Chocolate Percentages

"Something's wrong with the label Poppa."
"What do you mean, dear?"
"The percents add up to more than 100."

She was absolutely right...the percentages do not add to 100.  We have talked a lot about percentages, using them frequently in context, whether it was adding a tip, getting a discount or knowing how much battery life her iPod has.  It is this last context that provided the frame of mind surrounding the devastating 100%.

Her comments opened the door to what a percentage is...a ratio.  What she was confused about was what the units of that ratio were.
"What is the Vitamin C 15% of?".
"The whole thing?"
"What whole thing?"
"The whole container of chocolate milk."...jackpot!
"I think you're right.  If that's what the 15% means, the label couldn't possibly be right; that's too much stuff!"
"So the label's wrong?"
"Not quite.  The 15% is comparing the amount of Vitamin C in a serving of chocolate milk to the amount of Vitamin C that you need each day."
"Oh."
"What questions do you have?"
"So that's why they don't have to add up to 100?  Because each one is only used for itself?"
"Yep."

Definitely going to have to come back to this one, but love that she is initiating conversations!

Multiple Subtraction Methods

My daughter is in 2^nd grade and is currently learning about 2-digit subtraction.  It’s a skill we’ve been working on for a while, though I have very carefully stayed away from the standard algorithm.  During the last year, we’ve only talked about it in contexts, talking about time (friends are coming over at 3:30, it’s now 3:12), talking about sports (Duke has 54 points, UNC has 36….we’re winning by 18!), or any other place where math works its way into our lives.  Her preferred method is usually adding on in chunks, for example, 36 plus 4 is 40, plus 10 is 50, plus another 4 is 54, so Duke is up by 18.

Her elementary school uses a program that seems similar to Number Talks (if you haven’t read it, it is pretty much essential reading for elementary math!) in the sense that it promotes a variety of ways of attacking 2-digit subtraction.  Each day, she is explicitly taught a new method, then brings home homework to practice that particular method. 

Initially, I was displeased.  My daughter was able to follow the new procedures, but had difficulty explaining why they worked (my addition to the nightly homework assignments).  It seemed as though she was learning multiple algorithms for 2-digit subtraction without a regard as to why each of them work.  Disclaimer, I am not in the classroom with her, so I’m not entirely sure of the instruction surrounding this work.  I have the feeling that during class there is some explanation of why these methods work using a number line based on a few HW problems that I have seen, but it hasn't been sufficient enough for her to explain it to me, my ultimate litmus test.  

Number Talks changed my opinion...slightly.  Part of the joy of having a number talk with a class is not only seeing another way to solve a problem, but trying to make sense of someone else's method.  This is what helps improve flexibility when problem solving.  Understanding multiple methods to arrive at a solution allows you to add those methods to your repertoire.  The downside is that some of her assessments still "force" her to use methods that she may not have chosen on her own.  Indeed, this seems to be at the root of many of the anti-common core posts that have been circulated.  

I'm all for increasing mathematical flexibility, but I feel like just giving kids a bunch of algorithms and one day to digest them is not helping the flexibility.  Then I ran across this post by @TAnnalet.  It's pretty awesome!  I'm so glad to have people share their reflections and learning!  I love that students are learning multiple methods, but also are learning to create and test arguments.  That is the joy of mathematics and that will stick with these young mathematicians for a long time.