The Path Ahead

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

• Kindergarten
• 1st Grade
• 2nd Grade
• 3rd Grade
• 4th Grade
• 5th Grade

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

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

Source

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

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

Source

What a difference a month makes.

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

Back in July I closed my post with no answers:

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

Source

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

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

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

Kindergarten 2018-19

1st Grade 2018-19

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

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

Proposed Kindergarten 2019-20

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

Proposed 1st Grade 2019-20

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

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

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

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

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

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

Draft Kindergarten At-A-Glance 2019-20

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

Things to point out:

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

Parting Thoughts

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

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

Here’s to the path ahead.

The Annotated Numberless Word Problem

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

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

1. Start with a problem

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

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

2. Work backward

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

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

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

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

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

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

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

3. Plan purposeful questions

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

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

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

Slide 1

Ask for a volunteer to read the story.

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

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

Slide 2

Ask for a volunteer to read the slide.

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

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

Slide 3

Ask for a volunteer to read the slide.

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

Slide 4

Ask for a volunteer to read the slide.

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

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

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

Slide 5

What is the question asking?

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

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

4. The beginning and the end

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

The beginning

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

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

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

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

Image Source: https://pixabay.com/en/ice-ice-cream-milk-ice-cream-waffle-2367072/

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

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

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

The ending

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

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

Here’s a link to the video I included in this problem

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

Final thoughts

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

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

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