What is equity? The Danger of a Single Story.

As a Desmos Fellow, we’re asked weekly prompts as a way to get to know one another and dig more deeply into the mathematics we’re doing in our classrooms.  Last week when asked what our goals are for this coming year, I wrote:

“This year I hope to reflect a lot on teacher leadership and how to take what I have learned from 5 years of being an administrator into the classroom. I am deeply committed to equity and want to be better able to define what that means to me as a teacher leader.”

Yes, as I start my 20th year as a (white) urban, public school educator, my goal is to better define what equity means to me.  Over the past 5 years away from the classroom I have coached many teachers on related issues.  Now that I am returning to the classroom, this work becomes even more personal.   If asked what equity means to you as an educator, could you easily define it? I’m always surprised that it’s harder than I think.

Screen Shot 2018-07-26 at 8.36.20 AMMy working definition is that every child gets what they need every day.  I didn’t make this definition up myself, but really liked it when I heard it as part of district-wide coaching for equity work we did with Elena Aguilar’s team last year.  If pushed further I would talk about ending predictability in learning and student achievement by race, class, sexual orientation and other factors.  A more nuanced view of equity for me, also from Elena Aguilar is that equity involves what you see, where you look, who you listen to, and the self-awareness that you develop.

And yet,  I have a deep, deep fear that while we each could craft a public response to the question of what is equity, privately, inside of our minds, our definition is affected not only by our race but by the race of students who struggle at our schools.  This makes sense to some degree.  However, I want to admit some issues which give me pause and challenge us all to consider this notion of the power of a single story and how it might affect our teaching.

Chimamanda Adichie, Nigerian author of Americanah among many others, has an incredible Ted Talk entitled, The Danger of the Single Story.  If you have 20 minutes now, stop reading this blog immediately and go watch it.  If not, carve out time as soon as you can.

“Show a people as one thing, as only one thing, over and over again, and that is what they become… The single story creates stereotypes and the problem with stereotypes is not that they are untrue, but that they are incomplete. They make one story become the only story…The consequence of the single story is that it robs people of dignity.  It emphasizes how we are different instead of how we are similar.”

In my district, there is a huge achievement gap between white students and students of color, however the largest gap is between white and black students.  We talk about why and new programs and job positions are launched constantly to address these inequalities in outcomes.

As a white educator, if pressed, if really, really, really pressed, do I simply believe that equity in my district means getting black students to behave well?  To be on grade level?  I will admit that I have had moments when these thoughts creep into my mind.  I KNOW equity means a whole lot more to me, and yet, these thoughts have existed in my sub-conscious which illuminate the danger of the single story.

To repeat Adichie’s words, “The single story creates stereotypes and the problem with stereotypes is not that they are untrue, but that they are incomplete.”  

For example, when a colleague comes up to ask my advice on a struggling student whom I don’t know, do I immediately have an idea of the race of that student?  Developing the self-awareness, as Elena Aguilar discusses, to consciously NOT do this feels like a critical aspect of avoiding the danger of a single story.

The achievement gap between white and black students in my district is a deeply troubling issue with no one program, recommendation or ‘thing’ that will resolve it.  All of the work we do as a district should be rooted in understanding the complex web of issues at play and finding ways to measure the impact of the numerous programs we put into place to attempt to overcome this.  And yet, my fear is that in doing so, we have created a single story, that black kids struggle in our schools.  And while many do, I fear that it makes us look at black kids differently than white kids before we have ever taken the time to get to know them.  And if we look at them differently, we start to treat them differently because we believe we already know how their story in our schools will end.

I have no definite answers here and would love to explore this further with anyone who wants to.  But as I wrote in my goals to my Desmos Fellows colleagues, “…I am deeply committed to equity and want to be better able to define what that means to me as a teacher leader.”  I hope to be very intentional of what I see, where I look, and who I listen to so that I can continue to develop my own self-awareness and not allow a single story to define students.




TMC18 My Favorites: Making Relationships STICK

Blog revival!   On my honor, I will try, to prioritize my family, my work, my own professional development AND blog with more regularity.

My first TMC is in the books.  Twitter Math Camp, for those who are not yet a part of this fabulous community of math educators.  I hope to write a longer series of posts on how I learned and grew during these 5 days in Cleveland, but let me get this first post published before I start making grandiose plans for future posts.

One of my favorite parts of TMC was the time each day when we shared “My Favorites.”  So often, just hearing 5-10 minutes of a great idea was all the spark I needed to take an idea and run with it.  Mid-way through TMC I realized that I would really enjoy sharing one of my very favorite classroom management strategies…stickers.  It all started in 2012 when I read Sam Shah’s blog post on customized stickers.   WHAT A BRILLIANT IDEA!  I found an online coupon and ordered up stickers, notepads and postcards for positive notes home, stealing many of Sam’s phrases from his stickers.

After I gave my favorite no, many of you asked to have copies of the stickers I use and to hear more about how I use them.  You are welcome to steal any of the phrases, and better yet, make up your own and share them with me as I need to spice up my deck this year.

There are four themes to my stickers: Individual Positive Recognition, Positive Group recognition, Growth Mindset, and Redirection for off-task behavior.  Using these stickers, for me, is an equity strategy.  While there are many facets to my classroom management, a few key core values are represented through how I use these stickers.

  1. I work very, very hard to never say aloud, in front of the whole class, the name of a kid who is off task.  I do a whole lot to support those students to re-engage, and am quite strict with my expectations and consequences.  However I believe that as much as possible, we should say students’ names in front of the class only for positive things.  One aspect of the district-wide equity work we have done is understanding what it means to be a ‘warm demander’ as developed in Dr. Lisa Delpit’s book Multiplication is for White People.  This aspect of how I use stickers is one, of many ways I have interpreted this idea as I continue to understand this notion more deeply.
  2. I know that students need space when they are upset about grades or not doing well on an assignment from class.  I am always there to support and help them, yet recognize that teens are often not ready for this the very minute they receive the bad news.
  3. I try to stay very aware of status issues in my classroom and especially like to use the positive stickers for students who need that positive recognition and may not be getting it publicly or from their peers.  I most often use the positive stickers for students who have struggled and persevered or those who started class irritated at something and turned things around.
  4. As I mentioned when I did My Favorites at TMC, I always hand out these stickers by silently walking over to a student (or a group of students), sticking one on the outside of their hand, and walking away.  It may be followed by a squeeze of the shoulder or making eye contact as I walk away, and I may pull them aside later for a hug, or a private conversation.  However in the moment, whether I am beaming with pride or really frustrated at how off task they are, my demeanor is the same and I simply stick the sticker on their hand.
  5. THEY LOVE THEM!!  THEY COME AFTER CLASS AND BEG FOR MORE.  EVEN SENIORS IN HIGH SCHOOL.  I try to keep them on their toes, never knowing what other phrases I have stashed away.  So I use these sparingly, and NEVER when someone asks to get one.
  6. I made them on VistaPrint and it’s really easy to find coupons online.  If you make-em, share them with me including what you learn by using them.  Screenshots of them all are below.


group work

growth mindsetredirection

notepad and post card


A Tale of 2 Tweets

Earlier this week Eli Luberoff, Founder and CEO of Desmos, Tweeted to inquire how various Tech. Tools are used to support English Language Learners:Screen Shot 2017-12-18 at 8.54.02 PM

Although I’m not working directly with English Language Learners, I spent nearly the first decade of my career as a Spanish bilingual elementary school teacher and try to bring that lens to much of the professional development work I currently do.  While I had a lot I wanted to discuss around technology integration and supporting English Language Learners, I was at work when I read Eli’s post and threw out something quick:

Screen Shot 2017-12-18 at 8.55.23 PM

As soon as I hit Tweet, I regretted it.  Such a stock, somewhat thoughtless answer to someone who I knew was looking for more nuanced reflections.  Yes, both those tools are helpful, but successfully using any tool to scaffold and support the learning of English Language learners is far more complicated than this.


Fast forward a week: Monday, first day of winter break.  I take my 8-year-old son to my favorite homemade doughnut and coffee spot to hang out and play with one of my favorite weekend books: Anna Weltman’s  This is Not A Math Book.   (@annaweltman)

As we always do, I let him peruse the pages until he finds a set of imagery that appeals to him.  He chose this page and we spent the next while doodling.  We started on the same paper, taking turns building a drawing together, almost like a game of dots, but with doodles instead of closing boxes and gathering points.  He then grabbed his own paper, incorporating these mathematical sketches into his own artwork.


And then I had an ah-had moment.  We were doodling from Anna’s book, however THERE IS A DESMOS ACTIVITY which connects these doodles to formal, mathematical language:

Screen Shot 2017-12-18 at 9.35.27 PM

Heck.  There, in the center of Desmos’ introductory icon is the pinwheel my son had so gleefully turned into a flag in the hand of his creature.  It was a reminder of the power of informal before formal.  Language, reasoning, analysis…when first given opportunities for informal use, students are far more able to delve deeper into its formal use.  And while this pedagogical principle is a part of all good mathematical teaching, it’s especially important for English Language Learners and other students who may initially have difficulty accessing the more formal, vocabulary-rich language of mathematics.

So thank-you Anna for mathematical fodder during a lovely morning of doughnuts and coffee and for helping me move beyond my stock response about how Desmos can be a powerful tool for supporting the learning of English Language Learners.

Technology PD for our Subs

Last year around this time I blogged about using Robert Kaplinsky’s Sheep and Dogs in the flock problem as the opener for my session for our district’s substitute teachers on transitions to CCSS math.

District-wide our substitute teachers are invited to a full day of professional development each August.  I absolutely love leading a session as I believe being a sub is among the hardest and least appreciated work within a school district.  I love being able to personally thank them for the work they do as so often they never meet the teachers for whom they work.  Each year I learn of several subs in the group who have subbed for me in the past or who have subbed for my son’s elementary class and it’s always a really fun discovery.

This year I was asked to focus my session on ways teachers are using technology in the classroom and how subs can improve their classroom management when sub plans call for students to be using their Chromebooks.  I co-led the session with my fabulous TSA for Instructional Technology, Mia Gittlen, and we used Pear Deck as the basis for our presentation.  We all had so much fun!


If you aren’t familiar with Pear Deck, it’s an interactive software which allows participants to respond to questions posed during a presentation and their collective thinking is displayed.

Screen Shot 2017-08-26 at 10.01.18 PM.pngAfter some fun warm up questions, we asked them what their level of confidence is with technology.  These responses are from our first group of  teachers. Interestingly, our second group’s level of confidence was far more scared and confused.

screen-shot-2017-08-26-at-10-05-45-pm.pngAmong both groups, when asked how often sub plans expect them to use technology, the vast majority said sometimes and very few said never.  To me, this was the most surprising slide.  As the Instructional Technology Coordinator, it also made me quite happy that our classroom teachers have strong enough systems in place for students’ using Chromebooks that they frequently trust its use when they are out.

I love that students are doing research when they have a sub:


However ultimately, the biggest crowd pleaser was teaching tools one can use for classroom management.  The substitute teachers loved learning about the button that allows them to see all open windows on a students’ Chromebook.  Students occasionally use a keystroke to make their screen appear sideways and use it as an excuse for not being able to do work or to have to share a Chromebook with a friend.  Knowing how to undo this felt really powerful for the subs, especially those who were initially fearful of using technology when subbing.  I loved the feeling in the room as many who were VERY new to technology giggled with one another, trying out their new tools and tricks.

And having a district subscription to Pear Deck which long term subs can use was super exciting for the more techy ones of the bunch.  Pear Deck was obviously completely new to them all and they absolutely loved it and wanted more training on how to use it.  Such a successful session!Screen Shot 2017-08-26 at 10.12.29 PM

Finally, I saw this posted on Twitter this evening...about a Louisiana Senator who substitute teaches during his time off.  It’s worth bearing through the initial advertisement to watch this segment.  Wish more people felt the way he does about the work of teachers…

First Day Plans ->GOALS

What are the first day plans of this second-year administrator?  I am a big fan of writing down what I want to accomplish for the following day before leaving work each day.  So, on my last day of work in June, I wrote down what I needed to get done on my first day back.
Screen Shot 2017-08-06 at 9.55.05 PM

EVERYTHING on this list is an administrative/bureaucratic task.  And while this is actually not at all reflective of how I spend my time in a typical day, it’s a reminder of how easy it could be to fill my days with similar tasks.

Today, however, was my first day of work and I accomplished none of those tasks I placed on my calendar back in June.  Instead, I was at a Management Team retreat with all the administrators and managers in my district.  We reflected on and discussed our leadership styles in addition to district goals and our personal goals.  The highlight, however was this 3 minute video which I urge you to watch before beginning your own school year.

Which brings me to my goals.  I am in my second year of my role as Instructional Technology Coordinator for grades K-8 and my fifth year outside of the classroom.

  • Spend time every day either in a math classroom or doing math.  Working through all of the Desmos Activities which relate to middle school math is something which has helped me deepen my understanding of the connections between and progressions among concepts in these 3 grade levels.  The eight 8th grade math teachers at all 3 of our middle schools have all agreed to teach at least one Desmos Activity per week and I’d like to spend as much time as possible supporting them and teaching/modeling or co-teaching some of those lessons, especially for our new teachers.  Specifically, I want to spend enough time in all 8th grade classrooms to have qualitative and quantitative data around how student engagement and teachers’ ability to differentiate is affected by sustained, regular use of Desmos Activities which are aligned to our 8th grade curriculum.
  • Rethink and rework how I lead our department meetings.  In my district, Instructional Technology and Library and Media Services collaborate very closely as increasingly many of our Librarians and Library Media Specialists integrate teaching digital citizenship and online research ideas into their work with kids. Additionally, we have a strong integration of audio books into our readers’ workshop which is supported by our TSA Teacher-Librarians.  Often, our department meetings are spent reporting out on the work we’re each doing in classrooms and divvying up tasks which need to be completed for upcoming PDs we are leading.  Specifically, I want to begin the year by posting the names of all 3rd-8th grade teachers in our district (as those are the ones who have Chromebooks) and having us put a sticker next to each teacher we’ve provided classroom support for around technology integration.  I modified this idea from  Sara VanDerWerf’s  most recent post about how a school staff did this with their entire student body.
  • Collaboratively rework how my department provides classroom support.  Last year, the 5 of us in my department made nearly 600 classroom visits for observations with debriefs and/or teaching model lessons.  Spending such a large percentage of time in classrooms is rare for an administrator, but after watching the above 7-10 split video, I believe we can have a greater impact on instruction if we focus our efforts.  I’d like each of us to specialize in technology integration around specific student needs (of the 7-10 split): Special Ed. students, English Language Learners, Newcomers, Students needing additional challenges beyond the day-to-day curriculum, etc. etc.  In doing so, we can provide more targeted classroom support for technology integration.
  • Better support teachers to use student achievement data generated by software to differentiate in more powerful ways.  Using tech provides so much data. A single Desmos lesson collects tons of student responses.  Most teachers use a lot of tech tools, but rarely carve out time to use the data generated by those tools.  I want my team to learn to look at and use this data more efficiently to better support teachers to do this during our PD time with them.

I’m curious how others see their 7-10 split and how we can each adjust our teaching to better meet the needs of all of our students.  In the meantime, my 8-year-old LOVES bowling and we’ve learned that nation-wide, kids bowl free, once a week from April through August.  You’ve got 2 more weeks…For real!

Screen Shot 2017-08-11 at 9.55.21 PM

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…

Screen Shot 2017-04-15 at 3.08.09 PM

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:

Screen Shot 2017-04-15 at 3.22.44 PM

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.

Screen Shot 2017-04-10 at 8.04.44 PM

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.

Screen Shot 2017-04-10 at 9.23.02 PM

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.