Evolution of a Lesson

It started at our Department Chair meeting.  DCs had time to work together to plan a lesson together.  One group called me over because they were looking for ways to get students to understand two-variable inequalities.  Some of the specific difficulties were that students were having difficulty deciding what 0 < 2 implied about the graph and also trouble understanding WHY the graph of a inequality was shaded.  I suggested that a thin context may help students with reasoning through these two pieces, which quickly turned to assessing learning about the topic through Desmos.  My colleague Ellen took the reigns and said she would create an activity to send out.  The context was about buying canned juice ($0.75 each) and fries ($1.25 each) for a budget of $20.  Below is a screen shot from our first attempt (initial activity here; final product here):

I was pumped to have such amazing colleagues working together to create an engaging and thoughtful product for our kids...I also looooooove Desmos.  With the Desmos activity as a start, Chris Wright (@cwright4math) and I got to work.  We bounced a lot of ideas back and forth about what it was that we wanted the activity to do.  We decided that we wanted students to be able to pick points for the inequality and have them get feedback about whether their points were solutions or not.  Then, using the overlay feature, allows students to take a look at the pattern that was created.  We spent time learning how to code Desmos Computation Layer, eventually getting what we were looking for and sending it back out to our teachers. Here's the slide that took the most time learning, but created the most feedback from our teachers and students:

This is some of the feedback we got from the implementation:

"One of my GT 7 teachers did this lesson today. I had the opportunity to watch the students work through the Desmos activity. It was well received and worked well.

I would recommend having the class discuss some of the slides as a whole group before letting them move on to the next part.

One of the students also asked if we could change the colors. He didn’t like that the shaded area was red which showed correct answers and the points outside the shaded area were also red showing incorrect answers. He wanted all the correct answers to be the same color.

I LOVED the slide with the adjustable points. It was so cool when the message popped up to say if the point they selected was correct or not. I need to learn how to make this type of slide."

So back to work with some more CL.  We learned how to aggregate the student responses so that students could interact with their classmates' answers.  A big moment for us was defining what exactly the learning target was for our students.  This conversation had started with a desire to get students to graph linear inequalities, but as the activity evolved the purpose became clearer too.  We landed on "Students will make connections between individual solutions (ordered pairs) to a two variable inequality and the visual representation showing the solution set to the inequality by using a budget".  This new target was about creating an understanding about the connection between the symbolic and graphic representations, a fundamental precursor to graphing inequalities.  As a community we dove deep into the learning progression that takes place for the new learning of graphing 2-variable inequalities.

With this clearer conception of what we wanted students to know and be able to do, we got to the next step.  We want students to recognize that there is a line that divides the plane into solutions and non-solutions, so we put in this slide (this is the overlay, which reveals a lot of great student thinking - the conversations when students saw some of the solutions was great!):

We also wanted students to be able to tell if an ordered pair is a solution without substitution.  We changed to card sort so that instead of the symbolic inequality, we provided the graph so that students could build the connection between the solution set and the graph.

Collaboration and feedback (from students and teachers alike) was crucial to improving this activity.  Equally important was having a very clear learning goal for the task.  Thankfully, this goal became clear for us early in the process.  I'm grateful to have wonderful colleagues willing to try new things who have created cultures in their classrooms where their students are willing to provide honest feedback.  Here is a link to our final product.

Listening & Assumptions

K and I were walking in town and she asked, "Will I ever be as tall as you?"
JACKPOT!
I let her know that unfortunately, my height is on the decline :) and asked, "If you keep getting taller and I keep getting shorter, will we have to eventually be the same height?"
I was hoping to get into a conversation about systems, slopes, and ways to solve equations with variables on both sides.  Instead, I got schooled on assumptions...
K: "No we won't have to be the same height"
Me: "How can that be possible? Won't we eventually have to be the same height if you get taller all of the time and I get shorter?"
K: "What if I get 1 inch taller next year, then a half inch taller, then a quarter inch taller, then one one hundredth of inch and one one millionth of an inch?"
I was so caught up in my thoughts of linear systems, I got crushed by an 8 year old on thinking outside the box.  I'm so glad that I listened to what she said!

Listening

If you love board games I can't recommend Qwixx enough.  It's got some chance, some strategy and lots of math!

Without getting too much into game rules or strategies, it will be enough to know that a 7 on your first role (summing a white and any other die) is bad.

K: "What are the chances that you get a 7 on the first role?"

I'm super excited that she's even asking a probability related question, though I immediately recognize this would be a really tough one to answer.

Me: "Well first we have to know how many different outcomes are possible and then decide how many of them have 7s in them.  I think there are going to be some pretty big numbers.  Can we figure out how many outcomes there are first?"

K: "Sure"

Me: "Any thoughts on how many outcomes there are for six dice?"

K: "No."

Me: "How about 1 die?"

K: "6"

Me: "How about 2 dice?"

K: "11"

Me: "How did you get that?"

K: "Is it right?"

Quick aside here.  For years of our mathematical conversations, I haven't been telling K if her answers are right or wrong until after she answers "How did you get that?".  Lately I've dropped the question for arithmetic that I'm confident that she's mastered.  I almost said, no, but thankfully, this time I told her that I wanted her to explain where she got 11 from.

K: "Well you could roll a 2, a 3, a 4 and so on up to a 12. So there are 11 possibilities"

I'm so glad that I took a moment to ask where she got 11 from.  Based on her understanding of the word outcome and the types of questions she was asking 11, makes total sense. 

We continued to talk about how many outcomes there were for 2 dice, which had me acknowledging her answer as correct and offering another, different definition related to the actual dice as opposed to just their sum.  Then we did lots of thinking and listing, eventually working up to a general formula and an answer to how many outcomes exist for 6 dice.  We did not however answer the original question, but that's OK too!

Responsive Instruction in Math Class

Last week, I facilitated a session on responsive instruction in the math classroom.  The structure of the session was to start with a Which One Doesn't Belong, because we always have to start with some math. 

A little twist on this classic, I asked my participants to explain their choice via a creative response in goformative.com without using any words.  Here are some of the responses:

In these two responses, we made the connection between opening up and down and the coefficient of the x^2 term.

These two examples gave us an opportunity to talk about x-intercepts, as well as some representational limitations, with the quadrant picture highlighting that even though we can't see it in the picture, that the top right graph will eventually be in quadrant 2 as well as quadrant 1.  

After we discussed our WODB answers, we got in groups to get to the heart of the session.  Groups were asked to highlight areas that are consistently troublesome for students.  Part of my responsive instruction was to anticipate participant responses, so I guessed that fractions and signed arithmetic would be two areas that were common problems.  Sure enough, my participants agreed, adding also some trouble with creating and solving equations.

Our goal was to highlight trouble areas and come up with ways to help students before, during and after instruction.  The desired outcome was to go back to GoFormative and draw a representation of how to teach the topics with responsive instruction in mind.  To accomplish this, I provided participants with some resources to research their anticipated trouble spots.  We used the Progressions Documents and Nix The Tricks. 

The discussions that ensued with participants started to really blow me away.  Participants went back and forth between the Progressions and Nix The Tricks.  Questions surrounding signed arithmetic started popping up.  Teachers found a lot of usefulness in representing the number line vertically, thinking it would be more intuitive for their students.  Then came a struggle with how to use the number line to represent subtracting a negative.  Here I pivoted the conversation to the order of operations and how students make sense of that.  I referred participants to the PEMDAS section of Nix the Tricks.  We talked about the advantages of switching to GEMA and getting students to understand the definition of subtraction as adding opposites.  I was truly impressed by my participants openness and willingness to engage with these new thoughts on classic topics.  As we debated the merits of the two approaches, we also incorporated the other aspects of GEMA: the lack of division, the idea that we don't need to go left to right (and by extension the commutative and associative properties), the broader aspect of Grouping symbols versus Parentheses, and the connection to inputs in the calculator.  Here are some responses to this activity:

 Working with my colleagues, seeing their excitement about some new approaches, has really invigorated me at the beginning of this summer, and I can't wait to check in with them during the school year to hear how these approaches are going!

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 5^th 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.

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.  Acting as if I’ve never (or don’t still) givea worksheet to practice skills 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!

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!