Category Archives: Reflection

#NCSM17 #Sketchnotes Wednesday Summary

I’m attending the  National Council of Supervisors of Mathematics  2017 conference in San Antonio.  Here are my notes from Wednesday along with the session descriptions from the presenters.

Conferring with Young Mathematicians at Work:
The Process of Teacher Change
Cathy Fosnot

If children are to engage in problem solving with tenacity and confidence, good questioning on the part of teachers during conferrals is critical. Questioning must engender learner excitement and ownership of ideas, while simultaneously be challenging enough to support further development. Video of conferrals in action will be used for analysis, and a Landscape of Learning on the process of teacher change is shared as a lens for coaching.

Leading to Support Procedural Fluency for All Students
Jennifer Bay-Williams

Principles to Actions describes effective teaching practices that best support student learning. In this session we will focus on one of those teaching practices: “build procedural fluency from conceptual understanding.” Ensuring that every child develops procedural fluency requires understanding what fluency means, knowing research related to developing procedural fluency and conceptual understanding, and being able to translate these ideas into effective classroom practices. That is the focus of this session! We will take a look at research, connections to K–12 classroom practice, and implications for us as coaches and teacher leaders.

How to Think Brilliantly and Creatively in Mathematics: Some Guiding Thoughts for Teachers, Coaches, Students—Everyone!
James Tanton

This lecture is a guide for thinking brilliantly and creatively in mathematics designed for K–12 educators and supervisors, students, and all those seeking joyful mathematics doing. How do we model and practice uncluttered thinking and joyous doing in the classroom, pursue deep understanding over rote practice and memorization, and promote the art of successful ailing? Our complex society demands of its next generation not only mastery of quantitative skills, but also the confidence to ask new questions, explore, wonder, fail, persevere, succeed in solving problems and to innovate. Let’s not only send humans to Mars, let’s also foster in our next generation the might to get those humans back if something goes wrong! In this talk, I will explore five natural principles of mathematical thinking. We will all have fun seeing how school mathematics content is a vehicle for masterful ingenuity and joy.

Deep Practice:
Building Conceptual Understanding in the Middle Grades
Jill Gough, Jennifer Wilson

How might we attend to comprehension, accuracy, flexibility, and then efficiency? What if we leverage technology to enhance our learners’ visual literacy and make connections between words, pictures, and numbers? We will look at new ways of using technology to help learners visualize, think about, connect and discuss mathematics. Let’s explore how we might help young learners productively struggle instead of thrashing around blindly.

When Steve Leinwand asked if I was going to sketch our talk, I jokingly said that I needed someone to do it for me. We are honored to have this gift from Sharon Benson. You can see additional details on my previous post.

#NCSM17 #Sketchnotes Tuesday Summary

I’m attending the  National Council of Supervisors of Mathematics  2017 conference in San Antonio.  Here are my notes from Tuesday along with the session descriptions from the presenters.

Tracking, Equity, and the Many Paradoxes of Algebra II
Jason Zimba

What is Algebra II good for? For whom is it good? Phil Daro raised these and other questions for the Carnegie-Institute for Advanced Study Commission. Fast forward to now, and the debate about ‘what mathematics and why’ has found its way into the pages of the popular press. Is there anything district leaders can learn from this conversation? How might a district leader who prizes equity think about the question of tracking in school mathematics?

Routines for Reasoning:
Ensuring All Students
Are Mathematical Thinkers
Amy Lucenta, Grace Kelemanik

Instructional routines embody research-based best practices for struggling learners, especially when they focus on the Standards for Mathematical Practice and include ‘baked in’ supports
for special populations. Participants will explore a universally designed instructional routine, Connecting Representations, and learn how to leverage it to develop teachers’ capacity to ensure development of ALL students’ mathematical practices.

Letting Go: Cultivating Agency and Authority Through Number Talks in the Secondary Mathematics Classroom
Cathy Humphreys

In this session I share my dissertation study of two high school teachers as they learned to enact Number Talks. I wanted to know what the teachers found most challenging and how coaching supported their learning. In examining the videos of classroom lessons, I noticed marked differences in how agency and authority emerged in the two classes. I hope what I learned while searching for “Why?” will be useful for teachers and coaches alike.

Winning the Game in Mathematics Leadership
Matt Owens

Mathematics leadership is multifaceted in nature as we strive to intentionally impact students and educators in classrooms nationwide. Leadership pathways can be different from leader to leader, but ultimately curriculum/ content, instruction, activism, and assessment (CIAA) are all areas of evaluation for “PRIME” leaders in mathematics education. Discover the top seven practical strategies for overcoming the struggles that may arise in your role as a mathematics leader within your school/university, district, state, and national professional learning communities, while building the capacity of teachers’ leadership among mathematics educators in these respective communities.

Approaching Ten Tough Mathematical Ideas
for High School Students
Salmon Usiskin

The main purpose of this talk is to provide insights into mathematical content that many mathematics teachers may not have seen. By covering a broad range of content, from aspects of manipulative algebra through proof in geometry and in general, discussing language, applications, and representations, my remarks are designed for leaders to help in decisions they make in the professional development of their teachers.

#NCSM17 #Sketchnotes Monday Summary

I’m attending the  National Council of Supervisors of Mathematics  2017 conference in San Antonio.  Here are my notes from Monday along with the session descriptions from the presenters.

Knocking Down Barriers with Technology
Eli Luberoff

One-to-one. Accessibility. Personalization. Internationalization. Low oor. High ceiling. What do these all have in common? Each is targeted to making mathematics work for every student. Not just the con dent students with engaged parents, not just the struggling students, every student. We will explore the technology and techniques that can open doors, challenge the bored, empower the disempowered, and turn every student into a mathematics student.

Gut Instincts: Developing ALL Students’ Mathematical Intuitions
Tracy Zager

We’ve long misunderstood mathematical intuition, assuming it’s innate rather than developed through high-quality learning experiences. As a result, students who haven’t yet had opportunities to foster their intuitions are often denied access to meaningful mathematics. Through analysis of powerful classroom teaching and learning, we’ll explore three instructional strategies you can use to empower ALL students to grasp mathematics intuitively.

Problem Strings to Change Teaching Practice
Pam Harris

A problem string is a purposefully designed sequence of related problems that helps students mentally construct mathematical relationships and nudges them toward a major, efficient strategy, model, or big idea. We show how problem strings can be leveraged for changing teachers’ practice. Because it puts students’ ideas at the center, teachers are forced to listen deeply to kids and structure mathematics conversations around their thinking.

Rethinking Expressions and Equations:
Implications for Teacher Leaders
Michelle Rinehart

How are one- and two-variable expressions, one- and two- variable equations, and the standard form of a line connected in a powerful way? How might this progression support student learning of these “tough-to-teach/tough-to-learn” ideas? Explore the underlying theme that uni es these seemingly disparate topics using a technology-leveraged approach. Consider research and the role of teacher leaders in developing real understanding of these topics.

Talk Less and Listen More
Zachary Champagne

It’s a simple, and very complex, idea that great teachers
do and do well. Genuinely listening to students can yield incredible opportunities for teachers to not only know and connect with their students, but also increase the quality of teaching and learning that happens in the classroom. Join us as we explore the power of listening to students and using that information to inform our instruction. We’ll also explore strategies to help provide the supportive conditions and frameworks to help leaders support teachers in doing this work. We’ll do this through examining video clips of students sharing their mathematical ideas and consider what listening affords and what questions could be asked to further their mathematical thinking.

Productive struggle with deep practice – what do experts say

NCTM’s publication, Principles to Action, in the Mathematics Teaching Practices, calls us to support productive struggle in learning mathematics. How do we encourage our students to keep struggling when they encounter a challenging task? How many learners are accustomed to giving up when they can’t solve a problem immediately and quickly. How do we change the practice of how our students learn mathematics?

Effective teaching not only acknowledges the importance of both conceptual understanding and procedural fluency but also ensures that the learning of procedures is developed over time, on a strong foundation of understanding and the use of student-generated strategies in solving problems. (Leinwand, 46 pag.)

Low floor, high ceiling tasks allow all students to access ideas and take them to very high levels. Fortunately, [they] are also the most engaging and interesting math tasks, with value beyond the fact that they work for students of different prior achievement levels. (Boaler, 115 pag.)

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.)

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.)

Or to put it a slightly different way, experiences where you’re forced to slow down, make errors, and correct them —as you would if you were walking up an ice-covered hill, slipping and stumbling as you go— end up making you swift and graceful without your realizing it. (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.)

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.)

…help students slow down and really think about problems rather than jumping right into solving them. In making this a routine approach to solving problems, she provided students with a lot of practice and helped them develop a habit of mind for reading and solving problems. (Flynn, 8 pag.)

This term productive struggle captures both elements we’re after:   we want students challenged and learning. As long as learners are engaged in productive struggle, even if they are headed toward a dead end, we need to bite our tongues and let students figure it out. Otherwise, we rob them of their well-deserved, satisfying, wonderful feelings of accomplishment when they make sense of problems and persevere. (Zager, 128-129 ppg.)

Encourage students to keep struggling when they encounter a challenging task.  Change the practice of how our students learn mathematics.

Let’s not rob learners of their well-deserved, satisfying, wonderful feelings of accomplishment when they make sense of problems and persevere.


Boaler, Jo. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching (p. 115). Wiley. Kindle Edition.

Coyle, Daniel. The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. (p. 20). Random House, Inc.. Kindle Edition.

Flynn, Michael, and Deborah Schifter. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, ME: Stenhouse, 2017. (p. 8) Print.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy, Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series) (p. 136). SAGE Publications. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 46) Print.

Zager, Tracy. Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Portland, ME.: Stenhouse Publishers, 2017. (pp. 128-129) Print.

estimate and reason while dancing, singing, and playing

How might we promote peer-to-peer discourse that is on task and purposeful? What if challenge our students to estimate and reason while dancing, singing, and playing?

Andrew Stadel, this week’s #MtHolyokeMath #MTBoS Effective Practices for Advancing the Teaching and Learning of Mathematics facilitator, asked us to use visuals to engage our learners.  In his session, we used Day 127 How long is “Can’t Buy Me Love”?, Day 129 How long is “We will rock you”?, and Day 130 How long is “I feel good”? from Estimation180.

Here are my visual notes from class:

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Our homework was to estimate  How long is “I feel good”? and to try visuals with students.

I asked Thomas Benefield, 5th Grade teacher and FSLT co-chair for 10 minutes of class to try Day 127 How long is “Can’t Buy Me Love”? with 5th grade students.

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How might we make sense and persevere when making estimates? What is our strategy and can we explain our reasoning to others?

Students were asked for a reasonable low estimate, a reasonable high estimate, and then an estimate for how long the song is based on the visual. My favorite 5th grade response:

Well, you asked for a low estimate and a high estimate, so I rounded down to the nearest 5 seconds and doubled it for my low estimate. I rounded up to the nearest 10 seconds and doubled it for my high estimate.  For my estimate-estimate, I doubled the time I see and added a second since it looks like almost half.

#Awesome

It was so much fun that they let me stay for How long is “We will rock you”?, and How long is “I feel good”?, and they asked for Bohemian Rhapsody. Wow!

screen-shot-2017-03-04-at-7-42-45-pm

Andrew said that you know you have them when they start making requests.screen-shot-2017-03-04-at-7-43-06-pm

As you can see, it was a big hit. They were dancing in their seats. This quick snapshot of joy says it is worth it for our students.

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What if challenge our students to estimate and reason while dancing, singing, and playing? What joy can we add to our learning experiences?

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.

screen-shot-2017-02-03-at-2-50-42-pm

You can see the notes I started on paper.

mtholyokemath-2-zakchamp

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.

cyclingshop1

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

screen-shot-2017-02-03-at-3-08-56-pm

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.)

screen-shot-2017-02-03-at-4-03-03-pm

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.