Tag Archives: Principles to Actions

Number Talks: developing fluency, flexibility, and conceptual understanding #AuthorAndIllustrate

How might we work on fluency (accuracy, flexibility, efficiency, and understanding) as we continue to teach and learn with students? What if our young learners are supposed to be fluent with their multiplication facts, but… they. ..just…aren’t!?

It really isn’t a surprise, right? Children learn and grow at different rates. We know that because we work with young learners everyday.  The question isn’t “Why aren’t they fluent right now?” It isn’t. It just isn’t. The question should be and is:

“What are we going to do, right now, to make this better
for every and each learner in our care?”

In Making Number Talks Matter, Cathy Humphreys and Ruth Parker write:

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don’t know it yet), and over time, the properties come to life in students’ own strategies. (Humphreys, 62 p.)

Humphreys and Parker continue:

Students who have experienced Number Talks come to algebra understanding the arithmetic properties because they have used them repeatedly as they reasoned with numbers in ways that made sense to them. This doesn’t happen automatically, though. As students use these properties, one of our jobs as teachers is to help students connect the strategies that make sense to them to the names of properties that are the foundation of our number system. (Humphreys, 77 p.)

So, that is what we will do. We commit to deeper and stronger mathematical understanding. And, we take action.

This week our Wednesday workshop focused on Literacy, Mathematics, and STEAM in grade level bands.  Teachers of our 4th, 5th, and 6th graders gathered to work together, as a teaching team, to take direct action to strengthen and deepen our young students’ mathematical fluency.

We began with the routine How Do You Know? routine from NCTM’s High-Yield Routines for Grades K-8 using this sentence:

81-25=14×4
How do you know?

Here’s how I anticipated the ways learners might think.

Paper.Productive Struggle.195

From The 5 Practices in Practice: Successfully Orchestrating Mathematical Discussion in your Middle School Classroom:

Anticipating students’ responses takes place before instruction, during the planning stage of your lesson. This practice involves taking a close look at the task to identify the different strategies you expect students to use and to think about how you want to respond to those strategies during instruction. Anticipating helps prepare you to recognize and make sense of students’ strategies during the lesson and to be able to respond effectively. In other words, by carefully anticipating students’ responses prior to a lesson, you will be better prepared to respond to students during instruction. (Smith, 37 p.)

How many strategies and tools do we use when modeling multiplication in our classroom? It is a matter of inclusion.

It is a matter of inclusion.

Every learner wants and needs to find their own thinking in their  community. This belonging, sharing, and learning matters. We make sense of the mathematics and persevere. We make sense of others thinking as they learn to construct arguments and show their thinking so that others understand.

Humphreys and Parker note:

They are learning that they have mathematical ideas worth listening to—and so do their classmates. They are learning not to give up when they can’t get an answer right away because they are realizing that speed isn’t important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra. (Humphreys, 62 p.)

As teachers, we must anticipate the myriad of ways students think and learn. And, as Christine Tondevold (@BuildMathMinds) tells us:

The strategies are already in the room.

Our job is to connect mathematicians and mathematical thinking.

From NCTM’s Principles to Actions:

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.

And:

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

What if we take up the challenge to author and illustrate mathematical understanding with and for our students and teammates?

Let’s work together to use and connect mathematical representations as we build procedural fluency from conceptual understanding.


Humphreys, Cathy. Making Number Talks Matter. Stenhouse Publishers. Kindle Edition.

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

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Agenda: Embolden Your Inner Mathematician (10.24.18) Week 7

Week Seven 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 implement tasks that promote reasoning and problem solving. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them. (#SMP-1)

From Principles to Actions: Ensuring Mathematical Success for All

Implement tasks that promote reasoning and problem-solving: Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Learning Progressions for today’s goals:

  • I can implement tasks that promote reasoning and problem-solving. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them.

Tasks:

  • Poetry and watercolor (a.k.a., the beauty of mathematics)
  • Phases of the moon (See slide deck)

What the research says:

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.)

Equitable teaching of mathematics focuses on going deep with mathematics, including developing a deep understanding of computational procedures and other mathematical rules, formulas, and facts. When students learn procedures with understanding, they are then able to use and apply those procedures in solving problems. When students learn procedures as steps to be memorized without strong links to conceptual understanding, they are limited in their ability to use the procedure. (Smith, 93 pag.)

Evidence of 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 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.

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 (10.10.18) Week 5

Week Five 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 implement tasks that promote reasoning and problem-solving. (#NCTMP2A)
  • 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

Implement tasks that promote reasoning and problem-solving:Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Learning Progressions for today’s goals:

  • I can implement tasks that promote reasoning and problem-solving. (#NCTMP2A)
  • I can show my work so that a reader understands without have to ask me questions.

Tasks:

  • Read and “do the math” from Each Orange Had 8 Slices by Paul Giganti Jr. (Author), Donald Crews (Illustrator).
  • Select, read, and mathematize a book of your choice. Plan a lesson for your students.

[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.

Sheep Won’t Sleep #Mathematizing Read Alouds – implement tasks that promote reasoning and problem solving

How might we deepen our understanding of numeracy using children’s literature? What if we mathematize our read aloud books to use them in math as well as reading and writing workshop?

Have you read Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s by Judy Cox?

This week’s Embolden Your Inner Mathematician session is designed to learn and practice both a Mathematics Teaching Practice and a Standard for Mathematical Practice.

Implement Tasks that Promote
Reasoning and Problem Solving.

Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Jennifer Wilson and I use the following learning progression to help teachers and teaching teams calibrate their work.

From the Standards for Mathematical Practice,

Construct viable arguments and
critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

We choose to reword this for our students. Instead of I can construct a viable argument, we say I can show my work so a reader understands having to ask me questions.

We use the following learning progression to help students self-assess and reach to deepen their learning.

Now, Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s by Judy Cox gives away the mathematical thinking on some pages. We decided to read the book and ask our students to listen and take notes as readers, writers, and mathematicians.  Mathematicians notice and note details, look for patterns, and ask questions.  To support listening and comprehension (a.k.a. empower learners to make sense and persevere), we created visuals for quasi-reader’s theater and spelled sheep, alpaca, llama, and yak.  (Level  2; check.)

We also practiced a keep the pace up and get kids collaborating instead of relying on the teacher strategy we are learning from Elizabeth Statmore.

And every day I used 10-2 processing to keep the pace up and get kids collaborating instead of relying on me. For every ten minutes of notes, I gave two minutes of processing time to catch up and collaborate on making their notes accurate. (Statmore, n pag.)

Instead of 10-2 processing, we took a minute after every couple of pages to intentionally turn and talk with a partner with the express purpose of comparing and improving our notes and mathematical communication.

As teachers, we are striving to implement tasks that promote reasoning and problem solving.   Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s is a counting book so 1st graders can tackle the math. 2nd and 3rd graders can use this to connect skip counting and repeated addition to multiplication and to use and connect mathematical representations. 4th and 5th graders can use this to use and connect mathematical representations while attending to precision. (Level 1; check.)

Here’s a messy version of how we anticipated student work and thinking.

These read-aloud moments open up the opportunity for rich discussion and engaging questions. Students have the opportunity for more organic and deeper understanding of mathematical concepts thanks to the book that brought them to life, and it is an engaging way to look at math through a different lens.

As Professor of Mathematics Education at the Stanford Graduate School of Education Jo Boaler explains in her book Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching, “Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, flexibility, and multiplicity of ideas, the mathematics comes alive for people.”


Boaler, Jo. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching (p. 115). Wiley. Kindle Edition.

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

Standards for Mathematical Practice.” Standards for Mathematical Practice. N.p., n.d. Web. 15 Dec. 2014.

Statmore, Elizabeth. “Cheesemonkey Wonders.” First Week and AVID Strategies. 25 Aug. 2018.

Agenda: Embolden Your Inner Mathematician (10.03.18) Week 4

Week Four 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 implement tasks that promote reasoning and problem-solving. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them. (#SMP-1)

From Principles to Actions: Ensuring Mathematical Success for All

Implement tasks that promote reasoning and problem-solving:Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Learning Progressions for today’s goals:

  • I can implement tasks that promote reasoning and problem-solving. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them. (#SMP-1)

Tasks:

Anticipated ways to mathematize Sheep Won’t SleepSee the next blog post for additional details

[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.

Focus on Learning: Establish Mathematics Goals to Focus Learning

Worry in her beautiful, tired, sad eyes communicates so much. Strain across her face makes my heart ache. As we sit down for coffee with our children playing nearby, she blurts, “I don’t know how to make myself clearer, Jill. They just don’t, won’t, can’t – I don’t know – get it!” I sigh into my coffee which causes steam to fog up my glasses, and she laughs through her tears.  

Knowing that I am an evidence-interested educator, she pulls out her unit plans for me to see and offer feedback. “You were in our class yesterday. What I can I do better…? How do I help them learn?” Love and concern for her students is evident in her thoughtfulness, craftsmanship, and design.

I was in this class yesterday and had been for many days of the unit. I go again and again, because I am learning from her and with her students. This strong, organized, empathetic teacher is, in fact, a very good teacher.  

“What if we take your teaching up a level to a stronger focus on learning? Let’s look at the output that is causing you this worry and stress. Together, can we look at their work and identify what they, in your words, ‘just don’t, won’t, can’t’ do?’ And then, what if we establish mathematics goals to focus learning for you and your students?”

Sitting there on the bank of the Chattahoochee, occasionally interrupted, joyfully, by a toddler that needed to show us a valuable rock or other important discovery, we combed through student work. Outpouring concern and frustration, she talked about each learner, their strengths, and what surprised her about what they did not understand. I listened in awe of what she knew about her students in granular detail, and what she thought they knew but didn’t really. My notes highlighted every success she saw and the joy and pride she felt with every success.

How might we shift her work to increase the amount of success for her and her students? How might we empower learners to take action, self-assess, and ask questions early and often to improve their understanding and communication? What if we take what we just learned about her class and level it out to make her expectations and her thinking visible?

We found four categories or groupings:

  1. As soon I as finish explaining the task, they are all over me, Jill. They have no idea what to do or are too scared to get started. They want me to hold their hand. They are not empowered or safe enough to try.” They are splashing around in the shallow end, maybe even thrashing.
  2. They started, but cannot think flexibly when their first attempt gets them nowhere. They will not hear feedback or collaborate to think differently. They just shut down.
  3. “They are happily working along and find success.” They are willing to work in the pool, but need support build around them to know this is a safe, brave space to draft and redraft to think and learn. Mistakes are opportunities to learn; they do not define you.
  4. “They are first and fast and successful. They want and need more. I want to deepen and connect their learning, not broaden it.” They are willing to dive into the deep end confidently to explore new connections and representations.

This hard, important work helped us gain clarity about what is essential to learn in her classroom. Articulating frustration points as well as success points during her analysis of learning in her classroom revealed and organized a path for communication of learning intentions.

How might we empower and embolden our learners to ask the questions they need to ask by improving the ways we communicate and assess?

What if we make our thinking visible to our learners? What if we display learning progressions in our learning space to show a pathway for learners?

Great teachers lead us just far enough down a path so we can challenge for ourselves.  They provide just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out to the world, makes me want to step to the next higher level. Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

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

I can make sense of problems and persevere in solving them.  (CCSS.MATH.PRACTICE.MP1)

But, as a learner…What if I think I can’t? What if I’m stuck? What if I feel lost, confused, or discouraged? How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

NCTM’s recent publication, Principles to Actions: Ensuring Mathematical Success for All, calls us to support productive struggle in learning mathematics. How do we encourage our students to keep struggling when they encounter a challenging task? They 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?

How might we coach our learners in to asking more questions? Not just any questions – targeted questions. What if we coach and develop the skill of questioning self-talk?

Interrogative self-talk, the researchers say, “may inspire thoughts about autonomous or intrinsically motivated reasons to pursue a goal.” As ample research has demonstrated, people are more likely to act, and to perform well, when the motivations come from intrinsic choices rather than from extrinsic pressures.  Declarative self-talk risks bypassing one’s motivations. Questioning self-talk elicits the reasons for doing something and reminds people that many of those reasons come from within.” (Pink, 103 pag.)

Our coffee is cold and our children have lost interest in playing together. As we wrap up our reflection, feedback, and planning session, we agree to experiment the next week with her students. How might the work and learning change if we make a pathway for self-assessment and self-talk visible to the learners?

We plan to post #LL2LU SMP-1:  I can make sense of problems and persevere in solving them in the classroom and on the tables for easy reference.  Our immediate learning goal for the students is to make sense and persevere, to ask clarifying questions and try again, to show thinking for clarity and questioning, and to find multiple ways to solutions and find connections.

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.)


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.

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

Pink, Daniel H. (2012-12-31). To Sell Is Human: The Surprising Truth About Moving Others. Penguin Group US. Kindle Edition.