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.