Category Archives: Uncategorized

Oh, the Places We’ll Go!

The following sentence is not something I imagined writing or saying out loud in 2020. This spring my husband, daughter, and I are planning to move to Rochester, New York.

Up until two months ago, I was happily working at my job as a curriculum coordinator, and I’d just been elected as President-Elect of the Texas Association of Supervisors of Mathematics (TASM). That trajectory involved staying put here in Austin.

It’s funny how a global pandemic can radically alter your plans and priorities. (It’s not actually funny. I’ve been on an emotional roller coaster the past several of months as events have unfolded, and the ride isn’t over.)

So why the move and why Rochester, New York? While we had no immediate plans to move up until now, we had been entertaining the idea of eventually moving out of Texas. Our primary reason has to do with climate change. The summers here are brutal, and that’s only going to get worse, not better. It’s frustrating to me that when my daughter is out of school for the summer, going outside feels like a punishment because it’s so dang hot. It’s a huge missed opportunity to get out together as a family.

In mid-July when my husband tossed out the idea of moving, it didn’t take long to figure out where we would go. My husband and I have noticed over the past few years that whenever we go on vacation we invariably choose to visit extended family in upstate New York. Tom’s parents live in Syracuse and his brother lives in Rochester. My family lives outside Buffalo.

Our daughter has a close relationship with her grandparents and uncle. This summer she had almost daily video calls with her grandma and grandpa. I envy her because as a military brat I rarely lived anywhere near extended family while I was growing up and visits were sporadic.

This was a difficult decision because it means I’m going to have to resign from my job (though HR decisions are pushing me in that direction anyway), leave my coworkers and friends behind, and I won’t get to continue working with the great folks at TASM.

But it became a much easier decision after talking about our plans with our daughter. We were nervous about how she’d feel, and we were willing to reconsider if she had strong reservations. Turns out there was no reason to worry. She is over the moon about our move! She can’t wait to live close to family, and she is unbelievably excited about living somewhere that gets SNOW.

We’ll also be a short drive from Niagara Falls, which she loved visiting for the first time a few years ago.

Her excitement makes me excited. I’m excited about getting to live and explore a new place with her. I’m excited about milder summers where we can get outside and enjoy time together as a family. I’m excited to be close to extended family and have even more opportunities to spend time with them. I’m excited about eventually being able to go on vacation somewhere other than to visit family. I’m excited about living in a state where we can drive for 6 hours and actually get to a different state.

This may not be the direction I imagined or planned on going, but I’m excited and hopeful about what the future holds.

Join me at the Build Math Minds Virtual Math Summit!

In just over 2 weeks you have the opportunity to learn from over 20 amazing math educators…and me!

I’m excited to be a part of the Virtual Math Summit hosted by Christina Tondevold (Build Math Minds) and I’d love for you to join me.

The summit starts August 3rd (with a special pre-conference session that you won’t want to miss), so get registered now: 

It is three days of professional development specifically for elementary educators and it’s completely FREE!

I can’t wait to see how you take the knowledge you learn during the summit and run with it.

Remember that if you can’t be there when the sessions are released there will be a limited replay period through August 10, so that you can watch the sessions at a time that works for you.

Go here to get registered now:

Multiplication Number Talks Using Models

(This post is re-blogged from my other math blog.)

In my previous post I discussed the importance of planning number talks with the four stages of using models. I used a 1st grade example in that post, and almost immediately my colleague Heidi Fessenden shared this wondering.

Great minds think alike, because this is exactly what I’ve been thinking about lately!

In Round Rock ISD, we want our students to learn thinking strategies for multiplication, rather than attempting to memorize facts in isolation. Thinking strategies have the following benefits for our students:

  • There’s less to memorize because there are 5 thinking strategies to learn instead of 121 isolated facts.
  • They create consistent language across grade levels.
  • They afford a strategic mindset around how we think about computation facts.
  • Their utility extends beyond basic facts to computation with larger numbers.

The thinking strategies we want our students to learn are from ORIGO’s Book of Facts series. (Each strategy is linked to a one-minute video if you’d like to learn more.)

One-Minute Overview Videos

In our curriculum, students learn about these thinking strategies in their core instruction. We have two units in 3rd grade that focus on building conceptual understanding of multiplication and division across a total of 51 instructional days. In between those units, students practice these thinking strategies during daily numeracy time so they can build procedural fluency from their conceptual understanding. My hope is that planning number talks with the four stages of using models will facilitate this rigorous work.

I also hope it supports students in maintaining their fluency at the start of both 4th and 5th grade. Our daily numeracy time at the beginning of both of those grade levels focuses on multiplication and division. Even if every 3rd grade student ended the year fluent, it’s naive to think that fluency will continue into perpetuity without any sort of maintenance.

To help teachers envision what a number talk might look like at different stages of using models, I’ve designed a bank of sample number talks for each thinking strategy.

Each bank includes a variety of examples from the different stages of using models:

  • Stage 2 Referring to a complete model (Number Talks 1-4)
  • Stage 3 Referring to a partial model (Number Talks 5-8)
  • Stage 4 Solving the problem mentally (Number Talks 9-10)

You’ll notice some “Ask Yourself” questions on many slides. You’re welcome to delete them if you don’t want them visible to students. Ever since reading Routines for Reasoning by Grace Kelemanik and Amy Lucenta, I’ve been utilizing the same pedagogical strategies they baked into their routines to support emergent bilingual students and students with learning disabilities:

  • Think-Pair-Share
  • Ask Yourself Questions
  • Annotation
  • Sentence Stems and Sentence Starters
  • The 4Rs: Repeat, Rephrase, Reword, Record

Since not all of the teachers in our district might be aware of “Ask Yourself” questions, I embedded them on the slides to increase the likelihood they’ll be used by any given teacher utilizing these slides.


These sample banks are not designed to be followed in order from Number Talk 1 through Number Talk 10. Student thinking should guide the planning of your number talks. As Kathy Richardson shared in a tweet responding to my previous post, the four stages of using models are about levels of student thinking, not levels of instruction.

What these number talks afford is different ways of thinking about computation. A traditional number talk that presents a symbolic expression allows students to think and share about the quantities and operations the symbols represent. The teacher supports the students by representing their thinking using pictures, objects, language, and/or symbols.

A number talk that presents models, on the other hand, allows students to think and share about the the quantities shown and the operation(s) implied. The teacher supports the students by representing their thinking with language and/or symbols.

Trying It Out in the Classroom

For example, I led a number talk in a 5th grade class today, and I started with this image:

A student said she saw 10 boxes with 3 dots in each box. I wrote that language down verbatim, and then asked her how we could represent what she said with symbols. She responded with 10 × 3.

I asked the 5th graders to turn and talk about why we can use multiplication to represent this model. This was challenging for them! They’ve been multiplying since 3rd grade, but they haven’t necessarily revisited the meaning of multiplication in a while.

They were able to use the model to anchor their understanding. They said it’s because the number 3 repeats. This led us into talking about how there are 10 groups of 3 and how multiplication is a way that we can represent counting equal groups of things.

The number talk continued with this second image:

The first student I called on to defend their answer said, “I know 10 times 3 is 30, so I just took away 3.”

I recorded (10 × 3) – 3 = 27, but I didn’t let the students get away with that. I reminded them that multiplication is about equal groups. If we had 10 groups of 3, then we didn’t just take 3 away, we took away something else.

One of the students responded, “You took away a group.”

We continued talking which led to me recording (10 groups of 3) – (1 group of 3) = 9 groups of 3 under the original equation and then (10 × 3) – (1 × 3) = 27 under that.

I have to admit I screwed up in that last equation because I should have written 9 × 3 instead of 27. Thankfully number talks are an ongoing conversation. Students’ number sense is not dependent on any given day’s number talk, which means they’re forgiving of the occasional mistake.

What we did today is hopefully the start of a series of number talks to get students thinking about how taking away groups is one thinking strategy to help them derive facts they don’t know. Students don’t own that strategy right now, but our conversation today using the model was an excellent start.

Final Thoughts

I’m hoping these samples might inspire you to create number talks of your own based on the kinds of conversations you’re having with your students. Here is a document with dot images you can copy and paste from to create your own number talk images.

If you try out these number talks in your classroom, I’d love to hear how it went. Either tag me in a tweet (@EMathRRISD) or share your experience in the comments.

Planning Number Talks with the Four Stages of Using Models

(This post is re-blogged from my other math blog.)

A year ago I attended the Math Perspectives Leadership Institute led by Kathy Richardson. One idea she shared that really resonated with me was the four stages of using models.

Four Stages of Using Models

Stage 1 Moving the model. Students need to actually touch and move the model.

Stage 2 Referring to a complete model. Students can look at models that represent all the numbers in the problem.

Stage 3 Referring to a partial model. Students can look at a model and think about what would happen if a number was added or taken away or the model was reorganized.

Stage 4 Solving the problem mentally. The student can solve the problem mentally without a model but can also use the model to demonstrate their thinking or prove their answer.

Kathy Richardson went on to share the following points about the importance of models:

  • Models are used so the quantities become meaningful to the students
  • Models allow the students to look for structure, parts of numbers, and relationships between them
  • Every child has a way to work in the problem
  • Everyone can participate because they solve the problem in ways they understand

This got me thinking about number talks. Do we capitalize on the value of models when planning number talks? Or do we have a tendency to gravitate to stage 4 without considering whether each and every student is actually ready for it? If we spend the bulk of our time in stage 4, are we considering issues of access? Whose knowledge do we privilege when we consistently present problems symbolically and assume that students are thinking flexibly about how to manipulate the numbers mentally?

Don’t get me wrong, I want students to reach stage 4, but I wonder how we can ensure we’re taking the necessary steps to build each and every student up to this kind of thinking. If you revisit the four stages of using models, what it looks like to me is a progression of transferring the actions of computation from physical, hands-on actions to increasingly mental actions. If we want students to mentally compose and decompose numbers, then we can use these stages to build a bridge from physically performing the action to mentally performing the action, and each stage creates a pathway for this to occur.

Let’s look at this progression in 1st grade as students are learning and practicing the Count-On Strategy for Addition.

Stage 1

Moving the model. In this number talk, students build the count-on addition facts on a ten frame.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “Show 5.”


Teacher: “Add 1 more.”


Teacher: “What is 5 and 1 more?”

Students: “5 and 1 more is 6.”

Teacher: “How can we record what we did using an equation?”

Students: “5 + 1 = 6”

Follow up questions the teacher should ask to help students make connections between the two representations:

  • Where is the 5 in your model?
  • Where is this 1?”
  • Then what does the 6 mean?”
  • What do these two symbols mean, + and =?

Repeat to solve and discuss more problems as time permits.

Stage 2

Referring to a complete model. In this number talk, the teacher shows both addends using a visual model.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “How many on top?”

Students: “5.”

Teacher: “How many on the bottom?”

Students: “2.”

Teacher: “What is 5 and 2 more?”

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

While counting all is a valid strategy, I purposefully set our mathematical goal for this number talk for students to use what they know about addition to solve problems. I would accept counting strategies, but I would emphasize strategies involving addition, such as counting on 2 from 5.

Like the previous example from stage 1, I would also be sure we create and analyze an addition equation that this model represents.

Repeat to solve and discuss more problems as time permits.

Stage 3

Referring to a partial model. In this number talk, the teacher shows both addends using a visual model.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “How many dots are there?”

Students: “4.”

Teacher: “What if I added two more? How many would we have altogether?”

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

Like the previous example from stage 2, I would accept all strategies, but I would emphasize strategies that relate to our mathematical goal, which is using what we know about addition to solve problems. I would also be sure we create and analyze an addition equation that this problem represents.

Notice that throughout all three of these stages, the action still exists, “add 1 more” or “add 2 more.” The difference is that while students can physically perform the action in stage 1, they have to mentally perform the action in stages 2 and 3. In stage 2, they can see both quantities so they can refer to both and they can even mentally try to manipulate them, if necessary.

In stage 3, students are anchored with the first quantity, but now they not only have to imagine the second quantity, but they have to imagine the action as well. In the example above, while they cannot physically add two more counters to the ten frame, their repeated experiences with the physical action means they have a greater chance of “seeing” the action happening in their mind. The work through these three stages prepares students for the heavy lifting they have to do in stage 4.

Stage 4

Solving the problem mentally. In this number talk, the teacher shows a symbolic expression.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher writes the problem on the board and asks students to solve it mentally:

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

This stage flips the script with regard to creating and connecting representations. Now the teacher can select from a variety of models she can draw to illustrate a student’s strategy. She might draw a ten frame, draw a math rack, draw hands and label the fingers with the numbers a student said, draw a number line, or even write equations.

As Pam Harris says, the goal of creating a model to represent a student’s thinking during a number talk is to make that student’s thinking more “take-up-able” by the rest of the class. Just because we’ve reached the point where students can solve problems represented symbolically doesn’t mean we stop making connections to models. We don’t want to unintentionally send the message that the symbols “5 + 2” somehow mean addition more than all of the other representations students have created and used.

The advantage of moving to symbols is just that they allow us to communicate in more efficient ways. While the efficiency is less obvious in the case of 5 + 2 – recording 3 symbols vs drawing 7 dots – it is much more obvious with a 25 + 12 – recording 3 symbols vs drawing 37 dots.

Final Thoughts

As you continue to plan number talks this year, consider the four stages of using models, particularly how these stages can help create access to the critical mathematical ideas at your grade level for a wider range of learners in your classroom. The final goal may be solving symbolic problems mentally, but it doesn’t mean that’s where we have to start or even where we have to spend the majority of our time.

Not So Final Thoughts

After sharing this post, Kathy Richardson responded with the following tweet and I wanted to be sure to share it here since this post is heavily influenced by her work.

I also heard that her new book Number Talks in the Primary Grades is going to be released in January. I look forward to checking it out!

Sharing the Learning (2019 NCTM Annual Meeting – San Diego)

It feels like a dream, but this time two weeks ago, I was sitting in the opening session of the NCTM Annual Meeting in San Diego listening to Gloria Ladson-Billings opening keynote. (I’ll add a link to the video of her talk once it’s posted on the NCTM website.)

It was a whirlwind of a conference. I got to see my friend and former co-worker Meredith, hang out with countless #MTBoS colleagues, and attend so many great sessions! By the time the conference was over, my heart and brain were full to bursting. To get a taste, check out this Twitter Moment I created to capture many of my tweets from the conference.

I also took copious notes throughout the many sessions I attended. I’m not going to bombard you with all of my notes, but I did want to share short summaries and key takeaways from all of the sessions. I know it’s not the same as being there, but I’m happy to share the learning and spark some ideas for those unable to attend.

Here goes!

The Decision-Making Protocol for Math Coaching: Apply High-Leverage Practices and Advocate Change

Presenters: Courtney Baker (George Mason University) and Melinda Knapp (Oregon State University-Cascades) See tweets from this talk here.

In this session, the presenters shared the Decision-Making Protocol for Mathematics Coaching (DPMPMC). “A primary goal of the DMPMC framework is to increase the intentionality of coaching interactions by supporting the user to simultaneously consider mathematics content, coaching and teaching practices, and professional relationship building.”

If you’d like to learn more, check out the site linked above and specifically check out the two articles they’ve written about this protocol. The first is “Coaches Engage with Principles to Actions” from the September 2018 issue of Teaching Children Mathematics. The second is “The Decision-Making Protocol for Mathematics Coaching: Addressing the Complexity of Coaching With Intentionality” and Reflection from the March 2019 issue of Mathematics Teacher Educator.

One Takeaway: I like the dual-pronged approach to coaching. Whether you follow the protocol or not, I appreciate the challenge of picking just ONE mathematics coaching practice and ONE mathematics teaching practice to focus the work. There’s always so much we can do, but if we try to do too much, we decrease the coherence and impact for the teacher being coached.

Using Lesson Study to Empower All Students

Presenters: Kyndall Brown (UCLA) and Susie Hakansson (Retired)

In this session, the presenters shared an initiative from the California Action Network for Mathematics Excellence and Equity (CANMEE) to develop and implement a model of lesson study that places an emphasis on equity. The rationale behind this work is twofold. First, they want to make lesson study an integral part of professional learning and continuous improvement. Second, they believe equity and social justice the most urgent goal and challenge for mathematics education. You can access their slides and other materials in this folder.

One Takeaway: I really like the idea of using four focal students as a lens throughout the lesson study. “If we are to focus on equity, who do we select so that we shift our practices to impact positively students’ participation and their increase in mathematical proficiency?” Not only do you interview the focal students, but you also develop a profile of each one that:

  • is asset-based,
  • includes students’ prior knowledge (cognitive and affective),
  • includes student understandings,
  • includes outside of class attributes,
  • identifies learning goals, and
  • avoids deficit thinking.

Lesson Study and How to Generate Buy-in that Will Inspire Instructional Shifts and Evolve Teachers

Presenter: Chase Orton (Independent Consultant) See tweets from this talk here.

While the previous session focused on changes to the lesson study process, Chase focused on steps he takes to build buy-in and set teachers up for brave professional growth before the process even begins. The first step is the passion profile. According to Chase, teaching is a practice of identity. If we are going to ask teachers to undertake the process of lesson study, teachers need to reflect on their own identity – specifically their passions and their why – as well as get to know the identities of the others who will undertake the lesson study journey with them.

The second step is defining the ideal classroom. “Let’s say you’re teaching or witnessing the best math lesson ever. What does it look like? Be really specific, looking at what the teacher is doing, what students are doing, and what the classroom energy feels like.” This step creates a powerful pivot to establish focus for the lesson study as participants develop their research question. How does your ideal compare to reality? What forces are restricting your ability to create your ideal math classroom?

One Takeaway: I appreciate the effort Chase takes to do the very important work of investing in the people who are going to invest their time and energy into lesson study. How often do teachers feel like something is being imposed on them rather than feel like they are being included and part of a team effort? How often do they get the chance to reflect on their own experiences and beliefs and help set the goals for the work ahead? Chase has written extensively on his blog about his work with lesson. If you’re interested in learning more, check out these posts.

Minimizing the Matthew Effect

Presenter: Sara Van Der Werf (Independent Consultant) See tweets from this talk here.

If you ever get the opportunity to see Sara Van Der Werf present, take it! She is one of the most passionate and committed educators I’ve ever met. This session was effectively her throwing down the gauntlet that teachers can and must lead the way to change structures so that all students are successful. According to Sara, if we wait on superintendents and administrators, it will never happen. The great thing is that even if you can’t see Sara in person, she does a phenomenal job of writing about her beliefs and advice on her blog. For example, in her session she evangelized Stand and Talks as one of the best things she ever did to get students talking to one another, and for those who couldn’t attend, you can read all about them in this blog post. Be sure to also check out her posts on how she uses name tents to build relationships with her students and her post on why she loves cell phones in math classrooms.

One Takeaway: Sara mentioned that using color coding doesn’t get nearly enough attention as it should. She introduced us to the #purposefulcolor hashtag and shared an example of how she’s using color more intentionally to support students. For example, when doing a Which One Doesn’t Belong? she puts each image on a different-color background. Now students can say, “The red one doesn’t belong because…” rather than having to generate clunky language such as, “The one in the upper left corner doesn’t belong because…”

Leveraging the Predictable Design of Instructional Routines to Elicit and Use Student Thinking

Presenter: Danielle Curran (Curriculum Associates) and Grace Kelemanik (Fostering Math Practices) See tweets from this talk here.

It’s masterful how Grace Kelemanik and Amy Lucenta weave intentional and powerful pedagogical moves into instructional routines so they’re baked in from the start. Just take a look at the key teaching moves in the Try-Discuss-Connect routine:

What’s powerful about these teaching moves – individual think time, turn and talk, and the four Rs – is that they were intentionally chosen and embedded into the routine because of their alignment with research about how best to support emergent bilingual students and students with learning disabilities.

One Takeaway: I love learning about new instructional routines, but I was especially pleased with stepping back and making explicit connections between instructional routines, in general, and the effective mathematics teaching practices from NCTM’s Principles to Actions.

The Hierarchy of Hexagons: An Example of Geometric Inquiry

Presenter: Christopher Danielson (Desmos) See tweets from this talk here.

By this point in the conference, my brain was already feeling a little full. I chose Christopher’s session primarily to do something fun – exploring hexagons. However, I had previously read about this work on his blog, so I also wanted to experience it firsthand to help me bring this kind of activity back to do with my teachers.

The session did not disappoint! Collectively a room full of educators attempted to name, classify, and sort hexagons in meaningful ways.

  • What does it mean to say a hexagon looks like a comet? What are the defining attributes of all hexagons that are comet-like?
  • What do you mean when you say a hexagon is boxy? How many right angles are you saying it should have?

One Takeaway: During the session, Christopher centered our work around the van Hiele model for geometric understanding.

I’ll be honest that I only first heard about this model a couple of years ago and found it extremely useful when developing a progression of units and lessons across our grade K-5 curriculum. Interestingly, my colleague Edmund Harriss took issue with the van Hiele levels and started a lengthy, but insightful, Twitter conversation around these levels and geometry instruction in general. If you have a chance, I recommend perusing the thread sometime. My takeaway from the conversation mirrors this reflection from Christopher: “Yup. Not hard and fast developmental rules, but useful structure for describing student thinking and for planning instruction.”

More Than Turn and Talk: Supporting Student Engagement in Each Other’s Ideas

Presenter: Megan Franke (UCLA) See tweets from this talk here.

This was a fascinating session where Megan Franke shared research about the role of student participation in student achievement. A surprising finding in the research is that there isn’t an “ideal” or consistent profile of student participation or teacher support that is best for all students. Rather, the important thing is that teachers create a space where all students are able to participate in ways that work for them. For example in a classroom where there are whole class discussions, turns and talks, and collaborative problem solving, students have varied opportunities to participate.

One Takeaway: According to the research, student achievement is impacted if the student gets at least one opportunity every class to explain all the way through their ideas. If teachers only lead whole class discussions, this is unlikely to happen for all students but rather a small handful of students. This gives me a goal for next school year to share this research with our coaches, administrators, and teachers so they can evaluate their current classroom structures and adjust as needed to create opportunities in math class for students to find space(s) to participate that work for them.

Rethinking Mathematics Education (and Mathematics) through Neurodiversity

Presenter: Rachel Lambert (UC-Santa Barbara), Edmund Harriss (University of Arkansas), and Dylan Lane (Independent Researcher)

In this session, Rachel Lambert challenges the medical/deficit model of disability.

Differences exist, according to Lambert, not as deficits, but as part of natural human diversity. She went on to share research about people with dyslexia and dyscalculia. The medical/deficit model emphasizes the challenges these disabilities pose, but research has shown that people with these disabilities also have a set of strengths. She then ceded the floor to Dylan Lane and Edmund Harris. Dylan grew up with dyscalculia while Edmund grew up with dyslexia. They each shared their story, which emphasized the power of leveraging strengths rather than fixating on deficits.

One Takeaway: Often we oversimplify kids, especially when we see them struggling. There’s a false deficit binary of being high or low at math, but it’s not that easy or simplistic to categorize children that way. We are all a combination of strengths and challenges. If we can see all of each other, we can get past deficit thinking. We need to complicate the way we think about our kids, but also how we think about learning mathematics. Math has to have more ways for students to develop and demonstrate understanding – more linguistic for some, more visual for others.

Collaborative Coaching: How Can We Learn as a Team?

Presenter: Nicora Placa (Hunter College) See tweets from this talk here.

In this session, Nicora Placa talked about the important role of collaborative coaching as a different type of learning opportunity that allows all members of a team to learn together and take risks. When selecting coaching strategies to use in collaborative coaching, Nicora looks for tools that focus on foregrounding student learning and student thinking. In this session, she shared the plan for how she uses clinical interviews during collaborative coaching:

  • Background reading / Book study
  • Watch videos of interviews
  • Select tasks and anticipate misconceptions
  • Practice interviews with each other
  • Conduct and record interviews in team meetings / PD
  • Analyze interviews
  • Summarize and share what we learned

She also gave us an opportunity to practice conducting an interview in trios. One person acted as a “student” working on a math task, one person acted as the interviewer, and the third person recorded what the “student” and interviewer said. Afterward we reflected on the kinds of questions asked and alternatives that could have been asked to elicit more student thinking.

One Takeaway: I appreciate that Nicora shared the challenges of listening to student thinking:

  • Listening only for the right answer or particular solution path
  • Thinking about next instructional move instead of listening
  • Assuming students are thinking the way you are thinking
  • Not listening for what students know
  • Not trying to make sense of what students are doing

The sample questions as well as list of questions to avoid were extremely helpful.

The Whole-School Agreement: Aligning Across and Within Grades to Build Student Success

Presenters: Sarah Bush (University of Central Florida), Karen Karp (John Hopkins University)

The Whole School Agreement process aligns models, language, and notation across and within grades to that students see the regularity and familiarity in a cohesive approach to teaching mathematics. The presenters encourage centering this work around their articles:

One Takeaway: I’m excited to use this framework and these resources to support coaches and campuses. I was familiar with these articles, but I’ve never used them to center the work of creating whole school agreements. The presenters shared resources in these handouts that can help with the work:

Coaching Toward Common Ground: Creating a Shared Vision and Growing Professionally as a Team

Presenters: Delise Andrews (Lincoln Public Schools) and Beth Kobett (Stevenson University)

The presenters took us through a sped up version of a process they use to help teams create a shared vision and find common ground. First, we worked together to illustrate a picture of the “ideal” math classroom. Then we used our pictures to list qualities of our ideal math classroom. The presenters then posed a question to us, “If this quality isn’t there, what’s the opposite of that?” This led us to develop opposites for each one of our statements. Then we drew lines between them to create a spectrum, because often we’re not at one or the other. Rather, we’re somewhere in the middle.

Next, everyone in the group got to put a mark on each line to show where they are in their practice. This is very eye opening because patterns emerge. Perhaps as a team we are all doing really well on Thing #1, but Thing #3 is an area where we struggle. This can help us develop goals.

After picking one thing to focus on, we went through another exercise called 20 Reasons Why. Basically we had to come up with 20 reasons why that thing is the way it is right now. This is more challenging than it looks! It’s easy to come up with the first 5 or 6 reasons, but getting to 20 requires thinking beyond the usual suspects. Finally, if we had time, we would have sorted our 20 reasons and talked through the reasons for our sorting. For example, we could have sorted them into categories, “Things I can control” and “Things I can’t control.”

One Takeaway: I liked the idea of reversing assumptions. According to the presenters, breakthrough ideas happen when we challenge our original ideas and even reverse our thinking. What if the opposite is true? For example, if our team’s original reason was, “We don’t have time to plan these kinds of lessons,” we could turn it on its head and said, “What if we did have the time? How would we plan differently?”

Another example would be, “Our students who are struggling with 5th grade math don’t know basic math facts.” If we reverse our assumption, we come up with, “What if our students who are struggling do know some basic math facts?” (What? They don’t know any? Oh, they do know some. Good. We have a place to start.)

The End

Whew! Just going through all that makes my brain feel full all over again. If you attended NCTM what were your big takeaways? If you didn’t attend, but read through my tweets, this post, or other tweets, what piqued your interest or resonated with you?

I’m on a Podcast!

Recently I was honored to be a guest on my school district’s Teaching & Learning podcast. Here’s the blurb about the episode:

On this episode, we sit down with Elementary Math Curriculum Coordinator, Brian Bushart. Brian has led the direction of elementary math in Round Rock ISD for the last four years and his impact on teaching and learning for thousands of students continues to grow every day. In this interview, we talk about how ideas around math instruction have changed over the years, how “Sense-Making” and “Notice and Wonder” have helped students move beyond an over-reliance on memorization when it comes to understanding numbers, algorithms, and essentially how the world works. Be sure to scroll down to the notes below as Brian shared a treasure trove of resources for math teachers at all levels, parents of our students, and all of us who seek a better understanding.

After listening to podcasts for several years, it was a fun experience to be a guest on one. I appreciate Ryan Smith, our Executive Director of Teaching & Learning, inviting me to chat with him and share my passion for mathematics education. If you’d like to give the podcast a listen, you can subscribe in your favorite podcast app. Look for Round Rock ISD T&L Show. You can also stream the episode and find links to some of the resources mentioned in the episode on the show’s website.

While you’re at it, I highly recommend checking out the episode where Ryan interviews former Instructional Coach and current Curriculum Specialist Gina Picha about math anxiety.

The Path Ahead

Last spring I wrote about how over the past few years I’ve continually revised and refined the scope and sequence of elementary mathematics in grades K-5 in my school district. You can read those posts here:

The tl;dr version is that I concluded the series in May 2018 with these parting thoughts:

…what started as a blog series where I was planning to reflect on the changes I might make for next year has instead reaffirmed that the work I’ve done with my teachers over the past three years has resulted in six scope and sequences that make sense and don’t actually require much tweaking at all. I’m proud of what we’ve accomplished. Are they perfect? Probably not. But they appear to be working for our teachers and students, and at the end of the day that’s what matters.


Fast forward to this post I wrote reeling from my experiences at the Math Perspectives Leadership Institute in late June:

There is a HUGE disconnect between what [Kathy Richardson’s] experience says students are ready to learn in grades K-2 and what our state standards expect students to learn in those grades. I’ve been trying to reconcile this disconnect ever since, and I can tell it’s not going to be easy… I’m very conflicted right now. I’ve got two very different trajectories in front of me… Kathy Richardson is all about insight and understanding. Students are not ready to see…until they are. “We’re not in control of student learning. All we can do is stimulate learning.” Our standards on the other hand are all about getting answers and going at a pace that is likely too fast for many of our students. We end up with classrooms where many students are just imitating procedures or saying words they do not really understand. How long before these students find themselves in intervention? We blame the students (and they likely blame themselves) and put the burden on teachers down the road to try to build the foundation because we never gave it the time it deserved.


What a difference a month makes.

In May I was feeling proud and confident of the work I’d accomplished developing and revising our elementary scope and sequence documents. A month later I’m calling everything into question and having a crisis of conscience about whether the scope and sequences I’ve planned are actually creating some of the struggles I was trying to prevent.

Back in July I closed my post with no answers:

But how to provide that time? That’s the question I need to explore going forward. If you were hoping for any answers in this post, I don’t have them. Rather, if you have any advice or insights, I’d love to hear them, and if I learn anything interesting along the way, I’ll be sure to share on my blog.


This big question of how to reconcile the pace of learning for our youngest students with the pace of the state standards has been on my mind for months. Throughout the fall semester, I had countless conversations with colleagues in and out of my district. These conversations culminated in my taking a stab at revising our scope and sequences in grades K and 1 as well as proposing a new instructional model in grades K and 1. (Ultimately I made revisions to the scope and sequence documents for grades K-4, but I’m going to focus on K and 1 in this post.)

I’ve been sharing, talking about, and revising these document with teachers, instructional coaches, and curriculum specialists in my district for a couple of months now, and I feel like they’re finally in a shape that I want to share them here so you can see where all of this thinking has taken me since I last wrote about this in July.

As a point of reference, here are the Kindergarten and 1st grade units for the 2018-19 school year.

Kindergarten 2018-19

1st Grade 2018-19

Our curriculum is now open to the public, so if you’re interested in visiting any of these units to see unit rationales, standards, lessons, etc., you can do that here.

Contrast that with these proposed units for the 2019-20 school year:

Proposed Kindergarten 2019-20

  • Fall Semester
    • Unit 1 – I Am a Mathematician! (21 days)
    • Unit 2 – Beginning Number Concepts (30 days)
    • Unit 3 – Sorting and Classifying (30 days)
  • Spring Semester
    • Unit 4 – The Concepts of More, Less, and the Same (30 days)
    • Unit 5 – Joining and Separating Quantities (30 days)
    • Unit 6 – Building Number Concepts (30 days)

Proposed 1st Grade 2019-20

  • Fall Semester
    • Unit 1 – I Am a Mathematician! (15 days)
    • Unit 2 – Adding and Subtracting (30 days)
    • Unit 3 – Exploring Shapes and Fair Shares (27 days)
    • Unit 4 – Understanding Money (10 days)
  • Spring Semester
    • Unit 5 – More Adding and Subtracting (20 days)
    • Unit 6 – Collecting and Analyzing Data (10 days)
    • Unit 7 – Introducing Unitizing (15 days)
    • Unit 8 – Exploring the Place Value System (24 days)

Here are some of the changes and my rationale for them:

  • In Kindergarten we drastically reduced the number of units. Instead of 10 units, we’re down to 6. On top of that, the first unit has shifted from counting concepts to “I Am a Mathematician!” What does that mean? Here are the notes I took to describe this unit:
    • Exploring manipulatives
    • Exploring patterns
    • Reading books about counting, shapes, and patterns
    • Setting norms and expectations for engaging in a community of mathematicians
    • Establishing routines
    • Getting to know students’ strengths and areas of growth
  • I made the names of the units more vague. Rather than stress teachers out that their students should be counting to 5, then 10, then 20 in lockstep, I’m providing space for students to engage in number concepts in general. Teachers can differentiate as needed so students who need to work within 5 can continue to do that while other students are exploring 8 or 12 or 14.
  • I made the units in Kindergarten longer to give students time to “live” in the landscape of these concepts. This goes hand-in-hand with the new instructional model I’m proposing based on the work of Kathy Richardson. Now a typical day will include a short opening activity that’s done together as a whole class. The bulk of math time will be spent in an explore time where students self-select activities that are variations on the core concept of the unit. During this explore time, the teacher’s primary role is to confer with students and continually nudge them along in their understanding. Each day there is a short lesson close to help students reflect on their learning. Here’s a link to a sample suggested unit plan to help teachers envision what a unit might look like in grades K and 1. (Note: If you encounter a link you can’t access in the document it’s likely due to copyright that we don’t control.)
  • In 1st grade I reduced the number of units focusing on addition and subtraction. Similar to number concepts in 1st grade, I want to give students an extended amount of time to “live” in these concepts.
  • In 1st grade I moved place value to the very end of the year. According to Kathy Richardson, unitizing and place value topics are challenging for 1st graders. However, I have to include them because our state standards require it. In order to reconcile this, I want to give students as much of the year as possible for their brains to develop so they are working with the most up-to-date hardware when they start learning these critical concepts. Putting it at the end of the year also creates more proximity to when students will continue learning about place value in 2nd grade. I’ve even added a 2-digit place value unit to our 2nd grade scope and sequence to create a bridge and continue the learning.
  • In 1st grade, I created a unit just on unitizing and followed that up with a unit on place value. Using activities from Kathy Richardson’s Developing Number Concepts series, students will spend three weeks making, naming, and describing groups of 4, groups of 5, groups of 6, and eventually groups of 10. Then they’ll spend almost five weeks extending this as they learn how our place value system is built on groups of 10.

The units are just the tip of the iceberg. The math block in our district is 80 minutes and broken up across three components:

  • Focus Instruction (50 minutes)
  • Numeracy (10 minutes) – This used to be named Computational Fluency but I’m re-branding it because the names imply different goals.
  • Spiral Review (20 minutes)

So when I revised the scope and sequence documents, I also revised the learning across all three components.

Draft Kindergarten At-A-Glance 2019-20

Draft 1st Grade At-A-Glance 2019-20

Things to point out:

  • I’ve settled on a few anchor instructional routines across all grade levels – number talks, choral counting, and counting collections. That’s not to say that teachers can’t use other routines – I encourage them to – but my goal is to ensure that these three powerful, versatile routines are in everyone’s toolbox.
  • Kindergarten only has 60 minutes of math instruction in the fall semester so they don’t start spiral review until the spring semester.
  • In 1st grade the numeracy topics are fairly consistent across the year – skip counting, subitizing, making 10, and developing strategies for adding and subtracting within 20. My hope is that the consistency of topics across the year paired with the anchor instructional routines will allow the numeracy work to feel more like an ongoing conversation across the year.
  • In 1st grade creating, solving, and representing addition and subtraction problems is a spiral review topic over and over again. I want to ensure students have lots and lots of opportunities to engage with problems involving joining, separating, and comparing quantities.

Parting Thoughts

Now that I’ve started to get a plan in place, I have a lot of work ahead of me to create all the associated unit documents. I’m also going to be working on gathering teachers who want to pilot these new units. I’m wary of just dumping them on our teachers because they’ve already put so much work into learning the old units, and there are some heavy instructional shifts that might need to be made to make these units work. Thankfully I don’t think it will be too hard to find volunteers. Teachers who’ve looked at these plans and talked about them with me or their instructional coach have been really excited for the changes, so much so that I have an entire Kindergarten team trying out one of the new units right now!

While there are still a lot of unknowns and a lot of work ahead to support teachers, I do feel like all of the reflecting, conversations, and attempts at making a new plan over the past six months have brought me to a place where I feel like I’m moving in a good direction that I’m happy to follow for the time being.

Here’s to the path ahead.

The Annotated Numberless Word Problem

I recently modeled a numberless word problem in a 4th grade classroom. A few weeks later, I got an email about how the teachers were attempting to create and use some of their own, but they were encountering a problem…writing their own problems was harder than they thought!

They reached out to me for support, and I thought I’d share with you what I shared with them in case it’s helpful to anyone else creating their own numberless word problems.

1. Start with a problem

First things first, start with the problem you want to transform into a numberless word problem. Here’s the problem I started with for this example:

I type the problem on a slide, either in Powerpoint or Google Slides. You can create your problem on chart paper or on strips of paper if you’re working with a small group. I’m partial to digital slides because of some other features you’ll see later in the post.

2. Work backward

From here I create a copy of this slide and remove some of the information. Usually I start by removing the question.

Next I copy this new slide and again decide what information to remove. In this case I decided to remove the entire last sentence. That sentence dramatically changes our understanding of the situation. If you look at the slide below you’ll see that we know the total number of kids eating ice cream and the number of kids eating chocolate ice cream.

The situation is very open right now. The rest of the kids could be eating a variety of different flavors – vanilla, strawberry, chocolate chip. When I reveal the sentence that the rest of the students are eating vanilla ice cream, there’s a nice element of surprise because you aren’t necessarily expecting that the kids are only eating just two different flavors.

My next step is to remove one of the numbers. In this case I’ll take away the number of children eating chocolate ice cream.

Finally, I’ll remove the number in the first sentence to get me to the beginning of this problem. This is the first text students will read.

I structure my slides to minimize changes. I don’t want to overwhelm the students by revealing too much all at once. I will add new sentence, but I avoid changing language that’s already on the slide, if possible. More often than not I’m only changing a word like “some” into a specific quantity. There are rare instances where I’ll have to adjust a sentence as new information is added, but I try not to do that. I want the sentence structure to stay the same so that when the numbers are added that’s the only real change.

You might have noticed that I don’t include pictures on the slides with the text. This is intentional. I used to include pictures, but a colleague shared how distracting the pictures were for her students. Students were looking for meaning in them when they were only there essentially as decoration, with the intent that they would support visualizing. However, the pictures ended up confusing her students rather than helping because the students kept trying to make connections between the pictures and text. Since then I’ve avoided pictures on the text slides unless the picture is absolutely necessary.

3. Plan purposeful questions

The first step was to work backward to plan out each slide so that information is slowly revealed on each slide. Now it’s time to plan the questions I’m going to ask the students at each step along the way. I have two primary goals that I strive for in my questioning:

  1. I want students to visualize what the story is about as it unfolds. If they’re not “seeing” it, then they’re likely not making much sense of it.
  2. I want students to make guesses and estimates about quantities in the story using what they know about the situation and the relationships provided. I want them reasoning all along the way so that by the time they get to answering the question they are holding themselves accountable if their answer doesn’t make sense.

So now I go back through the slides in the order they will be presented and add the questions I plan to ask along the way.

Slide 1

Ask for a volunteer to read the story.

What are you picturing in your mind?
What do we know so far?
How many kids could be eating ice cream?
How many kids could be eating chocolate ice cream? Why do you say that?

Have students draw a quick sketch of the story so far.

Slide 2

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
What does this tell us about the number of kids eating chocolate ice cream?

When a new slide is presented, I always ask a question to get students to state the new information. I’ve also worded this as, “What changed? What do we know now that we didn’t know before?”

Slide 3

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
How does this number compare to our guesses? Does it make sense?
Are all of the kids eating chocolate ice cream?
What could the other kids be doing?

Slide 4

Ask for a volunteer to read the slide.

What do we know now that we didn’t know before?
What does this tell us about the number of kids eating vanilla ice cream? How do you know?

Have students draw another quick sketch of the story so far.

What question(s) could we ask about this math story?

Slide 5

What is the question asking?

Do you have all the information you need to answer that question?

Let students work on solving the problem. Confer with students as they work to look for strategies you want to bring up with the whole class.

4. The beginning and the end

Something I’ve been doing for the past year with numberless word problems is bookending them with visuals to add a little more texture to the experience.

The beginning

The first thing I do is find a high quality image or two to show the students and have them chat about before we dive into reading any text. My go-to website for images is Pixabay.

I type in a word or phrase related to the story problem, like ice cream, and more often than not I hit the jackpot:

I look for a photo that I think will capture kids’ attention and activate their prior knowledge of the context. It allows students who may be less familiar with a situation to hear the relevant language, such as ice cream, chocolate, vanilla, and cone, before we dive into reading the text.

Here’s the picture I ultimately chose to engage students at the start of this problem, along with some notes of how I’d facilitate the opening discussion with the students.

Image Source:

What do you notice? What do you wonder? Give students 20-30 seconds of think/write time. Then let students share 1 noticing and 1 wondering with a partner. Finally let students share a few of their noticings and wonderings with the entire class. You may choose to record these in a t-chart, but it is not necessary for this problem.

Tell the students that today they are going to read a mathematical story about ice cream.

When I paste the picture on a slide, I always go into the Notes section of the slide and paste the source of the picture(s), usually the URL where I found it. On Pixabay, more often than not the photos have licenses allowing reuse.  You can find the license information to the right of each photo. I know in the privacy of your own classroom it feels easy to get away with grabbing whatever picture you can find on Google Images, but it’s good habit to pull legal photos to avoid unforseen issues down the road. And with amazing sites like Pixabay and Wikimedia Commons available, there’s no reason not to at least start by looking for freely available photos.

The ending

I’ve been making it a habit to close each numberless word problem with a short video. This serves two goals:

  1. It further builds students’ knowledge of the situation discussed. In the case of the problem I shared in this post, it was about kids eating ice cream so I found a short video of a kid making ice cream. Even if you can only find longer videos, you don’t have to show the whole thing. You could just watch the first minute (or whichever section is most relevant or interesting).
  2. It serves as a pay off for all of the hard work students just did to make sense of and solve the problem.
Here’s a link to the video I included in this problem

I’m sure you can guess where I go to find videos. YouTube has such an endless supply of videos, that I haven’t yet encountered a situation where I couldn’t find a video worth sharing. Sometimes it’s the first video and sometimes it’s the tenth, but it’s always there waiting to be discovered.

Final thoughts

Now that you’ve seen me put together this numberless word problem in pieces, here’s your chance to see the finished product. This link will take you to the slideshow for the finished product.

In the Notes section on some of the slides, you’ll see references to students sketching in boxes. I created a recording sheet to try out when I modeled a different problem recently. If you want to check out the recording sheet, here’s the link. I don’t have a lot of experience using it yet so I don’t want to say more about it right now, but I do want to share in case it’s helpful.

If you have any questions, don’t hesitate to reach out in the comments or tweet me @bstockus. And if you create your own problem, please share it with us on Twitter using the #numberlesswp hashtag.

Origin Story

This summer I was invited to give an Ignite Talk at my school district’s Teaching & Learning SummeRR Camp.

SummeRR Camp

It’s a talk I’m proud of because in five minutes I was able to share why it’s so urgent to me that we ensure sense making is the focus of our work with students. Students deserve to develop positive relationships with what they’re learning now. It’s a disservice to assume they’ll learn to like it later. We never know what doors are closed to students because they learned to hate a subject or grow up thinking they’re not smart enough.

It’s the Great (Big) Pumpkin, Charlie Brown!

At this point it’s become an annual tradition that I make a batch of numberless word problems based on the results of the Safeway World Championship Pumpkin Weigh-Off. I’ve collected together the problems I’ve written based on the results from 2016, 2017, and now 2018 in this folder. [Update: Thanks to a request from my daughter, I’ve added some primary level pumpkin problems as well.]As with all the other files I share, you are welcome to edit them if you want to tweak them for your students. All you have to do is make a copy if you have a Google account or download the file in an editable format like Power Point. You will have full editing rights of your copy.

If you’re looking for some more mathematical inspiration as Halloween approaches, check out these three blog posts I wrote which include lots of photos and ideas for how to use them to spark mathematical conversations with your students.