The fire hose CAN drown you

When I started this Fellowship, I told myself that this would be a time of "YES" - a time to read professional books, journals, and blog posts, sign up to attend conferences webinars, and meet new people. And I've been doing a good job of this. Just one problem. With all the "yes," I fear I'm starting to lose the "why". Information overload is a real thing, and the fire hose can drown you.

I felt this way many times when I was teaching, and I chalked it up to being so busy that I did not have enough time to reflect. And here I am again, this time creating my own busy-ness, and I need a moment to turn off the hose and think quietly. So here are some things I'm learning ...

1. Taking time to reflect is an active process; I can't just wait around for that time to become available. I can and need to be in control of the fire hose.  (Kind of like making time to exercise, which I am also learning I need to do when working from home.)
2. Writing helps me reflect and think. Hence, this blog post.
3. I need to listen attentively when my students and colleagues of color are trying to tell me something, and I need to respond, thoughtfully and with an attitude of learning, not with defensiveness. Silence is not helpful, and sometimes the only response I should give is "thank you."

   

   Detail from Hotchkiss notebook
   

   
   
4. There are some really interesting notebooks on the Library of Congress website: George Brinton McClellan's math class grade book, with two months of student grades, lots of algebra scrap work, and diagrams and apparent to-do lists about fortifying embankments.  (I now know what fascines and gabions are.)  Also, Jedediah Hotchkiss's math notes, in which he writes, with excruciating detail and many exercises, about the meaning of numbers, fractions and money systems.  (And now I know that 1 Talent is the same as 60 Maneh or 3000 Shekels, which comes to $1659.609 in the currency of the time.  That .609 is 60 cents and 9 mills; I'm learning so much!)  Both documents were written between 1840 and 1850.  I don't know what to do with these, but despite the fact they are difficult to read for more than one reason, I still think they're cool.
5. I really need to spend more time thinking about what my current experiences will mean when I'm back with my students next year.
• How will I incorporate what I am reading about inclusion and social emotional learning?
• What am I going to do with the online tools I'm learning? How will I use them if we're meeting in person vs meeting remotely?
• Teaching for understanding and teaching the beauty of mathematics both require that I teach the skills required by the syllabus differently.  I'm not sure yet how to do that.
• Assessments should look different.  I'm not sure yet how to do that, either.
6. "The obstacle is the path."  and  "Pretend it's on the lesson plan."  I already know both of these things, but accepting that I am serving this fellowship remotely takes some work every day.
7. I find it very easy to get sucked into a sinkhole of time when checking social media (I need to remember #1 above instead) or when looking through LoC collections (see #4 above).  I can turn off both of those fire hoses if I choose. 

There are other important things I'm learning, and I definitely need to write about them, but I just realized I've been sitting in this chair for several hours, and I need to take a walk.  And drink an actual glass of water.

Choose your own adventure!

When I was in middle and high school, I came across a series of books called "Choose Your Own Adventure."  These were really exciting and I read many of them.  I realized this past week, that doing research at the Library of Congress and doing math have a lot in common with these books (including the excitement I felt as a kid.)

While wandering through some LoC collections of online books about games, I came across Games of skill, and conjuring: including draughts, dominos, chess, morrice[1].  "Skill and CONJURING"?!  My inner 13-year-old was immediately intrigued and I spent a bunch of time paging through.  I found the games "Morrice", also known as "Nine-Men's-Morris", and the game called "Fox and Geese".  I had heard of these games before, but never really played them, and since I was looking for things I and other math teachers might use, I found online versions of the games and played against the computer a few times.  I lost.  Every time.  The rules sounded simple enough, but the strategy is something I will need to think about a lot more!  The fact that a computer was always picking the best moves tells me there is some sort of algorithm that can be used, so I do plan on using these games in my classes.  Even though I don't know the strategy.  I'll learn something along with my students.

I continued through the book (choosing a different adventure), and found a bunch of puzzles.  One that intrigued me was this one, from page 91:

Based on the third figure, I realized that the rectangle (I decided that figure 1 was not just a parallelogram, but also a rectangle.) had to be cut into some sort of stair-step shape, and looking at the solution a few pages later, my thinking was confirmed:

Using the stair-steps to get what looked like a square (figure 2) was more intriguing, because I wondered if any rectangle can be cut this way in order form a square, or was there something special about the rectangle in the original problem.  So I posed myself this problem:

Is it possible to cut any rectangle into two equal pieces, so that the pieces may be rearranged to form a square?  I gave myself the constraint of the pieces being equal, and decided to stick with the stair-step division, so that I could at least answer that much of the question.

I needed some way to represent this problem, so I quickly sketched some diagrams and possible algebraic representations on a legal pad:

I didn't know exactly what I was doing, and the algebra was not obvious right away.  There were lots of "adventure paths" here.  I had to decide how many stairs and what variables might represent which aspects of the picture.  I knew that the area of the rectangle and the area of the square had to be the same.  I quickly decided that all the stair-steps needed to be the same size in order to fit the pieces back together.  I also figured out, based on the diagram in the book and some diagrams I drew (the bottom left one in particular) that there would be some number of divisions along the short side of the rectangle and one more division along the long side.  (As I write this, I am wondering if that's always true, or could I arrange the steps another way ...)  I didn't get very far with the problem, and I had other things to do, so I set it aside.

That night, while I was getting ready for bed, my mind wandered back to the problem, and I thought about focusing on the ratio of the two sides of the rectangle. If I divided the short side, x, into n pieces and the long side, y, into n+1 pieces.  Representing the two sides of the square using those variables, I could probably come up with a ratio of the sides.  Since I was tired, I made a note about this idea and went to sleep.  The next morning, I played with this idea and quickly drew a diagram of the rectangle and the resulting square using these pieces, and did some algebra to determine the ratio of y to x:

Since the number of divisions has to be a whole number, I made a table to determine some of the possible rectangles.  I realized that the ratios did not look right, as y/x should have been bigger than one, rather than less than one, and the the pink and black diagram showing the 3x4 rectangle certainly was not going to form a square.  My algebra looked okay, but something was seriously wrong.  When I met an untimely end in a Choose Your Adventure Book, I always backed up a page or two and tried again, so ...

I rechecked my diagrams of the rectangle and resulting square, was happy with the variables, and got a new sheet of paper to start the algebra again.  I immediately saw that I had changed the n+1 from the diagram to n-1 in the first run at the algebra, so redid the algebra on a new sheet, and made a corresponding table of values as well as a few diagrams to verify the work:

It does not surprise me that the base area of these rectangles/squares were perfect square numbers, and I did not think it unreasonable that the table was skipping the odd square numbers; but why was it skipping some of the even square numbers as well?  The resulting side lengths of the squares were increasing in a quadratic sequence. Why not a side length of 8? or 16? Maybe the fact that these are powers of two has something to do with it.  But then why is a side length of 14 not on the list?  Is there something about the factor pairs of the square numbers that worked?  Maybe there's a different way to divide up a rectangle to get these sizes.

As I started to pursue these ideas, more questions and ideas popped into my head, and I realized the adventure could continue:

• The numbers in each ratio are consecutive perfect squares. What if they're not consecutive?  What happens when I try to make a square?  What happens if they're not perfect squares, like 3/1; how close to a square can I get?  Maybe it's time to get out the ruler and scissors to create some models.
• I can take any square, and split the sides however I want, making the square 5x5, 6x6, or 14x14.  The units don't matter; it's still a square.  Can I divide the squares into two congruent stair-step shapes to get a rectangle?
• The limit of the ratio y/x approaches 1 as the number of divisions increases.  What does this mean?
• Are there other ways to cut a rectangle in order to get pieces that rearrange to a square?  I limited myself at the beginning to two congruent pieces that had a stair-step shape.  What if I removed or eliminated one of these parameters?
• As I was copying the pictures for this post, I realized that the original problem said "parallelogram", not "rectangle".  Is it possible to cut a parallelogram that is not a rectangle into shapes that could form a square? (I know how to cut a parallelogram to form a rectangle, so this should be possible ...)
• And what about the original solution shown in the book?  That showed five steps in one direction and six in the other, so according to my formula, the ratio of sides has to be 36/25, right?  The solution also mentioned dividing "a piece of card"; a 5x7 card has a side ratio close to 36/25.  Maybe that's what the puzzle was using.  Did they use 5x7 cards in 1865, when the book was published?

Now I have lots more to explore, including the history of stationery.  I don't know if I'll answer all these questions, because working on one of them may lead to other more interesting paths.  Or some other completely different and shiny math problem will show up and beg me for attention.  I'm always interested to see where the next path takes me ...

References

1. Games of skill, and conjuring: including draughts, dominoes, chess, morrice. George Routledge and Sons, London, 1865.

Lost in the Stacks

For the last 31 years, I have lived by a bell schedule and always had papers to grade or lessons to write.  All of a sudden, my days have become very free-form and my schedule nebulous, and I sometimes feel a little lost without the school-day structures.  While I am free to explore the Library's online collections, and wander through as many virtual stacks as I'd like (and go to the bathroom whenever I feel the need), I have found myself jumping on more tangible projects right away, because I am so used to producing or accomplishing something in a short amount of time.  A couple of my Learning and Innovation Office colleagues today reminded me that I can be patient, that they expect me to explore for now, and that the "production curve" should be very low at this point.  Part of the process is to dive into ideas, note what I'm thinking about them, and keep track of where I find them (since it's easy to get lost even when the stacks are virtual).  When something makes me want to share its discovery with someone, that's the time to start writing.  This takes some getting used to.*

On the other hand, I do have a couple small projects to work on.  One of them involves research on a particular topic, and the other two involve actual writing.  One of the latter is for a blog post introducing myself on the LoC blogs.  I'll link to that when it gets posted.  Which won't be until October.  Because nothing just "gets posted" to the LoC blogs.  There is a lot of editing that happens, and I have been warned that anything I write will come back to me with lots of "red ink" indicating what needs fixing and rewriting.  Three things jump to mind when I hear this.  The first is something my dad told me when I was a freshman in high school: "There's no such thing as 'good writing', there's only good rewriting."  And a now-retired colleague from the English Department, who was an informal mentor to me, told me "Peter, once you write something and send it off, it's no longer yours, but it's better to get it out there and let it benefit from the collaboration."  This is all good advice that I try to live by, but the third thing I think about was the time that another friend from the English Department and I sat in each others' classes for two weeks.  She did the math homework and took the quizzes on logarithms, and I completed the writing prompts and essays for Crime and Punishment.  She pulled no punches when grading those essays, and after I got over the shock of having that red ink on my essays (despite those earlier aphorisms), I did learn something about myself and my writing.  So, thanks, Laura; those two weeks talking with you about what it's like to be a student in our classrooms and growing as a teacher and a learner were really important to me.

So where in the stacks was I lost this week?

• How primary sources can be used to meet the needs of diverse learners, particularly the needs of differently-abled learners.  And a history lesson on the intersection of the Civil Rights movement and the Disabilities Act.  Here's the link to a video of that webinar if you're interested.  (Scroll down to the June 3rd webinar.)
• The Montana Farmer-Stockman newspaper from the late 1940s to early 1950s represented data about raising turkeys and sheep in all sorts of different graphs and charts.  The Chronicling America website is a gigantic treasure trove of newspapers, and I may be writing about what the various representations reveal about the data on turkeys and sheep.
• The Geography and Maps Division of the LoC has a mind-boggling array of items, from portolan charts and survey maps to data sets (and not just Census data!) from many countries and mapping COVID strains.  John Hessler, one of the specialists in that division provided me with so many avenues of research, I spent two days poking around in those collections, and I'm not alone.  Researchers for the courts and for Congress access this kind of material quite often.  Portolan charts are one of the things I've gotten really excited about, and I'll probably write about those at some point.
• The National Book Festival is going to be amazing this year.  (Not that it wasn't amazing in previous years, but the work that so many teams did this year to make it all virtual and available to everyone is incredible.)
• The LoC has a great many blogs with really interesting posts that I won't ever have the time to finish reading.  I did find one that described how a scientist relied on crowd sourcing to gather data about the Leonid meteor shower of 1833.  (That's not a typo; I really mean 1833.)  I forgot to copy the link for that one at first and had to retrace my steps to get it.  Took a while.  This is what I mean about getting lost in the stacks.

... and I have several dozen links to other ideas I still want to explore.  I am really grateful for this time I have to indulge my curiosity.  And I'm also grateful for the folks in my Office and the various other divisions who have been really kind about sharing information with me and pointing me in new and interesting directions.  Thank goodness this is all online for now, otherwise I really would need help finding the bathroom.

 

*To folks teaching or learning remotely, on a hybrid system, or even in-person right now, not having a bell schedule probably seems like a lame complaint.  It is, and I'm not really complaining about it, just noting that the shift in thinking exists.

Image Source: Library of Congress. Interior. Book stack. [192-?] Photograph. Retrieved from the Library of Congress, <www.loc.gov/item/2005681088/>.

Why I'm Interested in Primary Sources

As I was reviewing blog posts and webinars from the Library of Congress, I found a good community builder that might be useful, especially for remote learning.  The activity is called "Hide and Seek", and the example used the picture shown here.  You can find the entire webinar with a couple more activities and a bunch of ideas here.  (Scroll down to the May 27th entry.)

Brainstorming with a colleague, there are also a bunch of number-related questions of various levels we could ask about this picture: How many people are in the picture?  If the buildings shown have stores on the ground floor, and apartments above, how many apartments might be visible in this block?  The photo was taken in New York City around 1900; what was the population of NY at that time and what was the population density?  Could we use this picture to estimate the population density?  These questions are not necessarily the most profound things to ask, but maybe learning to ask questions could be the bigger idea here, rather than just finding the answer to a problem.

Solving problems is something we do quite a bit in math classes, but I (and I know many other teachers) try to incorporate reasoning skills and other practice standards into my lessons.  So I've been thinking quite a bit about how primary sources like this are related to math classes, and at this point, I've come up with a few general categories.  We can use primary sources to ...

• practice asking math questions (like we did for the picture above)
• jump into a modeling activity, where students have to solve an ill-defined problem by making and identifying assumptions, defining variables, and practicing an iterative solving process
• engage students with a story or historical context for the math ideas
• represent data and interpret data representations (and the biases inherent in those representations)
• connect math ideas to students' experiences by looking at documents through different lenses: mathematical, historical, personal, or social justice
• make clear connections between thinking skills they use in other classes and in life and ways of thinking mathematically, like identifying facts versus inferences or asking open versus closed questions
• spark inquiry-based learning activities.

All of this really excites me.  I don't have lots of details or examples of these uses worked out yet, so that's something I'm going to work out as I continue my research.  I plan to post some ideas here, and hopefully on one of the LoC blogs as well at some point.

Picture Source: Detroit Publishing Co., Publisher. Mulberry Street, New York City. Photograph. Retrieved from the Library of Congress, <www.loc.gov/item/2016794146/>.

A New Adventure

 I had forgotten that I started this blog until recently, when I needed to review some things I had written as part of my application to the Albert Einstein Distinguished Educator Fellowship.  So now I've started the actual Fellowship, working at the Library of Congress, and thought this would be a good time and place to record some of my thinking and learning.

I first heard about the AEF Program at an NCTM or ICTM annual convention about 15-20 years ago.  Having small children at the time, I thought taking a year off to work in Washington D.C. would not be a good idea right then.  About ten years ago, my wife found herself talking to a former Fellow, and we talked about the possibility of applying, but the timing did not seem right at that point either.

In the last few years, I've started to feel a little stale in my teaching practice.  I was feeling a little constrained in what I could do given the time and curriculum I was working with, and I had been teaching the same classes for several years.  Last October, I was talking with my wife about my feelings, and she suggested I apply for the Einstein Fellowship.  My older son had already graduated college and was out on his own, and my younger son was away at school himself, so the timing seemed good if I did get the appointment.  So now, nine months after submitting the application, here I am, working in the Library of Congress in the Learning and Innovation Office.  Well, not really in a Library building; COVID-19 has me and my colleagues working remotely for now.

And my brain is being stimulated in so many new ways!  I am learning lots of new acronyms (sometimes even the person using the acronym is not sure what it means), meeting some fantastic people who are Fellows working in other agencies or on Capitol Hill, browsing through the LoC's online collections, meeting more new people in the LIO* and beyond, and thinking about how the primary sources available online can help math teachers, and how a math teacher might view some of these sources through the "magic glasses" of math.  The information flood in the last two weeks has been like drinking from a fire hose, but at least the water's been warm.

As I browse the Library's online collections, read the articles and blog posts, and watch some recommended webinars, I've found myself deep in several rabbit holes of information.  It's been fantastic!  I already have ideas about some avenues I'd like to research, and articles I'd like to write.  I'll write more about those later.  In the meantime, here's a picture of the Library of Congress when it was still in the Capitol Building.  I think this is the version of the Library that was made of iron, to reduce the risk of losing the collection to fire for a third time.  The LoC is no longer in the Capitol (with somewhere around 200 million items in the collection, there wouldn't be room for the congress-folk), but has three buildings right across the street.  The Jefferson Building is the best known with its marble staircase, amazing murals and sculptures, and the iconic Reading Room.  When the office can return to the building, I'll be somewhere in the Madison Memorial Building or the Adams Building; both are just across the street and connected by tunnels.  (Getting to see the tunnels and the other non-public spaces is really exciting; I hope we can be back in the building soon!)

I'm really glad to have this opportunity, and tremendously grateful that the folks who oversee the AEF Program and the folks who I'll be working with saw something in my application that matched what they were looking for.  I look forward to learning with my fellow Fellows and with my LIO colleagues.  My plan is to post something here regularly about my experiences, and show my work.

Picture Credit: Chase, W. M., photographer. Congressional Library, U.S. Capitol. Photograph. Retrieved from the Library of Congress, <www.loc.gov/item/2004674579/>.

* LoC is the Library of Congress, and LIO is the Learning and Innovation Office.  This office is part of the CLLE division, but I'll have to look that one up to remember what it means.  See what I mean about acronyms?