Category Archives: Questions

PD planning: #Mathematizing Read Alouds

How might we deepen our understanding of numeracy using children’s literature? What if we mathematize our read aloud books to use them in math as well as reading and writing workshop?

Have you read Love Monster and the last Chocolate from Rachel Bright?

Becky Holden and I planned the following professional learning session to build common understanding and language as we expand our knowledge of teaching numeracy through literature.  Each Early Learners, Pre-K, and Kindergarten math teacher participated in 2.5-hours of professional learning over the course of the day.


To set the purpose and intentions for our work together we shared the following:

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Becky’s lesson plan for Love Monster and the last Chocolate is shown below:


After reading the story, we asked teacher-learners what they wondered and what they wanted to know more about.  After settling on a wondering, we asked our teacher-learners to use pages from the book to anticipate how their young learners might answer their questions.

After participating in a gallery walk to see each other’s methods, strategies, and representations, we summarized the ways children might tackle this task. We decided we were looking for

  • counts each one
  • counts to tell how many
  • counts out a particular quantity
  • keeps track of an unorganized pile
  • one-to-one correspondence
  • subitizing
  • comparing

When we are intentional about anticipating how learners may answer, we are more prepared to ask advancing and assessing questions as well as pushing and probing questions to deepen a child’s understanding.

If a ship without a rudder is, by definition, rudderless, then formative assessment without a learning progression often becomes plan-less. (Popham,  Kindle Locations 355-356)

Here’s the Kindergarten learning progression for I can compare groups to 10.

Level 4:
I can compare two numbers between 1 and 10 presented as written numerals.

Level 3:
I can identify whether the number of objects (1-10) in one group is greater than, less than, or equal to the number of objects in another group by using matching and counting strategies.

Level 2:
I can use matching strategies to make an equivalent set.

Level 1:
I can visually compare and use the use the comparing words greater than/less than, more than/fewer than, or equal to (or the same as).

Here’s the Pre-K  learning progression for I can keep track of an unorganized pile.

Level 4:
I can keep track of more than 12 objects.

Level 3:
I can easily keep track of objects I’m counting up to 12.

Level 2:
I can easily keep track of objects I’m counting up to 8.

Level 1:
I can begin to keep track of objects in a pile but may need to recount.

How might we team to increase our own understanding, flexibility, visualization, and assessment skills?

Teachers were then asked to move into vertical teams to mathematize one of the following books by reading, wondering, planning, anticipating, and connecting to their learning progressions and trajectories.

During the final part of our time together, they returned to their base-classroom teams to share their books and plans.

After the session, I received this note:

Hi Jill – I /we really loved today. Would you want to come and read the Chocolate Monster book to our kids and then we could all do the math activities we did as teachers? We have math most days at 11:00, but we could really do it when you have time. We usually read the actual book, but I loved today having the book read from the Kindle (and you had awesome expression!).

Thanks again for today – LOVED it.

How might we continue to plan PD that is purposeful, actionable, and implementable?

Cross posted on Connecting Understanding.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L. (2016-09-16). Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Norris, Kit; Schuhl, Sarah (2016-02-16). Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K-5 Learners (Kindle Locations 4113-4115). Solution Tree Press. Kindle Edition.

Popham, W. James. Transformative Assessment in Action: An Inside Look at Applying the Process (Kindle Locations 355-356). Association for Supervision & Curriculum Development. Kindle Edition.

Learner choice: using appropriate tools strategically takes time and tools

All students benefit from using tools and learning how to use them for a variety of purposes.  If we don’t make tools readily available and value their use, our students miss out on major learning opportunities. (Flynn, 106 pag.)

I’m taking the #MtHolyokeMath #MTBoS course, Effective Practices for Advancing the Teaching and Learning of Mathematics.  Zachary Champagne facilitated the second session and used The Cycling Shop task from Mike Flynn‘s TMC article.


You can see the notes I started on paper.


Jim, Casey and I used a pre-made Google slide deck provided to us to collaborate since we were located in GA, MA, and CA.  We challenged ourselves to consider wheels after working with 8 wheels.

Here’s what our first table looked like.


Now, I was having trouble keeping up with the number of wheels and the number of cycles.  So I did this:


This made it both better and worse for me (and for my group).

Here’s an interesting thing.  I’ve been studying, practicing, and teaching the Standards for Mathematical Practices. Jennifer Wilson and I have written a learning progression to help learners learn to say I can use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. (Sage, 6 pag.)

Clearly, I was not even at Level 1 during class.  Not once – not once – during class did it occur to me how much a spreadsheet would help me, strategically.


The spreadsheet would calculate the number of wheels automatically for each row so that I could confirm correct combinations.  (You can view this spreadsheet and make a copy to play with if you are interested.)

When making mathematical models, [mathematically proficient students] know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. (Sage, 6 pag.)

With a quick copy and paste, I could tackle any number of wheels using my spreadsheet.  I can look for and make use of structure emerged quickly when using the spreadsheet strategically.  (I want to also highlight color as a strategic tool.) Play with it; you’ll see.


[Mathematically proficient students] are able to use technological tools to explore and deepen their understanding of concepts. (Sage, 6 pag.)


There is no possible way I would have the stamina to seek all the combinations for 25 or 35 wheels by hand, right?

Students have access to a wide assortment of tools that they must learn to use for their mathematical work. The sheer volume of possibilities can seem overwhelming, but with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal. (Flynn, 106 pag.)

Important to repeat, “with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal.

For this to happen, we need to have a solid understanding of the kinds of tools available, the purpose of each tool, and how students can learn to use them flexibly and strategically in any given situation. This also means that we have to make these tools readily available to students, encourage their use, and provide them with options so they can decide which tool to use and how to use it. If we make all the decisions for them, we remove that critical component of MP5 where students make decisions based on their knowledge and understanding of the tools and the task at hand. (Flynn, 106 pag.)

To be clear, a spreadsheet was available to me during class, but I didn’t see it.  How might we make tools readily available and visible for learners to choose?

When we commit to empower students to deepen their understanding, we make tools available and encourage exploration and use, so that each learner makes decisions for themselves. In other words, how do we help learners to level up in both content and practice?

What if we make I can look for and make use of structure; I can use appropriate tools strategically; and I can make sense of tasks and persevere in solving them essential to learn for every learner?

How might we offer tools and time?

It’s about learning by doing, right?

Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

Flynn, Mike. “The Cycling Shop.” Teaching Children Mathematics, Aug. 2016. Web. 03 Feb. 2017.

Common Core State Standards.” The SAGE Encyclopedia of Contemporary Early Childhood Education (n.d.): n. pag. Web.

Deep understanding: visualize, connect, comprehend

We need to give students the opportunity to develop their own rich and deep understanding of our number system.  With that understanding, they will be able to develop and use a wide array of strategies in ways that make sense for the problem at hand.  (Flynn, 8 pag.)

Let’s say that the essential-to-learn is I can subtract within 100.  In our community we hold essential I can show what I know more than one way. 

Using our anchor text, we find the following strategies:

  • I can subtract tens and one on a hundred chart.
  • I can count back to subtract on an open number line.
  • I can add up to subtract on an open number line.
  • I can break apart numbers to subtract.
  • I can subtract using compensation.

What if we engage, as a team, to deepen our understanding of subtraction?

Deep learning focuses on recognizing relationships among ideas. During deep learning, students engage more actively and deliberately with information in order to discover and understand the underlying mathematical structure. (Hattie, 136 pag.)

In his Effective Practices for Advancing the Teaching and Learning of Mathematics class last week, Mike Flynn highlighted three advantages  of using representations to deepen understanding.

  • Representations build conceptual understanding and help assess comprehension.
  • Representations serve as a tool to make sense of the task and the mathematics.
  • Representations help develop proof of generalizations.

What if we, as a team, prepare to facilitate experiences so that learners can say I can subtract within 100 by deepening our understanding with words, pictures, numbers, and symbols?

Context: Annie had some money in her “mad money” jar.  Today, she added $39 to the jar and discovered that she now has $65. How much money was in the “mad money” jar before today?


Can we connect the context to each of the above strategies? Can we connect one strategy to another strategy?

If we challenge ourselves to “do the math” using words, pictures, numbers, and symbols, we deepen our understanding and increase our ability to ask more questions to advance thinking.

How might we use Van de Walle’s ideas for developing conceptual understanding through multiple representations to assess comprehension and understanding?

Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L. (2016-09-16). Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Van de Walle, John. Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2. Boston: Pearson, 2014. Print.

Read with me? Book study: Positive Discipline in the Classroom

How do we engage with and make meaning and connections from text? How might we notice and note big ideas from a text to capture what speaks to us? How do we show and share what we are thinking? When we cannot find time to meet, how will we connect, learn, and share? What if we try a slow chat book study?

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An invitation sent to members of our learning community:

In preparation for our continuing work with Kelly Gfroerer and Sarah Morgan Bonham, you are invited to learn and share using a “slow chat book study” of Positive Discipline in the Classroom: Developing Mutual Respect, Cooperation, and Responsibility in Your Classroom by Jane Nelsen, Lynn Lott, and H. Stephen Glenn. We will follow the schedule below to read and share ideas from a chapter per week.  

With a slow chat book study you are not required to be online at any set time. Instead, share your ideas and respond to others’ thoughts as you have time. This accommodates different schedules to allow for maximum community participation and for great conversations to unfold at a slower pace. We will use Twitter hashtag #TrinityReads to share and follow  the comments of others.

No need to sign up for the book study – just have a twitter account and search for hashtag #TrinityReads. And, when you post your comments please do include #TrinityReads so others can follow along and find your comments easily.

When you have more to say than 140 characters, we encourage you to link to blog posts, images, or other documents to share more fully.

The Book Study Schedule and Prompts

To help you think about what might be shared as you read we have established the following schedule and prompts to help with sharing and discussion.

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Each week the following prompts will be used to encourage sharing and discussion:

Sentence/Phrase – Share a quote that is meaningful to you, that captures the core ideas, that moved, engaged, or provoked you. Say more…

Connect – How do these ideas connect to what you already know, think, and study?  What text-to-self, text-to-text, text-to-world: connections can we make?

Extend – What new ideas extend or push your thinking in a new direction?

Challenge – What now is a challenge for you? What will/did you try?

I used to think… now I…

 How do we show and share what we are learning? When we cannot find time to meet, how will we connect, learn, and share? What if we try?

Join us.  We value your thinking, learning, and contributions.

Teaming: Deepen Understanding to Strengthen Academic Foundation

How might we learn and grow together? How do we connect ideas and engage in productive, purposeful professional development (aka learning experiences) around common mission, vision, and goals? What if we model what we want to see and experience in our classrooms?

Influenced, inspired, and challenged by our work at Harvard Graduate School of Education’s 2016 session on the Transformative Power of Teacher TeamsMaryellen BerryRhonda MitchellMarsha Harris, and I set common goals for faculty-learners.

We can design and implement a differentiated action plan across our grade to meet all learners where they are.

But, how do we get there?

For a while, we will narrow to a micro-goal.

We can focus on the instructional core, i.e. the relationship between the content, teacher, and learner.

For today’s Pre-Planning session, a specific goal. At the end of this session, every faculty-learner should be able to say

We can engage in purposeful instructional talk concerning reading, writing, and math to focus on the instructional core.

Here’s our learning plan:

8:00 Intro to Purpose
Instructional Core: Relationship between content, teacher, student

Explain Content Groups tasks

8:30 Movement to Content Groups
8:35 Content Groups Develop Mini-Lesson

9:05 Movement back to Grade-Level Teams in the Community Room
9:10 Share Readers’ Workshop Instructional Core ideation
9:20 Q&A and transition
9:25 Share Writers’ Workshop  Instructional Core ideation
9:35 Q&A and transition
9:40 Share Number Talk  Instructional Core ideation
9:50 Q&A and transition
9:55 Closure:  Planning, Reflection, Accountability

We also shared our learning progressions with faculty so they might self-assess and grow together.

Today’s goal:
Year-long goal:
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When  we focus on the instructional core and make our thinking visible, we open up new opportunities to learn and to impact learning with others.

How might we deepen understanding to strengthen learning?

NCSM 2016: Sketch notes for learning

NCSM 2016 National Conference – BUILDING BRIDGES BETWEEN LEADERSHIP AND LEARNING MATHEMATICS:  Leveraging Education Innovation and Research to Inspire and Engage

Below are my notes from each session that I attended and a few of the lasting takeaways.

Day One

Keith Devlin‘s keynote was around gaming for learning. He highlighted the difference in doing math and learning math.  I continue to ponder worthy work to unlock potential.  How often do we expect learners to be able to write as soon as they learn? If we connect this to music, reading, and writing, we know that symbolic representations comes after thinking and understanding.  Hmm…Apr_11_NCSM-Devlin

The Illustrative Mathematics team challenged us to learn together: learn more about our students, learn more about our content, learn more about essentials for our grade and the grades around us.  How might we learn a lot together?

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Graham Fletcher teamed with Arjan Khalsa. While the title was Digital Tools and Three-Act Tasks: Marriage Made in the Cloud, the elegant pedagogy and intentional teacher moves modeled to connect 3-act tasks to Smith/Stein’s 5 Practices was masterful.
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Jennifer Wilson‘s #SlowMath movement calls for all to S..L..O..W d..o..w..n and savor the mathematics. Notice and note what changes and what stays the same; look for and express regularity in repeated reasoning; deepen understanding through and around productive struggle. Time is a variable; learning is the constant.  Embrace flexibility and design for learning.

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Bill McCallum challenges us to mix memory AND understanding.  He used John Masefield’s Sea Fever to highlight the need for both. Memorization is temporary; learners must make sense and understand to transfer to long-term memory.  How might we connect imagery and poetry of words to our discipline? What if we teach multiple representations as “same story, different verse”?

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Uri Treisman connects Carol Dweck’s mindsets work to nurturing students’ mathematical competence.  Learners persist more often when they have a positive view of their struggle. How might we bright spot learners’ work and help them deepen their sense of belonging in our classrooms and as mathematicians?

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Day Two

Jennifer Wilson shared James Popham’s stages of formative assessment in a school community. How might we learn and plan together? What if our team meetings focus on the instructional core, the relationships between learners, teachers, and the content?

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Michelle Rinehart asks about our intentional leadership moves.  How are we serving our learners and our colleagues as a growth advocate? Do we bright spot the work of others as we learn from them? What if we team together to target struggle, to promote productive struggle, and to persevere? Do we reflect on our leadership moves?

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Karim Ani asked how often we offered tasks that facilitate learning where math is used to understand the world.  How might we reflect on how often we use the world to learn about math and how often we use math to understand the world in which we live? Offer learners relevance.

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Day Three

Zac Champagne started off the final day of #NCSM16 with 10 lessons for teacher-learners informed from practice through research. How might we listen to learn what our learners already know? What if we blur assessment and instruction together to learn more about our learners and what they already know?

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Eli Luberoff and Kim Sadler created social chatter that matters using Desmos activities that offered learners the opportunities to ask and answer questions in pairs.  How might we leverage both synchronous and asynchronous communication to give learners voice and “hear” them?

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Fred Dillon and Melissa Boston facilitated a task to highlight NCTM’s Principles to Actions ToolKit to promote productive struggle.  This connecting, for me, to the instructional core.  How might we design intentional learning episodes that connect content, process and teacher moves? How might we persevere to promote productive struggle? We take away productive struggle opportunities for learners when we shorten our wait time and tell.

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ASCD 2016: Sketchnotes for learning

Association for Supervision and Curriculum Development IASCD)
April 2-4, 2016
Atlanta, GA

Don Wettrick asked how might we offer time for learners to act on their imaginative ideas to create prototypes, products, and new outcomes.


Manny Scott challenged us to serve the whole child.  How might we offer hope to every learner?


Doug Fisher, Nancy Frey, and Stefani Hite shared their latest research and learning around purpose driven instruction and challenged us to collect evidence of impact.


Carol Ann Tomlinson and Michael Murphy encouraged us to level up in our ability to differentiate.  Do we have both short-term and long-term goals for our learners?


Carol Dweck asks how might we focus on learning and offer feedback that is observational instead of judgemental.


Shauna Peeples highlighted the power of positivity.  How might we send message containing love and purpose?


Nina Culbertson and David Griffith discussed CASEL’s social-emotional competencies and needs for all to thrive.


Kami Thordarson and Karen Wilson facilitated an interactive session for leaders embracing design thinking in their daily practice.  How might we use the DT iterative process to impact all learners in our care?