Moving On Before It’s Over (2nd Grade)

In this series of blog posts, I’ve been taking a look at each grade level’s scope and sequence for mathematics as I consider changes to make (or not) for next school year. So far I’ve written about Kindergarten and 1st grade. Today I’d like to tackle 2nd grade.

Here are our scope and sequences for the past three school years. What do you notice? What do you wonder?

2nd Grade – School Year 2015-16

2nd15-16

2nd Grade – School Year 2016-17

2nd16-17

2nd Grade – School Year 2017-18

2nd17-18

One thing that jumps out at me while analyzing the past three years is how different topics have bounced around throughout the school year.

  • 3-digit place value has shifted from the 2nd nine weeks to the 1st nine weeks to the 3rd nine weeks
  • Measurement has shifted from the 1st nine weeks to the 3rd nine weeks and back to the 1st nine weeks
  • Fractions was in the 4th nine weeks for two years and then shifted to the 3rd nine weeks
  • Multiplication and division were in the 3rd nine weeks and then they moved to the 4th nine weeks for the past two years

I might have come to these decisions on my own anyway, but I do feel like an important influencer in my work has been the Level 1 Curriculum Audit training I took in the fall of 2016. It really helped me think about all the different components of our curriculum and their purpose in supporting teachers in planning high quality instruction. One thing it really got me thinking about is being even more intentional about where topics appear in the curriculum, not just within one school year, but also looking across school years.

In my previous post about 1st grade, I talked about how the fall semester is focused on working within 20 while the 2nd semester introduces place value and numbers to 120. The purpose of spending the spring semester in 1st grade on place value is to create more proximity to 2nd grade where students are expected to use what they’ve learned about place value to start adding and subtracting 2-digit numbers.

I’ve divided 2nd grade in half in a similar way to 1st grade. The focus in the fall semester is building conceptual understanding of adding and subtracting 2-digit numbers. There are 80 days in the fall semester and half of them are devoted to this topic. Mastery is not expected by winter break, however. Check out the computational fluency column in the at-a-glance below to see how we start working toward procedural fluency with 2-digit addition and subtraction in the spring semester.

2ndAAGSpring

After unit 4, 2-digit addition and subtraction moves into our 10- to 15-minute daily computational fluency block for the remaining 97 days of the school year. I hear consistently from 3rd grade teachers that their students aren’t coming to them proficient with adding and subtracting 2-digit numbers, much less 3-digit numbers, so it’s my goal to ensure that our students leaving 2nd grade are solid on this.

So if the first half of the year focuses on adding and subtracting 2-digit numbers, what is the second half of the year focusing on? 3-digit numbers, namely introducing 3-digit place value and adding and subtracting 3-digit numbers. To beat a dead horse, I’m continuing to strive for sufficient instructional time for each and every one of our students. There’s no need to rush into 3-digit numbers in the fall semester as students are still trying to grapple with 2-digit number concepts.

I’m also trying to create a flow of addition and subtraction across 2nd and 3rd grade. Take a look at these addition and subtraction standards for both grade levels:

  • Second grade
    • 2.4B Add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations
    • 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

Notice how both grade levels are expected to add and subtract within 1,000. This is such important work that students are given two full years on it! I’ve noticed this is a theme across the primary grades of giving students ample time to engage with critical number concepts:

  • Kindergarten and 1st grade students spend two years getting to know the numbers to 20 really well through counting, representing, comparing, adding, and subtracting.
  • 1st grade and 2nd grade students spend two years getting to know 2-digit numbers really well through counting, representing, comparing, adding, and subtracting.
  • 2nd and 3rd grade students spend two years getting to know 3-digit numbers really well through counting, representing, comparing, adding, and subtracting.

It’s one thing to say that these topics spread across years, it’s another to ensure there’s some connective tissue to make sure it happens. That’s why I greatly appreciate our computational fluency and spiral review components of the math block. Here’s a look at the first semester at-a-glance. Look at the computational fluency topics. Why do you think I included what I did in those first several units? Then look at the spiral review topics. How are those intentionally placed in the timeline?

2ndAAGFall

The computational fluency block kicks off the year reviewing all of the basic fact strategies that were taught in 1st grade. We devote 10-15 minutes per day for the first 73 days of school to reviewing those strategies in order to build fluency of the basic addition and subtraction facts, but also because we want our students to “more than know their facts” (a wonderful phrase I learned from Pam Harris).

What I love about the Stepping Stones curriculum is that in 2nd grade it explicitly extends those basic fact strategies to addition and subtraction with 2- and 3-digit numbers. Not only does the computational fluency work in those early units reinforce fluency of basic facts, but it’s priming the pump to build on those strategies as students start adding and subtracting bigger numbers. I love that we’re modeling for students how powerful strategic thinking can be. We can use what we know about working with smaller numbers to help us work with larger numbers.

Looking at spiral review, I followed a similar structure to the beginning of first grade’s scope and sequence. If you look at the topics in spiral review, they are usually a review of 1st grade topics in preparation for learning the related 2nd grade concepts.

  • Unit 1 reviews 1st grade addition and subtraction standards in preparation for learning 2nd grade addition and subtraction standards in Unit 2
  • Unit 2 reviews 1st grade measurement (length and time) standards in preparation for learning 2nd grade measurement standards in Unit 2
  • Unit 4 reviews 1st grade geometry standards in preparation for learning 2nd grade geometry standards in Unit 5

Back when I started in this job, the feedback I got most from 2nd grade teachers had to do with either telling time or counting change. Based on that feedback, you’d think those are the two most important topics in 2nd grade. They’re not. I’ve worked hard over the past few years to convey what are and are not focal points via the scope and sequence.

You might have noticed that telling time appears in spiral review a lot. Learning to tell time is not always easy for students, but that doesn’t mean it should eat up a lot of instructional time. After focusing on it in Unit 3, we moved telling time to spiral review throughout the rest of the year as a reminder to keep reinforcing the skill.

We did something similar with counting change. The first thing I wanted to do was ensure our 2nd grade teachers understand they’re only slightly extending the work students did in 1st grade. Here are the two standards about counting change:

  • 1st grade
    • 1.4C use relationships to count by twos, fives, and tens to determine the value of a collection of pennies, nickels, and/or dimes
  • 2nd grade
    • 2.5A determine the value of a collection of coins up to one dollar

Pretty much the only difference between the two grade levels is that 2nd grade includes quarters. Throughout most of our 2nd grade curriculum, we review counting change in computational fluency as students practice skip counting by twos, fives, and tens. We finally bring quarters into the mix in Unit 9 as students learn about multiplication and division concepts.

2nd Grade – School Year 2018-19

For the most part I’m happy with this scope and sequence. However, there’s one thing that I’m curious about. I do wonder whether we should introduce 3-digit place value earlier in the school year. You might remember this is a topic that has bounced around our scope and sequence during the past few years. I still want to hold off on adding and subtracting with 3-digit numbers until later in the year, but I do wonder whether students have sufficient time to get to know 3-digit numbers before they have to add and subtract with them.

When I moved 2-digit place value to the spring semester in 1st grade, I didn’t worry as much because that’s all students have to do in 1st grade, place value. They don’t start adding and subtracting 2-digit numbers until 2nd grade. In this case, however, I’m squeezing place value, adding, and subtracting together into the spring semester of 2nd grade. I asked my 2nd grade curriculum collaborative, and they’re okay leaving 3-digit place value where it is for next school year, but I’m leaving it as an open question and something I’ll be keeping my eye on.

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 3rd grade’s scope and sequence in my next post.

 

 

 

Moving On Before It’s Over (1st Grade)

In my previous post in this series, I shared how our Kindergarten scope and sequence for mathematics has evolved over the past three years. Today I’d like to share our 1st grade scope and sequence.

Here are our scope and sequences for the past three school years. What do you notice? What do you wonder?

1st Grade – School Year 2015-16

1st15-16

1st Grade – School Year 2016-17

1st16-17

1st Grade – School Year 2017-18

1st17-18

It’s interesting to notice that the three units in the first nine weeks have remained fairly consistent with only some slight variations in number of days. We always start each year with a unit that looks back as it looks forward. The purpose of Unit 1 is to revisit number concepts introduced in Kindergarten while simultaneously introducing 1st grade data analysis concepts. Considering all the counting and comparing you can do while making and discussing picture and bar-type graphs, it’s a great fit. Even better, teachers and students tend to like making graphs at the beginning of the year as a “getting to know you” activity for the class.

One thing that’s been consistent across the years is that addition and subtraction are sprinkled throughout the school year. And by sprinkled I mean 5 units spread across the school year. In Kindergarten, students got to know the numbers through 20 really well as they counted, represented, and compared. In 1st grade, students get to know these numbers even better as they deepen their understanding of addition and subtraction.

It might seem like overkill to spend so much time on such a small span of numbers, but this work is rigorous for young children and there is a lot of ground to cover. No, really, here are all the critical learning phases students need abundant time to work through in Kindergarten and 1st grade (keeping in mind that they might need to pass through these phases more than once as the magnitude of numbers increases):

Understanding Counting

  • Counting Objects
    • Counts one item for each number
    • Keeps track of an unorganized pile
    • Notices when recounting a group results in a different number
    • Is bothered when counting a group results in a different number
    • Spontaneously checks by recounting to see if the result is the same
    • Knows “how many” after counting
    • Counts out a particular quantity
    • Reacts to estimate while counting
    • Spontaneously adjusts estimate while counting and makes a closer estimate
  • Knowing One More/One Less
    • Knows one more in sequences without counting
    • Knows one less in sequences without counting
    • Notices if counting pattern doesn’t make sense
    • Knows one more without counting when numbers are presented out of sequence
    • Knows one less without counting when numbers are presented out of sequence
  • Counting Objects by Groups
    • Counts by groups by moving the appropriate group of counters
    • Knows quantity stays the same when counted by different-sized groups
  • Using Symbols
    • Uses numerals to describe quantities

Understanding Number Relationships

  • Changing One Number to Another
    • Changes a number to a larger number by counting on or adding on a group
    • Changes a number to a smaller number by counting back or removing a group
  • Describing the Relationship Between Numbers
    • After changing one number to another, is aware of how many were added or taken aaway
    • Knows how many to add or take away from a number to make another number
  • Comparing Two Groups: Lined Up
    • Compares two groups that are lined up and determines which is more and which is less
    • When the groups are lined up, tells how many more or less, when the difference is 1 or 2
    • When the groups are lined up, tells how many more or less, when the difference is more than 2
  • Comparing Two Groups: Not Lined Up
    • Compares two groups that are not lined up and tells which is more and which is less
    • When the groups are not lined up, tells how many more or less, when the difference is 1 or 2
    • When the groups are not lined up, tells how many more or less, when the difference is more than 2
  • Using Symbols
    • Uses the greater than (>) and less than (<) symbols as a shortcut for the commonly used words (is more than, is less than) when comparing objects

Understanding Addition and Subtraction: Parts of Numbers

  • Identifying Parts of Numbers
    • Recognizes groups of numbers to 5 in a variety of configurations
    • Recognizes and describes parts contained in larger numbers
  • Combining Parts of Numbers
    • Recognizes and describes parts of numbers; counts to determine total
    • Knows the amount is not changed when a number is broken apart and recombined in various ways
    • Combines parts by using related combinations
  • Decomposing Numbers
    • Identifies missing parts by using related combinations
    • Knows missing parts of numbers to 10
  • Using Symbols
    • Uses equations to record combining and taking away parts
    • Interprets equations in terms of combining and taking away parts

Whew! Being a Kindergartner or 1st Grader is hard work!

You might be wondering how we spread out addition and subtraction across 5 units. I know some of our teachers have asked that same question! While we don’t follow a textbook verbatim, I do value the scope and sequence provided by our adopted resource, Stepping Stones by ORIGO Education. Here’s what we correlated from Stepping Stones with each of our addition and subtraction units:

Unit 2 – Introducing Count-On Addition Fact Strategies and Addition Properties

  • Stepping Stones, Module 2
    • Lesson 1: Identifying One More and One Less
    • Lesson 2: Counting in Steps of Two
    • Lesson 3: Counting On From Five
    • Lesson 4: Using a Number Track to Count On (to 15)
    • Lesson 5: Using the Count-On Strategy with Coins
    • Lesson 6: Using the Count-On Strategy
    • Lesson 7: Using the Commutative Property of Addition with Count-On Facts
    • Lesson 8: Using a Number Track to Count-On (to 20)

Unit 4 – Revisiting Subtraction Concepts and Introducing the Use Doubles Addition Fact Strategy

  • Stepping Stones, Module 4
    • Lesson 1: Reviewing Subtraction Language
    • Lesson 2: Using Subtraction Language
    • Lesson 3: Working with the Subtraction Symbol
    • Lesson 4: Writing Related Subtraction Sentences
    • Lesson 5: Working with Related Subtraction Sentences
    • Lesson 6: Solving Word Problems Involving Addition and Subtraction
    • Lesson 7: Writing Addition and Subtraction Number Sentence

Unit 7 – Introducing the Make Ten Addition Fact Strategy and Revisiting Equality

  • Stepping Stones, Module 7
    • Lesson 1: Exploring Combinations of Ten
    • Lesson 2: Using the Associative Property of Addition with Three Whole Numbers
    • Lesson 3: Introducing the Make-Ten Strategy for Addition
    • Lesson 4: Using the Make-Ten Strategy for Addition
    • Lesson 5: Using the Commutative Property of Addition with Make-Ten Facts
    • Lesson 6: Consolidating the Addition Strategies
    • Lesson 7: Applying Addition Strategies
    • Lesson 8: Adding Equal Groups
    • Lesson 9: Solving Addition Word Problems
  • Stepping Stones, Module 9
    • Lesson 1: Balancing Number Sentences (Two Addends)
    • Lesson 2: Balancing Number Sentences (More Than Two Addends)
    • Lesson 3: Working with Equality
    • Lesson 4: Representing Word Problems

Unit 8 – Relating Addition and Subtraction

  • Stepping Stones, Module 8
    • Lesson 1: Identifying Parts and Total
    • Lesson 2: Writing Related Addition and Subtraction Facts
    • Lesson 3: Writing Fact Families
    • Lesson 4: Introducing Unknown-Addend Subtraction
    • Lesson 5: Using Addition to Solve Subtraction Problems
    • Lesson 6: Working with Addition and Subtraction
    • Lesson 7: Counting On and Back to Subtract
    • Lesson 8: Decomposing a Number to Solve Subtraction Problems

Unit 10 – Applying Inequality and Comparison Subtraction to Measurement and Data

  • Stepping Stones, Module 8
    • Lesson 9: Solving Subtraction Word Problems
  • Stepping Stones, Module 9
    • Lesson 5: Working with Inequality
    • Lesson 6: Introducing Comparison Symbols
    • Lesson 7: Recording Results of Comparisons (with Symbols)
    • Lesson 8: Comparing Two-Digit Numbers (with Symbols)

Whether a teacher chooses to use any or all of these lessons in a given unit (along with other resources we provide) the chunking of topics is beneficial to help teachers plan out 5 unique, yet related, units of instruction rather than rehashing the exact same thing over and over again.

One major change that happened this school year was moving place value completely to the second semester. In the past we started teaching place value in the second nine weeks, but I feel like that sent a bit of a mixed message. Here I am saying that really getting to know numbers to 20 is critically important, but I was telling teachers to start teaching numbers to 99 after only a few months of school. What’s the rush? Learning unitizing and place value is important, but our standards don’t expect students to do anything with 2-digit numbers until 2nd grade.

So in effect, I split the 2017-18 school year in half. The first half of the year students get to focus on numbers to 20. As I said in my previous post in this series:

“One of my primary goals across each grade in grades K-5 is to ensure sufficient instructional time on core concepts for that grade level. I want students who need intervention later on to end up there because they truly aren’t understanding concepts, not because they weren’t given sufficient time to learn during first instruction.”

I can’t tell you how many times I’ve heard teachers tell me, “They’re in 5th grade, but they don’t even know combinations to 10!” This isn’t to say that teachers can’t differentiate throughout the school year by providing students opportunities to add or subtract beyond 20, but from an equity standpoint, we owe it to each and every one of our children to provide sufficient opportunity to grapple with and master grade level expectations.

The second half of the year allows students to continue learning about addition and subtraction within 20, but we introduce an additional focus of unitizing and place value in 4 different units across the second semester. Unitizing can be a challenging concept for young students, but it’s so important to so many concepts down the road. My hope is that holding off until after winter break allows those young minds a little longer to develop and be ready to tackle this important concept. I also hope that making it a focal point of the second half of 1st grade will create more continuity when students start 2nd grade in the fall where they start using place value concepts to add and subtract 2-digit numbers.

1st Grade – School Year 2018-19

Like Kindergarten, I’m pretty happy with our scope and sequence for 1st grade. I did ask my 1st grade curriculum collaborative if they were comfortable leaving place value only in the spring, and they had no complaints.

I’m still trying to decide what to do about spiral review for next year. I don’t want to dictate, but I know it can be helpful to have guidance about which topics to review throughout the school year.

1stAAGFall

1stAAGSpring

One thing you’ll see in 1st grade spiral review is something I’m also doing in grades 2-5, which is reviewing a concept from the previous grade level right before that concept comes up in the current grade level. For example:

  • Unit 1 spiral review is Kindergarten addition and subtraction concepts right before Unit 2 introduces 1st grade addition and subtraction concepts
  • Unit 4 spiral review is Kindergarten geometry concepts right before Unit 5 introduces 1st grade geometry concepts

I did this intentionally because a common complaint I hear from teachers is that students aren’t ready for instruction in the current grade level standards for whatever unit they happen to be in. The (non)-issue is that kids forget things. It’s natural. When learning ends, forgetting begins.

What we need to do is re-frame this experience. It’s not a fault of the children or of a teacher. Rather, it’s a normal human phenomenon. With the spiral review planned the way it is, teachers now have time to jog memories and re-solidify understandings of last year’s content before students are expected to tackle this year’s content.

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 2nd grade’s scope and sequence in my next post.

 

 

Our Venn Diagrams are One Circle

This past week my work life and my daughter’s school life came crashing together in the most wonderful way.

I.

On the way home from school on Thursday, she asked if we could practice “take away.” At first we practiced numerical problems like “What is 3 take away 1?” and “What is 5 take away 2?” Eventually I asked her if I could tell my problems in a story. The rest of the ride home we told “take away” stories. I told a few, and then she wanted it to be her turn:

  • “This one is sad. There were 2 cats and 1 of them died.”
  • “There were 6 oranges on the counter. A girl ate 2 of them and they died in her mouth.”
  • “There were 8 trees, and 3 of them got cut down.”
  • “There were 6 roads, and 2 of them fell down.” (I was able to figure out she was referring to overpasses because that’s what we were driving under at the time.)

Slightly morbid, but she’s 6 years old, so I roll with it, especially since she isn’t usually this chatty about anything related to school.

Anyway, as we were getting closer to home, I remembered that the math unit she’s currently in in school uses some numberless word problems, so I asked, “Have you ever had a problem about some geese and some of them stop to rest?”

(Stunned silence)

“How did you know that?!”

“What about a problem about a boy who checks out some books from the library and returns only some of them?”

(Stunned silence)

“Yes! How did you know that one!”

“Because I wrote them.”

“What do you mean?!”

“I’m the author of the take away stories you’ve been working on in math class.”

And thus our two worlds – my work and her school – came crashing together for the first time ever.

I’ve mentioned to her before that I work with and help teachers, but it’s always been in the abstract. Finding out that I was the author of specific problems she’s encountered in her classroom just blew her mind. She wanted to see some of them when she got home. Knowing she probably won’t always be this interested in my work, I was only too happy to oblige.

II.

As I was scrolling through the suggested unit plan to find the numberless word problems, I asked her about other tasks in the unit to see which ones she remembered. I asked about Bag-O-Chips, a 3 Act Task from Graham Fletcher, which was planned for the day after the numberless word problems, but she said she’d never seen it before. I have no idea how closely her teacher follows the unit plan, but lo and behold, the next day in the car when I asked what she did at school she said, “We did the bags of chips!”

We talked a little bit about the task in the car, and a little later as we finished up dinner I showed her the Act 1 video. Her eyes lit up. “That’s the video!”

We kept going back and forth between the image of what came in the bag and the image of what should have come in the bag. She happily used her fingers to figure out how many missing bags there were of each flavor.

I thoroughly enjoyed talking through the task with her, and what a pleasant surprise when she wanted to do another.

III.

I’m not one to pass up an opportunity talk about math with my daughter, so I quickly scanned Graham’s list of 3 Act Tasks to find one I know we didn’t include in our suggested unit plans. I settled on Peas in a Pod.

Peas01

Source: https://gfletchy.com/peas-in-a-pod/

First, we watched the video and estimated how many peas would be in each of the pods.

“I think there are 3 in this one, 4 in this one, and 10 in this one. No, 13 in this one.” (She estimated from right to left in case you’re wondering.)

“Hmm,” I said, “I think 3 is a good guess for the first one. I think there might be 4 or 5 in the second one, and I’m going to agree with your first guess of 10 for the third one.”

Estimation is a new skill for Kindergarten students. I talk about guessing and she talks about being right. She thinks the goal is to be the person who guesses the correct (exact) amount. I’m going to keep talking about being close and reasonable because over time I know her understanding of what estimation is will develop and refine.

Then we watched the reveal video.

Peas02

Source: https://gfletchy.com/peas-in-a-pod/

“I wasn’t right and you weren’t right!” She exclaimed.

“That’s okay. All of our guesses were pretty close, even though none of them matched the exact number of peas. I was surprised that this one only had 2 peas in it. I thought for sure there were more in there.”

“Me, too.”

“Hmm, I have another question for you. How many peas are there altogether?”

“Let me count.”

“I want to see if you can do it without counting on the picture. How many peas were in each pod?”

“8 and 7…and 2.”

“So how could you figure out the total?”

At first she tried using her fingers. She counted out 8 fingers, and then continued counting from there. I couldn’t really tell what she was doing, but at one point, after lots of ups and downs of fingers, she said, “18.”

Pretty close!

I didn’t say that though. Instead I said, “Hmm, I wonder if that’s the right amount. What other tool could we use to check your answer?”

She decided to get her Math Rack to check, and as a complete surprise to me she said, “Can you make a video of me?” Make a video of you solving a math problem? Why, of course!

Watching her first attempt, it was fascinating seeing her trying to keep track of two separate counts: (1) counting on from 8, “…9, 10, 11, 12, 13, 14,…” and (2) counting the 7 she was combining with the 8, “1, 2, 3, 4, 5,…”

It seems like she abandoned the double counting  when she was so close to being done. I wonder if she sort of gave up and just continued counting to 18 since that’s what she had thought the answer was before.

I had a split second to think about how to respond. I didn’t want to confirm whether the answer was correct, and I wanted to see if she would be willing to try combining the three quantities again.

There was definitely a lot more accuracy when she separately modeled each quantity! I was impressed with the double counting she was attempting earlier, but in the end she was more successful when she could show each quantity separately and then count all.

It was a proud dad moment when she didn’t just accept 17 as the correct answer. She decided we should look at the picture of all the open pea pods to check. And, sure enough, when I held up the phone with the image of all the open pea pods, she was able to count all and verify that there were in fact 17 peas.

All in all, I’m over the moon. All year long I’ve asked her about school (and math), but up until now her answers have been fairly vague. (“I’m so surprised,” said no parent ever.) The most I’d gotten out of her before was that they did Counting Collections.

But now we’ve actually had a full blown conversation about the work she’s been doing in school, specifically activities I wrote or helped plan for our Kindergarten units. I’ve always loved talking about counting and shapes and patterns with my daughter since before she ever started school, but to have our worlds collide like this was really special. I enjoyed getting to share and talk about my work with a very different, and more personal, audience than I’m used to.

 

Moving On Before It’s Over (Kindergarten)

This school year isn’t even over yet, but in my role as a Curriculum Coordinator, I’m already starting to look ahead to next school year. I feel like I’m cheating on the current school year, but if I don’t start now, there’s no way I’ll have everything ready when the teachers come back in August.

One of my responsibilities every spring is to analyze our instructional units to determine whether any changes need to be made for the upcoming school year. Over the past several years, I’ve made some pretty drastic changes to our scope and sequence, but each year I feel like it’s been less and less and that we’re settling on a coherent plan that works for our teachers and students.

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 – I thought I’d share our scope and sequences to give you a sense of what kinds of changes we’ve made over time and what we’re planning for next year. I have no idea whether this will be useful to anyone, but if I don’t share then I’ll never know.

Let’s start with Kindergarten!

Here are our scope and sequences of units for the past three school years. What do you notice? What do you wonder?

Kindergarten – School Year 2015-16

K15-16

Kindergarten – School Year 2016-17

K16-17

Kindergarten – School Year 2017-18

K17-18

Let me explain some of the big changes that have happened over the past few years as well as the rationale behind our scope and sequence.

Kindergarten starts with introducing students to the numbers through 5 and then the numbers through 10. This has been fairly stable over the past few years. At this early part of the year, the focus is on counting, counting, counting and representing, representing, representing. Students come to us with a wide range of abilities. We can’t presume their understanding so we want to ensure everyone has a solid foundation in the first month or so of the school year.

You’ll notice over the past few years that unit 3 on sorting and classifying jumped up from 11 days to 15 days to 25 days. Sorting and classifying are huge verbs in mathematics, and we wanted students to start engaging with them right away via our data and geometry standards. The jump in days came because the unit used to only include 3D figures. We used to introduce 2D figures later in the school year. Now this unit includes both 3D and 2D figures.

We circle back around to numbers to 10 in unit 4. Students continue to count, count, count and represent, represent, represent, but they also start comparing in this unit. This is followed by our measurement unit which extends the concept of comparison as students talk about things being longer or shorter, heavier or lighter, and more full or less full.

During the 2017-18 school year we made it so our addition and subtraction units are back to back, followed by our unit on numbers to 20. This is because the old scope and sequence confused teachers. For the first half of the year students engage with numbers to 10. After winter break, students used to work in a unit where they engaged with numbers to 20, only to encounter a subtraction unit afterward that suddenly said to only focus on numbers to 10 again. Teachers were baffled by this. If students were learning about numbers to 20, then why weren’t they subtracting with numbers to 20 in the next unit? The answer is because our standards explicitly state to add and subtract within 10.

We opted to remove the confusion by putting both the addition and subtraction units before the unit on numbers to 20. That way it maintains a flow of working within 10: They learn to count and represent numbers to 10, compare numbers to 10, and then add/subtract numbers to 10 (in contexts). Finally we extend to numbers to 20. Our unit on numbers to 20 is a long one because it takes the concepts of counting, representing, and comparing and puts them together all in one unit.

The year closes out with two units. The first is our personal financial literacy unit, which introduces skills such as identifying coins by name, identifying ways to earn income, differentiating money received as income vs gifts, listing simple skills required for jobs, and distinguishing between wants and needs.

The second unit to close out the year is our addition and subtraction unit that brings the operations together to give students an opportunity to start having to identify which operation is needed in a given situation. The earlier units focused on working through the language stages of addition and subtraction separately to help students connect those operations to the actions of joining and separating (as per our standards), but at the end of the year we want students to have the opportunity to problem solve and make decisions about whether a given situation involves joining or separating.

These last two units used to be in reverse order, but after some feedback from teachers I changed it for the 2017-18 school year. Basically we ran into an issue where teachers couldn’t give grades on the report card regarding the financial literacy standards because grades were due before they completed that unit. Since addition and subtraction were already introduced earlier in the school year, I moved that to become the final unit so that teachers could teach the entire financial literacy unit before they have to submit report cards.

Kindergarten – School Year 2018-19

I’m pretty happy with the Kindergarten scope and sequence from this school year. I’m going to meet with my Kindergarten curriculum collaborative in a month or so to see if they agree, but I’m not anticipating making any changes for next school year.

You’ll notice that our scope and sequence spends a TON of time on numbers to 10 because that is the focus of our Kindergarten standards. Students do extend these understandings as they work with numbers to 20, but numbers to 20 is actually the focus of the 1st grade standards. You’ll see what I mean in my next post on 1st grade.

One of my primary goals across each grade in grades K-5 is to ensure sufficient instructional time on core concepts for that grade level. I want students who need intervention later on to end up there because they truly aren’t understanding concepts, not because they weren’t given sufficient time to learn during first instruction.

One thing I am trying to decide about for next year is whether I’ll specify spiral review topics throughout the year. Here’s our at-a-glance so you can see how each unit is broken down into three instructional goals – focus TEKS (standards), computational fluency, and spiral review.

KAAGFall

KAAGSpring

In Kindergarten we don’t have spiral review in the fall semester because the math block is only 60 minutes – 50 minutes for core lesson and 10 minutes for computational fluency. In the spring semester we add in 20 minutes of daily spiral review to bring up our math block to 80 minutes daily.

I suggest topics to review during spiral review to help teachers out, but I am afraid that this creates a confusing message. I wholeheartedly want teachers to review the concepts their students need to review. For example, if a teacher knows some students are struggling comparing numbers to 10 in unit 8, then by all means, review that concept rather than sorting and classifying with 2-D and 3-D figures.

The only reason I list topics is to give some guidance to help teachers ensure that topics are coming up again throughout the year. I know from firsthand experience as a classroom teacher that I was often working at the day-to-day or, if I was extremely lucky, the week-to-week level. Now that I’m in a position that allows me to look at the level of the entire year, I try to provide as much guidance as possible for teachers to help them navigate the school year.

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 1st grade’s scope and sequence in my next post.

 

 

Rethinking Test Prep

I don’t know about you, but here in Texas we’ve got a state math test in grades 3, 4, and 5 coming up soon. The 5th grade test is taking place in mid-April followed by the 3rd and 4th grade tests in mid-May. In my school district, we used to stop instruction for one to two weeks prior to the test to focus on review. It’s always rubbed me the wrong way, and this year we changed that. If you want to read more about our rationale for doing that, I recommend reading Playing the Long Game, a post I wrote on my district blog. I also recommend checking out my Ignite talk from NCSM 2017. The work I’m sharing here has been a chance for me to put into practice the principles I shared in that talk.

If you don’t have time for all that right now and you’d rather check out the review activities I’ve created and get access to them for yourself, read on!

This year, with the help of our district instructional coaches, I put together collections of 15-20 minute spiral review activities that can be used daily for a month or so before the state test to review critical standards and prepare students without interrupting the momentum of regular math instruction. Here they are:

(Note: If you want to modify an activity, you are free to do so. Either make a copy of the file in your Google drive or download a copy to your computer. You will have full editing rights of your copy.)

When you look at an activity, it might look short. You might ask yourself, “How could this possibly take 15-20 minutes?” Good question! These activities are designed for student discourse. Students can and should be talking regularly during these activities. The goal is for students to be noticing, wondering, questioning, analyzing, sharing, and convincing  each other out loud. These discussions create opportunities to revisit concepts, clear up misconceptions, and raise awareness of the idiosyncrasies of the test questions, especially with regards to language.

Most of the activities are low or no prep, though here and there a few activities need some pages printed ahead of time. Be sure to read through an activity before facilitating it in your class so you don’t catch yourself unprepared.

Each collection of activities is organized around the Texas state standards (also known as TEKS). If you don’t live in Texas, you still might find these activities useful since there’s so much overlap between our standards and others. To help non-Texans navigate, I’ve added a column that (very) briefly describes the concept associated with each activity. If you’re interested in reading the actual TEKS each activity is aligned to, check out these documents:

If you try any of these activities out with your students, let me know how it goes in the comments. Enjoy!

What We Presume

I once heard an analogy that teaching is a lot like being a doctor…if the doctor had to diagnose and treat 25 patients all at the same time. It’s cute and helps drive home the point that the work of teachers is complex as they tackle the daily challenges of meeting the needs of many students simultaneously. However, this analogy hits too close to home as it reflects a shift in the profession I’ve been noticing over the past few years. The role of a teacher really has become more like being a doctor, and that bothers me.

These days, education is driven by capital D Data. Data, Data, Data. And why? Because like a doctor, we want to diagnose what’s wrong and help fix it.

And that’s where the problem lies. We presume illness.

This post from Tracy Zager exemplifies my concern. In the post, she recounts the diagnostic test her daughters each had to take on the very first day of 2nd and 4th grade.

Welcome to the new school year!

Unfortunately, nowadays teachers feel pressured to collect as much Data as possible as soon as possible so they can diagnose the illness and begin treatment right away. Does that really need to be our focus on day one? Or even day 2, 3, 4, or 5? As Tracy says in her post,

“On day one, I really don’t care if my students know the vocabulary word for a five-sided polygon, can tell time to the half hour, and can calculate perimeter accurately. I’d much rather know how they attack a worthy problem, how they work with one another, and how they feel about the subject of mathematics. I am much more interested in the mathematical practice standards than the content standards in the fall.”

The concern Tracy shares dovetails with the message Ken Williams gave in his keynote back in July at CAMT 2017. The overall talk was about disrupting the status quo with regards to labeling and limiting students. This message jumped out at me during his talk:

And yet this is exactly the kind of experience Tracy shared in her blog post! Ken Williams challenges this practice and the limits it places on our students:

When we presume there’s an illness – a problem with a student or group of students – then we’re setting our expectations about what we’re going to find. If we train ourselves to seek out faults and deficiencies, then that’s what we’re going to get good at finding.

Here’s what I’d love for us to presume instead. To quote Andrew Gael, let’s presume competence. Presume that when our kids walk in the door on the first day of school, they have funds of knowledge to draw on and the ability to learn even more. As we get to know our students, we’ll observe variation – it’s natural – and once we’re aware of what those variations are for individual students we can start brainstorming ways to accommodate to ensure each and every student can continue to have access to the learning in our classrooms.

When we presume competence, we aren’t looking for illness, we’re looking for strength. We’re sending important messages to our students from day one that we value who they are and who they can become as they journey with us through the school year.

 

 

Doing Math with #ElemMathChat

Last night we kicked off the fourth year of #ElemMathChat. Yay! It’s so exciting to spend an hour each week talking with and learning from so many passionate educators.

Birthday-Ad-2

One thing I’ve often heard from participants is that they like that we regularly do math together during our chats. I didn’t want to disappoint in our first chat of the year, so I dropped in a few tasks. I thought I’d collect them together in a blog post in case anyone missed the chat or wants all the pictures gathered together in one place. So let’s get started!

How Many?

This task actually appeared before the chat. I’ll admit that I sometimes try to cram a bit too much into our hour together – I want to do it all! – so I opted to move one of the questions out of the chat and instead turn it into something fun for folks to play around with during the day leading up to the chat.

How-Many

I saw two common answers to this question throughout the day:

  1. I assume you mean triangles. I see 4.
  2. How many what?

I owe Christopher Danielson thanks for turning me on to this deceptively simple question as well as for engaging with some of the folks yesterday who were tackling the question as it relates to this image.

I highly recommend checking out Christopher’s blog post where he talks more about this question and shares some images you just might want to use in your classroom. He only asks that you let him now what kids do with those images and ideas. You can share with Christopher on Twitter @Trianglemancsd.

Let’s Estimate!

For our first task during #ElemMathChat, I asked everyone to estimate the number of hats in this sculpture:

When I first saw this sculpture at the Fort Worth Convention Center at this year’s CAMT Conference, I was instantly curious how many hats were used to make it. It took some digging, but I finally came up with all the information I needed.

I asked participants to share their too LOW, too HIGH, and just right estimates. What I’m really looking for is the range they’re comfortable with. How risky are they willing to be with their estimates?

  • This is a low-risk estimate: “My too low estimate is 10. My too high estimate is 5,000. My just right estimate is 500.”
  • This is a riskier estimate: “My too low estimate is 400 and my too high estimate is 500. I’m pretty sure the number is somewhere in the 400s.”

Notice the difference? One person isn’t as comfortable limiting the range of their estimates while the other person has narrowed it down to “somewhere in the 400s.” I don’t really care about the just right estimate so much because I value helping students come up with estimates that make sense and are generally close rather than valuing whether or not they guessed the exact number. Helping students get better at estimating and be willing to make riskier estimates takes time and practice, but it’s valuable work.

Here’s the final reveal with some additional information about the sculpture, in case you want to do this activity with your students:

P06

Numberless Graph

As much as I love numberless word problems, I’ve been fascinated with numberless graphs this past year. I knew I wanted to include one in our chat! When I shared this first image, I asked my go-to questions, “What do you notice? What do you wonder?”

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The engagement was high and it was so much fun to see what people noticed and wondered as they looked at the graph.

NG01

NG02

We moved on to another question before coming back for the second reveal. Again, I asked, “What do you notice? What do you wonder?”

P09

Adding the scale and currency amounts just increased the wonderings about what this graph could be about.

NG03

Finally, after building anticipation and making everyone wait through another chat question, I finally revealed the full graph and asked, “What questions could you ask about this graph?”

 

 

 

The noticing and wondering didn’t stop! It was great!

NG04

In case you’re wondering, Pokémon GO is a game you can download on mobile devices. The game is free, but there are things you can buy within the game. So what this graph is showing is the average amount people spent buying things inside of the game. In Japan, for example, looking at all the people in the country who have downloaded the game, each of those people has spent $26 on average. In the US, on the other hand, the people who have downloaded the game have each spent $7.70 on average. The interesting thing about this is that the data is a bit misleading if you don’t know more details:

This leads to a great discussion to have with kids, “If US players aren’t spending nearly as much in the game as players in Japan, then how come the total amount earned from purchases in the US is over $100 million more than in Japan?”

A Lens Looking Forward

This isn’t doing math together, but I did want to share the final question of the night.

P13

My lens for a long time has been play, but I think I’m due for a new one. Not sure what it’s going to be yet. What about you? What word would you choose to use as a lens for the work (and fun!) ahead this school year?