The Color Game

hundred yardsMotivating the students in my Problem Solving class – convincing them that they are capable of thinking critically and crafting solutions, and getting them to care about the quality of the work they submit – this is the challenge I face this term, a challenge with which I am familiar. I’ve been coming at this challenge since I began teaching – trying to motivate through classes in Personal Finance, Geometry, and now under the broad umbrella of problem-solving strategies. I really enjoy the students in these classes – the juxtapositions of attitude and self-deprecation, street smarts versus humble acceptance [sadly] of their mathematical deficits. These teenagers are honest – sometimes hilariously so, and, sometimes painfully. They are always interesting, and unpredictable.

by Sara

by Sara

Most of them don’t believe they can solve problems, and I face the dilemma of demonstrating solutions for them versus getting them to work independently. When I model, the students faithfully copy my work. When I toss the ball back to them, they stop. We use individual whiteboards, large vertical whiteboards around the room (I encourage the students to get up and work on the boards whenever they want), partners strategies, table work; my plan (read: my hope) is that eventually the most reluctant of students will become comfortable enough to dip their toe in the water. The problem sets I am using are well tiered, I think, and there will always be an entry point, somewhere, for everyone.


from Jeremy’s mathography

One of my first assignments of the year was a Mathography, an assignment I love to read. Again, these kids are honest. Unlike my ‘gifted’ students, they do not feel they need to present their best face, and the results are compelling. Sometimes they are just so sad, like Jeremy. And then there is Gabby, who admitted that the teacher she liked the least challenged her the most. There are big red flags that get raised. Rekindling this child’s faith in her education is a tall order, and I hope I am up to the task; I do believe that the beauty of the patterns in math can inspire the disenchanted.

Gabby and her nemesis

Gabby and her nemesis

It strikes me as I am writing this that perhaps I may have a staggering amount of chutzpah, thinking I can ‘save’ these students. But I don’t; I sort of despair at my own powerlessness to change anything. My class is 45 minutes out of a teenager’s day, when they leave my room, when they leave school, everything is more importantthan my math class. I would just like to give a student a sense that they are mathematically capable, that there is interest and value in solving a puzzling problem, just because. Many of these students are given repeated tacit messages that the only value of their high school education is the accumulation of credits required to achieve graduation in four years.   I would be thrilled if the experience they had in my classroom added a little “and yet…” to that message. I hope that doesn’t sound like too low an expectation for my students or me, because the effort can feel both Sisyphean and Herculean. (Sorry – I couldn’t help myself, former English major that I am.)

color gameThis isn’t at all what I meant to write about. I want to write about the Color Game we played today, the game that half the class wanted to ignore, only to end up shouting out answers, arguing with each other, hands shooting up the minute I gave clues. It’s a pretty simple game; I found it in the introduction to the Teacher’s Edition of Crossing the River with Dogs. Students need to prove the location of colored squares on a grid given minimal clues. They are only allowed to claim a square when they can justify that it can only be one color based on the information they have been given. At first, the kids wanted to guess my pattern and have me tell them whether it was correct, and many were annoyed that I wouldn’t consider their offerings. But one by one, they realized that ironclad proof was required, and we played as a whole class right up to the ringing of the bell.   Of course, they begged to play again – frequently and soon.

Week 2 (if only 3 days counts as a week, that is)


I’ve been waiting for this wall to be built for 15 years.

Last night for a wide range of reasons, not the least of which is the layer of construction dust that can seep through plastic sheeting and is covering everything in my house, I had a bout of insomnia, and sometime around 2 am, I resigned myself to the inevitable being exhausted at work is never fun, but being exhausted while teaching five classes of high schools students on a gorgeous warm Friday can be brutally painful.

And yet it wasn’t.

In fact, some parts of the day were downright enjoyable.  What madness be this?

Although we are deep in procedural land in Algebra 2 (polynomial operations), the vast majority of the work on the boards around the room, as well as the explanation of that work, was done by students.  In fact, they spent most of the class time checking with each other, correcting each other, and discussing what order of operations was correct in a given problem.  My ‘hinge question‘ from yesterday’s lesson became today’s warm-up [my timing isn’t quite there yet…], but it led to a lot of fruitful discussion, and uncovered some widespread misconceptions.   The content right now is procedural to the nth degree [ouch! ; ) ] and I want to find some activities to spice things up, even though the loss of class time in September will most definitely create a painful shortfall later in the term.   My departmental curriculum, to which I am fairly strictly bound, unfortunately follows a procedural rather than conceptual route through Algebra 2; adding depth and meaning to the prescribed pacing is always a challenge for me.  That said,  I’m pleased with the classroom culture that has been created thus far by the random groupings and cooperative/supportive table work.

I reworked the Race Around the World activity last weekend; I was not satisfied with my presentation to the students, and hoped that clarifying the process would re-ignite some interest and activity in what I thought was a fun and engaging problem – how fast can you travel around the world given time and air flight restrictions?  I worked through the problem on my own, and redesigned the graphic organizer  to better support the work the students would need to do in order to complete their trip.  I demonstrated how I would proceed in planning andflat-earth-society1 calculating the time required for my trip.  It seemed as if a few more students understood the idea behind the activity, but many remained unmotivated to attempt the task.  I think that in missing the right hook from the outset, not even my revision could reignite the spark.    Very few students finished the task, and we were approaching the fourth day of class.  I decided it was better to move to the more substantive work of the course rather than continue to push this activity.  I’m not happy, but there were lessons learned for me, and for those students who chose to engage.   Next time around…

UntitledBut here’s the great part.
The day following this minor debacle, we began the major work of the class, exploring different problem-solving strategies.  I distributed notebooks (love those mid-July Staples sales!), and we jumped right into solving our first problem: Virtual Basketball bballLeague.  Using the Crossing the River book as a guide, I asked the students to attempt to solve the problem with a diagram.  Clearly, many of them are not familiar with Borg philosophy, and thus thought that resistance was a great idea.

“Why do I have to draw a picture?”

“I don’t like pictures.”i-d96f7f200b1dfc20c32e501406e775ef-LocutusOfBorg
“I just want to do math.”

“Can’t I do it my own way?”

IMG_6320And then some of them started to draw diagrams.  And others looked over at their diagrams and began to draw their own.  And students who hardly ever participated (I had many of these students last year in Geometry) agreed to put their pictures on the board.  And THEN they began to argue.  I refused to tell them what the correct solution was; I kept redirecting them to convince each other.    I thanked all of them for sharingIMG_6321their work, and reiterated several times that mistakes were GREAT, and that anyone who made a mistake publicly provided all of us with the opportunity to learn, and should thus be thanked.  This strategy seemed to appeal to the kids, because I got very little resistance (go Borg!) when I asked individuals to share their sketches.   In fact, I find that students who traditionally don’t do that well in math are more willing to share their work without knowing whether  it is correct than my ‘gifted’ track students.

We spent the entire period – in both sections of the class – talking about the Virtual Basketball League problem.  To be honest, I was surprised (very happily) by the success of the lesson after egg laid by Race Around the World.   When class began today, and I gave out the second problem in the series, only one or two students refused to draw diagrams (and I was able to coax one of those to sketch a visual of their solution after he arrived at an answer ‘mathematically’).    Once again, different solutions to the problem were shared, and discussion ensued.  I noticed that leaving the class completely to their own debate did run the risk of allowing a misconception to reign, and I asked some very pointed questions to draw their attention to the error.  I’m hoping that as we move through the problem sets  the class will uncover these mistakes on their own; I am concerned about ‘over-directing’.   I want to empower these students to find their own solutions, and find them within each other.

We finished the class today with the problem Pool Deck, which goes like this:

imgres-1POOL DECK Curly used a shovel to dig his own swimming pool. He figured he needed a pool because digging it hard work and he could use it to cool off after working on it all day. He also planned to build a rectangular concrete deck around the pool that would be 6 feet wide at all points. The pool is rectangular and measures 14 feet by 40 feet. What is the area of the deck?  

Surprisingly (to me, anyway), many students were mystified.  Stumped.  They struggled to make sense of the problem (I did not give them the photo).  They drew oval pools inside rectangles.  They found surface area; one student found volume.  Many who added on the deck did not subtract the area of the pool.  What threw them?  Was it that they needed to read about Curly (like from Oklahoma?) and his desire to go swimming before getting any pertinent information about the pool?   I thought a rectangular problem would be a piece of cake.  More to think about.

So the week ended really well despite my exhaustion.  And I can’t wait to see what these problem solvers are going to do next week.

[My week began with a lovely visit to my animator child (who is very animated, btw) in Baltimore at MICA. Rather than sharing images from Geo’s tumblr, here are some pics from my trip.]


Geo’s dorm, The Gateway

Love this kid.

Love this kid.

Landscaping at The Paper Moon Diner

Landscaping at The Paper Moon Diner

Seemed like a good start until….

auspciousThe school year started fairly well, I thought, as I reflected this afternoon.  Everyone – students and teachers alike –  was in an enthusiastic mood this afternoon, due in equal parts to the drop in temperature and humidity and the upcoming 4-day weekend.  The name tents and ‘Ms. Menard in Numbers’ were great introductions in my overcrowded Algebra 2 classes, and the Visibly Random Groupings – accomplished by handing out cards as the students enter the room – were actually met with way less resistance than I would have thought.  (I’ve already got kids lining

Instead of a question, this student gave me a math problem on the daily flap on his name tent!

Instead of a question, this student gave me a math problem on the daily flap on his name tent!

up to be the Card Shuffler each day.)  It’s such a simple strategy that I’ve been using it in all five of my classes, and it’s much less work than maintaining 5 seating charts.  I probably won’t mix the seating up every day all term, but I will continue with this procedure daily until the students are completely routinized.

Going over a basic algebra review sheet took more time than I anticipated in my trig classes, but it gave me a chance to see who my most willing participants are.   I wanted to distribute iPads and let the students explore Michael Fenton’s Desmos Bingo sheets, but time only permitted me to give a cursory demonstration of Desmos on the SmartBoard; I assigned the Bingo sheets as an exploration for the long weekend, with Des-Man as an extra credit assignment.  Even though it’s Friday night, I’ve received three messages from students working; the Desmos magic always hooks them!

My cat demonstrates how to make a name tent.

My cat demonstrates how to make a name tent.

In my Problem-Solving classes, we worked on 1-5-2-4-3 yesterday, with predictably high engagement when you give the kids playing cards, and some great empowerment for the quiet students who picked up on the patterns quickly.  I was thrilled to see one student, who warned me on the index card I collected with his contact information that he was ‘shy and slow’, become confident in front of my eyes as he took the challenge to successive levels.  Today I launched the Race Around the World activity somewhat less successfully; the students were confused by the task, and not nearly as enthusiastic as I had hoped.  In retrospect, it would have been more effective, I think, if I had posed the challenge to them as a question, asking “How fast do you think you can get around the world?” rather than distributing a piece of paper.  The students didn’t get very far into the task because the introduction was unclear to them, so I am going to continue with it when we return to school on Wednesday, giving myself and them a bit of a do-over; I will work on a hook for this activity over the weekend.

When I left school, I was feeling fairly positively about these first few days, although concerned that with all of my planning and thought, I had missed what now looks like an obvious opportunity to grab the attention of these students – students who are quite clear on the reality of being programmed into an off-track elective rather than a core math class.  I have had many of these children as students before – either in three term Algebra 1, or my

99 cent store decor

99 cent store decor

Geometry Fundamentals class – and have a good rapport with most, but I know that they are quite ready to give up, regardless of my belief in their ability to learn well in my math class.   So my positive reflection had a little shadow around the edges.

And then, at 11 p.m., I received this message from a student in one of the Problem Solving sections:

Screen Shot 2015-09-11 at 11.16.43 PM

Score!!!  Let the weekend begin.


Lights of Remembrance

T-48 hours

It may not look like it, but the open space in this basement is a miracle.

Okay – Against all odds, and in the midst of total chaos before the demolition begins in my house on Tuesday, I think I am ready!!  I’ve planned the first two weeks,  and am getting pumped – mostly because of the wonderful ideas I’ve gleaned and incorporated from the great big MTBoS.  I hope that SOMETHING I do helps SOMEONE as much as everyone out there helps me.

In Algebra 2, there is a weird schedule during the first week, with some periods  very short, and some classes meeting on different days, so the sequencing of activities may vary.  But the overarching goal is by the end of the week (before the 4-day weekend!), the students have an idea of who I am as a teacher, and that the expectation in my classroom is that they will be thinking and talking about math, rather than being passive receptacles of direct instruction.

Day 1 will be spent gathering student information (names, pronouns, contact info, favorites), making Name Tents (an idea I borrowed from Rachel Rosales three years ago and haven’t tired of yet, and revealing some information about myself using Heather Kohn’s activity – Ms. Menard in Numbers.

On Day 2, after completing a number talk, answering the wonderful question “What is 99 plus anything?” (from the wonderful book Building Powerful Numeracy for Middle and High School Students), we will discuss Growth Mindset and watch this wonderful video from a math major talking about fear (essentially more growth mindset and ‘grit’).

Day 3 – Lots of fun stuff!  The students will sign up for Remind, learn about the Manatee Squish signal (thanks, Darryl!), and play Desmos Bingo, courtesy of Michael Fenton.  And I’ve got a back-up if the wi-fi is down; we will play the Real Numbers Game I used last year (sample lesson designed in grad school before I had any idea what I was doing!).


When we return after the Jewish New Year, we will jump into Polynomials, and I will begin my Hinge Questioning, as I have promised.

Then there is my Problem-Solving class, and I’ve got some great things planned here as well.

On Day 1, in 25 minute periods, the students will also complete information index cards.  I know many of these students (I suspect a whole bunch of them will be coming from last year’s Geometry classes), so they won’t be making Name Tents.  The students will going want to know what the course is about, so we will discuss the many and varied benefits of improving their problem-solving abilities, and then work on a simple problem.  I am debating whether to use the aforementioned “What is 99 plus anything?” number talk (I think this will set a good tone for the class – a simple, accessible question that can lead to interesting mathematical ideas) or the oft-linked Noah’s Ark activity from Fawn Nguyen.  Again, an accessible activity with a strong visual component – I think it will be good for many of the students in the class.

Day 2 – We will discuss mindset and watch the math major video in this class as well, and then dig into our first problem – 1-5-4-2-3.  This problem, from Peter Liljedahl was written about from my new compadre, Lisa Winer.

I know the cards will be an instant hook for these kids, and again, a low floor and a high ceiling make for intense engagement and debate.

On Day 3 the students will share the results of the 1-5-4-2-3 investigation, if we run out of time the day before.  We will go through some of the administrative stuff that I pushed off from Day 1 (I have these classes in the afternoon; there is nothing worse florida-manatee-kings-bay-615than being the 4th or 5th teacher to give out a course syllabus on the same day) – classroom contract, Remind sign-up, and once again, just for funsies, the gentle reminder of the Manatee Squish.  After we’ve gotten the business out of the way, I will introduce A Race Around the World, one of the gems I found on Peter Liljedahl‘s website.  I modified the activity slightly to fit my class.

Over the long weekend, I am going to have the students write a Mathography.  I haven’t used this assignment in a while, and I am looking forward to reading them.  It’s wonderful to be teaching a class without a high stakes exam at its conclusion.


Next week, before I dig into following the Crossing the River with Dogs structure, I am going to borrow from the endlessly inspiring Alex Overwijk.  He has recently written about some incredible multi-day activities in his classroom; I’m thinking that next week’s 3 day week would be perfect for borrowing and adapting his Which One Doesn’t Belong sequence.

And there are more people I have to thank and credit for other strategies I am trying:

Visibly Random Groupings as described by beginnersmindmath

A Google Doc Planner thanks to Jessica, the algebraniac!

Screen Shot 2015-09-06 at 6.52.13 PMFinally, this: As I was writing this post, a brief glance at my Twitter feed showed this:

Glenn Waddell‘s response to the New York Times article about the TeachersPayTeachers website echoed exactly the sentiments I am feeling as I write this post.  My mindset swung from anxiety to enthusiasm today as I sifted through the rich mine of resources that my professional (and personal) community shares; I am completely honored to be included in the list in his post.

Wheels starting to turn

1837452I found out on Tuesday that I would not be teaching Geometry this fall – because of ‘department needs’, I will instead be teaching an elective course for juniors and seniors who need one more math credit and who have not been successful in other classes (along with my Algebra 2/Trig classes).  The structure of the class is completely up to me, which is kind of wonderful, or would be, if school wasn’t starting next week.

I’ve decided to teach this as a problem-solving class, using the structure of the book Crossing the River with Dogs; the book features a different problem-solving strategy in each chapter, strategies which are tiered and accessible.  I met a teacher at PCMI this summer – Evelyn Baracaldo – who developed and taught this class at the NYC iSchool, and generously shared her materials with me.  I probably shouldn’t be putting together YET ANOTHER NEW COURSE, but somehow I can’t resist this one.

The challenge – in the few days remaining before the start of school – is to envision the [realistic] classroom culture I want for this course.  So the ideas are spinning around in my head, giving me more than a bit of anxiety, although I know this is part of my process.  This is what I’m thinking about so far, and I would LOVE FEEDBACK.

  1. The weekly warm-up sheets will be perfect here – estimation180, Which One Doesn’t Belong, visual patterns, Would You Rather?.  I’m thinking that the students will save these in a folder all term, and then do an end of term reflection/assessment of their progress.
  2. This class will also lend itself naturally to Number Talks, which I have been reading about all summer.
  3. Our notebooks may not be full-blown INBs, but they can still have a structure – sections for each strategy (tabs!) with worked problems.
  4. Group work, large whiteboards, vertical non-permanent surfaces – these are modes of working which are excellently appropriate for this content!

Sounds great, right?  But here is what is still unclear to me:

  • How will students be accountable?  With all that group work, how do I ensure that everyone is working?
  • How will students be assessed? (I am responsible for grading them, and for developing a grading policy; this is engrained in how my school functions.)
  • How will I introduce new strategies?  Will we just work on problem sets?  One 3-act per strategy? Math Forum PoWs?

I’d like to NOT reinvent the wheel.  There are so many wonderful resources out there (and in my house and on my computer and in my files at school….); I want to pull together all the applicable nuggets of brilliance that I have already squirreled away and teach those kiddies some math!

I welcome your thoughts –

Asking the right questions

I was thrilled when I received this message from Le Sam Shah last week:

Screen Shot 2015-08-30 at 12.31.43 PM

At PCMI, I set a personal goal of improved formative assessment this year through ‘hinge point questioning’ (see this post), and had been wondering how I would maintain this new practice all year; many September initiatives, despite my best intentions and aspirations, fall by the way side as planning, grading, and college recommendations pile up, and as report cards, observations and parent conferences encroach on the regular day-to-day work.

But what Sam created a collaborative and open space, betterqs, in which teachers can share and reflect on their questioning practices, and by inviting me to participate, provided me with the checks and balances to stick to my goal.  And even better, the blog betterqs is open to anyone who wants to share.  As we all know, the more you participate in the #MTBoS, the greater rewards you reap.  So here are the details:


Hope to see you out there!

It all hinges on the question

hinges-1I had the privilege of attending PCMI earlier this summer, and the further privilege of attending a presentation by Dylan Wiliam on formative assessment.  One of the big takeaways from all of that awesome professional development, and something I have committed to as a personal teaching goal for this coming school year, is the implementation of regular hinge point questioning in my lessons.   In the ‘middle’ of the lesson (or the hinge point), the teacher poses a question which is designed to elicit evidence of understanding of the key ideas [presumably] taught thus far.  The students should be able to answer the question in 2-3 minutes, and the teacher should be able to evaluate the responses and make a decision on how to proceed in class in 30 seconds – whether to continue forward, clarify or re-teach, or remediate in some way (review a requisite skill, for example).   For further clarification or description, watch this video by the Great Dude of Formative Assessment, Dylan Wiliam.

I love this simple and brilliant idea, even though I’m uncertain of its implementation.  My classes are only 45 minutes long.  What happens if the evidence shows the students haven’t gotten the idea at all?  What if half of the students have gleaned the lesson goal?  My favorite idea from the video above is that we should design our lessons with hinges in them, rather than designing them as airport runways, building in feedback loops.  I teach in New York, and always have a Regents exam breathing down my neck.  But I know that this technique will help build better outcomes and deeper understanding for more students if consistently and intelligently used.

I committed this goal to the video archive of PCMI teacher goals (note no link shared here!), and am committing it to the InterBlag; thanks to Sam and Rachel for creating this collaborative space.