"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!

# Countably Infinite

## Thursday, February 9, 2017

### 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.

## Wednesday, August 24, 2016

### Priorities

For him to take risks without retribution. That he'll be given opportunities to explore his curiosity. That he will be encouraged to try 4/n— John Stevens (@Jstevens009) August 8, 2016

John Stevens' tweets (above) got me thinking and primed for day one back. This summer has made me think a great deal about what is important to teach our children. So far none of the truly important had anything to do with content. A brief list:Is taught the essentials while being given a platform to advocate for what is right, not to merely learn compliance. 6/n— John Stevens (@Jstevens009) August 8, 2016

1. Understanding and acceptance of all people

2. Respect for authority

3. How to highlight and rectify injustice

4. Respect oneself

5. How to debate an issue without resorting to character defamation

6. How to value someone else's argument even if you disagree

7. How to compromise

8. Critically interpret information and data

As I write this list, I can't help but be reminded of Robert Fulgram's

__All I Really Need To Know I Learned In Kindergarten.__In it, he creates a short list of everything people need to know to make the world's a better place. As a parent, I've tried hard to instill these characteristics in my child. There's always room to improve.

Just a reminder to myself that as we enter the school each day, these are the priorities. Curriculum should be supplied to support the goals of raising critical, involved, and educated citizens. We also desperately need more role models that can highlight these virtues.

## Wednesday, May 4, 2016

### Memory

There's an app on my phone that is a game of memory with 16 cards. Each cars has a point value. When you pair, the points go in your bank which can then be redeemed for free items. Pretty sweet deal!

You can miss four pairs. On the fifth miss, the game is over. Being a super nerd, I'm working on figuring out both the probability of clearing the board (matching all 8 pairs before having 5 misses) and the expected value of a play.

The 5 plays accumulate over time. Last night at dinner I introduced the game to a colleague, who started with 4 plays. I advised him to wait for the fifth play but grew played any way. He lost. No points. It got me thinking of the big difference between 4 and 5 plays, as well as the difference between 5 and 6 plays. At 4 plays, you are guaranteed nothing, at 5 plays you are guaranteed only a single pair and at 6 you are guaranteed to clear the board. Someone put some serious math into these probabilities! Well done app maker!

You can miss four pairs. On the fifth miss, the game is over. Being a super nerd, I'm working on figuring out both the probability of clearing the board (matching all 8 pairs before having 5 misses) and the expected value of a play.

The 5 plays accumulate over time. Last night at dinner I introduced the game to a colleague, who started with 4 plays. I advised him to wait for the fifth play but grew played any way. He lost. No points. It got me thinking of the big difference between 4 and 5 plays, as well as the difference between 5 and 6 plays. At 4 plays, you are guaranteed nothing, at 5 plays you are guaranteed only a single pair and at 6 you are guaranteed to clear the board. Someone put some serious math into these probabilities! Well done app maker!

## Tuesday, May 3, 2016

### Standard Algorithm for Subtraction

Kay is working on money in her math class. She had homework that required her to compute two people's money based on the coins they have, decide who has more and by how much.

She incorrectly calculated the amount of the difference for the two people. I asked her how she decided who had more money, since there were some markings on the paper. Using the comparison strategy, she explained, she was able to show that one person had an extra dime and penny. But there was a mistake that she was able to catch. One of the coins in her comparison was a nickel, not a dime. She then totaled the two and came up with 36 and 42 cents. She came up with an answer of 16 initially.

Standard response, "How did you get that?" She thought a moment, then said she did it wrong and it should be 6.

"How did you get that?"

"I know that 6 plus 4 is 10, and another 2 makes 42." Content she settled down and moved on. 30 seconds later...

"I could have done that so much easier I had just remembered my doubles facts...6 plus 6 is 12!"

Love this kid! She's making sense of problem, developing and evaluating strategies, and coming up with alternate paths to solutions. She is not, however, using the standard algorithm.

When is the right time for this introduction? She had a good conceptual grasp. Looking back to my post, she can use models to explain her thinking. She uses strategies that make sense to her. I'm torn on deciding if standard algorithm well become just one more tool for her or become a confusing abstract tool. How do you know 5 right time for the algorithm?

She incorrectly calculated the amount of the difference for the two people. I asked her how she decided who had more money, since there were some markings on the paper. Using the comparison strategy, she explained, she was able to show that one person had an extra dime and penny. But there was a mistake that she was able to catch. One of the coins in her comparison was a nickel, not a dime. She then totaled the two and came up with 36 and 42 cents. She came up with an answer of 16 initially.

Standard response, "How did you get that?" She thought a moment, then said she did it wrong and it should be 6.

"How did you get that?"

"I know that 6 plus 4 is 10, and another 2 makes 42." Content she settled down and moved on. 30 seconds later...

"I could have done that so much easier I had just remembered my doubles facts...6 plus 6 is 12!"

Love this kid! She's making sense of problem, developing and evaluating strategies, and coming up with alternate paths to solutions. She is not, however, using the standard algorithm.

When is the right time for this introduction? She had a good conceptual grasp. Looking back to my post, she can use models to explain her thinking. She uses strategies that make sense to her. I'm torn on deciding if standard algorithm well become just one more tool for her or become a confusing abstract tool. How do you know 5 right time for the algorithm?

## Sunday, May 1, 2016

### Kay 35 + 7

So Kay (6) became intrigued about the calendar. She started reading dates as we completed our monthly ritual of changing the dates. She told me that being able to count by 7's would be a good skill for keeping track of the date. She then went over to the calendar, paused a minute and started reading off, "7, 14, 21, 28..." She paused as she calculated the next number. "35". She stopped. To egg her on I told her, "If you add another seven, you get my favorite number." (Hitchhikers' Guide anyone?) She asked what it was to which I replied, "I'm not just going to tell you".

She came up with the correct answer of 42, but instead of confirming, I asked her how she got that. Matter-of-factly she stated, "I used my Make 10 facts". I assumed she did what I did, which is break the 7 into 5 + 2 and add the 35 and 5 to make 40. Just to be sure, I asked her what fact she used. She told me "7 + 3 is 10". I was totally confused.

She wanted to show me on a calculator, but since one was not handy, she ran over to where we keep our base 10 blocks. "I can show you". It was hard to contain my excitement! What ensued, was a pretty amazing explanation, I think.

It was about more than just the addition. Kay showed me a completely new way to think about decomposition. I never would have thought to take from the larger number to make my 10. Here I am, a math teacher in my 11th year, with a Bachelor in Math and a Masters in Math Ed, learning about addition from a 6 year old. I was truly and genuinely interested in her method, especially since she's my kid, but she was oh so eager to share it with me. It took her a lot of processing time, and she made a few mistakes. Once the ball was rolling, my job was to listen. As always though, whenever she shares her method, I try to share a different one, just for comparisons sake.

So how can we replicate this in the classroom? How can we make sure that students have a place to voice their thoughts? How can we give students enough time to make sense and explore their sense making? How can we be genuinely curious about the voice of each of our students on a daily basis?

She came up with the correct answer of 42, but instead of confirming, I asked her how she got that. Matter-of-factly she stated, "I used my Make 10 facts". I assumed she did what I did, which is break the 7 into 5 + 2 and add the 35 and 5 to make 40. Just to be sure, I asked her what fact she used. She told me "7 + 3 is 10". I was totally confused.

She wanted to show me on a calculator, but since one was not handy, she ran over to where we keep our base 10 blocks. "I can show you". It was hard to contain my excitement! What ensued, was a pretty amazing explanation, I think.

It was about more than just the addition. Kay showed me a completely new way to think about decomposition. I never would have thought to take from the larger number to make my 10. Here I am, a math teacher in my 11th year, with a Bachelor in Math and a Masters in Math Ed, learning about addition from a 6 year old. I was truly and genuinely interested in her method, especially since she's my kid, but she was oh so eager to share it with me. It took her a lot of processing time, and she made a few mistakes. Once the ball was rolling, my job was to listen. As always though, whenever she shares her method, I try to share a different one, just for comparisons sake.

So how can we replicate this in the classroom? How can we make sure that students have a place to voice their thoughts? How can we give students enough time to make sense and explore their sense making? How can we be genuinely curious about the voice of each of our students on a daily basis?

## Monday, April 18, 2016

### Connect 4 With My Kid

Inspired by Joe Schwartz's Connect 4 using a multiplication grid, I tried it out on my unsuspecting 6-year-old.

A little background. K's understanding of multiplication is in the frame of repeated addition. She can count by 2, 5 and 10.

The beginning of the game was quick. K was able to calculate multiples of ten quickly, even commenting "Oh, this is easy". When she got to her first entry in the 9 row, she interpreted the multiplication as 9 groups of whatever number is at the top of the slot. So she saw 9x4 as 4+4+... 9 times. She used this strategy for most of row of 9s, but when she did 9x7, she suddenly switched strategies and added 9 to 54. I asked her about the strategy and she said each spot is 9 more than the one before it. As the game progressed, she used a similar strategy vertically, recognizing that each column had its own progression.

Obviously, I was very pleased, as she was both enjoying the game and working on some important foundation arithmetic. Her strategies revealed a lot about how she interpreted the game. She noticed patterns and created her own meanings. I was also really intrigued by her choosing to consider the multiplication m*n as m groups of size n, but then proceed in rows by adding a group of m.

Overall, the game is a win for math, but I learned that my kid needs to work on her Connect 4 strategy...

A little background. K's understanding of multiplication is in the frame of repeated addition. She can count by 2, 5 and 10.

The beginning of the game was quick. K was able to calculate multiples of ten quickly, even commenting "Oh, this is easy". When she got to her first entry in the 9 row, she interpreted the multiplication as 9 groups of whatever number is at the top of the slot. So she saw 9x4 as 4+4+... 9 times. She used this strategy for most of row of 9s, but when she did 9x7, she suddenly switched strategies and added 9 to 54. I asked her about the strategy and she said each spot is 9 more than the one before it. As the game progressed, she used a similar strategy vertically, recognizing that each column had its own progression.

Obviously, I was very pleased, as she was both enjoying the game and working on some important foundation arithmetic. Her strategies revealed a lot about how she interpreted the game. She noticed patterns and created her own meanings. I was also really intrigued by her choosing to consider the multiplication m*n as m groups of size n, but then proceed in rows by adding a group of m.

Overall, the game is a win for math, but I learned that my kid needs to work on her Connect 4 strategy...

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