Is 1/2 always greater than 1/3?

Lately I’ve been reading the book Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense by Julie McNamara and Meghan Shaughnessy.

I posted the following picture to Twitter while I read during my daughter’s swim class.

My colleague, Hedge, replied about being challenged by a middle school teacher on this very issue.

I let her know I was also challenged about this idea several years ago when I was a digital curriculum developer. The argument I heard back then was that using contexts to validate the correctness of fraction comparisons ran counter to the fact that fractions are numbers. As such, 1/2 is always greater than 1/3 regardless of the context. At the time, I wondered about it, but I still felt that bringing context to bear was important.

Flash forward to now and I have been mulling this idea over all day. I think I may finally understand why we have to be careful what we say about the role of context when comparing fractions. I may be completely off the mark, but I’m going to share my thoughts anyway and let you decide in the comments if you’d like to challenge my thinking or share an alternative point of view.

Let’s start with whole numbers. If I told you to compare 3 and 6, you would probably tell me, “3 is less than 6,” or, “6 is greater than 3.” That is how the numbers 3 and 6 are related.

Now, what if I were to show you these two pictures of 3 and 6: (As illustrated by my daughter’s toys.)

Three large dolls

Six small figurines

Technically, the 3 dolls are larger and therefore they amount to more stuff, but does that really mean 3 is now greater than 6? In the end, the number of dolls my daughter has (3) is less than the number of figurines she has (6). The context doesn’t fundamentally change the relationship between the numbers 3 and 6.

In this case, I don’t even know how I’d justify that she has more when referring to the dolls. Sure, they’re bigger, but she may prefer to have more things to play with and choose the 6 figurines even though they are less in total size.
Let’s continue by looking at this from a fraction perspective. Now I’m going to take 1/3 of the dolls and 1/2 of the figurines.

1/3 of the dolls is 1 doll

1/2 of the figurines is 3 figurines

In keeping with the idea that context should dictate when one number is greater than another, I should be convinced that 1/3 of the dolls is greater than 1/2 of the figurines because 1 doll is so much larger than the 3 figurines. Oh wait, or is it that I should be thinking that 1/2 of the figurines is greater than 1/3 of the dolls because I end up with 3 figurines which is a greater number of things than 1 doll? It’s not so clear cut, even though I’m trying to let the context dictate how to interpret the fractions.

What it boils down to is that fractions represent a relationship. If I think about the relationships each fraction represents, then 1/2 is always greater than 1/3 no matter how I try to spin it. Looking back at my examples, taking 1/2 of the group of figurines means I am taking a greater share of that group (that whole) than when I take 1/3 of the group of dolls (a different whole, but a whole nonetheless). The size of the things in my group (whole) doesn’t matter because the relationship represented by 1/2 is greater than the relationship represented by 1/3.

Now, does that mean we should ignore contexts altogether? No. There are still rich conversations to be had about who ate more pizza when one person eats half of a small pizza and another person eats a third of a large pizza. Context is still interesting to discuss and helps students use math to interpret the world around them. However, if our goal is to compare fractions, then 1/2 is greater than 1/3 every time.

That’s the argument I came up with today as I tried to understand the criticisms I’ve heard. Now that you’ve read it, what do you think?

Sink your teeth into data. Don’t just nibble.

Looking for math all around started as a challenge I made for myself and I’m realizing it’s becoming a full-fledged theme for my year. When I had to think of a topic to moderate this week’s #ElemMathChat, I started by asking myself, “What’s a topic we haven’t talked about since the chat started in August 2014?” After some brainstorming, I eventually came up with analyzing data. What a great topic for my theme! I don’t think I could throw a rock without hitting some data in the world around me.

In fact, as I was fleshing out the topic for the chat, I was regularly checking some real-world data online. After a long dry spell, we finally got some rain in Austin. And by “some rain” I mean a deluge. On a couple days last month it just kept pouring and pouring. Throughout each day it rained, I found myself checking our neighborhood weather station on Weather Underground to see how much rain had fallen. By the time October was over, we had received 10.3 inches of rain in my neighborhood! That simple piece of data became the catalyst for tonight’s #ElemMathChat.

I started digging into rainfall data for October, then rainfall data for other months, and finally I expanded my data dive into other cities in and out of Texas. When I was done, I had a spreadsheet full of various tables of data that I wanted to share in my chat. To make this chat work, I realized I needed to be intentional about how I shared the data in order to tell a coherent story. I also wanted to create a variety of data displays that would match the various data displays students encounter across grades K-5. As an aside, I think #ElemMathChat sometimes leans a bit heavy on content for grades 3-5, so I was trying to be mindful to show some graphs that could be analyzed in a Kinder or 1st grade classroom.

It took several nights to research, create graphs, and pull it all together to make a story, and in the end I’m proud enough of the final result that I wanted to capture it on my blog.

Before starting my data story, I shared the following guiding questions that tied into my primary goals for the chat.

My Data Story

Our story begins with the piece of data that started it all. I asked the participants to tell me what they noticed and wondered about this statement.

What do you notice and wonder?

Many people wondered how this amount of rain compared to other cities. Funny you should ask.

What do you notice and wonder as you look at this pictograph?

One thing I noticed is that I accidentally left the key off the graph. Oops! Each picture is meant to represent 1 inch of rain. Despite my mistake, several people liked that the missing key invited students to wonder about what the pictures represent. That sounds like such a wonderful conversation to me that I opted to leave the key off when sharing the picture in this blog post.

I had a little fun with this graph because I had to decide which cities to include. I decided to focus on other state capitals, but the question became, which ones? When I noticed how many start with A, I decided that was more interesting than picking random capitals. It just so happens that all the other capitals on this pictograph are all on the East coast, so I wonder if it would have been better to choose capitals with greater geographic diversity. In the end this is just a fun way to get our story started so I’m okay with what I chose.

Next we moved from cities outside of Texas to cities inside of Texas, specifically along the I-35 corridor from San Antonio to Waco.

What do you notice and wonder as you look at the October rainfall totals for these cities?

Now that I shared two different graphs, what questions could you ask students about these graphs? What math skills can students bring to bear to interpret and further understand the data in these two graphs?

One thing that we often do with graphs found in textbooks and tests is ask one question about them and then move on. How unfortunate! There’s so much rich information to dig into here. One of my key points for tonight’s chat was reiterating something I read by Steve Leinwand about mining data. Ask a variety of questions about data displays. Sink your teeth into them; don’t just take a small nibble.

The one thing that stood out to me and many others in the chat was how little rain San Antonio received. The difference between San Antonio and New Braunfels is quite striking considering how close they are to each other.

Other people felt that Austin’s rain wasn’t fitting with a general trend in the data. I didn’t want to get into it in the chat, but I’ve noticed the rainfall in my neighborhood tends to be less than other parts of the city. Our weather station recorded 10.3 inches for October but others in Austin clocked in at closer to 13 inches of rain. I thought about using the larger number, but because the catalyst for this whole story was my weather station’s data, I opted to stick with that. By the way, I don’t think it’s an issue with our weather station’s rain gauge. Over the years there have been many instances of rainfall in other parts of the city while my neighborhood in north Austin remains bone dry.

Now that we’ve looked at rainfall in and out of Texas, it’s time to drop a bit of a bombshell. With this new information, what story is the data telling so far?

Here’s what I see as the story so far: Austin received 10.3 inches of rain in October, which was a lot compared to areas outside of Texas, but fairly common for our area in Texas. Not only was this a lot of rain, but it also fell in a very short amount of time, 6 days.

Next, I asked for help. Now that you know it rained only 6 days in October, which data display would you choose to represent October rainfall?


Option 1


Option 2

Most people preferred option 2 because it shows the full picture of October. That was surprising to hear. In my mind, because we just saw the picture graph showing that it only rained 6 days in October, I didn’t feel option 2 was needed. I already know it didn’t rain on very many days, so why waste the space with all those days showing 0 inches of rain? Option 1 puts the focus squarely on analyzing the rainfall on the days where it actually rained. In the end there’s no “right” answer, it all comes down to how you justify showing what you choose to show.

We’re nearing the end of our story. There are two more graphs remaining. What does this next graph add to our story? What is one question your students could answer based on this data?

I love looking for relationships so here are the questions I came up with:

  • Where do you see the relationship “three times as much” represented in this graph?
  • Where do you see the relationship “half as much” represented in this graph?

I especially like wondering what students will come up with because both questions have more than one correct answer.

And now for the last graph. How does this close out our data story?

Here’s a follow up question for you. What could be the sequel to the story I just told? How could you and your students explore and tell the sequel? What other data stories could your students explore and tell?

I closed the chat, and I’ll close this post, with two key points I want everyone to take away from this conversation.

Two Cats and Two Tortoise

Yesterday my wonderful co-workers threw us an adoption shower, and thanks to them our daughter is an even richer girl if you measure wealth in books.

One of the gifts was from Mary Beth. She said it’s a favorite thing to do with a favorite counting book, Rooster’s Off to See the World by Eric Carle.


Included with the book was a baggie that contained a small rectangular board and a bunch of small cards with animal pictures. My daughter pulled out the baggie and asked, “What’s this?” I said it was something we could look at while we read the book. My daughter didn’t want to wait to read the book, so this morning while I made breakfast, she plopped down on the kitchen floor to explore the baggie of cards on her own.

Within a few minutes I heard her say, “There’s two cats and two tortoise!” I looked over to see that she had filled her card with animal pictures. And sure enough, the card had two pictures of cats and two pictures of tortoises (or turtles, I’m not sure which yet). I like that all of the counting we do throughout our day has led her to notice and count things on her own without any prompting from me.

After she was sure I had seen the pictures, she cleared off the card and said, “I want to do more.” She put all the cards back in the baggie and started filling the rectangular board again. One thing that was really interesting to me was how she naturally made two rows of three pictures on her card.

When she finished filling the board up a second time I asked, “What do you notice?” I was curious if she would count again or if she would tell me something different. Her response was, “One chicken, one cat, and two fish.” A few seconds later she exclaimed, “And two frogs!”

Then she cleared the board again and started filling it with new animal cards. This time she chose only cards with fish on them. Her observation at this point was, “There’s a lot of fish on here!”

She started digging through the baggie for a few seconds before saying, “I need one more fish.” We haven’t really talked about “one more” very much so it was so interesting to hear her say that. Since she kept laying out the cards in the same arrangement, I’m assuming she could tell there was room for just one more card.

Picking randomly from the baggie wasn’t working so she pulled all the cards out and spread them out on the floor. She couldn’t find a fish so she changed the activity. She asked me, “Where’s the tortoise? Touch it.”

I dutifully touched the tortoise, and that was the end of the activity. She put all the cards back in the baggie and moved on to looking at another one of the books she received yesterday.

The one thing that came to mind as I watched all this unfold was Christopher Danielson’s message: Let the children play:

We adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas.

But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own.

I could easily have overtaken the activity by asking questions such as:

  • How many tortoises are on the board?
  • Are there more cats or fish in the baggie?
  • How many animal cards are not on the board?

All of these are great math questions, but they’re MY great questions, not my daughter’s. I want her to develop her own questions and curiosities to explore. In the end, it was much more fascinating and rewarding for me to see the ways she came up with to explore the cards and to share the things she noticed about them.

Thank you, Mary Beth, for the wonderful gift!

Looking For (and Finding!) Math All Around: Part 3

Welcome to the third and final post in this series of examples of math from the real world – specifically the Halloween aisle at Target. In the first post we looked at packages of 4 party favors, and in the second post we looked at packages of 6 or 8. Today we’re going to move into estimating using packages with larger quantities, and I’ll wrap up by sharing a few bonus images that didn’t fit anywhere else.

If you aren’t familiar with Andrew Stadel’s website Estimation 180, that’s what gave me the inspiration for taking and sharing the pictures in this post. He has a great Ed Talk from this summer’s California Teachers Summit that you should check out.

Here’s a quick rundown of Andrew’s main points:

  • Students often lack the access to opportunities to strengthen their number sense.
  • Estimation is a gateway to better number sense.
  • Estimation is important because it’s an opportunity to take numbers and make sense of things around us.
  • Have students estimate by giving them a visual, asking them a simple question, avoiding guesses, and justifying estimates with reasoning and context clues.
  • Make estimation accessible by having students create their own estimation activities.

When you visit Estimation 180, you’ll see that every estimation activity asks a simple question that fall into three categories:

  • How long…?
  • How many…?
  • How much…?

Then he guides students to estimate using reasoning, not guesses. He always asks students to make an estimate that is too high and an estimate that is too low before asking them for their actual estimate. Finally, students have to provide a reason they chose their estimate.

I say all of this because if you want to get the most out of the pictures I share in this post, then you’ll want to follow this same structure or something similar to ensure students are truly processing the activity and not randomly guessing. And with all that said, let’s get to the pictures! (Click a picture to see and/or save a larger version of it.)

Halloween Estimation 1

How many fingers are in the bag?

How many fingers are in the bag?

Before you look at the reveal, you may want to answer the following questions:

  • What’s too LOW?
  • What’s too HIGH?
  • What is your estimate?
  • What is your reasoning?

Okay, here’s the reveal:

24 fingers

24 fingers

Halloween Estimation 2

How many skull erasers are there in the bag?

How many skull erasers are in the bag?

I suggest asking the same questions that you did for the fingers. It may seem redundant, but what we’re going for is repeated reasoning through repeated questioning. What regularities will students begin to notice the more they estimate using those guiding questions?

I’d also like you to think about these two questions:

  • Do you think the number of skull erasers in the bag is greater or less than the number of fingers that were in the previous bag?
  • What is your reasoning?

And here’s the reveal:

60 erasers (Were you expecting it to be more than double the number of fingers?)

60 erasers (Were you expecting it to be more than double the number of fingers?)

Halloween Estimation 3

How many erasers in the pack?

How many erasers in the pack?

What strategies would you use to estimate here? Technically, your students could slowly count every eraser, so you may want to mesh this estimation with a quick images routine – show the picture long enough that students can get a mental image, but not so long that they can count one by one.

Here’s the reveal:

18 erasers

18 erasers

Halloween Estimation 4

How many party favors in the pack?

How many party favors in the pack?

This is another picture that could benefit from the quick images routine of showing the picture just long enough for students to get a mental image. You may even want to show it a second time to give students a chance to revise their thinking, but still keep it short enough that they can’t count one by one. I especially like how students can use color to help estimate with this picture.

Here’s the reveal:

50 party favors

50 party favors

Halloween Estimation 5

How many stickers in this pack?

How many stickers in this pack?

In order to make a better estimate, you might like some additional information:


How does the side view help you make a more reasonable estimate?

I can give you a bit more information:


How could the measurements help you estimate the total number of stickers?

And, finally, the reveal:

120 stickers (Were you close?)

120 stickers (Were you close?)

If you want to make your own estimation activities for your students, it’s really that easy. Find something that comes in a pack, cover or hide the total quantity if it’s written on the pack, and provide something for students to use as a benchmark. In the case of the previous pictures you could see all or some of the items in the pack to help get a sense of the size of each object.

As promised, I have some bonus pictures to share before signing off. These pictures didn’t fit with the other sections I wrote about, but I still wanted to share them.

Bonus Pictures


I like this picture because there are so many different ways students could find the total number of pumpkins. I also like that some students may notice the tall white pumpkin while others may only see the 3 by 3 array of pumpkins. It reminds me of a similar visual prompt Joe Schwartz shared in a post he wrote about the Notice and Wonder strategy. Scroll down to the section in his post that says “Grade 1.” What I liked was all the different number sentences the teacher recorded to show all the different ways students saw the quantities in his picture.


We saw several examples of arrays in the previous two posts. This is a much larger total than those examples. I like how the rows are spread apart from each other to draw attention to them. However, the columns also stand out because the color of gem is the same within each column. So much to talk about structure here, along with multiplication and fractions.


I considered putting this final image in with the estimation pictures. I didn’t hide how many bubble sticks are in each package, but that doesn’t necessarily tell you how many bubble sticks are in the whole box! I like that there are 10 packs in the two left columns and 1 pack all by itself. It’s such a natural way to show 24 × 11 broken apart into 24 × 10 and 24 × 1. (And I didn’t even plan it. This is how the box was arranged.)

That wraps up my blog series on finding math in the world around us. In addition to getting a slew of pictures you can use in your classroom, I hope it sparks your curiosity the next time you’re out and about, or even hanging around close to home. Maybe something will catch your mathematical eye. if so, snap a picture to share and discuss with your class. Then come back here and leave a comment to let me know about it. I’d love to hear your stories.

Looking For (and Finding!) Math All Around: Part 2

In my previous post, I shared images of various Halloween party favors packaged in groups of 4. Today I’m going to share packages of 6 and 8. (Click a picture to see and/or save a larger version of it.) Without further ado:


What sorts of things do you think students would notice as they looked at this picture? Here are some things I’m noticing:

  • There are 6 bouncy balls.
  • There are two groups of 3 bouncy balls if you look at the columns.
  • There are three groups of 2 bouncy balls if you look at the rows.
  • The top 4 bouncy balls have the same layout as the packages of 4 items from the previous post. It just looks like 2 more have been added at the bottom.
  • None of the colors repeat, so if I think about fractions I could say that 1/6 of the bouncy balls are yellow.
  • On the other hand, 0/6 of the bouncy balls are blue.
  • Half the bouncy balls are in each column.
  • One third of the bouncy balls are in each row.

This is a great time to mention something to be mindful of when using noticing and wondering. You can go in with a plan that your students will notice some particular mathematical idea you have in mind when you share an image, but that is not a guarantee that they will notice it. Especially if students are new to the practice of noticing and wondering, don’t be surprised or discouraged if their observations are not as rich as you were hoping. Perhaps they haven’t had a lot of practice noticing math before. Give them ample opportunities, and honor everyone’s noticings and wonderings even if they don’t match your desired noticings and wonderings.

I like these next two pictures because they show an arrangement of 6 in one column.



When we talk about multiplication, we often think of multiple groups/rows/columns/piles/etc. (emphasis on all those nouns being plural). However, we can’t neglect showing students models of one group of the quantity. If students can only model multiplication with 6 if there are 2 or more groups of 6, then there is a hole in their understanding of multiplication. If on a previous day you talked with your students about the bouncy ball picture being represented by 3 × 2 and 2 × 3, then these pictures are a great opportunity to talk about how to represent one column of mustaches or one column of bats using the multiplication expressions 1 × 6 and 6 × 1. I like that the package of bats also brings in other multiplicative relationships if you think about the number of eyes or the number of wings on all the bats.

It’s fascinating how differently a quantity of 6 can be packaged. This next one was the most interesting to me.


I wonder why they didn’t package the lizards separately like they did with the bouncy balls. Instead, you have two compartments with 3 lizards in each compartment. Whereas in the bouncy ball picture students might talk about rows and columns, this image likely steers conversation to the idea of groups instead.

And again, you can get into even more relationships if you think about the number of legs on all the lizards or the number of eyes. Would students realize the number of eyes is the same for both the package of lizards and the package of bats? How might they prove it to you? Would they understand the layout in the package isn’t affecting the total number of eyes since there are 6 creatures with 2 eyes each in each package? Often things that are obvious to adults are not at all obvious to young children.

Let’s move on to packages of 8. Here’s another example that’s great for talking about one group of a quantity.


Would students think of this as a row or column? In the images of the mustaches and bats, they were stacked one on top of each other in a column, but now we have one row of 8 pencils that can still be represented by a 1 × multiplication expression. Digging a little deeper, they might notice there are sub-groups of 2 within the package which can lead to more discussions about multiplication, or perhaps even fractions. Within this one package, each design makes up 1/4 of the package. If I bought 2 packages of pencils, would 1/4 of all my pencils have skulls?

Here are two final images of packages of 8.



I’ll leave you to notice and wonder about them individually and in comparison to the previous packages we’ve observed. What are you noticing and wondering? What might your students notice and wonder about them? What math topics could these pictures spark discussion of in your classroom?

After looking at all of these pictures, my final wondering for today is this: Why do party favors seem to always be packaged in even numbers like 4, 6, and 8? What products can you think of that are packaged using an odd number of items? Is there a practical reason to package using even or odd numbers?

In my next post we’ll look at packages with larger numbers of items and we’ll even get to do a bit of estimating. Stay tuned!

Looking For (and Finding!) Math All Around: Part 1

We often tell ourselves and our students that math is all around us, but that can ring hollow if you’re someone who looks around and, to be quite honest, you don’t really see it. I’ve been guilty of this myself. In the past I didn’t know what to look for – I didn’t know what “mattered” – so I didn’t really see it.

So recently I started challenging myself to find and share examples of math in the world around me in the hopes of showing others where to start finding it for themselves. I’ve been sharing pictures on my work Twitter account (@RRElemMath) so you can go there if you want to see the random pics I’ve shared so far this school year.

What got me kick started on this mission was taking part in the #mathphoto15 challenge that spanned this past summer. You can scroll through the hashtag to see a huge collection of photos people from all over the world shared on a variety of math topics throughout June, July, and August. You can learn more about the challenge on the official website. There’s even a section called In the Classroom where you can share how you’ve used some of the photos yourself with teachers or students.

Most of the math photos I’ve taken since school started have been at Target. Stores are such rich environments for math noticings, and walking through the store today, the Halloween section was a veritable cornucopia of math imagery. I took so many photos today that I decided to share them across a few posts rather than tweet them out randomly on my Twitter account where they might get lost in the noise. I also wanted to take the opportunity to share mathematical ideas I saw as well as ideas for conversations these photos might spark with elementary school students.

My hope is that browsing through these posts might inspire you to share some of these pictures with your students. Even better, I’d love for you to be inspired to start taking your own photos to share and discuss with your students.

One of the easiest things you can do with just about any of these pictures is to have your students spend time noticing and wondering about them. I recently wrote a post about this for my school district. You can read that here. I learned about the routine from Max Ray-Riek’s book Powerful Problem Solving. In case you don’t have his book handy, you’re in luck because you can read more about the strategy in this short PDF.

This first series of photos all have to do with the number 4. (Click a picture to see and/or save a larger version of it.)

4-Bracelets-Crop 4-Mazes-Crop 4-YoYos-Crop

As you look at the images, what do you think a primary grade student would say about them? Hopefully they would all be able to tell you there are 4 things in each package, but what do you think they would say if you asked them, “How do you know?”

Would they say, “I counted 1, 2, 3, 4”? Do you think any of them would notice the rows of 2 and say, “I saw 2 and 2, and I know that’s 4”? Would you be surprised to hear, “It looks like you could make a square out of the 4 bracelets or 4 yo-yos”?

The quantity may be small, but that doesn’t mean there isn’t room to notice, wonder, and discuss.

If you showed them one picture a day, they might start to notice how 4 is always arranged in roughly the same way. What are they going to say when you show them this?


And what if you show them this next image, but instead of worrying about quantity you ask them, “Which one doesn’t belong?” (h/t


Would they notice the cat notepad is the only one with an orange background? Would they notice the Trick Or Treat notepad is the only one with words? How else might they justify the other notepads not belonging?

Could you do this same activity with any of the other pictures of 4 objects? Don’t worry if you don’t necessarily have answers right away for why each one doesn’t belong. The point is to give your students a chance to think of and, more importantly, justify their own reason.

I have one final picture of 4 that I love because it offers up so much to talk about.


At first it looks just like the image of bracelets or yo-yos, but if students take some time noticing and wondering, one of them is bound to mention the spots on the back of the spiders. This might lead into a wondering about how many spots there are altogether. Someone else might notice the eight legs on each spider, which again might lead to a wondering about how many legs there are altogether.

The addition strategies for finding the total quantities of dots are excellent work for first and second grade students, and the multiplicative work determining the total number of legs is a great fit for second or third grade students. And all of this work is still perfectly appropriate for Kindergarten students because of the concrete image in front of them. Kindergarten students may not multiply 8 by 4 to find the total number of legs, but that doesn’t prevent them from finding the total all the same.

That’s all I have for this post, but come back next time to see some pictures of 6 and 8 in a variety of interesting arrangements.

Go Big or Go Home: Math Rocks Day 2

This has been a busy week, but I can finally sit down to write about day 2 of our Math Rocks class. (In case you missed the post about day 1, here it is.)

One thing that has kept me busy is reading and responding to all of the blog posts that our group has generated this week. Here are a few you should check out if you have a few minutes:

  • Leilani wrote about how one simple sentence led to rich problem solving and discussion last year.
  • Kari shared a story that sounds like it’s straight out of a teacher nightmare, but it really happened to her!
  • Carrie’s post is short and sweet, but I love that she chose to write about Counting Circles in her very first blog post.
  • Brittany shared an honest and touching reflection of an experience in Math Rocks this week.

I’m so impressed by the stories, reflections, and ideas already being shared. It makes me so excited to see what else we have in store this year!

We started Day 2 with some math. This is actually a problem we posed at the end of Day 1, but we never had time to discuss it because setting up everyone’s WordPress and Twitter accounts took quite a while!


This problem actually came from Steve Leinwand’s keynote at Twitter Math Camp 2014. The numbers involved are small, but I chose this problem because the relational thinking involved would likely stretch many of the educators in our group. This is the problem Brittany refers to in her blog post.

After giving everyone 5-10 minutes to solve the problem, I had them go around their tables to share their current thinking. I let them know before they started working that it was okay if they hadn’t finished solved the problem yet. The purpose of the discussion was to give them a chance to share either their solution *or* their current thinking about the problem. Both are perfectly acceptable. I wanted to model this specifically because it’s a teaching move I would like for them to try out in their classrooms. I got the idea from this Teaching Channel video. You’re welcome to watch the whole thing – it’s about introducing fraction multiplication – or you can skip to the 3:30 mark.

After sharing, most everyone was ready to jump into creating a solution together. I had them share their agreed upon solution on a blank piece of paper. Then they had to take a picture of it and tweet it out to our hashtag for the course, #rrmathrocks. As they worked, I walked around and talked to them about how their solution had to be convincing because anyone on Twitter would be able to see it, so the solution has to stand on its own.

I did this intentionally because after they tweeted out their work, I shared with them how they could do something similar in their classrooms by participating in the Global Math Task Twitter Exchange. Each week a class signs up to pose a problem to their grade level hashtag. Other classes from around the world solve the problem and tweet out their solutions. It can be very motivating to students because you’ve provided them a global audience for talking about and doing math. I wrote a post related to this a few weeks ago.

We didn’t talk about their solutions…yet. I have plans for them down the road.

After everyone tweeted out their solutions, we revisited our norms:

  • Share and take turns
  • Give each other time to think
  • Be open minded
  • Share far and wide
  • Be respectful of each other
  • Take risks
  • Always do your best

I’m especially proud of how much they’ve embraced being open minded and taking risks already.

We quickly moved on to reviewing first drafts of our new district common assessments. Our department has to write them, but we try to involve teachers as much as possible in the review process in order to get feedback and to be as transparent as possible. We want to assure teachers our goal is not to trick them or their students.

Since we had a group of educators from grades K-5, and our assessments are for grades 3-5, we paired up the primary teachers with intermediate teachers. The intermediate teachers were responsible for ensuring the primary teachers understood the standard correlated with each question.

Some wonderful discussions ensued. I talked to a few teachers about a question that they felt was one step too difficult for the students. They convinced me to make a change to the question so that it will be clearer from students’ work and answers whether students can truly do what the correlated standard says they should be able to do. Another group had questions about multiplication algorithms. We had a great conversation about the distributive property and the area model, and how these two things can support students up into middle and high school.

After they were done reviewing assessment items, we came back together to discuss ambitious math instruction. I love the phrase “ambitious math instruction.” I didn’t coin it of course. This came from Teacher Education By Design, a project out of the College of Education at the University of Washington. It’s one of my favorite places on the internet.

You should probably check out their page on ambitious math instruction for yourself, but here’s a snippet:

Developing a vision of ambitious teaching and putting it into practice is complex work. The instructional activities, tools, and resources offered by this project are designed to support teachers to learn about and take up practices of ambitious teaching and engage children in rich mathematics. The routine structure of the activities bounds the range of complexity teachers might encounter while creating space for them to learn about the principles, practices, and mathematics knowledge needed for teaching while engaging in the practice of teaching.

What I really like about this is the use of routine activities as a way to allow teachers to try out new ideas and practices within clear boundaries. They go on to share their core practices of ambitious teaching in mathematics:CorePractices

In Texas we have mathematical process standards that tell us what students should be doing to acquire and demonstrate understanding of mathematics. Now I have a set of practices I can share of what teachers can do to support their students in learning and using these processes.

We gave each table one of the core practices and asked them to create a semi-Frayer model that showed why the practice is important, example(s) of the practice, non-example(s) of the practice, and an illustration of the practice. Again, we had them take a quick photo and tweet them out to #rrmathrocks. This time we did pull their tweets up on the big screen and use them to talk through each practice.

Teacher Education By Design currently has 5 instructional activities on their site with more to come. Regina and I chose to share two of them – Quick Images and Choral Counting. Many of our teachers are already familiar with Quick Images, which is exactly what I wanted. Since they are already familiar with the routine, it meant they could focus on looking for the core practices in the videos we watched rather than trying to balance that with learning a new classroom routine. Choral counting was new for many of them, so we shared that activity second.

Before getting into either routine, I wanted to stop and think a bit about number sense. We did the Number Sense Trajectory Cut-N-Sort from Graham Fletcher.

As expected, there was a lot of interesting conversation about which concepts come first and why. I had wanted them to make posters and draw a quick sketch next to each concept, but we were pressed for time so I just had them do the matching and ordering. When they were done, I handed out the complete trajectory so they could self-check and discuss with the other members of their group. Because we ended up going through this activity more quickly than I had planned, I’m going to look for other ways to revisit the components of number sense at a later date. It’s a really rich topic, and I want to ensure our group has a good grasp of all it entails.

We finally went into the Quick Images activity. Regina modeled the activity with the group and did a little debrief before we watched two videos of Quick Images in action in a Kinder and 5th grade classroom. I think this routine is often considered a primary grades activity, so I purposefully showed both ends of the elementary spectrum to give them an idea of how robust it really is. When we discussed the videos, we specifically asked for examples of the core practices in action, and we talked about what math concepts can be explored through this activity.

I had wanted to end this activity by having everyone plan a sequence of 2-3 Quick Images that they could do in their classrooms at the start of school, but we were still trying to make up for some lost time. I’m sad that it didn’t happen because I wanted them to experience what it’s like to think through the planning of this activity. However, since this wasn’t a brand new activity for most of them, I felt like it was okay to let that go for now. Maybe we’ll revisit it in the future.

We then moved into Choral Counting. I led a count with them where we started at 80 and counted by 2s all the way up to 132. In the middle of the count, I stopped everyone and asked what the next number would be, and I asked how the person knew. In our debrief afterward, I admitted that I wasn’t intentional enough about where I chose to stop. I asked the group where I should have stopped, and they agreed that 98 or 100 would have been a better place to stop because students often have difficulty counting across landmarks.

I also asked whether we would say 216 if we continued the count. One person said yes, because all of the numbers are even and so is 216. I did my best to act like the surprised teacher: “Whoa! You just said all of these numbers are even. How in the world could you make that claim so quickly? There are 27 numbers up here!” She shared that the ones digit in each column was an even number. I told them it’s important to keep an ear out for grand claims like this. It’s easy to just accept the statement that all of these numbers are even, but to the untrained elementary school eye, that is not necessarily obvious nor do they necessarily understand why or how it’s true.

We watched a video of a 3rd grade class doing this activity, and again we debriefed with a focus on the core practices. I was so impressed with how intently they watched all the videos and all of the teacher moves they noticed. From conversations I had during the rest of the day, it sounds like some of them are inspired to be more intentional in their planning and carrying out of these types of activities.

Now that we had made up for lost time, I was able to have them practice recording some counts. One of the powerful pieces of choral counting is that how the count is recorded impacts the patterns students notice and the conversation that ensues. I had each person choose a count appropriate for their grade level and record it three different ways. This reinforced what some of them already noticed before about how intentional planning can make these activities that much more powerful.

At this point we were starting to run out of time, so all we were able to do with the remaining time is introduce the book Intentional Talk. We’re not going to read the whole book during this course. It offers so much, but I’d rather be selective and practice a few key strategies out of the book. We’re going to start with chapters 1 and 2 and add another one down the road if time permits. I really want to ensure everyone has the chance to process and practice the concepts from chapter 2 before trying to add more to their plate. If you’re wondering why, check out these posts I wrote about the first two chapters of Intentional Talk here and here.

After reading the first few pages of chapter 1, everyone tweeted out a key point that stood out to them.

IntentTalk1 IntentTalk2 IntentTalk3 IntentTalk4

We wrapped up our intense and amazing two days of learning by telling them about Math Rocks Mission #3. The gist of it is that they have to set goals for themselves and their students. They also have to anticipate the obstacles that might get in the way of meeting their goals. I’ve listed all of the Math Rocks blogs on the sidebar of the Math Rocks site. If you get a chance, you should take a look at their goal-setting posts. I’ve enjoyed reading about how excited they are for the upcoming school year as well as their thoughtfulness regarding their goals and potential obstacles. Not everyone has written yet, so you might wait until Tuesday which is their soft deadline because that’s when I launch Mission #4! We’ll be launching a mission per week up until school starts.

If you’ve made it this far, thank you for reading about our first two days together! It truly has been a privilege to spend 12 hours with this talented group of educators. I can’t believe this is just the beginning. We have 9 after-school sessions together throughout the school year and one half day session to wrap everything up in February. I’m looking forward to it!