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.



Math PD for Substitute Teachers


I was asked to lead a math session today for district-wide subs to help them better understand the math they’d find in CCSS classrooms.  Thinking about how to craft a PD on mathematics for K-12 substitute teachers who work in any classroom from kindergarten to high school PE and everything in between was a challenge.  Like many of you, when faced with a teaching topic on which I am stumped, I head to my friends in the blogosphere

Recently, a UC Berkeley undergrad in my math pedagogy course asked if it would be considered plagiarism to use a lesson his girlfriend had taught in his current teaching placement.  My reply was that if it was a great lesson that was appropriate for his students, it wouldn’t be plagiarism at all, and was actually what all great teachers do.

So thank you, Robert Kaplinsky, as  I very much stole your work today, though I gave you credit all along the way.  I am indebted to you for inspiring my session today which somehow left a room full of subs, with VERY diverse experiences, quite excited to teach math this  year.

Between  being a teacher in my district for a long while and now having been a parent here for a long while, I was surprised to realize how many of the subs in the room I knew.  Several of them had subbed for me and many more had subbed for my colleagues and even my son.  That made this a lot more fun for us all.

They started by predicting how 8th graders would respond to the question of:

There are 125 Sheep and 5 Dogs in a Flock.  How Old is the Shepherd?

They then watched this video which is further discussed in this blog post.

Man did I have a captive audience.  My favorite comment was from a woman who admitted that she hates math and would never sub in a math class. She said that had she been asked this question, she would do what many of the kids in the video do, write down the numbers and DO something with them.  She would assume that though the question made no sense, that it was she who didn’t understand and she doesn’t trust her math abilities.  She went on to say, however, that had this question been posed in an English class, she would have been the first to question the teacher saying that it made absolutely no sense.  She challenged the group to pose this question to both a group of middle school math students and again in a middle school English class.  She was confident that outside of a math classroom, more students would speak up about how the task makes no sense.  I found that hypothesis to be brilliant (while also worrisome that she’s right).

Following this conversation we did Robert’s lesson on questioning strategies which is explained in his blog post.  In short, teachers were in groups of 3 where one role played a teacher, one a student and one was an observer who wrote down all the questions asked by the teacher.  The teacher was given slip of paper with a math problem and knew the solution that the student had gotten.  The student was given a slip of paper with the math problem and a specific misconception.  The goal of the teacher was to ask questions to determine the misconception of the student.

For example:

Student: You are working on ordering decimals from least to greatest. The problem you are currently working on is ordering the decimals 0.52, 0.714, and 0.3. You correctly place them in the order 0.3, 0.52, 0.714.  However, the reason you put them in this order is because you look at the number after the decimal like a whole number (3, 52, 714) and do not understand the significance of place value. You are confident you are correct and don’t realize that you only accidentally got the correct answer.

Teacher: Your student is working on ordering decimals 0.52, 0.714, and 0.3 from least to greatest. Determine what understanding the student has by asking questions, especially questions that encourage elaborate responses.

I LOVED this activity and the rich conversations it generated. We easily spent 45 min. role playing 3 scenarios and debriefing each one and these teachers had so much to say.  We talked a lot about how developing questioning strategies could help a student unearth misconceptions even when the math the student was working on was too complicated for the sub.

However I realized that for this group of teachers, there were so many who feared math, that they couldn’t focus on the misconceptions and instead, for some, could only focus on explaining to each other HOW TO DO the problem.  In hindsight, I think I should have modeled the conversation one could have.  I could have role played the student and had the whole room of subs role play the teacher asking questions to me.  Only I would know the misconception and the group would ask questions to determine where I was confused.  I think had we done that for one scenario, these teachers would have been able to let go of their own fears of making math mistakes a bit easier.

What a neat afternoon for me and for them!



Next Summer…

Next summer, you say.  But why?  I haven’t even begin this new school year yet. I just cleaned the lens of my document camera and tested out last year’s white board markers to see which were still keepers.  And I did such fabulous PD this past summer that I’m itching to incorporate into this year’s curriculum and pedagogy.

And gee whiz, my husband and I bought a condo in Kauai this summer so why the heck would I do ANYTHING but go here next summer?

Screen Shot 2016-08-12 at 10.39.30 PM

Darryl Yong just published a blog post on how he and Bowen Kerins create the morning math sessions at the Park City Math Institute.  Read it now, before you read any more of my post.  If you haven’t attended PCMI, you should, and Darryl’s blog post leads you to more information about it and how to apply.

Today was my first day of work at my new job, Instructional Technology Coordinator for the Berkeley Unified School District.  I’m a teacher. I’ll always be a teacher.  So explaining or even stating what my new job is seems to only happen with some contortions as I try to explain that I’m a math teacher, but I guess I’m not a math teacher anymore, but I really am a math teacher.  This summer as I have met new people while on various travels, I’ve been asked what I do.  I can’t seem to say the title of my new job without a paragraph intro about how I was a middle school math teacher, then had a second child, then wanted more flexibility and became a TSA and then and then…and now I am…blah blah blah.

However, while a lot of experiences have shaped who I am as a teacher and now as an administrator, over the past 17 years, nothing has had as lasting impact as the summer of 2011 when my husband and I packed my then 13 year-old-step daughter, 2 year-old son, Bernese Mountain dog, two mountain bikes and who knows what else into our Subaru Outback, stopped at every McDonalds Playland between Lovelock and Winnemucca, and landed in Park City Utah.  Following my participation in PCMI’s 3 week course, I started a blog (yep, you’re reading it now), got connected to a bunch of regular math bloggers from whom I have bounced countless ideas off of, and found myself immersed in the online support world of math teachers.  Darryl’s post suggests that PCMI’s curriculum limits the amount of technology which is used.  While this is true, without a doubt, I wouldn’t have the job I have today (Instructional Technology Coordinator operating in the body of a middle school math teacher) were it not for Darryl and Bowen introducing me to Geometer’s Sketchpad and how dynamic mathematics software can allow students the opportunity to play around with visual representations of big mathematical ideas and  explore open-ended problems which elicit incredibly rich discussions.  My use of Sketchpad morphed into GeoGebra which more recently has morphed into Desmos, however the purpose of these tools remain the same.

So Thank You, Darryl, for your post and for reminding me of how I got here.  And speaking of gratitudes, Thank You Math For America, Berkeley, for urging me to go and paying my way.


Pokemon for Breakfast on the First Day of School

I’m tempted to post this without comment.  But comment I shall.

Since I’m not in the classroom, I can’t show this on the first day of school.  But if I were still teaching 8th grade math, I think I would.  Then I’d have them turn and talk, introducing themselves to a partner and discussing whatever the video made them think about.  No idea what this  video refers to?  Your partner will.  Or you’ll both lean over and ask the pair alongside you.

Somewhere along the way on that first week we’d do an activity with Desmos.  And when Friday rolled around, I’d have them do a quick write and pair-share to compare and contrast playing Pokemon Go to sharing a laptop with a partner when doing math as part of a Desmos Activity.  But sadly, I don’t have my own classroom filled with eager 13-year olds to build relationships with or affective filters to lower.

Since I now lead professional development more or less full time, I will instead show this video as part of my opening talk to the 100+ Berkeley K-12 teachers attending my department’s 3-day Instructional technology Institute.  Spoiler alert if you are a Berkeley teacher reading this…

And I do hope the Desmos folks forgive me for mentioning Pokemon Go and Desmos in the same sentence.

OH NO…video was taken down today 8/5 for copyright reasons.  So sad.