Category Archives: Connecting Ideas

Regularity in Repeated Reasoning through Choral Counting: Start at 6; count up by 5

Choral Counting gets to the heart of what we want for our mathematical communities. This activity creates space for all students to notice, to wonder, and to pursue interesting ideas. Students and teachers alike wonder together about patterns, and why and how numbers change or stay the same. [Franke, Kindle Locations1526-1528}

I wonder what can be learned from using a number line or ten-frames to shed more light on the patterns naturally found from members of the chorus.

Beginning with 6 and counting by 5s, we counted. Learners began adding “because…” to what they noticed. #Awesome

Choral Counting is an invitation; it provides an opportunity for each student to generate important mathematical ideas and for teachers to be curious about their students’ thinking. [Franke, Kindle Location 2057]

One learner said, “To move from one row to the next row, you add 30 because 6×5 is 30.” It is a regularity that repeats. Using the number line shows that to move from 6 to 36 there are 6 hops of 5 or a distance of 30.

The next comment was, “Each term on the diagonal going from the top left to the bottom right increases by 35 because 7×5 is 35.” Another regularity that repeats. Again, the number line shows 7 hops of 5 from 6 to 41, 11 to 46, 41 to 76, and so on.

Awesome that one “I notice…” that includes “because” inspires additional ones. Facilitating meaningful mathematical discourse invites students to develop and share important mathematical ideas.

What tools are within reach of learners as they deepen their numeracy and understanding? What is to be gained when we both author and illustrate mathematical understanding?

[Cross-posted at Author and Illustrate Understanding]

Franke, Megan L. Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom. Stenhouse. Kindle Edition

Learn, not memorize (within playing with sentences)

Playing with sentences begins with witnessing writing as performance. It’s a concrete way to reach out and engage our audience’s eyes and ears. (Anderson, 180 pag.)

Intent on learning more about sentence variation, my feedback partner helped me notice that I begin many of my sentences with nouns. Challenged to play more with my writing, I assigned myself the task of writing an 11 sentence paragraph using each of Anderson’s 11 Sentence Pattern Options from Chapter 8, Energy.

As a young learner, I was a memorizer. Doing what was expected of me, I learned the rules required for “the test”. Relieved and exhausted, I promptly forgot them. As concepts became more complex, my workload and anxiety increased. My favorite professor, Allen Smithers, noticed my lack of understanding. Dr. Smithers, patient and determined, challenged me to develop conceptual understanding. He challenged me to learn – not memorize. He expected me to confirm my understanding using drawings, graphs, tables, and equations. I grew as a mathematician, confident and capable. I learned, deeply. I am grateful.

Here’s the breakdown:

I know that I ended my sentence with an adverb instead of an adjective, but I choose to leave it as is.

Playing with sentences and ideas, I tried again.

As a young learner, I was a memorizer. Doing what was expected of me, I learned the rules required for “the test”. Relieved and exhausted, I promptly forgot them. As concepts became more complex, my workload and anxiety increased. Jill Lovorn, mathematician, was lost yet lucky. Success, assumed and shown, was shallow at best. Rote memorization – pages and pages of hidden work – masked missing conceptual understanding. I could use procedures, theorems, techniques, and algorithms. I got the right answers, mysteriously and remarkably.  No one knew, sadly. I survived.

Still ending that sentence with an adverb, I enjoyed playing with ideas and with sentences. Here’s the structure with a sentence checkup.

What do you think?

Anderson, Jeff. 10 Things Every Writer Needs to Know. Stenhouse Publishers, 2011.

Sentence Checkup: Mathematical Play with Sentence Patterns and Length

We know young writers will do what feels comfortable. They don’t play with their writing. They don’t try a sentence three different ways when it’s not working. They don’t explore what a varied sentence pattern or length can do for their writing’s rhythm and fluency. (Anderson, 178 pag.)

Blending a little math into writer’s workshop, what if we analyze and visualize our sentence patterns and lengths? Will learners play with their sentences after collecting and graphing a little data as described in 10 Things Every Writer Needs to Know?

Knowing how important visuals are to my learning, I used Google Sheets to “see” the variation in sentence length and to analyze the pattern of my sentence beginning.

Sentence checkup 1: Advance Your Inner Mathematician #TrinityLearns Session 5: Sequence and Connect

Wow! I am not worried about my sentence length. (Are they long? Is there an average number of words in great sentences, or is it about variety and rhythm?) However, I am appalled at the lack of interesting first words. It would have been so easy to write:

“Advance Your Inner Mathematician is a new course we are piloting this semester.”

And, the second sentence could have easily been,

“Anchored in Smith and Sherin’s ‘The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Middle School Classroom’, this course supports continued teacher learning after Embolden Your Inner Mathematician.”

Or the two sentences could have been combined into one sentence.

“Advance Your Inner Mathematician, a new course we are piloting this semester is anchored in Smith and Sherin’s ‘The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Middle School Classroom’, to support continued teacher learning after Embolden Your Inner Mathematician.”

Sentence Checkup 2: Fear of Imperfection; Deep Practice; Just Make A Mark 

I notice that this post is chock-full of questions (16 of 18 sentences) – a known trait of my writing.  I find the visual of sentence length interesting.

While I chose Google sheets as my tool, students can quickly graph this data by hand (please encourage the use of graph paper so that they attend to precision), and drop it in their writer’s notebook.

Will writers play more with their words and sentences if they see the patterns and frequency?

Anderson, Jeff. 10 Things Every Writer Needs to Know. Stenhouse Publishers, 2011.

Mentor Sentence: Notice, Emulate, Learn #LL2LU

As part of our Embolden Your Inner Writer course, Marsha and I drafted a learning progression for each chapter to help our writers when they feel stuck or need a push. However, these are just drafts. In order to feel confident, to have the courage to use them, we must use them ourselves, share them with learners, and seek feedback.

I’m trying out the following learning progression for Anderson’s chapter on Models, Chapter 2.

I can strengthen my craft, word choice, and mechanics by applying techniques from models and mentor texts.

Enamored with Daniel Coyle’s writing, I picked up my copy of The Talent Code, and found the following sentence.

The goal is always the same: to break a skill into its component pieces (circuits), memorize those pieces individually, then link them together in progressively larger groupings (new, interconnected circuits). [Coyle, 84 pag.]

Noticing the colon, I wondered if I am skilled at using them, knowing when to use them, and using them correctly.  (Ok…I’m not, but what can I learn?)

Another  Coyle book, The Culture Codeoffers this gem using a colon.

One pattern was immediately apparent: The most successful projects were those closely driven by sets of individuals who formed what Allan called “clusters of high communicators.”[Coyle, 69 pag.]

And, in 10 Things Every Writer Needs to Knowour anchor text,

Students need to know the truth: writing is cumulative. [Anderson, 9 pag.]

If I read and observe how these authors use a colon, I think I can use it myself to imitate the great writers.

Perseverance calls for action: show an attempt to think and question, ask and seek clarifying questions, try again with new information and actions.

What do you think?

I’m not sure I “read like a writer” as stated in Level 1, but I annotated well. I could find sentences that helped me think about using a colon. Maybe I read more like a writer than I thought. Hey, that’s one of the tips!  Then, I collected and recorded examples to imitate as suggested in Level 2. Curiosity caused me to want to know more.  I have asked questions, and I love how Jeff Anderson, in Mechanically Inclined, offers notes and a visual.

And, then…boom! I was struggling with a sentence in my previous post when it dawned on me: Use a colon! Here’s what I wrote:

The editor in my head – no, not the editor – the critic in my head convinces me to wait: wait until I know, wait for someone else, wait.

While I think I’m currently at Level 3 (maybe Level 4 when I press publish), I have more to learn and more work to do to be confident that “I can strengthen my craft, word choice, and mechanics by applying techniques from models and mentor texts.”

I do have the courage to continue.

Anderson, Jeff. 10 Things Every Writer Needs to Know. Stenhouse Publishers, 2011.

Anderson, Jeff, Vicki Spandel. Mechanically Inclined: Building Grammar, Usage, and Style into Writer’s Workshop. Kindle Edition.

Coyle, Daniel. The Culture Code: The Secrets of Highly Successful Groups. Random House Publishing Group. Kindle Edition.

Coyle, Daniel. The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

Coyle, Daniel. The Culture Code: The Secrets of Highly Successful Groups. Random House Publishing Group. Kindle Edition.

Fear of imperfection; deep practice; just make a mark

Do you know any learner’s that are stuck?  Are they convinced that they can’t?

“Fear of imperfection keeps us perched on the edge, afraid to dive in and start writing. If we sit and wait for the perfect words, they don’t come. Inertia sets in. Our mind halts. The clock slows. Much like hesitating at the edge of the ocean, afraid of the shock of cold, we wait. And in waiting, our anxiety spins.” (Anderson, 9 pag.)

Hesitating at the edge, afraid, we wait. How might we develop brave, bold learners who wonder – on paper – what they are thinking so that they might see it? What do we do to overcome the fear of the blank page? This fear, as real as it seems, is just a doodle away from getting your feet wet, right? The editor in my head – no, not the editor; the critic in my head convinces me to wait: wait until I know, wait for someone else, wait. What force is needed to overcome inertia? Is it just as simple as a doodle?

Are math and writing this closely related? Wow! Far too many students will not write the first step in math because they are not sure if they are going to be right? If they are going to be right, are they learning anything?

In Daniel Coyle’s “The Talent Code,” he writes about deep practice, working at the edge of your ability so that you make mistakes, learn, and repeat.

Deep practice is built on a paradox: struggling in certain targeted ways — operating at the edges of your ability, where you make mistakes — makes you smarter.  (Coyle, 18 pag.)

The second reason deep practice is a strange concept is that it takes events that we normally strive to avoid —namely, mistakes— and turns them into skills. (Coyle, 20 pag.)

In SMP-1, “I can make sense of tasks and persevere in solving them,” the first level asks for a visible attempt to think and reason into the task.

Are our young mathematicians and writers stuck due to inertia? Is it blank page fright? Is there space in class to draft and redraft, making revisions as you go? Are missteps celebrated and seen as opportunities to learn?

How can we help students dive – or tiptoe – in to get their feet wet? What if encourage learners to just make a mark and see where it takes them?

It doesn’t have to be perfect the first time… or does it?

Anderson, Jeff. 10 Things Every Writer Needs to Know. Stenhouse Publishers, 2011.

Coyle, Daniel. The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

Reynolds, Peter H. The Dot. Library Ideas, LLC, 2019.

Embolden Your Inner Writer – plans and resources

How can we strengthen and deepen understanding, confidence, and efficacy in the art and practice of writing? Joe Marshall, Marsha Harris, and I are facilitating a series for interested Trinity School faculty and staff.

Screen Shot 2020-01-18 at 7.50.13 AM.png

In this course, we will discuss, sketch, and write to deepen our flexibility, critical reasoning, and problem-solving. Writing should be anchored in reading rich literature and developed through a cycle of reading, writing, and feedback.

At the end of the course, participants should be able to say:

  • I can write more frequently and confidently.
  • I can heighten my awareness of the craft and conventions of writing.

Session structure:

  1. Motion and Models (01/17/2020)
  2. Energy and Words (01/31/2020)
  3. Focus and Form (02/14/2020)
  4. Cohesion and Frames (02/28/2020)
  5. Details and Clutter (03/13/2020)
  6. Celebration (03/27/2020)


Preparing for Session 1: Motion and Models:

Challenges to take up:

  • Read and write daily with the goal of building stamina
  • Read and write daily with the goal of strengthening this habit
  • Notice and note sentences that inspire or challenge and share in your journal so that others may learn alongside you

Joe, Marsha, and I have been planning this course for several months.  In one of my early journal entries I wrote:

I seem to have fallen into writing reports of PD and plans and posting them on my blog. I seem to fail to tell exciting stories associated with adult learning.

And now, I have done just that… again. For the next six sessions, I will share the agenda, learning goals, and tasks. AND, I will share my take on each session, my outcomes, learnings, ah-ha’s, and struggles.

Number Talks: developing fluency, flexibility, and conceptual understanding #AuthorAndIllustrate

How might we work on fluency (accuracy, flexibility, efficiency, and understanding) as we continue to teach and learn with students? What if our young learners are supposed to be fluent with their multiplication facts, but… they. ..just…aren’t!?

It really isn’t a surprise, right? Children learn and grow at different rates. We know that because we work with young learners every day.  The question isn’t “Why aren’t they fluent right now?” It isn’t. It just isn’t. The question should be and is:

“What are we going to do, right now, to make this better
for every and each learner in our care?”

In Making Number Talks Matter, Cathy Humphreys and Ruth Parker write:

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don’t know it yet), and over time, the properties come to life in students’ own strategies. (Humphreys, 62 p.)

Humphreys and Parker continue:

Students who have experienced Number Talks come to algebra understanding the arithmetic properties because they have used them repeatedly as they reasoned with numbers in ways that made sense to them. This doesn’t happen automatically, though. As students use these properties, one of our jobs as teachers is to help students connect the strategies that make sense to them to the names of properties that are the foundation of our number system. (Humphreys, 77 p.)

So, that is what we will do. We commit to deeper and stronger mathematical understanding. And, we take action.

This week our Wednesday workshop focused on Literacy, Mathematics, and STEAM in grade level bands.  Teachers of our 4th, 5th, and 6th graders gathered to work together, as a teaching team, to take direct action to strengthen and deepen our young students’ mathematical fluency.

We began with the routine How Do You Know? routine from NCTM’s High-Yield Routines for Grades K-8 using this sentence:

How do you know?

Here’s how I anticipated the ways learners might think.

Paper.Productive Struggle.195

From The 5 Practices in Practice: Successfully Orchestrating Mathematical Discussion in your Middle School Classroom:

Anticipating students’ responses takes place before instruction, during the planning stage of your lesson. This practice involves taking a close look at the task to identify the different strategies you expect students to use and to think about how you want to respond to those strategies during instruction. Anticipating helps prepare you to recognize and make sense of students’ strategies during the lesson and to be able to respond effectively. In other words, by carefully anticipating students’ responses prior to a lesson, you will be better prepared to respond to students during instruction. (Smith, 37 p.)

How many strategies and tools do we use when modeling multiplication in our classroom? It is a matter of inclusion.

It is a matter of inclusion.

Every learner wants and needs to find their own thinking in their community. This belonging, sharing, and learning matters. We make sense of mathematics and persevere. We make sense of others thinking as they learn to construct arguments and show their thinking so that others understand.

Humphreys and Parker note:

They are learning that they have mathematical ideas worth listening to—and so do their classmates. They are learning not to give up when they can’t get an answer right away because they are realizing that speed isn’t important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra. (Humphreys, 62 p.)

As teachers, we must anticipate the myriad of ways students think and learn. And, as Christine Tondevold (@BuildMathMinds) tells us:

The strategies are already in the room.

Our job is to connect mathematicians and mathematical thinking.

From NCTM’s Principles to Actions:

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.


Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

What if we take up the challenge to author and illustrate mathematical understanding with and for our students and teammates?

Let’s work together to use and connect mathematical representations as we build procedural fluency from conceptual understanding.

Humphreys, Cathy. Making Number Talks Matter. Stenhouse Publishers. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. Print.

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Multiplication Fact Fluency prototype – feedback requested

I am concerned that we are conflating automaticity with fluency.  How might we get clear on the difference?

From Assessing Basic Fact Fluency by Gina Kling and Jennifer M. Bay-Williams:

Think about how you assess reading fluency. Does your assessment plan involve listening and observing as children read as well as asking reading comprehension questions? Now imagine what you might learn about students’ reading fluency if you used only timed quizzes. How would your confidence in your assessment change?

When I use Google to search for sample “multiplication fluency assessments, worksheets from TPT show up.If you haven’t yet, it is important to stop and read three articles:

From Fluency: Simply Fast and Accurate? I Think Not! by Linda Gojak:

Building fluency should involve more than speed and accuracy. It must reach beyond procedures and computation.

From Fluency without Fear by Jo Boaler:

The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense.

Ok, so how do we assess fluency? Do we have common language for mathematical fluency?

From  Principles to Actions: Ensuring Mathematical Success for All

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

Let’s focus this 3rd grade standard:

Multiply and divide within 100.

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.


Do any of the fluency assessments from the above Google search help us assess fluency?

What if we try a different type of assessment? An assessment that involves listening and observing as children compute, reason, and respond.

What if we confer with learners individually, just like we do with reading assessments, to listen, record, and encourage learners as they think, reason, and compute?

I need, want, and invite your feedback on the following prototype.  I hope that I have constructed a viable argument and I seek your constructive critique.

Multiplication Fluency (within 100)
Conferring with Mathematicians

Section 1 asks students if they know their multiplication facts. We intentionally ask these questions because of John Hattie’s work on student expectations of self – effect size of 1.44.

Section 2 checks for accuracy and efficiency.  The teacher will code the student’s response as recall, uses a strategy, or skip counts.  The first 8 facts are common for all students. The next 8 facts can be customized for each student based on their responses to easiest and hardest.

 Section 3 checks for accuracy and flexibility.  In the previous section, the teacher gave the multiplication expression and the student responded with the product.  Now it is reversed. The teacher gives the answer and student states an expression and are asked if they know another way.

Section 3 also checks for accuracy and flexibility using images from #UnitChat.  Students are asked how many they see and how do they see them.  If they skip count, their answer is confirmed, and they are asked if they can say or write it again using multiplication.

If you open this Multiplication Fluency (within 100) pdf, you will find each image on its own page so students can touch to count when needed.

Will you take time to offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the assessment?

Thank you in advance.

Summer Learning 2019 – Choices and VTR

How do we learn and grow when we are apart? We workshop, plan, play, rest, and read to name just a few of our actions and strategies.

We make a commitment to read and learn every summer.

Below is the 2019 Summer Learning flyer announcing the choices for this summer.

In case you are interested, links to reviews of each book are shared below as well as the set of TED Talks for Voices from Diversity. #SoGood

Big Potential and The Power of Moments are repeats from last summer’s list because of faculty/staff engagement and enthusiasm. Blindspot and Developing Assessment-Capable Visible Learners are being used in book study groups during the current school year.  We hope to harness the power of the re-read and spread this ideas.

We will continue to use the Visible Thinking Routine Sentence-Phrase-Word to notice and note important, thought-provoking ideas. This routine aims to illuminate what the reader finds important and worthwhile.

Sentence-Phrase-Word helps learners to engage with and make meaning from text with a particular focus on capturing the essence of the text or “what speaks to you.” It fosters enhanced discussion while drawing attention to the power of language. (Ritchhart, 207 pag.)

However, the power and promise of this routine lies in the discussion of why a particular word, a single phrase, and a sentence stood out for each individual in the group as the catalyst for rich discussion . It is in these discussions that learners must justify their choices and explain what it was that spoke to them in each of their choices. (Ritchhart, 208 pag.)

What are you reading/watching/doing to grow as a learner over the summer? Please feel invited and encouraged to watch us (or join us) learn by following #TrinityLearns and #TrinityReads in June and July.

Ritchhart, Ron, Mark Church, and Karin Morrison. Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. Print

#TrinityLearns Leading Learners to Level Up as a TEAM (#LL2LU) Part 2

Continuing our work from last month, Trinity School’s Assessment Committee continues to grapple with the following questions.

As a team, how are we united (aligned) in our understanding and assessment of learning?  How might we grow our assessment literacy, understanding, and actions to focus on learning, assign competence, and empower learners to become agents of learning?

Under the leadership of Thomas Benefield (@yerlifeguard) and Becky Holden (@BHolden86), Trinity School’s Assessment Committee we continue our commitment to read and take action on Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill by Nancy Frey, John Hattie, Douglas Fisher.  

Below is our agenda for the April meeting where we have begun to grapple with growing our understanding together.

As a team of teachers representing all grade-levels at our school, we chose to analyze student work together and hold a norming meeting to explore and learn one way to help our grade-level teams calibrate and clarify expectations around collaboration and citizenship.

To ensure that all voices were heard, we started with quiet reading time to preview the draft of the learning progressions.  As we did last month,  we used a Google form (shown below) to record everyone’s initial thinking around the level of work based on the drafted learning progressions for Working Cooperatively and Displays Respect.

The artifact, in this case, was a two minute video that offers a glimpse of partner work. (The video is not shared in this post, but a screen shot of one second is shown below.)

Using the Google form continues to be critically important. Everyone’s initial thinking was made visible to the team. Look at the results from our initial thinking.

As you can see, we were all over the place in our interpretation of the meaning and expectations described in our learning progressions.

As a team of assessment leaders, we had anticipated this result. You can see how this might be problematic for students in different sections with different teachers, right?

High-functioning teams that focus on learning must calibrate their understanding of what is essential to learn so that all students are assessed fairly and equitably.

What happened next was nothing short of magical.

First, we discussed our leveling with one partner to explain our reasoning and understanding. It was quiet, calm, and intense.  As partners listened to each other, different interpretations and points of view were represented.  When enough time passed, we returned to the whole group setting and discussed. Again, magical! Everyone confidently shared their initial level assessment and then spoke of how their understanding was shifted by discussing it with someone else.

Then, we took time for individual reflection and leveled the same artifact again, based on our developing common assessment. Just look at the results.

Closer, so much closer to common understanding.

To hone our skills and understanding, we used the same two learning progressions for Works Cooperatively and Displays Respect using video from a different grade level. (The video is not shared in this post, but a screen shot of one second is shown below.)

Again, more closely aligned understanding.

What can be gained when all ideas are made visible to the entire team? How might we learn and grow together by sharing our thinking, seeking feedback, and calibrating with our team?

How do your school’s teams calibrate expectations, shared values, and common understanding?

What actions will we take to become stronger and clearer as a team?

Frey, Nancy, et al. Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill. Corwin Literacy, 2018.