What Shape Is It?

I was with friends in Vermont this week and shared this picture with them:

To my surprise, they were not as irate as I was over this mislabeling. It did, however, lead to an interesting conversation:

Eric: Why are you so angry about this?

Me: It's not a rhombus!

Sara: This is a parallelogram.

Eric: What's the difference?

Sara: A rhombus is a slanted square.

Eric: Then what's a parallelogram?

Sara: A slanted rectangle.

Eric: So what's a diamond?

Sara: A turned rhombus.

While maybe not mathematically precise, these definitions worked for two friends that don't live in math education. It got me wondering what all the fuss is about classifying quadrilaterals. With quadrilaterals, the "non-math" default is to give the most specific name possible, like 'square'. Calling a square a quadrilateral loses so much important information. We never label something 'up' the hierarchy like this in context. 

In other contexts though, going up and down the hierarchy is common. If I ask students if they have pets, they might say they have a dog, then I might ask what type of dog and they may say a Corgi, but it's totally acceptable to say dog instead of Corgi. 

So what is the skill we're attempting to instill when we work on hierarchies? It seems my friends have a pretty good grasp on defining shapes, even if they might not see a square as a rectangle, rhombus, parallelogram, kite, trapezoid, and quadrilateral, they would certainly see their Black Lab as a dog, a mammal, a vertebrate, and an animal. Is it the math specific vocabulary that is the stumbling block?

Desmos Fellowship Reflection (1/n)

This weekend I followed in the footsteps of Lauren Baucom, Daniel Luevanos, Sam Shah, Carl Oliver, Jay Chow, Sara VanDerWerf, Nick Corley, Jocelyn Dagenais (of Desmos French translation fame!) and over 100 others that are some of the most recognizable names in math education, as a Desmos Fellow. This year's fellows would be working with folks like Mary Bourassa and Christopher Danielson, who literally wrote the book on Which One Doesn’t Belong? I’m a pretty confident guy, but Impostor Syndrome set in way before the weekend started.  Before we ever met, my Cohort 4 fellows were making graphs in a potluck to see where our Desmos skills could take us (here’s mine) as a precursor to our weekend together.  I was blown away by the skill set of the people I was to meet shortly. Our first activity together upon arrival at HQ was to introduce ourselves (names and pronouns) and answer the question “What are you bringing with you this weekend?” in a metaphorical sense. Speaking in front of this amazingly talented group for the first time was nerve racking. I mumbled something about anxiety and excitement before highlighting my purpose for attending, working on social justice and environmentalism. The Desmos team was constantly reassuring us that we were picked for a reason, unfortunately, in a room of this talent, it’s hard to feel that way. Like, what do I have to offer Jay Chow? The man is a Desmos CL genius! But a summary of our responses and the genuine excitement of the Desmos team to have us all there helped set what would be the theme of the weekend: humanizing learning.

The intro activity was the start, but far from the end. At each step of the way, the team worked to ensure it was understood that we, and our opinions, thoughts, feelings, and well-being, mattered. The norms of the weekend asked us to

·       Stay strengths-based

·       Stay engaged

·       Embrace our growing edge

·       Attend to self-care throughout

We were constantly reminded of the importance of self-care, whether that meant standing, taking a walk, getting a snack, changing seats, or anything else that might make us more comfortable. Choices were provided in so many different ways, in terms of activities to participate in, topics to discuss, places to work together, people to collaborate with, or ways to engage with mathematics. By providing choices, Desmos was attending to my identity (what am I interested in) and power (getting to choose my own path).

These choices were not only provided as a structure for the entire weekend, but even within activities. When we were acting as students as Michael Fenton (not Michael Pershan 😊) led us through the activity Charge! we were invited to examine practices through a number of lenses: math learner, math teacher, math leader, or other. We were invited to use any tools we preferred to solve the activity. Some made tables, some made equations, some made graphs, some used Desmos, some used paper and pencil.  All pathways were acknowledged and honored. The same was true as Faith Moynihan and Lisa Bejarano led us through the activity of Point Collector: Lines. Faith and Lisa allowed us to pace through the first 4 screens to help us summarize the rules of the game, before allowing us to pace through the next 4 screens to attempts some challenges. She invited us to “stay on one challenge and get really good at it, or try all four challenges, or anything in between”. This clarification gave permission to us as students to choose what felt right for us. If we were determined to get the max points on one screen that was OK. What was important was the learning and the discussions. Those would happen regardless of the choice that we made.  Again, the decision to embed these choices empowered us as students by building on our identity, acknowledging and embracing the human element in learning mathematics.

The highlight (which is tough to achieve in a weekend that is like a SportsCenter Top 10 of professional learning) was the session offered by Lauren Baucom and Christelle Rocha. “Turning the Diamond on Desmos” built on the work of Rochelle Gutierrez and Laurie Rubel to help us all process what aspects of our students we tend to in our classrooms and schools’ structures: Access, Identity, Power, and Achievement. If you’ve made it this far in the post, you’ve probably noticed some of that language and that is a direct result of attending this amazing session. In fact, this session is what allowed me to make sense of all of the wonderful things that the Desmos staff did in their professional learning all weekend.  While we were well-tended to as adults all weekend, Lauren and Christelle challenged us to think about how we can focus more on identity and power in our classrooms.  The conversations started in this hour will go on for many years, I’m sure.

All in all, I’m amazed at how little time I spent in the Desmos environment. When I applied, I expected to learn all kinds of new things about Desmos, spend time coding in CL, building and critiquing activities builders, but my experience was far more transformative than that. This unbelievably talented group of educators pushed me to consider ways to humanize math class that emphasize impact over intent. Desmos is a tool that can help accomplish that goal, but is not an end, rather the means. For educators, transforming teaching and learning is the only path to achieving social justice. This acknowledgement by the company itself, their investment in the teaching profession, their desire to learn from the community, gives me hope. I will be forever grateful for this opportunity because of the relationships that were built and strengthened with like-minded educators!  I’m looking forward to seeing the work of my amazing Cohort 4 colleagues!

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!