Tag Archives: 5 Practices for Orchestrating Productive Mathematics Discussions

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.

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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.

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Sample timestamp from PD sessions.

Our intentions and purpose:

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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.

Mashup: #5Practices and #WODB

What if we engage in purposeful instructional talk as a team to focus on the instructional core? How might we design and implement a differentiated action plan across our grade to meet all learners where they are? What if we learn to integrate Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions?

Becky Holden (@bholden86), our EED math specialist, and I are working on formative assessment using anticipate and monitor, the first two of Smith and Stein’s 5 Practices.  While we don’t want a template, we keep using this sketch to plan, think, and share.

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It’s still a work in progress.  We’d love to know what you think.

Brian Toth (@btoth4thgrade) shared his learners and time with me so that I could play and work with our students on SMP-6, attend to precision.

The following three sketches are the notes and jots of what we anticipate our learners will think and say prior to the start of class.

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Our purposeful instructional talk set learning goals of I can attend to precision, and I can demonstrate flexibility to show what I know more than one way.  From our students, we are looking for complete sentences with strong vocabulary and word choice.  We want to see internal motivation to think deeply and a willingness to go past a surface initial answer. We know that we are growing toward constructing viable arguments. From our team, we are collaborating to learn more about our learners, to become more flexible ourselves, and to notice and note details of student answers so that we can design and implement a differentiated action plan across our grade to meet all learners where they are.

Toth-wodb

What if we learn and practice together? How might we grow in confidence, competence, precision, and flexibility?

Read with Me? Book study: 5 Practices for Orchestrating Productive Conversations

What if we study and practice, together, to embed formative assessment into our daily practice and learning?

After the success of the slow-chat book study on Embedding Formative Assessment we plan to engage in another slow chat book study.

A few years ago, as we embraced focusing our classrooms on the Standards for Mathematical Practice, a number of our community began reading and using the book by Peg Smith and Mary Kay Stein, 5 Practices for Orchestrating Productive Mathematics Discussions.

This book has been transformational to many educators, and there is also a companion book focused on the science classroom, 5 Practices for Orchestrating Task-Based Discussions in Science, by Jennifer Cartier and Margaret S. Smith.

Both books are also available in pdf format and NCTM offers them together as a bundle.

Simultaneous Study
: As our community works with both math and science educators, we are going to try something unique in reading the books simultaneously and sharing ideas using the same hashtag.

We know that reading these books, with the emphasis on classroom practices, will be worth our time. In addition to encouraging those who have not read them, we expect that those who have read them previously will find it beneficial to re-read and share with educators around the world.

Slow Chat Book Study
: For those new to this idea of a “slow chat book study”, we will use Twitter to share our thoughts with each other, using the hashtag #T3Learns.

With a slow chat book study you are not required to be online at any set time. Instead, share and respond to others’ thoughts as you can. Great conversations will unfold – just at a slower pace.

When you have more to say than 140 characters, we encourage you to link to blog posts, pictures, or other documents. There is no need to sign up for the study – just use your Twitter account and the hashtag #T3Learns when you post your comments.

Don’t forget to search for others’ comments using the hashtag #T3Learns.

Book Study Schedule
: We have established the following schedule and daily prompts to help with sharing and discussion. This will allow us to wrap up in early June.

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The content of the Math and Science versions line up fairly well, with the exception of the chapters being off by one.

We continue to used the following prompts to spur discussion.

SMP5: Use Appropriate Tools Strategically #LL2LU

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We want every learner in our care to be able to say

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

But…What if I think I can’t? What if I have no idea what are appropriate tools in the context of what we are learning, much less how to use them strategically? How might we offer a pathway for success?

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

We still might need some conversation about what it means to use appropriate tools strategically. Is it not enough to use appropriate tools? Would it help to find a common definition of strategically to use as we learn? And, is use appropriate tools strategically a personal choice or a predefined one?

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How might we expand our toolkit and experiment with enough tools to display confidence when explaining why the selected tools are appropriate and effective for the solution pathway used?  What if we practice with enough tools that we make strategic – highly important and essential to the solution pathway – choices?

What if apply we 5 Practices for Orchestrating Productive Mathematics Discussions to learn with and from the learners in our community?

  • Anticipate what learners will do and why strategies chosen will be useful in solving a task
  • Monitor work and discuss a variety of approaches to the task
  • Select students to highlight effective strategies and describe a why behind the choice
  • Sequence presentations to maximize potential to increase learning
  • Connect strategies and ideas in a way that helps improve understanding

What if we extend the idea of interacting with numbers flexibly to interacting with appropriate tools flexibly?  How many ways and with how many tools can we learn and visualize the following essential learning?

I can understand solving equations as a process of reasoning and explain the reasoning.  CCSS.MATH.CONTENT.HSA.REI.A.1

What tools might be used to learn and master the above standard?

  • How might learners use algebra tiles strategically?
  • When might paper and pencil be a good or best choice?
  • What if a learner used graphing as the tool?
  • What might we learn from using a table?
  • When is a computer algebra system (CAS) the go-to strategic choice?

Then, what are the conditions which make the use of each one of these tools appropriate and strategic?

[Cross posted on Easing the Hurry Syndrome]

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“The American Heritage Dictionary Entry: Strategically.” American Heritage Dictionary Entry: Strategically. N.p., n.d. Web. 08 Sept. 2014.

Encouraging and anticipating mathematical flexibility

From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

I wonder how many times I’ve taught “the one way” to solve a problem without considering other pathways for success. Yikes!

IMG_4846After completing Lesson 4 from How to Learn Math: for Students, A-Sunshine, my 4th grader, asked me to solve another multiplication problem.  I wondered how many ways I could show my work and demonstrate flexibility in numeracy.  The urge to solve this multiplication problem in the traditional way was strong, but how many ways could I show how to multiply 44 x 18? How flexible am I when it comes to numeracy? Is the traditional method the most efficient? Are there other ways to show 44 x 18 that might demonstrate understanding?

Screen Shot 2014-08-16 at 7.42.16 PMHow might offer opportunities to express flexibility? Will learners share thinking and strategies? How will we facilitate discussions where multiple ways to “be right” are discussed?  What if we embrace Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussion to anticipate, monitor, select, sequence, and make connections between student responses?

If my solutions represent the work and thinking of five different students, in what order would we sequence student sharing, and are we prepared to help make connections between different student responses?

How is flexibility encouraged and practiced? Is it expected? Is it anticipated?

And…does it stop with numbers? I don’t think so.  We want our learners of algebra to be flexible with

  • the slope-intercept, point-slope, and standard forms of a line and
  • the standard form, vertex form, and factored form of a parabola.

The list could go on and on.

How might we narrow what separates high achievers from low achievers? If number flexibility is a gateway to success, what actions are we willing to take to encourage, build confidence, and illuminate multiple pathways to success?

Summer Reading 2014 – Themes and Choices

As a team of 150 learners charged with a responsibility of developing and maintaining a learning community for ourselves and the 640 children that we love and care for, how do we learn and grow when we are apart? We workshop, plan, play, rest, and read to name just a few of our actions and strategies.  How do we model learner choice? We know student-learners need and deserve differentiated learning opportunities.  Don’t all learners?

There is so much to learn, practice, prototype, and consider.  How do we learn and share? What if we divide, conquer, and share to learn?


From: Jill Gough
Date: Friday, April 11, 2014 12:13 PM
To: All Trinity
Subject: Summer Reading 2014

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Hi,
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Summer reading for our community offers choices again this year.  We offer three themes from which to choose.  You may want to continue with the Art of Questioning, or you may want to explore Creativity or Social-Emotional as an interest.  Each theme offers three choices, and if available, you may choose to read using a traditional book, a Kindle book, or an audio book.
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A quick note and thank you: Last year Laurel Martin asked me why the books weren’t hyperlinked to Amazon so that we could quickly read reviews.  Great idea! (Hmm…I didn’t know how to hyperlink an image using Pages…but I do now. Thank you Laurel for pushing me to learn!) So, thanks to Laurel, if you want to read reviews, just click on a book in the flyer.
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We will use the 4As protocol to debrief during Pre-Planning.  We are also going to schedule a Wednesday afternoon so that our community can hear and share the big ideas from every book.
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Please see the attached flyer for information and links to additional information and a form to request your book.  Would you please select a book by Friday, April 25 so that we can have it for you before we leave in May?
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Thank you,
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Jill
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Here’s our flyer:

And, our version of the 4 As protocol worksheet: