(October 1 and 8, 2015) I’m combining 2 sessions into one report,  mathematical conversation topics  grouped by how they went over with the students.


  • Compasses: Some students are struggling with their (and my) compasses.  Parts get lost.  Positions slip.  The leads get lost.  No one agrees on which compass model (and we have about 8-10 of them in class) is best or worst.


  • Celtic Phalerae: I was so excited and this was such a flop. In week 1 some of the kids were excited about Celtic mandalas.  In my mind, mandalas are compass art.  But when I researched, I found that most Celtic mandalas seem to involve not just compasses, but the mathematics of knots.  I wasn’t ready to go there with this group.  So I found something so fascinating (to me, at least) instead – round metal objects that horses and sometimes soldiers would wear. In the Celtic Phalerae, there was a certain pattern to their design, compasses are used, they are beautiful, and their history is so interesting.  Unfortunately, I was the only one who thought this, though, so we soon returned to Euclid.  (Click on the hyperlinks if I’ve aroused your curiosity.)



  • Bisecting a line segment using compasses and straightedge only: Students were still trying to figure this out from week 2.  I finally ended up showing them.  Several students had gotten pretty close.  Some wanted to ballpark it, so we talked about the value of precision.  “Why didn’t people just measure the line divide by two?” someone asked.  No one knew.  “What math concepts do we have now that they didn’t have back then?” I asked.  After some interesting (and sometimes accurate) guessing, we talked about how they didn’t have zero, negatives, numbers that use a positional system, and most importantly, they didn’t have fractions/decimals.**


  • Pope Toasters***:  When I had first asked “What do we have now that Euclid didn’t have back then?” one student shouted out “Pope toasters!”  It was funny the first time.  And maybe a bit funny the second time.  But it got a student or two so excited that it was hard to focus on math for a few minutes.




  • Naming: I talked about some math websites I like.  Math is Fun ( has some nice gif (animated) files on how to do Euclidian constructions.  “That’s a terrible name!” said M, with S and some others concurring.  Others disagreed.  I took at vote.  6 for terrible name, 2 for great name.  The majority felt that the name demeaned mathematics, since math actually is fun, so people shouldn’t need to be told.  Interesting how naming is such a double-edged sword.  A name gives things realness and importance.  It gives us a sense of control over the unknown – if we can name it, it can’t be that bad.  But it can limit/constain.  Once something is named, infinite possibilities seem to diminish, right?  Some suggested replacement names for that site were “mathforkids” and “mathwhiz.”  But I’m not a kid and I love it.


  • Euclid cards: “Pick a card, any card,” I said, with my Euclid cards fanned out in front of me.  These cards were some items from the Definitions section of Euclid’s Elements.  When you choose your card, you can either (A) try to construct the thing with a ruler and straightedge, (B) attempt to define it, (C) both A and B, or (D) ask for help.  M* ended up with #6, the edges of surface.  This required, of course, first defining a surface.  That was fun.  L chose #22, a square.  Both situations ended up with everyone begging to get a turn to talk, draw, and pick cards.  This was really fun when we compared the student’s definitions with Euclid’s.  We’ll do more so everyone gets a chance.


  • Self-directed Euclidian constructions: During both weeks, everyone enjoyed using the ruler and straightedge to construct their own things.  See photo gallery.


  • Right angles: Students really get what these are, along with various terms and notations to indicate them.  My favorite right-angle moment was when A came up to the board, took the dry-erase marker out of my hand, and said “If this is a right angle, I can show you a left angle,” and proceeded to draw a backwards L.  So right she was.  So we talked about all the definitions of the word “right.”


Thanks to all of you for sending your kids for all this fun with Euclid.  Thanks too to Angie, who has been taking some great pictures throughout this course.  Be sure to peruse the photo gallery to see some wonderful creations from the kids.



*We have 3 M’s in the class, and I haven’t figured out a good way to indicate who is saying what in these reports.  Sorry!

**Read a nice paragraph at another of my favorite math websites:  Math Open Reference

*** For those of you from out of town, Pope Francis was recently in Philadelphia, and people were on the streets selling just about anything – shirts, flags, Pope dolls, and you could even pre-order a toaster that imprints his image onto your bread.

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