All posts by Jill Gough

Learner, Love Questions, Problem-finding, Math w/technology. Interests: Collaborating, PLC, Formative Assessment

Agenda: Embolden Your Inner Mathematician (11.07.18) Week 8

Week Eight 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)

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

Tasks:

What the research says:

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

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]

[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 (10.24.18) 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.

Collaboration – How might we level up again?

Last week, we drafted a learning progression for a team around collaboration and asked for feedback..   ICYMI: I wrote:

I’m curious to know what you think about the draft below. If we put this out in our classroom, will learners have a stronger opportunity to self-assess and level up?

I am grateful for all of the feedback we received.  Thank you.

The teaching team that I am coaching asked an important question.  “This works, Jill, for collaboration in our daily classroom learning. If we launch a team project, the Level 4 should really be the Level 3. How might we emphasize collaboration is co-creating something new together?

If we establish I can collaborate to co-create evidence of shared learning, work, and understanding as a goal, how can we level up to focus learning?

We want all learners in this community to be able to say

I can collaborate to co-create evidence of
shared learning, work, and understanding

At Level 1, learners are working side-by-side and periodically check-in with each other. While closer to collaboration, this is really parallel play. We are in the same place doing the same thing, and we at least acknowledge that other learners exist in our space.

At Level 2, learners exchange thinking and ideas as they discuss questions and actions to take together. At this level, learners add to each other’s thinking and make sense of new, different ideas and pathways.

At Level 3, learners listen and share deeply to riff and improvise, co-creating ideas, thinking, and learning.

At Level 4, learners reflect on what they knew and what they know now. They can articulate what is now possible because of shared thinking, learning, and working together.

Again, I’m curious to know what you think about the draft above. If we put this out in our classroom, will learners have a stronger opportunity to self-assess and level up?

I love what we learn when we make our thinking visible. Our students and colleagues help us learn, refine, and deepen our work.  Tell a colleague what you want next for and with your students. And don’t stop there. Teach. Help them learn even when you are learning too.

Brainstorm with colleagues.  Talk about you hopes and dreams for students and  level out what you see and want to see. Make your thinking visible to the learners in your care.

Teach.

Empower learners.

Lead learners to level up.

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.

Collaboration – How might we level up?

About 20 years ago, I worked with a wonderful, brilliant teacher who would tease me about collaborative learning. It was not his style. But, he tried. He would say to his class, “Pull your desks up close and uhh…collaborate. I’ll be back in a minute.”  Now, there were good outcomes from this opportunity. Students had a moment to breathe, catch up if behind (or confused) in their notes, and talk with classmates.

What is our definition of collaboration? In our teaching team or teams, have we established common language about collaboration? Have we shared it with the learners in our care?

What if the learners in your care are not meeting your expectations around collaboration?

  • Do we complain to colleagues or the learners that they are not collaborating?
  • Do we tell the learners that they need to collaborate without telling them how?
  • Do we assume that they <should> already know how? And, if they do not, are we frustrated and disappointed? Do we use our blame-thrower to put responsibility on someone else?
  • Do we take time to establish norms and common language around collaboration?

Teaching, telling, or complaining? Which one or ones are we stuck in? Problem-solving dissolves into complaining and venting when we fail to seek solutions and take action.  So, let’s brainstorm what it looks like and try something different.

Excerpts from a coaching session:

Teacher: I have no idea, Jill. They won’t collaborate. Do they not know how? They work in isolation, purposefully.  

Coach: Why is that important? Why should they work collaboratively? 

Teacher: Gosh, I think everyone knows that we collaborate to learn more, deeply. I think it is about perspective and listening to the ideas of others – even when you don’t agree. And, in math, it is about flexibility.

Coach: Tell me more about what you see and what you want to see.

Teacher: I see students sitting in groups, because that is how the furniture is arranged. But, they are not speaking to each other. Well, maybe…<sigh>…occasionally they check an answer. I want an exchange of ideas; I want them to learn from each other, together.  I hope that they will be curious about each other’s thinking and try to make sense of it instead of simply saying, “Oh, that’s not how I did it.” 

I’m curious to know what you think about the draft below. If we put this out in our classroom, will learners have a stronger opportunity to self-assess and level up?

If we establish I can collaborate to learn with and from others as a goal, can we use the above to focus learning?

We want all learners in this community to be able to say

I can collaborate to learn with and from others.

At Level 1, learners are working in isolation, perhaps racing to finish first.. Maybe learners plan to confer with others only after completing the task. Some might be trying to hide what they do not know; others are lapsing into teacher dependence.

At Level 2, learners are working side-by-side and periodically check-in with each other. While closer to collaboration, this is really parallel play. We are in the same place doing the same thing, and we at least acknowledge that other learners exist in our space.

At Level 3, learners exchange thinking and ideas as they discuss questions and actions to take together. At this level, learners add to each other’s thinking and make sense of new, different ideas and pathways.

At Level 4: learners listen and share deeply to riff and improvise, co-creating ideas, thinking, and learning.

All learners need independent think time to organize thinking, process the task, and gather resources.  AND, all learners need to learn from and with others in community because it promotes understanding, perspective taking, flexibility, listening, and critical reasoning.

So, when you are frustrated with how things are going, complain. Tell a colleague what your students are not doing. But don’t stop there. Teach. Help them learn even if they should already know it.

Brainstorm with your team. Ask hard questions. Describe what is going well and what is not.  Use this data to reframe and level out what you see and want to see. Make your thinking visible to the learners in your care.

Teach.

Empower learners.

Lead learners to level up.

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.