counting collections

The “Low Kids”

| Janaki Nagarajan

Disclaimer: this post refers to “low kids” and “high kids” as ways students are often talked about in schools. I hope as you read it, it’s clear that I am using these terms to describe how students are consistently categorized in narrow and dehumanizing ways, and that we as educators need to stop doing this – not only because it’s damaging but because it’s inaccurate.

I grew up thinking that intelligence, particularly in math, was both linear and elusive. On-demand fact recitation, no paper, fast and clever. Learn one thing first, and then the next. Your intelligence is reflected in above-average standardized test scores. That’s how you know you’re good at math. (By these standards, I was decidedly “mid”.)

Something about this fixed idea of smartness has never quite sat right with me, and I’ve always been drawn to ideas rather than performance – but truthfully, educators almost exclusively use these methods to categorize student achievement (the “whys” of that are complex and would take ages to unravel – but suffice it to say, this is how it goes). By now, I’m used to it, but it still makes my blood boil. Someone’s iReady score (sub in your standardized test) is the least interesting thing about them – even if the conversation is specifically about that student’s understanding of mathematics (or any subject).

Before school started, I did my customary go-around to my new students’ former teachers, excitedly sharing with them which kids from their class I get to work with this year.

“She’s sweet, but she’s really low,” said one student’s teacher from last year, without pause. (Blood boiling but biting my tongue – trying to tell myself it’s not this teacher’s fault, she’s part of a system that encourages this kind of categorization).

But what do you see in this photograph? 

Three days into the school year, these two students (who were labeled as essentially unable to understand grade-level mathematics) were tasked with counting a collection of objects. Every other group started counting by ones. These two decided on a “big rectangle”. Their instinct was to organize, prior to counting. A sense of how structure and patterns can be utilized to help us understand mathematics seems vital. Their low iReady scores do not reveal their aptitude to think this way. (Surprise?)

This year, as a math intervention, my grade level was told to create ability groups to meet with during small group time. While this is not my ideal teaching strategy, I began to think how these groups could serve as a humanizing and empowering experience for these students – capitalizing on their strengths, and nudging them to explore new grade-level content. Getting to work explicitly with students who may struggle to feel successful in mathematics – and giving them new ways to engage with it and show what they know is actually exciting.

One thing I discovered was that using Illustrative Mathematics’ resources in small group was a way to engage these students in mathematical inquiry and problem-based learning (because the absolute last thing we should be doing is gatekeeping that sort of thing for the “high kids”).

These two “low kids”, using an IM card sort, immediately began making connections between representations, situations and expressions. They worked together, collaboratively, bouncing ideas off of each other and defending their thinking. They were able to show that they understood the multiplication ideas they’d only been working on for a week.

Another opportunity we’ve had in our classroom this year is to engage in mathematical modeling. In the below example, students used a consensus board to attempt to figure out how many days our box of Ritz crackers would last our class for snack. They had to use information they knew (6 stacks in a box, 32 crackers in a stack), make some decisions (how many crackers should students get per day? How many students will eat crackers each day?) to attempt to find a solution.

The student whose work is represented on the bottom is one who was labeled “low” by other teachers at the school. What do you see? What I see is that she diligently represented each cracker in each stack to show exactly how many crackers we were working with. I see she made a decision about how many crackers each student would get per day (4). I see she got stuck thinking about how to incorporate the number of people eating crackers each day (most students did) and would benefit from a nudge to consider how she might show the crackers eaten by the class in one day.

To be clear – I think her work is phenomenal – equally as fascinating as the others in her group, who had their unique strategies and models, and their own places they got stuck.

And let’s talk about math quizzes.

My feelings about quizzes are mixed. Long opposed to most traditional assessments, this year what I’ve found is when used strategically, they can be a really interesting tool to uncover students’ understanding of the mathematics they’ve been learning. Some students might choose to complete their quiz independently – while I might ask others if they can show me their thinking about a question in a different way (especially if they’re struggling to represent their ideas on paper, and are hesitant to grab manipulatives).

What do you think this “low” student is doing?

He correctly represented his thinking on paper – he counted his grapes by ones and slightly miscounted – but when prompted to show his thinking with cubes, immediately realized he could count by 10s, physically moving the cubes to show what had and hadn’t been counted.

He then used his cubes to immediately solve the next question (which was more complex, and involved ideas such as measurement division that we hadn’t explicitly talked about at this point).

Another “low” student, when working one on one with him building and counting equal groups, would join his cubes together to make his groups, and then join the groups into one long unifix cube tower and count each cube. He meticulously recorded his answers to multiplication equations as he worked through the questions with only slight nudges from me.

If these students understand the mathematical concepts being discussed in class just as much as anyone else, why are they even labeled the “low kids”?

When a computer assessment is your primary way of determining someone’s aptitude, you’re going to be left with the impression that some in your class “can” and some “can’t”. And what about when a significant amount of your students who are scoring low are designated as multilingual learners? They are literally processing language differently than students whose primary method of communication since birth has been English. (Similar for students with learning disabilities).

As a result, they may struggle with reading English at the level that grade-level assessments assume competency in – be it with decoding, comprehension, and/or vocabulary – and when the majority of math tasks students encounter have a high cognitive load up front (read it, understand it, solve it), the barriers are built right in.

Reading comprehension is connected to mathematics only insofar as word problems are the primary source of determining mathematical understanding. Without multiple ways for students to access mathematics content, and multiple ways for teachers to assess it, certain students will always seem “low” while others seem “high”.

Beyond reading, the pedagogical structures of many math curricula (and if we’re really getting into it, school in general) simply aren’t developmentally appropriate for some students. Kids need different things! School systems lacking the ability to give a student an accommodation they need to show they can understand a concept should not be taken to mean “the student cannot understand the concept”.

Here are some questions I’m thinking and reflecting about:

  • Do we need to categorize students sometimes? It it even possible to categorize students in a humanizing way?
    • If yes, under what circumstances does it make sense to categorize students? When doesn’t it make sense? 
  • What kind of categories are helpful? How do we know?
  • I (and other teachers) have to teach a curriculum with fidelity. I have to give certain tests with fidelity. I have to show these tests to families, administrators, and coworkers, and decisions about these students are made as a result of these tests.
    • Given all that, how do we push back on labeling “low kids” and “high kids” (or sub your favorite euphemisms – “struggling learners”, “high fliers”)? 
  • How much of the change that needs to happen is about what we say – i.e. the words we use – vs. what we do – i.e. the structures, routines, and learning opportunities in our mathematics classroom?

Math in the Age of Distance Learning – Part 1

| Janaki Nagarajan
Peas are a mathematical powerhouse.

I’m naming this blog “Part 1” to hold myself accountable to write a “Part 2” in a couple months, when school is over and I can look back with a more summative lens on the various successes and challenges of teaching math to Kindergarteners in a distance learning setting.

For now, though, I’ll share my journey with these students, what we’ve done so far and how I’m thinking about the the rest of the year. I’ve been reading Amanda Jansen’s Rough Draft Math and it’s inspired me to share my partially-formed-almost-definitely-going-to-change thinking – as it’s still happening and evolving. So let’s get started.

But first, I’ll give you a little context.

The Beginning

It was a Saturday when I learned I would take over as a Kindergarten teacher for the rest of the school year for a teacher who was going on maternity leave. On a Tuesday morning, barely a week later, I came to the school for a professional development day and was told okay, it’s happening, and you’re the teacher now. On Wednesday, all the schools in our district closed, and by the next Monday I was teaching Kindergarten, fully online.

I think we were one of the first districts in the country to go fully online. My district had the time and resources to ensure every student had access to wifi and useable devices. I know many people are in contexts where they aren’t able to reach their students in this way. The inequity between school districts mere miles from each other is staggering.

The first week was intense – we were pushing out math, reading and writing material every single day. It was completely unsustainable for teachers and families – we were navigating new ways to use tech and overcomplicating absolutely everything. The one thing I do remember as a great was the opportunity for students to engage in what came to be known as #CountingCollectionsAtHome.

Students found and counted a collection of objects, and used Seesaw to document their estimation, counting strategy and total. Seesaw also lets you take a photograph and add it to a submission – and audio record over it. I was able to hear how students described their counts, and give audio feedback to push their thinking, such as: Was as your total more or less than your estimate? How do you know?

Phase 2

The following weeks, the situation in the country was evolving rapidly and we took a “pause” in our online instruction to reconfigure a more manageable teaching schedule. What resulted was a district shift to project-based or task-based learning, where we teach each main subject – math, reading, writing – on one designated day of the week. So instead of doing five assignments in each subject per week – we are now doing one.

We were encouraged to ditch prescriptive worksheets in favor of more open-ended tasks for all subjects – guided by the big grade-level standards. We tried to think about projects that were simple enough for students to accomplish without much difficulty but rich enough for those with more time to take further if they so wished.

Corn cob ten frames! What’s the same? What’s different? (Click the picture to see the intro video for this task)

My class meets live for 30 minutes three times a week, and each day we talk about a designated subject area (math, reading, writing). In each block of time, we think about what we learned the last week, share out, and explore some new ideas together. I make sure all students get an opportunity to contribute their ideas. Afterwards, a corresponding activity is posted on Seesaw, and students and families can work at their own pace to complete these over the course of a week.

This is the format I settled on for my class – but there are so many variations depending on so many factors. Our first week of trying this new iteration, when we having a bit of a “slow start”, I introduced the Same and Different routine by filming a video, and had students find their own things to compare.

A student example of the Same and Different image they created.

When we returned after spring break, I decided to split up the big Kindergarten math standards into chunks and spread them out over a certain number of weeks. First up – a two week chunk focused on counting. I took inspiration from Zak Champagne’s wonderful blog. We started by reading the book How Many? during our live time, looked closely at some of the pictures and talked about different things we could count in the world around us. Students took their own pictures and described all the things they could count in them.

A student tells us How Many? they see in a photograph they took.

The following week, we circled back on one of the student’s pictures and talked about what all we could count. Then we moved onto Choral Counting by 2s. To do this, I screen-shared my iPad and Apple Pencil and wrote in the GoodNotes app. I played around and found this easier to use than using the whiteboard feature or annotating.

Left: Choral Count by 2s we did together live. Right: After, students found patterns in a 5s Choral Count

I was hoping students would begin to see connections between these two counts – it would take a lot of time and days of thinking together (that we sadly don’t have) to really get to all the ideas – but looking at these counts in conjunction, they are able to begin to think about the relationship between 2, 5 and 10.

After exploring patterns of 2s and 5s, students found objects in their homes that come in groups of 2s or 5s (or 3s or 4s – inspired by Janice Novakowski). We emphasized representing how you counted all the objects. I have to say I’m extremely impressed with what students found – socks, shoes, doorknobs, chopsticks, chair legs, drawers, eyes, fingers, toes, pillows…the list goes on.

What I’ve Learned

  • Use tech minimally, consistently and only to enhance the experience rather than create it. It honestly shouldn’t really matter what platforms your district uses – all tech is ultimately just a tool to make things easier. Don’t let the bells and whistles drag you down. Ask yourself:
    • How can I make this experience the most like being in a classroom as possible?
    • What opportunities does distance learning afford us to learn more about student thinking than may normally be possible due to various in-school limitations?
    • How can I use tech to enhance student agency?
    • How can I use tech to let students see each other’s thinking?
  • Distance teaching math feels significantly more challenging to me than for reading and writing (which in some ways seem easier now). It’s hard to find ways to connect each week’s tasks (vs. a month-long book study or report).

Moving Forward

I have some ideas about what to focus on during the remaining eight weeks. Will those ideas change? Almost certainly. Nevertheless, here they are:

  • I’m hoping to move onto problem solving – thinking about addition and subtraction, decomposing 10, etc. for the next few weeks.
    • I’m thinking about interesting ways to do this, especially given that we only have math once a week. 3 Act Tasks? What else?
  • I’d like to then be able to explore geometry, measurement and data – and maybe end the year out with some math art and games.

Counting Collections

| Janaki Nagarajan

Also known as: My Absolute Favorite Activity to Do with Children

What kinds of pattern and structure do you see in this picture of students counting?

“Ms. Janaki! Come look at how we counted! We put thirty on each plate and now we’re putting five more on each plate until we run out.” I don’t think I’ve ever seen students as excited in math as when we were counting collections. But beyond just excitement – the value of children counting has been well-documented.

There is appeal in the simplicity of the launch (“go count!”), the flexibility and repeatability of the task (“count in any way that makes sense to you”; “count it in a different way”), and also the complexity of the outcome and what it can teach us about how children think and reason mathematically. Not to mention strengthening the connection between concrete, physical quantities and more abstract ideas of number.

A student demonstrating two groups five make ten and using that idea to count.

At first, most kids in my class immediately started grouping by fives. I had given them collections I knew were between 200 and 500 objects, and this turned into a time-consuming and arduous task. Many did not finish counting all their objects the first time around. Part of the beauty of this task is that is an exercise in patience for the teacher, as it allows them to actually see their students’ thinking, instead of immediately pushing for accuracy and efficiency.

A student who did not finish counting but discovered that making a “rainbow” of popsicle sticks with 5 sticks of each color could allow you to count by 30s.

Students also had to navigate the social situation of counting with a partner. Some pairs of decided to count separately and combine their totals at the end – others agreed on a mutual counting strategy and played different roles in keeping track of their collection.

Counting collections authentically introduces new mathematical ideas to students. The kids from the quote at the beginning are discovering partitive division. Others were using measurement division – if we put 50 objects on each plate, how many plates will we need?

Many also began utilizing arrays – and arrays within arrays – to more quickly count their objects. Creating a seeing structure within objects and quantities is fundamental for students’ understanding of number.

But what about accuracy? What can errors tell us?

Students ambitiously create arrays to count – the first time they’ve ever done counting collections.

Students frequently made errors in counting, recording and/or calculating. It was extremely tempting as a teacher to want to see if they got the “right” number, or jump in and point out mistakes. It took a lot of practice (and will continue to take more!) to be able to know what questions to ask to prompt students to check for accuracy themselves instead of “fixing” it for them.

This student made a slight calculation error – can you figure out why?

There is also a lot of value in seeing their mistakes. A student whose count is off because as they count by tens, they write “90, 100, 101, 110” is facing a different challenge than one who’s having trouble keeping track of what they’re counting or one whose groups are not equal. Errors show how to best support each student moving forward.

Extending the task – now what?

One day, as students were counting collections, a significant number of pairs had finished counting and recording quickly while others were still diligently working. I didn’t have enough collections for students to grab a different one, so I had to think quickly – I looked around the classroom to try and find some things students could count. Here’s what I came up with.

  • Count the books in the classroom library without taking them out of their bins.
  • Count how many crayons everyone has – you can open boxes, but don’t take them out.
  • Count how many dot stickers are on these sticker sheets without removing them. (Bonus – some were torn!)
  • Count how many keys are on all the keyboards in the classroom without moving the computers (please don’t move the computers!)

This was very successful, so the next time I had students do a “choose your own adventure” and count whatever they could find in the classroom and record how they counted.

Students were able to use the way objects that were too cumbersome to move were organized to help figure out how many there were.

Some lingering questions and things of interest

  • Seeing students negotiate how to count (or whether to count) parts of objects, such as the ripped stickers or broken clothespins was very interesting.
    • I kind of want to rip/break objects in each of the collections on purpose, just to see what they’ll do with it.
  • I wonder what students would do with objects that are the same type but have differing sizes? I’m picturing a giant tub of googly eyes we had in our classroom that were all different sizes.
  • I’m still learning how to navigate the social issues that arise between partners (or what to do when the “I’m going to start throwing my objects and laughing” starts). Anybody have tips on this?