Learning from my students

This week, I saw an algebra student try to find the factors of 15 by repeatedly multiplying numbers by 2 to see if the result might be 15.  I saw an advanced algebra student not remember how to simplify an expression using the order of operations.  And I saw the lesson of two very experienced teachers fail miserably.  (Full disclosure: the two teachers were me and my coteacher.)

But I also saw that algebra student excited to learn how to factor trinomials and differences of squares and know when she whether she was correct or not.  I saw that advanced algebra student smile when he realized he could understand function notation and its relationship with a graph when he used his calculator to do the arithmetic. And I saw two experienced teachers put their heads together to create a lesson that engaged their students, and uncovered some of the reasons why the previous one failed.

Teaching is full of large and small disappointments as well as tiny joys and enormous wonder.  And just like my students, sometimes I fail.  And when a lesson designed for an 85 minute block fails, that's a long 85 minutes.  Especially when you realize it's going off the rails in the first 15 minutes, and you keep scrambling to try to pull it together for the remainder of the block, but you keep failing for more than an hour.  Wednesday was rough.

At the same time, Wednesday had many moments of joy, like the two students who learned how to factor and use function notation.  It also included a group in one class becoming gleeful that they solved a problem unexpectedly by thinking about a different question.  And another class all gathered in one corner of the classroom to learn the definitions of sine and cosine.  (And I had forgotten about these last two moments until I was writing this post; it's not always easy to remember the good stuff when I'm trying to figure out where I went wrong.)

Anyway, my coteacher and I decided to revamp our thinking for Friday's lesson to figure out what went wrong on Wednesday.  We started the lesson with a non-curricular puzzle at the white boards (NPVS's for those of you following along).  The puzzle engaged the students, most groups came up with an answer they were happy with after a few trials, and four groups explained their solving process, two of them inventing notation to keep track of their work*.  Clearly, the students have the ability to think, communicate with each other, and problem solve.

The class then moved the desks out of the way, and put the chairs in a large circle.  We stood in a circle to acknowledge the power of seeing and hearing each other, which took a few minutes, as the kids were feeling a bit squirrelly, probably uncomfortable with the process.  (Who wants to be seen and heard in a math class?!)  After we sat down, my coteacher invited comments about how everyone thought the class was going.  We had asked questions three weeks ago in this format about what everyone hoped for and what success would look like for them in class.  This time, it was quiet for a moment, until one student asked if we really wanted to hear stuff, and could they be honest?  We answered yes, and another student quickly started the conversation with "I don't feel like I'm learning anything in this class."  And that opened the gates.  We talked about all kinds of issues:

• We don't like the random groups every day.
• We do like the groups we sit with (which were self-chosen).
• We don't like working at the white boards.
• After we work at the white boards, we don't have enough time to practice.
• We don't know everyone's name.
• We spend too much time on get-to-know you activities.
• There's too much homework.
• We need a break during class.

And so on.  There was one moment that stood out for me:  One of the students said that he didn't think the homework was too much, but he doesn't like doing it, so he just copies the solutions.  A few other kids jumped on him for his opinion about the length of the homework, and he started to retract the statement.  I interrupted to tell him that how he was feeling was perfectly valid, that he clearly spoke only for himself, and that I would not stand for other students ganging up because they shared a different opinion.  As the conversation continued, other students were making general statements about the class, and I continued to push them to speak only for themselves.  More students started using statements that started with "I feel ..." rather than "The class ...".

The circle discussion took longer than I had expected, and we did not get to any math content.  However, the students did agree that sticking with the same random group for the white board work for one week would be okay and that spending less time on the white boards each day and more time consolidating the ideas and then practicing them would be helpful.  The points about taking a break and homework length got tabled for a future discussion.  Everyone helped put the desks back into the pods formation, and we had a few minutes to hang out before the bell rang.  While students were having individual conversations, I checked in with a few students who had not spoken during the circle conversation.  A couple agreed with what was being said, and offered me their viewpoints.  Two students said that they liked the class the way it was, but didn't want to say anything in front of everyone else.  I also thanked the student who commented about his experience with homework for speaking his truth.

After the students left, my coteacher and I agreed that the conversation had been a good one.  While a few students checked their phones during the discussion, the engagement level was much higher that it had been on Wednesday, and the time on phones was significantly smaller.  We discussed the pros and cons of doing a white board problem on Monday, with the shorter class periods, and decided to go for it.  We believe the time at the white boards is some of the most important time we spend, and neither of us is willing to sacrifice that.  Given the length of the conversation with the students, we'll have to reschedule a couple things to account for the missed content time, but overall, we believe the trade-off will be worth it.

We'll see what Monday brings.

*One group used a series of shapes to track their work, another used series of arrows.

Constructive Arguments

Last week felt pretty good.  I'm trying to use "Non-Permanent Vertical Surfaces"* in my math classes on a daily basis, and I'm happy with what I am seeing.  In most cases, I present a problem, send the students to the whiteboards in their random groups, and watch to see where their thinking goes.  Some days, they make wonderful mistakes, and I gather everyone around a particularly interesting whiteboard to quickly talk about the good thinking, the correct paths, and the miss-takes that lead to the very interesting solutions.  Then, I send the kids back to boards, and watch as they discuss what they had been thinking and make revisions.  Some of the best days are when this process goes through a couple iterations, and students thoughtfully revise their work several times.  I always bring them back together afterward to focus on a couple of the solutions, and highlight the vocabulary and formalize the thinking.

It's been great fun to plan and watch, and the kids seem to enjoy the process.  One day this week, I needed some additional space to make some notes, and erased one of the whiteboards, and the students from that group complained that I did so.  They clearly took pride in what they had been thinking (as most kids appear to so, since they often take pictures of their work before the end of class when we erase all the boards).  I apologized to the group for erasing their work prematurely, and promised I would not so that again.

In my Precalculus class, we're starting our unit on the trigonometric functions.  To start class, I sat in the middle of the floor with the kids gathered around, and drew a picture of a bicycle.  Apparently it was a really bad picture, as the pedals were not connected to the frame, and no one was sure which side had the handlebars and which side had the seat.  After straightening that out, I indicated that the bike had ridden over a piece of gum which got stuck to the wheel, and rotated around.  I asked the students to make a graph showing the relationship between the height of the gum from the ground and the time.  Here are some of the results:

To start the conversation, I asked the students who drew the graph on the left to explain their thinking, as they had a great discussion on whether the horizontal axis represented time or distance traveled.  The other students agreed that I had asked them to graph the height of the gum in terms of time, but spending a little time on the drawings allowed us to preview a cycloid graph, which I plan to discuss later in the course.  

The other two graphs in the picture represent the work that appeared on all the other whiteboards.  The students with the pointy graph stated they believed it would be made of straight lines, since we had said the bicycle was traveling at a constant speed.  Great connection between constant speed and linearity!  That convinced most of the curvier graph groups that they had made a mistake.  A couple curvy graph groups stuck to their ideas and tried to explain that the vertical height of the gum was not traveling at a constant rate, even though the bike was.  It was a great few minutes of argument, after which I suggested we "do some math" to look at the evidence one way or another.

I drew a version of this diagram on the board, and everyone agreed that since the bike was traveling at a constant speed, the gum would rotate from point A to point C in the same amount of time it would rotate from point B to point D.  The pointy-graph folks were feeling pretty good.  Then, with some special-right triangle geometry (which we had to take a detour to justify, since that was something the students would have seen in Geometry, when they were doing school remotely) we showed that the vertical distance from Point A to point C was shorter than the vertical distance from point B to point D.  Since the BD distance was greater, the gum was moving faster in a vertical line there than along the AC distance.  A small existential crisis began arise among some of the pointy-graph-constant-speed kids, until another student pointed out that if you looked at the wheel edge on, you would see the gum moving only vertically.  Some discussions at the tables ensued, and everyone seemed more comfortable with the graph actually being curvy.

In previous years, these kinds of discussions and willingness to revise work and thinking occurred much later in the year, if they occurred at all.  Using the random groups with the NPVS's, and having a block schedule to give us time to explore made the rich conversations possible.

I can't wait to see what happens this week!

*Non-permanent vertical surfaces and visibly random groups are two of the strategies outlined in Peter Liljedahl's Building Thinking Classrooms in Mathematics.

Two weeks in

 Students have been back in my classroom for two weeks now, and I have a number of observations ...

• Most importantly, I don't have my "teacher stamina" yet.  After a day in the classroom, I get home exhausted, and have little energy for doing stuff I enjoy much less any more school work.  Thanks goodness my wife has been taking care of dinner!
• I'm also thankful for wonderful colleagues to talk to and share ideas with.  The time I spend talking to other teachers is not a lot, but it is good.
• I am really enjoying block scheduling!  I feel like we can really dig in and the students can have a chance to actually think.
• I still don't have everyone's first name solid in my mind yet, and last names are still a mystery for now.  Also, I mis-pronouned a student on Friday, so I'll need to apologize on Monday.  Learning is a process.
• I am really frustrated by the inflexibility of the online gradebook, and the expectations from parents and admin that it work like an ATM.  It's too hard to focus on learning (for me and the kids) when we have to track every last point.  On a related note, the students were visibly relieved when I suggested that we don't put quiz grades in the grade book.  Going gradeless will also be a process.
• Having students solve problems groups of three while standing at whiteboards has been really satisfying.  There has been lots of thinking and discussion.  There's still a focus on "Is this right?" and I need to reflect on how that's going and how my responses are affecting the kids' view on the process.  (I also need to remember to take a picture; it's been really satisfying to watch them work.)
• I am a little worried about "getting through the curriculum".  While we are spending time developing thinking, it does seem to be going slower.  On the one hand, we are not giving semester exams anymore, so there isn't that goalpost to get past, but then will the kids be prepared for "the next course"?
• Many of the kids (especially in my precalculus class) are worried that the pandemic has put them behind.  The fact that the phrases "learning loss" and "students are behind" really bothers me.  This is not a race, and not understanding everything we do in class quickly is not a sign that you are bad at math.

I have lots more thoughts and worries and ideas and fears and likes, but these are at the top of my brain for now.  We'll see how the next week plays out.

P.S. The department secretary got us Hagoromo chalk, and I love it!

Ideas circling in my mind

This week, I've be reading some articles on "Ungrading" and reviewing my notes about democratic classroom techniques.  (For a beginning article on ungrading, see https://www.jessestommel.com/how-to-ungrade/; and https://www.edutopia.org/article/power-democratic-classroom* for democratic classroom.)  I've also received an email from my department chair about how my electronic gradebook must be structured, with categories only for "Summative Assessment" (60%) and "Formative Assessment" (40%).  I have to think about this structure a bit, because I am worried that it still plays into the narrative that the grade matters more than the learning.  Also if "formative assessment" is assessment for learning**, I fear that putting points in the gradebook for this category, it becomes "assessment for points" instead.  My views on points in the gradebook have been evolving ever since I had to start using an online gradebook accessible to students and parents, and I need to think about this new requirement.  I'll write more on that later.

As I think about creating a classroom space where students feel like they can trust me to give good feedback and not play "gotcha" with grades, and where they are more interested in learning and doing math than in the points they can earn, I'm recalling the democratic classroom strategy of a classroom circle.  This is an activity that starts with everyone, including teachers and aides standing close together so that everyone can see everyone else, without having to do more than turning their heads.  Ideally, the furniture should be moved out of the way so that the space inside the circle is open; chairs can be placed around the outside so that everyone can sit down at some point.  Creating a good circle is a group effort, and everyone needs to participate to make it as round as it can be.  It's a good format to start a name game or other ice-breakers at the beginning of the year.  Creating the circle is also an opportunity to begin offering feedback, as the teacher (or ideally anyone) can offer comments, such as "The circle looks flat on that side" and the group can make adjustments accordingly.

Given the disconnectedness of the last couple years, I am wondering if creating some kind of script to use every time we move into a circle might provide some consistency and community.  Something along the lines of "We stand in a circle, where we can each see each other and be seen, with nothing to get in the way, with no one more important than another.  We can expand the circle to include others, and if someone is absent, while we can still maintain the circle, it will be diminished."  Still a work in progress ...  One thought I had for the first day, where we only have 15 or so minutes for each class, is to create a circle, and go around so everyone can say their name and pronouns and one thing they would like the rest of the class to know about them.  On the second day, also start in a circle, go around so everyone can say their name and something they know, think, wonder, or imagine about circles, then adjust the script to include some of those ideas.

The part of me that dislikes touchy-feely stuff is asking "Where's the math??!" but I'm willing to ignore that somewhat, for the first week anyway.  I also need to review the new curriculum documents I got this week to see what math I should be including.  More on that next week.

*Most of what I've learned about implementing a democratic classroom has come from the folks at Full Circle Leadership.  Thanks to Chris Fontana for continued encouragement!

**See Dylan Wiliam's Formative Assessment: Definitions and Relationships for more info about this.

About to start a new school year!

My first official day back at school is in two weeks; of course, I'll be there a few days earlier, with the boxes full of materials I took home when I left for my fellowship two years ago.  I am both excited and anxious to return to the classroom.  It's been a busy two years, and both the school and I have changed.

I feel a bit out of practice in front of students, perhaps even more than when I was department chair for three years.  Being away from the building entirely, I feel like I've forgotten what it's like!  I know that's not entirely true and I need to put that bit of worry aside.  Also, I will be teaching the same classes that I taught before the fellowship, so the classes should be familiar.  On the other hand, the school has switched to a block schedule, the curriculum for two of the classes have changed, there has been some turnover in the administration, and I will be coteaching a class for the first time.  So who knows what my year will look like?

I do have some goals.  I want to really work on creating a classroom community using some of the democratic classroom techniques I started trying a few years ago, and adding in some of the team- and relationship-building I have learned in the fellowship.  I also want to use more inclusive and student-centered pedagogy, including what I've learned while researching un-grading techniques, Liljedahl's building thinking classrooms, historical puzzles and games, and, of course, the see-think-wonder and related pedagogies from the Library of Congress.  My more personal goals are to write more about my own practice and about finding the joy of mathematics across the curriculum.  (I'm hoping to write a summary of my work at the end of each week on this blog.)  I also plan to maintain a work-life balance and find time to read and write for myself. 

Picture by Fernando FLeitas from Pixabay 

One big thing I've learned (and continue learning) is that I don't have to get it right the first, or even the nth time.  Living and working remotely (and remotely from home as well) has helped me be more comfortable with failure or with just being in the moment and not worrying about the end result so much.  So my last goal is to remember this when I'm in the classroom as well.

Will it all work out?
Will my worries become reality?
Will I accomplish my goals?
Will the whole thing fall apart?
Who knows?  But if it does come crashing down, I'll just yell "Jenga!" then pick up the pieces and start again.

Out of my comfort zone

There are reasons I chose to teach math.  I like the thought experiments, the proofs, the problems I can think about where all I need is my brain and maybe some paper and a pen.  And I don't expect to have to deal with this:

Okay, I only saw this* in one of the specimen rooms behind the scenes at the Smithsonian Natural History Museum.  I did actually get my hands on a buckler dory, however, as I was assigned that fish to notice and wonder about.  Our latest Fellowship professional development day had us learning about fish at the Museum, with folks who specialize in all things fishy.  So when we arrived at the classroom in the SNHS "Q?rious" classroom, we were each given a card which told us which fish we needed to find in the trays around the room to get to know up close and personal.

I found mine pretty quickly, because my card said "bukler dory".  Since it was named "Dory," I looked for something that might bear a resemblance to the character in the animated movie.  It wasn't bright blue, but it did have the right shape:

We spent some time noticing and wondering about our individual fish.  On my tray, there was a juvenile as well as an adult buckler dory.  Other trays had a puffer fish (not the poisonous kind), a sting ray (again, not poisonous, but still sharp), flounder, angler fish, different types of sharks, and something that looked like an alien brain sucker, that I wish I had written down the name of.

All the fish had been caught off the coast of Massachusetts as part of a fish population survey in the spring, and these had all been frozen until they were brought out for us to study.  They were on their way to be skeletonized and preserved for the museum's collections.

I was a little tentative at first about recording my observations of the dory, and just listed what I could see.  Then I got brave and poked it a bit (with a gloved hand) and found that one part of it was kind of like a balloon, while other parts were firm or downright bony.  I actually picked it up to see the other side, and I opened its mouth to peek inside (look at me being all brave!) and that's when one of the Museum's Fellows came over to see what I had found.  Matt just got his (bare!) hands on the thing and pointed out a number of really interesting features.  He opened the mouth all the way and explained how there are bony protrusions in the fish's throat that look a bit like molars and do the actual chewing of food.  (The tiny teeth around the mouth are all pointed inward to keep the food from swimming back out.)  There's also a mechanism to keep the food from passing over the gills.  Matt stuck his fingers in the gills to show me; they were really red because that's the most oxygenated organ on the fish.  (I decided I had already handled the poor thing enough and did not need to stick my fingers in there as well.)

Here's a picture of dory with it's mouth opened all the way.  It works kind of like a vacuum to suck in as much food as it can.  (I won't be able to watch that fish movie the same way again.)

I walked around a bit to take a look at the other fishes.  (If the group of fish you're talking about has more than one species, the plural is "fishes".)  The biology and life science teachers in our group were completely in their element, happily chatting away about their fish, and hugging them like old friends.  Okay, they weren't actually hugging, but they had no problems getting their hands on (and sometimes inside) their fish.  While I don't think I enjoyed the experience as much as some of the teachers, I did have a good time, I learned a lot about fish, and I got to see the Notice and Wonder strategy in a completely new setting.

I also got to look backstage in the specimen rooms, but I'll write more about that later.  In the meantime, here's a selfie of me and dory.  (I'm the one who took the picture.)

*I didn't catch the name of this particular specimen.  It's labeled "OH MY" because it's part of a group of items that curators bring out to show off, and that's the reaction most people have when they see it.  There were several OH MYs in the specimen room we got to see.  I think they did that on purpose.

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.

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!

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.