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.

Goals:
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)

Resources:

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:

81-25=14×4
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.

And:

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.