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.