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.

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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:

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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.

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

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

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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!

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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.

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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?

 

 

 

Take It Away – CGI National Conference 2017

At the end of June, I attended (and presented at!) my first CGI National Conference. I also visited the Pacific Northwest for the first time in my life. Seattle was beautiful and the learning was great. I know there are folks out there who aren’t able to attend many conferences, so hearing from attendees is one way they learn from afar. So, in case you weren’t there, let me tell you what resonated with me from the conference.

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One thing I especially liked about the conference was the essential questions. Speakers weren’t required to connect with them directly. Rather they were designed for participants to personally consider and reconsider as they attended keynotes and sessions:

  1. In what ways are your students allowed to bring “their whole selves” to the learning of mathematics in your classroom and school?
  2. What do you know about the cultural and lived experiences of the students in your mathematics classroom? (How can you broaden your knowledge?)
  3. How does your mathematics classroom interrupt and/or reinforce narratives of who is and who is not capable mathematically? (How could your classroom become more interruptive vs. reinforcing of these narratives?)

Not what you’d normally expect at a math conference, right? The focus on culturally responsive pedagogy was a breath of fresh air.

I also appreciated the emphasis on making connections – both in person and virtually.  A special thanks to Tracy Zager for giving folks a nudge as well as support. There were quite a few #MTBoS members in the audience, and I hope by the end of the conference that number increased.

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The Opening Keynote was a panel discussion called “Talking Math With Kids.” The panel included Christopher Danielson who blogs at the aptly named talkingmathwithkids.com; Allison Hintz and Tony Smith from the University of Washington; and Megan Franke, Angela Turrou, and Nick Johnson from UCLA. They told stories of their experiences working with young children around mathematics. The (extremely important) theme of their talk is that young children have mathematical ideas. We should listen to, value, and encourage them.

Then we moved into our first of six sessions. I happened to present during the first session. It was a little stressful, especially since the projector was not cooperating at first, but I was happy to get it out of the way right up front. 🙂 My talk was called “Numberless Word Problems in the Elementary Grades.”

In the talk we solved a numberless word problem together to create a shared experience. Then I shared the story of Jessica Cheyney using numberless word problems in her classroom to help students connect the act of separating to the concept of subtraction. Next I shared the story of Casey Koester, an instructional coach who used intentional planning and numberless word problems to help 2nd grade students make better sense of word problems. I closed by sharing resources teachers can use to implement numberless word problems in their classrooms.

Since we started in the afternoon, the opening keynote and session #1 were all we did on day 1. Day 2 opened with another keynote called “Equal Math Partners: Families, Communities, and Schools.” The keynote included Erin Turner, Julie Aguirre, and Corey Drake from the TEACH Math Project; and Carolee Hurtado from the UCLA Parent Project.

I loved this keynote! We often talk about what teachers and students are doing in schools and gloss over or ignore the role parents can and should take in their children’s mathematical development. We also ignore the role that students’ family, community, and culture play in their learning of mathematics. The two projects shared in this keynote were inspiring to listen to and so important for us to hear.

The first story was about the UCLA Parent Project, a multi-year project that invites parents in to become partners in their children’s math learning. It also builds up the parents into leaders.

The second project was the TEACH Math Project. Pre-service teachers were required  to take a community walk to interview people and learn more about the community in which their students lived. We often ask teachers to create tasks and problems based around student interests, but this often leads to generic problems around what we assume the students’ interests are. In this project the pre-service teachers had to get to know their students, their lives, and their interests for real. Then they had to use what they learned to create relevant tasks and problems. I loved it.

After the keynote we attend session #2. I went to Megan Franke’s “No More Mastery: Leveraging Partial Understanding.” This resonated so much with me because it matches my current thinking about how we should be analyzing and interpreting student work.

According to Megan Franke, mastery learning “breaks subject matter and learning content into clearly specified objectives which are pursued until they are achieved. Learners work through each block of content in a series of sequential steps.” The trouble with mastery learning, however, is that actual learning isn’t that clean. Further, it sorts students into two groups – those who’ve got it and those who don’t – which contributes to inequality.

A partial understanding approach, on the other hand, looks at understanding as something we can have varying amounts of. What’s important is finding out what students’ current understanding and capabilities are and build from there. Megan shared an example of a preschool counting task where students had to count 31 pennies. According to the mastery approach – they either counted to 31 correctly or they didn’t – only 2.5% of the students demonstrated mastery of counting. However, when they scored the students on a range of numeracy criteria – knowledge of the counting sequence, 1-to-1 correspondence, cardinality, counting the whole collection, and organization – the picture changed completely. Only 12% of the students demonstrated little to no number knowledge while 64% of them demonstrated understanding of multiple criteria.

For session #3 I got to attend Christopher Danielson’s “The Power of Multiple Right Answers: Ambiguity in Math Class.”

I especially love the power of the phrase, “Well, it depends…” and hope to help teachers in my district see the power in crafting questions and tasks that lend themselves to some ambiguity. I also love this thought by Allison Hintz retweeted by Christine Newell:

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If you haven’t seen Christine Newell’s Ignite Talk from NCSM 2017, “Precision Over Perfection,” check it out because it touches on this very idea.

During session #4 I went to lunch, and I’m going to skip talking about session #5 because it didn’t really resonate or push my thinking very much.

Session #6 was fantastic though! I saw Jennifer Kolb and Jennifer Lawyer’s talk “The Importance of Counting in Grades 4 & 5 to Support Complex Ideas in Mathematics.” I noticed that counting in general and counting collections specifically appeared across the conference program. I have made the counting collections routine a mainstay in my primary grade curriculum materials. I was especially intrigued to hear stories of how intermediate grade teachers are using the routine. The two Jennifers did not disappoint!

In the example above, counting groups and then groups of groups helped nudge these 5th grade students into an understanding of the Associative Property of Multiplication.

This same idea of “groups of groups” led students to explore groups of 10 in a way that led to deeper understandings of place value and helped introduce exponents:

Counting is a skill we naively think students “master” in the early grades, but taking a partial understanding perspective, we can open up the concept to see that there’s so much more to learn from counting in later elementary grades and beyond!

On day 3 of the conference we opened with another enlightening keynote “Anticipatory Thinking: Supporting Students’ Understanding of How Subtraction Works.” This keynote was led by Linda Levi from the Teachers Development Group and Virginia Bastable from Mount Holyoke College.

Linda Levi’s portion of the talk reflected on the meaning of computational fluency. She reminded us that while many people think of fluent as being fast, the definition is much broader and more nuanced than that.

“Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands.” (Principles and Standards for School Mathematics, 2000, p. 152)

We started with a video example of a student solving 5,000 – 4,998 using the standard algorithm. Is this an example of computational fluency? According to the above definition, no, it’s not. Producing an accurate answer like a calculator is not the same as demonstrating computational fluency. In this example the student did not demonstrate flexibility in the methods he chose, he didn’t understand and couldn’t explain his method, and his method is not based on mathematical ideas that the student understands.

We then watched videos of two other students who used subtraction strategies they invented. Were these students demonstrating computational fluency? The students clearly understood their strategies and they were based on mathematical ideas the students understood. However, we then watched these same students solve another problem and realized that these students were not flexible in their thinking. They used the same strategies for subtracting even though other strategies would have been more efficient for the new problem. It’s really important to remember how multi-faceted computational fluency is and attend to all facets as we work with students.

One of Linda Levi’s main messages was that understanding how operations work is the foundation for computational fluency. She shared with us how we can use equations that represent students’ strategies as objects of reflection for discussing why a strategy works and to help make explicit important mathematical ideas.

Virginia Bastable followed up with a talk about mathematical argument which was along the same theme of helping students understand how the operations work.

One thing that resonated with me from her talk was the important work of opening up mathematics learning beyond the narrow focus of answer getting. Rather, mathematics is a landscape that also involves sense making, exploring, wondering, and even arguing.

After the keynote I attended Kendra Lomax’s session “Learning from Children’s Thinking: A CGI Approach to Formative Assessment.” This session dovetailed nicely with Megan Franke’s session on partial understandings because the whole point of the CGI assessment is to get a sense of where the child is at in a variety of ways rather than a binary “yes, they have it” or “no, they don’t.”

If you’re interested in this assessment approach, then I have good news for you! A slew of assessment resources are available at Kendra’s website, Learning From Children. Look at the resources under “Listening to Children’s Thinking” in the menu at the top of the page.

For my final two sessions I went to hear more from Linda Levi and Virginia Bastable. Linda’s talk “Understanding is Essential in Developing Computational Fluency” gave us practice recording student strategies using equations as a way to make explicit the properties and big ideas embedded within the strategies.

Virginia’s talk “Support Math Reasoning by Linking Arithmetic to Algebra” dove more deeply into the role mathematical argument can play in helping students develop a deeper understanding of the operations. When I think back to the skill-based worksheets of my youth, I’m jealous of the deep thinking elementary students are given the opportunity to do in classrooms today.

We came back together for a closing session and that was the end of the conference. Spending three days with like-minded educators who care so deeply about mathematics education and nurturing children’s mathematical ideas helped recharge my batteries before coming back to work for the 2017-18 school year. It will be another two years before the next CGI conference – this time in Minneapolis – and I can’t wait to attend!