Humanizing the 2020 Election

I have been a math teacher for 22 years. And during that time, far too many times, I haven’t paused our regularly scheduled pacing guide to have deep conversations with my students about the world they are living in. Today I made sure to carve out that space.

Actually, I made the decision to carve out space on September 30. It started with Jess Lifshitz’s Tweet. Thank you Jess, for sharing such powerful books and pushing me to constantly think about humanizing my teaching.

Though I teach 8th grade and Jess teaches elementary school, I have purchased so many books that Jess has recommended from her classroom as kids are never too old to be reading and discussing themes from children’s’ books. When my copy arrived, I eagerly read and discussed it with my two (white) sons and was fascinated how it resonated with both my 6 and 11 year olds. I already felt so much uncertainty and fear around the election and knew that whoever won, a whole lot of uncertainty and fear would remain in my students’ lives. I decided then to read it the day after the election. I’ve been holding onto it for today.

In my class we have been talking for weeks about the importance of getting everyone they know to vote. How it doesn’t matter who you vote for, but we all deserve to be heard and all our voices matter. We did math earlier this week to understand the power of the electoral college system and why swing states ultimately decide the election. And we looked at various data visualizations to build understanding of the US population and how it impacts how states vote:

But today we needed no numbers, just our humanity.

Let me take a quick thematic detour here to say that this is the first read-aloud I have done this year. OMG, the neck craning one has to do on Zoom to be able to read a book aloud while holding it close enough to the camera for students’ to see the pictures is NO JOKE. K-5 Teachers doing 100% distance learning, my heart goes out to you. It is TOTALLY not as easy as these authors make it seem.

As I introduced the book I said that I had been saving it all year to read today, the day after the election. Long before we’d know who would win (and ironically even now we don’t know), I knew that the election season would bring up a lot of hatred and fear and that all of us would feel some anxiety about what comes next, especially as we’re living through this pandemic. I shared that a huge reason why I teach is because youth give me so much hope and I love that year after year I can build a trusting, caring space that doesn’t only revolve around doing mathematics together, knowing that when the year ends each and every student will go into the world shining bright. I shared that I hoped that through talking about our uncertainty from the election we’d all feel a bit closer to one another and I was confident that this closeness would also allow us to better support one another when doing math.

After reading the story aloud, I explained that we’d spend the next 10 minutes doing 3 things:

  1. Everyone would write in the chat what this story makes you think about.
  2. Everyone would spend time quietly reading other students’ comments in the chat, scrolling back to read at whatever pace they wanted.
  3. Everyone would respond to at least 2 other students’ comments using the @ symbol and addressing them by name.

What they wrote was endless, powerful, brilliant, and wise. Their words, their humanity and their care for one another is so apparent. I’m only sharing bits and pieces, a mash-up of all my classes, to protect all my students’ privacy.

  • I’m not a threat, I’m a child with a life.
  • Yeah, me too, it’s like when I go to synagogue and there’s security guards with guns.
  • I am so sorry! That is awful.
  • I think about that all the time, that if EVERYONE did little things to make the world a better place there could be so much change
  • I want to know more things I can do so that everyone feels safe wherever they go.
  • Climate Change is really scary for me and I don’t feel that enough adults are doing anything about it.
  • I hate being scared of existing in society. I’m upset that I have to be afraid.
  • I’m thinking a lot about racism.
  • Yes! You are entirely right. It is super messed up that people are killed and treated unfairly for who they are.
  • I’m glad my whole family voted.
  • I hate being scared of going out in public because I’m Black.

We will know the results of the election in the coming day or two. No matter who wins, our students are desperate for these conversations. May we always work to create the space and truly listen.

Distance Learning Tips from Berkeley’s Middle School Math Team

Several years ago I was named Berkeley’s Instructional Technology Coordinator.  As part of my work as a new administrator, I wanted to get to know teachers and teaching styles in my district as well as I could.  That first year, along with my DigiTech team of 3 others, we made over 700 classroom visits.  My biggest takeaway: there are SO MANY incredible teachers doing SO MANY incredibly varied things in their classrooms and we should showcase and learn from them as often as possible.  From then on, I was committed to the idea of encouraging teacher-leadership and teacher-led PD.  I have many mentors and collaborators within my district for this and am honored to work with each of them.  Two years ago, when I decided to return to the classroom, I made a promise to myself to continue this theme of supporting teacher-leadership and teacher-led professional development.

Screen Shot 2020-08-09 at 2.17.42 PMMy gratitude for The Berkeley Public School Fund runs deeper than words could ever express.  You can read about their extensive work on their web site, but I want to especially thank them for being such a powerful supporter of teacher-led initiatives and professional development.  For the past 2 summers, King Middle School teacher Geeta Makhija and I have received large grants to run middle school math summer institutes.  At both, our idea was to develop teacher leadership and strengthen our middle school math team’s collaboration.  Although our original grant was for an in-person summer institute to focus on middle school math intervention, we pivoted to organize a virtual institute focused on sharing & reflecting from distance learning in the spring and improving it for the fall.  We have 3 middle schools and around 25 middle school math teachers.  We offered everyone the opportunity to be paid to learn something new over the summer to improve distance learning and then present on it to our entire team.  Eleven teachers said yes and the presentations were incredible.  Our institute spanned 4 days and each day had presentations plus small group collaboration time to dig deeper into any one of the topics we had learned about that day.

I know I don’t have the time to carefully write up summaries of each of the presentations, so instead will share bullets of what I want to remember.  I don’t have permission to share each presenter’s slide show, so I removed the links, but here are the presenters and their topics.  I’m so incredibly proud of our team, the unique and powerful topics that each teacher took on for their summer learning.

    • Have a slide show template.  Students build a slide to show their work and record a Screencast explaining their thinking.  Cap the time they are allowed to record so that you are not listening to them for too too long.
    • Give students leveled choices using choice-boards for asynchronous work and a rubric for how you’ll evaluate that work.
  • DESMOS: Escape rooms are an incredibly fun way to do team-building activities.
  • HONORING BIPOC MATHEMATICIANS: I blogged about it here.
  • IXL:
    • Explicitly teach students how to learn from an example so that when they get something wrong and an explanation pops up, they have learned what to do with that.
    • If you use the diagnostic, don’t give it all at once.  Spread it out over several days and teach students how to find and click “I don’t know this yet” when they honestly have no idea how to do a problem.
    • Explicitly teach students how to go back and forth between their Zoom screen and their Pear Deck screen.
    • Use for Quick Writes & journal responses.
    • You can run a lesson synchronously one day, then create a new link to have it available asynchronously for students who were absent.
    • Use different types of mood meters to address how students are feeling, socially/emotionally on different days.
  • 3-ACT TASKS:
    • When showing an image and you want students to think of a story to describe what may have happened, have them use the chat.  Call out students names and highlight stories as they go.
    • Have students use the @ button to reply to comment on others’ stories in the chat.
  • ZOOM:
    • Use a sand timer when putting students into breakout rooms.  Show them the timer and say you’ll flip it when you click to have them zapped into breakout rooms and will call them back when the sand runs out.  Use the sand timer for short ‘turn and talk’ type situations.
    • Connecting a document camera means you can have a permanent writing surface without having to rely on Apple Pencil/iPad.
    • If you can get physical white boards into kids’ homes, they can show work and hold it up to camera.  Very easy to see it.  Trying to see kids’ written work on paper over the camera is nearly impossible.
    • Berkeley Everett’s web site Math Visuals  is a great one for Zoom-led number talks.
    • Kids on Chromebooks cannot rename themselves, so be ready to do this as the teacher/host when a kid’s Zoom name isn’t their real name (as often happens when signing in on a family member’s device).  Host can change a students’ screen name by going to participants->More
    • All students (and teacher) can add pronouns to their name by going into Zoom settings and editing their Profile.  Your pronouns will then permanently show up on any Zoom session alongside your name.
    • “Leave Computer Audio” turns off all Zoom audio.  Great to use when you are giving quiet work time to kids while they are on Zoom.  A kid can turn that on when they want quiet work time and off when they want to come back onto audio to ask teacher a question.  But that way, others won’t have to listen to the questions.  It’s found by clicking the arrow next to the microphone/audio button.
    • In breakout room settings you can set it to have a countdown timer broadcast into the screen in breakout rooms so participants always know how much time is left.

I hope these tips are as helpful to you as they were to me.  I wish I had time to write up a full summary of all 11 presentations, but this is the best I can do for now.  Our math institute was far more powerful than a list of bullet points.  There were several times when teachers were brought to tears by the incredible depth of diverse knowledge and strengths among our team.  We learn SO MUCH from one another and have such profound respect for one another.  Our teaching styles and strengths differ from one another, yet we also have so much in common.

And a huge thanks to Geeta Makhija, who wrote this year’s grant and didn’t let me walk away at times this spring when I wasn’t sure I could hang on to the work any longer.  Her unrelenting confidence in our ability to make this dream a reality in the midst of all the uncertainty about this coming school year (and us both having very young kids at home) is something I will always be grateful for.   The Berkeley Unified School District’s Middle School Math team is truly remarkable.








Celebrating Blackness and Mathematicians of Color

Before I share how and why I celebrate Blackness and the Achievement BIPOC Mathematicians in my classroom, take a few minutes to watch this video of Alexis Scott, an electrical engineer.

After watching this video, think about your reaction to the clip or what Professor Scott’s have to do with us as math teachers.  When I asked the middle school math teachers in my district to share their reactions (in the chat on Zoom since that’s where we live for professional development nowadays), they wrote things like:

  • I wonder if staff members at my school feel like this?
  • I am angry.
  • I worry this is how families of my students feel when they come to my school.
  • This makes me so profoundly sad.
  • Do I perpetuate this feeling in my classroom?

I shared this video to help explain how my commitment to celebrating BIPOC mathematicians has evolved over the past few years.

I first started thinking about this after reading high school math teacher Annie Perkins’s blog post and resources around mathematicians are not just white dudes.  Soon thereafter I discovered Math-Teacher Educator Dr. Kristopher Childs’ slides on Black Mathematicians and Latinex in Stem and the website Mathematically Gifted and Black.

Screen Shot 2020-08-07 at 11.00.22 PMIn February, for Black History Month, I would use the last 5 minutes of class twice a week to feature a Black Mathematician on a slide.  As a class we’d read the information, then I’d ask students what fascinated them about this person and/or what questions they had.  We’d use our final minutes to Google the answers to their questions.  At the end of February I made a bulletin board with the printed copies of those slides.

However, over the past year, my thinking on WHY it’s important to feature BIPOC mathematicians in my classroom has evolved.  It’s not just my BIPOC students who need to see themselves reflected in the successes of professionals in STEAM fields.  My white students need to know of their incredible accomplishments as well.  As Professor Scott so poignantly reminds us in her Ted Talk,  BIPOC mathematicians are constantly made to feel invisible by people not believing they have accomplished all that they have.  And I want to do everything in my power to ensure that this eraser does not get perpetuated in my classroom.  My hope (which my students know well) is that they never make assumptions of who can or cannot be successful in their math and science classes and that my job as their teacher is to make each and every one of them believe that they belong, not just in my math class, but any math class.

So now I do the work of celebrating the accomplishments of BIPOC in STEAM fields all year long,  I aim for once a week during the last 5-7 minutes of class.

Here are a few examples of how this can be done.

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Take a moment and think to yourself: what you notice and wonder about this photograph.

When I asked our middle school math team (again in the chat on Zoom), their responses included:

  • I wonder if her earrings have great significance to her.  They are striking and beautiful
  • Her posture shows her pride.
  • She is wearing a lab coat from Harvard Medical School.
  • She looks like she knows something we don’t.
  • Her eyes tell 1000 words.

I then gave teachers 3 minutes to read this article about Lash Nolan, the first Black President of Harvard Law School.  I let everyone know ahead of time that 3 minutes wouldn’t be long enough to read the entire article, so not to worry.  The goal was just to learn a bit more about her.  Afterwards, I again asked them to use the chat, this tine writing something they would remember about Ms. Nolan.

As a final example, I showed teachers this video of Mimi Aung, a Burmese-American engineer at NASA’s jet propulsion laboratory.

Again using the Zoom chat, I asked teachers to share what interested them about Ms. Aung’s work or what questions they have about her work.

Several teachers asked factual questions which we were quickly able to answer through a Google search.  However the majority of comments focused on a central theme: what microagressions does she endure working with what appears to be exclusively men and nearly exclusively white men?

You could share any of these photos, articles or videos similarly to this.  And the conversation you have with students can be as brief or extended as you have time for on any given day.  Now that I have these conversations weekly, I know that some weeks we will go deeper into issues that arise and other weeks we won’t have time.  But over the course of the year, these conversations becomes an integral part of my math classroom and are used explore how ALL my students define a mathematician or someone working in STEAM.

I am happy to share my resource list of texts and videos with one request: that you will email or send me a Tweet as you discover more resources I can add.  This list allows me to do this work over the course of a school year, but honestly barely scratches the surface of the incredible work in STEAM fields happening by the BIPOC community.  I treasure your feedback.   The resource list is below:

Celebrating BIPOC and Women professionals in STEAM



Compliance vs. Actual Communication


Know Better. Do Better.

I have read this in various forms on Twitter threads and it’s a notion often shared by my district’s PD Coordinator whom I greatly admire.

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In his talk, How to Overcome Microwave Equity in the Leading Equity Summit’s Virtual Conference, Cornelius Minor adds a nuanced lens to this idea.  When asked what he wishes he had known when he began his career as an educator, he replies, “Instead of what I wish I’d known, I think about what I wish I’d listened to.”

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How often have we felt that we needed to hear a new idea several times before it actually sinks in or before we actually believe it? I know at times I have felt this sentiment.  I feel challenged now, more than ever, to check myself.  Am I actually listening?  Especially when what am being told pushes or challenges my own beliefs or ways of seeing the world.

But he didn’t stop there.  What REALLY sticks with me is his challenge to educators to broaden what forms of communication we listen to.

“Everything that a child does is a form of communication.  Our first missteps are that we assume that all of those messages are going to come verbally or in a written form.”

As a teacher, I feel I always listen when kids come to talk to me about something.  And I often begin class with some quiet writing time where I’ve asked questions to check in with how they are doing, or to reflect on some aspect of my class.  Check.  Done.

But what if we open our eyes to this notion that every aspect of a student’s’ behavior is a form of communication with us.  What happens when we deeply listen there too?

“The way a child walks into the room; the way that a child wear’s their hoodie; the way they respond to my requests.  All of those things are forms of communication.  And if I am not paying attention to those things, I am missing the message…A student’s behavior is a form of communication.”

What if instead of being frustrated that kid #1 takes off his hood before entering class and kid #2 always wears it and then refuses to take it off when I ask him to, I instead spent time reflecting on what each of those students is communicating with me and built my relationship with each kid around learning more about that?

We are challenged to constantly ask ourselves,

“What is the difference between compliance and actual communication?” 

He explains that we have to intentionally work towards decentralizing power because,

“No kid is going to tell me their truth if I am holding all the power.”

Finally, a call to action to each of us:

“What messages do we communicate to students through our affective responses to them?  Often we articulate words that say we value kids, but then engage in behaviors that do the opposite.  WE must work to ensure that our words are in alignment with our actions.”

This notion that often, unknowingly, our words our not in alignment with our actions reminds me of the notion of “discretionary spaces” which Deborah Lowenberg Ball and Amber T Willis discussed at CMC-North this year in their talk, “(How) Can Mathematics Teaching Disrupt Racism and Oppression.”

Though a detailed analysis of student/teacher/class interactions, they show how any of our common “best” practices as teacher, whole class discussions, calling on students to show their thinking at the board, etc., will can reproduce patterns of racism and marginalization unless we are constantly and consistently reflecting on the impact of our words and affective responses to students-not just the one who is speaking, but everyone else as well.  In this whole class discussion there were 59 student/student/teacher interactions in just over 2 minutes.

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As educators are we thinking constantly and intentionally about status, student agency, and the spoken and unspoken messages students are communicating with us (and to each other) in each of these countless discretionary spaces?

As I return to the classroom for the first time in 2020, I enter with a new definition of what it means to deeply listen to my students.  This quote, engraved in the National Museum of African-American History and Culture in Washington DC, applies to changes large and small that I hope to make this year as an educator.  I hope you will each question and challenge me and hold me accountable as this new year marches on.

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Fa-La-La: Pre-Winter Break Math

The week leading up to winter break for a middle school teacher is a unique mix of giddy, feisty, whining, forgetful, happy, flirtatious and most of all exhausted teens.  Oh yeah, and lots of absent ones who left early for holiday travels or trips back home to see family in other countries.  Every year I seem to make different choices about how much to embrace this perfect storm of challenging teaching and how much to ignore the reality and just carry on.  This year I decided I wanted our last two days of school to feel as joyful as possible so that everyone would leave simply loving the beauty of math.

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But how to achieve that elusive goal?  I tried my luck at a Twitter post.

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I owe a huge thank you to everyone who responded.  If you are looking for art projects suitable for both 8th Grade Math and Math 1, click here to see the incredible ideas gathered from my post.  And most certainly click if you are looking for art projects for yourself or your family for winter break.  Since posting my request, I have done nearly every one of the recommended projects at home with my kids and have a few more we’ll try over winter break.

I chose to do Susan Russo’s (@Dsrussosusan) Kaleidoscope Project because it was the one which literally made me squeal in delight when she shared it with me.

Since I wasn’t sure all my students would know what a Kaleidoscope was, I started with this video from inside the world’s largest, located on a resort in upstate NY.  I showed 10-15 seconds, then paused it and asked students what they noticed and wondered.  They talked about repeating shapes, wondered how it was made, or if mirrors are involved, and how the video was taken from inside it.  We played and paused a few times, to see it through various iterations.  After the video I mentioned a few fun facts from this travel web site.

Building a model in GeoGebra requires both a general understanding of reflection and also what y=x and y=-x looks like.  We haven’t yet studied linear equations, so wanted to do a brief introduction.  Using this graph in Desmos,  I asked table groups to come up with as many points as they could which would be graphed by each equation.  After graphing their coordinate pairs, I had them predict what the graph would look like.  I then turned each graph on.

We were then ready for GeoGebra. My students had used GeoGebra a few times during our transformation unit, so they were already comfortable with how to create rigid motions.  Creating a Kaleidoscope is surprisingly simple.  Using both axis plus y=x and y=-x, you add a single point, then reflect it 7 times over both ends of each lines.  Once you have 8 points, they all move in unison since they were created as reflections from point A.  To create the drawing effect, you highlight all 8 points and turn on the trace tool (through settings)

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Some students were satisfied with a single color (which is easy to change by highlighting  points and changing their color).  However helping them understanding dynamic colors became quite thrilling.

The idea of dynamic colors is that the color changes depending on how far the points are from (0,0).  To do this, you add (0,0) to your graph, then add a segment between point A and (0,0).  Once again, select all the points, then right click and open settings.  Choose the advanced tab and play around with values for red, green and blue using multiples of your segment length (called h in my graph).

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Students played around for a LONG TIME with both colors and drawing and redrawing designs in their kaleidoscopes.  I simply let them explore and run around the room checking out each other’s designs.  Eventually, I asked that they choose a design they wanted to use for a permanent display.  I had them take a series of screenshots of their design as it morphed and grew.  They used to create gifs with their screenshots.  ***I learned that having students make their own accounts in gify is a nightmare so I eventually just signed everyone into my account which was far easier as now I have all their work in 1 place.  I also (thankfully) learned that lots of people can be logged into a single gify account at once, all creating and saving gifs. ***


They’re each so proud of their gifs and I cycled them together into a slide show which will play on the monitor in our school’s hallway when we return from break.  And with that, I’m good and ready for vacation!

13 Seconds of Mathematical Bliss

Got 13 seconds?   Good.  Watch this.

I used this silent video today for a stand and talk as part of 8th grade rigid motions.  I asked if anyone knew what the sum of the angles of a triangle are.  Maybe 1/4 of the class raised their hands and knew it’s 180 degrees.  I said that there are a lot of things we know in math, but often we don’t ask why.  I asked if anyone knew why the angles add to 180.  Crickets.

Before we had any conversation, I showed the above video 3 times in a row.  Students ooed and awed.  Some busily started talking to a neighbor, bouncing up and down with, “I get it, I get it.”

Then we broke it down into a whole series of pair-shares.

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“What do you assume is true here?

They concluded that L1 and L2 are meant to be parallel.  I threw a few extra points on the image, at each intersection point, telling them that these points would help them describe what they saw with more mathematical precision.

Screen Shot 2019-09-30 at 8.53.05 PMI then paused here and pairs described everything they had seen.  Students talked about corresponding angles, congruent angles, translating an angle along a vector.


We repeated the process pausing here and again here:

The rich mathematical language this 13 seconds of video evoked was so exciting to witness.  It felt like nearly everything they had learned over the past two weeks of rigid motions was giddily talked about.

Are you a math coach?  A technology TOSA?  Or someone who loves having students talk about visualizations?  If so, I so want more of these gems.  For the second year in a row, I am teaching a new grade level, so I don’t have a whole lot of time to seek them out as I can barely keep up with tomorrow’s content, so I am selfishly blogging in hopes that some of you will send me more of these resources connected to my course (8th grade math) or any course before mine.



Informal to Formal: Building Confidence

I teach a math support class (called Math Studio in my school) which is a second period of math for a small group of students.  For me, there is a magic one can create in this class.  With just the right amount of preview and scaffolding in math support, students arrive in regular math class later in the day ready to shine.  And many have never felt that shine in math class before, so it’s incredibly rewarding for me to watch and for them to experience.

Next week we begin our 8th grade unit on transformational geometry and today in Math Studio we developed an informal understanding of various transformations, going especially deeper on translations.

First, students did Shifting Shapes, a relatively new Desmos activity which provides a visual and conceptual look into rigid and non rigid motion.  They are given a square with draggable vertices and asked to make their own shape.  In subsequent slides their shape undergoes various transformations which they are asked to describe in words.

Here is an example of one student’s shape (original in grey).

I so love both the informal and formal language they used to describe it:

“It went woop and to the side”

“It did a backflip”

“My shape rotated”

“It was turned sideways.”

Here is a second transformation (from a subsequent slide–again grey is original and red is after the transformations)

Students descriptions are again quite elegant:

“It went smaller then to the left”

“It got small then ran away”

“The shape got smaller then moved to the side”

I have to admit, I LOVED the idea that a translation is when a shape runs away from itself.  I ended up naming this idea after the student who said it and for the rest of the period, when we talked about translations, we talked about it being a slide, but also it running away from itself.  I’m smiling thinking about this idea as I type this.

While Shifting Shapes was a great, quick way to get some informal language out about transformations, I then went deeper into the mechanics of translations.  Again, I wanted to spend time looking at them informally first.

Using GeoGebra, I drew a polygon and a vector. Screen Shot 2019-08-29 at 10.55.17 AM Students then talked about what they saw and made predictions of what the arrow on the vector meant.  I explained that a translation was the same as the shape ‘running away” and the vector showed the direction and distance that it would ‘run’ (or slide).

Screen Shot 2019-08-29 at 3.51.29 PMWith this drawing on my LCD, I gave students 2-3 minutes to go to a vertical white board, draw the original shape, then draw their prediction of where the translated image would be.

They then came back to their seats where I showed in GeoGebra what the translated figure landed, and we discussed their drawings.  At first, their drawings had the translated image in the right general area, but they were considering only the direction of the vector, not distance.  I had them predict where the end of the vector would be if I moved the beginning of it to A.  I modeled this on several vertices as we discussed, like this:

We did this back and forth between vertical white boards and watching GeoGebra about 5 times with different figures and vectors.  By the end, maybe 15 minutes later, nearly every student could accurately draw the translation, label its vertices with A’, B’, etc, and explain whether the translated figure would overlap the original one or not.

I am hoping this deep conceptual dive, rich with visual models and lots of time to both practice and refine ideas will provide the scaffolding my Math Studio students need to feel confident and skilled when we cover this concepts in 8th Grade math beginning on Tuesday.  By no means will I eliminate the visual models in my regular math classes, but the reality is that we’ll move to the informal a bit more quickly and with a few less examples.  This blend of GeoGebra and Desmos within a single lesson felt seamless to me as the 8th grade Common Core transitional geometry work blends so well with the strengths of each of these tech tools.

Exploring Identity in Math Class

“Why are we doing this, it’s not history class?”

This student’s critique of today’s lesson spent on identity is one which will sit with me for some time.  I want to deconstruct it in my own head and discuss it tomorrow in class.

For now, here is a very brief summary of how and why we talked about our identities today in 8th grade math,  honoring the work of Jess Lifshitz and Sara Ahmed

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Today was Day 2 of school.  Students arrived into class with this on the board.  Their questions and observations were so rich and in 2 class periods students assumed it was a play on words for “square root” which cracked me up as it was something I hadn’t even considered.

One thing each class period noticed was how rare it is to see a tree’s roots and al the ways this photo is very out of the ordinary. I used that as an opportunity to talk about how we each bring to class a huge amount of roots, that define us and make us who we are.  But often, that person sitting next to you in class only sees you in the chair, but knows nothing about all the factors that make you who you are.  I explained that feeling safe and respected in math class means that the people around you know enough about you to respect your roots and who you are inside.

IMG_7202I then read The Day you Begin which is such a wonderful book about our identities, assumptions we make about one another and how often we have more in common with people than we realize.  But we have to have the courage to talk about ourselves so people can get to know us.  My students were REALLY into the story. I joked about ‘when life was fun and easy’ in 2nd grade and we reminisced about coming together on a rug for story time when their bodies were much smaller and there were far less of them in a classroom at the same time.

From there we did this activity from Teaching Tolerance which had been recommended to me by Shira Helft.  I modeled it by writing my name in the center circle, and talking about how arrows pointing outward are places to write things about yourself that you’d like for other people in math class to know about you.  Arrows pointing inward are assumptions people make about you (whether true or false) without really knowing you.  Their work was thoughtful and powerful as evidenced here.


Finally, we used visible random grouping for the first time this year.  Students were paired up and asked to do the following:

  1. Introduce yourselves if you don’t know each other’s names.
  2. Using your paper, read 1 thing you wrote.  Choose one sentence starter to use:  “One thing you should know about me is…”     OR  “One assumption people make about me is…”

I changed pairs several times so that they had the opportunity to do this with more than 1 partner.

Later this week, I have a slide show of images that are up around my room which represent my values as a teacher.  We’ll use this in conjunction with writing and sharing along this prompt:

What I need from my classmates to be successful in math class is…

What I need from my teacher to be successful in math class is…

Which gets me back to the critique of today which was shouted out by a student: “Why are we doing this, it’s not history class?”

I plan to honor this question and ask it to the class later this week, hoping that together we have a more solid understanding of why…


Where does Chocolate Milk Come From?

[This post is a part of the fabulous Virtual Conference on Humanizing Mathematics organized by Hema Khodai and Sam Shah. More information found here.]

Look at these photos for a few seconds.  What do you notice?  What do you wonder?

Last weekend my family and I were driving 2+ hours north of the San Francisco Bay Area to a tiny town on the Russian River, near the coast where the river hits the ocean.  It was a gorgeous drive, though lots of farmland.  We passed numerous fields of grazing cows, all black and white, with no comments from the peanut gallery in the back seat.  Then, we passed a field of brown cows.  My 5-year-old literally started screaming, “I knew it, I knew it!  Chocolate milk DOES come from cows.  THERE THEY ARE!  The chocolate milk cows!”

It took me a minute to realize how the farmland had changed to prompt this giddy comment.  My 10-year-old seized the opportunity to start lecturing his younger brother on the impossibility of this conjecture, however I cut him off to share my own commentary.

“Isn’t it fascinating to realize what ideas someone has in their head and we have NO IDEA of those ideas without something like this (passing brown cows) to open a window onto their thinking?”  My 10-year-old sighed and went back to sorting his Magic Cards.

I spent much of the rest of the car ride pondering this idea, thinking about how to ensure that my math classroom opens that window onto my students’ brutally honest thinking and reasoning as frequently as possible.

Earlier in the week, I found myself engaged in deep philosophical [mathematical] discussion with my chocolate-milk-pondering child.  We were reading a book entitled “How many” which had a photo of a pizza with 6 slices of basil under the heading “half dozen.”  After a lengthy discussion of all the ways to count those 6 slices, he asked what a full dozen would be.  We talk about cutting things into 2 parts and each being a half.  When I realized all my talk was confusing, we tried with fingers.

“Hold up 6 fingers,” I requested.

I did the same.   I asked him how many fingers we had up all together and he looked at me confused about what to count first.

“Let’s put our hands together,” I suggested. And we did.  Like this:

Screen Shot 2019-07-29 at 9.10.19 PMWe pressed our hands together and he knew right away that 5 and 5 makes 10.  Then we pressed our thumbs together and I asked him how more that was.  He paused and said, “10-11-12.  It’s 12!” And then he paused again, became frustrated and said that it couldn’t be 12.

I asked why.

“Because 1 and 1 is 11,” he exclaimed, now with a sigh of relief and glee at having discovered the answer.  He grabbed the book from me, flipped to page 11 and showed me how 1 and 1 makes 11.Screen Shot 2019-07-29 at 9.10.19 PM




Our debate of 12 vs 11, of which this bedtime moment was only the beginning, reminded me of something Tracy Zager discussed in her workshop on ways to assess students’ mathematical understanding at OAME (Ontario Association of Mathematics Educators) this past May.  She talked about doing math with Christopher Danielson, how it made her nervous at first because he has his PhD in math and she is an elementary school math coach.  But she quickly realized that Christopher is endlessly fascinated by learning how other people think about the math and that when solving problems together, he is constantly delighted by watching and learning from others’ thinking.

As a parent, I felt so incredibly lucky to witness my son’s rough draft thinking (to coin a math teacher term) around how chocolate milk is made and how to quantify 11 vs 12.  And because I both witnessed these moments and took the time to ask more questions, left some unanswered and delight in our conversations, my son now has a far deeper understanding of 11 vs 12.  (Cuz I ain’t touching that chocolate milk notion as it’s simply too precious).

Since then, I have spent the past several days thinking about how many of these equally delightful moments I miss with my students and how can I shape my classroom community this year so that we all don’t simply value this rough draft thinking, but delight in it and know that it’s necessary for deeper understanding of the mathematics we’re learning together.

My challenge to myself, and to us all, is to develop this delight in ourselves and each and every one of our dozens (and in my case, and many of yours too,  100+) students; a delight in watching and learning from each other’s thinking.

And after pondering that challenge, if you find yourself with extra time, Google “brown cows.”  Apparently my 5-year-old isn’t alone.

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Joyful Mathematical Noise

Screen Shot 2019-06-26 at 9.18.50 PMTwo decades ago, I began my teaching career as an elementary school teacher.  Pairing students up to collaboratively read these poems was always a highlight.  Diverse talents, coming together to blend their voices into a unifying theme.

Thanks to a very generous Strategic Impact Grant by the Berkeley School Fund and additional funds from Susanne Reed, BUSD Professional Development Coordinator, Berkeley Unified math teachers spanning grades 4-9 had the opportunity to spend a week together creating some incredibly joyful mathematical noise.

All three middle schools have elective classes for students who need additional support in math.  This summer work was the culmination of a year-long, teacher-led focus on creating more collaboration and coherence between the content of these courses at all 3 sites.


Monday through Wednesday was an intense focus on how we support struggling students in middle school.  Geeta Makhija, King 7th Grade Math and Intervention teacher co-led these days with me.  As we have built relationships with our struggling students we have learned that many previous teachers have not believed that they could succeed in grade level math because of lower grade skills they lacked.  Teachers continue teaching without carving out targeted instruction to support these necessary foundational skills because we often don’t know how these elementary math skills were taught in the first place.  This 3-day institute allowed us the opportunity to learn from elementary experts so that we can better support students who still need time to learn this foundational math.

Our mornings were led by Ana Delgado, former K-5 BUSD Math coach and current Math TSA at two of our elementary schools.  We wanted to more deeply understand how multiplication and fraction concepts are taught in 4th and 5th grade as they are the biggest gate-keepers to success in higher level mathematics.  


Ana used this graphic to discuss the importance of using three types of representations when teaching math.  She said the word “cat” and asked us to share what came to mind.  Teachers verbally shared the mental images which came to mind, “Grey; puffy, striped, cuddly…etc.”  No one shared the letters c-a-t.  Ana contrasted this with the teaching of mathematics where we often ONLY discuss the abstract, symbolic representation of an idea, without also discussing the concrete and visual representations.  I loved this notion and am excited to think more deeply about how to build students’ mental models of math concepts for everything that I teach.

Ana then gave us the opportunity to compare and contrast how we solve elementary math problems with how students are taught to do them.  We practiced partial quotients for division and explored both an area model and number line for fraction multiplication.

In addition to Ana, we had presentations by several teachers on classroom routines and pedagogical strategies which have been successful in their Math Support classes.

  • Wendy Lai (King) presented on how she uses math stations.
  • Robert MacCarthy (Willard) presented on what he has learned from Jo Boaler’s work on Mathematical Mindsets and how he incorporates mindset work into his classes for Mindset Mondays.
  • Geeta Makhija (King) led us in a 3-act task focused on multiplication and fractions (The Big Pad by Graham Fletchy) and discussed the power of using rich tasks in our Math Support classes
  • Joshua Paz (Longfellow) presented on how he has students doing error analysis and how he sets up his grade-book for students to understand which concepts they are strong in and which they need to continue working on.
  • Kinjal Shah (Willard) taught us how she spirals the math in her course, with explicit times each week to review material from earlier grades and other times to preview the math coming up in the grade level course.  One resources she uses for students to independently practice & keep track of their progress in foundational skills is a web site called That Quiz.
  • I presented on how Desmos’ Snapshot’s Tool can be used to bring multiple perspectives on problem strategies into class discussions and allow more students’ voices to guide class conversation.

Our end-goal was for our courses to be more aligned, so our afternoons were spent collaborating, putting shared resources into folders and co-planning our courses.   In addition to Ana, we were joined by elementary RTI teacher Vanessa Sinai.  This was truly a collaborative effort. Tons of ideas and links were shared which are all found at the bottom of our agenda here. 

We used some of our grant funds for a daily raffle where we had tons of incredible math resources as prizes including, of course, a #mathgals t-shirt.

As our team planned for this summer work, we realized that a broader outcome for this grant was to bring math teachers, together, from upper elementary through middle to 9th grade to more deeply understand students’ needs.  Thursday and Friday were led by our 6-8th grade Math Coach, Ryan Keeley.  One day was a collaboration between 7th grade math teachers and the following day was 4th-7th grade teachers.  The goals of both days were similar: develop deeper professional relationships between sites and grade levels in order to increase our common pedagogy and mathematical approaches to the teaching of big ideas.  We talked about how an idea introduced in one grade level (integer addition in 6th and 7th grade, for example) leads into ideas in later grades (vector problems involving distance and time) and the importance of using models in the lower grade which will inform those bigger ideas.  IMG_6520

For me, this week demonstrated the power of teacher-leadership and the need for sustained partnership between elementary, middle and high school teachers.  Every day we were reminded that we teach the SAME students and that improving student learning depends on us continuing to learn from one another’s expertise.  I have such gratitude to the Berkeley School Fund and Susanne Reed, Professional Development Coordinator for valuing and funding this summer work.

Evidence of this learning was evident throughout our evaluations:

“All of my learnings will go directly toward informing instruction for my students that is more relevant, engaging, thought-provoking, and personalized, and that will promote deeper conceptual understanding. I have spent the past few days excitedly building my toolbox, reflecting, and recharging to begin a deep dive into reshaping what support class can look like this year. I believe that with these new additions students will grow even more in enjoyment of the class and growth in mindset and as problem solvers, and increase retention and understanding of topics covered. I’m confident that my students will benefit because even as a participant in this work I’ve grown in these ways – I can’t wait to see where my students will go with these additions.”

“I had a blast participating and am taking away so much that I’m eager to put into play for this upcoming school year. To have this kind of energy and momentum going into summer is awesome. I can’t wait to roll up my sleeves and start weaving together all that I’ve learned, all of these best practices, to make a more effective and impactful support program for our students. Truly a wonderful professional development time.”

I am so looking forward to returning to teaching next year with these amazing colleagues and so many more not captured in this lunchtime photo.  I’m honored to be a part of this powerful group of reflective teacher-leaders.