In my previous post (Link), I shared how I’ve recently starting doing math with my daughter to help her get warmed up for the start of 4th grade. In that post I talked about how I’m using the centers from the Illustrative Mathematics K-5 curriculum (Link) to revisit and practice working with multiplication and arrays.
In the six and half years I worked as a district math curriculum coordinator, a common concern I heard from 4th and 5th grade teachers is that their students don’t come in knowing their multiplication facts. I can attest that my daughter learned a lot about multiplication and division in 3rd grade, but I’ll be honest, she hasn’t done a whole lot of multiplying or dividing this summer (not to mention fluency is something that tends to develop over a period of years, not months). It comes as absolutely no surprise to me that she’s rusty, particularly with knowing her multiplication facts. I’m going to go out on a limb and claim that a lot of kids are rusty at the start of a new school year. We need to give them grace, which means not saying things like, “Didn’t your teacher teach this last year?” We also need to intentionally build in opportunities to practice and dust off the mental cobwebs.
Today I’d like to share how my daughter and I have been practicing multiplication facts. What I like about what we’re doing is that (1) it only takes a few minutes a day, (2) it reinforces flexible use of strategies, and (3) it gives her a second chance everyday. I got this idea from a free math intervention called Pirate Math Equation Quest (Link), developed by Dr. Katherine Berry and Dr. Sarah Powell from the Meadows Center (Link) at The University of Texas at Austin. Their intervention includes a component called Math Fact Flaschards that goes like this:
Student completes two trials of Math Fact Flashcards, each for 1 minute
Teacher and student count cards after each timing
Teacher monitors and provides feedback as needed
After 2 trials, student graphs the higher score
Rather than use traditional flashcards, I created flashcards that show two facts per card, the initial fact and its turnaround. For example, the card with 2 × 5 also shows 5 × 2. I got this idea from the 4th grade Investigations 2nd edition curriculum. It reinforces the idea that every time you know the answer for one fact, you really know the answer for two (with the exception of square numbers).
Before we start a trial, I always remind her that she is going to “just know” some of the facts because she’s so familiar with them, but for the ones she doesn’t “just know” she can use one of the multiplication strategies she’s learned. The following poster is hanging on the wall next to where she’s sitting so she can turn and reference it as needed.
These are the thinking strategies developed by Origo Education (Link). If you’re not familiar with them, check out this YouTube playlist that includes one-minute videos explaining each strategy. (Link) If you want to see how a child uses one of the strategies, here’s a link to a short video of my daughter talking through the Build Down strategy she used to solve 9 × 7. (Link)
Please note, you can’t just throw strategies at your students. They have to be intentionally introduced and practiced, but it is well worth the time! Students who lack a robust toolbox of strategies have to rely solely on memorization (which is a big ask!) or inefficient strategies like skip counting. If you’re interested in learning more about how to teach these strategies, Origo has a great series called The Book of Facts that shares activities and games for teaching a set of fact strategies for each of the four operations. (Link)
During each trial, I present the flashcards one at a time. I put all of the ones she answers correctly in a pile and any she answers incorrectly in another pile. After the minute is over, she counts the number correct, and then we discuss the ones she answered incorrectly. Sometimes her incorrect answers are because of a simple mistake, and I reinforce that it’s fine because she has been able to recognize the error herself. However, sometimes it’s more than a simple error. I was able to pick up very quickly that she’s also rusty with doubling 2-digit numbers that involve bridging a ten. For example, to solve 4 × 7, she can easily double 7 to get 14 and double 14 to get 28. However, to solve 4 × 8, she can easily double 8 to get 16 but she gets stuck doubling 16. Her answer might be 26 or 36.
Based on this observation, I’ve added in practice with doubling 2-digit numbers. This practice is untimed for now, though I might eventually add these cards into the deck of multiplication flashcards.
At the end of the two trials, we graph her higher score for the day. I really love this because if she blows the first trial for whatever reason, she knows she’s going to get a second chance to get a higher score. It really takes the pressure off.
We’ve only been doing it for a week, so there’s not a lot of data to look at, but I’ve already used her graph to talk about how we all have good days and better days. I also reinforce that while some days are lower, her rate of incorrect responses is consistently low. She only ever misses 0, 1, or rarely 2 cards during a trial. She’s also been really good about stopping and thinking of an appropriate strategy whenever she gets stuck, and she is doing a great job of executing her chosen strategy accurately.
For full transparency, her deck of flashcards includes all of the facts including the “easy” ones like 0s and 1s facts, and I’m okay with that. They’re still facts and she needs to know them. The important thing is that I continue to monitor to uncover any issues where I can support her, like with doubling 2-digit numbers. Eventually I might ween the deck down to the ones that need more intensive practice.
I like that this practice doesn’t take a lot of time, only about 3-5 minutes. If you’d like to try this out in your classroom, you might consider doing it in small groups, which is an idea shared in the Pirate Math Equation Quest intervention I mentioned earlier. During the one-minute trial, the teacher goes around the group round robin style, showing one flashcard to each student. All of the flashcards are placed in one pile and the total correct is the group’s score. The goal as a group is to try to get more and more correct each time. I like that this allows for a bit of a tradeoff. The teacher doesn’t have to feel pressured to run this activity individually with every student, but at the same time, she can learn something about each student as she conducts these trials in small groups. I’m doing this with my daughter everyday, but a teacher might be able to make small groups such that she ends up seeing every student every 3-4 days.
As I was reading over the small group directions, I realized they recommend letting the student continue trying until they get the answer correct. If the student answers incorrectly, the teacher intervenes with a suggestion such as a strategy a student might use. I think I might try that with my daughter rather than setting aside incorrect answers. Helping in the moment seems much more powerful than helping at the end. It also does a better job of validating the power of identifying and correcting mistakes. I like forward to seeing how it goes next week!
This week I’m starting to do a little math with my daughter everyday to dust off the cobwebs before 4th grade starts in September. One of the resources I’m using is the centers from the Illustrative Mathematics K-5 curriculum (Link to Kendall Hunt’s version of IM K-5 Math).
We kicked things off on Monday with a center called Can You Build It? (Link) One thing I like about the IM centers is that they often contain multiple stages within the same center, so you can choose just the right starting point within a given concept. Since my goal was to revisit arrays and the meaning of multiplication, we started with Stage 1. In the original IM version, one person builds an array secretly and then describes it to their partner and the partner tries to recreate it.
I changed this stage into a cooperative game that turned out to be really fun for my daughter. Here’s how it works:
Draw a target area card. (I created a deck of cards that have the numbers 10 – 27 on them. This means there are 18 possible target areas, which feels like a good range. The numbers are also small enough that you won’t spend all your time counting out the tiles you need before making your array.)
Each player secretly makes an array with that target area.
Share your arrays. If you made the same array, you collectively earn 1 point. If you each made a different array, you collectively earn 2 points. (To clarify, a 2 by 6 array is the same as a 6 by 2 array.)
Earn 5 points in as few rounds as possible.
If you don’t have square tiles handy, you could use a free app like Number Frames from the Math Learning Center (Link) which can be used in a browser or downloaded onto a tablet.
Or if you still want something hands-on, you could always use some crackers!
After a couple of days playing Stage 1 and revisiting how to build and describe arrays, we moved on to Stage 2. There are a couple of key differences here:
Instead of secretly making only one array, the goal now is to make as many different arrays as possible with the target area.
The game is competitive now. The player who makes more arrays earns 2 points and the other player earns 0. If both players make the same number of arrays, they both earn 1 point. The winner is the first to 5 points. (The original IM center used a slightly different scoring scheme. I opted for something similar to the game we played for Stage 1.)
My daughter immediately started bumping into ideas related to prime numbers. Here are some highlights from our conversation as we played for the first time:
1/ Daddy: Today our game is slightly different. This time when we draw a target area, our goal is to make as many different arrays as possible. If we get the same number of arrays, we each earn 1 point. If one of us makes more than the other, that person earns 2 points.
2/ Daddy: (draws card) Our first target area is 20. (both make arrays in secret) Daddy: I made a 2 by 10 and a 4 by 5. How about you? Me: I made those, and I made a 1 by 20. Daddy: Oh! I forgot that one! Me: You have to remember you can *always* make a 1 by array!
3/ Daddy: (draws card) Okay, this time our target area is 13.
(both make arrays in secret)
Me: Ugh! I can only make one.
Daddy: Me, too. What did you make?
Me: 1 by 13.
Daddy: Hmm, I wonder why we could only make one array.
Me: Maybe because it’s an odd number.
4/ Daddy: (draws card) Now our target area is 11.
(both make arrays in secret)
Me: No! You can only make one again.
Daddy: Huh, is this an odd number, too?
Daddy: That’s interesting.
5/ Daddy: (draws card) Ok, our target area is 10.
Me: I’m just going to write down the 1 by array. I don’t even need to make it.
(both make arrays in secret)
Daddy: What did you make?
Me: A 1 by 10 and a 2 by 5.
Daddy: Same here. Is 10 odd?
Me: No, it’s even.
6/ Daddy: You made two really interesting observations today. Do you remember what they were?
Me: …if a number is odd you can probably only make one array?
Daddy: What else?
Me: …and you can always make a 1 by array for every number!
I love how this game has a simple premise – make arrays – but it creates opportunities for students to notice deeper ideas about numbers and multiplication. If you woudl like to try this game out with your own child or students, here’s a link to the center. (Link)
If you work in a grade level that introduces prime and composite numbers, I also recommend checking out 4th Grade Unit 1 of the IM curriculum for well-designed, ready-to-go lessons. (Link)
[UPDATE]Alyson Eaglen shared a great idea on Twitter. She said that instead of using cards with pre-printed target areas, she suggests rolling three 9-sided die and the sum is the target area. What a great way to bring in some bonus addition practice! If you don’t have 9-sided dice, you could always use five 6-sided dice or whatever combination of dice yields the range of target areas you’re interested in for the game. If you don’t have physical dice handy, Polypad’s free virtual manipulatives (Link) include a variety of dice under the Probability and Statistics menu.
Recently on Twitter I asked for advice about how other people engage with professional reading.
I got a lot of great advice. (Thank you to everyone who contributed!) A recurring theme was how much more people get out of their reading when they interact with others. One way I’d like to try to spur some interaction while I read is to use my blog as a place to share my thoughts on my professional reading. At worst, no one will respond, but I’ll still have done some reflecting on my own so that’s not so bad. At best, folks will comment and I’ll get to engage more with the ideas from whatever I happen to be reading.
I’m not quite ready to dive into a professional book at the moment, but I did want to spur myself to get started, so today I reread over the material at TeachingWorks (Link) on high-leverage teaching practices to refresh my memory. I jotted down notes while I read that I’m putting in this post. If you read through them and they spark any thoughts, feel free to share in the comments! I’m particularly interested in the topic of high-leverage teaching practices. I’d love to hear what others think about them or hear what other resources on this topic that you think I should read!
“The goal of classroom teaching is to help students learn worthwhile knowledge and skills and develop the ability to use what they learn for their own purposes.” – I’m curious how “worthwhile” is defined. Worthwhile to whom? Deemed worthwhile by whom? I do like the idea of using “what they learn for their own purposes.” Schooling isn’t about what anyone else thinks a student should ultimately do, but about the knowledge and skills a student learns and their agency to make choices about how they use them.
“All students deserve the opportunity to learn at high levels.” – I just listened to an episode (Link) of the podcast “Teaching While White” where Tim Wise talks about the history of schooling in this country and he shares a quote by Thomas Jefferson where he says 6 years or so of schooling should be provided to all (white people) in order to elevate those with talent from the “rubbish.” Clearly the goals of public education in this county from its inception have not been to ensure that all students learn at high levels, but rather to find a small population who we deem capable of learning at high levels and letting them rise to the top.
“Learning is an active sense-making process.” – This is the nature of human brains. Even if we could provide every student the same inputs, our brains are making sense of them against the background of our own unique experiences which is why the outputs can be so vastly different for each person. Regardless of what those outputs are, it is the sense that each person’s brain has been able to make of what they’re experiencing. It’s no wonder that you can have such a broad range of skills and abilities within a single classroom. It also demonstrates the challenges teachers face in identifying and responding to what their students have learned.
“Teaching is interactive with and constructed together with students.” – If you’ve ever tried to teach the same lesson to different groups, this will make sense. What stands out to one group vs. another may impact the conversation you have and where you focus your time and what the ultimate learning is for a given group. One group may need a different way of interacting than another group in order to be successful. Even if you teach just one class (like an elementary teacher), you’ll notice year to year differences between groups of students. Those variations and your interactions are the basis of constructing knowledge together. There might be similarities about what’s learned between groups, but there will inherently be differences.
“The contexts of classroom teaching matter, and teachers must manage and use them well.” – This reminds me of the porous boundary between the classroom and the surrounding environments Dr. Deborah Ball talks about in her AERA 2018 Presidential Address (Link). According to Dr. Ball, these environments aren’t just physical, they also include historical racism, the legacy of slavery, colonialism, whiteness, housing policies, segregation, school structure, teaching as an occupation, the enormous health and wealth disparities in our country, and curriculum. It’s a big “multivariate soup” within which teaching and learning take place. “Environments permeate the classroom and have no bounds themselves.”
“The goal has been to identify a small set of instructional practices that are crucial for beginning and early career teachers to be able to do well, and a small number of topics and ideas that they should understand and know how to teach.” – From the elementary lens, this is a powerful idea because elementary teachers are required to “do it all.” They are expected to teach every subject well, and while there are unique challenges to teaching each content area, how might it benefit teachers (particularly new and early career teachers) to focus on a core set of skills that can be applied across content areas? It feels like a much better use of their time, especially when you consider all the professional development opportunities teachers can be bombarded with that are often siloed by content. Each of these PD opportunities may be amazing, but if they aren’t helping teachers develop big picture understandings about teaching and learning, the impact may be smaller than we’d hope. I’d much rather focus professional learning on these core skills and then look at how they can be applied in different content areas.
“…striving to isolate those aspects of the work of teaching that matter most for the quality of students’ educational opportunities.” – From reading this page it sounds like their group has done a lot of work to involve a variety of stakeholders in order to create and refine their list. I wonder how others who weren’t part of this work can create buy-in with teachers that these practices “matter most for the quality of students’ educational opportunities.”
“We also seek to identify the highest-leverage content knowledge needed for teaching. High-leverage content is particular topics, practices, and texts that are both foundational to the K-12 curriculum in this country and important for beginning teachers to be able to teach.” – This would be useful to connect with standards at a given grade level to help teachers understand where and how to focus their time and attention with their students.
“These practices are used constantly and are critical to helping students learn important content. The high-leverage practices are also central to supporting students’ social and emotional development.” – I like how this acknowledges that we’re teaching people, not just content, and so the skills of teaching need to include skills related to building relationships and working with people.
Here’s the list of high-leverage practices
Leading a group discussion
Explaining and modeling content, practices, and strategies
Eliciting and interpreting student thinking
Diagnosing particular common patterns of student thinking and development in a subject-matter domain
Implementing norms and routines for classroom discourse and work
Coordinating and adjusting instruction during a lesson
Specifying and reinforcing productive student behavior
Implementing organizational routines
Setting up and managing small group work
Building respectful relationships with students
Talking about a student with parents or other caregivers
Learning about students’ cultural, religious, family, intellectual, and personal experiences and resources for use in instruction
Setting long- and short-term learning goals for students
Designing single lessons and sequences of lessons
Checking student understanding during and at the conclusion of lessons
Selecting and designing formal assessments of student learning
Interpreting the results of student work, including routine assignments, quizzes, tests, projects, and standardized assessments
Providing oral and written feedback to students
Analyzing instruction for the purpose of improving it
It’s overwhelming when you look at it all at once, especially when you consider there’s quite a bit of depth to each of these statements, but I like the idea that if these are the things that matter most, then this list provides solid avenues teachers, instructional coaches, and administrators can pursue to help improve the quality of students’ educational opportunities.
“Although many teaching capabilities are used across subject areas, some are subject-specific.” – This is where I’d like to see subject-area PD focus. The high-leverage practices keep us focused, but we can learn the nuances of how to use them successfully in each subject area without feeling like we’re always learning something brand new or disconnected from previous learning.
TeachingWorks hasn’t provided a list of high-leverage content yet. It says they began the work of identifying high-leverage content in 2011. I’m curious where they are 10 years on.
I guess everything gets a list on this site. This page shares 10 critical features of practice-based teacher education – “professional training that is deliberate about making sure that novice teachers can use specific practices of teaching” in an effort to create “a more just society, achieved through classroom instruction that disrupts racism and attends to all students as individuals and as members of multiples communities.”
Here is their list of critical features of practice-based education
Models of skillful teaching
Opportunity to practice
Ambitious learning goals for children
Deliberate attention to Black and brown children
Content knowledge for teaching
Coherence, sustainability, and continuous improvement
Part of why they share this list is because there’s not one model program or way to teach teachers. Rather, we need to create programs for particular contexts and students, but these critical features can help shape that work.
One thing they talk about in this section is how they decompose the high-leverage practices and provide opportunities for teachers to learn and practice individual parts of each practice. This makes sense given how dense the high-leverage practices are.
If you want to see some of these critical features in practice, I recommend watching the entirety of Dr. Deborah Ball’s talk that I mentioned earlier. In particular she demonstrates deliberate attention to two Black children in her class and the power we have as teachers to build up or tear down these students with decisions we have to make in-the-moment.
I’ve been drawn to this idea of high-leverage practices for several years now. Having worked as a curriculum coordinator in a school district with 34 elementary schools with over 1,000 elementary teachers, I constantly bumped into the limits of teachers’ time. Teachers are pulled in many directions from administrators, instructional coaches, the curriculum department, state requirements, not to mention teachers’ own interests about what they’d like to learn. I feel like we can work smarter, not harder, by centering our professional learning efforts around a set of common practices like those shared by TeachingWorks. It would create common language and would reassure teachers that any professional learning they are doing is tied into the bigger picture of what it means to provide quality instruction to all students. Unfortunately I wasn’t a very good salesman because I never found any traction with the idea in my district, which is fine, but it doesn’t mean I’m letting it go. I don’t know how or when I might be able to work with these ideas further, but I know that I’d like to.
I was recently invited on the Complementary Angles podcast to share about numberless word problems. Over the course of an hour we talked about:
The research that supports the use of numberless word problems
How numberless word problems empower all learners in working with problem situations
Advice for teachers implementing numberless word problems for the first time as well as advice for veteran users
How numberless word problems have evolved over time
How numberless word problems can be used in remote learning environments
That’s a lot of ground to cover, so they broke up the interview into two episodes. Check them out below.
If you have any advice for teachers using numberless word problems, particularly if you’ve been using them during remote learning, share them in the comments and/or tweet them using the hashtag #numberlesswp. We’re all better together!
A week ago, I closed out six and a half years serving as the Elementary Mathematics Curriculum Coordinator for Round Rock ISD. I wrote a blog post on my last day where I reflected on my accomplishments. If you’re interested, you can check out that post here.
Today I’d like to share the lessons I learned while doing this job.
Lesson #1 – Know your “why”.
Early on in my job, district leaders were given a copy of Simon Sinek’s book Start With Why. I’ll be honest that (corporate) leadership books tend to rub me the wrong way, but from time to time I find value in a message and it sticks. In this case, the idea of knowing your “why” resonated with me – why do I do this work? Why does it matter to me?
I gave an Ignite talk a few years ago where I shared my “why” and how I came to know what it is:
My “why” is driven by my evolving relationship with math, from the time I was a student until now. When I was still in school, I got the feeling that math was supposed to be making sense, but despite all the procedures I memorized and accurately reproduced, it just never did. To borrow a term from Robert Kaplinsky, I was a math robot. I did what I was told, but I only knew how to do what I was told. Despite earning good grades in my math classes, I finished high school feeling like an imposter.
My relationship with math took a sharp right turn years later when Pam Harris led PD at the elementary school I was teaching at. She re-introduced a room full of elementary school teachers to mathematics in a way that finally made sense, and it completely changed the trajectory of my career. I mean, just look at the past 12 years or so. I led a team of people designing digital math curriculum for grades 4-7. Then I served 34 elementary schools as a district Curriculum Coordinator. These are not jobs I ever imagined having before attending that PD!
So what is my “why”?
Sense making should be the focus of what we do for each and every one of our students so that they develop a positive identity toward mathematics today.
If you’re interested in this idea of knowing your “why” but you’re hesitant to add another book to your already too tall “to-read” pile, check out Simon Sinek’s TED Talk instead:
This quote from his talk resonated with me:
People don’t buy what you do, they buy why you do it. The goal is not to do business with everybody who needs what you have. The goal is to do business with people who believe what you believe.
Over time I came to understand that my goal as a Curriculum Coordinator wasn’t to “sell” my curriculum documents to my teachers. Rather, I learned there’s power in sharing my “why” through my curriculum documents, through my professional development sessions, and through my communication with administrators, instructional coaches, and teachers. This is important because of the next lesson I learned…
Lesson #2 – Systemic change is hard.
When I became the Elementary Mathematics Curriculum Coordinator for an entire school district, I saw it as an opportunity to bring about systemic change in mathematics education. I didn’t know right away exactly what changes I was going to make yet, but I was passionate and excited at the possibilities. In other words, I was naive. I had been entrusted with the keys to the car…but little did I realize just how many other drivers had their hands on the wheel steering our schools, teachers, and students. There are so many people in a school district vying for attention and trying to move the system in one direction or another. Just because something was a priority to me did not mean it was a priority to everyone else.
I also quickly realized how little power came with the title. My primary responsibility was to develop and maintain all of the elementary math curriculum units for grades K-5. However, I had no power to make anyone teach them. While there were loose district expectations, campuses and teachers had a lot of leeway to make their own instructional decisions.
Don’t get me wrong. when I say I didn’t have the power, it doesn’t mean I wanted the power necessarily. I wasn’t looking to be a dictator telling everyone exactly what they needed to be doing in their classrooms. However, there’s no denying there is definitely a different feel to the job and your ability to affect change when everything you do is essentially a suggestion.
After learning that systemic change is hard, I tackled the problem of figuring out how to do it anyway.
Lesson #3 – Change what you can change. Influence the rest.
Rather than bemoan the fact that I couldn’t make anyone do anything, I turned my attention instead to what I did have the power to change – my curriculum documents. This is where it’s important that I knew my “why”. It influenced all of the decisions I made as I continually developed and revised our curriculum documents year after year.
Some of the notable ways my “why” influenced and permeated my curriculum documents are:
Writing a rationale for each unit so that teachers could understand the goals of the unit as well as why those goals are important
Embedding links to articles, blog posts, books, and videos at the end of each unit rationale so that teachers had the option at their fingertips to deepen their understanding of the concepts in the unit.
Restructuring the elementary math block to include 10 minutes of daily numeracy work and 20 minutes of daily spiral review.
Embedding three anchor numeracy routines throughout the school year across all of the elementary grades – number talks, choral counting, and counting collections.
Eliminating the 10-day test prep unit at the end of the year for grades 3-5 and instead implementing daily spiral review throughout the entire school year. I wrote a blog post about this on my district math blog. If you’re interested in reading more about my rationale for this decision, you can read that here.
Creating yearly at-a-glance documents (3rd Grade Sample) that showed how math concepts wove through all three components of the math block across the year – core instruction, numeracy, and spiral review.
Completely redesigning the Kindergarten and 1st grade math units to provide more time for students to explore and engage hands-on with math concepts. I wrote about these changes on this blog. You can read about them in more detail here.
Looking back on it now, this is, to a degree, systemic change. By changing (and continually refining) what was within my control – the documents my teachers engaged with on a daily basis as they planned instruction – I changed the system in which they worked. It’s important to note that I didn’t do this work in isolation. In addition to being driven by my “why”, many of these changes were also driven by teacher feedback. I regularly consulted our instructional coaches and brought in teachers to help plan units and create resources.
But changing documents isn’t enough, especially if not everyone uses them. While I didn’t have the power to make anyone use these documents, I did learn over the years that I had the power to influence them.
During my six and a half years as a Curriculum Coordinator, I had the opportunity to either lead or help plan so many PD sessions: summer PD, new teacher PD, after school PD, online PD.
Some of the ways I shared my “why” through PD include:
Developing a 7-month long program called Math Rocks that was designed for teachers to dive more deeply into their practice and build positive identities around mathematics for themselves and their students. I’ve written several posts about it on this blog which you can check out here.
Creating a uniform set of slides to introduce teachers to our Elementary Mathematics Beliefs document at all summer PD and new teacher PD sessions.
Creating a session called Maximizing the Math Block to share with teachers how the elementary math block is structured and the rationale behind each component. This session was given at New Teacher Orientation, on district PD days, and after school at various campuses.
Regularly highlighting the work of educators around the district who demonstrated practices that aligned with our beliefs about teaching and learning mathematics. The pictures below are from a session I led for campus principals to help them better understand our process standards using classrom examples from their own campuses. I always loved seeing a principal sit a little taller whenever I shared the work of a teacher from their campus.
Getting in front of principals, coaches, and teachers turned out to be an ideal way to share my “why” and get others on board with my vision for teaching mathematics. It’s not a quick fix, that’s for sure, but it’s effective if you’re willing to put in the work over time. Systemic change is as much about the cultural change you can influence as it is any technical changes you put in place.
Lesson #4 – I can’t please everyone, but I can listen to them.
The more people in an organization, the more people you’re inevitably going to disappoint. As much as I worked to bring as many people as possible on board with changes I made, it’s just not possible to please everyone, and that’s okay.
What’s not okay is not listening. Even if the eventual change isn’t exactly what they want, I’ve found that if people feel heard they are more likely to accept the change (perhaps begrudgingly), or at least not vocally oppose it quite so much.
And if you take the time to listen, you might even find some common ground or an idea you wouldn’t have considered otherwise. As long as I kept my “why” at the forefront of my thinking, I found it easier to make compromises rather than getting hung up on needing something to be “my” way.
Lesson #5 – Learning is a marathon, not a sprint.
I’m referring to the K-12 experience of learning here. It’s easy to get wrapped up in a particular lesson within a particular unit within a particular grade level and feel like you’ve failed as a teacher or the students have failed as learners because they didn’t learn the thing they were supposed to by the end of the lesson. Learning targets and daily goals are all well and good for keeping us focused, but the variability among people is so high it’s naive to think everyone will achieve the goals you’ve set for them every single lesson, every single day.
Learning is about bringing about incremental change over time in the ways students think, their dispositions, and the skills they possess. One of the reasons I broke up our elementary math block into three components – core instruction, numeracy, and spiral review – was to give more space for concepts to weave throughout the school year so that the pressure wasn’t on any given lesson or even any given unit for success for every single student. Rather, we have all year long to help lift up each and every student.
Don’t get me wrong, there are forces from above telling (or even demanding) teachers and students that they should be achieving learning goals on a rigid schedule, but somebody wanting something to be true doesn’t make it true. And just because someone wants something doesn’t mean they get to have it.
Lesson #6 – Forgetting is normal. Expect it, don’t fight it.
Related to the previous lesson, I learned that forgetting is a normal part of the process of learning. If your students forget previously learned material, you haven’t failed your students and your students haven’t failed you. As soon as you stop teaching something and move on to a different topic, the brain does a very normal biological process of forgetting what was just learned as time and attention are given to the new topic.
All is not lost, however. Intentionally waiting and returning to a topic later gives the brain a chance to go, “Wait, you still wanted me to know that?” As you review and practice, students relearn what was lost, but more importantly pathways are strengthened in the brain so that future forgetting will be lessened because now the brain knows this is information that needs to be held onto.
So the next time you revisit something like types of quadrilaterals and your students look at you like you just spoke a foreign language, don’t have a heart attack. Stop, breathe, and tell yourself, “This is totally normally.” Then do the work of helping them remember what they learned before in order to lesson future forgetting.
Lesson #7 – There is a lot of redundancy and inequity across school systems.
Think about it – No matter how big or small a school district is, their charge is exactly the same. Whether you have one 5th grade classroom or 300 5th grade classrooms in your district, every single one of your students is expected to learn the 5th grade standards. What varies is the level of support the district can provide its teachers.
A district with 300 5th grade classrooms likely has a larger budget and can afford a robust curriculum department with one or two people overseeing each subject area. A district with just one 5th grade classroom, on the other hand, likely has a very small budget and may not even have a curriculum department. Or if they do, it might consist of one person overseeing all subjects for K-12. The level of support these districts can provide their teachers is inequitable, despite the fact that both districts are required to provide the exact same service – educating all of their students.
Or think about this. I worked for a district with 34 elementary schools, and I developed and maintained around 90 elementary math units for grades K-5. My colleague in another district of similar size was in charge of developing and maintaining her own set of elementary math units for grades K-5. And another colleague in another district was doing the exact same thing. And my colleague in another district…and so on. There is a lot of redundancy in education. It doesn’t help that while some districts are open and happy to share resources, others are locked down and protective. More collaboration could save a lot of time and energy, not to mention result in more high quality resources for all.
I haven’t learned a solution to this problem. I’ve just become acutely aware of it. I do appreciate that social media sites like Twitter and Facebook have facilitated the sharing of ideas and resources among educators. These online communities are organic and unsystematic, but they’ve shown that we can be better and achieve more when we erase school district boundaries and work together.
I guess the final lesson I’ve learned is that I crave new experiences and challenges to learn from. I left my digital curriculum writing job in 2014 seeking a new challenge, and I found it in Round Rock ISD. Serving as the Elementary Mathematics Curriculum Coordinator and figuring out how to be successful in the role was a tough nut to crack. While I didn’t solve every problem and while not every effort I made was a success, I learned so much over the past six and a half years in no small part thanks to the leadership in my department who trusted me to not just to do my job but to do it well. I look forward to bringing the lessons I’ve learned with me into whatever role I take on next.
I have to say, I so appreciate that I’ve had this blog going for as long as I have. It was so nice as I was reflecting over the past couple weeks that I had so many blog posts to look back on where I captured various aspects of my work. I know blogging isn’t for everyone, but it sure is a great way to capture your thought process at a particular point in time that you can return to later. Don’t worry about your “audience”. Write for yourself and if it resonates with anyone else, consider it a bonus.
Today marks my last day as the Elementary Mathematics Curriculum Coordinator in Round Rock ISD. After serving in this role for six and a half years, I’m resigning so that my husband, daughter, and I can move to Rochester, New York this spring to live closer to family. I wrote previously about our reasons for moving here.
Leaving this job is strange because it feels like my tenure in the role is just a blip in the life of our school district. Someone did my job before me and someone will carry on now that I’m gone. That’s the same for all the folks who take on a position in a complex system like a school district. We all have a window of opportunity to make an impact in the time that we’re a part of that system. The question I’m asking myself right now is what did I accomplish in the time that I had the privilege of serving in this role? Or more simply, did I use my time well?
Today I’d like to share what I consider to be my accomplishments over the past six and a half years. I’m also going to follow up with another blog post about lessons I learned along the way. I’m not one for bragging or tooting my own horn, so writing this post has been uncomfortable, but on the other hand it does leave me with a sense of satisfaction that I did make good use of the time I was entrusted with leading elementary mathematics instruction in RRISD.
Accomplishment #1 – I survived my first year on the job.
If you’ve never worked in curriculum, it’s important to know that it has a life cycle based around the adoption of new standards and instructional resources. It’s really busy at the front end when standards and resources are new, but it gets way more chill as time goes on. Everyone becomes more familiar with the standards. Units become more settled. Resources become more fleshed out.
As luck would have it, I happened to join the district and start in this role at the front end when everything was brand new. I started in July 2014 and that August, teachers were expected to teach for the first time ever:
using the newly launched Google site that housed our curriculum documents.
Needless to say, teachers were stressed! I got a lot of frustrated (and some outright angry) emails and phone calls that year. Just a few months prior in the previous school year, teachers were using curriculum units that had been around for several years. They were comfortable with those units. Now it felt like suddenly those familiar units were snatched away, replaced with brand new units, with brand new lessons, in a brand new platform. With our new standards, the order of some topics got shifted around while others were completely removed from a grade level.
It was a shock to the system. It was a shock to me joining that system. It’s like I was dropped onto an airplane as it was taking off. Oh, and it wasn’t even a complete airplane. It was still being built as it shakily launched into the air.
It was a hectic year, to say the least. I told myself if I could survive that school year, every year after would feel easier by comparison. And for the most part, that was true. Every year posed challenges, but nothing as tough as that first year. Did I mention that in the first month of the school year I was told to develop and deliver a yearlong PD program for over 100 elementary math interventionists, on top of all the work I was already rushing to complete to get the remaining curriculum units and assessments written?
I should point out that I left my previous job to take this one because I was looking for something more challenging. And boy did I find challenge! So even though it was hectic, it was also exhilarating.
It’s also important to note that I didn’t do it alone. I survived that first year in large part thanks to Regina Payne. She was a life saver! At that time in our Curriculum Department two people were in charge of elementary mathematics, the Curriculum Coordinator (me) and the Curriculum Specialist (Regina). I am eternally grateful for Regina’s help and patience that first year. She knew where all the documents were, what work still needed to be done to get the curriculum completed for that school year, and she had a wealth of knowledge about how things worked in our district.
Coming into the role I didn’t know what I didn’t know. I never knew what was on the horizon until after I’d lived through a full year as the Curriculum Coordinator. It was a wild ride that first year, but I survived and went on to thrive.
Accomplishment #2 – Suggested Unit Plans
A tension I felt early in my time as Curriculum Coordinator was about how much and what kind of support to provide teachers through our curriculum. Some teachers wanted me to leave them alone, feeling like I was stepping on their toes by providing lesson plans in the ARRC. (That’s the name of our district curriculum. It stands for Aligned Round Rock Curriculum.) Others felt like we didn’t provide enough support because we didn’t have a lesson ready for every single day of the school year.
At first I leaned toward providing less. Teaching is a craft. I didn’t want to interfere by presuming I could tell teachers which lesson to teach everyday. Nor did I want to interfere with their ability to be responsive to the needs of their students. I felt that a good balance was to provide a sampling of lessons and ideas in each unit, but I wasn’t going to write daily lesson plans.
One day I was forced to reconsider my feelings about providing daily lesson plans. In a meeting, one of our district leaders shared how challenging it was for some teachers to plan quality lessons for every subject when their entire team was made up of teachers with only 1-2 years of experience. Even more problematic, these teams of novice teachers are often located at campuses with high turnover rates, which happen to be campuses with more students of color and more students living in poverty.
As I considered what to do, I thought about the metaphor of a teacher as a restaurant employee who not only serves as the waiter, but also as the chef, meal planner, grocery buyer, etc. If it’s unreasonable to expect one person to do every aspect of running a restaurant, much less do it well, then the same goes for a teacher running a classroom. If my goal was for all teachers in my district to teach high-quality, standards-aligned lessons, then I decided it was my responsibility to provide those lessons for teachers who were not in a position to plan them on their own, especially considering the variability of experience and all of the many other responsibilities on their plates.
That’s easier said than done, of course. This was a mammoth undertaking! As Curriculum Coordinator, I’ve been responsible for developing, revising, and maintaining about 90(!) math units across 6 grade levels (K-5) along with 3 grade levels of an accelerated curriculum for talented and gifted students in grades 3-5. Not only is that a lot of units, but it also meant writing A LOT of lessons.
Before I took this job, I spent 5 years developing digital math curriculum at a private company. I know what it takes to develop high quality daily lessons: it takes teams of knowledgeable people and it takes time. When I took on this project, I didn’t presume I could get it all done in a year. Rather, in my first year I brought together teams of 3-4 teachers to develop only 3-4 suggested unit plans per grade level.
That was it for year one, and it was still almost more than we could handle! At first Regina and I led planning meetings together, but quickly it turned into leap-frogging. She would lead one meeting and I would lead the next because as it turned out, each meeting created a lot of work she and I had to continue doing on our own after the teachers went back to their classrooms.
On each collaborative planning day, we tackled just one unit. The teachers worked with us to do the heavy lifting of unpacking standards, developing a flow of concepts across the number of days in the unit, and planning ideas for lessons.
Here are notes from one of the very first units we planned with our 3rd grade collaborative in the fall of 2016. We wrote these as we unpacked the standards at the start of our planning session. If you’d like to see the whole planning document for this particular unit click here.
After unpacking standards and planning the flow of the unit, we would examine existing lessons that were written in-district as well as lessons from our adopted resource to see if they could be used as-is, whether they needed revisions, or whether we needed to write a brand new lesson from scratch. Each planning meeting ended with a lengthy to-do list of resources and lessons Regina and I had to create ourselves.
It was a ton of work, but it was all worth it once we started getting feedback from teachers on the first units we posted. Very quickly teachers started asking for more! They were loving the daily lesson plans, especially when we included outside resources created and shared by members of the greater math education community, such as:
One of the teachers in that 3rd grade collaborative came to our next planning meeting and said, “I think that’s the first time I’ve ever hit every standard in a unit before. It felt so good!”
The final product looked something like this. You can check out the full suggested unit plan here.
It took 3 years, but we now have suggested unit plans for almost all 90 elementary math units. There are a few shorter units that I just never got around to because of time and/or budget, but we’re close!
Accomplishment #3 – Spreading the Word of Numberless Word Problems
As I mentioned earlier, in my first months on the job I was tasked with developing a yearlong PD program for over 100 math interventionists. At one of our first sessions together, Regina Payne introduced the interventionists (and me!) to numberless word problems. I was so taken by the idea, I wrote this blog post to capture the story she shared about how she got the idea and how she implemented it for the first time.
As we planned PD for the interventionists, we used the IES Practice Guide “Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools.” One of the recommendations in the guide is to “include instruction on solving word problems that is based on common underlying structures.” I decided in my free time to create sets of numberless word problems organized by the CGI problem types. I posted them on this blog so that anyone could use them. I also continued writing more blog posts and eventually presented about numberless word problems at various conferences. I now have a page dedicated to numberless word problems on this blog which you can visit here.
Today my numberless word problem resources are accessed hundreds of times daily, and I love to read all the tweets I’m tagged in and the ones tagged to #numberlesswp where people share their experiences using them in their classrooms. I’m thankful to Regina Payne for introducing me to numberless word problems and I’m proud that we’ve been able to get the word out about them not just in our district but far beyond as well.
Accomplishment #4 – Redefining Math Instruction in Grades K and 1
My first day on the job, way back in July 2014, I stopped by a summer PD session led by one of our veteran instructional coaches, Mary Beth Cordon. She was leading a session on teaching number concepts in the primary grades. As I sat down, she handed me a copy of Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases. I read a little and was intrigued so I borrowed the book to continue reading. Little did I know how much Kathy Richardson would influence my work over the next six and a half years.
The biggest influence she had on my work in RRISD was after I attended her leadership institute in the summer of 2018. After a week of learning from her, I was left with a profound feeling of disequilibrium about teaching math in the primary grades. Here’s an excerpt from a blog post I wrote shortly after the institute:
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.
If you’re interested, you can read the full blog post here. I spent the next six months mulling over these ideas and talking about them with anyone who would listen. I eventually came to the conclusion that I needed to restructure our primary grade math units to give students more time to really dig in and explore mathematics concepts. Here’s an excerpt from a later blog post where I talked through the changes I was planning to make:
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.)
If you want to read more about the changes I ended up making in Kindergarten and 1st grade, you can check out that post here. These were fairly revolutionary changes compared to the way math had been taught, so I didn’t spring them on everyone. Rather, I talked with our Curriculum Director, Darrell Emanuel, about getting teams of teachers to pilot the units. He was gracious enough to sweeten the deal by offering to buy additional manipulative kits for each participating classroom so teachers would feel like they had everything they needed to teach with these units.
I launched the math pilot units in grades K and 1 in the 2019-20 school year. I hosted professional development sessions in the summer to help the pilot teachers understand the “why” behind the new units and to familiarize them with the changes to instruction. I also connected with the instructional coaches at campuses teaching the pilot units to ensure there was additional support on site. Within the first month I visited classrooms, and I met with teachers to answer questions and offer support. I created collaborative documents where pilot teachers could drop in ideas, tips, and questions, to try to create a sense of community among the teachers as they utilized these new units. I also shared a digital photo album so they could see what the lessons and activities looked like in other classrooms and at other campuses.
Even with all of that support, there were still bumps in the road, but after observing classrooms and talking to teachers, I feel affirmed that we’re moving in the right direction. Here’s an excerpt from an email I received from a Kindergarten pilot teacher last spring:
I hope you are doing well! I wanted to offer some quick feedback on the timing and activities of the pilot while it is relatively fresh on my mind.
I felt like there was a strong variety of activities in the units in the first semester which made conferring easier. In the second semester I had trouble getting as many independent activities going in the More, Less and the Same unit. Overall we ended up spending more of our time on whole class activities during that unit which made conferring more of a challenge. My students did show strong mastery of the more, less and the same concepts in the unit and I felt like that unit could have been shortened by at least a week to leave more time for Joining and Separating quantities.
Overall I feel like our kids will be headed into first grade with a very strong number sense.
Accomplishment #5 – Math Rocks
I planned and facilitated many, many, many, many hours of professional development over the past six and a half years.
Out of all of it, I’m most proud of Math Rocks, a 7-month long professional development program Regina and I designed for teachers to dive more deeply into their practice and to build positive identities around mathematics for themselves and their students. Math Rocks had two goals:
Goal #1 Build relationships
We wanted participants in Math Rocks to build relationships around mathematics with one another, with their students, with colleagues at their campus, and with educators outside of our district. In addition to many in-person meetings, we also asked each participant to create a blog and a Twitter account so they could share the great things they were doing in their classrooms and so they could connect with other educators.
Goal #2 Be curious
We also wanted participants to be curious throughout the program: about mathematics, about their students’ thinking, and about their own teaching.
One of my favorite moments that exemplifies these two goals was leading a book study of Making Number Talks Matter in our first year. Each participant got two copies of the book, one for themselves and one for a book study buddy, a colleague on their campus they could invite to read and discuss the book with. In addition to fostering relationships on campus, we also built relationships outside of our district as we joined a national book study hosted by the Teaching Channel. One of the facilitators, Kristin Gray, even hosted a virtual session on number talks with our Math Rocks teachers.
Math Rocks ran as a district-level course for four years. I’ve written several blog posts about about our experiences which you can check out here. Regina and I led it for the first two years, and a team of fantastic instructional coaches led it the next two years. Word of mouth about it was so positive that I was invited to lead it at individual campuses, which is one of the reasons instructional coaches stepped up and took over leading it at the district level. I ended up leading Math Rocks at three campuses over the past few years, which was a great experience because I learned the importance of adapting the course to meet the needs of individual campuses.
Accomplishment #6 – Amplifying Educator Voices
We have amazing and passionate educators in RRISD, and it was important to me to help get their voices heard within and outside of our school district so that others can learn from them.
One of the ways I did this was by regularly encouraging them to apply to speak at local, state, and national conferences. For some, this was outside their comfort zone, but I’m so proud of them for taking a risk and doing it anyway.
Not everyone wants to lead professional development sessions and not everyone would get to attend anyway, so I looked for other avenues for sharing educator voices such as inviting teachers and instructional coaches to write guest posts on my district math blog. These became the Teacher Talk and Coaching Corner features on the blog.
Finally, every two years elementary teachers are eligible for the Presidential Award for Excellence in Math and Science Teaching (PAEMST). In 2016, I nominated Deb Swyers, a 3rd grade teacher from Elsa England Elementary School, for the award. Not only did she complete the lengthy application, but she ended up becoming a Texas state finalist for the award. I was so proud and happy for her!
This year two of the teachers I nominated for PAEMST completed the application. That alone is a cause for celebration considering they did it while teaching during a pandemic. Our two applicants this year are Jessica Cheyney, a Kindergarten teacher at Double File Trail Elementary School, and Haillie Johnson, a 2nd grade teacher at Elsa England Elementary School. It will be some time before we find out who the Texas state finalists are, but I’m rooting for both of them!
Accomplishment #7 – Serving the Greater Math Community
As much as I’ve loved serving the teachers and students of Round Rock ISD over the past few years, I’ve also valued the opportunities I’ve been afforded to serve the greater math community within and beyond Texas.
In the fall of 2015, I was invited to become a newsletter editor for the Global Math Department, a group of mathematics educators who put out a weekly newsletter and host a weekly professional development webinar. I went on to serve on on the Board of Global Math Department from 2016-2018.
If you’re unfamiliar with the Global Math Department, it is an amazing FREE resource for the mathematics education community. You can check out previous newsletters here and subscribe here. Sign up for upcoming webinars and watch previously recorded webinars for FREE here. It’s good stuff!
Over the past few years, I’ve also been invited to join the planning committees for several math education conferences:
NCTM Innov8 2017
NCTM Regional in Seattle 2018
While I’ve learned that large-scale event planning is not my favorite thing to do, I treasure the opportunities I’ve had to work with other dedicated educators to plan and put on these conferences.
And finally, closer to home, I’ve valued the time I’ve been part of the Texas Association of Supervisors of Mathematics (TASM). I started as a TASM member way back in 2014, and after a few years I was invited to serve as their NCTM Representative.
That same year I was nominated and elected to serve as TASM Vice-President, a position I held from 2018-2020. During my time as TASM Vice-President, I planned our fall and spring professional learning events. I also made it my mission to find ways to create more opportunities for our members to interact professionally and socially at our in-person meetings and between meetings. At our October 2019 meeting, I launched the TASM Events Committee and invited members to join. That December we hosted the first ever TASM Power Hour, which has become a monthly virtual hangout where TASM members vote on hot topics to discuss. A couple months later, at our spring 2020 meeting, we hosted our first ever game night to give members an opportunity to socialize together.
Last spring I was elected President-Elect of TASM, and I was looking forward to serving in this role. Unfortunately, the pandemic led me and my family in a direction I hadn’t foreseen. When we made the decision to move to New York, it meant resigning from the TASM Board and saying good-bye to my colleagues. I valued my time as a member of TASM. Being a Curriculum Coordinator can be a lonely job at times. TASM was an invaluable resource for connecting me with other wonderful math leaders from around the state.
As I mentioned at the beginning of this post, I’ve been reflecting on whether I made good use of my time in my role as Elementary Mathematics Curriculum Coordinator (that title will forever be a mouthful). Looking back, I’d have to say yes. The job wasn’t always easy, and there’s plenty I didn’t get to do because of time or budget constraints, but in the end I’m proud. I had remarkable colleagues all along the way who supported me in my work and helped me accomplish some great things over the past six and a half years. While I focused on accomplishments today, I’d like to take some time in my next post reflecting on what I’ve learned in my time in this role. The accomplishments I’ll be leaving behind in my district, but the lessons I’ve learned I’ll be able to carry with me as I move on to new adventures.
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.
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.
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.)
Use a Rule Strategy for Multiplication – There isn’t a video for this one. This thinking strategy involves multiplying with 0 or 1.
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:
Ask Yourself Questions
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.
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.
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.
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.
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?”
Teacher: “How many on the bottom?”
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.
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?”
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.
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.
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!