Tag Archives: Standards for Mathematical Practice

Goal work: learn more math, study the Practices

The math committee met this week to work on our goals. We agreed that, for the rest of this school year, we would spend half of our time on learning more math and the other half studying to learn more about the Standards For Mathematical Practice.

We met this week to learn more math and to discuss Chapter 1, Mathematical 1: Make Sense of Problems and Persevere in Solving Them in Beyond Answers: Exploring Mathematical Practices with Young Children by Mike Flynn.

Yearlong Goals:

  • We can learn more math.
  • We can share work with grade level teams to grow our whole community as teachers of math.
  • We can deepen our understanding of the Standards For Mathematical Practice.

Today’s Goals:

  • I can make sense of tasks and persevere in solving them.
  • I can reason abstractly and quantitatively.
  • I can look for and make use of structure.


Learning Plan

3:05 5 min Quick scan of Jo’s YouCubed article (pp. 2, 11)
3:05 20 min Solving equations visually to make sense of the algebra
(Learn more math)

productive-struggle-4 productive-struggle-3

3:25 5 min Book Club warm-up

3:30 20 min Use Visible Thinking Routines to guide discussion of Chapter One: Make Sense and Persevere
(deepen our understanding of the SMPs.)

3:55 5 min Feedback – “I learned…, “I liked…,”I felt…

Read Chapter 2: Reason Abstractly and Quantitatively

Update on PD (Goal: Scale our work to our teams.)

When we set purposeful team goals, we help each other make progress, and we use our time intentionally.

Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

Van de Walle, John. Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2. Boston: Pearson, 2014. Print.

VTR: Sentence-Phrase-Word to dig deeper into Standards for Mathematical Practice

From Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners:

Sentence-Phrase-Word helps leaners to engage with and make meaning from text with a particular focus on capturing the essence of the test or “what speaks to you.” It fosters enhanced discussion while drawing attention to the power of language. (Ritchhart, Church, Morrison, 207 pag.)

Screen Shot 2014-10-19 at 7.26.45 PMWhat if we read and learn together, as a team? How might we develop deeper understanding?

Screen Shot 2014-10-19 at 7.29.46 PMAs a team of learners, we first read Make sense of problems and persevere in solving them independently and highlighted a sentence, phrase, and work that resonated with us.  In round robin fashion, we read aloud our selected sentence so that every member of the team heard what every other member of the team felt was important.  Just the act of hearing another voice read and callout an idea was impactful.

After completing the Sentence-Phrase-Word Visible Thinking Routine for Make sense of problems and persevere in solving them, we asked everyone to take another Standard for Mathematical Practice to read and markup, highlighting a sentence, a phrase, and a word.

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We divided into teams where each of the remaining Standards of Mathematical Practice were represented.  Each learner shared the SMP that they read highlighting a selected sentence, phrase, and word. My notes are shared below. I was amazed at the new ideas I heard from my colleagues when using this routine.

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Seek diversity of thought. Listen to others.  Hear differently. Promote engagement, understanding, and independence for all.


Ritchhart, Ron, Mark Church, and Karin Morrison. “Sentence-Phrase-Word.”Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. 207-11. Print.


Deep Dive into Standards of Mathematical Practice

As a team, we commit to make learning pathways visible. We are working on both horizontal and vertical alignment.  We seek to calibrate our practices with national standards.

On Friday afternoon, we met to take a deep dive into the Standards of Mathematical Practice. Jennifer Wilson joined us to coach, facilitate, and learn. We are grateful for her collaboration, inspiration, and guidance.

The pitch:

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The plan:


  • I can anticipate Standards for Mathematical Practice that learners will employ during this lesson.
  • I can begin to design lessons incorporating national standards, a learning progression, and a formative assessment plan.


  • Safe space
    • I can talk about what I know, and I can talk about what I don’t know.
    • I can be brave, vulnerable, kind, and considerate to myself and others while learning.
  • Celebrate opportunities to learn
    • I can learn from mistakes, and I can celebrate what I thought before and now know.


Learning Plan:

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The learning progressions:

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The slide deck:

As a community of learners, we

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SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 2)

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

I can reason abstractly and quantitatively.

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?
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A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?
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In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?

Level 4:
I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations.

Level 3:
I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.   

I can identify and connect the units involved using an equation, graph, or table.

I can attend to and document the meaning of quantities throughout the problem-solving process.

I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion.

Level 2:
I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:
I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

What evidence of contextualizing and decontextualizing do you see in the work below?

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[Cross-posted on Easing the Hurry Syndrome]

SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

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

I can reason abstractly and quantitatively.

I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively.  (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)

Level 4:
I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:
I can reason abstractly and quantitatively.

Level 2:
I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:
I can define variables and constants in a problem situation and connect the appropriate units to each.

You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively.  Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway.  How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.

[Cross-posted on Easing the Hurry Syndrome]


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.

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]


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

Visual: SMP-3 Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

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

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How might we facilitate learning and grow our culture where critique is sought and embraced?

From Step 1: The Art of Questioning in The Falconer: What We Wish We Had Learned in School.

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

This paragraph connects to a Mr. Sun quote from Step 0: Preparation.

But there are many more subtle barriers to communication as well, and if we cannot, or do not choose to overcome these barriers, we will encounter life decisions and try to solve problems and do a lot of falconing all by ourselves with little, if any, success. Even in the briefest of communications, people develop and share common models that allow them to communicate effectively.  If you don’t share the model, you can’t communicate. If you can’t communicate, you can’t teach, learn, lead, or follow.  (Lichtman, 32 pag.)

How might we offer a pathway for success? What if we provide practice in the art of questioning and the action of seeking feedback? What if we facilitate safe harbors to share  thinking, reasoning, and perspective?

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Level 4:
I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:
I can construct viable arguments and critique the reasoning of others.

Level 2:
I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:
I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

How might we design opportunities for intentional, focused peer-to-peer discourse? What if we share a common model to improve communication, thinking, and reasoning?

[Cross-posted on Easing the Hurry Syndrome]


Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.