Saturday, May 28, 2022

Still Angry, but working on it.

Well, the last week has again been pretty upsetting for me as a teacher.  A school shooting and the political responses to it (or lack thereof) have been really disheartening.  Why does there seem to be a political will to do nothing about the gun epidemic (especially from those proclaiming to be pro-life), even as it affects teachers and children?  Why are teachers not trusted to teach children well, but we're hailed as heroes when we go above and beyond to do more for our students?  Why is it that in some places in this country, incorporating social-emotional learning into math class is seen as unnecessary, when we are having lockdown drills at the same time?  Is teaching the racist parts of U.S. history really so much more traumatic than having a shooting in the classroom?  I'm feeling pretty disrespected right now.

I know I am not alone.  I have talked to colleagues who are leaving the profession because of the toxic culture in education.  And a school can say, "we're not like that", but when administrators are more worried that parents will complain about bad grades than they are about their teachers' well-being or credibility, it's clear there is a link between the culture at large and the individual experiences.  It certainly makes me question my decision to go back to the classroom in August.

I hate that I feel this way.  There have been so many teachers in my family, and I've had so many teachers who have inspired me, that I can remember wanting to be a teacher since I was in third grade.  Being a teacher is so much a part of who I am, that it's hard to imagine doing anything else.

So, I am writing this blog post, reading about building Thinking Classrooms*, researching "ungrading" practices, and working on myself to provide a welcoming classroom for all my students.  I've also written my representatives in Congress about gun control.

*Specifically, Peter Liljedahl's Building Thinking Classrooms in Mathematics, Grades K-12.

Image by athree23 from Pixabay with modifications by me.

Sunday, May 22, 2022

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.

Monday, May 16, 2022

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.

The word "Coexist" written with symbols of different religions and genders, including a peace sign.

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.  

Black and white photograph showing a hand-drawn sign that reads "Racism is not Patriotism".
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.

Sunday, May 15, 2022

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?!

Saturday, May 7, 2022

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 (, 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:, and more on "Notice and Wonder" here:  The Library of Congress uses a variation called "See, Think, Wonder" or "Observe-Reflect-Question"  

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:, and this page has another example of addition: