Category Archives: Reflection

Enhancing Growth Mindset in Math – Learning together

We asked:

How might we, as a community of learners, grow in our knowledge and understanding to enhance the growth mindset of each of our young learners?

As a team, we have completed Jo Boaler’s How to Learn Math: For Students and have shared our thinking, understanding, and learning.

Blending online and face-to-face learning, we worked through the Stanford units outside of school so that we could explore and learn more when together.

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Here are some of the reflections shared by our team.

As a teacher my goal is to help children approach math and all subject areas with a growth mindset. It is of utmost importance that my students truly know that I believe in them and their ability to succeed!

Everyone my age should know that you should never equate being good at math with speed. Just because someone is a slower problem solver does not mean that they are a weak math student. Rather, sometimes the slower math thinkers are the strongest math thinkers because they are thinking about the problem on a deeper level. Being good at math is about being able to think deeply about the problem and making connections with it.

When talking to yourself about your work and learning new things, reminding yourself that you can try harder and improve is critical to potential success.  People are more willing to persevere through difficult tasks (and moments in life) when they engage in positive self talk.  

Mistakes and struggling, in life and in math, are the keys to learning, brain growth, and success.

Thinking slowly and deeply about math and new ideas is good and advantageous to your learning and growth.

Taking the time to think deeply about math problems is much more important than solving problems quickly.  The best mathematicians are the ones who embrace challenges and maintain a determined attitude when they do not arrive at quick and easy solutions.  

Number flexibility is so powerful for [students]. I love discussing how different students can arrive at the same answer but with multiple strategies. 

Working with others, hearing different strategies, and working strategically through problems with a group helps to look at problems in many different ways.

“I am giving you this feedback because I believe in you.”  As teachers, we always try to convey implicitly that we believe in our students, and that they are valued and loved in our class.  However, that explicit message is extraordinary.  It changes the entire perception of corrections or modifications to an essay–from “This is wrong, you need to make it right” to “I want to help you make this the best it can be,” a message we always intended to convey, but may not have been perceived.  

Good math thinkers think deeply and ask questions rather than speeding through for an answer.

Math is a topic that is filled with connections between big ideas.  Numbers are meant to be manipulated, and answers can be obtained through numerous pathways.  People who practice reasoning, discuss ideas with others, have a growth-mindset, and use positive mathematical strategies (as opposed to memorization) are the most successful.

We learn and share.


VTR: Sentence-Phrase-Word to dig deeper into Standards for Mathematical Practice

From Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners:

Sentence-Phrase-Word helps leaners to engage with and make meaning from text with a particular focus on capturing the essence of the test or “what speaks to you.” It fosters enhanced discussion while drawing attention to the power of language. (Ritchhart, Church, Morrison, 207 pag.)

Screen Shot 2014-10-19 at 7.26.45 PMWhat if we read and learn together, as a team? How might we develop deeper understanding?

Screen Shot 2014-10-19 at 7.29.46 PMAs a team of learners, we first read Make sense of problems and persevere in solving them independently and highlighted a sentence, phrase, and work that resonated with us.  In round robin fashion, we read aloud our selected sentence so that every member of the team heard what every other member of the team felt was important.  Just the act of hearing another voice read and callout an idea was impactful.

After completing the Sentence-Phrase-Word Visible Thinking Routine for Make sense of problems and persevere in solving them, we asked everyone to take another Standard for Mathematical Practice to read and markup, highlighting a sentence, a phrase, and a word.

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We divided into teams where each of the remaining Standards of Mathematical Practice were represented.  Each learner shared the SMP that they read highlighting a selected sentence, phrase, and word. My notes are shared below. I was amazed at the new ideas I heard from my colleagues when using this routine.

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Seek diversity of thought. Listen to others.  Hear differently. Promote engagement, understanding, and independence for all.


Ritchhart, Ron, Mark Church, and Karin Morrison. “Sentence-Phrase-Word.”Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. San Francisco, CA: Jossey-Bass, 2011. 207-11. Print.


SMP7: Look For and Make Use of Structure #LL2LU

Screen Shot 2014-08-24 at 4.43.58 PMWe want every learner in our care to be able to say

I can look for and make use of structure.

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

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 integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

Are observing, associating, questioning, and experimenting the foundations of this Standard for Mathematical Practice? It is about seeing things that aren’t readily visible.  It is about remix, composing and decomposing what is visible to understand in different ways.

How might we uncover and investigate patterns and symmetries? What if we teach the art of observation coupled with the art of inquiry?

In The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators, Dryer, Gregersen, and Christensen describe what stops us from asking questions.

So what stops you from asking questions? The two great inhibitors to questions are: (1) not wanting to look stupid, and (2) not willing to be viewed as uncooperative or disagreeable.  The first problem starts when we’re in elementary school; we don’t want to be seen as stupid by our friends or the teacher, and it is far safer to stay quiet.  So we learn not to ask disruptive questions. Unfortunately, for most of us, this pattern follows us into adulthood.

What if we facilitate art of questioning sessions where all questions are considered? In his post, Fear of Bad Ideas, Seth Godin writes:

But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones.  Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed.

How might we create safe harbors for ideas, questions, and observations? What if we encourage generating “bad ideas” so that we might uncover good ones? How might we shift perspectives to observe patterns and structure? What if we embrace the tactics for asking disruptive questions found in The Innovator’s DNA?

Tactic #1: Ask “what is” questions
Tactic #2: Ask “what caused” questions
Tactic #3: Ask “why and why not” questions
Tactic #4: Ask “what if” questions

What are barriers to finding structure? How else will we help learners look for and make use of structure?

[Cross posted on Easing the Hurry Syndrome]

Dyer, Jeff, Hal B. Gregersen, and Clayton M. Christensen. The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators. Boston, MA: Harvard Business, 2011. Print.


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?

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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]

Peer-to-Peer PD: Sharing our practices to build capacity – feedback

We met this week for Peer-to-Peer PD: Sharing our practices to build capacity.

I love when our learners connect ideas.  One of Ashley’s participants connected social media with formative assessment.

I learned how I can incorporate student Twitter in the classroom. I found her idea of having a “CTO” in the classroom is genius. Great window into what he/she is learning that day. She said they are more engaged as a result, and I believe it! So cool.

Another of Ashley’s participants outlined action steps, student engagement, and connected learner globally.

I was able to reflect and talk through some ideas of how to use Twitter in Kindergarten. I was inspired! My next steps are to have a discussion with my class about Twitter (what they know, etc.?). Then I will start using Twitter in the classroom by having the photographer for the week take pictures during recess of creations, children playing, nature, etc. and the last 5 minutes of recess we will choose a picture to tweet!
I also plan to use Twitter as a form of pen pals when we study Mexico. My plan is to connect with an elementary school classroom in Mexico that is Tweeting and exchange ideas about culture.

One of Janet’s participants connected using DropBox with our My Learning e-portfolios.

We will use Dropbox with our kids now! This will be very helpful for My Learning.

One of my participants connected Leading Learners to Level Up formative assessment to our My Learning e-portfolios.

My Learning is greatly about students taking ownership of their learning through the analysis of development (big picture). Being able to see the learning objective and self-assess progress towards the goal is a necessary skill for learning progressions (steps along the path).

A participant in Erin and Karen’s session

I learned about four different iPad apps that offer students opportunities to express themselves in different ways. Story Wheel focused on recording children telling stories. I liked the fact that this could be used in a small group lesson, which would give the children a creative way to organize and sequence their thoughts.

I will use these apps in my classroom, but I think it would be especially useful to have access to Skitch for Evernote since we are already using Evernote to house digital portfolios. One of the biggest obstacles to using iPads to record things for me is figuring out how to transfer saved projects into Evernote.

And, one of Melissa’s participants actually commented on the session being student driven.

I learned SO much about Haiku in a short amount of time! Melissa was such a good, patient ‘teacher’ and it was ‘student’ led, everything she taught was recommended/requested by us teachers.
Also, the class size was perfect and I really enjoyed that it was a hands-on class. I learn best by doing, so instead of listening or watching a ‘lesson’ on technology and then having to go back and do it later, I was able to do it right then. Wonderful!

When we dare to ask learners what they need and want to learn and match these up with what they can contribute, we learn and grow together. We connect ideas and learning. We build a stronger team by leaning in.

How might we spread this learner driven work throughout our school?

PD: Reading and Assessment – Learning Together – empathy and feedback

Yesterday, we met together as a community to begin to study assessment and reading. As is my practice, I also want to share what really happened and the feedback.

As noted in our agenda, we meet twice on Wednesday’s with faculty because of our schedule. At 12:30, we meet with teachers of our 3s and Pre-K children.  At 3:30, we meet with teachers of 1st-6th grade.

Our slide decks used to deliver the overviews of assessment and reading are shared below.

I began with the idea that we focus on learning and asked faculty to do a quick think-pair-share on how we focus on assessment. I asked if, in the think-pair-share, assessment of learning was discussed. I also asked if assessment for learning was discussed.  (Rhetorical questions that caused some head nodding.)  Learners in our care want us to follow their learning, to know where they are, and to help them move to the next level. Can we offer our learners feedback that they can understand and act on to learn and grow? When we offer feedback of learning, do we follow it with feedback for learning? Do we have a feedback system? Can even our summative assessment be used as formative assessment?

I thought my presentation was stronger at 12:30 than at 3:30.  At 12:30, when I asked for a 2nd think-pair-share, it was clear that more learning was needed.  I took the opportunity to model (and discuss) formative assessment.  I adjusted, right then, my instruction to “feed up,” to offer clarity of the goal.  At 3:30, the discussion of formative assessment was stronger, and I did not need to adjust my plan.

Maryellen followed with a photo journal of ways and things we read.  It was awesome! She began with the goal “I want to become a reader” quote posted on a Kindergarten bulletin board.  Every child has this goal.  She highlighted the mystery of the alphabet and the patterns readers need to learn. What happens when the patterns fail as illustrated in the word pseudonym? She took as through an entire series of things that are important to read.  And then, the big whammy! Maryellen read a passage from her iPhone users manual.  While we could read every word, what did we understand? Awesome! The iPhone passage offered instant empathy for young readers.  Just because we can recognize every word, it does not mean we can read. She wrapped up her overview with the following quote:

Reading gives us somewhere to go when we have to stay where we are.  ~Mason Cooley

With these two overviews, faculty divided themselves into study groups and met to begin sharing our common practices.  We want to know about our current reality. What are we doing well? How can we do more of it?

At 12:30, Kathryn Nevin shared a formative assessment technique she used that day.  As kindergarteners arrive for music, they enter the room singing the same song every meeting.  Kathyrn is looking for walking with a steady beat, melody, doing the right thing, self-regulation, and a sense of space and direction.  She realized that these learners might be struggling to understand the expectations. So, on Wednesday, she used her iPad to capture their entrance to music on video. After completing the start of class routine, she played the video for her learners.

How awesome is that? The children naturally took the opportunity for self- and peer-assessment while in Kindergarten! Assessment for learning…even our youngest learners can participate, learn, and grow.

We will continue to grow and learn, together.  I’ve included the entire set of feedback comments at the end of this post, but I want to share a few specific comments.

I gained knowledge in understanding how to use continuous Formative Assessment in teaching EED Science.  It really helped me to realize the differences in Summative  and Formative.  I will strive to use Formative assessment.


I liked the format of today’s session.  Having a whole group explanation of both areas was helpful for a lot of people.
Jill– you did a great job of explaining assessment!  Lots of examples were really helpful to everyone.  I liked the feed up, feed back, feed forward explanation.
Maryellen– your presentation was so inspiring!  It’s wonderful to think of all of the ways we touch our children through reading.


Assessment in Specials classes is different than assessment in the base classroom. Sometimes I struggle with the “how” of assessment in more formal ways since I only see my students once or twice a rotation. I want to continue to work at finding ways to formatively and summatively assess all of my students, across grade levels…in meaningful ways.


I thought it was helpful to discuss different assessment options with other grade levels, reading instruction generally across grade levels, but I think it could have been helpful to incorporate some reading about reading, some study, possibly some kind of writing about reading and the possible article.

As is my practice, I emailed a copy of all of the feedback from this session to all participants.  It is another way to model formative assessment.  Everyone is informed about what everyone else offered as feedback.

#TrinityLearns integrated studies (week 3)

When we have the opportunity to see what happens in other parts of our community, we begin to connect ideas and experiences.

Alpin Hong and Jun-Ching Lin surprised our 6th graders with visit and a brilliant lesson on harmony, color theory, and superhero theme music.

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It was Fitness Friday in First Grade. How great is it to combine math and fitness? Don’t you just love that two of our PE team lifted the work and learning of both the student-learners and the teacher-learners in 1st Grade?

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If you read #TrinityLearns Community (week 2), you know that we are teaching each other new ways to communicate learning.  Last week many of us learned about the app Pic Stitch which quickly combines multiple images into a collage.  (I asked Amanda Thomas, and Joe asked Jedd Austin.)  Notice how Kathy bright spots Brian’s work with our 2nd Graders.

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This week, Melissa Walker embedded her class’s twitter feed on her Haiku page.  This seems to be spreading through the 5th and 6th grade Haiku pages so that our families have another view of what happens at school. Amanda bright spots Melissa’s work.

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We’ve also seen our young learners making connections between math and science.

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Perhaps two of my favorite tweets – because they helped me connect in person – were from our 3s and Pre-K classrooms.  I could sit down with these young learners at carpool and ask them good questions.  They could describe details about their day, their interests, and their learning. I now know we need a rocket ship to rescue the balloons that got away.  I learn more and more each day about the interests of our learners.

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What will we learn about, for, and with each other as we continue to learn and share?

There were many more beautiful, rich learning experiences for all learners in our community.  A digest of our Tweets from the 3rd week of school is shown below.