# as shown by Anna and Cate, daughters of Barbara Maclay Cameron

### "Calculus By and for Young People (ages 7, yes 7 and up)", which was published in 1988.]

On Don's facebook page, July 17, 2012: Hi Mr. C.! Thinking of you this morning as I started my day talking to Anna (7) about multiplying positive and negative numbers. We got out the chalkboard and I used postman stories to explain it to her! [Don learned it from Dr. Robert B. Davis in the '70s, Barbara learned it from Don in the '80s, and her girls learned it from Barbara in 2011(Cate) and 2012(Anna)- what a connection!]

### Barbara: "It can be hard sometimes to trust that unschooling really works. And then you find yourself having a conversation with your (very excited) children about multiplying positive and negative numbers. I'm not sure I would believe this if it weren't happening right in front of me".

There are 4 problems above:

Problem 1 (Monday) says the postman brings you 3 checks (+ 3) for \$2 each ( + 2) ; are you richer or poorer as a result? The checks are for your birthday, that adds  \$6 to your                                               bank  balance  = (+ 6). So we write that as + 3 x + 2 = + 6  (you have \$6 more)

Problem 2 (Tuesday) says the postman takes away 3  ( -3 ) (they really belong to the lady upstairs), checks for \$2 each ( +2) ; are you richer or poorer as a result?  you                                                         have \$6 less than before he came. So we write that as -3 x + 2 = -6

Problem 3 (Wednesday) says the postman brings you (+ 3) bills for \$2 each ( -2 ); are you richer or poorer as a result? You have \$6 less than before he came. So we                                                             write that as +3 x -2 = -6.

Problem 4 (Thursday) says the postman takes away 3  ( -3 ) , bills for \$2 each ( -2) (they really belong to the lady upstairs). So taking away the \$6 in bills, adds \$6 to                                                            your  bank balance. So we write that as -3 x -2 =+6.

### Another quote from Barbara " thinking of you again yesterday when I was doing math with Cate--we were taking the 'if this problem feels hard, how can we make it into a problem that's easy?' approach."

"Mr. Cohen, I felt like I was channelling yesterday when Maggie and I got involved in figuring out how many pieces there were in the huge lego pyramid she had built.  My favorite part was when she realized that there was a quicker but less interesting way to do the calculations, and she said, "Let's do it the longer way!  It's more fun!"

I posted a photo of the pyramid and tagged you in it. I'm not sure if it's readable, but that white piece of paper says "10.404'

Donald Cohen: your pyramid is a marvel, Maggie! It looks like an Egyptian pyramid. I like the idea of doing something the longer way, but more fun in the process. Is the pyramid made by the squares of even numbers? Like 2^2=4, 4^2=16, 6^2=36?

Jennifer Olinger (a friend): Wow! That's impressive! Is that number (10,404) the number of Lego bricks she used?

Barbara: Not the number of bricks, the number of dots. A spontaneous unschooling math project.

### I posted it to share with my beloved childhood math teacher.

Barbara wrote: "Mr. C, the bottom square side is 48 dots across, with each layer above being 4 dots shorter. We started by figuring out 48 x 48 (which in and of itself was new, since M. had never done double-digit multiplication), and then 44 x 44, and then 40 x 40 (easy!), and then used those three numbers to predict what the next one (36 x 36) would be without doing the multiplication. I have to run now, but I'll send you an email with the details.. And emailing or skyping with Maggie would be amazing. I was just about her age when you taught me two important things: (1) math isn't just arithmetic, and it can actually be fun and (2) that I'm actually pretty good at math! I can't imagine anything better than Maggie getting to get some Mr. Cohen math magic. I'll email you asap!