People see mathematics in very different ways. And they can be very creative in solving problems. It is important to keep math creativity alive.

and

When you learn math in school, if a teacher shows you a method, think to yourself, what are the other ways of solving this? There are always others. Discuss them with your teacher or friends or parents. This will help you learn deeply.

I keep thinking about mathematical flexibility. If serious about flexibility, how do we communicate to learners actions that they can take to practice?

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?

Filed under: #LL2LU, Algebra, Connecting Ideas, Questions Tagged: #LL2LU, How to Learn Math, How to Learn Math: For Students, Jo Boaler, math flexibility, mathematical flexibility ]]>

As a team of interested math learners, we will spend 10 hours (1 PLU of credit) learning together using the following outline as our course of study.

In order to share our reflections, we will use a copy of the Enhancing Growth Mindset in Math Google doc to record, expand on, and share the reflections from the Stanford MOOC and our thoughts and connections to the CCSS.

It is my hope that each teacher-learner will share their reflections with everyone in the group or at least one other member.

How vulnerable will we be? What if we share what we know and don’t know and learn together?

Filed under: Professional Development Plans, Questions Tagged: growth mindset, How to Learn Math, How to Learn Math: For Students, Jo Boaler, reflection ]]>

…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!

After 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?

How 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?

Filed under: Connecting Ideas, Learning Progressions, Questions Tagged: 5 Practices for Orchestrating Productive Mathematics Discussions, How to Learn Math, How to Learn Math: For Students, mathematical flexibility, number flexibility ]]>

I 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]

Filed under: #LL2LU, Algebra, Ask Don't Tell, Learning, Learning Progressions Tagged: #LL2LU, doodling, learning progression, learning progressions, Standards for Mathematical Practice ]]>

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?

Filed under: 21st Century Learning, Professional Development Plans, Questions Tagged: agenda, digital tools, pre-planning ]]>

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]

Filed under: #LL2LU, Learning Progressions, Questions Tagged: #LL2LU, CCSS, learning progression, Standards for Mathematical Practice ]]>

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?

Filed under: Assessment, Cafe Reading, Connecting Ideas, Creativity, Learning, Professional Development Plans, Questions, Social-Emotional Tagged: 4As Protocol, Art of Questioning, assessment, professional development, Summer Reading ]]>

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

“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.*

“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.*

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]* *

Filed under: 21st Century Learning, Ask Don't Tell, Assessment, Learning, Questions, Social Media Tagged: #LL2LU, #MVPschool ]]>

Everyone.

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

**Observation:**

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.

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.

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?

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.

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]

Filed under: Conferences, Peer Observations, Questions, Reflection Tagged: Jennifer Wilson, NCSM, Peer Observations ]]>

There is so much to learn, practice, prototype, and consider. How do we learn and share? What if we divide, conquer, and share to learn?

From: Jill Gough

Date: Friday, April 11, 2014 12:13 PM

To: All Trinity

Subject: Summer Reading 2014.Hi,.Summer reading for our community offers choices again this year. We offer three themes from which to choose. You may want to continue with the Art of Questioning, or you may want to explore Creativity or Social-Emotional as an interest. Each theme offers three choices, and if available, you may choose to read using a traditional book, a Kindle book, or an audio book..A quick note and thank you: Last year Laurel Martin asked me why the books weren’t hyperlinked to Amazon so that we could quickly read reviews. Great idea! (Hmm…I didn’t know how to hyperlink an image using Pages…but I do now. Thank you Laurel for pushing me to learn!) So, thanks to Laurel, if you want to read reviews, just click on a book in the flyer..We will use the 4As protocol to debrief during Pre-Planning. We are also going to schedule a Wednesday afternoon so that our community can hear and share the big ideas from every book..Please see the attached flyer for information and links to additional information and a form to request your book. Would you please select a book by Friday, April 25 so that we can have it for you before we leave in May?.Thank you,.Jill.

Here’s our flyer:

And, our version of the 4 As protocol worksheet:

Filed under: Assessment, Creativity, Questions, Social-Emotional Tagged: 5 Practices for Orchestrating Productive Mathematics Discussions, and Engineering in the Classroom, Bully Nation: Why America's Approach to Childhood Aggression is Bad for Everyone, Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation, Creative Confidence: Unleashing the Creative Potential Within Us All, David Kelley, David Streight, Diana Baumrind, Elizabeth A. Hebert, Five Practices, Gail Boushey, Gary Stager, Invent To Learn: Making, Joan Moser, Larry Nucci, Marilyn Watson, Marvin W. Berkowitz, Mary Kay Stein, Parenting for Character: Five Experts, Peg Smith, Steve Long-Nguyen Robbins, Summer Reading, Susan Eva Porter, Sylvia Martinez, The CAFE Book: Engaging All Students in Daily Literary Assessment and Instruction, The Power of Portfolios: What Children Can Teach Us About Learning and Assessment, Thomas Lickona, Tim Brown, Tinkering, Tom Kelley, What If?: Short Stories to Spark Diversity Dialogue ]]>