Encouraging and anticipating mathematical flexibility

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

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

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

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

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

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

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

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

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

The list could go on and on.

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

Posted in Connecting Ideas, Learning Progressions, Questions | Tagged , , , , | 2 Comments

Visual: SMP-1 Make sense of problems and persevere #LL2LU

What if we display learning progressions in our learning space to show a pathway for learners? After Jennifer Wilson (Easing the Hurry Syndrome) and I published SMP-1: Make sense of problems and persevere #LL2LU, I wondered how we might display this learning progression in classrooms. Dabbling with doodling, I drafted this visual for classroom use. Many thanks to Sam Gough for immediate feedback and encouragement during the doodling process.

Screen Shot 2014-08-16 at 1.21.17 PMI wonder how each of my teammates will use this with student-learners. I am curious to know student-learner reaction, feedback, and comments. If you have feedback, I would appreciate having it too.

What if we are deliberate in our coaching to encourage learners to self-assess, question, and stretch?

[Cross posted on Easing the Hurry Syndrome]

Posted in #LL2LU, Algebra, Ask Don't Tell, Learning, Learning Progressions | Tagged , , , , | 6 Comments

Back to school: Pre-Planning 2014 agenda and opportunities

How do we celebrate our culture and show quick snapshots of what we value to new members of our community? How do we leverage digital tools to communicate, collaborate, and take control when we have choice?

As a Leadership Team, we designed an agenda for celebration, learning, and teaming.

We use Google docs and spreadsheets to communicate, collaborate, and choose time slots for learning.  The tweets shown are linked back to the source if more detail is wanted.  There is a quote from each of the nine summer reading books to offer a snippet from the books not chosen by any member of our community.

How do we celebrate our culture? How might we leverage digital tools to communicate, collaborate and offer choice? What if we up the ante on our infusion of the 4 Cs?


Posted in 21st Century Learning, Professional Development Plans, Questions | Tagged , , | Leave a comment

SMP-1: Make sense of problems and persevere #LL2LU

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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…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?

  • Level 4:
    I can find a second or third solution and describe how the pathways to these solutions relate.
  • Level 3:
    I can make sense of problems and persevere in solving them.
  • Level 2:
    I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.
  • Level 1:
    I can show at least one attempt to investigate or solve the task.

In Struggle for Smarts? How Eastern and Western Cultures Tackle Learning, Dr. Jim Stigler, UCLA, talks about a study giving first grade American and Japanese students an impossible math problem to solve. The American students worked on average for less than 30 seconds; the Japanese students had to be stopped from working on the problem after an hour when the session was over.

How might we bridge the difference in our cultures to build persistence to solve problems in our students?

NCTM’s recent publication, Principles to Action, in the Mathematics Teaching Practices, 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?

[Cross posted on Easing the Hurry Syndrome]

Posted in #LL2LU, Learning Progressions, Questions | Tagged , , , | 3 Comments

Summer Reading 2014 debrief: learner choice and voice (part 1)

We know student-learners need and deserve differentiated learning opportunities.  Don’t all learners?  How do we model learner choice? As a team of 150 learners charged with a responsibility of developing and maintaining a learning community for ourselves and the 650 children that we love and care for, we read and learn over the summer.  As we return to school, how do we share what we’ve learned?

We have planned a two-part summer reading debrief and sharing experience.  Part One has adult learners meet and share what they gleaned from a common read.  Our plan is shown below.  Notes and resources are collaboratively documented using Google docs so that all notes are available to every adult learner.

Here’s a sampling of our collective aspirations:

  • be flexible with everyone in the classroom,
  • know that we’re all on different stages as learners.
  • sequence in a logical order to facilitate understanding for all.
  • select high quality (low floor, high ceiling) tasks that offer many pathways for success.
  • use terminology like “yes and” instead of “no but”.
  • draw to get it on paper.
  • illustrate what they want to say and for comprehension.
  • plan for more prototyping and creating.
  • offer students enough time to think things over.
  • allow the kids to give feedback, there’s always time to make improvements.
  • use I like… I wish…
  • incorporate more nature into learning.
  • make time for students to create and problem solve without much instruction.
  • develop growth-mindset and reflection.
  • continual consciousness of modeling, building respect, patience.

Part Two will be a session in September where each group will share from their common read so that each member of our community has a sense of the salient points from each book.

What if we share what learn with others? How might we leverage communication tools to learn and share? What action(s) will we take based on what we learn?

Posted in Assessment, Cafe Reading, Connecting Ideas, Creativity, Learning, Professional Development Plans, Questions, Social-Emotional | Tagged , , , , | 1 Comment

#LL2LU Fractions – we are smarter than me & modeling C’s – #MPVschool & #TrinityLearns

A new definition of strength: Can we learn together? What if we collaborate, ask for feedback, and lean in to leverage expertise and perspective of others?

If we truly believe in communication, collaboration, and the other C’s, how are we – as lead learners – modeling and taking action?

<Note the timestamps in the following communication, collaboration, critical thinking, and problem-solving.>

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“Hear” snippets of Nicole’s thoughts as she is developing the assessment shown above:

    •  I’m  writing a mathematics unit for a grade level that I have never taught to learn, to  help my team, to help our young learners.
    • This is hard.
    • I’m trying to model backwards design unit planning (Grant Wiggins hung the moon, most recently evidenced by his math blog post today). Stage 2 (How will I know when they have learned it?) must come before Stage 3 (the learning plan). Teachers should have access to the assessments (formative and summative) at the beginning of the unit.
    • Our learning outcomes are all I have to work with.  Reading these standards in depth helps me some, but I need feedback.
    • I heart Google.
    • The “I can…” statements need to be student-friendly. They will be directly related to the standards-based rubric we will need to create.
    • I’ve worked through several leveled assessments as collaborations with classroom teachers, but I have yet to write one independently.
    • Wait, why am I writing this independently? It’s nearly midnight. I’m sending this to Jill.

Screen Shot 2014-04-24 at 7.09.54 PMScreen Shot 2014-04-24 at 7.09.34 PMScreen Shot 2014-04-24 at 7.08.44 PM Screen Shot 2014-04-24 at 7.08.56 PM

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“Hear” snippets of Jill’s thoughts as she gave feedback and edited the assessment shown above:

      • Wow…Such good work.
      • Level 1 “I can decompose a figure into equal parts. I can name each part.”  
        • I wonder if decompose is a 3rd grade word. (I do not know.)  I also wonder about “partition” as a 3rd grade word.
        • I wonder if you are having a resolution problem with the shapes in Level 1. The image shown is a rectangle, not a square.
        • I wonder how successful a child can be partitioning the circle without having the center marked and using a compass.
      • Level 2 “I can represent a fraction on the number line when some fractions are given to me.“  
        • Can we eliminate the word “some” and/or simplify?
        • What if we say I can represent fractions on a number line?
        • What if we add number lines to identify fractions before asking students to take action on number lines? Just this month, Jennifer Wilson and I presented on conceptual understanding of fractions and the new way to convey a consistent story using number lines. 
        • My TI-Nspire software and the fraction lessons will give me number lines. I’m not sure about mixed numbers and partitions past 1, but Nicole will know.  At least adding a visual might help.

Screen Shot 2014-04-24 at 7.10.26 PM

Nicole thinking:

How on earth did Jill create this fancy number line in a Google doc? I like her train of thought here but think the visual at it stands now will be too hard for grade 3 students.

Jill’s thinking:

Right. Number lines too hard. Would it be easier if we think together now that we are both awake?

Below is a copy of the next iteration of this assessment after a Google hangout discussion and co-learning conversation.

How might we collaborate, ask for feedback, and lean in to leverage expertise and perspective of others?

A new definition of strength: We are stronger than me. Learn and share!

[Cross posted on Curriculum Reflections] 

Posted in 21st Century Learning, Ask Don't Tell, Assessment, Learning, Questions, Social Media | Tagged , | 3 Comments

JGough observes JWilson – #NCSM14 Developing Conceptual Understanding Through the Progressions

Peer-to-Peer Observations…who learns?


Piloting an Observation of Practice with Jennifer Wilson (@jwilson828), I observed Jennifer during our NCSM presentation, Developing a Conceptual Understanding of Fractions. Below is a record of what I observed and then my reflection.  I am grateful for the opportunity to observe Jennifer and to learn from and with her.

Jill Gough observes Jennifer Wilson
NCSM: Wednesday, April 9, 2014, 10:00 – 11:00, New Orleans, LA
Session:  Developing Conceptual Understanding Through the Progressions:  Fractions, Ratios, and Proportions


Jennifer began her session with Essential Learnings for the hour.  By the end of this session, everyone should be able to say:

  • I can describe a fraction a/b as a copies of 1/b.
  • I can construct questions that push and probe student thinking about questions.
  • I can explain the role that technology plays in deepening student understanding of fractions.

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I like that Jennifer began her session this way, because it offered participants focus and direction. I wonder if the audience appreciated this intentional guiding of thinking.  I like that there was a content essential learning, an assessment essential learning, and a process essential learning.  What if all PD lessons were launched this way? I wonder if we would see more engagement and thought and less nervousness about which button to press.

I appreciated the situation Jennifer experienced with technology as the session began.  The same situation happened to me in Sacramento a couple of weeks ago.  After playing her PowerPoint slide, Jennifer returned to the Navigator software to find all the traditionally black text was now white making is difficult to read.  In addition, when she returned to the slideshow, it was frozen.  (Same for me in Sacramento, but I was using Keynote.)

I like that Jennifer pressed forward, abandoning her slide deck to keep the conversation going with the participants.  I wonder how she would handle this in such a short session if she was presenting alone.

I liked the formative assessment question sent via quick poll to engage the audience right away.  While I was not quick enough to capture the results, they were varied. Each choice had some takers.  I liked the use a warm-up question using the Navigator. I liked that Jennifer took action based on the results.  Good modeling.  I liked that the Navigator was used as a tool and that fractions were the featured event that anchored the discussion.

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I love watching Jennifer facilitate peer-to-peer discourse, because she requires individual thinking time prior to group discussion.  I aspire to use her protocol. I need this when I am learning. What if we established this as a goal or must do for all instructors?

Screen Shot 2014-04-21 at 5.30.40 PM

Jennifer used the following files during her session:

Using appropriate wait time, Jennifer offered hands-on experience and investigation time for participants to explore each of the files.  She prompted participants to shoulder-partner up and brainstorm push and probe questions that might be asked of students.  Jennifer used the Navigator to capture class responses to comment on and visualize quality questions from the audience and the authors.  I like that she modeled push and probe questioning through this discussion.  She honored participant questions and offered advancing questions to help push their thinking.

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I observed participants struggle with what questions to ask and what response to hope for from learners.  One participant said “I don’t know how to explain these true/false questions.  I know which ones are true and which are false, but how do I explain in words?” Evidence of probing for understanding and that thinking was being pushed.  I liked that I had the opportunity to witness and then facilitate peer-to-peer discourse as these teacher-learners grappled with the why of fractions.  I wonder if we should incorporate the 5-whys protocol and theWhat Makes You Say That Visible Thinking Routine to facilitate rich discussion and uncover possible misconceptions.

Using the Navigator class capture, Jennifer complimented the audience on their approaches to Fractions and Unit Squares and that none of their methods were the way she thought.  “My students continually surprise me. You too!” I like that she modeled her way only after showing several correct ways.

Jennifer closed by returning to the Essential Learnings to remind participants of their experience and what the target was for the session.

Jill’s reflection:

 As a result of this observation of practice and feedback loop, which aspects of my teaching do I feel are bright spots?

I like starting sessions with Essential Learnings written as I can… statements.  Observing Jennifer launch and close her session with the I can… statements was the first time I’ve seen someone other than me leverage them in a professional development session.  In my own practice, I send the agenda ahead of the session when possible. I wonder if I should declare them as the beginning of the session. I thought it was a powerful anchor for our work.

I like starting with a warm-up question that turned into formative assessment.  In my own practice, I have intentionally described how I am shifting gears based on a warm-up question to model adjusting (not abandoning) the plan to meet learners where they are.  I wonder how many teacher-learners miss the adjustment when we are not direct about the shift we are making.  At the Social Media PD Saturday, Sam commented on the shifts I made based on the learners’ questions and the fact that I still accomplished everything on the plan.  What if we are more intentional when we shift to make the shift transparent? Based on these results, we are going to shift to X, and we will get to our destination. This connects to Jo Boalers’ I am giving you this feedback because I believe in you. What if we are direct? I/we are shifting the plan because we need to do X to accomplish XX.

I like using the Navigator to highlight multiple pathways to success.  I appreciated Jennifer using the Navigator to intentionally highlight more than one way to accomplish a task.  I wonder if zooming in to show 2 screens from the class capture helps participants sitting in the back to see.  What if we regularly and intentionally honor work by showing multiple correct pathways?

As a result of this observation of practice and feedback loop, what questions do I have about my own teaching?

I wonder how to include more participants when using the Navigator.  At Jennifer’s session, there were 30 handhelds and 40 participants.  I wonder how to engage all participants. Do we need more resources? Should we partner? I want to try more participants connected to Navigator in my work with large groups.  Even though every participant (40-90 participants) had a handheld, we observed those not connected and displayed were less engaged.

What if I am more intentional about Think-Team-Share? I appreciated Jennifer’s direct and intentional let’s take 30-60 seconds to organize our thoughts silently before sharing. What if I add this to my facilitation toolkit? What if I add let’s take a minute to quick-write. Describe your thinking on paper to offer a visual of your idea when sharing with others. How might I model this as a strategy for any learning episode?

As a result of this observation of practice and feedback loop, what new ideas do have?

Well, one is in the paragraph above.  “What if I add this to my facilitation toolkit? What if I add let’s take a minute to quick-write? Describe your thinking on paper to offer a visual of your idea when sharing with others. How might I model this as a strategy for any learning episode?” I hear myself talk about teaching learners visual note taking, but do I model it for teacher-learners? What if I write a blog post that show what is in my head so that it is clearer to me and to others?

I am inspired to work on my closings.  I like how Jennifer cycled back to the Essential Learning for the session to have participants do a quick mental self-assessment.  I believe the way she closed the session improved the chances that the message will stick.

[Cross posted at Easing the Hurry Syndrome]

Posted in Conferences, Peer Observations, Questions, Reflection | Tagged , , | Leave a comment