Wednesday, December 5, 2018

Discourse, Clarity, and Being Intentionally Vague

Problem-based learning is a great way to develop students' conceptual understanding while at the same time providing them with real-world contexts in which to apply the math they are learning. While it's tempting to rush through the early stages of the problem-solving process and get into students solving the problem independently so we can get to the mathematical discourse we want to engage our students in, I'd like to encourage you to slow down and invest the class time needed so that every student has a clear understanding of what the problem is that they are solving.

Being Intentionally Vague
A three reads strategy like those employed by Ready Math is a great place to start the problem-solving process. Such methods provide students with different lenses through which to read and understand a problem. Stanford has also published a report in which it advocates for the use of such practices to support English Learners.

The challenge that many teachers I work with have is that students often jump right into grabbing the numbers and using whichever operation the unit's focus is on and doing the calculation. Jumping the gun like this surely doesn't help students develop the mathematical concepts we want them to develop through the problem-based learning in our classrooms. So how do we prevent this form of answer getting that our students have developed?

Being intentionally vague by incorporating numberless word problem concepts and practices into your problem-solving activity is one way to begin shifting the culture in your classroom to meaning-making from answer getting. Block out the problem's question and important information ahead of time not only eliminates the infamous number grabbing, but it also shifts the focus to developing an understanding of the context of the problem. By doing a close read of a numberless problem, all students can begin to develop a movie of the problem in their head. Who is this problem about? What are they doing? Where are they? These are key questions for all students to have a grasp of if they are eventually going to capture the mathematical concept of the problem.

Once the context of the problem is established, the class can then begin to discuss possible "mysteries" that this story might incorporate. What might happen to the characters? What might they be doing? By anticipating the possible mysteries of the problem, we draw students into the narrative of the problem and at the same time, they develop mental images of the mathematical actions/concepts they are learning.

Clarity Applied
Too often when students rush through making sense of the problem, it leaves some of the students behind while others sprint ahead and finish early. When we rush through understanding the problem we tell students that math is about answer-getting rather than applying concepts or mathematical understanding. High achieving students never develop the discipline needed to identify and think through a problem, while at-risk learners don't have a place to start because they don't understand enough of the problem to apply their developing thinking.

When students have a clear understanding of the problem they can at least begin to engage in the problem using their current mathematical understanding and skill. This is where productive struggle happens! While the skills and strategies they employ to solve the problem may not be elegant or sophisticated, they are their own. By starting with what they understand, students are then able to connect their developing thinking to increasingly sophisticated and efficient ways of thinking about the mathematical concepts inherent within the problem.

Discourse: It's Where the Learning Happens
We often think that mathematical learning happens while students are solving the problem and therefore it is necessary to remediate and support students during the independent work time. This is not true. The independent work time is where productive struggle begins - it's where students apply their understanding in a way that makes sense to them.

Instead, the real growth and development of efficient mathematical thinking happens during discourse. The presenting of student work samples, analysis of those samples, and connecting between samples is how the community of learners collectively supports one another in moving the mathematical thinking forward towards efficiency. Through this process, students connect their developing thinking with the thinking of their classmates.

By starting out slow and being intentionally vague, we invite students into meaning-making while we unveil new and important information about the problem we're exploring together. When students have a clear understanding of the problem, they can apply their understanding, and therefore are able to join the "math party" as a valued contributor. Slowing down in this manner makes the discourse, and therefore the learning, all the more rich and beneficial for all students.

Saturday, December 1, 2018

Book Report: Great by Choice

Great organizations have a recipe for their success and that recipe doesn't change much over time. In fact, when times are hard, rather than looking to make a change to the recipe, great organizations first look internally to see if the organization needs to recalibrate and recommit to the systems, beliefs, and practices that made them great in the first place. This is my big takeaway from Jim CollinsGreat by Choice, which I recently listened to on Audible.

How does the idea of committing to a recipe for success relate to education? Think of all the initiatives that you've experienced in your career. Does it seem like every year (quarter or month!?!) someone has something amazing that they're bringing back from a conference, webinar, or article and wanting everyone to jump on board? This constant barrage of initiatives is exhausting and can contribute to teacher burn-out.

Instead of constant change, Collins talks about the idea of a 20-mile march: a consistent, informed, and reasonable effort in the same direction. Rather than fits and starts of action, a steady and common pace is established and worked through over time.

So what is your recipe for success? What are you and your school or district committed to? What do you know works and will have the greatest impact on student learning? For my work, I stay focused on just a few things:

  1. The development of a learning-centered community with commonly understood and developed rules, rituals, and routines are committed to by all.
  2. The implementation of a common Standards aligned curriculum.
  3. Instruction that is both cognitively demanding, inclusive, and engaging.
  4. Professional Learning Communities wherein classroom-based assessments are agreed upon and the results are analyzed together in order to adjust and improve instructional practice.
  5. Clearly communicated learning objectives and criteria for success.
Before adding or changing any of these, I first look into the following:
  1. Has everyone bought into the culture of the classroom and is everyone a part of the community?
  2. Are we using the curriculum according to its design and intent?
  3. Is our instruction engaging, inclusive, and cognitively demanding?
  4. Are our assessments calibrated to our learning objectives? Do we understand the objectives of the lessons?
  5. Do students know what success looks like for any given lesson?
Knowing your recipe for success, giving it time to develop, and recalibrating to it when things start to go south is foundational to the health of any organization and contributes to its long-term success. I hope you'll take some time to identify your district or school's recipe for success and find ways to further develop each component - go deeper with your commitment to the foundational beliefs and practices of your organization for the benefit of your students and community!

Saturday, November 24, 2018

What am I doing in Your Classroom?

One of my goals for the 2018-2019 school year is to conduct regular learning walks. During these learning walks, I'm gaining a general idea of some of the common successes and struggles teachers are having in the implementation of our new math curriculum. These trends help me to plan for in-person professional development, newsletters topics, screencasts, and coaching opportunities when they arise.

I also leave a note for teachers at the end of my learning walk. The note contains some of my observations while I was in the classroom. I don't leave any sort of evaluation of what I see. Instead, my note states what I saw the teacher and/or students doing and why what I saw mattered. For example, I might write something like, "When your students weren't able to respond to your prompt, you didn't rescue them. Instead, you rephrased the question and asked students to turn and talk about what they knew or didn't know. By doing this you supported productive struggle and communicated to your students that you believe in them - you said they were capable of doing hard things."

My district has an emphasis on increasing student engagement and we've relied heavily on the work of John Antonetti, and James Garver. When I'm in a classroom, I start by looking for the engaging qualities found in their Powerful Task Design Rubric. By using this common language in my notes, I help teachers connect the professional development we've already done with the rubric to what they're doing in the classroom.

Achieve the Core's Instructional Practice Guide, especially Core Actions Two and Three, has also been helpful in finding language that reflects instruction that is aligned to the Shifts of College and Career Standards. Core Action Two reflects intentional teacher moves that support productive struggle and mathematical meaning-making that our curriculum is built on. Core Action Three reflects the Standards for Mathematical Practice and when I see evidence of Core Action Three in the classroom, it indicates that the teacher and students are engaging math as a meaningful discipline rather than a set of rules and facts to be memorized.

These two tools have been helpful in helping me to narrow my focus during the learning walks while at the same time helping me to connect what teachers are doing in the classroom to district initiatives.

Saturday, September 9, 2017

Math Curriculum Adoption

This year my school district is going through the process of adopting a new K-5 Math Curriculum. Our goal is to run an objective and transparent adoption process with a diverse and representative group of educators and community members.

Rather that reviewing curriculum through the lens of which one we like the best - through our own bias - our goal is that we implement a curriculum that aligns to broad research based and standard focused criteria that includes:

We also want to ensure that our adoption team is diverse and representative in its makeup. Aside from having representatives from each of our 10 elementary schools and balance across grade levels, we also sought to include Special Education teachers and community members. It was also important for us to include teachers who have had past experience with our District Math Leadership Team as well as those who have not. Our thinking in this decision was that while we value the experience of our teacher leaders within the realm of math - aka "The Math People" - we also believe that people who have not been as involved in math leadership could provide us with invaluable perspectives throughout this process. We also sought to diversify the team in terms of race, gender, teaching experience, and language.

In gathering this team, I created a Google Form for teachers to apply for the team and provide some background information about themselves and why they want to be on the team. We then used these applications to create our team.

With our goals established, we then sought an adoption process that would assist us in meeting our goals. With the help of a two day seminar hosted by the Washington State Office of the Superintendent of Public Instruction (OSPI), we were able to build our process off of their four research based practices for curriculum adoption. These four hallmarks are:

  1. Identify Parameters - Who are your stakeholders, what are their needs, what are your goals, and what constraints do you have?
  2. Analyze Existing Course Design - What results are you currently getting? Is there a need for revision or replacement?
  3. Select Course Materials - Which curriculum will you select?
  4. Implement Course and Determine Effectiveness - What are your short-term and long-term goals? How will you know if you're achieving them? What support will be provided to achieve those goals?
In future posts I'll flesh out these four practices in more detail as well as provide a description of how we're living the practices out.

Thursday, January 19, 2017

Number Talks Week

My phone rang. It was Robert, one of my district's Instructional Coaches. He also happens to be our resident Number Talks guru.

"You know how there's a 'Shark Week?'" Robert asked. "What if we did something like that, but with Number Talks? Kind of like a 'Number Talks Shark Week?'" Thus began our Number Talks campaign in the fall of 2016.

Going with the Goers
We designed a Google Form that we sent to the Elementary Teachers in our district asking for volunteers. Aside from the usual logistical information, we also wanted to know how often people were using Number Talks and what kind of Number Talk they wanted to observe.

Divide and Inspire
As the replies rolled in, Robert, Robin (another Instructional Coach we asked to help with our venture) and I gathered and divided the respondents amongst ourselves. While building our cohort, we considered the following:
  • Relationships - Who do we already have positive working relationships with?
  • Location - Are there a group of teachers at a building that could support each other when we're not there?
  • Grade level - Which grade level(s) do we feel we have a level of expertise in?
  • Availability - What does the rest of our day/week look like?

Once our cohorts were created, we began emailing them to set up specific times and to clarify our focus for the Number Talk.

Not a Sales Pitch
It was important to us that our model lessons were not just a "song and dance" - a sales pitch where we try to give teachers "a taste" of our "product." After all, these teachers showed that they value Number Talks by signing up - they'd already bought in. Rather, our goal was to partner with teachers in deepening their use of Number Talks. Therefore, our format for the model lessons included a "Pre-brief," a classroom demo with students, and a debrief session.

Our pre-lesson discussion focused on our own planning of the Number Talk. As we walked teachers through our planning sheets, we discussed the lesson arc, our prescribed teacher moves and questions, and what we anticipated students would say or do. We also asked teachers to record notes using  Achieve the Core's Coaching Tool. Rather than just checking the descriptors on the scale, we asked that teachers look for evidence that students demonstrated Key Indicators A-C on Core Action 3.

During the model lesson, we asked that teachers just be an observer - that they not redirect student behavior, ask questions, or make comments. We wanted them to notice the instructional moves we'd already talked about and wonder about the decisions we made that were outside of the script.

Afterwards, the teacher and coach left the classroom to debrief the lesson. The coach talked through any rationale behind the unscripted instructional decisions, pivotal points within the Number Talk, or any other observations and wonderings. There was also time for the teacher to share his or her observations and questions.

When it was possible, other Instructional Coaches or Principals covered classes so we could meet with the teacher immediately before and after the lesson. Other times, we were able to meet with the teacher during his or her prep time, or before/after school.

Following Up
We sent a follow up Google Form to see how our efforts impacted teacher practice. We only had about half of the people who took part in Number Talks Week reply, but two thirds of respondents said they are using Number Talks more since the model lesson.

Teachers also responded that they noticed increased confidence, communication, and making connections during Number Talks.

Three teachers said they'd like us to follow up with them in person.

We modeled about 40 Number Talks between the three of us over 5 days and it was a ton of fun! There's so many stories that could be told from this effort. One of my favorite stories was in a Kindergarten classroom where the teacher told me that one young man only speaks Spanish and his first contribution to ANY lesson had been during our Number Talk!

Despite all the good, there's some things I'd do differently if I had to do it over again.
  1. Get subs - in the future, I'd like to have roaming subs to hangout with us so they could cover the classes for the pre and debriefs. Relying on already busy people made these critical conversations difficult.
  2. Following up - this is where most of the revisions would be. I think the survey was good - we got lots of informative feedback and data. But, I also think a personal reaching out would have elicited further coaching conversations. Teachers are busy and a form is so depersonalized. I think that the personal, "Hey! I'm still here and am in your corner! How can I help you" would have helped to continue the work.
Number Talks Gatherings
Another layer we added to Number Talks Week was an opportunity for teachers from around our district to get together, hear a short talk from us on an aspect of Number Talks (for the first one we chose to focus on planning), and then give teachers time to plan, create, collaborate, or gather materials. This was the first of our Number Talks Gatherings and we plan on hosting them once a month for the remainder of the school year. (I've written about the second Gathering where our focus was on assessment).

Monday, January 16, 2017

Number Talks Gatherings: Planning

Number Talks are a powerful instructional routine to facilitate mathematical discourse and discovery. Because they are naturally differentiated, they also provide entry points for a variety of learners.

My colleague Robert Hansel and I started a monthly gathering to provide a time and space in which teachers could share their experiences and expertise with Number Talks in order to deepen each other's own understanding and implementation. These Number Talks Gatherings are a monthly event held after school. Each Gathering has a 30 min. "focus lesson" where one of us shares some aspect of Number Talks. The rest of the time is for teachers to collaboratively plan, problem solve, or gather resources.

Our first Gathering's focus was on planning for a Number Talk.

Selecting a Problem Set
The Achieve the Core Focus Documents give teachers a snapshot of what their grade's major work is. They also communicate expected fluencies for each grade. Choosing Number Talks problem sets and strategies that reflect the major work of your grade is how we encourage teachers to begin.

To better help teachers understand the math at their grade, and therefore select appropriate problems, we also encourage teachers to dig into the Progression Documents (specifically those for the Counting and Cardinality, Operations and Algebraic Thinking, and Numbers and Operations in Base Ten Domains).

Once teachers have identified the major work at their grade and reflected on how that work is developed, they are better equipped to select Number Talks that will be meaningful for their students.

Reflecting on what students may say is a key step in preparing for a Number Talk. It's where our understanding of the math inherent within the NumberTalk and our knowledge of our students' (see Component 1b Danielson's Framework) converge. Through anticipating our students' responses to the math at hand, we begin to tie where students are in their own development of mathematical concepts and skills to their possible future selves. We begin to link mathematical concepts and representations along the road to efficiency. Anticipation is where the hard work of facilitating mathematical discourse begins.

During the anticipation stage of planning is where I've also begun integrating visual representations into my planning process. Jo Boaler and Cathy Williams, in conference I attended, pointed to a wealth of research supporting the use of visuals in maths class. So, when I write out a possible student strategy for solving a problem, I also demonstrate their thinking through a visual of some sort. For addition and subtraction problems, I use an open number line, while I use an array for multiplication and division problems.

If anticipation is where the hard work of facilitating discourse begins, then questioning is anticipation's natural outflow. Typically the questions, "What's the solution?" and "How do you know?" are incorporated into every Number Talk involving the four operations. (For Number Talks using visual models and/or tools like rekenreks, the standard questions are "What do you see?" and "How do you see it?") But, these two questions only get the ball rolling. Other questions to facilitate discourse may include:
  • Who solved it a different way?
  • Who agrees/disagrees? Why?
  • How is____ similar/different from _____?
This is by no means an exhaustive list. (Robert Kaplinsky has a great post on the art of asking questions, with some really good questions for different situations). Through your questioning, your goal is for students to communicate their reasoning, interact with the reasoning of others, make conjectures, ask questions of their own, etc. Therefore, your list of possible questions should flow from your anticipation of students' interactions and be a reflection of that anticipation with an eye on student discourse with each other. Because of this dynamic interplay between students and math your questions may (should?) very from talk to talk.

Last Thoughts
Planning for a Number Talk requires thoughtful examination of grade level standards, knowledge of various progressions of mathematical concepts, and a deep understanding of our students. And, to be honest, this is just a start. There's still the issue of measuring student progress, instructional arc and strategies, planning responsively, knowing when to move to a different strategy, and a whole lot more.

Friday, November 11, 2016

Number Talks Gathering: Targets and Measures

"What is the goal?"
"What should my learning target be?"
"This is great, but how do I assess student learning?"

These are some of the more common questions teachers in my district have when it comes to deepening their use of Number Talks. In our October Number Talks Gathering, I talked through some options for answering these questions. You can find the Slides for the Gathering here.

Learning Targets for Students

When I model Number Talks, I use the target, "I can think flexibly, efficiently, and accurately when solving math problems." The language of the Standards (and research around numeracy, for that matter) seems to value students' use flexible thinking around quantity and place value. That flexible thinking should become ever more efficient as students analyze, play with, and incorporate student generated strategies based on the properties of the operations and their own flexible thinking.  I love the student friendly language of flexible, efficient, and accurate. As an advocate for Guided Language Acquisition Design (Project GLAD) I've even incorporated story and Total Physical Response (TPR) into the learning target as well as Scout Awards that incorporate visual cues to
anchor students' thinking back to our target.

The challenge with this learning target is that, while it's student friendly, it's somewhat vague and unhelpful to teachers as they analyze student achievement of the Standards. How do we know if a student is being flexible, efficient, and accurate? To what degree? To what end?

Scaling the Target

Select a problem set
First, I select a problem set and plan for my Number Talk by anticipating possible student strategies, drawing visual models I may use to demonstrate student thinking, and reflecting on questions I'll use to delve into student thinking as well as help students persevere when they struggle.

After planning for the Number talk, I then use this template to define and scale the learning target for the Number Talk. I begin by using the required fluency shown on the Achieve the Core Focus Document for that grade. In this case, I'm planning for a First Grade Number Talk.

Which standards are evident within my chosen problem set?
From there, I use the Coherence Map to find concepts and skills related to my chosen problem set. I use the fluency outcome as my focus standard within the Map. By doing this, I can see the progression of mathematical skills and concepts that are evident within the problem set I've chosen.  I then read through, analyze, and synthesize the standards related to this Number Talk and make notes of how they're relevant to the problem set I've chosen.

Creating "look-fors" within the Number Talk

After reflecting on the standards, I then write descriptors of  a student who is "proficient" with the learning target. From there, I write indicators that the student is approaching my learning target. The last section describes things that a student may do that would cause concern.

Gathering Data

Gathering data becomes another issue. The February 2015 issue of Teaching Children Mathematics has a wonderfully concise and instructive article on "Classroom-Based Formative Assessments" in which two types of formative assessment are described.

"Show Me Tasks" are those in which the teacher uses observation within the flow of a lesson to gather evidence of student learning. Within a Number Talk, observation of student thinking can be accomplished by "Mining for gold" while students turn and talk with an partner before a class discussion. The practice of "Mining for gold" is also helpful in selecting students to share their thinking during the whole class debrief, much like the Second and Third Practices of the Five Practices. Aside from listening into student conversations, a teacher can collect observational data during the whole class discussion and debrief. When I was in the classroom, I used a spreadsheet similar to the one pictured below to record wether a student had met or was approaching my target, or if the student needed additional support.

"Hinge Questions" are those that a teacher poses at a pivotal point in the learning progression. After collecting observational data, a teacher may want confirm his or her observations through an exit ticket. One way I've done that is by writing four different problems on index cards (see image below). These problems are reflective of the problem types we've been working on within our Number Talks. Students return to their groups, receive a problem different from the other students in their group, and solve in a way that makes the most sense to them. I then collect the samples and analyze them to determine the direction of future Number Talks. I've also seen a similar routine using sticky notes and a door chart like the one pictured below.

Three ways to collect formative assessment


One of the things I love about Number Talks is that the class creates a safe and respectful environment in which risks, mistakes, and learning are valued above speed and the right answer. At the same time, the demand placed on teachers to quantify student learning is a reality that encroaches on the safety students find within a Number Talk. By focusing students' attention and reflection on their ever increasing and unique flexible and efficient strategic thinking, while the teacher works in the background to scale learning targets and uses observational data for instructional decision making, I hope teachers can preserve the safety and success students find in Number Talks, while at the same time increasing teacher intentionality through our analysis and use of learning progressions and formative assessment.

I hope this process is helpful to you in your use of Number Talks! Let me know how it's working for you! Or, if you have any constructive feedback, let me know!