Things That Make Me Happy Right Now

These are some of the things that make me happy right now.   They are in no particular order.

  1. On Tuesday, I watched a fifteen year old young woman sit patiently,  encourage,  and guide an eleven year old girl as she debugged an issue with the code for her robot.   She stuck with this girl for a good thirty minutes until the problem was resolved.  It was such a generous thing to do.
  2. On Wednesday,  I watched as eleven year olds huddled together helping each other try to figure out how to do new things as they were learning to code.   I loved that these kids would jump in and offer help to each other.   Watching them together was such a lovely sight.
  3. A colleague brought his new puppy for a visit during student-led conferences on Friday.   Playing with this little Bernadoodle for 20 minutes filled my heart with joy.  Time stopped and there was magic in those few minutes.
  4. During one of my student-led conferences, a toddler tagged along.    He discovered the wonder of a funny stuffed elephant in my room that plays music.   Every time the music stopped, he piped up “again”.   Watching his wonder and joy as he played made me happy.   Having the opportunity to see wonder and joy in a child’s eyes is such a gift.
  5. It is the fall and that means that morning skies are filled with hot air balloons.  Looking up into the deep blue sky that is only present in this part of the country and seeing the hot air balloons is one of my favorite things about fall.
  6.  I got a phone call from my daughter  who had recently returned from doing pro bono work as a physical therapist in Guatemala.   As I listened to her talk about the work she did with a 24 year old man whose spine was injured as a result of gun violence, I could hear the impact she had made in his life and the lives of the other patients that she treated.   It makes me happy that I have children who are working to make the world a little better.
  7. One of my student aides decided that since I bring baked goods for them from time to time, he would bring something for me.   It became a bit of a joke that there must be some sort of curse.   The first time he tried to bake for me, the oven broke and his parents had to replace it.   The second time he tried to bake for me, his fish tank broke and he had to get a new one.   I told him he needed to stop trying to bake for me because it was putting a pretty heavy financial burden on his parents.   He didn’t listen to me.   On Monday, he brought me a package of the best chocolate chip cookies that I have ever eaten.  He is an amazing cook and the fact that he wanted to share that with me makes me happy.
  8. At the beginning of the year, I asked my students to write one thing that makes them smile.   It has been such a gift to be able to glimpse into their lives.   One of them wrote that it makes him happy to help the homeless.   When I first read it, I wondered about it because it is not what one normally sees from an eleven year old boy.   As the year has progressed, I have seen the truth in his statement every single day.   He is unfailingly generous to everyone that he encounters and he is the happiest kid in the room.   It makes me happy to see his truth.

The Distributive Property and Rational Numbers – A Desmos Lesson

One good thing.   That has been my professional mantra for the last five or six years.   I try to find one good, research-based thing that I want to bring into my instructional practice in a given year.   That doesn’t mean it is the only thing that I change, but I try to really focus on making one piece of my practice better.   This year, I decided that bringing Desmos into my classroom would be my “One good thing.”

Over the last month, I have used Desmos five times.

  1. The first time, I used a lesson that I created matching stories to graphs.   It was a good lesson, but I had no idea what I was doing as I facilitated it.   The students loved it and learned despite my unease.
  2. The second time, I used a Desmos-curated lesson on graphing inequalities on the number line.   It went much more smoothly.  A little experience went a long way.
  3. The third time, I again opted for a Desmos-curated lesson.   This was the Battle Boats lesson in which students plot points in four quadrants and essentially play Battle Ship.   It was a huge success.   At this point students were walking into class and excitedly asking if we were doing Desmos.
  4. The fourth attempt was a lesson that I created based on Cathy Yenca’s (@mathycathy) Twin Puzzles Desmos lesson.   She had a great lesson on Order of Operations using twin puzzles.   I wanted to incorporate negative values into my puzzles, though, and she had only positive values.   So, I built a copycat of her activity and used expressions with negative values.   It was great, but I had a hard time figuring out how to input the puzzle into the activity builder.  I had not yet figured out that there was a copy function in Desmos.   I ended up creating the puzzles in Word, printing them out, and then taking photos that I inserted.   This only worked so well.   I had the puzzles in a stack when I took the photos and you could see shadows of the other puzzles underneath.   By the time I figured it out, there was no time to redo the photos so we lived with it.   Once again, the students were engaged and learning despite my imperfections.
  5. After the success of attempt number four, I went home and completely reworked my lesson for the next day to be a Desmos activity.   The lesson was focusing on the Distributive Property.   My students had already built a fairly solid understanding of it during a previous unit.   This lesson was going to build on that prior knowledge/experience and extend their understanding to include negative values.

There is a quick summary of the Distributive Property activity below and a link to the lesson if anyone would like to look at it or use it.   I included some teacher notes and notes on anticipated error points in the slides.

The lesson started with a card sort comparing visual representations and algebraic expressions.   All the values are positive at this point.   (I still need to find a better way to add diagrams into Desmos).   After students completed the card sort, we debriefed it as a class.


Next, students  explored the Distributive Property and subtraction as they rewrote expressions in expanded form.   I had students work independently and then discuss it with their table groups.   Only after table group discussions did I direct students to hit the “Share with the Class” button.   At that point, we had a whole class discussion.   During the discussion, I asked students what they noticed about the second and third elements in the table.   This was how I drew out the discussion of whether the Distributive Property works with subtraction.   After the discussion, students  explored the Distributive Property and negative values as they rewrote expression in expanded form.   Once again, I had students work independently and discuss with their table groups before sharing their responses with the class.   I focused the discussion on elements two and four in the table.

Next, students rewrote expressions in expanded form into expressions in factored form.  Some students had trouble remembering what “factored form” meant, so we went back to the first slide in the activity and reviewed which expressions were in factored form and which were in expanded form.   Once again, students worked independently and in table groups before sharing responses with the class for a whole group discussion.


Then, they applied the Distributive Property on a couple of word problems.

Finally, they wrapped up with a card sort that was a formative assessment.


Incorporating Desmos into my classroom is still a work in progress.   There are still things for me to learn and there are still things that I can do better.   That said, I’ve made a few interesting observations along this short journey.

  • Kids walk into the room and see the chromebooks and excitedly ask if we are doing Desmos.   They love it.
  • Kids who don’t see themselves as  being “as good as everyone else” at math suddenly are deep in the thick of things and are getting ideas as quickly or more quickly than others.   Math is not about being fast, but it is interesting to see how a student is suddenly grasping new ideas more quickly than he or she usually does.
  • There are lots of opportunities to introduce cognitive dissonance and to suddenly hear outbursts  of “Oh!” followed by furious typing as a student suddenly makes a big connection.
  • Students who are absent for all or part of a class can go  home and do the Desmos activity.   The days that I did the Twin Puzzles and the day that I did the Distributive Property activities, some of my students were pulled out of class to have a skype session with a scientist in Antarctica .   It’s not the sort of thing that I want them to miss so it was nice to be able to just give them the class code and tell them to do the lesson at home.


A Desmos Lesson That Felt Like a “Fail”

Sometimes, despite diligent planning and preparation, a lesson does not go quite as planned.   That was definitely true of my first Desmos lesson.   In fact, at least in my eyes, it was a bit of fail.

After attending a session on Desmos at the NCTM annual conference last April, I could see the power and promise of the platform to help students visualize math.   After returning home from San Diego, I looked at a bunch of lessons on Desmos and found several that I wanted to try with my students.   I also found one that was fairly similar to a lesson that I already teach that matches the story to the graph.   I decided that this particular lesson would be a good place for me to start.   (The version of the lesson that I usually teach is described in this blog post .)

As I prepared the lesson, I invested hours figuring out how to build piece-wise graphs on Desmos and how to build a card sort.    Once I had the lesson built, I created a class code on Desmos and  had one of my student aides enroll in it and do the lesson to test it out.    Based on his feedback, I made a few tweaks to improve the lesson.   Going into the lesson, I felt like I had done everything that I could to prepare but still felt the usual nerves when I try out a new platform for the first time.   When it comes to teaching a lesson, I like to have anticipated every possible eventuality and have thought about what I want to do in response.    That can be hard to do when I’m learning a new platform.

As I taught the lesson over the course of the day, it got a little smoother with each iteration.   I made a few changes after each period and it was fairly smooth by the end of the day.     My struggles were not with the lesson or pacing, but with how to facilitate a lesson on the Desmos platform.    All of my planning didn’t prepare me for how students screens would look when I used Pause or for the fact that when I displayed my screen on the Promethean board every student in the room could see what every other student was typing (even before they hit Share With The Class).   Hence, I did not anticipate the fact that the impulsive boys in my second period class would see it as a great opportunity to type silly messages and pull other kids off task.   After a few trials and missteps, I figured out that I needed to make kids Pause, turn to the screen to introduce the task/question, freeze the screen on the Promethean Board before kids started, and tell kids not to hit Share With The Class until I gave the direction to do so in order to ensure that each student had the opportunity to think about the task/question without being influenced by someone else’s response.

I walked away from the lesson feeling that most of my cognitive energy/focus was on how to manage facilitating the technology rather than on facilitating the discussion the way that I normally would have done.    As I looked at student work during the lesson and the ending exit ticket, I could see that students had grasped the necessary concepts.  When I graded the unit test that was the day after the lesson, the class average was 97% with the lowest score above an 80%.   So, objectively speaking, it was a success.   It definitely felt like a fail though.

So, where do I go from here? Despite the bumpy road, my students said they liked using Desmos and I still believe their is power in it.   So, I will live outside of my comfort zone.   I have two lessons planned for this week using lessons created by Desmos to teach about plotting points on the coordinate plane and graphing inequalites .   I also spent several hours this morning creating another lesson of my own on Desmos modeled after @mathycathy ‘s Twin Puzzles Desmos activity to explore order of operations with rational numbers.   (Her activity was great, but I wanted to incorporate some rational numbers rather than just using whole numbers).

Every day, I tell my students that  you get better at things when you work at them.   Hence, I am going to work at getting better at facilitating Desmos lessons by doing some more of them.


International Day of the Girl 2019


16839b616fc57ba94a01508efd925f96Today is International Day of the Girl.   It is intended to focus on the needs and challenges that girls face.   In some places around the globe, those challenges are central to their very survival.   In other places, the challenges are not quite as large and all-encompassing but they are never-the-less very real.

Today, I’ve been thinking a lot about what girls need in a math classroom.  My list derives from the lessons that my girls have taught me.

Girls need each other.   There is a lot of research about the effects of stereotype threat and how it plays out in terms of performance (I highly recommend Claude Steele’s book, Whistling Vivaldi).   Every year, I see the truth of it in my classes, no matter how hard I work to mitigate it.

Somehow, every year the magic of the master schedule hands me one class that is 75% female and several classes where the class is 70% male.   Every year, there is magic in the air of that mostly girl class that produces tremendous growth in each and every one of the students (including the boys).    I think that the girls feel safer taking academic risks and growth comes with that venturing forth.   I also think that the class becomes an incredibly collaborative place during that hour.

While I can’t give each of my girls the gift of a classroom dominated by girls, I try to give them as much of that magic as I can.     I could choose to mix up my table groups and use the girls to help manage classroom behavior, but I don’t.   I choose instead to make my table-groupings single-gender groupings.   A few years ago, I started asking the girls in the boy-dominant classes if this was something that they would prefer.   Every single time, the girls have chosen to stick together.

Girls (all kids, really, but especially girls) need the chance to make sense of math.   It is not enough for most of them to learn a set of procedures or sequential steps.   They need to see how ideas fit together and why things work.  If they are given the chance to explore ideas and to look for connections, they discover that they are good at math.   Too many of them (even though they are Gifted) have to learn the oh,so important lesson that everyone can be good at math.

Girls need time to do math.   When I take away the pressure to work quickly, girls are free to think more deeply and they perform better.   They thrive when they get the message that “good at math” does not equate to speed, especially when I back that message  up with instructional choices (e.g., letting them stay after class to finish a test or come in during lunch to get an early start so that they don’t feel so much pressure during the test).

Girls need to know that I believe in them.   Girls need to be asked the hard questions, not just the easy ones.   They need to know that I know they can answer them and will stick with them until they do.   They need to know that I know just how capable they are.

Girls need to know that math makes a difference.   Seeing how math makes a difference in the world, how it can make the world a better place, makes math more meaningful for girls.

Girls need to know that it is OK to make mistakes.  Girls have often received the message that they have to be perfect.     They need to know that perfection isn’t all it is cracked up to be.   They need to know that making mistakes and then figuring out where the miss-step or misunderstanding is can be incredibly powerful.   They need to learn that mistakes can be fixed.   They need to learn to be brave, sometimes even fearless.

Today, on International Day of the Girl, and everyday, here’s to all the  girls that are learning to be brave and bold and strong and discovering that they can indeed do anything.

Who do you think you are?

Middle school is a time when kids start to ask some fundamental questions about themselves.   They grapple with big ideas.   “Who am I really?,”   is not something they utter aloud but it is something that creeps into their world.   They grapple with this question as they navigate changes in friend groups and explore new extracurricular activities.   They ponder it as they walk into classrooms and dive into curriculum.   It rattles around in the recesses of their mind as they walk the tightrope between childhood and adolescence, figuring out the push and pull of their relationship with their parents, trying to grow up but not yet quite ready to do so.

As I get to know students each year, I grapple with the same question about each of my students.    “Who do you think you are and who are you really?”    To begin to find the answer to that question, I give my students a survey during the first week of school.   I ask them a number of questions, but there are two that I always find most telling.   I ask them whether they like math and give them a continuum to select (make an x on a line scoring themselves from 1 – I hate math to 10  – I love math).   I also ask them whether they think they are good at math and give them a continuum to select (make an x on a line scoring themselves from 1 – I am terrible at math to 10 – I am amazing at math).

This year, the responses from two of my students on these two questions stood out.   Both of the students were girls.

The first student indicated that she sees herself as good at math but that she hates it.  This struck me because most 6th grade students equate “liking” math with “being good at math” or “hating” math with “not being good at math”.    I don’t agree with the coupling of these two things, but I understand why most entering 6th graders tie them together.   I find myself wondering why this particular young woman feels this way and wondering how it came about.    I wonder if she doesn’t like it because someone made it tedious and boring rather letting her discover its richness.   I wonder if she was the only girl in group of boys who made her feel uncomfortable.   I wonder whether her experiences as a student of color has impacted her feelings.   I wonder what she likes and how I can bring that into our class.

The second student indicated that she sees herself as not very good at math (this young woman has been identified as being “gifted” in math)  and as not liking it.    I wonder what experiences brought her to this conclusion.    I wonder if she thinks that being good at something means it should be easy or if something happened to shake her belief in herself.   I wonder if she has somehow gotten the idea that fast means good.   I wonder if she knows how deeply she is thinking when I ask her why and she explains her thinking.

As I consider who these two girls think that they are, I wonder if I will be successful in helping them find a slightly different answer to that question.   I certainly intend to try.

What Mathematicians Do…..

During the last five minutes of class today, I asked my students to write.   I asked them to write about what mathematicians do.   Here are a few of the things that I saw when I was peeking over shoulders.

Make mistakes but stick with a problem until they get it. 

Notice patterns.

Explain their thinking.

Figure out new ways to solve a problem. 

Find creative ways to solve a problem. 

Talk to each other. 

Help each other.  



Not a bad start.   Not bad at all.


Dabbling with Desmos

Desmos has been something that I have wanted to explore for several years.   I have had good intentions to put aside some time and play with it.   That time has never materialized, though, for a number of reasons.  When I looked over the schedule for NCTM’s Annual Conference last spring and saw Ivan Chang’s (@drivancheng) session How to Desmoify Your Math Lesson to Promote a Growth Mindset, I made sure to build it into my schedule.

The session was really well done.   The presenters built in time to play within the software and gave great guidance on building  a lesson for Desmos.   They talked through the idea of starting the plan for the lesson on paper, making it dynamic, and doing multiple iterations.   They talked through good pedagogy (building the lesson to start with discovery and sense-making, moving on to solidify understanding, and then formalizing the concept).   They demonstrated how student work can be projected in real time and how it can be anonymized to protect student privacy and ensure that there is academic safety.

When I returned home, I decided that I want to begin by using some Desmos lessons that have already been curated.   I picked three lessons that pertain to CCSS standards relevant to my course from their “favorites” list to try.

  1.   Battle Boats – This is a lesson on graphing on the coordinate plane.   Usually, I have students play Battleship when I teach this concept, so this is an extension of that idea that uses Desmos.   I think it will be an easy entry point for our first foray into Desmos.
  2. Inequalities on the Number Line – This is a great lesson that enables students to construct an understanding of how/why the graph of an inequality comes about as they plot points on the number line that fulfill the inequality and then see how the graphs change when all the points their classmates also plotted are added to the number line.  I really like the way that it builds the idea of a ray as the solution set and the way that it builds the idea of a closed or open circle.
  3. Graphing Stories – This is a nice lesson matching graphs to stories.   It is similar in concept to a lesson that I already teach.   After looking at this lesson, I decided to use this as a model of sorts for creating my own first Desmos lesson.     I like the idea that if it crashes and burns, there is a back up plan for second period.   Thus far, I have built the introduction to the lesson and a card sort activity.   I still need to build a closing formative assessment.

In building my first Desmos lesson, I learned how to build graphs with complicated step functions using their software in order to get the graphs that I wanted.  I also learned how to build a card sort in Desmos.   Given that, even if I decide the lesson isn’t as good as the existing lesson in the Desmos “favorites”  and that it would be better to use their lesson rather than my own, I feel like the time was well-spent.