The Sieve of Eratosthenes

Somewhere in my past I learned about the Sieve of Eratosthenes – it was presented as a method for finding out prime numbers. The end.

In my last reading of Introduction to Arithmetic, Nicomachus mentioned the sieve in Book 1 chapter 13 when talking the relationships between odd numbers, specifically finding out whether any two numbers have a measure in common with one another. We would call this a “common denominator.” Nicomachus says there are three classes of odd numbers, which I won’t go into here because it’s complicated, but he introduces the sieve as a method for producing the three classes, and finding out their “common measure” at the same time. When you mark each odd number as he suggests, their relationships appear.

I stopped at 31 because I was running out of ideas for symbols ;-)

I was taught simply to start with 3 and cross out any number that was divisible by 3, then move to 5 and do the same. This means that you’ll skip over 15 and 45 because they were crossed out when you were doing 3s. Then you do the same with 7, 11, 13, and so on (you skip 9 and all the rest of the numbers you’ve already crossed out). After a while, the numbers you’re left with are the primes – the ones that aren’t divisible by anything other than 1 and themselves.

Doing it Nicomachus’s way is so much richer though, because you’re not just solving a single problem – you’re noticing the relationships between the numbers, and that’s what Arithmetic really is. Not just performing operations with symbols, but understanding the relationships with the reality beyond the symbols.

Canonical hours and The Tempest

The Tempest is so interesting. It’s the only one of Shakespeare’s plays where he invented the plot instead of using one from history, myth, or romance. Instead, it’s inspired by tales of ancient voyages, like Aeneid and Jason and the Argonauts, but also by contemporary accounts of voyages to the New World. The characters are based on stock characters from the popular improv theater commedia dell’arte, only the story isn’t improv because Prospero is directing the whole thing. 

Also, it’s one of only two where Shakespeare observes the Aristotelian unities of time and place, which means that the time it takes to act the story on stage is the same as the amount of time that passes for the characters within the story.

In Act 1, scene 2, we meet Ariel for the first time when Prospero calls to him to come and give a report of the work he’s done that day, carrying out Prospero’s orders regarding the storm and bringing the ship’s passengers to land. Ariel has been having a great time doing all that, and gives an animated account of the storm, the wreck, and the passengers’ behavior.

But then Propsero mentions the time -- it’s almost three p.m. -- and says, “The time ’twixt six and now / Must by us both be spent most preciously.” In other words, there’s more work to be done.

Three p.m. is the canonical hour of Nones, the ninth hour after sunrise. It’s the hour at which Jesus died on the cross. It’s the time in the afternoon when the day is drawing to a close, but your work isn’t necessarily done yet, and you’re tired. It’s the hour of temptation. Since it’s connected to death (both Jesus’ death and the approaching death of the day with its memento mori) it’s also the hour for growing in wisdom and maturity, and it’s the hour of forgiveness -- both seeking and giving.

As soon as Prospero mentions more work, Ariel becomes fractious, complaining about the work, and reminding Prospero of his promise to set him free soon. Prospero scolds Ariel, who repents and obeys quickly and enthusiastically the rest of the play.

From here on out, all the characters in this play will face trials and temptations, will be reminded of their sins, will need to seek or offer forgiveness.

Miranda may be the one exception to that -- I’ll need to be watching her as I read the rest of the play with my class.

My math collection

 

I haven't done a math post in ages! Several years ago I got really busy with other studies and had to set my math studies aside, but I've been able to take them up again recently, reading Introduction to Arithmetic by Nicomachus of Gerasa with my friend Esther, who blogs at Dappled Things. She suggested I return to the topic here and I have a few ideas for future posts, but in the mean time I thought I'd share a peek at my shelves, and mention a few favorite titles.

My youngest is a senior this year and is using The Teaching Textbooks, so a lot of these books and games are things we used when she and her siblings were a lot younger, but I've kept them out because I still refer to them from time to time.

The middle shelf is mostly manipulatives and decks of cards. The basket is full of Math-U-See blocks. The cardboard box has pattern blocks. We hardly ever use the manipulatives any more -- they're just here because I don't want them separated from everything else. We don't really play "math games" much any more either, but we do play card games sometimes. The white bottle in the red sleeve is a bottle full of pennies for using when we play our family's favorite card game, Continental. 

The top shelf is mostly books for my own study and use. I pulled the Ruth Beechick book out so you can see it better. It's just a tiny thing, but so important. If you're just getting started, I can't recommend it enough. Behind it are the textbooks and CDs I got from The Teaching Textbooks, when we were doing it that instead of using their subscription service. Asimov's Realm of . . . books are kind of a history of the development of math. They make great read alouds with middle school and older kids, if taken in fairly small doses. Give you lots to talk about. 

 

 

The bottom four are books for me on child development and teaching as it relates to math. The top four are Denise Gaskins' excellent series of math games for all ages. The yellow one is a program that I started then abandoned -- it seems like it would be really good for young kids but I got it too late to use with my younger set. I'm thinking I should revisit it with my special needs son, and see if he takes to it. It starts with counting on the fingers, based on some interesting brain science -- there's actually a part of the brain that connects the fingers with numbers. The ones on the right are all "living" math books -- a few biographies, a history of counting. Picture books for younger kids but still very interesting.

 

 

Bottom, three ancient college textbooks -- I don't remember where they came from. The blue paperback is the text to a Great Courses class I took several years ago. Standing on top of the stack is Horace Grant's Arithmetic for Young Children, out of print, but an excellent resource. It's standing on his Second Stage of Arithmetic, also excellent and out of print -- I sent the google doc of the book to a custom printing service and got it that way, which wasn't as expensive as it sounds. The only other option would have been to print and bind it myself and I didn't want to spend time on all the formatting. The colored books are Life of Fred, which are fun and helpful.

See my math label for earlier posts describing some of the things I was doing with my children and learning on my own.