Author: Janaki Nagarajan

The Number Corner

| Janaki Nagarajan

I’m not quite sure how I’ve managed to end up in this very specific situation twice thus far in my fairly short career, but here we are. (For context, here is my post from 4 years ago).

The Beginning (redux)

Slightly too accurate school bathroom decor

It was a Thursday when I learned I would take over as a 3rd grade teacher for the rest of the school year for a teacher who was going on maternity leave. On a Wednesday, a few weeks later, I was assigned to the school as a substitute paraeducator, and at noon, very unexpectedly, was told okay, it’s happening, and you’re the teacher now. There was some back and forth on whether this was actually true, whether the teacher would return for a couple more days, but come 11pm that night, it was clear *I* was now in charge of this class. While a global pandemic did not shut down our schools the next day, as it did the last time I was in this situation, I still felt far from entirely prepared.

All that to say – this past year (and especially past week) has been a whirlwind.

That first half day, going in with absolutely no idea what I was supposed to teach, was an interesting ride, to say the least. Following the general plans as best I could, trying my best to keep a handle on what I’d been told over and over was a very challenging class, I saw that the one academic routine that had been planned for that afternoon was the “Number Corner”.

The Corner of Numbers

Literally, numbers (kind of?) in a corner

I wasn’t super familiar with this routine, which is part of the Bridges in Mathematics curriculum – I had come across it a few times as I subbed across the district, though I spent most of this year teaching & subbing in the subset of schools in this district that use Illustrative Mathematics instead – so again, my understanding was something like – “there’s a calendar with patterns”.

I’m not going to claim to know much more about the idea of Number Corner than that in this post either – this story is really about eliciting and marveling at kids’ mathematical thinking.

Anyway, back to the story. I told the students it was time for Number Corner, and watched as they all filed into this little corner of the room – some on the floor, some standing behind, some at desks in the vicinity. And then I stepped back to watch them communicate their ideas – the excitement behind the patterns they noticed both in the visual representations on the calendar and the numerical ones on the chart (they hadn’t done it in a few days due to testing, so they were turning around quite a few of the squares).

And so, for a brief moment in the middle of a very, very hectic day, I got to listen to my favorite thing – kids’ thinking about math.

What happened Next

That afternoon, as I was debriefing my day with the principal (and trying to come up with a plan for the next one), she mentioned that the kids had been really engaged in the “math thing” we were doing (she had been in the class watching as we did this routine). At the time, I hadn’t given it much thought – it was on the schedule, I tried my best, kids seemed to enjoy it, but my understanding was that this routine was usually just a quick blip in the day, and sometimes skipped altogether.

It made me think though – what if I approached this routine with more intention? What would I find?

Sometimes, as teachers, we have things that we do in our classrooms because we love them so much, and they give us something in our days/weeks/months to look forward to. For me, that’s always been both “free art with interesting materials” and “free mathematics with interesting materials”. While this doesn’t quite fall into the latter category, “opportunities to see what kids think about math” is definitely a close second.

So then, what did they notice? A student I’d struggled to connect with immediately offered her prediction of a cyclical pattern with the fractions (we hadn’t reached the end of the first ‘cycle’ yet). The shock that came with the uncovering of the first orange shape was palpable. There was a lot of conversation attempting to predict when the next orange shape would come – some students thought that every 8 days, there would be a new color introduced. In the days to come, students noticed that the diagonals could be used to predict color and fractions, that every two days the total area added up to 24 (minus a couple outliers), and imagined how the outline of the shape might be manipulated in future dates to preserve the area of 24.

My favorite part is that during this time I can step back, listen, and facilitate/probe when necessary, but otherwise, the students have got it. They’re building on each other’s ideas, calling on each other, turning and talking, and are absolutely devastated if we have to wrap it up before they get a chance to share their idea.

What I’ve found particularly fascinating is that students, as a whole, have been significantly more engaged in our number corner routine than any math lesson we’ve done together* – even with lessons that are meant to be “hands on”, “exploratory”, or “fun”.

*Sidebar to say, they also did love when we played Illustrative Mathematics’ “Mystery Quadrilateral”.

To me, at least, this is a very interesting issue – and I’ll come back to it again in the next section.

For the days remaining in May, I’ve thought about how this routine they’ve already bought into can facilitate further mathematical thinking. Building off this as a warm up or cool down in math, thinking about the possibilities for the next new shape that will appear? Look more closely at the data we’ve collected for new patterns we see? (I’ve been really curious if they’ll make the connection that green < 1/2, orange = 1/2, and purple > 1/2 – or if that’s even an important thing to notice). We’ve tried a few of those ideas, as documented below.

A student’s prediction for 5/21
Another prediction
Calendar grid updated with colors to highlight patterns
What possibilities are there for the next shape?

As much as I love being wildly creative in my math routines and lessons, digging deeper, thinking about what new ideas I can try out – it’s also important to step back and realize the power in kids being excited about the math they’re doing – and go from there.

the epilogue (i.e. some random, possibly connected thoughts)

I’ve been reading texts from two math-y authors lately – Is Math Real? by Eugenia Cheng and Ben Orlin’s series of Math with Bad Drawings books. Both authors offer their thoughts on how doing mathematics is quite different than doing mathematics in school. While that sentiment is not new to me, and I’ve certainly made attempts to consider what bringing more of “the approach to mathematics that actual mathematicians use” into my classrooms, there are a couple ideas they put forth that have really elegantly stretched and also simplified my thinking on the math we do in classrooms.

First, Dr. Cheng says that the crux of what mathematics is about, and what mathematicians actually do, is justification (p. 73). As she says, this is the difference between “what is 6 x 8?” and “why does 6 x 8 = 48?” I think students love this routine in part because it requires them to justify their ideas, rather than calculate (similar to a good Which One Doesn’t Belong?). Any task they’re given during class (via the curriculum) is generally asking them to calculate (and sometimes, as a sort of bonus, have some sort of justification for that calculation – but more along the lines of “make sure your answer is correct). I think it’s also a time for students to be able to meaningfully talk to each other about their math ideas in a decidedly non-evaluative and curious way.

Second, in the introduction of the book Math Games with Bad Drawings, Orlin writes that “the secret to our brilliance is that we never stop learning, and the secret to our learning is that we never stop playing.” (p. 11) During the Number Corner, I see students playing with ideas – making conjectures, justifying ideas, creating generalizations out of patterns.

I’m excited to see what June will bring.

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 2

| Janaki Nagarajan
Image credit: Math With Me MN; annotations show student thinking

Recently, I came across a Twitter question that asked Kindergarten teachers about their favorite virtual routines or ways of interacting with students online. It got me thinking about what felt the most successful in my distance learning with Kindergarteners. I’d been spending so much time thinking about the fall, and about what hadn’t worked or needed to be better that I had not paused and simply reflected on what did feel successful.

I was surprised the answer came to me so easily. It starts with a picture.

The Power of a Photo

I hadn’t realized so many of our math routines followed the same structure: show some kind of photo on Zoom, look at it together and talk about a specific question (“how many?”; “what comes next?”; “what shapes do you see?”; “which one doesn’t belong?” amongst others). Then students would take their own pictures, in their own contexts, and answered or talked through the same questions we’d examined as a group.

What do you notice? What do you wonder? What shapes do you see? How might you describe a shape that you don’t know the name of? Image link.

As a class, we examined the photo on the left for shapes. We’d already been talking about shapes – both 2D and 3D, though some students were less sure about 3D shapes. My goal in selecting this picture was to give students 1) the opportunity to examine a lot of shapes, some that have clear names and some that are perhaps more complex, and 2) to show a picture with a context that might be meaningful or familiar to some students.

The photo appears to be an outside kitchen/stand in South Asia – many of my students are Indian and have travelled there, and I recognized the metal pans and colorful beans as part of my own childhood. Embedding students’ math exploration in a variety of contexts shows that math is everywhere, and pushes back on the defaulting to the same objects and contexts over and over.

When students described the shapes they saw, they found circles, “small circles”, cylinders, squares, and a box or cube. One student said that if you turn the jars sideways, you’d see a rectangle (note the slight angles present on one of the jars!) To follow up, students looked for shapes around them. They found square walls, rectangular mirrors and round (spherical) balls. I missed an opportunity here for students to grab some paper and draw a representation of a shape from the photo, or from around them – but we did get to see students’ thinking and representations live while exploring visual patterns, as I’ll show below.

Student work – In The Moment

A student holds up their drawing while explaining their thinking. They’ve extended the visual pattern & numbered each step.

With a little Zoom finagling, this setup works pretty well. We were exploring a visual pattern, and students had been told to bring something to write on and something to write with. I was sharing the image from my iPad, and annotating it, while looking at all my students in the “gallery view” on my laptop. After I annotated some of our noticings onto the pattern, I asked, “what comes next?” Students drew the pattern as they saw it, and what they thought the next step would be. What was interesting about this pattern is that the “next” could come in a few different places, so students’ drawings did not necessarily look identical.

Students were able to hold their work up as they spoke so their classmates could see what they’d drawn. This method would work fairly well for certain types of routines, including representing patterns, shapes and counting collections. Another option would be for students to come prepared with a collection of objects and count, recreate a pattern or identify shapes live. This gives the opportunity for students to directly respond to each other’s thinking in the moment.

Some examples of visual patterns students created afterwards that match (and extend on) our original pattern we looked at together. Students could also have done this live!

Student Work – Follow Up

If the live work is meant to extend on a single image (or group of images) we all examine together, then the follow-up is meant to give each student an opportunity to carefully select and explain their thinking behind our focus. One of the first activities our class did was to look at a scene or collection of various things and ask “How Many?” (I wrote about this more in my previous blog).

This image has an empty alt attribute; its file name is 2020-05-15_79c1ee15-4506-4b2b-8d49-6988e2a78ce4.jpg
A student’s photo for the prompt, “How Many?”

Giving students the opportunity to create or find their own image, and explain in their own thinking all the different things they might count in it gives teachers (and students) a better idea of how well they can take what they learned live and apply it to their own context. The student on the left, for example, found a collection of legos and other objects. There are SO many things we could count in this picture – and when it’s captured in an image, it’s easy for both teachers and students to revisit and learn from.

The follow-up doesn’t have to be a photo – photos are great, but so are videos of students explaining their thinking, or showing how they counted or built something. Students might also submit work that shows how they represented what they counted, for example.

Students also created their own “Which One Doesn’t Belong?” images, after exploring the book together on Zoom. Students recorded themselves and a family member each explaining which one they think doesn’t belong and why.

Some student examples of Which One Doesn’t Belong? In the video, a student and his family talks about which one they think doesn’t belong from a student-created set. Students submitting follow-up work allows them to engage family members or caregivers with the math.

The Rest of It

I want to make sure I mention some of the other activities and tasks we did that I think are interesting and worth revisiting, but in the interest of keeping this blog as concise as possible, I will be brief and include links to student work around these ideas.

First of all, during a couple of our Zoom meetings, we tried an Esti-Mystery and a Splat!, two routines from Steve Wyborney. Esti-Mysteries involve students estimating how many objects are in a container, and revising their estimates as clues are revealed. Splat! requires students to think about how many objects were hidden under the “splat” using what they know about how many there were to start with, and how many are left.

To see more of what this looked like, follow these Twitter links:

We also thought about how chickens could be arranged inside and outside of a coop. This activity was adapted from one that Bethany Lockhart uses with her Kindergarten class.

Towards the end of the year, students built structures out of paper and talked about what shapes they saw, how many sticks they used, and how they would use their structures. This was an idea from Chapman University’s #MathPlay series, and it was amazing to see their creativity!

What’s Next?

This fall, I’m teaching 3rd grade, so I’m adapting what I’ve learned and thinking about the needs of slightly older students, and I’m also waiting to hear specifics of what fall teaching will look like in my district (aren’t we all?!) But in the meantime…

  • Routine is key. I want to remember to set up a routine (just like in class!) where students engage in a warm up, problem solving and discussion, even if we are virtual.
  • I’m very curious about having a virtual “Hands Down Conversation” (from this book) with students about math (and literacy!)
  • I want to do 3-Act-Tasks both synchronously and asynchronously with students. The fact that photos worked well as a jumping off point for mathematical discussions shows me that with proper preparation, watching a video together could be even more interesting! And maybe students could create their own…
  • Social Justice and math are intertwined, and I want to explore this much, much more next year. A good jumping off point might be having discussions around slow reveal graphs and students collecting data which we can then analyze together.
  • As I mentioned above, I’m trying to make sure that when I bring in an image, or story problem to students, the contexts are meaningful and go beyond what we think of as “neutral” or “default” (i.e. White, Western, European). However, I need to get better at spending just as much time with students unpacking the contexts and how they connect to them – because just throwing in a “diverse” picture (cringe) does absolutely nothing.
  • I want to make counting collections and choral counting more of a regular routine, even in 3rd grade. Going back and really thinking about place value and base ten will be important for students, especially in light of inconsistent experiences this spring.

I’d love to know what you are thinking about for this coming school year!

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.

#DebateMath in 3rd Grade

| Janaki Nagarajan

Last month I spontaneously took part in a Twitter chat about Chris Luzniak’s book, Up for Debate! – about using techniques from debate to encourage engagement in mathematics. The book says it’s for grades 6-12, however, as a 3rd grade teacher I was curious anyway and wanted to learn more.

I won the book through the chat & look forward to diving into it deeper (what I’ve read so far has been great!) – but I was inspired by what I learned and wanted to try a little bit with my students right away.

Warming Up

The next day, the students’ morning task was to write a CLAIM (their position) and a WARRANT (a justification for that position) to the question “What’s the best pizza topping?” The goal was to get students familiar with the claim/warrant terminology and push them to think of justifications others might find compelling (rather than “I like it”, “It tastes good”, etc.). I also hoped to familiarize students with justifying their answer to a subjective question, which we would be exploring more of later.

Some especially creative answers:

  • Artichokes because they add a nice salty flavor
  • Ice cream on a “polar pizza” because it goes nicely with the oreos

I wanted to give students a chance to practice thinking of claims and warrants to solve the type of math question we’d been working on, while also switching up the prompts from “best” to “coolest” (Chris has a lot of variations in the book, but I particularly liked the idea of starting with these).

How do you think students might have answered these questions? One student said the best way to solve #1 was 36 + 36. Another said the coolest way to solve #2 was to cut the rectangle in half to make two squares. They were still working on coming up with justifications.

Ideas like “best” and “coolest” are subjective, so there’s a lot of room for disagreement – but students still have to work to convincingly justify a subjective claim. I also wanted to get them thinking about if the “best” way to do something might be different from the “coolest”.

DEBATING COUNTING COLLECTIONS

All of this was a warm-up to extending #DebateMath to Counting Collections. It was a routine to count on Wednesdays, and this week I asked students (in partners) to come up with either the best way to count their chosen collection, or the coolest. I wanted to keep the vocabulary the same from the morning but bring it into a new (but familiar) context.

Once students chose their collection, decided how they’d count & began counting, we paused partway to make sure everyone wrote down their claim and warrant so we could use this language when we came back together as a class.

In thinking about how to best share our claims and warrants (and then discuss them), I decided to have students come to the rug & sit down, with one pair at a time sharing their claim/warrant – those who disagreed would stand & push back on the idea/justification presented while those who agreed remained seated. (Side note: doing this again, I’d re-think this a bit. It’s important to have those who ‘agree’ be part of the conversation as much as those who disagree.)

One pair said the best way to count their collection was to split it into groups of 100, because there were less groups to count, and counting by 100s is easy. Others disagreed and said it takes too long to count to 100, so making groups of 10 would be better, because that’s also an easy number to count by. Some disagreed – you have to count more groups of 10 so it’s just as difficult. Some thought maybe groups of 50 would be a good compromise.

In particular, one pair of students decided the coolest way they could count their popsicle sticks was to use rulers to measure the length of a stick. They determined 2 popsicle sticks = 9 inches. For every 2 sticks, they added 9, then divided the total by 9, and finally doubled the total. They argued this was the coolest way because it got measurement involved and that’s unique!

Some ideas to push their thinking further…

  • A lot of the justifications for why one way was “best” seemed to be that it was “easiest”. Are the “best” and “easiest” ways to solve a math problem always the same?
  • Could we write an equation or equations to represent the way the popsicle stick group counted? What do we notice about it?

DEBATING VISUAL PATTERNS

The following week, we did this as our morning warm up (we got in the habit of independent think time, small group talk, then whole class discussion for a variety of math warm-up tasks each day – including a few different visual patterns.)

To be completely honest…we never got around to a whole group discussion about our claims and warrants (we did this over a couple days right before winter break and only got through #1 as a group). Some students did individually come up with ideas and justifications for the “easiest” way to find the area in Step 43, but unfortunately the whole class wasn’t quite ready for this discussion).

So…what do you think the easiest way is to find the area of Step 43? What do you think students might say?

What I love about #DebateMath

As stated above, concepts like “coolest”, “best” and “easiest” (and others mentioned in the book/on the Twitter chat) are fairly subjective, which means using this kind of language can elicit a range of strategies from students, and also push them to keep looking for more (could another way be even easier?) The language scaffold of “My CLAIM is _________” and “My WARRANT is _________” is immensely helpful for students to clearly express their thinking in a structured way that 1) helps the teacher understand their thinking and 2) orients other students to their thinking and helps them respond to a classmate’s argument.

I also appreciate the way this routine really de-centers the teacher and leads to authentic engagement amongst students. When leading our #DebateMath discussion post-Counting Collections, I was sitting back and observing as students shared their argument and called on others to respond, interjecting only to pause occasionally to slow down and dissect an argument or response more.

Given the proper tools & scaffolds, students can essentially lead this discussion with little to no input from the teacher. The teacher may be strategic in who they call on, or the specific task students will be debating and the question around it, but otherwise, it’s really the students who are engaged in debate with each other.

Play in Math: “Math Recess” in 4th Grade

| Janaki Nagarajan

So, I’ve begun my adventures in subbing. I spent most of the summer panicking that I didn’t have a job yet, and when one finally came along, it wasn’t quite what I expected. I’m grateful for a job in any form, but it’s been a process sort of mentally giving up plans I’d made in my mind over the last few months of what my classroom would look like, how we would start the year and the mathematical learning trajectory I was excited to be on with my students. Trying to make the best of an unexpected situation, I found myself asking:

what can I do as a substitute?

And specifically, what can I do in terms of engaging with and eliciting students’ mathematical thinking? To be honest, sometimes not much. There have been days I go in knowing my job is to do whatever is written in the sub plans, which is often to open the curriculum book and attempt to “teach” something I’ve never seen before. Once, we had a little extra time so I decided to do a choral count to reinforce ideas about place value, but for the most part, students have just independently done worksheets.

I was excited when I found out I got to sub in a friend and former classmate’s 4th grade classroom, and even more excited when she told me I was welcome to add or change anything about the schedule, if there was something I particularly wanted to do. They had extra time in math, and I thought for awhile what we could do in the 30-45 minutes we would have after doing the planned quick image activity.

I decided to bring in a variety of games, books, puzzles and manipulatives that would provoke mathematical thought and allow students to freely play around with them and see what happened. I thought of what we could call this time and settled on ‘Math Recess’, the title of a book by Sunil Singh and Christopher Brownell encouraging math play. I was also inspired by hearing Tracy Zager advocate for “D.R.E.A.M” – “Drop Everything And Math” time in schools, where students have time to playfully explore mathematics. (I am sure there are many people who have been proponents of math play in school – these are just two sources I was thinking about as I was planning what this time would look like).

If I implemented this as a routine in my own classroom, I’d be inclined to come up with “Math Recess” norms together with my students instead of imposing rules on them. But as a sub, sometimes I have to be more controlling than I’d like.

I tried to have a good mix of activities students could explore – I brought 21st Century Pattern Blocks and found dominoes in the classroom, we had games from Math for Love and books such as Which One Doesn’t Belong? and How Many? and a collection of Open Middle problems students could work on independently or with partners.

Students were immediately drawn to dominoes and Prime Climb (pictured above), and many eventually became intrigued with the pattern blocks – some made characters, such as a “person in the snow”; others made new shapes (stars!) or worked to figure different ways to fit the blocks into the hexagon cutout.

(As an aside – a student and I had an interesting and rather philosophical conversation regarding his creation of “person wearing a coat in the snow”. I couldn’t guess that was what he was making, but once he told me what it was, I remarked how it was interesting that I couldn’t see it until he told me what he had made, and then I could see it, and I wondered why that is. He said because when you know what you’re looking for, it’s easier to find.)

Part of play is embracing the unexpected

Despite the success of the above activities – the ones that were more independent or required more think time to really get into (the books and the challenging math problems) sat mostly untouched. Realizing that students were not likely to sit and think about the mathematical ideas presented in these choices while their friends were building and playing with each other, I decided to follow the students regarding other things they were interested in pulling out and playing with. For example, one student wanted to build with unifix cubes and was meticulously making sure each stack had the same number of cubes in order to build a “house”.

Other students asked to draw or read. At the time, since this wasn’t my class, it was the end of the day and no good seemed like it would come from requiring that kids only engage with the math activities (especially since this was likely a one-time experience for them), I said okay. About eight kids spent the time drawing with each other or reading independently. I wasn’t sure if this was the right choice to make – I really wanted kids to be engaging with math – but then at the end, one of the kids who I thought had been drawing came to me with the following question:

Student (hiding the above paper): If you guess the number on this paper, you’re the master of the universe.

Me: Infinity? (Student turns around paper). What number is that?

Student: I don’t know!

Another student: That can’t be a real number, because there are no commas.

(Students continued discussing this idea as they packed up.)

Turns out that giving students room to be creative beyond what I had planned allowed for them to come up with really interesting mathematical ideas on their own!

What is the goal of math play?

  • Moving forward, I’m thinking a lot about the goal of this kind of math play.
    • Should it just be a total free time for students to do whatever they like (relating to math)?
    • Should teachers use this as a time to assess specific ideas students have about mathematical concepts?
    • Should the options change based on the unit being studied?
  • What does it look like to have a routine of math play in the classroom?
    • How often does it happen?
    • How long do kids get each time?
  • What separates “math play” from other kinds of inquiry-based math?
    • And in what ways does it or does it not matter to understand this difference?

3 Act Tasks

| Janaki Nagarajan

“All those ways are correct, it’s just what you feel most comfortable with.”

-Student reflecting on different strategies to solve a 3 Act Task

The first time I tried a 3 Act Task was in a school I’d never been in before, in front of a group of unfamiliar students at a grade level I’d never taught. Not to mention, this was before I took the lead student teaching, so I’d only even been at the front of my own class a handful of times.

The concept of a 3 Act Task was totally new to me – I discovered them while searching around for something interesting I could do with these students. I quickly became enamored with the idea and committed to bringing it back to my second graders. I was especially interested in finding tasks with an interesting “Act Three” – where students could see how mathematical modeling might need to be adjusted for the “real world”.

“Rows of Oranges” (source)

Part one

The first task we tried was “Rows of Oranges” from Kendra Lomax’s blog. At the time, students were working on adding and subtracting within 1000, and this task may fit more into adding and subtracting within 100, however the structure and array represented an interesting way for second graders to also begin thinking about multiplication.

The task shows ten oranges arranged in two rows, and one by one they are peeled, with their slices arranged in columns of five. The video stops with two oranges peeled, and I asked students to notice and wonder with a partner, finally settling on the question, “how many slices are in all the oranges?” I then asked what information students already know to answer the question, and what more they might need. Then I showed them the picture above where the first two oranges are shown to have contained 19 slices, and students got to work thinking about the question.

Most students came up with the idea of 19 slices/2 oranges and had various methods for adding or multiplying to find five groups of 19, such as the student on the left. One student (right) had an interesting idea – that only one of the oranges had 9 slices, and the rest each had 10, therefore there should be 99 slices in all.

One of the most interesting things about the “finale” of this task is that the 10 oranges contained 98 slices – which is not an answer students came up with based on the information they were given. When I asked them what they thought about this – does it make sense? – a student said that it did make sense because maybe two of the oranges were a little smaller and only had 9 slices each instead of 10.

Part two

“Downsizing Tomatoes” (source)

A couple weeks later, we returned to 3 Act Tasks with a task from Graham Fletcher called “Downsizing Tomatoes“. This time, students are practicing measurement division by figuring out how many small ketchup bottles the large bottle will fill. Almost all students had estimates between 4 and 6 small bottles, based on Act One, where the video pauses and students said they noticed about one and a half small bottles filled, and a little more than half of the large bottle left.

After asking students what information they have/need to answer the question, I showed them the picture above. Their task was to figure out how to use this information to figure out how many small bottles the large bottle can fill.

The student on the left showed they understood the idea of figuring out “how many 64g’s are in 397g’s”. On the right, a student added until they got to 384, writing that this is the “closest I got”.

Another student (above) chose to subtract 64s from 397 using hundreds, tens and ones until they ended up with a number they could not subtract 64 from.

While watching the “reveal”, students noticed that as many of them had calculated, 6 bottles were filled completely. They also noticed a seventh bottle was filled a little bit. We had a conversation about this – how would we represent that amount? Students came up with ideas like “six point one”, “six point two” and “six and a half”, reasoning that the bottle looked either about half full or slightly less than half. It’s especially interesting that students said that 6.1 and 6.2 are less than 6 and a half, without any formal conversations about fractions or decimals in school.

One thing I am wondering about this task in particular – students are using the ketchup’s weight to determine something about the volume. While this does work, I think it would be valuable to think with students about why this works. I wasn’t sure how to incorporate this conversation into our lesson but I’d be interested in doing so in the future.

going forward

  • One of the things about most of the 3 Act Tasks I’ve found available online is that they’re often set in White, Western, English, mid/upper class contexts (e.g. the types of food/houses/activities often shown) – which makes sense, because that’s the context most of the people who are creating and sharing these tasks exist in. But what would it look like to create 3 Act Tasks centering other experiences?
    • This is something I’d love to explore more. However I, like everyone, am limited by my own lived experiences, so I am curious what kinds of collaborations could happen to create a more diverse library of 3 Act Tasks.
  • Jenna Laib has an amazing blogpost where students created their own 3 Act Tasks. I love this idea and hope to incorporate it into my own teaching.

Mathematical Art: Kolam

| Janaki Nagarajan
(Source)

A few months ago I came across this article that explores an ancient Indian art form, Kolam (written கோலம் in Tamil), and its connection to mathematics, gender and culture. This piece was particularly interesting to me as I grew up watching my aunts put kolams outside their doors in the morning when I’d visit India. I even remember having a small practice book to learn how to make different designs.

After reading the article, I became interested in how I might bring this into my classroom as a student teacher. On Twitter, Simon Gregg very helpfully gave me some suggestions.

I kept his idea in the back of my mind until I had a chance to use it. Originally, I was hoping to make the whole “traditional math art” thing much bigger, and involve other student teachers at my school, with us each bringing in math art from our cultural backgrounds, but ultimately that didn’t happen. (How is it that we never have as much time as we think we have?!) I was determined, however, to do something with kolams before the end of the year, and so on June 20th, the last full day of the school year, I made it happen.

(Source)

I was excited to be able to co-teach this lesson with one other student teacher who was next door and also teaching 2nd grade. We brought our classes together in one room while we introduced the lesson. She talked a little bit about math art she grew up with – Rangoli – which is an Indian art form similar to Kolam, although kolams are unique in their use of pulli (புள்ளி) – the dots around which designs are drawn.

Students spent time noticing and talking with those around them about the designs above – they noticed shapes within the kolams and how the lines went around (and not through) the dots. They noticed symmetry (a new word for many of them), and quite a bit about the number of dots in each row and how that affected the design. Then it was time for students to go back to (or stay in) their respective classrooms and try drawing their own kolams.

This was challenging for students. Even if they could see the symmetry, it was hard for them to actually draw it. Some students struggled with the idea of drawing around the dots instead of connecting them, or drawing straight lines instead of curved. I think students could have used a lot more time exploring the rules behind kolams before attempting to draw one themselves. (The ones pictured above are from a few of the kids who did seem to get the idea!)

I think, given more time, we could have gone much deeper into this activity (not to mention the dotty paper I gave them does not represent the pulli arrangement for all kolams). I felt slightly frenzied and disorganized and not entirely sure what my goal was for the students (especially considering school was over and I couldn’t extend or build on it). However, it was important to me that I teach this lesson even if it wasn’t perfect. My hope is that in teaching about culturally-embedded mathematics, students begin to see the world around them – in all of their contexts – as mathematics, and as mathematics that matters.

Further reading

Here are a few other interesting reads about Kolam and math.

“Learning New Ways to Play Games”

| Janaki Nagarajan

Mancala, Nim and the Social Aspects of Mathematics

DIY egg carton mancala

What is math?

I asked this question to eight surprised second graders on a Tuesday in late April. “Um…numbers? Plus or minus…like one plus one equals two?” I was trying to individually interview a few of my students to better understand what they thought about math and where they saw themselves within it.

The goal of these interviews was to gather data about these students’ mathematical ideas and experiences before I started full-time student teaching, and before they embarked on our four-week adventure of playing the mathematical strategy games mancala and nim.

The logic behind this controlled structure was so that I could research how my students engaged with these games as a part of my student teaching capstone project. Here’s one of my favorite answers to this question:

“Math is where teachers like you, well assistant teachers like you who are about to become real time good teachers, it’s where they teach you numbers and stuff so like plus, multiplication, times, subtraction and other stuff using numbers and that can help you find out how to think about coordinations and stuff.”

-2nd grade student
When I was 9 years old, my uncle in India taught how me to play Pallanguzhi, a Tamil mancala game. I still play it with friends and family.

Mathematics is often presented in school as static, objective, and independent of cultural contexts. Someone long ago discovered or invented how it works, and our job is to learn the rules and get the answers right. But this ignores the mathematics that is embedded in our daily lives, that spans across cultures and experiences.

While I am still learning how we as educators can be more conscious and inclusive of the intersection of mathematics and culture in our classrooms, I did want to at least acknowledge some of my own experiences of mathematics in childhood. I showed my students the picture above to explain that mathematics is all around us, and within us in ways we are not always aware of.

Thes rules for Nim are based off of this lesson from Dan Finkel/Math For Love.

During my time student teaching, I was committed to trying a variety of different math tasks that differed from how math was generally taught in our class, with the goal of creating a math culture that valued and was responsive to all students’ thinking. For this particular project, I wanted to focus on playing games (and these two games in particular) because:

  • Games are objectively fun. Everyone can get excited to participate and play, even if they may otherwise experience math anxiety.
  • These games provide a really simple entry point for all students – the rules are simple and the language demands are limited – all students can participate equally.
  • They are open-ended, allow students to decipher strategies and alter the game rules. To be successful in these games requires both fluency and deeper understanding.
  • They are entirely student-led, and allow teachers to step back and observe how students interact with each other, to help understand the social aspects of mathematics.
  • These particular games do not require basically any money to be spent at all – games do not have to be fancy or colorful to be fun and mathematically rich.
  • These games are transcultural and can be used to push back on the many ways we center White, Western culture in math class.

I decided on baseline rules for the games, generally for ease of explanation (more common forms of nim, especially, are more complicated to explain). However many students started pushing back on the rules once they got the hang of the games, and tried altering some parts to see if that affected what strategies they were using.

Students played the games in (the same) pairs of two (so four kids in a game) in an effort to push them to utilize each others’ ideas and thinking in terms of how they would play. This is a strategy I heard from multiple sources – if the goal is to promote thinking over skill, strategically pairing students creates a collaborative (and not just competitive) environment.

Half of the class each played one game the first two weeks, then switched for the next two. After two weeks of playing each game, students video interviewed each other to ask their partner what they had learned playing that game. I was particularly blown away watching what they’d done given this freedom – sure, they got a little silly sometimes, but the ways students came up with to explain their thinking with basically no direction was fascinating.

Some students decided to use the physical game objects to demonstrate what they meant as they described what they’d learned. Some wrote themselves scripts of what they wanted to say. Many mentioned how they’d learned from their partner, and what it meant to use sportsmanship. Students got really into being the “interviewer” as well, adopting a professional tone of voice and coming up with their own questions to ask their partners – “What was the biggest difference between mancala and nim?” “Which did you like better? Why?”

I thought it was especially interesting to provide multiple ways students could demonstrate their learning and engagement. Students could explain their ideas verbally in a more formal manner during their partner interviews (as well as when I interviewed them). They could also write or draw on their recording sheet whatever made sense to them, or chat more informally with their partner and group as they played the games. These multiple avenues allowed me to get a fuller picture of what each student was thinking, understanding and learning.

There was a lot of learning for me as well, going through the research process. I was trying to figure out how students engaged with the games, but also trying to see if and how students’ ideas about math were impacted through this experience. I think that is a difficult question to isolate – how kids feel about math and how they talk about it are affected by so many things – but one student did feel their definition of math was expanded because of what we had been working on. As they said:

“[Math] can also be fun, like it can also be counting and learning new ways to count, or learning new ways to play games.”

-2nd grade student

Cue happy tears!

Going forward

  • This whole experience felt slightly artificial because of all the work I was doing around it to collect data and research. I am wondering how to more authentically incorporate games into math class (instead of just a special, four-time experience).
  • What are other games that meet the criteria of low tech, cheap, simple, mathematically rich, either transcultural or work towards centering knowledge of marginalized students?
  • I’m always a little wary of “math games”. Sometimes they just seem so flashy, fancy, or only promote fact fluency (i.e. “tricking” kids into thinking math is fun – I know that’s a cynical take, but still). What’s a good way to discern which games are worthwhile?
  • How can games open up a conversation about the cultural and social aspects of mathematics often overlooked in school? And then…what do you do with that?

This blogpost is part of the The Virtual Conference on Humanizing Mathematics.