44 Reflects on 17

Recently, as seems to happen about once a year, a conversation or two, probably more, threads the needle of Twitter about online math community aka #mtbos.  I remember it first happening here  and here in 2013.  These conversations are healthy and rich.

Each time this thread reappears, I am overwhelmed with the desire to say thanks.

In the summer of 2011, with generous funding from Math For America, I headed to the Park City Math Institute with my husband, our two mountain bikes, my then 13-year-old step-daughter and 2-year-old son, and our Bernese mountain dog.  Though I enjoyed endlessly fun family time, the real gift was discovering myself as someone who was good at math and welcomed into a community of smart, funny, creative, serious and goofy mathy folks.  At the time I was a middle school math teacher ( I really think I snuck in the back door when I got my single subject credential), it was the first time in two decades that I had studied math.

When I was a kid, I was really good at learning math facts.   In 2nd grade I was literally kicked out of class for being too good at my times tables, forced to memorize my 13s-25s when the teacher ran out of work for me to do.  In 3rd grade I got moved to a 4th grade class and in 5th grade I walked to our local middle school to take pre-algebra with 8th graders.  I took Algebra 1 in 6th grade, BC calculus as a sophomore, followed by MV Calc and Linear Algebra.  Throughout much of this I had the most amazing teacher, John Benson, who used Mathematica in the late 1980s better than most use Desmos in the late-2010s and who never stopped believing in me, even when I had given up on myself.

But give up on myself I did.  Senior year of high school, I promised myself that as soon as I got to college, I would never take another math class.  Sadly, I kept that promise for nearly 20 years.

I had gotten to a point where I could do a whole lot, but understood very little of it.  And I hated, really, truly hated, being seen as a math nerd.  There were maybe 16 of us in my MV Calc class and 4 of us in my Independent Study Linear Algebra class.  My memories (which are probably not entirely accurate) are of me, being the only girl, with a bunch of pimply, nerdy boys who were extremely competitive, quick at math, and uninterested in slowing down for me who needed to carefully draw, model and see every new idea.  I desperately needed Jo Boaler to whisper in my ear when I left for college, but thankfully I found her as an adult.

Although I occasionally get glimpses of this stereotype in my adult life, it’s pretty darn rare.  Starting in Park City and again when I did Math For America and again now within the #mtbos community, I have found my people.  And chances are, like me, you were a nerdy math kid.  But somehow you persisted when I did not and I am so thankful to have found you.  In Park City, among many others, I met Kate and Ashli,  Sam and Tina along with the wise teaching of Bill and Darryl and Cal.  And now, as a Desmos Fellow, I feel so lucky to be surrounded by such creative, silly, brilliant people like Suzanne and Jenn (and so many others!) who do the creative work I wish I had been encouraged to do when I was 17.  THIS is the math I would have loved at 17 and THESE are the people I wish I knew then.  Thankfully, I’ve found them now along with so many more.

Desmos Potluck for Lunch

One of the endless new ideas I have gleaned from being a Desmos Fellow is that of a Mathematical Potluck.  I’ll let Shelly explain…

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Fellows post graphs of a new calculator idea they’re tinkering with and we collaboratively try to recreate it, learn from it, chat about it, etc.  The conversations we have on Slack are pretty amazing and I have been really fascinated at how much math one can learn simply from playing with other people’s creations.

I’m surprised it took me this long, but after 7 months of Desmos Fellow potlucks it finally dawned on me that I should be creating this same learning environment with my own students.  My teaching of students nowadays comes in fits and starts, but one consistent source is a weekly class at UC Berkeley.  It’s a Cal-Teach intro course on math and science pedagogy for undergrad math/science majors who are considering getting their teaching credentials.  The course is just 2 units, meets just once a week, and touches on everything from equity to teaching English Language Learnings to learning to lesson plan to pedagogical content knowledge to integrating instructional technology.  So I share my potluck with the caveat that our time to develop Desmos calculator skills and ideas is a bit fleeting.

However, I found hosting a Desmos potluck to be an incredible way to build mathematical ideas while also building a sense of community among students in my class.  In class we briefly played around with the calculator and I purposefully left their homework very open-ended:

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The next week in class as our warm up, I had them look at all the submissions, choose one which interested them for any reason, play around with it and try to recreate it or use ideas from it to inspire something new.  I encouraged them to walk around the room and speak to the author of the item if it had mathematical features they wanted to know more about or aesthetic features that they hadn’t yet discovered.    Their homework was to continue their potluck contribution:Screen Shot 2017-04-15 at 3.26.25 PM

As mathematically fun as the potluck structure was, what was most exciting to me was seeing their beaming faces and the pleasant surprise of so many students to learn that others had found inspiration in their graphs.  For example, here is Danielle’s original graph:

Danielle’s work inspired multiple potluck contributions including Aubrianne’s contribution: “I was inspired by Danielle’s graph and played with using multiple variable coordinates on a polar graph and with variable shading.”

Meanwhile, Aubrianne’s original contribution was this.

Which also inspired several graphs including Wes who explains, “I was inspired by Aubrianne’s graph. I enjoyed how hypnotizing it was and wanted to create something similar. I tinkered with parametric functions and element lists in desmos. It was basically a lot of trial and error.”

If you’re interested in seeing more, The entire potluck is here.

I am currently helping to teach a 6th grade RTI math class and going to launch a Desmos potluck with them as well.  This notion that students should create something, and later realize that their ideas have inspired new ideas is such a powerful one and I’m looking forward to playing around with how to use this notion to deepen students’ learning of mathematics.

A Tale of Two Cities

Palm Springs and San Antonio.  Two warm escapes from the Northern California rain.  Each hosted some of my favorite teaching conferences: CUE (Computer Using Educators) and NCTM (National Council of Teachers of Mathematics).  I am able to go to each every few years.  This year I brought a small team to CUE.  However thanks to to Twitter, I did a decent job of learning from many of this year’s NCTM workshops.

I tweeted a thought last night which I regretted sharing nearly immediately afterwards.  At risk of sticking my foot in my mouth even more, I’m sharing it here.

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Soon after sending this, I realized that while my intent was to think more deeply about my own role in the EdTech community, my tweet unintentionally came off sounding like a criticism of CUE.  So, to both clear the record and ensure that I can sleep better tonight, let me be clear: I admire and respect the work of both organizations and have learned a ton from each.

At CUE I fell in love with George Couros Greg Montague, my techy neighbors in East Bay Cue  and many of the fabulous teachers whose presentations I attended.   I re-fell-in-love with Jo Boaler  and was filled with gratitude that I could spend the week with my Berkeley colleagues in Instructional Technology and Professional Development.

I have plans for our entire Instructional Technology Department to read & reflect on this inspirational gem of a book during our department meetings.

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However, now that conference season (for us math & tech folks) is paused, I am doing a lot of reflecting on my role as someone who spends nearly all my working hours planning for or leading professional development in instructional technology.  For me, it’s year 4 out of the classroom.  Year 4 in the Instructional Technology world.  And to respond to my own tweet, I do believe that too often, Instructional Technology PD becomes a list of tools and tricks without allowing time for deep thinking about pedagogy.

Much of my first three years in this work were spent doing just this: teaching tools. Why? To reluctant teachers I replied, “Because they’re cool, engaging, exciting…” It is only this year, year 4, where my own focus has dramatically shifted.

Few teachers have time for technology when it’s seen as a separate subject.  I now collaborate with my department to think about the integration of instructional technology around district-wide themes.  While we used to exclusively focus on doing model lessons in classrooms and PD around teaching how to use a tool such as Google Slides, Plickers, Screencastify or how to use Desmos, we now focus on integrating technology in support of our four K-8 district-wide equity strategies: High Help/High Perfectionism; Opting-In; Bringing Multiple Perspectives into the Classroom; and Bringing Students Lives into the Classroom.  The TSAs and I are working with teachers to plan units where we specifically think about how to leverage technology to support students whose needs are not currently being met.  And in doing so, slowly, we’re supporting changes in teachers’ pedagogy.  I feel incredibly lucky to have other administrators who share these values and regular time tScreen Shot 2017-04-10 at 8.02.26 PMo connect with the TSAs from all the content-areas.  Together we’re reflecting on our role as coaches using Elena Aguilar’s work as our guide.  At his CUE keynote, George Couros remarked, “Every single one of you in this room can make me better at my job,”  Now, more than ever, I feel this way about both the EdTech and Math online worlds that I inhabit as often as I can.  So though you won’t ever find me leading a workshop on the Top 20 MathEd Tech Tools, my journey straddling these two worlds continues.  My response to ‘Why?’ has changed and continues to evolve.  “Equity,” is always part of my response.  Not equity of access, but equity of experiences and opportunities.  I leave you with this touching video, again, part of Couros’ keynote, as a reminder that teachers and students, using technology strategically, in support of equity, has transformational potential.

Dueling Banjos: Desmos-Style

Working out of the District Office can be a very lonely job because unlike being surrounded by math teachers at my old school who meet regularly and collaborated monthly with all the middle school math teachers in the district, my department is now tiny and it’s hard to find the same sense of community within the district office as one does at a school site.

I was honored to be chosen as a Desmos Fellow this year and it has been one of several places where I have worked to foster connections between myself and others who are looking at instructional technology through the lens of math and equity pedagogy. Although the fellows often collaborate as a group, there have been several fellows with whom I have worked with individually on projects.  Jenn Vadnais, a former math coach and newly minted TSA for Instructional Technology in Redlands Unified School District is one of them.

Today, Jenn blogged about teaching a 4th grade Geometry lesson using a Desmos Polygraph Activity she had written.  Her post is here and involves a group of extremely curious students and a tech-curious teacher, both of which are an exciting audience when you’re leading a model lesson.

Two weeks ago, I too aught a 4th Grade Geometry lesson, on similar standards using a Desmos Activity based off the Which One Doesn’t Belong ideas from Christopher Danielson’s book and Mary Bourassa’s web site, both with this same name.  Though we live in opposite ends of a very large state, I couldn’t help but smile at our Dueling Banjo mathematical work.  **SIDENOTE: As a banjo player, I can assure you that there is a fabulous Dueling Banjo Pandora station where musicians take turns improvising from  the riff played by the other, always returning to the same original riff before the next banjo player takes a turn. This style of music shares a name, but nothing thematic with the film of the same name.  I promise.**

When writing this activity, in additional to mathematical content goals, I was encouraged by a math coach in my district to have specific language goals.  Together, we wrote sentence frames, increasing in complexity, using conjunctions that student wouldn’t necessarily think to use in a math activity.  The Desmos Activity Which One Doesn’t Belong: 4th Grade Geometry is here and the half-sheet of sentence frames that students used is here.

My intention was to have them do this activity three times within the hour, each with a different lens.  The big idea behind a Which One Doesn’t Belong (WODB) activity is that any of the 4 images can be the one which doesn’t belong, depending on your reasoning.

ROUND 1: Give it your best shot.  I first wanted students to work through the math, not focusing on language.  So after doing slides #1-3 together at the rug, discussing that there was no one right answer, that their reasoning mattered, a lot, the students returned to their seats to complete the activity.  My intention for this first part of the lesson was for them to complete all of the slides, with minimal support or reframing from me.  The class was at the end of their geometry unit, and I had written this activity as a review of the major concepts.  After about 3/4 of the class was near the end, taking their time drawing lines of symmetry on the final slides, I called the class back to the rug.

ROUND 2: Mathematical Reasoning.  Did you use 2 of the words from the vocabulary bank on each slide?  Did you hastily choose the image that you believed didn’t belong or did you spend time thinking or discussing the different options?

changing-answersI used the histogram feature for multiple choice questions to show screens where zero or just 1 student had chosen a certain answer. Students turned and talk to explain which figure they thought didn’t belong and why, focusing on including at least 2 vocabulary words.  When a pair finished, I challenged them to choose a figure which hardly anyone had chosen and determine why it could be the one which doesn’t belong.  We did this for two sets of figures, and then students returned to their seats to read over the responses to each slide, check that they had used at least 2 vocabulary words, and improve their answers, mathematically, as much as possible.  Again, I encouraged those who finished quickly or said that they didn’t need to improve any to choose a figure which few others had chosen and develop a response for it.  The histograms for each slide changed dramatically which was fun for students (and I ) to watch.

ROUND 3: Use more sophisticated academic language.  After improving their work mathematically, students returned to the rug and I passed out these sentence frames.

screen-shot-2017-02-28-at-9-01-37-pmTogether we looked at student responses from this slide:


Specifically, the student responses for the letter A:


I had one student choose his favorite ‘mathy’ response that he thought we could improve.  He chose, “The A is different because it is not a parallel”  As a class, we used the sentence frames to come up with three versions of that response to make it sound more, “like what a mathematician would say.”

Here is what they came up with as a class:


Students then worked with a partner at the rug, chose one of the sentence frames, and improved a second response from those displayed on the board.  With the final 10 minutes of class, they returned to their seats and used these sentence frames to improve as many of their own responses as they could.

Quite sadly, I don’t have a screen-shot of this same set of Red-A responses after they improved them.  When I run Desmos Activities in someone else’s classroom, I always do it logged into the classroom teacher’s account so that the student work stays with them.  The downside of this is that if I forget to take a screenshot before leaving the class, poof, the work is gone for me once I leave.  From walking around the room, and showcasing a few responses on the big screen before recess, I saw an incredible amount of focus on academic language.  Separating the language lens from the mathematical lens seemed to especially encourage struggling students to take the time to truly consider both.

Desmos For Social Justice Part Deux

Blogging seems to have slipped away from me this winter.  I was inspired to blog tonight about using Desmos to discuss how to support kids post-Trump’s election  and realized that the last time I blogged, it was on the very same topic.

Tonight was the first class session of the Spring Semester of the math and science pedagogy course I teach at UC Berkeley.  My step-daughter is a 1st year student at UCSB and has remarked how rarely, if ever, her professors have discussed the impact Trump has or could have on students’ lives or the fear his proposed policies create for many students. I found this somewhat surprising, but understandable.

However, in a pedagogy course, where students have a teaching placement, it feels of vital importance that these aspiring teachers reflect on how to lower students’ affective filters, whether that be around the math and science content, or simply ensuring that every student feels safe when at school.  Knowing that students, especially students of color, may be coming to school fearful of how Trump’s presidency could profoundly affect their families is important, whether or not your personal politics align or differ from his.

Tonight, day 1 of class, I posed this question via Desmos Activity Builder: Graph your emotions over the past week (the inauguration was 3 days ago and the Women’s Marches across the country were 2 days ago).  I made absolutely no mention of any of this, I simply posed the question via Desmos. **As a side note, out of 23 students, 4 had used Desmos in high school and 2 had used it in a UC Math course**

Graphs included:

I love the various interpretations of ‘graph.’  Some used the time/distance axes while others graphed a picture as a physical representation of their emotions.  The best part, however, were the discussions following the graph-making.  In pairs, students had 3-4 minutes with each prompt where they first introduced themselves and then discussed each.

  1. Choose a graph that’s interesting to you.  Discuss your interpretation of it.
  2. Choose a graph that you have questions about.  Discuss what questions you would ask of its author (these were displayed anonymously on the teacher dashboard).
  3. Choose a graph which you believe includes the President Trump’s inauguration. Discuss why you’re confident of this.
  4. Do you think this is an activity which connects math and social justice?  Why or why not?  What kinds of conversations do you think teachers should (or should not) be having with young kids about politics right now?

A class, which up until then had been nearly silent, roared to life.  Although question #3 felt like it came out of left field, I could tell by looking at the many grins and nodding faces when I read it that there were many who had included the past days’ political events and were excited to have me acknowledge them explicitly.   I can’t wait to continue this thread throughout the semester.

Desmos for Social Justice

Undergrads walked into my math pedagogy class at UC Berkeley this evening as emotional as I am about the results of the election.  So we talked about it.  But first, we graphed it.  Using Desmos Activity Builder I asked students to graph time versus their emotions over the past 48 hours.  We spent the next 90 minutes analyzing graphs, points of interest, and ended by pushing the tables out of the way, standing in 2 concentric circles with them paired up, talking about everything from how the election affected them, to how it affected students in their teaching placements, to what their role, as future teachers should be in supporting their students to be mathematicians and scientists committed to social justice.

Fluidity of Expert Status

In a recent post about Great Classroom Action, Dan Meyer mused about how to use the feedback features in Desmos’ Activity Builder to allow students to self-assess while not exacerbating issues of status in the classroom.  He is referring to Cathy Yenca’s post on how she uses the teacher screen during card sorts.


Let me first say that since Cathy’s 9/21/16 post, I instantly stole this idea.  I had been meaning to blog about it.  A whole post just to thank her for refining how I was teaching with Desmos Activity Builder.  I appreciate Dan’s musings as now I can post about his thoughts to.  First, I LOVED Cathy’s ideas.  She is one of my two classroom heroes.  Though we have never met and have only an occasional virtual teaching exchange, she, along with Julie Reulbach are two women who I feel most similar to as a teacher.  I see myself in them ALL THE TIME.  They are my heroes because unlike me, they are still teaching in the classroom.  Though I am now an administrator, I glean ideas from their blogs which constantly make my work stronger.  And I know, were I to return to the classroom, we’d have a whole lot more to share with one another.  And although there are tons of bloggers who influence me, they are the two who I really connect with as teachers. If you teach math, are curious about technology integration, and don’t read their blogs, please do.  You’re in for a treat.

Now, onto my thoughts on expert status.  As a middle school math teacher (which honestly wasn’t all that long ago, though I often start to feel like it was another lifetime ago), I pulled groups of students who had similar mathematical needs several times a week.  I gave exit tickets at least 3 days a week and the following day, I’d pull 1-2 groups of students.  Sometimes it was students who made a similar error. Other times it was students who got everything correct and I wanted to push their thinking on a certain concept or better understand how deeply they understood.  Sometimes it was students who got the same problem correct, but didn’t show any work and I wanted to hear from them how they approached the problem.  These groups were fluid and whomever was pulled was pulled for a specific reason which only corresponded to their work on yesterday’s exit ticket.  What amazed me, was how the membership of the groups changed from day to day.  It was never the same students pulled day after day.  Why?  Both because my criteria for how I pulled groups changed constantly based on what I wanted to know from students and because when you pull groups for a very specific reason, different kids fall into that category on different days as the reason is always changing.

However, I often received A LOT OF critique from colleagues in the math department. They said I was exacerbating status in the classroom.  Me mentioning that I was pulling groups of students with a similar error made many assume that the same struggling kids were in my group day after day and that gave them a lower status in the classroom.  While I completely understood their concerns, I knew that it was different kids every day and felt confident that I was aware of the harm of always pulling the same kids.

Cathy’s idea of having kids see this screen while they are doing a card sort in Desmos Activity Builder made so much sense to me.


Recently, while doing a card sort in a 5th grade class, I too had students strive for expert status.  But as Dan mentions, I don’t want this to create status issues and anxieties.  There were more subtle things I did while this was happening to have several pathways to “expert” besides having all your cards become green (correct).

* At one point, there was a pair of students who I could see from the screen had no cards correct. However they were very focus, having intense mathematical discussions, and showing each other their conflicting reasoning using scratch paper.  I stopped the class (thank you, “Pause Screen”) held up their scratch paper and made a very big deal about how impressed I was that they were having a “Math Fight” (my favorite kind of fighting) and that they were trying to resolve their math fight by using scratch paper to convince the other of their thinking (and yes I did mention the math practices that they were displaying).  When the class returned to work, everyone wanted scratch paper.  So now, “expert status” was defined by not only getting all green cards, but by convincing your partner with scratch paper (as students were paired up each Chromebook).

* I also am a lover of personalized stickers from Vistaprint. I have about 10 different phrases, all fun like this:fullsizerender-3

 I stole this brilliant idea many moons ago (2012) from Sam Shah.  Did you get a sticker because your group was one of the first ones to get all your cards sorted correctly?  Nope.  Since I already had a reward for those who were faster to get everything accurate, stickers (which are VERY high status) were only given to groups to I saw debating, struggling, and making sense of their misconceptions either within their pair, or by getting help from today’s ‘experts’ who were going around offering help.  And somewhere in there I stopped the class to talk about how you earned a sticker and that if you had one, you had done the hardest work of all.  Having a sticker with a positive message from me on your hand, as these girls do, becomes a very sought-after status as well.



I think having experts is fine.  However like Dan (and Cathy, I’m sure) we have to think about how to have the kids who are experts be defined in multiple ways.  Tomorrow’s ‘experts’ on a card sort should be defined in a different way.  By constantly changing your definition of ‘expert’ we open up the title as one which can be earned by different kids on different days.