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!
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.
The Decision-Making Protocol for Math Coaching: Apply High-Leverage Practices and Advocate Change
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.
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:
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.
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.
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
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
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?
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
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
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:
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.)
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?
[Update – I’ve added a four more Quizlet study sets to my Multiplication Facts Practice folder. The three “Practice Doubling” study sets are designed to provide students practice doubling a number, a necessary skill to be able to efficiently use the Doubling Multiplication Fact Strategy The “Practice Halving” study set is designed to provide students practice halving a multiple of ten, a necessary skill to be able to efficiently use the Use-Ten Multiplication Fact Strategy.]
As a member of NCSM, I get a weekly email called the Marshall Memo that shares summaries of a variety of education-themed articles. What I like about the Marshall Memo is that I get exposed to articles I may never have encountered on my own. Even better, while many articles are on topics that aren’t math-specific, I’m still often able to able to make connections to my own work.
It also connected to something I’ve been thinking a lot about lately, which is the strong research evidence that retrieval practice promotes learning:
“Retrieval practice” is a learning strategy where we focus on getting information out. Through the act of retrieval, or calling information to mind, our memory for that information is strengthened and forgetting is less likely to occur.”
“…quizzes or tests that require students to actively recall specific information (e.g., questions that use fill-in-the-blank or short-answer formats, as opposed to multiple-choice items) directly promote learning and help students remember information longer.”
IES Practice Guide, Organizing Instruction and Study to Improve Student Learning, page 21
This also brings to mind “Rachel,” a thought-provoking blog post from Michael Pershan that has had me thinking about the interrelationships between deriving and recalling facts.
Suppose a student has just derived 9 x 4. If they’re confident and successful, they might have an opportunity to share that solution with the class — I might ask them to share their solution, and they might have a moment where they ask themselves, “wait, what was 9 x 4 again?” This is recall practice. Or, maybe, they are working on a larger problem in which 9 x 4 is merely a step, and their later work calls on them to remember the product of 9 x 4. They derive it, and then turn back to the problem and ask themselves, “what was 9 x 4?” Or perhaps, while working on a large set of multiplication problems, a student derives 9 x 4 and is then asked to derive 90 x 4. They ask themselves: what is 9 x 4?
All of this thinking got me inspired to give Quizlet a try for creating study sets that provide students practice both deriving and recalling multiplication facts. I organized my study sets around the thinking strategies shared in The Book of Facts: Multiplication, published by ORIGO Education.
“Research show that the most effective way for students to learn the basic facts is to arrange the facts into clusters. Each cluster is based on a thinking strategy that students use to help them learn all of the facts in that cluster.”
The Book of Facts: Multiplication, ORIGO Education
If you’re unfamiliar with these thinking strategies, ORIGO has kindly created a one-minute overview video of each one:
For each strategy I created three levels of study sets in Quizlet. Level 1 focuses on reinforcing the thinking strategy. As students practice the flashcards, they are presented a pictorial representation of the multiplication fact that reinforces the thinking strategy. For example, if students are solving 8 × 5, the reverse side of the flashcard shows the product as well as a visual that reinforces the idea that each fives fact is half of the related tens fact. In this case, the array model shows that 8 × 5 is half of 8 × 10.
Level 2 focuses on a verbal reminder of the related thinking strategy. The front of the card remains the same, but the back of the card includes a reminder of what students can think about to help them derive the fact. Here’s the back of the 8 × 5 card in Level 2:
Finally, in Level 3, the focus is on recalling the multiplication facts. The back of the card does not include any reminders; it just shows the product. If students get stuck, the teacher can ask the student to recall the thinking strategy they’ve learned, otherwise students should focus on recalling the facts.
In addition to the strategy-focuses study sets, I’ve also included three study sets that practice a variety of multiplication facts when students are ready to focus on recalling across all of the facts. Version 1 focuses on the x0, x1, x2, x3, x4, and x5 facts. Version 2 includes a wide variety of all facts. Version 3 focuses on the x6, x7, x8, and x9 facts.
You can access all 21 study sets on Quizlet. If you’re not familiar with Quizlet, there is a free version and a paid version. I’d recommend starting with a free account. If you’re a teacher, be sure to indicate it when creating your account because teachers get extra features.
Some words of advice, Quizlet offers a wide variety of modes for practicing study sets.
I’ve noticed that many of these activities show the product and students are supposed to answer with the multiplication expression. If you want to start by presenting the multiplication fact to the students, all you have to do is click the Options button and then change “Answer with” to “Definition” instead of “Term.” I recommend doing this because generally we want students to recall the product not the multiplication expression.
In the Flashcards activity, I recommend turning on Shuffle. If students are at a point of focusing on recall rather than deriving each fact, then I also recommend turning on Play. This will make the flashcard automatically turn over after a few seconds. This prevents students from falling back on counting strategies.
In the Learn activity, I recommend going into the options and deselecting “Multiple choice questions.” For retrieval practice, research does not recommend multiple choice questions. Rather, the “Flashcards” and “Written questions” are preferable Question Types for this activity.
In the Test activity, I recommend only the “Written” and “True/False” question types. Again, in all of these activities, don’t forget to change the “Answer With” option from Term to Definition.
And finally, if your students are not familiar with the thinking strategies in these study sets, then they may be very confusing and unhelpful to students. In The Book of Facts series, ORIGO recommends four teaching stages:
Introduce the strategy – Hands-on materials, stories, discussion, and familiar visual aids to introduce the strategy or sub-strategy
Reinforce the strategy – This stage make links between concrete and symbolic representations of the facts being examined. Students also reflect on how the strategy or sub-strategy works and the numbers to which it applies.
Practice the strategy – This stage aims to develop accuracy and increase ‘speed’ of recall. In this stage, a range of different types of written and oral activities is used.
Extend the strategy (to greater numbers) – Students are encouraged to apply the strategy to numbers beyond the range of the basic number facts. The activities in this stage are designed to further strengthen students’ number sense, or “feel” for numbers.
The Quizlet study sets I created fall within the Practice stage. If you’d like to teach these strategies to your students, I do recommend checking out The Book of Facts: Multiplication because it provides several activities at each of the four stages for each strategy.
If you try out these study sets with your students, let me know how it goes! I’m excited to be able to share this resource for retrieval practice to the teachers in my district. If I hear feedback from them, I’ll be sure to let you all know how it goes.
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.