K: "Poppa, what's the opposite of 1 million?"

Me: "What do you think it is?"

K: (thinks for 10 seconds or so) "Negative 1 million?"

Me: "Right. What does opposite mean?"

K: "You just flip it around....what's the opposite of 'c'?"

Me: (confused, but somewhat excited that we may be talking algebra!) "Hmmm...what do you think?"

K: "x"

Me: "How is that opposite?"

K: "a, b, c, so z, y, x"

Me: "I never really though about opposites of letters."

K: "Every letter has an opposite."

Me: "Does every number have an opposite?"

K: "Yes...(long pause) except there's one number that doesn't."

Me: (intrigued) "What number?"

K: "Zero...unless you count zero."

After I grabbed some paper to write down this deliriously wonderful conversation, I asked some follow up questions:

Me: "What's the opposite of x?"

K: "c...is that right?"

Me: "I guess that depends. What would you say is the opposite of a?"

K: "z"

Me: "What would you say is the opposite of z?"

K: "a"

Me: "So what is the opposite of x?"

K: "Oh, it has to be c. Opposites come in pairs"

I love these conversations! I never would have thought that the opposite of c was x!

# Countably Infinite

## Thursday, October 5, 2017

## Friday, August 11, 2017

### Geometry & Trig & Coherence

The year I taught Geometry and Trig in the same year, I probably learned the most I ever have about vertical coherence (across grades rather than within a course/grade). There were so many connections.

For example, we spend so much time in Geometry working with congruent triangles, so much time proving triangles congruent. This proof process is important for truly understanding the unit circle definitions of the trig ratios. When I taught Trig previously, not at the same time as Geometry, I would always just have students tell me that two triangles in a unit circle were congruent, but we never took the time to explain truly WHY the triangles were congruent. Understanding why the reference triangle for a 120 degree triangle and a 60 degree triangle are congruent is a powerful tool. Students could tell me that the triangles were congruent, but not how they knew. None of them made the connection to the triangle congruence that they all spent so much time with in Geometry.

This coherence is a two-way street. In Geometry class, we need to spend more time on topics and items that will likely enhance understanding in later classes. Instead of doing a bunch of congruent triangle proofs like this...

For example, we spend so much time in Geometry working with congruent triangles, so much time proving triangles congruent. This proof process is important for truly understanding the unit circle definitions of the trig ratios. When I taught Trig previously, not at the same time as Geometry, I would always just have students tell me that two triangles in a unit circle were congruent, but we never took the time to explain truly WHY the triangles were congruent. Understanding why the reference triangle for a 120 degree triangle and a 60 degree triangle are congruent is a powerful tool. Students could tell me that the triangles were congruent, but not how they knew. None of them made the connection to the triangle congruence that they all spent so much time with in Geometry.

This coherence is a two-way street. In Geometry class, we need to spend more time on topics and items that will likely enhance understanding in later classes. Instead of doing a bunch of congruent triangle proofs like this...

to proofs of things that will help their conceptual development later like this....

I'm not saying we need to abandon problems like #2, but rather shift the balance. Most books and curricula that I've seen have put a heavy focus on triangle congruence like #2 that will not likely show up again for many HS math students. If we shift the balance to do the majority of problems that students will likely encounter in their future, we will help enrich their comprehension when dealing with more sophisticated math.

Conversely, in Trig, we need to make sure that we are spending the time to go through the proof of why sin(Θ) = -sin(180 + Θ). Using problem #4 above, we can help students make the explicit connection of why this identity MUST be true, even when we don't know the value of Θ.

There was another area where I saw this overlap, and again, was underutilized in the Geometry class, which was Translation Symmetry or Frieze Patterns. We spend a lot of time on reflection and rotation symmetry, but not so much on translation symmetry. We also spend much of this time on 2-D shapes and next to none on applying this to functions. It is really no wonder why students struggle with even and odd symmetries, as well as periodicity, in Trig when they are asked to apply symmetry to functions for the first time, while at the same time learning about trig functions.

Look at this example from my class several years ago

While the definition is good, all of the examples are 2-D shapes, no graphs. Adding a parabola, absolute value, or cosine graph would help students understand that this definition applies to graphing as well as 2-D shapes. The connections between algebra and geometry can be strengthened by figuring out WHY parabolas are symmetric.

Students were then asked to calculate values for x equal distances to the right and left of the min/max x-value. They began to notice relationships, were asked to explain those relationships, then connect to the symmetry of the graphs.

In my opinion, this is much better than marrying algebra and geometry in the traditional sense, i.e., "a segment has length 2x - 5 and the distance from its midpoint to its endpoint is 3x - 10, find the length of the segment".

One of my goals this year will be to see where else in our curriculum we can focus on coherence throughout the grades and courses. I'd love more ideas on where we can strengthen the coherence of our curriculum, especially in high school where the connections can frequently get lost between the courses.

## Tuesday, July 25, 2017

### The #MTBoS Reach

Doing some curriculum work this summer and I had an amazing experience. Besides learning in-depth about sixth grade standards and their role in the larger picture of secondary math, I gained a new perspective on how the #MTBoS is starting to influence the mainstream of math education. Our charge was to help write learning progressions, complete with resources for teachers to use to help them move their students through though progressions to fully understanding the standards.

After our first day, we gathered to compile a list of common resources. I was excited to see Illustrative Mathematics as a common response. It was also exciting to hear people in our math office discussing Desmos, Andrew Stadel, Graham Fletcher and Annie Fetter. We had some amazing discussions about the intent of the standards and making sure that we are progressing through them to help students truly understand the mathematics that they are doing, not just cranking through rote procedures.

I've been following the Twitter conversation recently about the direction of the #MTBoS. It's interesting to see the evolution of a place that has helped me implement a lot of the changes that I wanted to occur in my class, but struggled to find the support and resources to pull off. This collection of people has produced some amazing resources that are directly in line with my view of what math and math education are all about. I moved out of the classroom a year after I discovered this wonderful community, and have since struggled with my place in it. My (infrequent) blogging has centered around the place I do most of my math, which is at home. Professionally, I find it tougher to find relevant work to blog about.

Then this came across my Twitter feed.

Leeanne helped me to more clearly define my post-classroom #MTBoS self. A "redistibutor of others cool ideas."

This came after an incident in the previous week, where two teachers I was working with were discussing how much they love Desmos: Polygraph. I am preparing a PD for our math office when we get back to school in August and was curious because this was a resource that I was planning on incorporating. I asked them where they had first found out about it and it turns out it was from teachers that had attended a smaller version of the PD I'll be presenting in August.

So now, here I am, redistributor of others cool ideas. Hopefully, the #MTBoS will keep them coming!

After our first day, we gathered to compile a list of common resources. I was excited to see Illustrative Mathematics as a common response. It was also exciting to hear people in our math office discussing Desmos, Andrew Stadel, Graham Fletcher and Annie Fetter. We had some amazing discussions about the intent of the standards and making sure that we are progressing through them to help students truly understand the mathematics that they are doing, not just cranking through rote procedures.

I've been following the Twitter conversation recently about the direction of the #MTBoS. It's interesting to see the evolution of a place that has helped me implement a lot of the changes that I wanted to occur in my class, but struggled to find the support and resources to pull off. This collection of people has produced some amazing resources that are directly in line with my view of what math and math education are all about. I moved out of the classroom a year after I discovered this wonderful community, and have since struggled with my place in it. My (infrequent) blogging has centered around the place I do most of my math, which is at home. Professionally, I find it tougher to find relevant work to blog about.

Then this came across my Twitter feed.

Leeanne helped me to more clearly define my post-classroom #MTBoS self. A "redistibutor of others cool ideas."

This came after an incident in the previous week, where two teachers I was working with were discussing how much they love Desmos: Polygraph. I am preparing a PD for our math office when we get back to school in August and was curious because this was a resource that I was planning on incorporating. I asked them where they had first found out about it and it turns out it was from teachers that had attended a smaller version of the PD I'll be presenting in August.

So now, here I am, redistributor of others cool ideas. Hopefully, the #MTBoS will keep them coming!

## Thursday, February 9, 2017

### 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!

"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.

## 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!

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