Tag Archives: Tracy Zager

Deep Practice: Building Conceptual Understanding in the Middle Grades

2017 NCSM Annual Conference
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.

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. 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.)

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 pag.)


Coyle, Daniel. The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. (p. 18-20). Random House, Inc.. Kindle Edition.

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.

#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.