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."