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.

- 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.
- 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.
- 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.

]]>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).

]]>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.

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.

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 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.

]]>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.

]]>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.

]]>Puzzles and games can be a great way to build mathematical thinking. Kids will sometimes fall into doing math without realizing it because they see the challenge as a form of play. This is something that I want to encourage, so I have been looking for puzzles and games that will build math capacity. Today’s post is about one of those “finds”. This particular “find” is a great way for kids to use the “solve a simpler problem” problem-solving strategy.

The Tower of Hanoi is a puzzle that consists of a base with three pegs and a set of discs of different sizes that fit onto the pegs. At the beginning of play, the discs are all placed on a single peg in ascending order. The smallest disc is on the top and the largest one is on the bottom. The goal of play is to recreate the stack of discs on one of the other pegs. Only one disc can be moved at a time. A disc on the top of a stack is moved onto either an empty peg or onto the top of another stack. However, a larger disc can’t be placed on top of a smaller disc. The goal is to recreate the stack on a different peg in the fewest number of moves.

When I have watched kids play with this puzzle, I have seen two different outcomes.

- Some students will play for a while and then hit a point of frustration. When they hit a point of frustration, a good strategy might be to solve a simpler version of the puzzle – take away some of the discs. They can start with just three discs and see if they see a pattern. Once they think that they saw a pattern, they can add another disc and see if the pattern holds.
- Some students will play for a while and find a solution method. They can be challenged to predict the number of moves it would take to solve the puzzle for a different number of discs. This can then lead to the idea of testing out smaller numbers of discs and looking for an algebraic pattern (how many moves for n discs). If further extensions is necessary, it can then be extended to a programming challenge. (create a program that works recursively to solve the task).

Because of the nature of the puzzle, the challenge can be made more concrete or more abstract to match the needs of different students. In both cases, “solving a simpler problem” is an important stepping stone in finding a solution path. Students will have walked away from the puzzle having gained experience with an important problem solving strategy that they may then consider using on other complex problems.

This is a fun way to introduce students to the “solve a simpler problem” problem-solving strategy. It can be used in a classroom, as part of a Family Math Night, and as a fun addition at home for family fun. I really like it because it has so many entry points and so many exit points.

You can purchase a Tower of Hanoi puzzle from retailers like Amazon. If you prefer to make your own, here is link with instruction on how to make the puzzle. The pieces are fairly small so it should be doable with scrap wood.

]]>

Miss Malarkey Doesn’t Live in Room 10 was one of many favorite books when my children were young. It tells the story of a bunch of children who were convinced that the teachers *lived* at school. The teachers were there all the time. They ate in the cafeteria and slept in the teachers lounge. That vision of teachers as one-dimensional beings is not that far-fetched from how students often see their teachers and sometimes, it doesn’t seem that far from our reality. However, I do want my students to see me as a little more human than Miss Malarkey’s students saw her. To that end, I am trying to bring little bits of myself into my classroom.

This year, I have decided to use a quilt that I am making to introduce myself and some of my hopes for our classroom community. In the process, I want to also talk to my students about what mathematicians do. (This is an outgrowth of @TracyZager ‘s Becoming the Math Teacher You Wish You’d Had .which is one of the books that I have been reading this summer. ) I spent some time last weekend putting together this quilt top with a fractal design. The quilt top is done, but quilting has not yet commenced.

These are some of the things that I want my students to know about me, about mathematics and about life.

- I’m pretty mathematical, but don’t feel particularly artistic. Designing a quilt top always takes me out of my comfort zone. I dive into that uneasy feeling because I like the process of quilting and I like having the quilt when I am done. Just as I was willing to dive into something that is challenging for me, I want my students to be willing to do things that are challenging because
**m****athematicians take risks.** - When I was putting together the quilt top, I didn’t have space to lay the pieces out before sewing it unless I put them on the floor. Knowing that my dog was likely to think the quilt blocks were some wonderful new toy, I didn’t want to chance having some of them buried in the back yard. So, I had to envision it in my head as I worked. Not surprisingly, I ended up reversing part of the fractal. I had to rip out some seams and fix it. That’s OK. It’s OK to make mistakes if you go back and figure out what went wrong and fix them. Just as I examined my mistakes and fixed them, I want them to be willing to make mistakes and to learn from them because
**mathematicians make mistakes.** - There was a lot of precision in creating this quilt top. If calculations, measurements, and sewing were not precise, the design would not create a fractal. (I also want to expand on the idea of what precision means but that part of the conversation will be saved for another day.) Just as I put in the effort to be precise, I want my students to be precise in their work because
**mathematicians are precise.** - While I piece the quilt using a machine, I do all the quilting by hand. This means I will be spending a fair amount of time and effort sketching out the quilt design on the quilt top and many, many months actually quilting the design. (While I will use this to talk about perseverance and the idea of sticking with something for a long time, I will also give my students the chance to experience this another way on another day.) Just as I am willing to put in the time and effort to tackle this big project, I want my students to be willing to stick with something for the long haul because
**mathematicians rise to a challenge.** - As I was putting together this quilt top and I discovered my mistake, I discussed what I was seeing it with my family. Talking about mistakes and asking others what they are seeing helps to clarify things. I want my students to be willing to ask questions in order to better understand things. I want them to be brave enough to not be perfect. I want them to know that
**mathematicians ask questions.** - I love math. I also like quilting. This was my first attempt at connecting quilting to math in this way. I want my students to know that there are connections between mathematical ideas and between mathematics and the rest of the world. I want my students to see that mathematics is about seeing patterns and connections and relationships and to know that
**mathematicians make connections.**

This conversation about who we are individually and how we see math is just beginning. I’m looking forward to getting to know who each of these children are and learning more about what they think mathematicians do (and maybe changing some of their thinking about that).

]]>

These are the lessons that my father taught me. They are the same strategies that I used in college, graduate school and as an engineer. They are the lessons that I try to teach my students.

- Read the problem carefully. You can’t solve something if you aren’t clear what is being asked. For me, this means reading the problem three or four times. I read the problem the first time to get the gist of it. (This is similar to teaching students to pre-read a non-fiction text by looking at the headings/subheadings and visuals within the reading section before they actually read the text so that they have a high level view of where the text will take them.) I read the problem a second time to pull out all the details. (This is similar to close reading that is taught in ELA classes.) If there are a lot of details, I read it a third time to make sure I didn’t miss anything. When I think I am done solving the problem, I read it again just to make sure that I actually answered the question. There is nothing magic about reading the problem four times. However, when I encourage students to read a problem at least three times, they are forced to slow down a little bit. They spend some time thinking about the big picture and some time thinking about the details. This gives them time to think, to process, to make sense of the problem rather than simply reaching for some formula that may or may not be appropriate.
- Sort out what you know and what you need to know. When I read through the problem a second time (and third time), I write down what I know and what I need to know. Seeing the information on paper sometimes helps see patterns or connections. When I teach students to do this, I use something similar to a Freyer model that is used to teach vocabulary. In the middle, students write the question. In the top left rectangle they write what they know. In the bottom left rectangle, they write what they need to know. In the top right rectangle, they draw a picture, make a model, or create some other kind of visual representation to help make sense of the problem. In the bottom left rectangle, they actually solve the problem.
- Draw a picture or make a model to help make sense of the problem. Drawing a picture and making a model are both ways to help visualize what is happening in the problem. It is an important part of sense-making.
- Organize the information that you know and look for a pattern or patterns. Oftentimes, this can mean making a table and looking for patterns within the table. It is sometimes easier to see patterns if the information is organized.
- See if there are bounds (upper and lower bounds) This narrows the scope of the problem.
- Break the problem into a simpler/smaller problem. Once you have solved the simpler problem, you can use that insight to tackle the more complicated problem. This year, I will have my students solve the Tower of Hanoi puzzle in the early days of school to see the power of this technique. The Tower of Hanoi is essentially a recursive task. Seeing the pattern/relationships with the larger puzzle is daunting. Seeing the pattern with a subset of the puzzle is less daunting. The algorithm can then be tested on larger parts of the puzzle. Teaching this technique with a puzzle sometimes enables kids who think that they aren’t “mathy” discover that they are in fact quite good at mathematical thinking. (This was illustrated when I challenged my family to solve the puzzle. My husband, the theoretical computer scientist, messed with it a little and said that he could see it was recursive. He didn’t want to solve it because he could just write a program to do the work. My daughter who is math-phobic because a math teacher once told her she was stupid sat down and solved the problem. She wasn’t afraid of it because it was a puzzle. She is great at math but has a hard time seeing it because of those harmful words once spoken. )
- Compare the problem to other problems you have solved (how is it the same/how is it different). In order to teach students this technique, I model this thinking in class debriefs of problems. As we discuss solutions, I regularly ask students how this particular problem is similar to or different than something that we have done on a previous day.
- Work backwards. Teaching students to use this technique is a natural outgrowth of using Kagan Cooperative Learning Structures. When students in groups discuss their responses, it seems that the beginning of their discussion is always “What did you get?” Inevitably, there is someone who got a different answer. That means that they have to work backwards to find the error.
- Take a break. Sometimes, you just need to walk away from a problem in order to think more clearly about it. When I was in school, I would go bake cookies. We didn’t have a fancy stand mixer and the dough was too thick to use a hand mixer so I had to mix the dough with a big wooden spoon. Beating the dough by hand was a good way to work through my frustration and clear my head. I tell my students about this. I am also intentional about telling them about the times when I was in college and I would wake up in the middle of the night with the answer to some problem from one of my engineering classes. I would get up and write it down and then go back to sleep. It was the best feeling.
- Talk through the problem with someone else. Sometimes, talking through a problem with someone helps to clarify things and helps you to see a path forward. I teach students to do this as an integral part of my instruction. Every day, students talk to each other about problems. We work through problems as a whole class. They work together with a partner. They work with the members of their table groups. The daily experience of discussing math shows students the power of this technique. Because my math teachers were largely fans of sitting silently in rows, I had to learn this technique at home. I try to make sure this is not true for my students.