Category Archives: Questions

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.


Empower learners.

Lead learners to level up.

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.

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.


Agenda: Embolden Your Inner Mathematician (09.05.18) Week 1

Week One of Embolden Your Inner Mathematician

Teacher-learner-leaders will engage in 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.

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)


Slide deck:

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

Gough, Jill, and Jennifer Wilson. “#LL2LU Learning Progressions.” Experiments in Learning by Doingor Easing the Hurry SyndromeWordPress, 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.

Establish goals to focus learning – Reading Workshop 5th Grade

What if we design a lesson to orchestrate productive discussion, critique the reasoning of others, grow as readers and writers, and deepen understanding through reflection?

The 5th grade team invited me to co-labor with them to help our young learners deepen their understanding of reader’s response journals. As a team, they are focused on implementing and deepening their understanding of Wiliam and Leahy’s  five strategies in Embedding Formative Assessment: Practical Techniques for K-12 Classrooms :

  • 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

From our Instructional Core work during Pre-Planning, we are working to  establish goals to focus learning.

The 5th Grade team drafted the following learning progressions to make their thinking visible to our new students. As a team, they have established these goals for students. (Level 3 for I can establish goals.)

How might we use these established goals to focus learning? What student outcomes should we anticipate, and what teacher moves should we plan based on prior experience?

At their invitation (#soexcited), I facilitated a lesson on using the drafts above to improve and strengthen reader’s response journal entries while modeling the use of assessing and advancing questions to focus student learning. (Level 4 for I can establish goals and Level 3 for I can focus learning.)

Here’s the plan:

And, the slide deck:

These learning progressions are in each student’s reader’s response journal so they can use them in class and at home.

It was a crisp 30-minute lesson. All of our anticipated outcomes presented during the mini-lesson.

We wanted our students to learn more about

  • making their thinking visible to another reader,
  • adding text evidence to support their ideas,
  • including details that support understanding,
  • participating in productive discussion,
  • critiquing the reasoning of others,
  • growing as readers and writers,
  • using learning progressions to improve their work.

After reading one of my reader’s response entries, our students’ frustration at not having read Bud, Not Buddy by Christopher Paul Curtis surfaced during  their feedback loop to me. This offered me the opportunity to ask their teacher if he or she would have read every independent reading selection made by his or her students. It was a strong “ah ha” moment for our students.

The students’ comments could be categorized in themes. Samples of our students’ reflections are shared as evidence of effort and learning.

  • An ah-ha for me is that my teacher has not read every single book in the universe.
  • I learned to pay attention to text evidence and explaining my text evidence so the reader understands why I added the quotes and page numbers.  I also learned to pay attention to visuals and formatting.
  • I don’t know what an ah-ha moment is. (Oops! Needs more instruction and time to learn.)
  • I know that everyone has not read the book and that I need to add enough detail for people who haven’t read the book.
  • An ah-ha for me is that I think that adding the definitions was smart because I didn’t know some of the words.
  • I learned to pay attention to science experiments. (Yikes! Needs more instruction and time to make sense of the task.)
  • I learned to ask myself if it makes sense and if another person could understand.
  • I learned to ask myself “how can I improve this? What details should I add?”

We know this is not a one-and-done event for our students and our team. We learned about our students and know what me should work on next. We must continue to practice making our thinking visible and hone our skills to use goals to focus learning.

Our school’s mission calls for us to deepen students’ educational experiences and empower students as agents of their own learning while we help them build strong academic foundation.  We strive to make our thinking visible to each other and to our students.

What is to be gained when we make our thinking visible to our students and use established goals to focus learning?

Wiliam, Dylan; Leahy, Siobhan. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. (Kindle Locations 493-494). Learning Sciences International. Kindle Edition.



Using number lines to build strong, deep academic foundation

Many students struggle with algebraic ideas because they have not developed the conceptual understanding (Hattie, 129 pag.)

Are you a “just the facts ma’am” mathematician, or do you have deep conceptual understanding of mathematics? How did Algebra I, Algebra II, and Calculus go for you? Did you love it,  just survive it, or flat-out hate it?

What if we focus on depth of knowledge at an early age? How might we change the future for our young learners?

Imagine you are back in Algebra I, Algebra II, or Calculus working with polynomials.  Do you have conceptual understanding, procedural fluency, or both?

Learning has to start with fundamental conceptual understanding, skills, and vocabulary. You have to know something before you can do something with it. Then, with appropriate instruction about how to relate and extend ideas, surface learning transforms into deep learning. Deep learning is an important foundation for students to then apply what they’ve learned in new and novel situations, which happens at the transfer phase. (Hattie, 35 pag)

What if, at the elementary school level, deep conceptual numeracy is developed, learned, and transferred?

Our brains are made up of ‘distributed networks’,and when we handle knowledge, different areas of the brain light up and communicate with each other. When we work on mathematics, in particular, brain activity is distributed between many different networks, which include two visual pathways: the ventral and dorsal visual pathways (see fig 1). Neuroimaging has shown that even when people work on a number calculation,such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing. (Boaler, n pag.)

Screen Shot 2018-08-26 at 6.50.50 PM

Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. (Heath and Heath, 106 pag.)`

A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success. (Boaler, n pag.)

Well, that’s worth repeating, huh?

A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success.

Often, we rush to efficiency – to “just the facts ma’am” mathematics. Surface knowledge – memorized facts – is critical to success, but that is not the end goal of learning.  The goal of all learning is transfer.

When we use number lines to support conceptual understanding of number quantity and operations, we deepen and strengthen mathematical foundation.  Our young students are learning that multiplication is repeated addition, that 4 x 5 is 5 four times, which lays the foundation for being able to transfer to the following polynomials.

a + a + a +a = 4a
 a + 3b +a + 3b = 2a + 6b

Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air. (Heath and Heath, 106 pag.)

How might we focus on deep learning and transfer learning by studying and learning visually? What if we embrace seeing as understanding so that we learn to show what we know more than one way?

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.

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) (p. 35). SAGE Publications. Kindle Edition.

Heath, Chip. Made to Stick: Why Some Ideas Survive and Others Die (p. 106). Random House Publishing Group. Kindle Edition.

Focus on Instructional Core: establish goals to focus learning

As part of our school’s Pre-Planning, Marsha Harris and I facilitated a faculty-teams workshop to continue our work and learning in the Instructional Core.

Here are my notes from the session.

The agenda, shared ahead of the meeting, looked like this:

The slide deck that accompanies this plan looks like this:

As seen in the slides, we checked in with John Hattie’s research around teacher clarity.

Teacher clarity involves the instructional moves a teacher makes that begin with carefully planning a lesson and making the learning intentions for that lesson or unit clear to herself and her students. 

It extends to consistently evaluating where students are in the learning process and describing the success criteria on which students can assess their own progress and on which the teacher bases her evaluation of a student’s progress with a idea or concept. (Hattie, 38 pag.)

To model teacher clarity, we looked at two drafts for

I can establish goals to focus learning.

First, establish goals:

Then, focus learning:

How might we partner together to establish learning goals? What if we by “do the task as a learner” to notice and note needed prerequisites and anticipate potential learning obstacles? Can we deepen learning experiences by connecting to prior learning standards and strategies?

What if we make learning goals visible so that learners are able to identify what they know and need to know next?  How might we team to anticipate needed questions to assess and advance learning? What if we teach learners to ask more questions to forward and deepen learning? How might we empower learners to level up?

When we focus on learning,
we strengthen the Instructional Core.

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) (p. 38). SAGE Publications. Kindle Edition.