student teaching

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.

Which One Doesn’t Belong?

| Janaki Nagarajan
via Stenhouse Publishers

One of my favorite activities I did regularly with my 2nd graders in the past couple months was inspired by Christopher Danielson’s “Which One Doesn’t Belong?” posters.

These are an incredible resource! Find them here.

The posters came in a set of 8 – a perfect amount for us to explore one per remaining week of school. I found some unused magnetic name tags and decided on this set up.

I had no idea what was going to happen – I gave no verbal directions (only vaguely pointing out we had something “new” in our classroom). It took a couple days, but students did start to engage.

Students’ thinking about the equilateral triangle not being “stretched” shows that they’re beginning to think about the relative size of angles.

Each week, I would put up a new poster on Monday and we would have a whole-class discussion about our ideas on Friday.

As we continued with this routine week after week, students’ reasoning became more sophisticated and their engagement increased as well – students would sneakily move their name to a shape during circle times and transitions. They would congregate around the poster, asking about and pushing back on each other’s choices.

Nobody chose the blue square because it was the most “regular”. What might make a shape “regular”? Do some properties of shapes matter more than others?

Here are some of my favorite noticings and conversations we had.

  • The top right hexagon can be turned into a drawing of a cube by drawing three lines. (Do you see it?)
  • The other three hexagons can be “squished” a bit to all look the same. (What does this tell us about angles?)
  • The top left shape (in the poster on the right) could be straightened out into a very long, skinny rectangle.
  • What is a shape? Are letters shapes? Is everything a shape? Do all shapes have names?
  • What is a polygon? Can we come up with a definition from this poster (below)? Could we explain it to a Kindergartener?
    • Ideas – a polygon might be: a shape with a lot of sides, a shape with at least 4 sides, a shape with any number of sides that are “line segments”.
The top left doesn’t belong “because it is a square and I don’t know what a polygon is but I know it is not a square!”

Now for some questions moving forward…

  • How might you use these posters in your classroom the whole year? With younger kids? Older?
  • How might you support students to create their own WODB images, perhaps moving beyond just shapes?

How do you use WODB in your classroom?

Let me know in the comments!

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?