Category Archives: Reflection

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.

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You can see the notes I started on paper.

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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.

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

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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.

8wheelsspreadsheet

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.

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

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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.” Nctm.org. 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?

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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.

OAME 2016: Sketch notes for learning

OAME Annual Conference – Barrie 2016 – Leap Into Math
hosted by MAC2 at Georgian College, Barrie, ON

Robert Lang challenges us to open possibilities for every learner. Start with art and find the structure. Seek connections through creativity.

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Catherine Bruce highlighted the importance of fractions, the lack of clarity on the anatomy of a fraction, the need to attend to and understand unit fractions, and to help learners find clarity and understanding.

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Paul Alves modeled powerful pedagogy as he empowered participants to code.

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Melissa Poremba challenges us to use literacy to further develop a stronger culture of numeracy.

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Steven Strogatz used his New York Times series to highlight the importance of humor, empathy, relevance, and visualizations. His breast cancer article, Chances Are, connected, for me, to Catherine Bruce’s earlier talk. Fractions often bring more clarity and understanding than percents and decimals.

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Chris Suurtamm challenges us to honor algebraic thinking, visualization, and flexibility in learners of all ages.

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Never ever miss an opportunity to learn with Graham Fletcher.

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There were two sessions of Ignite talks at OAME. I was a speaker for the first session, therefore, no sketch notes.  Here are the highlights from the second session.

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HGSE Teaming: Sketch notes for learning

Our team (Maryellen Berry, Rhonda Mitchell, Marsha Harris, and I) attend the Harvard Graduate School of Education’s 2016 session on the Transformative Power of Teacher Teams taught by Katherine Boyles and Vivian Troen.

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

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Teams that lack open conflict are dying entities.

Boyles and Troen challenge us to level up from a “culture of nice” to a collaboration.

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Elizabeth City joined us to make the case for teacher teams and introduce intentional talk around the instructional core.  How might be build collective efficacy?

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Boyles and Troen then facilitated a session to help teams set norms and change the sense of what is possible.  The instruction core was again emphasized as well as task focus.

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Daniel Wilson started our second day with a session on cultivating collaboration.  How might we have communication, coordination, cooperation, and collaboration.  His definition of collaboration, coming together to create something new, inspired our team to co-labor and set new goals?

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Monica Higgins used the Mount Everest case study as a catalyst for discussion around leadership, responsibility trust, and teaming.

Changing your mind can be a show of strength.

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Aviya Schler and Jacob Pinnolis discussed implementing faculty rounds at their school.  How might we build a culture of inquiry where we are curious about each other’s practice? What if we share our questions and help each other “see” what happens during class?

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Jodi Doyle and her team creating and sustaining collaborative, committed teaching teams.  How might we grow together to serve all learners in our care? What if we structure team meetings to embrace the power of positivity, have serious task focus around students learning, and be product oriented?

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Tina Blythe began our last day with using protocols to learn from student and teacher work.  How might we support deep learning and thinking?

Many eyes looking helps us learn and notice more.

How will we team? norm? collaborate? support? become more curious?

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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#ShowYourWork: words, pictures, numbers

In her Colorful Learning post, Learning: Do our students know we care about that?, Kato shared the following learning progression for showing your work.

What if we guide our learners to

I can describe or illustrate ow I arrived at a solution so that the reader understands without talking to me?

Isn’t this really about making thinking visible and clear communication?  Anyone who has taught learners who take an AP exam can attest to the importance of organized, clear pathways of thinking. It is not about watching the teacher show work, it is about practicing, getting feedback, and revising.

Compare the following:

What if a learner submits the following work?

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Can the reader understand how the writer arrived at this solution without asking any questions?

What if the learner shared more thinking? Would it be clearer to the reader? What do you think?

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How often do we tell learners that they need to show their work? What if they need to show more work? What if they don’t know how?

How might we communicate and collaborate creatively to show and tell how to level up in showing work and making thinking visible?

How might we grow in the areas of comprehension, accuracy, flexibility, and deeper understanding if we learn to communicate clearly using words, pictures, and numbers?

Flexibility and sense-making to build confidence and long-term memory

In his TEDxSonomaCounty talk, The Myth of Average, Todd Rose (@ltoddrose) challenges us to consider and act to leverage simple solutions that will improve the performance of our learners and dramatically expand our talent pool.  (If you’ve not seen his talk, it is worth stopping to  watch the 18.5 minute message before reading on.)

There are far too many students who feel like they are no good at math because they aren’t quick to get right answers. (Humphreys & Parker, 9 pag.)

Efficiency must not trump understanding.  How often do we remember the foundation once we’ve mastered “the short cut?” Were we ever taught the foundation – the why – or were we only taught to memorize procedures that got to an answer quickly?

Of course, students must be able to compute flexibly, efficiently, and accurately. But they also need to explain their reasoning and determine if the ideas they’re using and the results they’re getting make sense. (Humphreys & Parker, 8 pag.)

How might we design and implement practices that help our young learners make sense of what they are learning?  In Brain-Friendly Assessments: What They Are and How to Use Them, David Sousa explains how necessary sense-making and meaning are to transfer information from working memory into long-term memory.

The brain is more likely to store information if it makes sense and has meaning. (Sousa, 28 pag.)

Dr. Sousa continues:

We should not be measuring just content acquisition. Rather, we should be discovering the ways students can process and manipulate their knowledge and skills to deal with new problems and issues associated with what they have learned.  (Sousa, 28 pag.)

The first chapter of Making Number Talks Matter highlights the importance of number talks.  We want our young learners to develop flexibility and confidence working with numbers.

Listen to Ruth Parker and Cathy Humphreys discuss Number talks:

From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

What if we take action on behalf of our young learners?  What if we offer multiple pathways for success?

How might we dramatically expand our talent pool?


I am grateful to Kristin Gray (@MathMinds) and Crystal Morey (@themathdancer) for their leadership and facilitation as a dozen #TrinityLearns faculty participate in an online book club (#mNTmTch) for Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding Grades 4-10 along with over 600 educators across the globe.


Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.