Lobster Mobsters (best math game ever!)

Years ago, my colleague Cathy introduced me to a game that she learned from her mother in the UK.  We were pooling our seventh grade math classes together to play a game the last day before a vacation, and she enthusiastically explained this "lobster pots" game that sounded really complicated to me.  I had my doubts, but she insisted it was great.  Then we played, and I was amazed by how good it was.  It became an instant favorite.  It engaged all learners, but was challenging.  It involved higher order math applications and connections, yet allowed students to work on developing number sense, too (keep those calculators away!).

With a few revisions, the game became an excellent "low floor, high ceiling" game capable of inviting in and sustaining all learners.  I have played the game with students as young as second grade, and it would certainly challenge high school students, too.

How to play

Game players are cast as "lobsterpeople" who trap lobsters in "pots" to sell them at the lobster pound.  You can play as a whole class (this is how we prefer it), in small groups or individually. Students can work in a group as a part of a crew or solo.  They start the week with a certain amount of money and a certain number of pots.  Each day, students decide where to place their pots: how many in the harbor and how many off shore?

We've Been Sharma-ed - Mahesh's Visit to Grade 3

It's always exciting (for me, at any rate) to have the opportunity to come together with colleagues to improve our understanding of math and math instruction.  This week we had a special guest, Mahesh Sharma, who came to WCSU for clinical rounds.  Lots of folks were able to attend various "rounds," but since I was able to attend all four rounds, I thought it might be helpful to document and share what went down... 

Monday, December 7 

After some delicious beverages and treats (and I spilled an entire cup of tea all over the lovely spread because obviously, someone needs to have caffeine before leaving home), we started with a discussion of place value...

Ever wonder why the names billion trillion, quadrillion, etc.?  Pictured at left, Mahesh explained that the word "million" comes from mille which means "one" and llion which means "group."  

I was unable to substantiate this (fact checking found that the etymology of million is that mille means "thousand" and  the addition of the suffix -ion changes the meaning to "a great thousand"). 

Fitting It Together: Another Take on Number Visuals

A visual number pattern created by artist and scientist Stephen Von Worley forms the backbone of day two of youcubed.org's week of inspirational math.   

The lesson envisioned by the youcubed team is here and starts with a video of Jo Boaler talking about the importance of "brain crossing" - connecting visuals with symbols - followed by exploration of the number pattern, finishing with summarizing and sharing out.  The whole shebang in one day (possibly followed with an exploration of consecutive number patterns).  Whew!  

I had the opportunity to test this out with various classes ranging in age from 3rd grade to 9th grade, and some adventurous 3rd grade teacher colleagues of mine tried it out with their classes, too.  Low floor high ceiling problems like this have great potential to encourage creativity, build stamina, make and test conjectures, and apply and connect math concepts.  Although we found the youcubed lesson fun, and yes, inspiring, it felt a bit... unwieldy and unfocused.  

We found ourselves asking: How could we use it to hone our conceptual goals?  In this blog post, I am putting forth a possible answer to this question with this overall plan:

An overall instructional plan for Tier 1.  From beginning to end this could span about 2 weeks (more or less depending on how solid and flexible the students' grip on additive reasoning is). 

Low Floor, High Ceiling - Math Coach Attempts Math Art

How often do we (as teachers) hear about something that sounds pretty great, but then we can't picture how it should fit in with all of the other great stuff that we do?  

Last week, I had the opportunity to hear David MacAulay (illustrator and author of visual feasts such as The Way Things Work) speak, and he said something that resonated with me. 

In order to really see something, I need to draw it.

 Well, I'm no David MacAulay, but I thought I'd give it a try... So, for a training session yesterday, I made this poster to explain the difference between high floor, low ceiling and low floor, high ceiling math activities and questions:

Then, I created a visual of what integrating low floor, high ceiling activities into a math class could look like.  

I based this on my own experiences threading these problems into my teaching, but I tried to build it bigger and better.   Then, I made it into a "Thinglink" to add some depth to it. Check it out below (here's a link in case you can't see the "thing").  Each information "i" should be clickable and will provide a little more information about the parts. 

This isn't the only way to do it though.  It's just one way.  How would you improve it?  Thoughts?  Questions?  Comments?  Please send them in... edorsey@u32.org.

Meanwhile, here are some resources I created to analyze and revise our questions and see here for my original post on how to do a low floor, high ceiling makeover.  

Hope you enjoy!

Engaging in a Math Debate

"One train may hide another."

Recently, Lisa, a sixth grade teacher sent me a link to this Amazing 6th Grade Math Work! blog post that she had written up to document a math debate that she released into her classroom around the difference between ratios and rates.  It's definitely worth reading her post, but if you don't have time, here's a synopsis...

Lisa began by admitting that, although she knew the official definitions of "ratio" and "rate," she wasn't sure what the practical difference was.  So, she put that out there and had her students research the terms, debate the differences and come up with a working definition that they summarized in a table.

After she sent this to me, I had some thoughts, too, and then the two of us debated back and forth over email a bit (for those of you who are riveted by such things, I'll paste these in at the end of this post).  Then, Lisa brought these ideas back to her students.

Making Number Talks Happen

I mentioned in my post Does Automaticity = Timed Tests? that number talks are a great tool for building number sense.  Number talks can be used regularly to engage our students in mental math that puts the focus on developing efficient strategies.

In the past month, I've had the opportunity to facilitate number talks in a few different schools with teachers and with students using the visual cluster of dots shown to the right here.  I am excited to report that teachers have been having success using this strategy with their classes.  In this post, I'll share some general guidelines (nuts and bolts) for facilitating a number talk, and share one teacher's experience using them with her third grade class. 

Math Practices vs. Habits of Mind vs. Positive Norms

The term "math habits of mind" was first used by Cuoco, Goldenburg and Mark in a great 1996 article (linked here).  The habits of mind that they envisioned are process and disposition standards rather than content standards.  The general idea was that the content of mathematics will evolve, so we want to give students "a set of tools that they'll need to use, understand and make mathematics that doesn't yet exist" (Cuoco, Goldenburg, & Mark, p. 2).  The habits of mind are the tools they proposed.

The Common Core Standards for Mathematical Practice (also know as the 8 Math Practices) were influenced by the habits of mind as well as the NCTM problem solving standards and the National Research Council's 2001 Report "Adding it up" (linked here)  Like the habits of mind, the Math Practices address thinking processes, and dispositions that help students develop "deep, flexible, and enduring understanding of mathematics" (Briars, Mills, & Mitchell, 2011, p. 20).

WCSU Non-negotiables vs. Common Core in the current state

A few years back, the WCSU Math Steering Committee spent a significant amount of time and effort articulating student achievement non-negotiable skills for each grade level.  The driving idea is that the Common Core States Standards, although significantly more focused and coherent than past standards, are still too wide. To focus our curricula, we needed to pick skills (informed by CCSS) at each grade level that all students must be able to know, understand and do at the deepest (or highest, depending on your perspective) levels of math knowing.  Monitoring student progress in relation to these skills forms the backbone of our multi-tiered system of support for math.

After we articulated those skills and began to use them to guide our curricula we saw that it wasn't clear what it meant to be able to know, understand and do those skills.  So, we had to define levels of knowing for each non-negotiable.

First, we developed a general rubric (click here) to guide this work.  Then we began applying the rubric to each of our non-negotiables.  It took us an entire day to do this for just the kindergarten non-negotiables.  The process is arduous, but very enriching.  Completing this work for all grade levels k-9 is the Math Steering Committee's top priority right now.

Untangling Assessments: What are they all for?

At WCSU, we use the assessment terms and definitions outlined in the Vermont Multi-tiered System of Supports Response to Intervention and Instruction (MTSS-RtII) Field Guide. The goal is to provide a supervisory union wide "balanced and comprehensive assessment system" that:

• Identifies students who require a closer look (screening); 
• Investigates and analyzes learning difficulties (diagnostic); 
• Informs "core" instruction (formative); 
• Monitors progress (progress monitoring); and 
• Verifies learning over time (summative).

Various common assessment tools are used to provide a full picture of students’ academic and/or behavioral knowledge, abilities, and dispositions. This is the point in developing the WCSU Local Common Assessment System, which is admittedly a work in progress. It's important to note that currently we are at the beginning of this process, especially in mathematics.

When we look at how math is being assessed across WCSU, we see a host of assessments. Not all of these assessments are common across the supervisory union. Not all of them are aligned with our non-negotiables. As we continue to move forward, we will be replacing and revising these assessments to create more cohesion and alignment.

The aim of this blog post is to take a look at some of the assessments that we are using right now and to provide a clear message for those of us who use them. What are these assessments? How should they be used? How should they influence our instruction? How should they influence grades and reporting?

How to do a Low Floor High Ceiling makeover...

There is a loft at the top of my house.  Only the limber and sure footed can climb the rickety ladder to get up to it.  Once there, only the very small in stature can stand or sit up comfortably.  Pretty much, the only people who go up there are children between the ages of 5 and 14.  Every so often, I go up there with the vacuum hose, but I almost never climb all the way in...   

This space is a classic high floor, low ceiling space - only serving the needs of a narrow group.  Other areas of our house have low floors and high ceilings (relatively speaking).  Those are the rooms in which we spend most of our time together.

In math, there are high floor, low ceiling problems.  In fact, if we look with a critical eye, we might see that most of the math we do is high floor, low ceiling.  The issue with these problems is that very few of our students can approach them independently and those students finish them in no time.  So we have to have different kids working on different problems, and none of them are independent for long.  The result is a heck of a lot of management and more paper work than any teacher can reasonably keep up with.

Several years ago, I became very interested in low floor, high ceiling problems.  I noticed that certain problems just worked with everyone.  Rather than having a class fragmented by ability, they allowed me to have a class that was individualized but unified.  Students were independent enough that I could work with individuals or small groups based on formative assessment.  And these problems had legs... even my "speediest" students could be coaxed to see these problems as an opportunity to enter the sandbox and play.  Low floor, high ceiling problems encourage stamina and creativity.

Is it okay to use a textbook?

My first year of teaching at U-32, I taught Algebra 1, Algebra 2, and IMP (Interactive Mathematics Program).  Since I began teaching without ever having taken a math education class, I was unclear about the practical differences between Algebra 1 and 2, and the math standards that we were using at the time didn't specify them.  We had no written curriculum, and the textbooks were museum relics.  I spent a lot of time creating resources.  I'm not sure how coherent they were...  IMP, on the other hand, was a program.  It was easy to follow.  The focus and direction were clear to me.  It was research-based.  I trusted it.  It felt purposeful, good.

Why am I attending this PD? I don't teach math.

One of the recommendations that came out of the WCSU Comprehensive Math Review was:

If the district concludes that teachers have a difficult time delivering high level rigorous mathematical content, we advise the district to investigate the use of “math specialist”/compartmentalized mathematics teachers, such as what is done in middle and high school (National Mathematics Advisory Panel, 2008). The district may need to investigate the feasibility of such compartmentalization across all grade levels.  

We haven't moved to a math specialist model everywhere, but for various reasons more and more teachers are finding that they aren't "the math teacher" for their students.  It is a valid and natural question to ask why those teachers still need to attend "math PD."

Loree and Mahesh don't completely mesh...

For those of you who are new to the scene, please allow me to introduce you to Loree and Mahesh...

Loree Silvis is a former primary-grade teacher who co-led the development of the PNOA and has consulted on math instruction and assessment with elementary teachers in WCSU in the past.  

Mahesh Sharma is a former Professor of Mathematics who is known for his work on math learning problems and has an ongoing relationship providing professional development for math instruction for all grade levels in WCSU.

Does "automaticity" = timed tests?

Bring up timed tests for math facts at a dinner party or on a soccer sideline and you're likely to start a debate.  There are those who feel that timed tests put the focus on rote learning which is obsolete in this day and age in which we can rely on calculators and computers for these calculations.  Others laud the timed tests as they are seen as taking us back to the basics in a good way.

Last year, my own third grade daughter was time tested each week.  On Fridays, she was given a table of ten facts to complete within 20 seconds.  She "passed" by completing the test correctly on time for two weeks in a row.  Then, she would move on to a new set of facts.  As I lamented in my "Coverage vs. Mastery" blog post, I feel that this practice emphasized speed over depth for her.  She memorized to pass, but I was not impressed by the depth of her understanding (granted, I'm a hard case).  The timed tests had a much more demoralizing effect on a friend's son who would make himself physically ill on Thursday nights anticipating these tests the next morning.

Sorting Out the Messages

Recently, I went with a group of folks from WCSU to a training facilitated by Loree Silvis called PNOA* and Common Core.  Sitting in this workshop, I encountered ideas that pulled with my current practice and others that pulled away from my current practice.  As a result, I experienced some internal conflict.

I find that I often feel this way when I attend workshops and receive different messages about math instruction, curriculum and assessment.  It takes me some time and thought to sort it out, and I know that I am not unique in this regard.  Recently, I have had conversations with teachers that highlight the pervasiveness of this feeling.  Teachers perceive mixed messages about what to teach and how to teach it.  The messages come from a variety of sources, and teachers feel pulled in different directions.  They want focus, clarity, and communication.  

To address this, Jen Miller-Arsenault, Bill Kimball and I are collaborating with the goal of unpacking and reconciling some of the conflict and to provide WCSU teachers with a consistent message about math curriculum, instruction and assessment.

We will address the mixed messages in separate posts that are linked below.  For the in-service on October 13, we will spend some time "browsing" the posts.  To honor your individuality, we'll use this menu differentiation strategy (that I've used with homework and practice activities in class).


Main Dishes (Read and comment on each of these posts.):

• WCSU Math Non-negotiables vs. the Common Core States' Standards
• Untangling Assessments: What are they all for?

Side Dishes (Read and comment on at least one of these posts.)

• Math practices vs. Habits of Mind vs. Positive Math Classroom Norms
• Is it okay to use a textbook?
• Why am I attending this PD?  I don't teach math.

Desserts (Yummy, but optional.  You can put these in your doggie bag for later, if you want.)

• Loree and Mahesh don't completely mesh...
• Does "automaticity" = timed tests?
• How to do a Low Floor High Ceiling makeover...

If you have another mixed message you'd like addressed, please write it in as comment below on this post or send me an email (edorsey@u32.org).  



*"PNOA" stands for Primary Number and Operations Assessment.  It is an interview format assessment that was developed by a team of Vermont educators as a multipurpose tool for grades k-2.

Teaching Place Value: Keep Calm and Transmogrify!

Earlier this week, I received the following email from a colleague:

I am looking to teach place value and its pattern (multiply by ten every time you move to the left, divide as you move to the right)... I need to make it hands on and am not sure how to do it, as you can only show from ones to tens to hundreds. Beyond that it's too big... 

I immediately thought of a slight of hand that I have used with my students to teach place value:  the Transmogrifier invented by Calvin (A.K.A. Bill Watterson).  It was a "machine" that Calvin invented that could vary the appearance of a thing, without altering its relationship with other things.  For example, it could change Calvin into a bug, but the bug would still interact with the world as Calvin would.  In other words, the Transmogrifier made Calvin's appearance variable.

Here is a video I made to explain how I have used this idea to teach place value:

I might also play them a video like this so they get an idea of the real magnitude we're trying to convey:

So remember...

And here's the link if you want to buy extra 1000 cubes.  Or you and/or your group can try this flat pattern and have them build the 10 by 10 by 10 cubes.

Any thoughts?

Math Mindsets

Over April vacation in 2011, I read a book called Mindset: The New Psychology of Success by Stanford professor and developmental psychologist Carol Dweck.  It shook my foundation.  I re-examined my teaching, my parenting and my self identity.  Probably many of you are familiar with the book which (in a nutshell) states that:

• Intelligence is not fixed, and the brain is strengthened through mistakes and challenges.
• Our self beliefs affect the way our brains work.
• It is possible to change our self beliefs (and therefore make our brains work better).

When I returned to my classroom after vacation, I put all of my plans on hold.  Fostering growth mindset became my first priority.  I developed this lesson for my seventh grade students: Ellen's Mindset Lesson.  My stated instructional goal was this:

The most important thing for me to teach you is that you are capable of learning math at deep, deep levels if you open yourself to mistakes and challenges. 

A Math Idea Worth Checking Out

I've been meaning to post this for a while, ever since I was visiting Heather Robitaille's fifth grade classroom last fall at Rumney.  As an ongoing math thread in her class, Heather has her students use checkbooks use to track various debit and credits that she devises for them.  In her own words, here's how she works it:

I give the students deposits each day and throw done withdrawals in there as well.  For example, students have to rent their chair...  If I rent it to them for $3.75 a day, then how much would they be charged for the week?  Some students decide to do a daily withdrawal, while others pay the sum up front for the week.  At the end of the week students have to have a certain balance.  If their balance is off, then they go back and find their mistake and correct it just like you and I would have to do.  I really try to make this very realistic for the kids. Adding and subtracting decimals with real life skill.

Math Steering Update: Making Sense of Mastery

Greetings, math nerds!  I hope you have all had a fabulous summer... Last spring, on the WCSU Math Coaching Blog, I left off with a discussion of mastery and number concept.  Obviously, we want mastery, but what does it mean?  What does it look like?  How do we teach for it?  The math non-negotiables give us a sense of what will be "covered," but to focus on mastery, we need more...


This summer the WCSU Math Steering Committee had the opportunity to spend a few days working collaboratively on a document that (we hope!) will be an excellent resource supporting WCSU math instruction.  We worked together on the format and content for kindergarten and first grade.  I will be working with individual Math Steering folks to fill in other grade levels as soon as possible starting with 5th and 6th grade (because much of the work was done last year as a focus of math coaching, but just needs to be curated).

Number Concept, Number Sense & Numeracy

In my last blog post, I posted a video of Dave Willard explaining that taking the time to focus on mastery of number sense made grade level concepts more accessible. However, it occurs to me that it might not be clear what "number sense" is. We have various terms that we may throw around: number concept, number sense, numeracy. But what do these terms really mean? And how do they relate to mastering grade level concepts? Let's unpack these terms a bit...

Coverage vs. Mastery: Slowing Down with Dave Willard

A couple of weeks back, we had the opportunity to come together around math instruction at East Montpelier Elementary School for clinical rounds with Mahesh Sharma.  Mahesh worked with a group of fourth grade students who are taught math by Anne Carter and Dave Willard.  After working with the students, Mahesh remarked how well they did.  As I watched the class, I was seated next to a fifth grade teacher who was also favorably impressed.  Dave and Anne spent a little time with us afterwards reflecting on the changes that they made to their math instruction this year.  These changes were inspired by the training they received together last summer at "Lab School" - a week long course focusing on math instruction that was taught by Mahesh. 

Dave was adventurous enough to agree to let me capture some of his thoughts and ideas on the video embedded above (if you can't see the video, here is a link).  He's very articulate (even under duress), so be sure to watch Dave discuss how he changed from a "coverage" mindset to a "mastery" mindset this year.  And he feels that this change has made his students stronger math learners. 

Calais Unplugged

I spent this past week in lovely Calais. While we weren't literally unplugged, connectivity issues lent a bit of a "post-apocalyptic" feel to the first half of the week (minus the mayhem - everyone was making the best of it).  No Google Drive access, no internet, no network, no printing, no cell phone coverage, etc...  

Since my essential tools for working as I travel from school to school are my Chromebook, iPhone and a set of manipulatives, this was a challenge.  It forced us to go "old school," and highlighted the power of good manipulatives - especially when there's no power!

By the end of the week, all systems (with the exception of cell service) were back up and running, and there is a lot of good stuff to share out...

Tech Assisted Communication and Application of Division

When I left EMES, fifth grade students were working to get to the communications level on long division (blog post: Math Coaching at EMES: The Sequel).  As you would expect, students were in different places with some ready to communicate and apply and others needing more conceptual work.

Tech Assisted Communication

Students have been using videos to practice the communication piece and document their learning as they became ready.  Fearlessly, one teacher used a cell phone to record the videos and enlisted the media specialist/technology guru to upload it to Teachertube.

Here is one student's video.  Kindly, she was willing to share it with all of us.  I love it for so many reasons, but especially because of how proud of herself she looks.  If you can't see the video below, click on this link: http://bit.ly/1GrotF5.

Meanwhile, what to do with the students who have the communication piece down???

Math Coaching at EMES: The Sequel

Ellen Shedd's freshly renovated room at EMES is to die for.

A cancelation left me with an open week, so we carved out some time to work with the fifth grade teachers at EMES since our last attempt at a coaching week was foiled by snowfall and programming conflicts.  I had a blast (love those fifth graders!).

A couple of things I really love about EMES: the "Acts of Kindness" announcements and the gorgeous classrooms (see left).

Focus

We focused entirely on fifth grade this time.  As readers may recall a couple of weeks back, I spent a couple of days at EMES and the focus of the fifth grade was division (Division, Order of Operations and Ugly Fraction Problem... post).

This time, our goal was to move students from the intuitive to concrete/pictorial to abstract/symbolic understanding of division using the standard algorithm efficiently.  We had three days (Monday, Wednesday and Friday) to spend.  For each of the three days that I was there, our schedule was the same:

7:45 - 9:00 -     I was available for planning/debriefing.  
9:00 - 10:00 -   I worked in one fifth grade class.
10:00 - 10:30 - The teacher and I debriefed.
10:30 - 12:45 - I was available for meetings.
12:45 - 1:45 -   I worked in the other fifth grade class.
1:45 - 2:15 -     The teacher and I debriefed.

The focus of our collaborative time was looking at the evidence of student understanding collected using formative assessments to decide how to mold our instruction to improve understanding.

Math Coaching at U-32 and Dealing with Apathy for Deep Math

I was back at U-32 this week, which made me feel nostalgic.  Much has changed in middle school math from last year to now: heterogeneous grouping, math on core, a few new babies, a new schedule...  but I am appreciative that our administrators and superstar representatives on the scheduling committee (that's right, CG!) have made it so that the middle level math teachers still have common planning time.  We were able to pull together a coaching schedule that involved planning, modeling, debriefing, data analysis, and more planning. All of this on a (somewhat) half-time coaching schedule!

Focus

The prior week, the middle school teachers and I sat down on at our Thursday math huddle to map out our week of math coaching.  After batting around a few ideas, we decided to focus on seventh grade content and instruction (with our next week of math coaching focusing on eighth grade content).  The content would be operations on fractions starting with multiplication.  The modeling would involve using the "Square 1" concrete area of a rectangle.  The focus of our collaboration would be using formative assessment to inform instruction because we know that this is something we all need to do better.

Division, Order of Operations and an Ugly Fraction Model! Math Coaching at EMES and Beyond

Well... A snow day on Monday, in-service on Friday and Fitness on Thursday gave us only two days to focus on math coaching this week at East Montpelier.  So, we did the best we could with the time we had.

Goals

Fifth grade teachers, Robin and Ellen S. were looking to work on teaching long division using the area of a rectangle model so that they can move onto Divisibility, Least Common Multiple and Greatest Common Factor.  Our math coaching work this fall focused on models for multiplication and using the area of a rectangle to model the partial product method and the standard algorithm for multiplication.

Fractions Galore! The Week of January 26 - 30: Math Coaching at Doty

I had so much fun at Doty this past week working with Lisa (who teaches 6th grade math) and Sonya (who teaches 5th grade math) and their students.  Thanks to their careful questioning, I had a couple of "Eureka!" moments.

Right after Thanksgiving, we combined the 5th and 6th graders to teach the area of a rectangle model for multiplying fractions using the instructional plan:

• Outlined here.
• Illustrated in this blog post.

Since then, Lisa continued on to generalizing a conjecture for multiplying any two fractions and worked on developing contexts for multiplying fractions.  Then she moved on to teach multiplying mixed numbers, which I demonstrate in this video:

From there, she taught division of fractions and mixed numbers.

Multiplication of Fractions (And January's Math Practice of the Month!)

In this "two-for-one" post, I am sharing some videos that I have been experimenting with to assist WCSU teachers as they plan their instruction around multiplication of fractions while simultaneously emphasizing CCSS Math Practice 3 (illustrated at left).

Math Practice 3 states that students must be able to construct viable arguments and critique the reasoning of others.

As you watch the videos, think about how students are being asked to present their arguments and analyze the reasoning of others.

What instructional practices foster growth of this practice?

About the lesson...

Typically, this is a fifth grade concept, however, if students do not have deep conceptual understanding that they can communicate using an area model, we need to "go back to square 1."  For seventh grade students, an excellent place to revisit this concept is between studying probability of compound events and ratios and proportions.