## Monday, January 8, 2018

### Permutations Of Minions

We’re getting ready for my daughter’s Minion themed birthday party.  She’s turning 8.  Where does the time go?  Since we had a snow day today, we were all home during the day and making preparations.  My wife made the poster below.

Kayla couldn’t figure out which minion was which.  She kept saying their names in different orders.
“Kevin, Stuart, Bob”
“Bob, Kevin, Stuart”
Not expecting much conversation I asked how many ways the three could be arranged.  Without shying away from the challenge, she sat and thought for a minute.  Deciding she needed to write down her thinking, she got some paper and made the lists below before confidently announcing her answer of 6.

I praised her for both showing her thinking and the organization of her method.  She had been stating aloud, “well there are 2 ways that they could stand with Kevin in front, then there are two ways that they could stand with Bob in front, and then there are 2 ways that they can stand with Kevin in front”.
Curious how far I could I push, I asked her about including another minion, Dave.  She struggled more with this one, initially listing 17 possibilities.  When I told her that she missed some possibilities, instead of going back to her list, she surprised me by trying to go into some arithmetic.  She started with a “4 times 3 times 4….48”.  I asked her how she got that and she started explaining that there are 4 minions and then you multiply by 3 because there are three left, but she couldn’t explain where the other 4 was from.

She revised her thinking to just the 4 times 3, but then realized that she had already created more than 12 possibilities.  I asked her what she knew about arranging 3 minions.  She said she knew there were 6 ways to do it.  I asked what happens when we add another minion.  She connected the dots and was able to realize that there would be 4 times 6 possibilities for arranging 4 minions, where the 4 was “the number of minions that could be first and 6 was the number of ways that the other 3 could be lined up”.

To build on the momentum, I asked about adding a fifth minion, Jerry.  Here she struggled again, so my wife asked her what patterns she noticed.  We started listing the ways to line up 2 minions, then 3, then 4.  Kayla recognized the recursive nature of the pattern that a 5th minion would mean 5 times as many possibilities as 4 minions, so she set to work on multiplying 5 by 24, which was immediately followed by 6 times 120, at which point her desire to do multiplication stopped.

My wife then gave Kayla a brief explanation about what a factorial is, how to calculate it and how to represent it.

Besides being ridiculously excited that I could bring Kayla into this conversation, I reflected that this is exactly the type of You do, We do, I do model of teaching that I espoused in my Ignite talk

You Do – Kayla had a prompt about lining up minions and wrestled with the task with no input from me.

We Do – In the extension problems of 4 and 5 minions, my wife and I asked Kayla questions to understand her thinking and refine her thought process.  By asking about the origins of the values in her arithmetic, she was able to independently identify flaws in her reasoning.  Some direct guidance was also provided.  I told her 17 was not correct, but didn’t offer any additional strategies.

I Do – Once a pattern was identified, Kayla was in a place where she could attach some new vocabulary (factorial) to an experience.

## Thursday, December 14, 2017

### I gave an Ignite

My nightmare came true!  I was asked to give an Ignite speech in front of all the middle school principals and instructional coaches.  Despite being in front of children all day and frequently presenting to staff at my school, public speaking is one of my…least comfortable areas.  I was free to choose any topic, but the theme of the meetings is “I used to…, but now I…”.

I was asked to present by my coach and resource teacher.  Her confidence in my ability and her consistent support made it possible for me to overcome my discomfort.  After my initial reluctance, I accepted and had to pick a topic.  After reaching out to my coaching colleagues, I landed on math and coaching.  Initially I made a presentation that came off as very preachy.  I’ve come by some hard-earned wisdom about math pedagogy that I thought I could share with instructional leaders in a way that would empower them to feel confident with discussing math pedagogy.  What I found instead was that by sharing my personal journey, I could accomplish the same goal in a way that may be more well received.  I’ve made a lot of poor choices throughout my career, but have tried hard to learn from them is counterproductive to a true narrative and a realistic vision of math education.

An underlying push in this talk is the influence of the #MTBoS (the Math Twitter Blog-O-Sphere, an online community of math educators) on my teaching, personal and professional growth.  Many of the techniques were either acquired directly from this group of amazing people, or, at the very least, I discovered I was not alone in my findings about how students learn math best, reassuring me that my techniques were following a good path.

While the talk may not be as inspiring as Annie Fetter’s Notice &Wonder or DanMeyer’s Math Class Needs a Makeover, I was happy to be able to share my journey with my district and am hopeful that it carries some positive influence and change behind it.  I am also so thankful to have so many people, both in real life and on the internet to both challenge and support me.  I’m a lucky guy!

## Thursday, October 5, 2017

### Every Letter Has An Opposite

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!

## 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...

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!

## 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!

### Multiple Subtraction Methods

My daughter is in 2nd 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.