Category Archives: Sketch Notes

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.

9_wheelsspreadsheet

[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?

2ndgrade65-39

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.

#GISAConference 2016: notes and quick reflection

2016 GISA Annual Conference
Monday, November 7, 2016

Wendy Mogel (@DrWendyMogel) encourages us to raise wildflowers instead of bonsai trees.  She challenges our community to help our children through the journey to independence instead of hoping to skip over the struggles that come with the journey.

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Justin Cahill (@justybubPE), Brian Balocki (@BrianBalocki), and John Turner were serious about Keeping Kids in Motion. While originally scheduled into a traditional classroom, they encouraged everyone to check in and join them outside of experiential lessons to implement in PE and in base classrooms.  They taught the why, the how, and the what of keeping kids (of any age) in motion. Best GISA session EVER!

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Mary Cantwell (@scitechyEDU) facilitated a conversation around design, STEM, STEAM, and Design Thinking.  My big, lingering take-aways are the following questions.

How might we impact our learners and their approach to solving problems every day?

and

If the users of our assessments are our learners, how might we design with them in mind and design using an empathetic lens?

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Rich Wormeli (@RickWormeli2) sent the message that students will only be creative, courageous, and persistent if they have teachers willing to be creative, courageous, and persistent. Sense-making is a worthy goal, but don’t stop there; strive for meaning-making. Relationships first.  Use assessment to reveal the story of learning.gisa2016-wormeli

#TLC2016 Day 2 Notes

Sharing my day two notes from the Teaching Learning Coaching conference:

Partnering for Impact: Realizing Our Best Potential 

How might we learn the art and the science of receiving feedback? Sheila Heen asks

Will you take the easy path or the more difficult one?

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Reflect and share your “guide to working with me” to help our teams learn to help each other learn.

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You cannot lead if you are not learning. ~ Michael Fullan

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Presentations that make an impact have 7 principles of partnerships. Know your core beliefs.

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I am grateful that Marsha Harris (@marshamac74) shared her notes from Sheila Heen’s keynote, Michael Fullan’s keynote, and Jim Knight and Kristin Anderson’s breakout session.  Her notes add context, commentary, and detail to my sketches.

#TLC2016 Day 1 notes

Sharing my day one notes from the Teaching Learning Coaching conference:

Partnering for Impact: Realizing Our Best Potential 

Every student deserves a great teacher not by chance, but by design. ~Doug Fisher

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Never place a strategy or assessment tool higher than student learning. ~Doug Fisher

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Jim Knight – #ItStartsWithUs – Listen. Have courage. Find common ground.  Have empathy. Love one another.

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Kristin Anderson – Build trust. Without trust, influence diminishes..kristinandersontlc2016


I am grateful that Marsha Harris (@marshamac74) shared her notes for Doug Fisher’s keynote session and Jim Knight’s keynote.  Her notes add context, commentary, and detail to my sketches.

#NCTMRegionals Day 1 notes

Sharing my day one notes from the NCTM Regional Conference in Philadelphia:

Peg Cagle: Teacher Leadership – Advocating for ourselves, our students, our profession.

“The difference between listening and pretending to listen, I discovered, is enormous. One is fluid, the other is rigid. One is alive, the other is stuffed. Eventually, I found a radical way of thinking about listening. Real listening is a willingness to let the other person change you. When I’m willing to let them change me, something happens between us that’s more interesting than a pair of dueling monologues.” – Alan Alda

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Becky Holden: Building Understanding – Meeting Students Where They Are

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Kristin Gray: Lesson Planning That Begins with Student Thinking
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#NCTMRegionals Inspire, Innovate, IGNITE! sketchnote

I had the opportunity to attend, live tweet, and sketch the 2016 Regional Conference & Exposition in Philadelphia opening session Inspire, Innovate, IGNITE! delivered by Chike Akua, Jimmie L. Davis, Matthew Larson, Sara Moore, Tom Reardon, Michael Steele, and Rose Mary Zbiek.

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I am grateful to Bob Lochel for sharing an artifact of evidence of my work.

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If interested, you can check out the presenters’ slide decks to add context to my notes at NCTM’s blog.