At the end of June, I had the pleasure of spending a week learning from Kathy Richardson at the Math Perspectives Leadership Institute in Hutto, Texas. I’ve been a fan of Kathy Richardson ever since my first week on the job as elementary math curriculum coordinator in Round Rock ISD. That week I sat in on a summer PD session on early numeracy led by Mary Beth Cordon, one of our district instructional coaches. She had us read a little out of Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases. I was hooked from the little I read, so I asked if I could borrow the book.

I devoured it in a couple of days.

Since then I’ve purchased multiple copies for all 34 elementary campuses, led campus and district PD sessions on the critical learning phases, and led a book study with over a hundred math interventionists. The book is so eye opening because it makes tangible and explicit just how rigorous it is for young children to grapple with and learn counting concepts that are second nature to us as adults.

I was so excited for the opportunity to learn from Kathy Richardson in person this summer, and she didn’t disappoint. If you’d like to see what I learned from the institute, check out this collection of tweets I put together. It’s a gold mine, full of nuggets of wisdom. I’ll probably be referring back to it regularly going forward.

As happy as I am for the opportunity I had to learn with her, I also left the institute in a bit of a crisis. There is a HUGE disconnect between what her 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 wanted to share about it in this blog post, and I’ll also be thinking about it and talking to folks a lot about it throughout our next school year.

So what’s the disconnect?

Here’s a (very) basic K-2 trajectory laid out by Kathy Richardson:

  • Kindergarten
    • Throughout the year, students learn to count increasingly larger collections of objects. Students might start the year counting collections less than 10 and end the year counting collections of 30 or more.
    • Students work on learning that there are numbers within numbers. Depending on their readiness and the experiences they’re provided, they may get this insight in Kindergarten or they might not. If students don’t have this idea by the end of Kindergarten, it needs to be developed immediately in 1st grade because this is a necessary idea before students can start working on number relationships, addition, and subtraction.
  • 1st Grade
    • Students begin to develop an understanding of number relationships. After a year of work, Kathy Richardson says that typical 1st graders end the year internalizing numbers combinations for numbers up to around 6 or 7. For example, the number combinations for 6 are 1 & 5, 2 & 4, 3 & 3, 4 & 2, and 5 & 1. Students can solve addition and subtraction problems beyond this, but they will most likely be counting all or counting on to find these sums or differences rather than having internalized them.
    • Students can just begin building the idea of unitizing as they work with teen numbers. Students can begin to see teen numbers as composed of 1 group of ten and some ones, extending the idea that teen numbers are composed of 10 and some more.
  • 2nd Grade
    • Students are finally ready to learn about place value, specifically unitizing groups of ten to make 2-digit numbers. According to Kathy Richardson, she says teachers should spend as much time as possible on 2-digit place value throughout 2nd grade.
    • Students apply what they learn about place value to add and subtract 2-digit numbers. By the end of the year, students typically are at a point where they need to practice this skill – which needs to happen in 3rd grade. It is typically not mastered by the end of 2nd grade.

And here’s what’s expected by the Texas math standards:

  • Kindergarten
    • Lots of number concepts within 20. Most of these aren’t too bad. The biggest offender that Kathy Richardson doesn’t think typical Kindergarten students are ready for is K.2I compose and decompose numbers up to 10 with objects and pictures. If students don’t yet grasp that there are numbers within numbers, then they are not ready for this standard.
    • One way to tell if a student is ready is to ask them to change one number into another and see how they react. For example, put 5 cubes in front of a student and say, “Change this to 8 cubes.” If the student is able to add on more cubes to make it 8, then they demonstrate an understanding that there are numbers within numbers. If, on the other hand, the student removes all 5 cubes and counts out 8 more, or if the student just adds 8 more cubes to the pile of 5, then they do not yet see that there are numbers within numbers.
    • My biggest revelation with the Kindergarten standards is that students are going to be all over the map regarding what they’re ready to learn and what they actually learn during the year. Age is a huge factor at the primary grades. A Kindergarten student with a birthday in September is going to be in a much different place than a Kindergarten student with a birthday in May. It’s only a difference of 8 months, but when you’ve only been alive 60 months and you’re going through a period of life involving lots of growth and development, that difference matters. It makes me want to gather some data on what our Kindergarten students truly understand at the end of Kindergarten compared to what our standards expect them to learn.
  • 1st Grade
    • Our standards want students to do a lot of adding and subtracting within 20. Kathy Richardson believes this is possible. Students can get answers to addition and subtraction problems within 20, but this doesn’t tell us what they understand about number relationships. If we have students adding and subtracting before they understand that there are numbers within numbers, then it’s likely to be just a counting exercise to them. These students are not going to be anywhere near ready to develop strategies related to addition and subtraction. And then there’s that typical threshold where most 1st graders don’t internalize number combinations past 6 or 7. So despite working on combinations to 20 all year, many students aren’t even internalizing combinations for half the numbers required by the standards.
    • The bigger issue is place value. The 1st grade standards require students to learn 2-digit place value, something Kathy Richardson says students aren’t really ready for until 2nd grade. And yet our standards want students to:
      • compose and decompose numbers to 120 in more than one way as so many hundreds, so many tens, and so many ones;
      • use objects, pictures, and expanded and standard forms to represent numbers up to 120;
      • generate a number that is greater than or less than a given whole number up to 120;
      • use place value to compare whole numbers up to 120 using comparing language; and
      • order whole numbers up to 120 using place value and open number lines.
    • I’m at a loss for how to reconcile her experience that students in 1st grade are ready to start putting their toes into the water of unitizing as they work with teen numbers and our Texas standards that expect not only facility with 2-digit place value but also numbers up to 120.
  • 2nd Grade
    • And then there’s second grade where students have to do all of the same things they did in 1st grade, but now with numbers up to 1,200! Thankfully 2-digit addition and subtraction isn’t introduced until 2nd grade, which is where Kathy Richardson said students should work on it, but they also have to add and subtract 3-digit numbers according to our standards. Kathy Richardson brought up numerous times how 2nd grade is the year students are ready to begin learning about place value with 2-digit numbers, and she kept emphasizing that she felt like as much of the year as possible should be spent on 2-digit place value. If the disconnect in 1st grade was difficult to reconcile, the disconnect in 2nd grade feels downright impossible to bridge.

I’m very conflicted right now. I’ve got two very different trajectories in front of me. One is based on years upon years of experience of a woman working with actual young children and the other is based on a set of standards created by committee to create a direct path from Kindergarten to College and Career Ready. Why are they so different, especially the pacing of what students are expected to learn each year? It’s one thing to demand high expectations and it’s another to provide reasonable expectations.

And what do these different trajectories imply about what it means to learn mathematics? 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.

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





Opening Our (Virtual) Doors

A really exciting thing happened today! My school district launched our brand new, open-to-the-public curriculum site. Not only can you peruse all of the elementary mathematics curriculum documents I’ve been writing and revising for the past four years, but you can access any and all courses taught in my district, PreK through 12th grade. I’m so excited that my district has taken this step!


As someone who’s been part of the Math Twitter Blog-o-Sphere for 6 years this August, the idea of openly sharing ideas and resources is a deeply held belief of mine. This is not a zero sum game. If I share a lesson I make with others, it doesn’t in any way diminish the learning of my own students. For the past six years I’ve had to live two lives, my work life where everything I make is locked behind file permissions, and my home life where I make and share things like numberless word problems for anyone and everyone to use. Considering how much the content of my two lives overlapped, it always felt strange, but now I’m free to share the resources I’m making at work while continuing to share anything I’m inspired to make at home in my spare time.

I’m in a good place.

While our curriculum site launched today, this has been a project in the making this entire school year. Basically our hand was forced because our existing curriculum site was built in Google Sites…the old Google sites. Google released a new version of Google sites, and they’re so happy with it, they’re discontinuing the old Google sites. Basically in a year or so, our existing curriculum site will cease to exist. There is no magic button to convert our existing site pages to become new Google site pages, so no matter what, we were going to be making something from scratch.

Our department took this obstacle and made it into an opportunity to redesign our curriculum. We spent the fall semester gathering feedback from teachers about what they wanted from the ARRC (that’s the name of our curriculum) – what they wanted to keep and what they wished it could offer. We held focus groups across the district and gathered feedback through a district-wide survey.

After we debriefed all of the feedback, we created four sample sites using four different platforms – new Google sites, WordPress, Build Your Own Curriculum, and Ogment. Then we held another round of focus groups where we invited teachers in to try out all four sample sites. Their feedback, along with several other factors, led to the final decision to build our new curriculum site in WordPress.

During the spring semester we finalized the design, created sample units across courses, and invited teachers in for an open house to provide a final round of feedback before we pulled the trigger on the herculean task of creating all of our units for every course. We debriefed after the open house and made our final decisions. Finally, at the end of April, we started creating curriculum units in our new site.


When the site launched today, our goal was for each course to have the first few units loaded and ready to go. And now that we’ve launched, we’ll continue adding units throughout the summer. I’m hoping to have the entire year of math curriculum for grades K-5 and three grade levels of TAG up by mid-September. I’m going to be so busy for the next few months!

This weekend I’ll try to write up a post with some highlights of what’s available in our elementary math curriculum. There are some things accessible only by district employees because of copyrights we don’t control, but there’s still plenty to dig into. If you’re interested, head on over and have a look around.


Moving On Before It’s Over (5th Grade)

I’m finally reaching the end of my blog post series which has been a retrospective of my elementary curriculum work for the past three years. If you want to read the previous posts in this series, here they are:

Today I’ll be sharing our 5th grade scope and sequences. 5th grade is a strange beast, at least in Texas. It’s a Student Success Initiative grade which means students are required to pass the state reading and math tests (STAAR) in order to move on to 6th grade. If students fail the test, they are given two more chances to pass before a grade level placement meeting is held. In order to accommodate three testing dates, 5th graders have to take their test nearly a month and a half earlier than they did in 3rd and 4th grade.

The implication is that despite having a year’s worth of curriculum to teach like everyone else, 5th grade teachers have to teach all of their standards by late March rather than early May. This makes planning the scope and sequence a perpetual challenge because you’re always working with less time than other grade levels. And with all of the meaty rational number topics in 5th grade, teachers are definitely not clamoring for less time to teach.

With that in mind, here are the scope and sequences for the past three years. What do you notice? What do you wonder?

5th Grade – School Year 2015-16


5th Grade – School Year 2016-17


5th Grade – School Year 2017-18


Looking back, I’m noticing a trend across grade levels where I used to split topics up into chunks that ended up being too small for teachers. In 2015-16 I split the following topics into two units each:

  • Volume – The first unit focused on multiplication and the second on division
  • Addition and Subtraction, Data Analysis, and Perimeter – The first unit focused on fractions and the second on decimals
  • Multiplication and Division – The first unit focused on fractions and the second on decimals
  • Geometry – The first unit focused on coordinate geometry and the second on classifying 2-D shapes

That year we crammed in 11 units before the STAAR test (12 if you include a short review unit). Needless to say, 5th grade teachers let it be known that year that they felt like they were flying way too quickly through their units.

The next year I think I over-corrected. Instead of many short units, I offered fewer, longer units. We went from 11 units to 6 units before the STAAR test. I wanted teachers to feel like they could really dig into the topics for an extended period of time. I also tried to make moot a debate about whether fraction multiplication/division should come before or after decimal multiplication/division. Since they were combined into one unit, it meant teachers could choose their preferred teaching order.

Like I said, I think this was an over-correction. That year teachers let me know that the units were too long. Some of the feedback was that by the time they got to the end-of-unit assessment 25 days later, students had forgotten content from earlier in the unit. Other feedback was that teachers didn’t know how to utilize their time within the unit. They felt like there was too much to cover, even though they had a longer block of time in which to teach.

Last spring I sat down with my 5th Grade Curriculum Collaborative to (hopefully) find a sweet spot. First we talked about which units to keep combined and which to separate. They decided that the unit on volume, multiplication, and division could stay together. They also felt that the geometry unit didn’t need to be broken up.

What did need to be broken up were the units on rational number operations. They said these are the topics where students have the greatest struggles. Namely, students need dedicated time to work on addition and subtraction with fractions that have unlike denominators. They also wanted to introduce fraction multiplication and division earlier to give students even more time to encounter related word problems. One of the biggest struggles our students have is knowing when to multiply or divide fractions in a word problem, and if it’s a division problem, in what order to divide – unit fraction divided by whole number or whole number divided by unit fraction.

After deciding that we would have 8 units and the order in which they would be presented, I asked them to identify the unit that they felt was the most critical. They decided Unit 2 on adding and subtracting fractions is the most critical. I asked how many days they needed to ensure success with that unit. They decided on 20 days.

We repeated this process to identify their second priority unit, which was Unit 3 on multiplying and dividing fractions. Again, we talked about the amount of time needed to teach this topic well. Because we considered these two units to be our top two priority units, it was non-negotiable to steal days from these two units as we created the rest of the scope and sequence.

Have we found the sweet spot? I think so. I’ve received minimal feedback from teachers about the 5th grade scope and sequence this year. It helps that this year Texas shifted the date of the STAAR test a little later than it was in 2015-16 and 2016-17. Teachers ended up with two additional weeks of instruction than in years past which definitely gave them a bit of breathing room.

This impacted our post-STAAR units however. In 2015-16 and 2016-17, after the first STAAR administration, we had two mirror units (Units 13a and 13b in 2015-16 and Units 8a and 8b in 2016-17). The rationale was that based on all the data collected that year, teachers should have had a pretty good idea of which students would pass on the first administration and which students would not. When scores are returned 3 weeks after the test, campuses tend to scramble to create intervention groups and provide intense intervention. My philosophy is, why wait?

Once the first administration was done, we wanted teachers to start providing that intervention and support immediately so that they could intervene for a full 6 weeks instead of just 3. This might involve mixing students around across classes so that some students would learn from the Going Deeper Unit while others learned from the Enriching Connections unit. Both units had the same standards, we just provided different instructional resources.

Once the second STAAR administration was over in May, all of the 5th graders got to take part in the final unit of the year which focused on personal financial literacy. This is a unit students tend to enjoy so I wanted to make sure everyone got to take part. If we had offered this unit after the first administration, some students might have gotten yanked out of it when scores came back, which isn’t fair.

As I mentioned previously, things changed this year when Texas moved the first administration two weeks later. They also moved the second administration a week later as well. That had a big impact on my 3rd, 4th, and 5th grade scope and sequences because now there isn’t enough time after STAAR in May to warrant a complete unit. That meant the personal financial literacy unit had to move immediately following the first administration of STAAR or else it wouldn’t get taught at all. It also meant the two mirror units are much shorter this year. Considering students got more time for first instruction this year, I’m not complaining.

5th Grade – School Year 2018-19

Based on the lack of feedback, I’m going to keep the scope and sequence the same for next year. Over the past two years, my 5th Grade Curriculum Collaborative has worked with me to develop suggested unit plans for 7 of the 8 units before STAAR. Teachers have been really happy with these model plans. Once we write the 8th plan, it will be nice to go back and start making revisions to the existing plans now that they’ve been in use for a couple of years.

I’m not sure whether I’ll make any adjustments to the computational fluency and spiral review topics this year either.



You’ll notice that, like 4th grade, 5th grade also starts with a review of multiplication facts. I’ve done the math and across grades 3, 4, and 5, we’ve incorporated nearly 75 hours of instruction on multiplication and division facts across these three grade levels. Here’s the breakdown:

  • 3rd Grade – Nearly 50 hours focused on conceptual understanding and more than 10 hours of procedural fluency practice spread across the entire year
  • 4th Grade – Nearly 10 hours of procedural fluency practice spread across the first semester
  • 5th Grade – Five hours of procedural fluency practice in the first nine weeks

I have some work to do to help see this enacted in the way that I envision, but I feel good about the structure we’ve put in place to intentionally teach and reinforce this skill across the intermediate grades.

Looking at the spiral review topics, I’m pretty happy with their flow, especially at the beginning of the year. In Unit 1 we focus on 4th grade fraction topics. 5th grade is really a fraction- and decimal-heavy year. I totally get that students might have forgotten some of what they learned in 4th grade, but we just don’t have the luxury of time for excuses. The students need to hit the ground running if they’re going to have sufficient time to grapple and become proficient with the 5th grade material.

I like how the Unit 2 spiral review parallels the focus TEKS topics, but using whole numbers instead of fractions. We do the same thing in Unit 3. However in Unit 3 it’s doing double duty because in addition to being a parallel to the focus TEKS work, it’s also revisiting whole number multiplication and division which will be a focus in Unit 4. Then in Unit 4 the spiral review topic is decimals which is in preparation for Unit 5.

Got a question about our scope and sequence? Wondering what in the world I’m thinking about planning things this way? Ask in the comments. Otherwise, that just about wraps up this blog series.

Parting Thoughts

Over the course of this blog series, I’ve really appreciated the experience of reflecting on my past three year’s worth of curriculum work in each grade level. It’s been interesting to see how my thinking has changed over the past few years and how much of it has been influenced by the feedback from and collaboration with our teachers. I greatly appreciate that they’re willing to share what’s working and what’s not. I don’t teach in a classroom day in and day out, so I would be handicapped in my work without their expertise and insight.

Looking back over three years and six grade levels, I’m struck by how complex this work is. There are so many moving parts in terms of the numerous standards within a grade level, how they interconnect across grade levels, how to bundle standards meaningfully into units, and how to decide the appropriate amount of time for any given unit. If you ever find yourself in the position of doing this work, my advice is to invite a group of colleagues to work with you, do your best, and expect that you won’t get it “right” the first time.

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

Oh, and I have an exciting announcement for those who’ve read all the way to the end! My school district is currently in the process of making our entire curriculum for all subjects K-12 an open education resource for anyone to access. Copyrighted lessons will still be restricted to district employees, but all of our curriculum documents and a wide variety of non-copyrighted resources will be freely available. We’re currently in the process of transferring everything into our new curriculum site. I’ll be sure to share the link (and probably blog about it!) once the site goes live. I’ve always been a big fan of sharing because we’re better together, and I’m so thankful to work for a school district that shares this philosophy.




Moving On Before It’s Over (4th Grade)

After taking a break to prep my session for the 2018 NCTM Annual Conference, it’s time to get back to this blog series on my spring curriculum work as I prepare for the 2018-19 school year. If you’re just joining us, here are the previous posts in this series:

Today I’ll be talking about our 4th grade scope and sequence. Here they are for the past three years. What do you notice? What do you wonder?

4th Grade – School Year 2015-16


4th Grade – School Year 2016-17


4th Grade – School Year 2017-18


The first thing I notice are my efforts to figure out how I wanted to break up multiplication and division across two units. Here’s what we’ve tried over the past three years:


  • Unit 3
    • Multiply 2-digit numbers by 2-digit numbers
    • Divide 2-digit numbers by 1-digit numbers
  • Unit 6
    • Multiply 3- and 4-digit numbers by 1-digit numbers
    • Divide 3- and 4-digit numbers by 1-digit numbers

What was our rationale?

In the spring prior to the 2015-16 school year, our adopted resource, Stepping Stones, underwent a revision to more closely align to the TEKS. In March 2015 we were sent preliminary scope and sequences of the revised courses. While doing our curriculum work that spring, we decided to try to follow certain topics in the order presented in the revised scope and sequence documents to ensure that students would see lessons in order. Our thinking was that presenting lessons out of order could lead to problems if later lessons assumed knowledge of earlier lessons.

In those preliminary documents, 2-digit by 2-digit multiplication was taught first followed by 3- and 4-digit by 1-digit multiplication. I was a little concerned about this, but we decided to stick to our plan. Lo and behold, when the revised Stepping Stones launched that summer, the order had been reversed. Our curriculum documents were already completed and posted for teachers to use by that point so we stuck it out for that year. However, as you can see below, we changed things up for the next school year.


  • Unit 3
    • Multiply 3- and 4-digit numbers by 1-digit numbers
    • Divide 2-, 3-, and 4-digit numbers by 1-digit numbers
  • Unit 5
    • Multiply 2-digit numbers by 2-digit numbers

What was our rationale?

So we flip-flopped the multiplication topics, but we also merged all of the division into Unit 3. Our thinking was that students already did a lot of dividing with 2-digit numbers in 3rd grade, so really division with 3- and 4-digit numbers was just extending that. That allowed teachers to solely focus on 2-digit by 2-digit multiplication in Unit 5.

However, this came back to bite us in the butt because of a tricky little standard about interpreting remainders. Teachers emailed to let us know that they really wanted to revisit division in Unit 5 because their students were still having difficulty with interpreting remainders. This was great feedback, which leads us into the current school year.


  • Unit 3
    • Multiply 3- and 4-digit numbers by 1-digit numbers
    • Divide 2-digit numbers by 1-digit numbers
  • Unit 5
    • Multiply 2-digit numbers by 2-digit numbers
    • Divide 3- and 4-digit numbers by 1-digit numbers

What was our rationale?

Multiplication remained untouched this year, but we did spread division out across Units 3 and 5. Our hope is that by only doing division of a 2-digit number by a 1-digit number in Unit 3, teachers can focus more of their energy on interpreting the remainder. Then in Unit 5 they can extend division to larger numbers while reinforcing what students learned about interpreting the remainder.

We’re in a good place now, and I don’t foresee changing this for next school year, but it gives me pause to think about the fact that this simple rearranging of a few topics was a 3-year process. It’s important to note: There are no absolute right answers in this work. I can consult teachers. I can read professional journals and books. I can read up about how other curriculums structure their scope and sequences. In the end, I have to use my best judgment…

…and then wait and see what happens when teachers and students interact with these units. I can (and do) iterate and revise, but by the nature of the work, it’s over a scale of years, not days or weeks. No pressure! It makes me think of how if I think back to my first year teaching, I feel bad for that group of students because I know so much more about teaching than I did back then. What I have to remind myself is that regardless of the specific decisions I’ve made each year, I’ve always been striving to do my best for the students (and now teachers) that I serve. And I’d rather know that I’m improving each year than continuing to make the same mistakes time and again.

Another noticing I have about the 4th grade scope and sequence is how I had too many units in the 2015-16 school year. Similar to 3rd grade, I tried breaking some topics up over multiple units to create opportunities for them to spiral back. Most notably I did this with fractions (Unit 4 and 7) and decimals (Unit 9 and 11). Teachers didn’t like this. They specifically requested one fraction unit and one decimal unit, which we created in 2016-17 and continued in 2017-18.

The last thing I’ll point out is how our angle measurement and 2D geometry unit shifted from the second semester in 2015-16 to the first semester in 2016-17 and 2017-18. This was influenced by the Level 1 Curriculum Audit training I attended in the fall of 2015. One of my big takeaways from that training was that topics that are absolutely brand new to students should be introduced as many months as possible prior to the first time students will be assessed on them. Angle measurement is completely new to students in 4th grade. Introducing it a couple of months before the state test doesn’t give students sufficient time to learn and reinforce it, so I moved it earlier in the school year. This gives plenty of time to revisit it between first instruction and the STAAR test.

4th Grade – School Year 2018-19

Here I am sharing the curriculum work I’m doing this spring, and it turns out I’m not really changing our courses that much. I don’t anticipate reordering any of the units in 4th grade. Our computational fluency and spiral review topics seem pretty solid as well.



Looking at computational fluency and spiral review, the first four units basically serve as review of 3rd grade concepts. Notably, teachers have a whopping 59 days of computational fluency to work on multiplication and division fact fluency, which was a HUGE focus of the 3rd grade scope and sequence. This amounts to about 10-12 hours of practice at the start of 4th grade. We specifically organized this work around the thinking strategies taught in 3rd grade to create common language across grade levels.

The spiral review concepts in the first few units are critical, especially reviewing 3rd grade geometry concepts in Unit 3. As I was working with my 4th Grade Curriculum Collaborative this year to plan Unit 4, we talked about the 3rd grade geometry standards. The 4th grade teachers were surprised to hear that their students should come in already knowing a lot about a variety of quadrilaterals – parallelograms, trapezoids, squares, rhombuses, and rectangles.

I talked about how they have their own heavy work to do introducing angle measurement and parallel and perpendicular lines; they don’t have time to “teach” those quadrilaterals in their 4th grade unit. There was some resistance, but I pushed back that they have to utilize that spiral review time in Unit 3 to revisit all of those polygons and attributes that the students learned in 3rd grade. Otherwise they’re setting themselves up for some real (and potentially avoidable) challenges in Unit 4.

I am excited to check in with our 4th grade teachers next year because this school year our 3rd grade teachers were able to use a newly created suggested unit plan for their geometry unit that was chock full of amazing lessons. I’m hoping the 4th grade teachers will be pleasantly surprised by the level of thinking students bring next year. I made it a goal this year with all of my Curriculum Collaboratives to plan all of our geometry units K-5 to ensure students are always engaging with grade-appropriate standards and building the levels of geometric thinking they need across these six years.

Got a question about our scope and sequence? Wondering what in the world I’m thinking about planning things this way? Ask in the comments. I’ll wrap up this blog series with 5th grade’s scope and sequence in my next post.







You Didn’t Hear It From Me

On Thursday, I’ll be sharing about numberless word problems at the NCTM Annual Conference in Washington, D.C.


In preparation for my session, I reached out to fellow educators on Twitter, asking them, “How have numberless word problems impacted student sense making in your classroom?” I’ll have plenty to say about this in my talk, but I wanted to take this opportunity to let a wider variety of voices share their thoughts and reflections on using numberless word problems.

Macy, Math Interventionist, Arkansas:

“They no longer see two numbers and add. They think about the problem and what a reasonable answer could be.”

Julie Bourke, 2nd grade teacher, Michigan:

“I can actually watch my students shift from plucking out numbers and adding them to reading the problem and visualizing what is happening. This helps them solve and understand exactly what the quantities in the problem represent.

I also saw a shift in my repeated direction of “don’t forget the unit/label” has disappeared because the students aren’t thinking of the numbers as separate from the problem. They are making sense of the context, deciding on the best strategy to solve and the numbers in the problem aren’t really the focus.

This has also improved my own teaching. I was a “circle the number and underline the key words” teacher and I was teaching students to follow directions. Now I am teaching mathematicians who make sense of problems, develop strategies and discuss solutions within a context.

This has been an important shift for my career and my own understanding of teaching math.”

Kristen Mangus, Math Support Teacher, Maryland:

“I have shared these with teachers in my school, K-5. K teachers started using these when they began teaching word problem standards and they instantly noticed a difference in how their students solved problems compared to when they taught problem solving without numberless word problems.

Numberless word problems also reduce “number plucking” because students have time to think about the problem, make connections and ask questions so that they are ready and confident when the numbers are introduced.”

Kjersti Oliver, Middle School Instructional Facilitator, Virginia:

“These are great for MIDDLE SCHOOL TOO! Especially students that are EL or struggle with word problems! They can work for equation word problems, systems, proportions, etc.! Great entry point for students!”

Carrie DeNote, Math Interventionist, Florida:

“The student Notice/Wonder about everything now. I’ve seen N/W t-charts on their assessments where they have used it to help them make sense of a question.”

Jana Byrd, K-2 Elementary Specialist, Alabama:

“The very first time I used numberless story problems, I was amazed at the amount of math vocabulary that naturally surfaced during the discussion; greater than, less than, the same, equal, etc.

Without the numbers in the problem, I noticed that students focused on finding the relationship between the quantities even though they weren’t there. That prevented them from just grabbing numbers and doing something with them.

When the numbers were presented within the story problem, they did what made sense to them. They were able to decide on a strategy and discuss their thinking in a more clear manner. I’m sold on numberless word problems, especially when introducing new situations to students.”

Jordan Hill, 2nd grade teacher, Alabama:

“It has allowed the students to stop and make sense of the situation before attacking the problem.”

Wendy Wall, Mathematics Support Teacher, Virginia:

“Thank you! You have created an opportunity for students to talk and reason. You have created a resource teachers love!”

Deepa Bharath, Math Coach, Florida:

“Focus is on understanding the context, considering what is asked and possible strategies – students can notice structures and similarities, this is like the other one we did, when numbers are shown students tend to think less and just compute. Also helped students to be less afraid of fractions and large numbers – we solved the same problem with whole numbers before working with fractions, almost like a number string estimating first how the answer would be affected.”

Nicole Grygar, 1st grade teacher, Texas:

“When solving word problems, they are not jumping to conclusions. They are working all the way through the problem to make sure they are solving the right question.”

Christine Mauer, Special Education Resource & Inclusion Instructional Assistant, Texas:

“Taking the numbers out of the questions has allowed them to become immersed in the story first.”

Jenna Laib, K-8 Math Specialist, Massachusetts:

“Students are willing to think deeper and slower about world problems; they don’t shy away from a block of text as much, and they have a greater awareness of problem types (CGI style) which helps them determine their strategy. I have noticed the biggest change in students with disabilities, especially students with language-based disabilities like dyslexia.”

Melanie Tindall, Elementary Math Specialist K-5, New Jersey:

“Numberless word problems help students think about and visualize the problem. They help students think about what information they know and what information they need in order to solve the problem. They also help students think about what question(s) can be answered with the given information.”

Kristine Venneman, Elementary Mathematics Specialist, Middletown:

“Students are essentially forced to consider the context to begin their solution path without simply adding or multiplying.”

Rose Scullion, K-5 Mathematics Specialist, New Jersey:

“Before numberless word problems became part of regular instruction students would take the numbers they saw in the problem, cross their fingers, have a hope and a prayer, and perform some type of procedure or algorithm, with no sense if they were correct or not. Now, students are relying more on visualizing the mathematical context, planning out their solutions, and choosing strategies to solve.”

Anonymous, Math Coach, Connecticut:

“The use of them have increased students focusing on the context and sense making.”

Shawna Velt, Special Education Math Consultant, Michigan:

“I share this strategy with special education teachers to support students in understanding word problems. We use cubes to model along with each step”

Brian Buckhalter, K-4 Math Coach, Mississippi:

“Traditionally, the “goal” of math class is to find the answer. Numberless word problems take the attention away from finding the (usually) one correct solution. Instead, they open the door for discussion among students to share their interpretations and reasoning about problems. Then the focus shifts from following steps or other procedures to reasoning, examining relationships, extending patterns, doing what “just makes sense” (as my students would say) and other hidden beauties of truly understanding mathematics.”

Thank you to everyone who took the time to share their feedback and experiences! It was so heartwarming to read how numberless word problems have impacted other classrooms across the country. As someone whose mission it is to help students develop identities as mathematical sense makers, it means a lot that this strategy has helped so many of you foster that with your own students.

And to those of you able to join me at the NCTM Annual Conference in D.C., I look forward to seeing you in a couple days!

A Little Preview

Next week I have the privilege of presenting a session about numberless word problems at the 2018 NCTM Annual conference. Even if you don’t teach in grades 3-5, I still invite you to join us because there will be lots of ideas shared of interest to multiple grade levels.


During the session, I’ll be referencing a few numberless word problems used over the course of several months in a 3rd grade classroom in my district. I thought it might be fun to share them before my session so folks could take a peak (and possibly even try one or two of them out before my session!).

The Collie and Chihuahua Problem – This is a comparison problem where the difference is unknown.

The Ancient Penguin Problem – This is another comparison problem. This time the larger quantity is unknown.

The Sand Castle Problem – This is an equal groups problem with an unknown product.

The Minecraft Problem – This is a multi-step problem involving multiplication and addition.

The Piano Practice Problem – This is a multi-step problem involving addition and subtraction.

The Pie Problem – This is a multi-step problem involving multiplication.

Enjoy! And if you’ll be joining me next week at NCTM, I look forward to seeing you in Washington, D.C.!

Moving On Before It’s Over (3rd Grade)

If you’re just joining us, I’ve been writing a series of posts as I embark on my spring curriculum work to prepare for the 2018-19 school year. I’m sharing how our scope and sequence has evolved over time, rationales for why things are the way they are, and thoughts on what changes I might make for next school year. If you’d like to back up and read about an earlier grade level, here are the previous posts in this series:

Today I’ll be talking about our 3rd grade scope and sequence. Here they are for the past three school years. What do you notice? What do you wonder?

3rd Grade – School Year 2015-16


3rd Grade – School Year 2016-17


3rd Grade – School Year 2017-18


Remember back in my first post in this series when I said, “Now that I’ve been doing this for a few years – and I’m starting to feel like I actually know what I’m doing…“? Yeah, 3rd grade is a prime example of how I have learned a lot over the past few years. I’m a little (maybe a lot) embarrassed to show you what it used to look like back in 2015. I had good reasons for what I attempted to do, but this was just a tough nut to crack.

So what was going on several years ago when I put our 3rd grade teachers through the wringer with 18 units in one school year? If you look at the 2015-16 scope and sequence closely, you’ll notice that one topic appears waaaaay more frequently than the others – multiplication and division. There were a total of 7 units just on multiplying and dividing.

This was very intentional. Just like I have specific numeracy goals in the previous grade levels, my goal in 3rd grade is to ensure students leave the school year as strong as possible in their understanding of multiplication and division. Specifically, I want to ensure students have the chance to develop mental strategies for multiplication and division.

Before I became the Curriculum Coordinator in my district, a team of folks analyzed fluency programs and ultimately decided that ORIGO’s Book of Facts is the one we would purchase for our entire district. After that decision, but still before I started working in this role, our district went through the adoption process for a new math instructional resource. Teachers selected ORIGO’s Stepping Stones program.

This turned out to be a wonderful fit because the mental strategies from the Book of Facts are baked into the lessons in Stepping Stones. (If you want to learn more about these mental strategies, check out these awesome 1-minute videos from ORIGO.) I didn’t want to rush students through the strategies, so I followed the Stepping Stones sequence of multiplication and division lessons. This gave each strategy its due, but it also resulted in 7 units on just this one topic.

Unfortunately, this meant squeezing in everything else in between all of those multiplication and division units. To my credit, I did share this scope and sequence with a team of six or eight 3rd grade teachers to get their feedback before putting it in place. I must be a good salesman because they thought it made sense and wanted to give it a try.

I’m sure you can imagine, it was tough that year. Just as teachers started a unit, it felt like it was ending. This happened to also be the year that our district started requiring teachers to give a district common assessment at the end of every unit. That decision was made after I’d already made all of my scope and sequences, otherwise I might have thought twice….maybe. The teachers felt like they were rushing through unit after unit and assessing their kids constantly. It was too much.

The next year we tightened things up quite a bit. We were able reconfigure concepts to end up with five fewer units than the year before. Without sacrificing my ultimate goal, I do feel like we ended up with a scope and sequence that has a reasonable amount of breathing room.

A major change that happened between last year and this year is that we removed the 10-day STAAR Review unit. We took 5 of those days and gave them to teachers at the beginning of the year to kick off with a Week of Inspirational Math from YouCubed. We took the other 5 days and gave them to units that needed more time. My rationale is that teachers often tell me they don’t have enough time to teach topics the first go round. If that’s the case, then I can’t justify spending 10 days at the end of the year for review. Those days should be made available earlier in the year to ensure there’s enough time for first instruction. If you’re interested, I shared additional reasons for this change along with an alternative to the traditional test prep review unit in this post on my district blog.

As embarrassed as I am to share the scope and sequence I inflicted on our 3rd grade teachers for an entire school year, looking at it now, I am proud of what we attempted and proud of the revisions we’ve been able to make over time. It’s finally a wieldy scope and sequence!

My reason for sharing this is to let people to know this work isn’t easy, especially people who are in the same boat as me or considering moving into this kind of role. There are a lot of moving parts within and across years, and you’re bound to make some mistakes. The important thing is to always have an eye for continuous improvement, because there is always something that could use improving. And if you can enlist the help of great teachers to provide their expertise and feedback, even better. This is not work that should be undertaken solo.

3rd Grade – School Year 2018-19

So what’s the plan for next school year? One area that’s been nagging me is addition and subtraction. If you read the 2nd and 3rd grade standards on this topic, you’ll notice the first half of each standard is identical except for one word: fluency.

  • Second grade
    • 2.4C Solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms
  • Third grade
    • 3.4A Solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction

One of the 8 effective teaching practices from NCTM’s Principles to Actions is that we should build procedural fluency from conceptual understanding. I see this happening in in our 2nd grade curriculum:

  • We build conceptual understanding of multi-digit addition and subtraction across 60 days in 3 units
  • And this helps us build fluency of 2-digit addition and subtraction in our computational fluency component across up to 97 days in 6 units

What about in 3rd grade? We kick off the year reconnecting with 2-digit addition and subtraction in our computational fluency component for 30 days in Units 1 and 2. This overlaps with our efforts to reconnect with the conceptual understanding of adding and subtracting 3-digit numbers in Unit 2.

Starting in Unit 3, our goal becomes moving students toward fluency. We strive to achieve this by having it as a computational fluency topic for up to 64 days in 4 units. Problem solving with addition and subtraction, and later with all four operations, also appears throughout the year in 41 days of spiral review in 3 units.


When I write it all out like that, I feel pretty good about it, but I do wonder if it’s enough. I hear from 3rd grade teachers, especially in the fall, that their students are having a really difficult time with addition and subtraction, a much harder time than they are with multiplication and division.

I’m not sure I want to make a change to 3rd grade’s scope and sequence though. They have enough on their plate. I want their kids to begin building multiplicative thinking, build a strong understanding of how multiplication and division are related, and, oh yeah, build fluency with all of their multiplication and division facts. That’s a lot to accomplish!

What I really want to do is look at how our 2nd and 3rd grade teachers are teaching addition and subtraction. My gut tells me the problems I’m hearing about have something to do with the standard US algorithms for addition and subtraction.

In case you’re wondering, the phrase “standard algorithm” does not appear in our addition and subtraction TEKS until 4th grade. And that makes sense. When you’re adding or subtracting 2- and 3-digit numbers, that can be done fluently in your head, given practice. However, once you hit 4th grade, and you start adding 6-, 7-, and 8- digit numbers, you’re going to want to pull out a calculat…er…I mean algorithm.

Despite my best efforts, I know there are some 2nd and 3rd grade students being taught the standard US algorithms which might be causing some of the issues I’m hearing about. As I like to say in this sentence I just made up, “When standard algorithms are in play, number sense goes away.” If teachers are still teaching standard algorithms despite everything in our curriculum pointing to the contrary, then I’ve got some work to do to shift some practices, including providing professional development. Thankfully I’ve already got some lined up this summer! I also need to work more with our instructional coaches on this topic so they’re better equipped to support the teachers on their campuses.

Got a question about our scope and sequence? Wondering what in the world I’m thinking about planning things this way? Ask in the comments. I’ll continue with 4th grade’s scope and sequence in my next post.