Exit Tickets...hmmm

Exit tickets are making me rethink the end of class. In a good way. Here is how my 45-minute classes typically go:
1. Warmup (sometimes related to the lesson, sometimes not) = 5 min
2. Launch (explain the exploration that students will be engaging in) = 5 min
3. Explore (students explore in groups while I circulate, facilitating discussion and discovery via questioning) = 30 min
4. Sum up = 5 min

This year, I have vowed to leave time for exit tickets. Though my classes are small enough to allow for conversation with every group and most students, I was curious to know what I could learn from asking each and every one of them a question at the end of class. Rarely is it a question where I am looking for the same answer. Sometimes it is verbal feedback, sometimes it is a choice of problems to solve, sometimes I ask them to create a problem based on what they learned.

Verbal feedback - opinions
This allows me to have brutally honest feedback from my students. In the 2nd week of school,  I asked my new 6th graders to tell me one word that they were feeling about math this year. Here are some of their words:
- Enlightened
- Fun!
- Intimidated
- Excited
- Ready.
- ok?
- Awesome and Challenging (they cheated, but that's ok)

I knew right away that most kids were psyched/thirsty to learn more and that a few kids were already feeling overwhelmed or unsure. This actually allowed me to worry less - I thought many more kids were feeling the difference from elementary math in a scary way. I now knew who my timid kids were, which allowed me to better target my support / challenge of all 6th graders. 

Numerical feedback
I have also given my students exit tickets with two options of problems to try - pick the one that shows your level of understanding. After 1 day of integer addition with 6th grade, I gave them some problems as an exit ticket and I was astounded to find that all but one student aced the problems and more than half of them chose the more challenging option. There was one student who turned in a blank exit ticket. Had I done my normal discussion-style "sum up" at the end of class, I would never have known that he was struggling. I was able to find him at study hall, catch him up, and move on with the class at a faster pace than I would have without the exit ticket.

I have also used numerical feedback to do some "My Favorite No"work. With 7th grade, we reviewed percent calculations, which was still so challenging for them. We used work from exit tickets as a warm-up the next day in class, examining where students were on the right track and where they were led astray.


Verbal feedback - conceptual
When I am feeling like the "sum up" would be too forced / too soon, I have used the exit ticket to assess conceptual understanding. I recently asked my 8th graders to define a function at the end of a class where they were exploring this concept (the lesson got cut short due to picture day). Though they understood how it worked for different representations, no one had quite the right generalization. This allowed me to plan and augment the previous exploration to do a "part two" the following week where they were much more able to articulate a fantastic definition. (I asked them to define it in groups the second time around).

Do I use exit tickets daily? No. There is no replacement for listening and actively questioning student thinking in that discovery phase. But in the first few weeks of trying them out, I see how I might use exit tickets as a way to allow each student to "sum up," to pause the aha moment and delay it from the end of one class to the beginning of another. Because if they truly understand, they should be the ones saying it, not me. 

The Joy (and importance of) of Mistake-Making

The longer I teach, the more I am struck by what happens in the first few weeks of school. There is so much purpose layered into those first moments together, be it in classes, time in community, in advisory, out at recess. When I think back to my first year teaching math (I was teaching high school Algebra 2 and Geometry), I was eager to dive in to the curriculum and to avoid all the messy stuff that comes with new beginnings, new faces, new feelings. Now in my seventeenth year of teaching (and most of that at the middle school level), I know full well that we must spend time crafting the culture of our classrooms at the outset of the year. That culture comes from mindfully, purposefully, explicitly laying out what we want learning to look like.

One of the most important pieces of math culture I want to encourage is collaboration and sharing one's thinking. That means being willing to take risks and share ideas that may be wrong, so we need to be comfortable talking about our mistakes. Without explicitly teaching that, I realized that in subtle ways, this message of being comfortable making mistakes has permeated so much of my early lesson plans and collaborative work across grades 6 through 8. And just two weeks into school, I have witnessed several moments of downright joy in mistake-making.

My 8th graders have been with me the longest - most of them are starting their third year with me. They are steeped in the culture of mistake-making, but I am still blown away by their non-attachment to mistakes, their willingness to dive back in after a setback. In their first encounter with Desmos activities last week, students worked on creating graphs that precisely matched the rate of filling of different glasses:

gif courtesy of Desmos

But then, they got to create their own glasses, graph the rates of filling, and challenge their classmates to try their glasses. And what they made for one another was WAY more challenging than the original activity:

And they were super excited to try one another's creations, even if it meant going back over and over again to revise and refine their matching graphs.  Their tenacity in attending to precision was astounding. You could have heard a pin drop in the room as kids were poised over their computers. And there were several instances of joyful jazz hands when they were able to achieve the ranking of "very precise."

In 7th grade, students are doing corrections from their end of year quiz last year. Many of them are looking at the question that asked them to find the volume of a trapezoidal prism and wondering why they multiplied l*w*h. This is a powerful misconception that must be unlearned and reinforced with cognitive dissonance though problems like these. And while it may be a strong statement to say their corrections bring them joy, they are certainly satisfied with themselves for completing a complex problem like this, comprised of many sub-problems.

In 6th grade, my newest students have impressed me right out of the gate with their willingness to engage in debate over the "right" answer. During a "Which One Doesn't Belong" activity (where students find ways that each number may not belong), they stated that 16 in the following image doesn't belong because it is the only one with an odd tens digit.

image from wodb.ca

There was then a very involved debate about whether 0 was even, odd, or neither, with many arguments formed on either side. 

Later that week, we did the Four 4's activity and someone put up the following expression that equalled 5:  

As we closed class, I put it up on the board and asked the class what they thought it equalled. There were 2 answers: 5 or 17. Again, both sides weighed in and we talked about order of operations which settled it for 17.

In both of the cases with 6th grade, I assured them that even math teachers have trouble with these types of questions. My mentor teacher Henri used the term "mathematical consensus" to describe how we as a community of mathematicians have decided to agree upon such "right" answers, and I love this term. It makes math a dynamic subject, one where logical arguments are vetted together rather than answers handed down from the holy textbook on high.

As we embark on another year of messiness (in the best way) in middle school, I also must take a moment to pay homage to those teachers that have come before me, laying the groundwork for kids to feel safe taking risks, to feel valued for their deep thinking rather than their speed of computation, to have the courage to question things we seem to tacitly accept, to know that mistakes are part of the learning process. For beyond each mistake, each frustration, each wrong solution path lies the aha moment. The joy in that is audible, palpable, contagious. Which why I love teaching middle school.

Essential Questions: the ingredient that flavors all of my math classes

This blog post is part of Sam Shah's "Virtual Conference of Mathematical Flavors", which asked "How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?

Above the board in my classroom, there are 4 questions posted. They are:

• What is math?

• What is the pattern, and how many ways can I represent it?

• How does math connect to the world around me?

• How am I a mathematician?

These questions are my Essential Questions, which I developed during the Project Zero Summer Institute in 2007. They are touched upon constantly in my 6th, 7th, and 8th grade math classes. They are hard to answer concisely, yet we are constantly asking them and our answers will evolve over time. 

Here I will share some reflections on each of these questions and how they have helped me "move the needle" for my students.

What is math?

I ask my students this question on the first day of school. They write their answers up on the giant whiteboard wall in my room. I start with 6th grade and then 7th and 8th contributes. I love seeing what the 6th graders come in with, but also the complexity (and poetry!) that 8th graders (who I've taught for 2 years at this point) use to talk about math: 



What is the pattern, and how many ways can I represent it?
In exploring this question, we drive home the key ideas that 1) math is about describing patterns, not just problem solving or calculations and 2) there are multiple ways to make sense of a problem, and by taking time to look at these, we come away with a deeper understanding. While we look at patterns constantly in my classes, one of my favorite routines for doing this is using Fawn's Visual Patterns website. After staring at a pattern for a bit, my 7th graders come up and circle how they see the image growing, taking turns using different colored whiteboard markers. I love watching them have an aha moment as they realize how to use the growth to create an explicit formula for the number of objects based on the figure. After doing one pattern per week for the first half of the year, students create and analyze their own visual pattern. 


How does math connect to the world around me?
We do a lot of projects in my classes, and I always try to have the students use the math in a real-world way as a professional would (this was another big takeaway from Project Zero - see David Perkins Whole-Game Learning). My students are architects, cartographers, financial advisors, artists, engineers, and amusement park owners. They use the math in ways that are fun and practical (though I think there should also be room for appreciating the beauty and elegance of math for math's sake - see below). They learn how to use the tools of math and how to apply what they know to solve open-ended, creative problems. They look at math in the news once a week and learn to view statistics with a scrutinizing eye.

6th graders attending to precision as they build giant candy boxes.

8th graders use slope to find the steepest staircase in the school

How am I a mathematician?
This is perhaps the most important question of all. It is crucial to me that each student sees themselves reflected in the curriculum and feels empowered to not just do math, but feels like they are a mathematician. I do this by making frequent opportunities for creativity, collaboration, and connection. I want my students to see math as a dynamic, human subject that they have the power to influence and inform. Through the mathematician project, students research (and later present as) a mathematician, examining not only their person's math contributions but how their identity shaped their experiences with math. In collaborative groups, students work together to articulate their understanding and to move collectively towards the fulfilling aha moment. Students have opportunities to connect math to what they find relevant and interesting, be it music, baseball, feminism, food, or cars. Finally, I allow space for students to find beauty in math, honoring and displaying their creations. My hope is that when they go to high school, they will have a bank of positive math moments to come back to when they face a new challenge, as well as a deep appreciation for this nuanced and varied subject.

Reflections on my first Twitter Math Camp (better late than never)

Good morning, Cleveland.

Twitter Math Camp exceeded my expectations. It was undoubtedly the most welcoming group of educators I have ever seen (which is saying a lot - teachers are pretty nice on the whole). The conference is capped at 200 people, which keeps it small enough to facilitate a bunch of social meetups in addition to workshops and keynote speeches. I arrived on Wednesday for the Desmos pre-conference. I had signed up for this before I knew I was chosen for the Desmos fellowship (which was the weekend before!), and wasn’t sure if it would be redundant, but it wasn’t at all. I got a chance to  learn about the Desmos geometry tool and how to get the graph to do statistics calculations. These are two things I will definitely use next year in my quest to Desmos more in my 6th and 7th grade classes. The Desmos precon ended with a happy hour at a great local brewery, and then a bunch of us ended up playing games at a local game bar/restaurant (they had the biggest collection of games I have ever seen!).

Not a math game.

The next morning, I had my first of 3 consecutive mornings learning the math and art behind Islamic geometry design. This workshop was taught by Annie (@anniep_k), Megan (@veganmathbeagle), and Stephen (@sweimar). It was so relaxing and fun to create these 6-fold and 8-fold symmetric shapes!

6-fold symmetry with an extra point (oops)

8-fold symmetry with (sloppy) weaving

After lunch, we settled in for Marian Dingle's keynote called "Measures of Center." She spoke poetically and powerfully about her and her children's experience as black teachers, learners, and people. It was a call to action to all of us in the room and beyond, questioning who we truly serve when we teach, and begging us to consider the experiences of those who are not in the "center". By end, many of us were in tears. When she left the stage, Marian saw me crying and stopped to give me a hug!



The following day, there was a fun and empowering keynote from Julie Reulbach. She came bursting through a banner with pom-poms, as if we were in a pep rally!  She proceeded to be our cheerleader as she encouraged teachers with the following mantra (complete with stickers for each of us):

After Julie's keynote, it was time for my presentation on Social Justice Projects for Middle School. After Marian's call to action the day before, I was disappointed that only 6 people came to my talk! This left me wondering what I can do differently to get the message out to a broader audience about ways to incorporate social justice into our math curricula. I am still wondering about this.

Overall, the conference was extremely well-run. It really did feel like camp, what with all of the ways to connect with other MTBoS folks - speed dating, newbie dinner, and trivia night. I left on Saturday before game night and the writing of the annual TMC song! I definitely hope to go back next year when TMC is in Berkeley, CA.

Goodbye, Cleveland.

Reflections on the Desmos Fellows Weekend

The night before the Desmos Fellowship weekend started, I couldn't sleep. I had no idea what to expect. I knew literally no one, except for a few names from the MTBoS. I wasn't sure what they wanted out of me - would we be asked to give input on the software? Test new features? Show how we use Desmos in our classrooms? Surely, we'd be asked to "prove ourselves" in some capacity. I mean, after all, they'd paid my way out to SF, so I assumed I'd need to add value in some way. I was nervous, but also excited. As an island unto myself in my school (the only 6-8 math teacher), I was super psyched to talk with other math teachers about math.

The scene when I arrived was a little intimidating, even for this extrovert. Everyone was standing around talking, introducing themselves, and it was loud. I met a lot of new cohort members, and I was surprised to learn that there were lots of Fellows who had returned from cohorts 1 and 2. 

That's me on the far right, arms folded. Not sure why I look so annoyed! (Photo credit: Eli)

Pretty soon into the evening, Dan Meyer assured us that we had already given Desmos everything we needed to - that our participation in the weekend was "enough." He also explained why we were chosen (which I am sure was a question on a lot of our minds - how were we the 40 teachers selected out of over 400 applicants?). He explained that we were connectors - we were teachers who present at conferences, who lead PD's with other math teachers, who connect over Twitter to share ideas. His message made it clear that I needed to merely be myself to be of value. This was both relieving and liberating.

Desmos is an amazing software tool, but it is also an amazing company. Nearly all of its ~16 employees came to the entire weekend of events, many of them staying at our hotel to participate fully (despite living nearby). It is clear that everyone, from Eli the founder/CEO, to the teaching faculty, to the tech team, to the business team, are all totally committed to the mission of creating technology that allows ALL students to have greater access to and success in learning math. There were several times that I was blown away by the integrity and commitment of these folks: their precepts on the wall, Eli's anecdotes about starting the company on his own, Michael's willingness to sit down and teach me about how to make manipulatives on the graph, Shelley's warmth and hospitality all weekend, and Cori's crazy-fast responsiveness to new feature requests. This feeling is contagious when you work with this team, both online and in person. And the Fellows take this commitment to the next level, offering advice, positivity, and feedback in their spare time.  

After our short 2.5 days together, I left with an overall feeling of incompleteness. I felt like I just wasn't able to fully contribute in a room of 50+ math teachers, all working on their own projects. I have more to give! More to receive! More people to meet, more brains to pick! But I now realize that the Fellows weekend was the just tip of the iceberg. What lingers is feeling like when you start reading a really good book and can't wait to read more. Unfortunately, 4 days at TMC plus being a full-time mom during the summer means that I have had to put this book down as of late. I can't wait to finish the activity I started during the Fellows weekend, to try out some of the ideas we generated over casual conversations, and to stay connected to this community of math teachers who are amazingly committed and curious. 

Put simply, Desmos is great at what they do and wants to keep doing it even better. I feel privileged to be part of the group that helps them do that.

Desmos Fellows doing a Math Mingle on the first evening - no tech, just good problems! Photo credit: Hanako

Big Questions about Big Numbers

Last month, my eighth graders worked on the Big Numbers project. This project is the opening part of my unit on exponential functions. The goal is get students to be more comfortable with the notion of million, billion, and trillion, making sense of the magnitude of large numbers. We start by reading David M. Schwartz's How Much is a Million?. Then I give them their assignment: to make a 2-page spread in the sequel to that book. They need to think of a huge amount of something and make sense of it. One page is an illustration page that shows how they made sense of their quantity, and the other page shows all the math they did to reach their conclusion.

I love doing this project with 8th graders because they come up with the most creative questions! There is such a wonderful balance between intellectual "rigor" and childlike creativity at their age.

Below are some of the questions that my students came up with (including the followup question which is how they made sense of their number (my definition of "making sense" was that their answer needed to be something within their direct experience, eg. something they could directly could see or experienced (such as the distance to fly between 2 cities). Here is the book they made.

• How Big is the Earth's Circumference? How long would it take a horse to canter around once?
• How much snow has fallen in Boston over the past 5 years? If you packed the snow into Empire State Buildings, how many would you need and what amount of space would they cover on the ground?
• If the moon were really made out of cheese, how long would it take the entire Earth's population to eat it?
• How many Roxys (my dog) does it take to get to the moon? How many times would those dogs cover MA? 
• How many Chipotle burritos will stretch across the US? How long would it take everyone in the school to eat them?
• If you made a giant burger out of all the cows in the world, how big would it be? How long would it take Earth's population to eat it?
• How many M&M's are made each year? How many US Pentagon buildings would they fill?

First Blog Post!

Woohoo - my first blog post (as a math blogger)! I have been intending to start this blog for years, with the idea that it would be a place to record all of the aha moments from my teaching. This year, my school has focused on WONDER, and it has been making me think a lot about how questions are greater than answers, yet so often are not the focus of a math class. But when students wonder, they come up with some truly amazing questions. This year I have been focusing on truly listening to them, without an agenda of steering their thinking toward my desire end goal. True, we sometimes get derailed and end up talking about infinity when we "should" be talking about fractions, solving equations when we "should" be adding polynomials, or the meaning of e when we "should" be simplifying expressions with exponents. But I often find that when I follow the students' train of thought, we end up accomplishing more because of their buy in and understanding! (By the way, these are all examples of things that came up in the last week!)

So here's to this blog - may I actually update and maintain it!