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.




How to Talk So Students Will Listen and Listen so Students will Talk

This week I messed up.  I got so frustrated at a student who was opting out that I didn’t stop to think about why.

Let me take a step back.

Our school’s mental health counselor often tells me that I should become a social worker.  She is always amazed at what my students tell me about their personal lives.  While flattered, in reality it makes me angry that this is seen as rare from a math teacher.  I am constantly thinking about what it means to teach for equity.  Creating a safe and trusting classroom where students are willing to risk telling me about the hardest parts of their lives is a really important part of their academic success.  If you spent only a few minutes in my classroom, you might never know that this happens because as I am teaching, I rarely pause the academics.  However, it’s the in-between moments that make the difference.  I believe these in-between moments are an important part of Culturally Responsive Teaching.

Screen Shot 2019-02-02 at 10.01.48 AMA student is doodling on his arm while the rest of the class is taking notes.  In the moment, I tell the student to stop and not get behind on their notes.  He picks up his notebook and gets back to work . But later in class, I go over, sit next to the kid, and ask if everything is ok.  I say that I noticed you’re really distracted today and want to check in.  Sometimes he says, “Nah I’m fine.” and sometimes he says, “My parents are thinking about getting a divorce and I’m scared.”  (And yes, too often he says, “I stayed up too late playing Fortnight last night.)

Screen Shot 2019-02-02 at 10.00.16 AMA girl is chasing someone down the hallway when she’s supposed to be lined up outside my door.  In the moment, I yell and tell her to get in line. But when the class comes in, and I’m giving high 5s at the door, I pull her aside, wait til everyone is in, and ask her if everything is ok.  Sometimes she says, “Nah, I’m fine” But sometimes she says that this kid just called her a bitch or that this other girl posted a picture of her on social media which she wants deleted.  And sometimes she says, I really like this boy and I don’t want my friends telling anyone.

When students misbehave, in small and large ways,  I am very strict in the moment to get them back on track, and try to later follow up to better understand why.  And often, kids tell me all sorts of personal things.  And often, I don’t have time in math class to really listen.  So when they tell me things, big or small, I then offer a time during my prep or at lunch or after school when they can come back to tell me more.

But this week, I messed up.  I have a student who has a lot going on at home including a relative who is dying.  And this student is opting out of being a student, a learner, a community member, in every way possible.  He has been opting out all year.  I have tried so many things, meeting with his mom; daily texts to his mom giving her a positive about his behavior; daily behavior goals with positive incentives chosen by the kid; checking in with his other teachers and his counselor; having a circle of support with him, led by the restorative justice counselor, to help him see what a wonderfully talented kid he is and helping him see how he has a community of adults at school who value him and are here to support him.

I messed up, because every day this week I got upset about his behavior, and the various ways he’s opting out.  I gave consequences.  But in those in between moments, I didn’t ask him how he’s doing.  I used the in between moments to tell him how disappointed I was in him instead of asking him how it’s feeling to be back at school after missing a bunch of it.  He’s a kid who has told me nothing about his life outside of school.  While I know a lot about him, it’s not from him telling me.  And that is my failure.  I need to take a deeper breath.

Taking more of these deep breaths, focusing on the ‘in-between moments’ in my math classroom as much as the teaching of mathematics, is my focus for the second half of this year.   For #TMC19 (Twitter Math Camp 2019 which is in Berkeley in July…yes!) I am thrilled to be presenting on this very topic.  Marian Dingle, an amazing elementary school teacher in Georgia who I met last summer at TMC, suggested the idea of interviewing students and families to learn about what they need from their teachers in order to be successful in school.  It feels so powerful to do this in a Berkeley school to present as an equity-strand workshop at an international teacher conference hosted in Berkeley.  I am so excited to be doing this work with my students this spring and hope that their experiences and their voice can become the beginning of a deep conversation with teachers from all over the world.

In the meantime, please watch this video, “Our Culture Our Schools-Culturally Responsive Education in New York City” made by the Coalition for Educational Justice 

This work is so important.  While I will still mess up, I will hopefully mess up less often.  And while it’s easy to talk about how to best teach that upcoming lesson on fractions, the work expressed in this video has just as big of an impact on how much students actually learn from that fraction lesson.

Unwanted Subject

Last week a man was asked by police to leave a Menchie’s Frozen Yogurt store in Kirkland, Washington.  He had been in the store for a while and hadn’t ordered anything.  He made the two employees nervous and they called the cops.  Turns out here was there for work.  He’s a court supervisor and was supervising a court-sanctioned outing between a mother and her son.  Which is why he didn’t order anything and was sitting in Menchie’s.  Would you like to guess his race?  He’s Black.  Would you like to guess the race of the 2 employees?  They’re White.Screen Shot 2018-11-20 at 2.25.13 PM

Here’s how Byron Ragland, the court-supervisor who was escorted out of Menchies by the police reacted:

“You want to stand up for yourself, as a man, or as someone who was just doing his job, and say ‘hey, this isn’t right,’ ” he said. “But in the moment I’m thinking: ‘I’m a black man, and If I start emoting, I might not walk out of here.’ And so you rationalize to yourself: ‘What’s the big deal, it’s just Menchie’s, just leave.’ But then later, you realize that you gave in — that you consented that this is the way it’s going to be, to always be.’

“Living this kind of mental life will drive a person insane,” he added.

I urge you to read the entire article from the Seattle Times, found here.

And as you are reading, ask yourself if he could have been one of your students (he could have).  And could a parallel situation have happened to one of your 4th graders, or 7th graders, or 11th graders (it could).  And then ask yourself if that student of yours would have told you about it and how you would have reacted.

Screen Shot 2018-11-20 at 2.26.38 PMFor me, this story brings up a reaction that I had to Julie Reulbach’s keynote at Twitter Math Camp last summer.  Julie eloquently brought up the countless ways that teachers are maligned by non-educators.  I absolutely agree.  She argues that student engagement shouldn’t be the sole responsibility of their teachers.  I also absolutely agree.  And yet, as math teachers (who I believe most people reading my blog are), we have to build relationships with our students so that we have ways to talk with them about what it feels like to grow up in the world in their skin.  Because that unengaged student of color sitting in my classroom may very well have had numerous experiences just like the one which Byron Ragland had as a grown man.

And as a teacher, while I can’t ultimately prevent this from happening, I can have deep relationships with my students so that they will confide in me.  Trust me with their truth.  And trust me to be a white ally as best I can.

The other day as students were walking into my class, one asked if he could speak to me privately.  Once the rest of the class was underway on their warm up, we stood in the doorway.  He confided in me that he and some friends had tried vaping and he wanted someone to know.  He was nearly in tears because it felt so good to tell an adult whom he trusted. He repeated over and over, “It just feels so good to tell you.  It just feels so good that someone knows.”

Student engagement IS my responsibility, and my work has to involve supporting students as they experience their world both inside and outside of school.  Their engagement so often is unrelated to the quality of my math lesson.  Many are facing countless challenges at home, with friends and out in the world.  I know this is part of what Julie was saying.  And that we can reach out to them so many times and there will still be issues facing teens that all of our best intentions can’t overcome.

And yet, part of my job HAS to be building and supporting my students as they grow up and become a greater part of the fabric of their world. A white student of mine may go on to become an employee in a frozen yogurt shop who calls the police because he/she feels nervous about the black man in the shop who is using his phone and hasn’t ordered anything.

And while I can’t walk in the shoes of a black student of mine, to know how it feels, I try to always assume that an unengaged student is one who I can work to make deeper connections with, to learn more, share more, and hope that I can become that teacher where, “It just feels so good to tell you.  It just feels so good that someone knows.”  And while that is part of my work with students of all races, it feels especially important with students whose race is different from mine.  THAT is part of my work as a white ally.  Screen Shot 2018-11-20 at 2.55.21 PM

Because “Living this kind of mental life will drive a person insane.”

There is some reaction happening in Seattle.  Here is a second article explaining the police department’s apology.  

What I hope is that the cops who escorted a black man out of a yogurt shop for simply doing his job will return to that shop to have an equally serious conversation with the two white women who called them in fear.  We all need to talk more about race and our implicit bias.  People’s lives (literally) depend on it.