

This week, I've been working on a webinar that a colleague and I are making about teaching with primary sources.  We presented our draft to another team member, who asked at the end, "So, what does this look like in your classroom?  What advice do you have for how to implement these ideas?"

Oh.  Right.  With actual students.

I had a sudden flashback to when I was the department chair and not teaching any classes, but still expected to be the "instructional leader" for the department.  At that time, I felt like I was losing touch with what it meant to be in front of students, and it's one of the reasons I returned to the classroom.  Now, I've been away from the classroom again, and I'm surprised (and a little disappointed) that I'm sliding past thinking about the actual teaching experience.  Again.

It was good to have this reminder, not only as I prepare for this webinar, but also as I approach the end of my Fellowship and look forward to returning to the classroom in August.  All the content, strategies, and new ideas I have experienced won't go very far until I seriously consider what it all might look like, away from the sterile professional development environment and plopped down in the middle of a wonderfully personal, messy, and exciting classroom.

So I'm remembering using an individual to group to classroom discussion strategy for starting a Notice and Wonder routine.  I'm thinking about the different colored sticky notes for students to write their reflections and questions on.  And I'm reviewing all the checking for understanding routines I use to take the temperature of the class.  Since the upcoming webinar I'm giving is not specifically about this kind of stuff, and it's difficult to model some of these strategies in a remote situation, I have to think creatively about how to at least tell the story of how I've used them.

But that's the beauty of teaching for me -- figuring out how to tell the story of my subject in such a way that the students become part of that story.  Writing and presenting webinars about ways to tell the story continues to be fun.  But actually getting my hands, heart, and imagination in contact with students is something special.  It's far too easy to forget that (and too many people making decisions about education seem, like me, to forget).  

It was good to be reminded.

Image by Pexels from Pixabay

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!