Tag Archives: The Talent Code

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.

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.

Deep practice: building conceptual understanding in the middle grades

Deep practice:
building conceptual understanding in the middle grades
2017 T³™ International Conference
Friday, March 10, 10:00 – 11:30 a.m.
Dusable, West Tower, Third Floor
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.

[Cross posted at Easing The Hurry Syndrome]

Growth mindset = effort + new strategies and feedback

What if we press forward in the face of resistance?

For me, the most frustrating moments happen when a learner says to me I already know how do this, and I can’t learn another way.
Me:  Can’t or don’t want to? Can’t yet?

A growth mindset isn’t just about effort. Perhaps the most common misconception is simply equating the growth mindset with effort. Certainly, effort is key for students’ achievement, but it’s not the only thing. Students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve. (Dweck, n. pag.)

What if we offer a pathway for learners to help others learn, and at the same time, learn new strategies?

What if we deem the following as essential to learn?

I can demonstrate flexibility by showing what I know more than one way.

I can construct a viable argument, and I can critique the reasoning of other.

The trick is to choose a goal just beyond your present abilities; to target the struggle. Thrashing blindly doesn’t help. Reaching does. (Coyle, 19 pag.)

How might we provide pathways to target the struggles to learn new strategies, to construct a viable argument, and to critique the reasoning of others?

MathFlexibility #LL2LU

ConstructViableArgument

What if we press forward in the face of resistance and offer our learners who already know how to do this pathways to grow and learn?

How might we lead learners to level up?


Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc. Kindle Edition.

Dweck, Carol. “Carol Dweck Revisits the ‘Growth Mindset’” Education Week. Education Week, 22 Sept. 2015. Web. 02 Oct. 2015.

Listeners: evaluative, interpretive, generative

What type of listener are am I right now? Do I know what modes of listening I use? How might I improve as a listener? What if I actively choose to practice?

Listening informs questioning. Paul Bennett says that one of the keys to being a good questioner is to stop reflexively asking so many thoughtless questions and pay attention— eventually, a truly interesting question may come to mind. (Berger, 98 pag.)

I’ve been studying a paper Gail Burrill (@GailBurrill) shared with us a couple of weekends ago.  The paper, Mathematicians’ Mathematical Thinking for Teaching: Responding to Students’ Conjectures by Estrella Johnson, Sean Larsen, Faith Rutherford of Portland State University, discusses three types of listening: evaluative, interpretive, and generative.

The term evaluative listening is characterized by Davis (1997) as one that “is used to suggest that the primary reason for listening in such mathematical classrooms tends to be rather limited and limiting” (p. 359). When a teacher engages in evaluative listening the goal of the listening is to compare student responses to the “correct” answer that the teacher already has in mind. Furthermore, in this case, the student responses are largely ignored and have “virtually no effect on the pre-specified trajectory of the lesson” (p. 360).

When a teacher engages in interpretive listening, the teacher is no longer “trying simply to assess the correctness of student responses” instead they are “now interested in ‘making sense of the sense they are making’” (Davis, 1997, p. 365). However, while the teacher is now actively trying to understand student contribution, the teacher is unlikely to change the lesson in response.

Finally, generative listening can “generate or transform one’s own mathematical understanding and it can generate a new space of instructional activities” (Yackel et al., 2003, p. 117) and is “intended to reflect the negotiated and participatory nature of listening to students mathematics” (p. 117). So, when a teacher is generatively listening to their students, the student contributions guide the direction of the lesson. Rasmussen’s notion of generative listening draws on Davis’ (1997) description of hermeneutic listening, which is consistent with instruction that is “more a matter of flexible response to ever-changing circumstances than of unyielding progress towards imposed goals” (p. 369).

If you’d like to read about these three types of listening the authors continue their paper with a case study.

Evaluative listeners seek correct answers, and all answers are compared to the one deemed correct from a single point of view.

Interpretive listeners seek sense making.  How are learners processing to produce solutions to tasks? What does the explanation show us about understanding?

Generative listeners seek next steps and questions themselves. In light of what was just heard, what should we do next? And, then they act.

For assessment to function formatively, the results have to be used to adjust teaching and learning; thus a significant aspect of any program will be the ways in which teachers make these adjustments. (William and Black, n. pag)

“Great teachers focus on what the student is saying or doing,” he says, “and are able, by being so focused and by their deep knowledge of the subject matter, to see and recognize the inarticulate stumbling, fumbling effort of the student who’s reaching toward mastery, and then connect to them with a targeted message.” (Coyle, 177 pag.)

What if we empower and embolden learners to ask the questions they need to ask by improving the way we listen and question?

Unless you ask questions, nobody knows what you are thinking or what you want to know.” (Rothstein and Santana, 135 pag.)

How might we practice generative listening to level up in the art of questioning? What is we listen to inform our questioning?

How might we collaborate to learn and grow as listeners and questioners?


Berger, Warren (2014-03-04). A More Beautiful Question: The Power of Inquiry to Spark Breakthrough Ideas . BLOOMSBURY PUBLISHING. Kindle Edition.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

Davis, B. (1997). Listening for difference: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education. 28(3). 355–376.

Johnson, E., Larsen, S., Rutherford (2010). Mathematicians’ Mathematicians’ Mathematical Thinking for Teaching: Responding to Students’ Conjectures. Thirteenth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education Conference on Research in 
Undergraduate Mathematics Education. Raleigh, NC. Retrieved from http://sigmaa.maa.org/rume/crume2010/Archive/JohnsonEtAl.pdf on September 12, 2015.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan, and Paul Black. “Inside the Black Box: Raising Standards Through Classroom Assessment.” The College Cost Disease (2011): n. pag. WEA Education Blog. Web. 13 Sept. 2015.

Yackel, E., Stephan, M., Rasmussen, C., Underwood, D. (2003). Didactising: Continuing the work of Leen Streefland. Educational Studies in Mathematics. 54. 101–126.

Intersection of struggle and hope (TBT Remix)

The trick is to choose a goal just beyond your present abilities; to target the struggle. Thrashing blindly doesn’t help. Reaching does. (Coyle, 19 pag.)

When learners are thrashing around blindly, how might we serve as refuge for support, encouragement, and a push in a new direction? (And, what if one of the learners is me?)

Many days we stand in the intersection of struggle and hope.

We can observe our children carefully and look into their eyes and say, “Can I tell you what a great person you are?” and follow-up with concrete examples of the way they give amazing hugs and how kindly they treat their friends.  This is the stuff of our most important relationships: Aiming to understand and be understood. (Lehman, Christopher, and Kate Roberts)

… some teachers preached and practiced a growth mindset. They focused on the idea that all children could develop their skills, and in their classrooms a weird thing happened. It didn’t matter whether students started the year in the high- or the low-ability group. Both groups ended the year way up high. It’s a powerful experience to see these findings. The group differences had simply disappeared under the guidance of teachers who taught for improvement, for these teachers had found a way to reach their “low-ability” students. (Dweck, Carol)

Move the fulcrum so that all the advantage goes to a negative mindset, and we never rise off the ground. Move the fulcrum to a positive mindset, and the lever’s power is magnified— ready to move everything up. (Achor, Shawn.)

To pursue bright spots is to ask the question “What’s working, and how can we do more of it?” Sounds simple, doesn’t it? Yet, in the real world, this obvious question is almost never asked. Instead, the question we ask is more problem focused: “What’s broken, and how do we fix it?” (Heath, Chip and Dan Heath)

And so the challenge of our future is to say, are we going to connect and amplify positive tribes that want to make things better for all of us?  (Godin, Seth)

Move the fulcrum. Pursue bright spots. Amplify to make things better.

Aim to understand and to be understood.


Intersection of struggle and hope was originally published on December 10, 2014.


Achor, Shawn (2010-09-14). The Happiness Advantage: The Seven Principles of Positive Psychology That Fuel Success and Performance at Work (Kindle Locations 947-948). Crown Publishing Group. Kindle Edition.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc. Kindle Edition.

Dweck, Carol (2006-02-28). Mindset: The New Psychology of Success (Kindle Locations 1135-1138). Random House, Inc.. Kindle Edition.

Heath, Chip; Heath, Dan (2010-02-10). Switch: How to Change Things When Change Is Hard (p. 45). Random House, Inc.. Kindle Edition.

Lehman, Christopher, and Kate Roberts. Falling in Love with Close Reading: Lessons for Analyzing Texts and Life. N.p.: n.p., n.d. Print.

Transcript: Seth Godin – The Art of Noticing, and Then Creating.” On Being. N.p., n.d. Web. 09 Dec. 2014.

What we don’t remember about the foundation…

I wonder if, when the house is finished, we forget the foundational infrastructure required for function.  How does water get into and out of my house? Who ran the wires so that our lamps illuminate our space? Who did the work, and what work was done, prior to the slab being poured?

When we recall a basic multiplication fact, it’s like flipping a light switch in our house. The electrical wiring allowing us to turn on the light is linked to sound, safe, and deeply connected infrastructure. (K. Nims, personal communication, August 30, 2015)

Just like the light switch is not part of the foundation, memorization of multiplication facts is also not foundational. It is efficient and functional.  Efficiency must not trump understanding.

We need people who are confident with mathematics, who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve. (Boaler, n. pag.)

Screen Shot 2015-08-30 at 7.45.22 PMStudents who rely solely on the memorization of math facts often confuse similar facts. (O’Connell, 4 pag.)

Students must first understand the facts that they are being asked to memorize. (O’Connell, 3 pag.)

What if we have forgotten all the hard work that came prior to the task of memorizing our multiplication facts?

Do we remember learning about multiplication as repeated addition? Have we forgotten the connection between multiplication, arrays, and area?

Conceptual understanding of multiplication lays a foundation for deeper understanding of many mathematical topics.  Memorizing facts denies learners the opportunity to connect ideas, exercise flexibility, and interact with multiple strategies.

The goal is to have confident, competent, critical thinkers. Let’s remember that a strong foundation has many unseen components.  What if we slow down to develop deep understanding of the numeracy of multiplication?

Second, going slow helps the practitioner to develop something even more important: a working perception of the skill’s internal blueprint – the shape and rhythm of the interlocking skill circuits.”  (Coyle, 85 pag.)

How might we serve our learners by expecting them to show what they know more than one way?


Boaler, Jo. “The Stereotypes That Distort How Americans Teach and Learn Math.” The Atlantic. Atlantic Media Company, 12 Nov. 2013. Web. 30 Aug. 2015.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

O’Connell, Susan, and John SanGiovanni. Mastering the Basic Math Facts in Multiplication and Division: Strategies, Activities & Interventions to Move Students beyond Memorization. Portsmouth, NH: Heinemann, 2011. Print.