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.


Gravity Maze – Learning to Work Backwards

This year, I am attempting to teach my students problem solving strategies in a slightly different way than I have in the past.   They will be playing with games and puzzles that uncover specific problem-solving strategies.   As we debrief the game play, my hope is that we will be able to de-privatize their thinking and everyone will gain experience with these different problem-solving strategies.

Gravity Maze is a game by ThinkFun that challenges players to construct a three-dimensional maze utilizing a specific set of parts.   The goal is to put a marble in the  “start” position and have it run through the maze in such a way that it ends in the “end” position.    For each challenge, there is a specified start point and end point on the board.  Players then must determine how to construct the three-dimensional maze going from start to finish using the specified parts for that particular challenge.   When they think they have accomplished the task, they test the maze by depositing a marble in the “start” piece and seeing if it ends up on the “end” piece.

As is true of many problem-solving challenges, there is more than one way to complete the task.   However, “working backwards” from the end back towards the start is pretty effective for many of the challenges.   As the class debriefs play with this puzzle, I am hoping that the discussion will bring out the “work backwards” problem solving strategy (among the other strategies that students may also use).

Scars Left Behind


Curiosity and excitement fill their young faces.   There is a growing look of confidence and increasing sense of calm.   There are also a few traces of fatigue.  ( Starting middle school can be exhausting.)   On some faces, I also see the traces of scars left behind from days and years  past.

Some of the scars, they have begun to share.   Some have written that they don’t think they are good at math even though they clearly are if I can see it after only three days together.   Some have written that they think they are good at math but that they hate it.   I find myself awake in the small hours wondering what must have happened to make them feel this way.    Others have written bits of the stories that have left scars that won’t soon heal.

Some of the scars, they have kept hidden probably out of self-preservation.      Some of those scars are from a time so long ago that they may not even remember and some are so fresh that I know they must still be raw.   They don’t know that the events that left those scars in their lives also touched me.   I have known their families on some level for a very long time.   I have seen them as preschoolers and elementary children trailing along behind as their parents came for IEPs or conferences.   I am sure they don’t remember me from those early days, but I remember them.   I remember them “before” and I am only beginning to get to know them “after”.   They don’t know that while my scars from those events don’t cut as deep, the events that caused those scars will never leave me.

As this new year dawns,   I am left with an overwhelming sense that this year is important.  Every year is important, but this one feels especially so.    I feel the weight of the responsibility before me.

A New First Day for A New Year

Today, we went on a quest.    It seemed like a good way to start our year together.   I hope my students will see this whole year as a quest, filled with puzzles and challenges, problems to solve, twists and turns and surprises, a grand adventure.   So, we began as I hope we will continue.

Our quest began with a puzzle to solve.    Each table group of four was a given a set of five envelops (one for each person and one to be shared by the table).   The challenge was to use the pieces in the envelops to collaboratively build five equal-sized squares.   The catch is that no single envelop contained the pieces to build a square.   They must work together to achieve the goal.   There were a couple of strings attached to the puzzle.  First, a team member can’t take a piece from another person.   They can, however, offer a piece to someone.   Second, each team member has to play a specific role.   The role is determined by the card on their desk.   There are four different roles:   Look Ma, No Hands (you can’t use your hands during the challenge);  Speak No Evil (you can’t talk during the challenge) ;  See No Evil (you must do the challenge while blindfolded); Mean Girls (everything you say must be mean).

(You can read more about this non-verbal problem-solving task here.  )  When a team successfully completed this challenge, they received this clue.


They had to decipher the clue in order to find their second task, which was to discover the ground rules for our class.   IMG_0409

Here, they found the task tucked behind the poster.


Once they completed the task and deciphered this clue , they arrived at the next challenge.


After solving the Tower of Hanoi and receiving the clue to next station,  students arrived at the next station.  After finding the clue and completing the task (collecting the parent letter and survey),  they then had to find the next clue.


This directed them to their final destination for today’s class.


Students learned some of the lessons of how our class will work and discovered some of the important places in our classroom.   They did the exploring, the thinking and the talking.   I was just there to give them a little coaching when they got stuck.     It was a good day and a good beginning on our much larger quest.




“I’m not smart enough”

“I’m not smart enough for this school.”   Those words lurched out in a whisper between halting gasps, tears trickling down his cheeks as he ducked his head down so that his hat would hide his face in the hopes that his peers would not notice.   Twenty minutes had passed since the assessment had started. I glanced down at his assessment and saw that not a single question had been answered.   There was not a single mark on his paper.

I had been out in the school lobby, talking to parents.   When I walked into the room where students were taking the assessment, another teacher came up to me and told me that there was a boy who was about to cry.   After watching him for a few minutes, I went over to talk to him.   That was when I heard those heartbreaking words.    I reassured him and tried to calm him.   I proffered tissues and plied him with a drink of water.   I whispered words of reassurance that he was definitely smart enough for this school and reminded him to breath.   I reassured him that he should just do his best and that his best was enough.    When I walked away, he had started working through a problem.

I don’t know this child or his story, but I am so saddened by our brief exchange.   School won’t start for this eleven year old boy for almost a week.   He  decided he isn’t good enough before he has even begun.   No child should ever feel this way.   I don’t know this child or his story or where he will go next.   He will not be in my class, so I will have limited opportunities to impact his outcomes.   I am going to choose to see today, though, as one good thing.   Today, I got to tell an eleven year old boy that he is good enough and he believed enough of what I said to at least start, to at least begin.   Tomorrow, I will talk to the teacher that will share this year with him so that she will know to encourage him and help him to discover all the things that he can do.

El Paso

The images filling the news are images from my childhood and adolescence, but they have been bloodied by prejudice and by hate.   At least twenty dead.   Twenty-six more wounded.  A young man’s head and heart have been filled with hate and he poured it out onto a sea of families who were preparing for the promise of a new school year.   Too many  did not live to see that promise.   Those that did survive will carry the trauma with them like a weight upon their backs and in their hearts.

The children of the city will return to school in a short while, but they will be forever changed by the hate that was planted and fed and permitted to grow.    Their teachers will welcome them to a new class, a new school year.   They will provide them with normalcy and routine.   They will do their best to fill their world with a sense of safety and security and love and they will do it while they themselves are reeling from the hate that they have seen, the hate that has traveled the width of a state and the width of a country to break hearts in ways that can not be mended.

It cannot be denied.  Prejudice and hate have been planted.   They have been fed over and over.   They have grown.   They have become deadly, over and over.

Like most of my friends, my work and my life have taken me from El Paso.   Like most of them, I have spent the last hours checking on those that we love who are still there.   Like most of them, I have waited to hear the names of the dead and wounded, waited to hear if the families of our friends are safe.   We have waited, knowing that El Paso is  just a really big small town, knowing that the odds are good that somehow we will know someone or know someone who knows someone whose life has been forever changed.   All the while, I have seen the usual  offers of thoughts and prayers.   Today, I ask for more.     I ask for people to have the courage and conviction to speak truth in the face of prejudice and hate, to not turn away and pretend it is not there.   I ask for people to fight it with every fiber of their being.    Once again, we have seen that lives depend upon it.

Math Has a Front and a Back

What is mathematics really?   As the year progresses, I hope my students will discover some of the answer to that question.    While reading  Tracy Johnston Zager’s book How to Become the Math Teacher You Wish You’d Had, I came across an idea that I found both profound and completely obvious.   Math has a front and a back.   Zagar shared this idea from Reuben Hersch’s work What Is Mathematics Really?   (Now, I want to go back and read Hersch’s work.)

Math is simultaneously formal and intuitive, precise and incredibly creative, finished and unfinished, partly public and partly more private.    The front of mathematics is what we usually think of when we think of math.   It is “finished”. It is complete and precise and formal.    It has order.   It is filled with definitions and theorems and proofs. It is what is presented publicly in the form of lectures, books, and papers.    The back of mathematics is often hidden from sight, but it is nonetheless present.    It is “unfinished”.   It is what happens in one’s head or in informal discussions in offices or labs or in cafes.    The “back” of math is messy.   It is incomplete and intuitive. It is the beginning of an idea to be explored and tested.     It is ripe wit false starts and  uncertainty.  It is full of “maybe” and questions.  It is full of beginnings that sometimes lead to answers and sometimes lead in new directions.

The front of mathematics is what students think math is.   The back of mathematics is often hidden from their sight.   This year, I want my students to discover both the front and the back of mathematics.   I want the invisible to become visible.   I want them to embrace the messiness of an idea they are just beginning to grasp.   I want them to be willing to dive into that uncertainty and to explore it, to see where it leads.   (I think this is what some people might refer to as “rough draft” thinking.   I’m not sure I fully embrace that term here though.   Sometimes this is rough draft thinking.   Sometimes, though, I think it isn’t far enough along to be a rough draft.  I worry that calling it that might put ideas about how formed an idea has to be in order to be considered.)   I want students to also embrace the formality and precision of the public side of math.  I want them to be able to clearly and precisely communicate their mathematical thinking so that someone else can make sense of it.   I want my students to know that there is a front and a back of mathematics and to revel in both of them.