All posts by Jill Gough

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

Advance Your Inner Mathematician #TrinityLearns Session 5: Sequence and Connect

This semester we are piloting a new course, Advance Your Inner Mathematician, for teachers to continue learning after Embolden Your Inner Mathematician. This work is anchored in Smith and Sherin’s The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Middle School Classroom.

In session 5, a dilemma presented itself, and I have yet to resolve what I should have done.

The goal is to present a formidable task and monitor learner work and thinking so that the learners in the room participate in productive mathematical discourse to learn from and with each other.

Before session 5, I selected learning goals and a task, and I anticipated learner thinking.

  • Learning goals for session 5:
    • I can use ratio reasoning to solve tasks.
    • I can look for and express regularity in repeated reasoning.
    • I can show my work so a reader understands without asking me questions.
  • The task for session 5: Jim and Jessie’s Money from Illustrative Mathematics 
  • Anticipated *ways learners might think and work:

Paper.Productive_Struggle.18

*Note: I had not anticipated using a number line, but it was used well in the session, so I added it to my notes after the fact to capture it for future classes.

During the session:

  • I established the first goal:  I can look for and express regularity in repeated reasoning.
  • I launched the task using the Three Reads Routine to support learners entering and thinking in the task.
  • I monitored student thinking and collaboration. I noticed and took notes on strategies and tools being used.
  • I facilitated the whole group discussion by sequencing learner’s work to build understanding and flexibility, and to help learners persevere in making sense of other’s thinking.
  • I did not have to connect the mathematics because the learners did it for themselves. #Awesome.

Here are notes about the sequence during the class discussion and a bit of the narrative:

The pair of teachers, Group 1, using the number line shared first leading with an initial guess of Jim and Jessie each having $150 at first. The next to share, Group 2, was the pair that used a table where the guesses were anchored in how much money Jim and Jessie had at first; their initial guess was $100.  Group 2’s comment to Group 1 was that the number line made it so much easier to understand the numbers in their table, and Group 2’s replay was that the table was so helpful to see the pattern.  #Nice

The next pair, Group 3, shared that they found a difference of $21 in how much money Jim and Jessie had after their expenses, and they were still grappling with how that would help them. They asked for a little more time to think.

The final pair, Group 4, to share started with how difficult it was to find a second pathway after using algebra.  The struggle, they said, is real when trying to think of how to start over with a different tool or strategy.

This is what makes anticipating as a team magical.
We expand each other’s thinking and flexibility
just by showing up and bravely sharing our thinking.

As Group 4 finished their algebraic solution, I noticed that a member of Group 3 was standing… well, bouncing… with an urgent need to share.  “The connections,” she said, “has been here all along. It is with the number line reasoning.” This excited learner connected all of the shared thinking and learning together.  #Awesome

Now, here is my burning question. I have not resolved this dilemma.

My way – the way I thought first – was not shared.

Learning happened and was articulated. Connections were made in a rich discussion.

Do I care that there is another way to think about this task and that it was not taught? No one’s idea, other than mine, was left out of the discussion.  I like my way, and I think it is easier. (Isn’t that the way it is with teachers?) Am I being selfish wanting to share my way too? Or, do you think it might continue the learning to see yet another way?

I’d really love to know what you think.

Number Talks: developing fluency, flexibility, and conceptual understanding #AuthorAndIllustrate

How might we work on fluency (accuracy, flexibility, efficiency, and understanding) as we continue to teach and learn with students? What if our young learners are supposed to be fluent with their multiplication facts, but… they. ..just…aren’t!?

It really isn’t a surprise, right? Children learn and grow at different rates. We know that because we work with young learners every day.  The question isn’t “Why aren’t they fluent right now?” It isn’t. It just isn’t. The question should be and is:

“What are we going to do, right now, to make this better
for every and each learner in our care?”

In Making Number Talks Matter, Cathy Humphreys and Ruth Parker write:

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don’t know it yet), and over time, the properties come to life in students’ own strategies. (Humphreys, 62 p.)

Humphreys and Parker continue:

Students who have experienced Number Talks come to algebra understanding the arithmetic properties because they have used them repeatedly as they reasoned with numbers in ways that made sense to them. This doesn’t happen automatically, though. As students use these properties, one of our jobs as teachers is to help students connect the strategies that make sense to them to the names of properties that are the foundation of our number system. (Humphreys, 77 p.)

So, that is what we will do. We commit to deeper and stronger mathematical understanding. And, we take action.

This week our Wednesday workshop focused on Literacy, Mathematics, and STEAM in grade level bands.  Teachers of our 4th, 5th, and 6th graders gathered to work together, as a teaching team, to take direct action to strengthen and deepen our young students’ mathematical fluency.

We began with the routine How Do You Know? routine from NCTM’s High-Yield Routines for Grades K-8 using this sentence:

81-25=14×4
How do you know?

Here’s how I anticipated the ways learners might think.

Paper.Productive Struggle.195

From The 5 Practices in Practice: Successfully Orchestrating Mathematical Discussion in your Middle School Classroom:

Anticipating students’ responses takes place before instruction, during the planning stage of your lesson. This practice involves taking a close look at the task to identify the different strategies you expect students to use and to think about how you want to respond to those strategies during instruction. Anticipating helps prepare you to recognize and make sense of students’ strategies during the lesson and to be able to respond effectively. In other words, by carefully anticipating students’ responses prior to a lesson, you will be better prepared to respond to students during instruction. (Smith, 37 p.)

How many strategies and tools do we use when modeling multiplication in our classroom? It is a matter of inclusion.

It is a matter of inclusion.

Every learner wants and needs to find their own thinking in their community. This belonging, sharing, and learning matters. We make sense of mathematics and persevere. We make sense of others thinking as they learn to construct arguments and show their thinking so that others understand.

Humphreys and Parker note:

They are learning that they have mathematical ideas worth listening to—and so do their classmates. They are learning not to give up when they can’t get an answer right away because they are realizing that speed isn’t important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra. (Humphreys, 62 p.)

As teachers, we must anticipate the myriad of ways students think and learn. And, as Christine Tondevold (@BuildMathMinds) tells us:

The strategies are already in the room.

Our job is to connect mathematicians and mathematical thinking.

From NCTM’s Principles to Actions:

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

And:

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

What if we take up the challenge to author and illustrate mathematical understanding with and for our students and teammates?

Let’s work together to use and connect mathematical representations as we build procedural fluency from conceptual understanding.


Humphreys, Cathy. Making Number Talks Matter. Stenhouse Publishers. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. Print.

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Multiplication Fact Fluency prototype – feedback requested

I am concerned that we are conflating automaticity with fluency.  How might we get clear on the difference?

From Assessing Basic Fact Fluency by Gina Kling and Jennifer M. Bay-Williams:

Think about how you assess reading fluency. Does your assessment plan involve listening and observing as children read as well as asking reading comprehension questions? Now imagine what you might learn about students’ reading fluency if you used only timed quizzes. How would your confidence in your assessment change?

When I use Google to search for sample “multiplication fluency assessments, worksheets from TPT show up.If you haven’t yet, it is important to stop and read three articles:

From Fluency: Simply Fast and Accurate? I Think Not! by Linda Gojak:

Building fluency should involve more than speed and accuracy. It must reach beyond procedures and computation.

From Fluency without Fear by Jo Boaler:

The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense.

Ok, so how do we assess fluency? Do we have common language for mathematical fluency?

From  Principles to Actions: Ensuring Mathematical Success for All

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

Let’s focus this 3rd grade standard:

Multiply and divide within 100.

CCSS.MATH.CONTENT.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

 

Do any of the fluency assessments from the above Google search help us assess fluency?

What if we try a different type of assessment? An assessment that involves listening and observing as children compute, reason, and respond.

What if we confer with learners individually, just like we do with reading assessments, to listen, record, and encourage learners as they think, reason, and compute?

I need, want, and invite your feedback on the following prototype.  I hope that I have constructed a viable argument and I seek your constructive critique.

Multiplication Fluency (within 100)
Conferring with Mathematicians

Section 1 asks students if they know their multiplication facts. We intentionally ask these questions because of John Hattie’s work on student expectations of self – effect size of 1.44.

Section 2 checks for accuracy and efficiency.  The teacher will code the student’s response as recall, uses a strategy, or skip counts.  The first 8 facts are common for all students. The next 8 facts can be customized for each student based on their responses to easiest and hardest.

 Section 3 checks for accuracy and flexibility.  In the previous section, the teacher gave the multiplication expression and the student responded with the product.  Now it is reversed. The teacher gives the answer and student states an expression and are asked if they know another way.

Section 3 also checks for accuracy and flexibility using images from #UnitChat.  Students are asked how many they see and how do they see them.  If they skip count, their answer is confirmed, and they are asked if they can say or write it again using multiplication.

If you open this Multiplication Fluency (within 100) pdf, you will find each image on its own page so students can touch to count when needed.

Will you take time to offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the assessment?

Thank you in advance.

Summer Learning 2019 – Choices and VTR

How do we learn and grow when we are apart? We workshop, plan, play, rest, and read to name just a few of our actions and strategies.

We make a commitment to read and learn every summer.

Below is the 2019 Summer Learning flyer announcing the choices for this summer.

In case you are interested, links to reviews of each book are shared below as well as the set of TED Talks for Voices from Diversity. #SoGood

Big Potential and The Power of Moments are repeats from last summer’s list because of faculty/staff engagement and enthusiasm. Blindspot and Developing Assessment-Capable Visible Learners are being used in book study groups during the current school year.  We hope to harness the power of the re-read and spread this ideas.

We will continue to use the Visible Thinking Routine Sentence-Phrase-Word to notice and note important, thought-provoking ideas. This routine aims to illuminate what the reader finds important and worthwhile.

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

However, the power and promise of this routine lies in the discussion of why a particular word, a single phrase, and a sentence stood out for each individual in the group as the catalyst for rich discussion . It is in these discussions that learners must justify their choices and explain what it was that spoke to them in each of their choices. (Ritchhart, 208 pag.)

What are you reading/watching/doing to grow as a learner over the summer? Please feel invited and encouraged to watch us (or join us) learn by following #TrinityLearns and #TrinityReads in June and July.


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

#TrinityLearns Leading Learners to Level Up as a TEAM (#LL2LU) Part 2

Continuing our work from last month, Trinity School’s Assessment Committee continues to grapple with the following questions.

As a team, how are we united (aligned) in our understanding and assessment of learning?  How might we grow our assessment literacy, understanding, and actions to focus on learning, assign competence, and empower learners to become agents of learning?

Under the leadership of Thomas Benefield (@yerlifeguard) and Becky Holden (@BHolden86), Trinity School’s Assessment Committee we continue our commitment to read and take action on Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill by Nancy Frey, John Hattie, Douglas Fisher.  

Below is our agenda for the April meeting where we have begun to grapple with growing our understanding together.

As a team of teachers representing all grade-levels at our school, we chose to analyze student work together and hold a norming meeting to explore and learn one way to help our grade-level teams calibrate and clarify expectations around collaboration and citizenship.

To ensure that all voices were heard, we started with quiet reading time to preview the draft of the learning progressions.  As we did last month,  we used a Google form (shown below) to record everyone’s initial thinking around the level of work based on the drafted learning progressions for Working Cooperatively and Displays Respect.

The artifact, in this case, was a two minute video that offers a glimpse of partner work. (The video is not shared in this post, but a screen shot of one second is shown below.)

Using the Google form continues to be critically important. Everyone’s initial thinking was made visible to the team. Look at the results from our initial thinking.

As you can see, we were all over the place in our interpretation of the meaning and expectations described in our learning progressions.

As a team of assessment leaders, we had anticipated this result. You can see how this might be problematic for students in different sections with different teachers, right?

High-functioning teams that focus on learning must calibrate their understanding of what is essential to learn so that all students are assessed fairly and equitably.

What happened next was nothing short of magical.

First, we discussed our leveling with one partner to explain our reasoning and understanding. It was quiet, calm, and intense.  As partners listened to each other, different interpretations and points of view were represented.  When enough time passed, we returned to the whole group setting and discussed. Again, magical! Everyone confidently shared their initial level assessment and then spoke of how their understanding was shifted by discussing it with someone else.

Then, we took time for individual reflection and leveled the same artifact again, based on our developing common assessment. Just look at the results.

Closer, so much closer to common understanding.

To hone our skills and understanding, we used the same two learning progressions for Works Cooperatively and Displays Respect using video from a different grade level. (The video is not shared in this post, but a screen shot of one second is shown below.)

Again, more closely aligned understanding.

What can be gained when all ideas are made visible to the entire team? How might we learn and grow together by sharing our thinking, seeking feedback, and calibrating with our team?

How do your school’s teams calibrate expectations, shared values, and common understanding?

What actions will we take to become stronger and clearer as a team?


Frey, Nancy, et al. Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill. Corwin Literacy, 2018.

#TrinityLearns Leading Learners to Level Up as a TEAM (#LL2LU)

As a team, how are we united (aligned) in our understanding and assessment of learning?  How might we grow our assessment literacy, understanding, and actions to focus on learning, assign competence, and empower learners to become agents of learning?

Under the leadership of Thomas Benefield (@yerlifeguard) and Becky Holden (@BHolden86), Trinity School’s Assessment Committee made a commitment to read and take action on Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill by Nancy Frey, John Hattie, Douglas Fisher.  The committee has met approximately once a month to study, discuss, and learn more about growing our young learners as capable, independent, self-correcting, and self-reliant learners.  

Below is our agenda for the March meeting where we have begun to grapple with growing our understanding together.

As a team of teachers representing all grade-levels at our school, we chose to analyze student work together and hold a norming meeting to explore and learn one way to help our grade-level teams calibrate and clarify expectations.

To ensure that all voices were heard, we started with quiet reading time and used a Google form (shown below) to record everyone’s initial thinking around the level of work based on the given learning progressions for Making Thinking Visible and Using Text Evidence.  

Screen Shot 2019-04-10 at 7.18.23 PM

Using the Google form was critically important. Everyone’s initial thinking was made visible to the team. Look at the results from our initial thinking.

As you can see, we were all over the place in our interpretation of the meaning and expectations described in our learning progressions.  It was eye-opening.

As a team of assessment leaders, we had anticipated this result. You can see how this might be problematic for students in different sections with different teachers, right?

High-functioning teams that focus on learning must calibrate their understanding of what is essential to learn so that all students are assessed fairly and equitably.

What happened next was nothing short of magical.

First, we discussed our leveling with one partner to explain our reasoning and understanding. It was quiet, calm, and intense.  As partners listened to each other, different interpretations and points of view were represented.  When enough time passed, we returned to the whole group setting and discussed. Again, magical! Everyone confidently shared their initial level assessment and then spoke of how their understanding was shifted by discussing it with someone else.

What can be gained when all ideas are made visible to the entire team? How might we learn and grow together by sharing our thinking, seeking feedback, and calibrating with our team?

How do your school’s teams calibrate expectations, shared values, and common understanding?


Frey, Nancy, et al. Developing Assessment-Capable Visible Learners, Grades K-12: Maximizing Skill, Will, and Thrill. Corwin Literacy, 2018.

Book study: #ChoralCounting and #CountingCollections – session 2 #TrinityLearns

As a community, we are focused on high-quality instruction that leads to deep understanding.  The teachers of our youngest learners take action to develop young, strong mathematicians.  Together, we are studying Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom to deepen and strengthen our understanding of learning and teaching early numeracy.

Counting is a vibrant part of early learning about mathematics. Young children are constantly counting as they make sense of their world. [Franke, Kindle Locations 490-491}

For our second session together, we turned our attention to choral counting and what we learn when we listen to the diverse thinking of the learners in the room.  As you can see from the tweets below, our teachers are working with our young students to confirm a sense of belonging while strengthening our culture of being seen, known, and heard by teachers and peers.

Choral Counting gets to the heart of what we want for our mathematical communities. This activity creates space for all students to notice, to wonder, and to pursue interesting ideas. Students and teachers alike wonder together about patterns, and why and how numbers change or stay the same. [Franke, Kindle Locations1526-1528}

In addition to deeper work with choral counting, we continue to empower young learners to count, record, and think.

Learning about counting and cardinality are big ideas in the early grades. Having a collection of items invites children to count to find the total number of objects. As children count, they come to understand the relationship between numbers and quantities and connect counting to cardinality. [Franke, Kindle Locations 501-503}

How do we strengthen and deepen understanding, confidence, and efficacy? Who do we help when learners persevere, show their work, and . . . ? What are ways to empower learners to become self-correcting, self-reliant, and independent?

#TrinityLearns


Franke, Megan L. Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom.. Stenhouse. Kindle Edition.