Tag Archives: 5 Practices for Orchestrating Productive Mathematics Discussions

Agenda: Embolden Your Inner Mathematician (10.17.18) Week 6

Week Six of Embolden Your Inner Mathematician

We commit to curation of best practices, connections between mathematical ideas, and communication to learn and share with a broad audience.

Course Goals:
At the end of the semester, teacher-learners should be able to say:

  • I can work within NCTM’s Eight Mathematical Teaching Practices for strengthening the teaching and learning of mathematics.
  • I can exercise mathematical flexibility to show what I know in more than one way.
  • I can make sense of tasks and persevere in solving them.

Today’s Goals

At the end of this session, teacher-learners should be able to say:

  • I can use and connect mathematical representations. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them. (#SMP-1)
  • I can show my work so that a reader understands without have to ask me questions.

From Principles to Actions: Ensuring Mathematical Success for All

Use and connect mathematical representations: Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

From Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5

In ambitious teaching, the teacher engages students in challenging tasks and collaborative inquiry, and then observes and listens as students work so that she or he can provide an appropriate level of support to diverse learners.  The goal is to ensure that each and every student succeeds in doing meaningful, high-quality work, not simply executing procedures with speed and accuracy.(Smith, 4 pag.)

Learning Progressions for today’s goals:

  • I can use and connect mathematical representations. (#NCTMP2A)

  • I can show my work so that a reader understands without have to ask me questions.

Tasks:

  • Visual representation of multiplication, exponents, subtraction. (Connect 2nd-5th grade with Algebra I and II.)
  • Apples and Bananas task (see slide deck)

What the research says:

Not only should students be able to understand and translate between modes of representations but they should also translate within a specific type of representation. [Smith, pag. 139] 

Equitable teaching of mathematics includes a focus on multiple representations. This includes giving students choice in selecting representations and allocating substantial instructional time and space for students to explore, construct, and discuss external representations of mathematical ideas. [Smith, pag. 141]

Too often students see mathematics as isolated facts and rules to be memorized. [Smith, pag. 141]

\Anticipated work and thinking:

Slide deck:

[Cross posted at Sum it up and Multiply it out]


Gough, Jill, and Jennifer Wilson. “#LL2LU Learning Progressions.” Experiments in Learning by Doingor Easing the Hurry Syndrome.WordPress, 04 Aug. 2014. Web. 11 Mar. 2017.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5. The National Council of Teachers of Mathematics, 2017.

Agenda: Embolden Your Inner Mathematician (09.19.18) Week 3

Week Three of Embolden Your Inner Mathematician

We commit to curation of best practices, connections between mathematical ideas, and communication to learn and share with a broad audience.

Course Goals:
At the end of the semester, teacher-learners should be able to say:

  • I can work within NCTM’s Eight Mathematical Teaching Practices for strengthening the teaching and learning of mathematics.
  • I can exercise mathematical flexibility to show what I know in more than one way.
  • I can make sense of tasks and persevere in solving them.

Today’s Goals
At the end of this session, teacher-learners should be able to say:

  • I can use and connect mathematical representations. (#NCTMP2A)
  • I can show my work so a reader understand without asking me questions.

From Principles to Actions: Ensuring Mathematical Success for All

Use and connect mathematical representations:Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

Learning Progressions for today’s goals:

  • I can use and connect mathematical representations.
  • I can use and connectmathematical representations.
  • I can show my work so that a reader understands without have to ask me questions.

Tasks:

What the research says:

From Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5:

High-level tasks not only hold high mathematical expectations for every student, one aspect of equitable classrooms, they also “allow multiple entry points and varied solution strategies” (NCTM 2014, p. 17).

…Positioning students as valuable contributors to mathematical work, even as authors and owners of mathematical ideas, supports the development of positive mathematical identities and agency as mathematical thinkers. [p. 72-73]

Too often students see mathematics as isolated facts and rules to be memorized. … students are expected to develop deep and connected knowledge of mathematics and are engaged in learning environments rich in use of multiple representations.

Mathematics learning is not a one size fits all approach …, meaning not every child is expected to engage in the mathematics in the same way at the same time. … the diversity of their sense-making approaches is reflected in the diversity of their representations. [p. 140]

Examples of Anticipated thinking and outcomes:

[Cross posted on Sum it up and Multiply it out]


Gough, Jill, and Jennifer Wilson. “#LL2LU Learning Progressions.” Experiments in Learning by Doing or Easing the Hurry Syndrome.WordPress, 04 Aug. 2014. Web. 11 Mar. 2017.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5. The National Council of Teachers of Mathematics, 2017.

Smith, Margaret, and Mary Kay Stein. 5 Practices for Orchestrating Productive Mathematics Discussions.The National Council of Teachers of Mathematics, 2018.

 

I can elicit and use evidence of student thinking #NCTMP2A #LL2LU

We strive to grow in our understanding of the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All. This research-informed framework of teaching and learning reflects a core set of high leverage practices and essential teaching skills necessary to promote deep learning of mathematics.

Elicit and use evidence of student thinking.

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

In order to support our teaching teams as they stretch to learn more, we drafted the following learning progressions. We choose to provide a couple of pathways to focus teacher effort, understanding, and action.

When working with teacher teams to elicit and use evidence of student thinking, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Dylan Wiliam’s Embedding Formative Assessment: Practical Techniques for K-12 Classrooms along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around eliciting evidence of student thinking, we anticipate multiple ways learners might approach a task, empower learners to make their thinking visible, celebrate mistakes as opportunities to learn, and ask for more than one voice to contribute.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, anticipate what learners will produce, and brainstorm how we might select, sequence, and connect learners’ ideas.

How will classroom culture grow as we focus on the five key strategies we studied in Embedding Formative Assessment: Practical Techniques for F-12 Classrooms by Dylan Wiliam and Siobhan Leahy?

  • Clarify, share, and understand learning intentions and success criteria
  • Engineer effective discussions, tasks, and activities that elicit evidence of learning
  • Provide feedback that moves learning forward
  • Activate students as learning resources for one another
  • Activate students as owners of their own learning

We call questions that are designed to be part of an instructional sequence hinge questions because the lessons hinge on this point. If the check for understanding shows that all students have understood the concept, you can move on. If it reveals little understanding, the teacher might review the concept with the whole class; if there are a variety of responses, you can use the diversity in the class to get students to compare their answers. The important point is that you do not know what to do until the evidence of the students’ achievement is elicited and interpreted; in other words, the lesson hinges on this point. (Wiliam, 88 pag.)

To strengthen our understanding of using evidence of student thinking, we plan our hinge questions in advance, predict how we might sequence and connect, adjust instruction based on what we learn – in the moment and in the next team meeting – to advance learning for every student. We share data within our team to plan how we might differentiate to meet the needs of all learners.

How might we team to strengthen and deepen our commitment to ensuring mathematical success for all?

What if we anticipate, monitor, select, sequence, and connect student thinking?

How might we elicit and use evidence of student thinking to advance learning for every learner?

Cross posted on Easing the Hurry Syndrome


Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

Wiliam, Dylan; Leahy, Siobhan. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. (Kindle Locations 2191-2195). Learning Sciences International. Kindle Edition.

I can establish mathematics goals to focus learning #NCTMP2A #LL2LU

We strive to grow in our understanding of the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All. This research-informed framework of teaching and learning reflects a core set of high leverage practices and essential teaching skills necessary to promote deep learning of mathematics.

Establish mathematics goals to focus learning.

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

In order to support our teaching teams as they stretch to learn more, we drafted the following learning progressions. We choose to provide a couple of pathways to focus teacher effort, understanding, and action.

When working with teacher teams to establish mathematics goals to focus learning, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning by John Hattie, Douglas Fisher, and Nancy Frey along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around establishing mathematics goals, we anticipate, connect to prior knowledge, explain the mathematics goals to learners, and teach learners to use these goals to self-assess and level up.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle.

Once prior knowledge is activated, students can make connections between their knowledge and the lesson’s learning intentions. (Hattie, 44 pag.)

To strengthen our understanding of using mathematics goals to focus learning, we make the learning goals visible to learners, ask assessing and advancing questions to empower students, and listen and respond to support learning and leveling up.

Excellent teachers think hard about when they will present the learning intention. They don’t just set the learning intentions early in the lesson and then forget about them. They refer to these intentions throughout instruction, keeping students focused on what it is they’re supposed to learn. (Hattie, 55-56 pag.)

How might we continue to deepen and strengthen our ability to advance learning for every learner?

What if we establish mathematics learning goals to focus learning?

Cross posted on Easing The Hurry Syndrome


Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

Overview: Embolden Your Inner Mathematician

Taking action on known national goals, 15 Trinity School teacher-learner-leaders will begin a semester-long professional learning journey to deepen our understanding of NCTM’s Effective Mathematics Teaching Practices.

We commit to curation of best practices, connections between mathematical ideas, and communication to learn and share with a broad audience.

Goals:

At the end of the semester, teacher-learners should be able to say:

  • I can work within NCTM’s Eight Mathematical Teaching Practices for strengthening the teaching and learning of mathematics.
  • I can exercise mathematical flexibility to show what I know in more than one way.
  • I can make sense of tasks and persevere in solving them.

Facilitators:

Weekly schedule of topics:

Sep. 6 Subitizing and Number Talks:
 Elicit and use evidence of student thinking
Sep. 13 Numeracy and Visual Learning:
 Elicit and use evidence of student thinking
Sep. 20 Make sense of tasks and persevere in solving them:
 Facilitate meaningful mathematical discourse
Sep. 27 Attend to Precision and Construct a Viable Argument:
Facilitate meaningful mathematical discourse
Oct. 4 Strengthen Mathematical Flexibility:
Use and connect mathematical representations
Oct. 11 Visual Patterns – Strength Mathematical Flexibility:
Use and connect mathematical representations
Oct. 18 Mathematizing Children’s Literature (part 1):
Implement tasks that promote reasoning and problem solving
Oct. 25 Mathematizing Children’s Literature (part 2):
Implement tasks that promote reasoning and problem solving
Nov. 1 Designing Intentional Number Strings:
Building Procedural Fluency from Conceptual Understanding
Nov. 8 Using Appropriate Tools Strategically:
Building Procedural Fluency from Conceptual Understanding
Nov. 15 Empowering Learners:
Establish mathematical goals to focus learning
Nov. 22 Thanksgiving
Nov. 29 Deep Practice – challenged and learning
Support productive struggle in learning mathematics
Dec. 6 The Art of Questioning or Making Sense of Tasks part 2
Support productive struggle in learning mathematics
Dec. 13 14 Review and Reflection:
Pose purposeful questions

Anchor Resources:

Norms:


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

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-. The National Council of Teachers of Mathematics, 2017.

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6-8. The National Council of Teachers of Mathematics, 2017.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

Anticipating @IllustrateMath’s 6.RP Overlapping Squares

To anchor our work in differentiation and mathematical flexibility, we use NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein.

Kristi Story, Becky Holden, and I worked together during our professional learning time to meet the goals for the session shown below.

From  NCTM’s 5 Practices, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle.

The learning goals for students include:

I can use ratio reasoning to solve problems and understand ratio concepts.

I can make sense of tasks and persevere in solving them.

I can look for and make use of structure.

I can notice and note to make my thinking visible.

Kristi selected Illustrative Math’s  6.RP Overlapping Squares task for students. Here are the ways we anticipated how students would approach and engage with the task.

This slideshow requires JavaScript.

Our plan for helping students who are stuck includes providing and encouraging the use of a graphing tool such as graph paper or TI-Nspire software installed on their MacBooks. We also intend to use the following learning progressions.

I can make sense of tasks and persevere in solving them.

I can look for and make use of structure.

Finally, we also want our learners to work on how they show their work.

#ShowYourWork Subtraction

When mathematics classrooms focus on numbers, status differences between students often emerge, to the detriment of classroom culture and learning, with some students stating that work is “easy” or “hard” or announcing they have “finished” after racing through a worksheet. But when the same content is taught visually, it is our experience that the status differences that so often beleaguer mathematics classrooms, disappear.  – Jo Boaler


Boaler, Jo, Lang Chen, Cathy Williams, and Montserrat Cordero. “Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning.” Journal of Applied & Computational Mathematics 05.05 (2016): n. pag. Youcubed. Standford University, 12 May. 2016. Web. 18 Mar. 2017.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

PD Planning: Number Talks and Number Strings

As we begin the second part of our school year and as the calendar changes from 2016 to 2017, we review our goals.

The leaders of our math committee set the following goals for this school year.

Goals:

  • Continue our work on vertical alignment.
  • Expand our knowledge of best practices and their role in our current program.
  • Share work with grade level teams to grow our whole community as teachers of math.
  • Raise the level of teacher confidence in math.
  • Deepen, differentiate, and extend learning for the students in our classrooms.

Our latest action step works on scaling these goals in our community. The following shows our plan to build common understanding and language as we expand our knowledge of numeracy.  Over the course of two days, each math teacher (1st-6th grade) participated in 3-hours of professional learning.

Jan10-13Agenda.png
Sample timestamp from PD sessions.

Our intentions and purpose:

Screen Shot 2017-01-15 at 8.35.21 AM.png

Screen Shot 2017-01-15 at 8.35.31 AM.png

We started with a number talk and a number string from Kristin Gray‘s NCTM Philadelphia presentation. We challenged ourselves to anticipate the ways our learners answer the following.

kristingraynumbertalk

We also referred to Making Number Talks Matter to find Humphreys and Parker’s four strategies for multiplication.  We pressed ourselves to anticipate more than one way for each multiplication strategy to align with Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions.

Screen Shot 2017-01-15 at 7.23.12 PM.pngFrom our earlier work with Lisa Eickholdt, we know that our ability to talk about a strategy directly impacts our ability to teach the strategy.  What can be learned if we show what we know more than one way? How might we learn from each other if we make our thinking visible?

Screen Shot 2017-01-15 at 8.46.22 PM.pngAfter working through Humphreys and Parker’s strategies (and learning new strategies), we transitioned to the number string from Kristin‘s presentation.

Screen Shot 2017-01-15 at 7.41.14 PM.pngThe goal for the next part of the learning session offered teaching teams the opportunity to select a number string from one of the Minilessons books shown below.  Each team selected a number string and worked to anticipate according to Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions.

To practice, each team practiced their number string and the other grade-level teams served as learners.  When we share and learn together, we strengthen our understanding of how to differentiate and learn deeply.

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.
—John Hattie, Doug Fisher, Nancy Frey

As we begin the second part of our school year and as the calendar changes from 2016 to 2017, what action steps are needed to reach our goals?


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) (p. 136). SAGE Publications. Kindle Edition.

Humphreys, Cathy; Parker, Ruth (2015-04-21). Making Number Talks Matter (Kindle Locations 1265-1266). Stenhouse Publishers. Kindle Edition.

Norris, Kit; Schuhl, Sarah (2016-02-16). Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K-5 Learners (Kindle Locations 4113-4115). Solution Tree Press. Kindle Edition.

Smith, Margaret Schwan., and Mary Kay. Stein. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics, 2011. Print.