What do you think when you see … ?

I’ve fallen behind in the MTBoS blogging initiative, but my one day weekend turned into a four day weekend thanks to some well-timed snow, and it’s time to catch up!


This year, I have been preoccupied with how students forget so much, so quickly.

This year, I am teaching AP Calculus, which means that I am preparing my students for a cumulative final exam. This exam presents a daunting challenge: how do I get my students to remember the material that I taught in September when we get to May?

However, I have realized that Calculus presents a similar but even more daunting challenge: how do I teach my students Calculus when they do not remember their Algebra? It is a common refrain among Calculus teachers that “Calculus is easy, Algebra is hard,” but time constraints force me to be selective about when I explicitly teach the supposedly prerequisite Algebra content. Sometimes I can use the review to my benefit. For instance, I knew that none of my students remembered the “log laws” and by reteaching it, I was able to help students discover the derivative of \log_bx and a^x, and we explored how you can find the derivative of complicated logarithmic functions in more than one way. But other times, the inability to recall content has derailed my lessons. For example, when my students used the first derivative test to discover that x=-\frac{b}{2a} is the x-coordinate of the vertex of a parabola, they were unimpressed because they did not recall the formula.

In my Mathematical Foundations class, I have struggled with the same problems but on a different level. This course is meant to support students who are concurrently taking Algebra 2, so at the beginning of the year, I decided to start with a unit on linear functions. I wanted to deepen their understanding of linear functions by focusing on point-slope instead of slope-intercept and by doing a lot of modeling. However, I immediately hit a ton of roadblocks. I found that not only did students struggle to solve linear equations, most were still adding and subtracting using their fingers. How did these students get to Algebra 2 with such huge gaps in their mathematical understanding and skills? How can I best help students with Algebra 2 skills when they have yet to master elementary level content? 

I have posed these questions to coaches and administrators, and I believe that vertical alignment is likely an important part of the solution, but I can also start doing a better job on my own in my classroom. As a teacher, I play a huge role in what my students remember – I message what is worth remembering, I teach students how to study, and I create situations where they need to remember. I want my students to remember what I teach them, and I have found one question that I ask over and over again that has helped me toward that goal …

What do you think when you see … ?

In October, I went to a two-day workshop for AP math teachers, and one idea stuck with me. In his session, Jamil Siddiqui was going over the solutions to some problems by asking the “class” a series of questions. At first glance, I thought Jamil’s questions were just a more practiced version of the “what do I do next?” that I see first year teachers ask all the time. These questions are not open-ended and they do not encourage student-to-student discourse, yet I discovered that there is something subtle and clever about Jamil’s approach.

Instead of modeling how to solve the problem, Jamil was modeling how to analyze the problem. For example, if the problem was asking about speed, Jamil would ask, “If you see ‘speed,’ what do you think?” We were expected to respond with the “absolute value of velocity.” I got the sense that this call and response was rapid and habitual in Jamil’s classes; each question he asked was an exercise in recalling essential information, every time strengthening the memory of that content.

Then Jamil would ask, “If you see ‘absolute value,’ what do you think?” I thought: distance from the origin? A “V” shape? Jamil corrected us – were expected to respond with “piecewise.”  That answer to that last question fascinated me. Piecewise is not the definition of absolute value. However, in Calculus, piecewise is often the key to analyzing an absolute value function. Each key word is linked to the most important thing to remember, not the definition.

After the session, I followed up with Jamil and he shared his seven page Calculus “Key Words” glossary with me. I took it as a starting point and launched my own version in my class soon after.

How do I use this question?

In each of my classes, we created a glossary in the back of their notebooks. One column is labeled, “When you see/hear …” and the other column is labeled, “You think …” In the first column, I chose words that my students run across often. For calculus, this meant integral, derivative, maximum, etc, and in Foundations, this meant x-intercept, factor, product, area. For the second column, I brainstormed the most essential piece of information to associate with that word, and in most cases, this was not the definition. For example, when you hear derivative, you think instantaneous rate of change or slope of the tangent line. When you hear x-intercept, you think y=0. And so on. I included vocabulary that is introduced in my class and from previous years; I include whatever students need for my class.

Every day in class, I take two minutes to practice recall with these key words. I call the activity a “definition check,” and we do it everyday right after the warm up. I project a word, randomly cold call a student and ask, “what do you think when you see ___?” By letting students look at their glossary or pass if they don’t remember, I provide a low-stakes opportunity to practice recall. I ask the same question to 2-3 students, and then move on. And I often go back to words from much earlier in the year to make sure they continue to remember those words.

During class, I use this question like a hint. When I’m circulating and a student needs help, I point to the notation or the directions, and say, “what do you think when you see ___?” Sometimes they need their glossary to answer my question, but once they do, they’re usually off and running.

My Favorite: Completing the Square


Grad School: Take 1

I essentially learned how to teach math by thinking about how to teach completing the square. In my high school methods class with Yolanda Rolle, I was paired up with two of my classmates (Mary Feeley and Amanda Miles) to prepare a lesson on completing the square which we would then teach in a mock lesson to the rest of the class. We did some research and started to realize that there were a lot of different ways to approach this. Our concept map is a nice picture of this complexity.


We thought hard and long about the sequencing of a quadratics unit. In particular, we were wondering if students would have already seen the quadratic formula (and completing the square would be used to make sense of it) or if the quadratic formula was coming later. We looked at textbooks and saw both approaches. We decided that for our lesson, our “students” – who were really our classmates – had already solved quadratics using square roots, factoring, and the quadratic formula, although the quadratic formula made little sense to them at the time.

Our lesson started with the following worksheet.


We hoped that there would be some variance in solution strategies and we could follow up with a discussion about the pros and cons of each strategy. Since this was a grad school class, there were even more strategies that we had hoped: one “student” (Kate) used difference of two squares to solve these equations. We concluded that the quadratic formula was annoying but always worked, and that both factoring and square roots were pretty easy … but did they always work? We tried to show that square roots always worked by starting a discussion about how to solve x^2+4x=7. I was glad that our method lent itself to a discussion about a big idea (comparing different methods), but we didn’t really get into why we need to add (\frac{b}{2})^2.

Grad School: Take 2

I had originally thought that completing the square and quadratics were purely algebraic skills. After some reflection and more planning, my group taught the “class” a second lesson on completing the square. This time, we picked up where we left off and tried to focus on why adding (\frac{b}{2})^2 created a perfect square. And we used Geometry to do it!

We led the class through folding a piece of paper whose area represented the left side of our equation: x^2+4x=7. Therefore, the area of this paper was 7.


We asked the class: how can we make this paper into a square, so that we could then use the square root method. Someone suggested that we add 4x to the bottom, but we concluded that this would not be helpful because then the area would be 7+4x and we would be no closer to using square roots. We concluded that we could only add a real number. We eventually led our clever “students” to the discovery we wanted: split the 4x in half,


and move one half to the other side.


Which smoothly transitioned us to a new question – what do we need to add in order to complete the square? – and we had no trouble answering with 4. We later generalizing this approach with x^2+bx. I loved the use of geometry, but there was something forced about the discovery, and the entire lesson relied pretty heavily on the background knowledge of our “students.”

Second Year of Teaching – Algebra 2

In my first year of teaching, I tried using that lesson in my Precalculus Honors class, and it fell flat. The “discovery” turned into a demonstration, and I realized that I had to go back to the drawing board.

By my second year of teaching, I had developed a new approach to quadratics. Everything started with vertex form: y=a(x-h)^2+k. Vertex form is easy to graph (find the vertex) and solve (use square roots). When I introduced quadratics in standard form: y=ax^2+bx+c, my students didn’t know what to do. So we came up with a plan: change the equations back into vertex form. But how do we do that? We have to complete the square and factor.


While I was still doing a lousy job of motivating the (\frac{b}{2})^2, I had made an important shift: instead of just teaching my students how to solve equations, I was trying to get them to understanding the structure of a quadratic function. This algebraic trick revealed the vertex of a parabola and facilitated finding the x-intercepts.

Fourth Year of Teaching – Mathematical Foundations

This year, I teach 2 sections of an Algebra 2 support class. While my students have definitely been taught completing the square (probably twice), it hasn’t stuck. These classes have a lot of students with low English proficiency and/or learning disabilities, who hate math and/or see themselves as terrible at math. In this last unit, I was focused on teaching multiplication, first with integers and then with algebraic expressions. We almost exclusively used an area model to multiply. Then, we started to use algebra tiles and the area model to factor. In this context, completing the square was a problem-solving rich application of multiplying and factoring. 

In this activity, I asked my students, how many little squares (i.e. whole numbers) do we need to add to make a perfect square out of x^2+2x, x^2+4xx^2+6xx^2+8xx^2+10xx^2+12xx^2+100xx^2+bx, and x^2+5x. And their approach was fascinating.

When the b-value was a multiple of 4, most students had the same approach: spread the x-tiles (the long thin rectangles) evenly around the four sides of the big square (the x^2 tile).

But when the b-value was not a multiple of 4, students were pretty stuck. Many students came up with solutions that were not actually squares, which led to a pretty good teaching moment.


Some students came up with some really wacky and fun solutions.

And when a couple students came up with an elegant solution, they were so confident that: (a) they could calculate the number of little squares without drawing them, and (b) they could explain their approach to their peers.


In this class, completing the square is not about solving equations or the structure of quadratic functions; completing the square is about problem solving and how area connects algebra and geometry.


Completing the square has a special place in my heart. By trying to teach completing the square, I have learned so much about a topic that I thought I understood as a teenager. I have taught completing the square 7 times – twice in grad school, twice in Algebra 2, twice in Precalculus, and now once in Mathematical Foundations. I don’t think I’m anywhere close to finding the best way to teach this topic, and I love that challenge. 

A Day In the Life


My school’s new schedule splits eight periods into an A day and a B day with four periods each (plus advisory on Mondays and Tuesdays). My A days are very different from my B days; I teach two blocks on A days, and only one on B day. I decided to write about Thursday (which was a B day) since I had a mix of teaching and lead teacher stuff and philosophy club. So just keep in mind that this is only kind of representative of my day-to-day job experience.

Before School Starts

My alarm goes off at 5:40, I snooze once, take a shower, get dressed, pack my bag, eat breakfast, and get out the door by 6:30. I walk to the bus stop, take the bus to Chelsea, and walk the rest of the way to school. Before taking off my coat and hat, I walk into the mailroom. I put the keys to the math department Chromebook cart in Ms. Yu’s mailbox. I grab some blue paper and get in line for the copier. I make copies of quizzes (always on blue paper) for my Mathematical Foundations classes and make some copies of my do now. I head to my classroom, and settle in by logging into my computer, opening up my slides for my first period class, and having some coffee. A few Calculus students drop by. A couple students work on their problem sets (due on Friday) and ask me a question or two, a couple students ask me to retake quizzes and I print them out a retake. A couple students just use my classroom to meet before walking to class, and one student uses my counter as her second locker, dropping off her jacket every morning.

First Period – Mathematical Foundations (8:00–9:27)

The bell rings, but I hardly notice that I’m supposed to start class because so many of my students are running late. Students grab their notebooks from the counter, and I pass out two copies of the do now to each table in the hopes that students will work together. Students call me over to help or to ask, “Mister, is this right?” When they get it right, I flip it over to reveal a slightly harder problem. They were practicing multiplying linear binomials using the area model, which was a big part of this unit. By the end of the do now, most of my class has arrived, and I transition to the next activity, which I call a definition check. I use an app on my phone to randomly call on a student asking them, “What do you think when you hear _____,” filling that blank with a vocabulary word. They respond with the answer from their glossary. I call on another person and ask the same question. I repeat with a few vocab words and move on. Today, the vocab was: multiplication, factor, and product, and we connected this back to the do now: the factors are the side lengths and the product is the area. Next I passed up some algebra tiles and projected a timer with 20 minutes to show how much time students had to finish their problem sets. They had been working on their problem sets for a couple days now, and most groups had discovered a formula for completing the square. When the timer went off, I collected the problem set, and passed out a handout. Students glued/taped the handout in as page 52 in their notebooks, which took a few minutes. Then, I asked students to make a rectangle with an area of x^2+6x+9 using their algebra tiles. Once they had, I clarified that when they are asked to factor, I am looking for the side lengths, which were (x+3) and (x+3). I reminded them that algebra tiles are just one of three methods we have used to factor, and showed them how I would solve the same problem in two other ways. I passed out some practice problems on factoring and circulated as they tried them out. Finally, I passed out a quiz, which was their third and final attempt of the quarter to demonstrate their ability to multiply and factor polynomials. And with that, 2nd quarter came to a close.

Second Period – Intensified Algebra / Algebra 1 PLC (9:30–10:56)

When B days fall on a Wednesday or Thursday, I have two meetings called PLCs (Professional Learning Communities), and I start with the the Intensified Algebra / Algebra 1 PLC, which has five math teachers, an assistant principal, an instructional coach, and me. Today is an unusual meeting: our representative from Agile Mind has come to support us around unit planning the next unit in our Intensified Algebra courses. Our Intensified Algebra courses meet every day instead of every other day to try and give extra support to our highest-need 9th graders. This course tries to cover Pre-Algebra and Algebra 1 topics in order to get students ready to take Geometry as a sophomore. This course is the only one we offer with a curriculum that we pay for. The curriculum was designed by the Dana Center and is called Agile Mind. As part of the program, we are provided with two on-site visits by a program representative, with the today being the second. So instead of our instructional coach, the Agile Mind representative is running the show today. She passes out the test and standards for Unit 5, which is about solving linear equations and inequalities. In partners, we take the end of  unit test, and then as a group, we discuss what content the test is designed to cover. I found taking a test to be pretty boring and tedious. I also found it discouraging that some of the questions were meant to preview the material in the next unit without providing a need for it. For instance, students were asked to solve a simple two-step linear equation by setting each side equal to y, making a graph/table, and finding when the graph/table had the same y-value. But there was no need to solve these problems this way, and it felt unnecessarily convoluted to me. On the other hand, taking the unit test before starting the unit is a good way to think about the big picture of the unit, which I think can be really hard the first time you teach a course.

Third Period – Algebra 2 PLC (10:59–12:51)

For my second PLC, I moved next door to work with a different assistant principal, a different instructional coach, four Algebra 2 teachers, and Wesley, who is the only one who teaches AP Statistics and Discrete Math. Right now, the Algebra 2 teachers are working on planning lessons individually for the upcoming unit, and next time, we will give each other feedback on these lessons. Hopefully, after that, all of the Algebra 2 teachers can implement all of the lessons we created and we can observe each other in the process. While they work through their lessons, I help one teacher understand some of the resources and work on planning my upcoming unit in Mathematical Foundations, which is supposed to support their work in Algebra 2. I’m thinking of focusing on radicals, and while I have some ideas, I’m still not sure how I will make that happen. After the meeting, I have 25 minutes for lunch – I eat a tuna sandwich and chat with two other math teachers.

Fourth Period – Prep (12:54–2:20)

Usually, during fourth period of PLC days, I type up some notes from the two PLCs so that the teachers who aren’t in the meeting can hear about what is happening, but today, I have a few distractions. For once, my classroom is available, and it is great to be able to work in there, but a bunch of my seniors (who don’t have a class) come by to hang out and do some work at a leisurely pace. They distract me a bit, but they are pretty entertaining too – they talk presidential politics and take an online citizenship test for fun, and I can’t help but chime in from time to time. Meanwhile, I finish writing quiz retakes for Calculus on Friday, and make some copies. I answer some emails and start writing PLC notes but don’t finish. I do some last minute prep for philosophy club and run across the street for some iced coffee, getting back to my classroom right before the bell rings at 2:20.

After School – Philosophy Club

For the last three years, I have spend my Thursday afternoons running a Philosophy Club. My students (mostly seniors and mostly my Calculus students) insist on getting food before the meeting, so we start around 2:45. We always warm up with ethical dilemmas, usually pulled from the New York Times Magazine. We discussed Should My Rich Friends Apply For Financial Aid? followed by Is My Neighbor Obliged to Report Me to Immigration? Most of my students have low-income, immigrant families, so these dilemmas were an exercise in playing Devil’s advocate, and after expressing some initial outrage and indignation, my students did a pretty good job. Next, I asked When Does Bread Become Toast? because I saw that article and it seemed like a fun way to circle back to a recurring Philosophy Club question: where do we draw the line? Their reaction was way more intense and polarizing than I could have possible imagined, and they considered the same extreme questions that I had: what about the crispy bread that isn’t yet browned? What about the burnt to a crisp bread? Are either/both of those toast? Finally, we moved on to our main topic: free will. We had a really fun conversation about this question last week, and this time, I added a new wrinkle: if we do not have free will, can we still be held accountable for our actions? After some careful consideration of that question, we read a chapter from Dostoevsky’s Notes From Underground that I though connected to the discussion about free will. We usually end philosophy club whenever the number of tangential discussions reaches a critical mass, and this week, that happened at 4:25, which is a bit later than usual. I packed up and headed home, getting there around 5:15.

At home, I took it easier than unusual, watching some TV and relaxing on the couch. I also adapted Sam Shah’s delightful activity into a Calculus problem set to give my students Friday – I won’t see them for 9 days and want them to start thinking about inflection points and have some fun doing some math. I get to bed around 10:30.

#MTBoS 2016 Blogging Initiative


The school year thus far has been rather challenging. I have been constantly working and yet always falling further and further behind. Blogging felt like a vanity project that I could not afford to prioritize over lesson planning, grading, supporting other teachers, writing recommendation letters, and so on. But I worked hard over vacation, and I’m starting to feel like I have my head above water enough to dust off the blog.

Therefore, I, Joey Kelly, resolve to blog in 2016 in order to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS.

The first prompts are now posted, so stay tuned …

Hope & Despair

I recently saw this article, and I thought the “reason for despair” and “reason for hope” conceit was good food for thought to get me blogging again. This year, I find that I am constantly vacillating between hope and despair. Instead of talking about the state of education in general, I want to talk about the different aspects of my job.


Teaching AP Calculus AB

Reason for Despair: My school does not traditionally do well on the AP exam (last year, 20% of students passed the exam, and in the last five years, only 21% have passed). This year (my first year teaching Calculus), the class size has increased by 40% from 25 to 35 students (compare to two years ago, when we had 37 students and TWO sections). My school switched schedules this year and as a result, I have about 15% less class time. Almost all of the AP-specific professional development has been designed assuming everyone teaches with a lecture-based, test-prep, rely-on-the-book mindset. In lieu of that approach, I have been developing a lot of my own resources, and it can be exhausting. My students are strained by jobs, other classes, and applying for colleges, and I sometimes struggle to get them to put the necessary time and effort into their coursework.

Reason for Hope: I moved with my students from Precalculus to Calculus, and I love these kids. When I ask them to quiet down, they do! When we discover something, some students act like their minds are blown. And even though we have long block periods, the bell sometimes sneaks up on us, and some students groan when they have to leave. We’re working on class shirts, and I regularly get to spend time with my students talking about Calculus after school.

Calculus is beautiful and fun and designing this curriculum has been a surge of creative energy for me. There are some great opportunities for discovery-oriented activities (e.g. the power rule), but there are tons of opportunities for surprising connections. I have deviated from the traditional sequencing in a few significant ways, and the response has been really positive from my students. I started the year with integration (using area) and derivatives (using slope) side-by-side, which made the Fundamental Theorem of Calculus feel truly Fundamental. We did Split 25 and discovered that e is weird and magical. We did the chain rule one day and u-substitution the next. We skipped limits at the beginning of the year, and instead we’ve felt the need for limits by acknowledging that “dx” – the magical calculus number that is both greater than zero and approximately zero – doesn’t really make sense. I want my students to see Calculus as something that makes sense AND is wonderfully confounding, and I am feeling successful thus far.


Teaching Mathematical Foundations

Reason for Despair: These students are all also enrolled in Algebra 2, but are in my class for extra support. I don’t know how many of them are passing Algebra 2 when they struggle with basic arithmetic. I sometimes struggle with why they have to take Algebra 2. I have never had students so new to English before, and I speak no Spanish at all. I have never had students who hate math so much or are so comfortable being blatantly disrespectful. Our Algebra 2 teachers are rarely on the same page, but I have a mixture of their students. I constantly struggle to find a balance between giving students support in the basic skills they lack and helping them become fluent with Algebra 2 skills/concepts. There is no curriculum for this class, and I am exhausted trying to create materials all the time. I am not sure that my efforts are always worthwhile because I don’t know if this class will or should continue to exist in the future.

Reason for Hope: I am learning so much about how to work with students who limited English proficiency. I am gaining some classroom management confidence. I get to experiment with teaching elementary and middle school skills mixed as embedded remediation. I have become fascinated by how students do and don’t retain information over time, and in my district’s Math Vertical Team, we have started to map out a conceptual strand from k-12 and are going to think about how to remediate along the way. Since I don’t have a curriculum, I can be responsive to student strengths/weaknesses. My students have definitely learned that a fraction bar doubles as a division sign, and that a line has a constant slope. I’ve gotten to work with Algebra tiles and the area model as a way to think about multiplying and factoring polynomials, and I love the puzzle-y nature of the work and the strength of the connection between Algebra and Geometry. I’ve started to think deeply about how to best intervene and support students with learning gaps, even if I don’t have answers yet.


Being Lead Teacher

Reason for Despair: In a department of 21, we have 15 new teachers, many of which are younger than me and new to teaching. We started the year still looking for two math teachers, we have already had a teacher quit, and we were not fully staffed until Thanksgiving. While other departments are refining unit plans and performance tasks, we are still just starting from scratch in a lot of places, especially in Algebra 1. I feel like I’m repeating and restarting a lot of the work that I did last year. We simply do not have enough collaborative time, and we have not had a department meeting since October. Most of our PLCs – the only scheduled collaborative time – are not structured to include all teachers of a given subject.

Reason for Hope: For the first time, we have an administrator who used to be a math teacher. Our department meetings have had an really awesome energy. Our PLCs are way more productive with long blocks, and I have gotten a lot better at collaborating with coaches/administrators and sharing what happens in a meeting with the people who weren’t there. Because of my work in Foundations, I feel really tapped into Algebra 1 and Algebra 2 skills/content. I purchased a ton of notebooks with the department budget, and lots of teachers have tried interactive notebooks with positive feedback. I finally figured out a coaching cycle structure that works for me: unstructured group co-planning. I feel really good about the materials I left behind for precalculus teachers.


School Culture

Reason for Despair: I could go on and on about this, so I’ll keep it brief: initiatives and pressure comes top down; overworked teachers with too little prep/collaborative time; not enough teachers; too much teacher turnover.

Reason for Hope: I could probably go on and on about this too: new teachers are open and energetic; we’re more focused on understanding mathematics than passing tests; I have adequate autonomy in my classroom and curriculum.

NCTM Lessons: Randomness, Verticality, and Spiraling Activities

A few months ago, I went to my first math ed conference: the NCTM Annual Conference, which was here in Boston this year. I really enjoyed myself, and I think I got a lot out of it. One session in particular gave me the final push I needed to change my teaching in three significant ways. That session was: Improving Student Success by Spiraling Activity-Based Learning with Bruce McLaurin and Alex Overwijk. I went to a lot of smart and interesting sessions at NCTM, but I think that this session had such a major impact on me because of a sort of synchronization between 1) their experiences and 2) my ongoing struggles. Below, I try to explain why I was able to actually act on what Bruce and Alex were talking about, and what these three changes looked like in my classroom this past spring.

Change #1: Daily Visible Random Groupings

I was primed for this change.
I have always struggled with how to assign seats. I want students to be happy and buy into the seating arrangements. I often let students choose their own seats, but that causes some other problems: kids can be cruel and exclude their peers, and working with your friends is a recipe for distraction. However, my biggest problem is that no seating arrangement works for the entire year – things just get stale after a while. I’ve tried to assign new seats periodically to shake things up. Sometimes, I keep some of the kids who I know are friends together, and other times, I’ve tried randomly assigning seats. But inevitably, kids tell me: “I can’t work with him/her” or “I can’t see from here,” and whatever I had planned falls apart.

This session gave me the final push.
At some point during the session, this slide went up:Screen Shot 2015-04-20 at 8.55.15 PM

And something clicked. Visible Random Groupings are a simple and systematic solution, and I tried it for the rest of the school year. Everyday, as students came into class, I held out a partial deck of cards. Students choose one at random, went to the corresponding group, and got to work on the warm up activity. Because students are picking their own card, they believe me when I claim the seats are random; I think that is what’s meant by “visible.” While I have randomly assigned seats before, doing it everyday has been a revelation. If they don’t like it, I tell them, “Too bad. Maybe it’ll be more to your liking tomorrow.”

After about three weeks, I sought some student feedback, and it was mostly positive:

“I like the new do now system and seating arrangements. It forces us to work with new people everyday which means new ideas everyday.”

“The way we start class now is a good way for everyone to meet each other.”

“I like your changes because you get placed with different people and you get to work with them. It’s better than talking to your friends all class.”

“I think daily seat changes were good because they didn’t let you rely on one person to do the work.”

Change #2: Vertical Non Permanent Surfaces

I was primed for this change.
My school has a surplus of small individual whiteboards. During my first year of teaching, these were a game changer. Using a whiteboard instead of a handout was like a magical injection of engagement … for about 20 minutes. They sparked some new ideas for lessons, but fundamentally, individual whiteboards are good for individual work. I had read about how easy it is to get big whiteboards from Home Depot or Lowe’s several times: from Andrew Stadel, Nathan Kraft, Fawn Nguyen. It sounded like a good idea, but it seemed like a lot of work for a meager change.

This session gave me the final push.
This session was first thing in the morning, and it started with a notice/wonder on some chart paper spread around the room. I took the bait and got up and jotted down some ideas. It helped wake me up and got me talking to some of the other people at the session. Later, I saw myself in action on twitter:

If that had been all, I would not have given it a second thought. But later in the session, Alex mentioned that his students are standing about 80% of the time! That statistic blew me away. It reminded me of grad school and how the phys ed teachers always harped on how all the research says that our students do too much sitting. While I agree with the sentiment, I never had any good ideas on how to execute it. I’ve tried stations activities, but asking students to get up and move to another table during class was like pulling teeth.

That statistic – vertical 80% of the time – convinced me to try this. The wonderful Ms. Harding drove me to Home Depot, and they cut the boards in half for me. I attached five whiteboards to the walls of my classroom, and it was transformative.

Now, each table gets their own whiteboard with the day’s warm up activity taped to it, and the increase in engagement and collaboration has been both significant and obvious.

An example of Estimation 180:

Whiteboard 7

An example of Which One Doesn’t Belong:

Whiteboard 9

A Mobile from from solveme.edc.org:

Whiteboard 10

An example of Notice/Wonder:

Whiteboard 11

And another Notice/Wonder:

Whiteboard 12

An example of 3 Reads:
Whiteboard 13

Change #3: Spiraling Activities

I was primed for this change.
I was initially drawn to this session by the phrase “by eliminating units” in the session description. In my first two years of teaching, I kept looking for someone to help me understand what a “unit” is and how long/short or broad/specific they are supposed to be. At different times, I have been criticized for having units that are too short and for having units that are too long. I have been able to deduce that a unit should be between two weeks and two months long, but that still leaves a pretty wide range. For example, “sine/cosine graphs” feels like a discrete topic, but does that mean it is a unit or is it just one part of a larger “periodic functions” unit, which would also contain application/modeling problems and maybe even tan/sec/csc/cot, inverse functions, or the unit circle? Or maybe all of this is part of a giant Trigonometry unit that also includes  SOHCAHTOA, identities/proofs, and the laws of sines and cosines? How do we benefit from breaking all of this content into units, and how do we make sure that we don’t lose the connections between the units by creating a distinction?

This session gave me the final push.
The session started with a Notice/Wonder about this image.Screen Shot 2015-04-20 at 8.36.30 PM

I was a little taken aback by the onslaught of color (I’m colorblind), but I took some time to process and found this fascinating. This image shows two different ways to organize the same course: 1) by units, and 2) by activities. At the top, the course is organized into seven units. This course organization squishes all of the work on similar triangles into the first 1/7 of the course. The courses at my school are all organized by units.

The array with the colored circles shows how the same course could be organized by activities. Here, the activities and assessments are on the x-axis and the standards/objectives on the y-axis. Now similar triangles are spaced (i.e. spiraled) throughout the course, showing up in activities 1, 6, 14, 21, and 28. This session told the story of how and why Alex and Bruce shifted from a unit-based course to a spiraled, activity-based course. I love this structure: it feels more authentic and I bet they are seeing much better retention of content learning.

My school is committed to unit-based design, and I do not have the freedom to abandon units. We have relatively new year long plans (or “pacing guides” or “scopes and sequences” or whatever you want to call them) that outline the units into which we need to divide each course, and the standards that compose each unit. So while I love the changes that Alex and Bruce have made in their course, I didn’t know how I could bring that work into my classroom.

At the end of the session, I asked Alex if he had any advice. He suggested that I could still implement activities within my course’s units. Alex used “activity” to mean something different from “lesson.” While lessons teach content, activities teach content by doing mathematics, which is a perfect instructional pairing with my school’s performance task initiative.

I have tried a variety of activities and 3 acts in the past, but I was mostly trying to use them as a hook beforehand or an application after-the-fact – I did not realize that they could be the driving force for the entire instructional sequence. I had just adapted Ferris Wheel for a performance task, and I tried to build on that by formalizing our study of periodic functions with an activity: Dan Meyer’s Scrambler 3 Act. Here’s a quick summary of what that looked like.

1) We started with the Act 1 video and a notice/wonder on the whiteboards:Whiteboard 1Whiteboard 3

Whiteboard 2

2) Then we came to a consensus about what question we were going to try to solve: How high is the red oval when the clock hits 0?

3) I challenged students to figure out the answer without using trigonometry. We had been able to do that with the ferris wheel, but not this time. Some groups were able to figure out that it would be somewhere in the first quadrant, but not exactly how high. We NEEDED trigonometry and to write some functions.

4) So we paused the scrambler, and we focused on developing content knowledge. I gave each group a graph of a sine/cosine function, and asked them to write an equation for it. Our previous knowledge helped us find the parent function, amplitude, and midline, but students struggled to turn the period into the “b-value.” As I circulated, I gave groups some hints, and they started to get a feel for why and how to use the period=2pi/b formula.

Whiteboard 8Whiteboard 5

Whiteboard 4

5) After another example, we came together as a class and wrote equations for the movement of the red circle and the red oval using the Act 2 video. Temporarily, we ignored the midline. To find the equations, we had to apply the practice and learning we had just done.

6) Finally, we discussed how to combine these two functions and how the midline was involved. This discussion required clarifying and being more precise about what “h” meant in each equation. By the way, this was one of my favorite discussions of the year. Typically, one student has the insight that the midline of one function is the other function, and I love pushing them to share that insight with their peers.

Whiteboard 6

I loved how one activity was able to be:

  • the hook that generated student perplexity and mathematical questioning,
  • the need to teach some specific content,
  • a vehicle for developing mathematical practices, and
  • an application that communicates that the content we just learned is useful.

Now to figure out how to repeat this process with other activities/content/courses.

NCTM Takeaways

A few months back, I attended the NCTM annual conference here in Boston. I wanted to blog about every session that I attended, and while I’m trying to catch up on a bunch of blog topics from this year, I just haven’t been able to devote the time and energy to write about every interesting idea from every session. Instead, I want to write just a sentence or two about each session. For the most part – with so much information being thrown at me – I can only absorb a couple ideas from each session anyway.


Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone)
Elizabeth Green
There is only one 21st century skill – being able to learn – and that is what we need to communicate to our students. This was the first instance I saw of a recurring theme: supporting teachers as learners.


Improving Student Success by Spiraling Activity-Based Learning
Bruce McLaurin, Alex Overwijk
More on this session coming soon. After this session, I made three important changes in my classroom.

The Focus on Mathematics Academy
Shannon Hammond (with Glenn Stevens, Tracia Fung)
NCTM sessions are way more fun when they involve doing math. Shannon had mentioned the rat problem to me before, but now that I actually did it, I understood what it had to do with polynomials and why it is such a great task.

Adaptation: Creating Open Problems from Closed Curricula
Geoff Krall


Back to the theme of supporting teachers as learners. We have tons of old textbooks at our school, and while I’m not fond of using the textbook as the curriculum, I would like our teachers to have somewhere to look for support. With Geoff’s framework, I have some concrete strategies I can use to help teachers bring out the potential within the textbook.

Contexts for Complex Numbers
Michael Pershan, Max Ray
More on this session soon. I tried out some of their activities and adapted their work for my precalculus final.

Exhibition Center
With a few exceptions, I found walking around the giant room full of vendors to be really boring. One of the exceptions was the MTBoS booth, where, with some great luck, I won a class set of radian protractors!

Explore MTBos Tweet

Lots of great stuff here. Two snippets stuck with me:

  • At my school, we talk a lot about performance tasks and tasks that require students to do the work of a mathematician. But what exactly does a mathematician do? In one ignite talk, we got some definitions, including: “Mathematicians are people who enjoy the challenge of problem … people who see beauty in a pattern, a shape, a concept, a proof … people who share ideas.” I love that definition.
  • Reading and literacy strategies are effective for making sense of word problems. We need to stop “cracking the code” and saying things like “total means add” and so on.

Shadow Con
Again, lots of great stuff here, but again, the two ideas that stuck:

  • Lesson study, co-planning, and/or coaching cycles often involve peer observation. The observer does not have to be a silent observer – they can call a “teacher time-out” and talk to the teacher. I like how this makes the lesson implementation a collaboration and how Elham stressed that co-ownership of the lesson during the planning stages was crucial. These lessons are percolating as I think about my coaching role for the upcoming year.
  • Michael Pershan talked about why our hints aren’t good enough and how they need to get better. I’ve tried to put this into action by 1) before the lesson, adding a column into my planning template to brainstorm hints and 2) after the lesson, jotting down the hints that I ended up giving and didn’t plan (so that I have them for next time).

Desmos/Mathalicious Trivia
I was part of the winning team: Strength in Numbers!


Building Student Understanding of the Mathematical Practices through IN-formative Assessment
Matt Mcleod
“Using structure” seems a lot simpler in the standards than it is in practice. We talked about two different components of structure: chunking (seeing something as a single object) and hidden meaning (rewriting something in an equivalent form). The challenge is to get students to use a structural approach instead of a computational approach. While the session’s example focused on early algebra, we were given time to brainstorm for our content areas, and I found these ideas definitely translate to logarithms and trigonometry.

With Respect for Teaching: Making Mathematics Instruction Explicit
Deborah Ball
I really liked the distinction between explicit instruction and direct instruction. One of her videos inspired a new warm up activity that I’ve been trying: I give students a problem and I ask them to give me an answer that they KNOW is wrong. Then we talk about how they know that.

Assessing Conceptual Understanding
Chris Chung, Adam Lavallee
These guys were my age, maybe younger, and I looked at them and thought: I can do this … which in a manner of speaking I did at a different, much smaller conference a few weeks later. Hopefully, I will blog about that experience soon.

Transforming Practice: Organizing Schools for Meaningful Teacher and Leader Learning
Elham Kazemi, Allison Hintz, Lynsey Gibbons
After seeing Elham at Shadow Con and considering their work was primarily at the Elementary level, I’m not sure this session was a great choice. Mostly, I was just jealous of their PD structures and the functionality of their PLC and lesson-study experiences.

My Favorite Math Fun Facts
Francis Su
Lots of fun and surprising results here. My favorites were 1) a guessing game about which enormous quantity was the biggest, and 2) The Spherical Pythagorean Theorem.

Emphasizing Experimentation and Discovery in the Teaching of Geometry
Matthew Chedister
Again, sessions are more fun when I get to do math. Deductive reasoning is hard, but it is easier if we prepare with inductive reasoning aka experimentation/discovery. I learned that there are a ton of different ways to find the sum of the angles in a pentagram, and none are trivial to articulate.

10 To-Do’s for Converting Principles to Action into Tangible Improvements
Steven Leinwand
Some stuff I agreed with and some I questioned, but wow does this guy speak with a lot of energy. The most interesting piece for me: the “jump in and participate” model of coaching. He argued that you shouldn’t just take notes and talk about it in a debrief later but instead should jump in and ask a question or draw a picture. I’m not sure I agree, but it is making me question my previous assumptions.


Martin Gardner and the Mathematical Practices
Michael Serra
This session was just so much fun. I think a highlight was that he gave out prizes by playing the “phychological game,” where everyone has to pick a positive integer, and the lowest unique positive integer wins. I believe the winners were #1 the first round, then #9 and #5, and none of them were me.

Strategy in Sports: Conditional Probability and Expected Value
Josh Tabor
By going to a session about probability, I was trying to go outside my comfort zone a bit. I think I was disappointed because I felt so comfortable and followed the math so easily. Oh well.

When Am I Ever Gonna Use Math in Real Life?
Adam Poetzel
Adam talked about which students ask “million dollar question”?

Screen Shot 2015-04-20 at 6.59.28 PM

I like how the image implies that students ask this question more often when they are frustrated or disengaged then when they are curious. Adam presented three ways to “sell” math (brainpower, opportunity, and relevant tool). I agree with all of these answers, but I worry that these responses are only good ways to answer the curious student. Soon, I hope to write more about how I try to approach these students.

9. Four Fours

Play 9

I meant to blog about this problem about 10 months ago, but better late than never.

Highlight: We got student-supplied answers for 1-100.

Lowlight: Those 100 answers did not come from 100 different students. In fact, one student was able to come up with about a third of them himself. We got some but not nearly enough female representation. There was no faculty representation.

Highlight: For the first few days, students were really excited and interested. Their submissions were a bit overwhelming to keep track of.

Lowlight: That engagement level disappeared entirely after a few days. We had 30 or 40 left to figure out, and only 1-2 students were working on it.

Highlight: The leaderboard. Some students really took pride in having their name publicly displayed.

Lowlight: The start of the school year was so crazy that this was the only problem that we had up for the first month.

Highlight: There was a nice unintentional synchronicity between this task and my class. In precalculus, we covered piecewise and step functions, and a couple students noticed that this allowed for crazy new opportunities in this problem. Now they could write 0 or 1 with only one 4 using decimals and the floor/ceiling function.

Lowlight: It was unclear whether or not using e was a violation of the rules. Generally, I like this ambiguity – it allows me to ask the student what they think it fair. Unfortunately, they decided it was fair, and they used it over and over again with the floor function, making things just a bit too easy. In hindsight, I should have pushed him to realize that this shouldn’t count in the hopes he would persist and find a more interesting solution.

The final product (click to enlarge):09-Solutions

What would we change for next time?? Maybe it would be more interesting to focus on a smaller batch of numbers and try to find as many ways to write them? Perhaps that would lead into questions about what makes two equations “different” or not? Maybe a leaderboard with a bigger font so student names are even more public? We definitely need some new ways to publicize our problems and offer different ways for submissions …

13. Thirteens

Play 13. Thirteens

Source: NRICH Maths – Elevenses

Why did we choose this problem?
Good Play With Your Math problems:

  • involve some sort of “play” before choosing a specific strategy.
  • have a low floor (accessibility and entry point)
  • have a high ceiling (need for more complex mathematics)
  • have a succinct, accessible, intuitive wording and visualization

This problem meets all four criteria:

  • You have to experiment, observe patterns, struggle, and adapt before finding a more elegant strategy. For us, this is what mathematical play looks like.
  • This problem is extremely accessible, requiring just integer addition and division in order to test each pair.
  • The need to organize the results and systematically test all potential options is no easy task, creating a need for certainty and the desire for a more efficient and elegant strategy. While this problem did not prove all that difficult for me or Ms. Yu, we conjectured that the solution strategy would prove particularly counterintuitive to our students.
  • This problem was already succinct and even presented as a poster. We knew it would be possible to present this problem visually, we just needed to decide how much to change.

Modification #1: 13 instead of 11. 
We opted for the Thirteens version of the problem for a few reasons: because it was time for Play 13, because we released it on Friday the 13th, and in order to make the problem more challenging. We included the Elevens version on our handout as a scaffold.

Handout Front

Handout Back

Modification #2: Ditch the table. 
There is no apparent reason why the numbers should be organized in a square grid. Is this intended to help someone solve the problem? Or might it distract people from the essential features of the question? Instead, we opted to organize the numbers haphazardly.

Early Highlights

  • One of my students took one look at the problem and cleverly asked if he could use the 13 at the top. Then we looked to see if it would make a difference.
  • I had a Problem Set work day in one of my classes on Friday, and a few students worked on this problem for the entire period.
  • Several students kept track of their answers by connecting the circles with a line; however, this became difficult after a while. Perhaps we could have made this easier by arranging the numbers in a circle … or maybe this is the perfect result, creating a need for systematic counting and organized results.
  • On a similar note: Ms. Harding solved the problem and organized her results in a much different way than I did.
  • One student found all but one of the pairs. When I asked him if he thought he had them all, he said he thought he did, but he wasn’t sure. Exactly!

12. Space Race

On Friday, we shared Play 12. Space Race with our students.

Play 12: Space Race

Here is a sketch of how this problem developed.

1. Find a fun problem.
We met Ben over the summer at a BU discussion workshop, showed him some of our Play problems, and talked about our approach to the project. Later, Ben sent us a problem that he thought would make a good Play problem. The problem was of his own creation, and had been posted in Math Horizons (April, 2011):

April’s Sandbox problem comes from Benjamin Dickman of Columbia University Teachers College. On the first day of math class, thirty-six desks are arranged in a circle for eighteen boys and eighteen girls. The teacher says that students can take turns choosing their own seats as long as the gender of the students alternates between turns.The last two students to pick will be Amanda and Bill (not necessarily in that order).

Bill has a crush on Amanda and is determined to sit next to her. Amanda, on the other hand, can’t stand Bill and is determined not to sit next to him. All the boys want to help Bill, and all the girls want to help Amanda, which sets up Problem 258, You Can’t Please Everyone: which gender should pick first, and why, if the decision is up to Amanda? If the decision is up to Bill?

Ben added two suggestions: 1) that we could choose a smaller number of desks (e.g. 12), and 2) rephrase the context to get rid of the gender elements.

We played around and solved the problem ourselves, and we agreed with Ben – this was a fun problem that our students would enjoy … if we could make it accessible.

2. Re-write the problem so that it is visual and concise.
This problem was wordy, and for a long time, it sat on the shelf as a candidate that wasn’t yet ready to become a Play problem. I took forever to wrap my head around how to adjust this problem, but one day, inspiration struck: it’s a game. Bill and Amanda both want to “win,” but they have different criteria for success. And with that, Amanda and Bill became Player A and Player B. Desks became squares. Students became X’s.The rectangular shape ensured an even number of “desks.” Stripped of the gossip-filled classroom context, it retained the same mechanics and underlying mathematics.

3. Design a draft
I sketched a draft on paper, CiCi (aka Ms. Yu) masterfully brought it to life in InDesign, and we sat down together over February break to tinker and refine.

4. Debate every word
We tried out several different versions of the wording of the problem. Are the squares “boxes” or “spaces”? What should the title be? Are the last two spaces “adjacent” or does that vocabulary need defining in order for the problem to remain maximally accessible? How do we phrase the prompt at the end to show that this is not simply a game, but a problem that can be solved?

5. Design a handout
When we sent the draft back to Ben, he misinterpreted what the X’s in the board meant. The X’s are intended to show a game currently being played. He thought that they represented the starting board, adding a new constraint to his original problem. We made a handout so students could play with the problem, and made sure that the board was blank, hopefully erasing the potential confusion.

Play 12 Handout Page 1

On the back, we initially planned to add a smaller version with only 12 spaces as Ben had originally suggested, but that only takes up so much space. CiCi did some quick copy/pasting/resizing and all of a sudden we had a beautiful variety of playing board options.

Play 12 Handout Page 2

6. Print and give students a chance to play
We posted it on Friday and used it in our classes as a Friday Warm Up. It wasn’t easy to make them stop so I could start class. I encourage you to print out the board and play along with us!