*I completely credit my children for one of the most successful teaching moves to date, and one I never used purposefully enough with my own students. As most children do, my son discovered his ability to ask questions. “Why?”. “How come?” and “Will it ever…?” became common appearances in our conversations. What would start as a simple request could become an abbreviated lecture in thermodynamics or animal classification. He asks to understand. He asks to discover. He asks because he thinks I know, and he wants to know, too.  How can you ask “Why?” to change the course of your lesson? Or your lesson planning? Or maybe even your coaching? Maybe even decide to share what happened when you asked why: #Iaskedwhy

One of the teachers I was working with casually walked by my desk one day and asked, “Will I be getting at G.SRT.4 by doing this?” She pointed to the proof laid out neatly in the student edition of the textbook.

“Well, it looks like a reasonable proof using similar triangles, so it looks like it. Why?”

“Because I’m working on my lesson plan for Friday, and I want to make sure that’s all I have to teach.”

“Hmm. What’s your plan look like so far?”

“Well, I’m going to show them the proof, then maybe give them a partial copy of it and have them fill in the blanks. And then, maybe by the end of class they can replicate the whole thing from scratch.”

Yup, that’s when my heart sunk. Not only did the lesson not meet my vision of high quality instruction, I also felt like I was looking directly into my past, and it was haunting me. I had actually done a very similar lesson in my own class not too many years before. Not only was it not effective in helping the students “prove” anything, it didn’t help students think mathematically, make mathematical connections, or see beauty in mathematics.

I’m sure you could talk to former students and they would tell tales of how I taught math that would make us all cringe. I was ignorant. I didn’t ask myself, or my students, “why” nearly enough. I often thought my job was to impart mathematical knowledge instead of helping students to create mathematical understanding. I didn’t engage students in true discourse regularly, and when I did, I’m not sure that it was truly in service of the mathematical goal. Having been pulled out of the classroom to support teachers, I thought my chance to fix things, to make it right for students might never be realized.

So when this teacher proposed a lesson plan from my past, I thought of this as my chance for redemption. Here was a lesson plan, begging to be brought into our new reality and my desire for high-quality math instruction that offers opportunity for students to engage in the mathematical practices.

So I asked, “Why?*” And that’s the moment things started to change.

Here’s what we began thinking about together:

• What did students already know about proving mathematical theorems?
•  How far could students get on their own?
•  What information did students really need to be provided for them?
•  What questions could we ask that would be the “least helpful” for students?
•  How could we help students persevere?
•  Should students work by themselves or with others? (If others – how many in a group?)
•  Was it worth the risk to try something different?

As a result, here is what we planned:

• First, we were committing to be less helpful. Students knew enough about what it meant to prove something to let them loose on the problem before helping them.
• Different students might be able to get to different places on their own, so we wanted to design different supports along the way, depending on where the student(s) get stuck. One size does not fit all.
•  If we wanted students to prove the Pythagorean theorem using triangle similarity, we needed to start with some kind of similar triangles and ask them to get to the Pythagorean theorem.
• We anticipated student responses and misconceptions along the way, and wrote down questions that we could ask that would be the least helpful but still move students forward.
• We decided to call out that we were asking students to think like a mathematician. We knew this wouldn’t be easy, so we decided to put a spin on it. We called our lesson – “The Pythagorean Challenge,” and we promised the groups that were able to come up with a proof could be Pythagoreans for the day (wearing a laurel crown and all).
•  This teacher already had her student desks situated in groups of four, and we thought collaboration would be key in trying something new. We added a caveat that students who wanted help, even of the least-helpful-type, needed to make that decision as a group.
• The teacher agreed that the end result in both her original plan and in our new plan was the same, but that students were more likely to be able to prove the theorem on their own (or other theorems for that matter) in the future using our new design. Sounds like a risk worth taking.

Our thoughts afterwards:

•  Students were more engaged than the teacher imagined, and it wasn’t all about the silly crowns. Students challenged themselves to think like a mathematician, and really pursued excellence when asked to be a mathematician. Note to self: Students rise to the level of your expectations, so set them appropriately high!
• Having anticipated what students would do, and where they would get stuck, before-hand was extremely useful. Between the two of us, we came up with all but one of the possible misconceptions and needs that presented themselves during class. Knowing what could possibly come up, and having least helpful questions to move students forward, made facilitation with the larger classes easier (a quick note – The teacher ran the class by herself, like she normally would, with each of her classes at 33 students or more). Note to self: Anticipating with a colleague got us closer to reality than if we had done this alone.
•  Group thinking was powerful in this situation. Students relied on each other for ideas, suggestions, and help which allowed the teacher to facilitate the larger group and empowered students that they have within themselves the knowledge to figure it out. Note to self: Collaboration works best when students have to rely on each for ideas.

If you’re interested, you can check out this for some guiding ideas.

Score: For this lesson, I felt redeemed.

I remember the sweat dripping from my brow as I worked last minute to complete my grant proposal for funds competitively available within my district. With an infant at home, and a full teaching load at the high school, I had almost missed the deadline and my chance.

The grant’s only stipulation was that the funds had to be used to support students within the district, but otherwise, the sky was the limit. My neighboring science teacher took that to heart and was writing a grant for a telescope to aid his students’ discovery of our universe. I sat trying to find the right words to explain why graphing calculators were necessary for my Algebra 1 students. How could I convince self-proclaimed “non-math” types that the tools could potentially open up their mathematical skies just the way a telescope literally could?

Not so far away, some very smart people must have realized the same thing. Enter Desmos, the graphing calculator. I hadn’t yet entered the age of smart-phones. (I am not quite a laggard, but I’m close. I was shamed into getting a smart phone by my 75-year old grandma, who chastised me for not being able to send her pictures of her great-grandkids via my not-so-smart-phone.) But Desmos was different for me. I was impressed from the beginning, and sold on the usefulness from the instant I saw it. What I didn’t know was that Desmos was the cocooned caterpillar, and teacher.desmos.com was the beautiful butterfly about to emerge  My mathematical skies opened up with each Polygraph, Marbleslide, and Activity Builder I explored.

Without a classroom of my own, I found an unsuspecting and eager first-year math teacher and offered to co-teach her class for a day. She thought the offer was too good to be true, and so did I. As we sat and planned how to move from linear to exponential functions, I slyly suggested that we check out Polygraphs. The panic in her face made me think she was stealing school supplies. Not that kind of polygraph, I assured her. Her panic turned to joy when I showed her what Desmos had created.

Here are some of the things we thought of ahead of time:

•  How would we introduce computers into the class? (A cart of Chrome Books would be available, from this point forward, but hadn’t yet been used in the class)
• Did we want students to work in partners or alone? What would be the benefit of either case?
• Did we want students to write anything down?
• What did we want the big mathematical takeaway to be?
• How could we use their responses and actions to move students towards our learning intention?
• How could we use the Polygraph to further our instructional focus on student discourse?

Here’s what we decided:

•  If there were going to be two of us in the room, it could be the perfect time to also learn a new structure for getting out (and putting away) technological resources. We picked a day with longer class periods to try this for the first time.
•  We decided we wanted students to work in pairs the first time, but we wanted to make sure both students were accountable for working. We were hoping this would help with the student discourse piece (see more below).
•  I wanted to go straight for the technology and ditch the paper/pencil, but my co-teacher (the first-year teacher) reminded me that we needed to teach students how to interact with both technology and paper/pencil at the same time. Wise beyond her years is an understatement.
•  We wanted to focus on math practice 6 – attend to precision – with students’ language in discussing slope of lines. We thought this might be particularly helpful when we moved into exponentials.
•  We decided when and where we would stop students and ask them to reflect on the language their classmates were using, the precision of the language, and engage in math practice 3 – critique the reasoning of others.
•  We wanted students talking, arguing, debating…if we want to increase the quality of student discourse we have to begin with some kind of measurable quantity.

And here’s what we found:

• Students quickly came up with a better structure than we had for getting out technology and putting it back. They had been using Chrome Books on and off in other classes for years, and were very willing to help with ideas to make it run smoothly. Notes to self: Students are worthy collaborators. Not only do I need to be willing to listen to their ideas, I need to solicit their thoughts.
• Students working in pairs worked better than we thought. I’m pretty sure that someone in #MTBoS first suggested it, but it worked wonders. Since this time I’ve tried Polygraphs numerous times, and students working in pairs seems to work better every time. Note to self: Students need a chance to try their thoughts out on a partner before committing them to an activity. Give students space to think aloud.
• In hindsight, we didn’t think through enough what we wanted students to write down, and why. We wanted the paper/pencil part to aid the technology part and instead it became just a compliance piece. There are times to use both, in hopes of bridging the gap, but Polygraphs probably isn’t the best one. In this case, I recommend letting the technology speak for itself. Note to self: Never make a worksheet without a clear mathematical purpose. Think twice before printing!
• Starting and stopping students in the midst of Polygraphs was way more complicated than it sounded. The students get hooked quickly, and they don’t like being interrupted. (Would you?) We started saving these conversations for small groups and the flow seem to go much better. When we saw students in between a round, we often popped in for a quick chat to help them reflect on their language and precision. Note to self: Discourse needs to be in service of learning. If it's not, rethink when/how you’re asking for student talk.
• Polygraphs far exceeded our expectations for getting students talking. One of our favorite features is the “What your classmates have asked…” which scrolls through questions other students have asked. These questions spurred students to refine their questions, offer examples when they had none, and became our source for other high-ceiling tasks (Ex: What is the least number of questions you could ask to identify the correct line?) Note to self: Always think through how I can use students’ thinking/work to further their classmate’s thinking/work before I step in and use my thinking/work. Be like Desmos.

 Moral of the story: Skip the sweaty grant-writing and get on board with Desmos Polygraphs!