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 wodb.ca)


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!

Go Big or Go Home: Math Rocks Day 1

My brain is full! I just finished two amazing days of PD with about 30 educators in my district. I promised I’d blog about it, and I need to because I just have so much going on in my head right now. Like I said, my brain is full!

This school year, I’m leading an advanced course with elementary teachers in my district. I didn’t really have any guidance beyond that, so it was left to me and my co-worker Regina to set some goals and make a plan. All we started with was a name: Math Rocks. And that’s only because our district already offers an advanced course called Reading Rocks.

Back in May, Regina and I put together an application and asked teachers to apply for this course that has never existed before. Amazingly enough, about 36 people took the time to apply. We read through their applications and selected 24 educators to be in our inaugural class. What I like about it is that we have a wide variety of folks – general education teachers K-5, a few instructional coaches, a TAG teacher, and a few interventionists. And within that group we have dual language teachers and inclusion teachers. They are so diverse; I’m excited about the varied perspectives they’ll bring to our work.

We kicked off the course yesterday and today. We’ll continue our work online for the next month before school starts. Once the school year begins, we’ll meet every other Thursday after school throughout the fall semester. We’ll continue into the spring semester with a final meeting in early February. It’s going to be awesome!

But let’s get back to the first two days. This is the most we’ll ever be together in one place: 12 intense hours across two days.

We opened the first day with a little estimation from Andrew Stadel’s Estimation 180. We of course did the task that started it all: How tall is Mr. Stadel?

After everyone made their estimates, we had them take a walk. Every time we asked a new question they had to find a new partner and introduce themselves. We went through the usual Estimation 180 questions:

  • What is an estimate that is too LOW?
  • What is an estimate that is too HIGH?
  • What is your estimate?

We also added some questions of our own:

  • Where’s the math?
  • Which grade levels could do this activity?
  • Which process standards did you use?

Take A Walk

This was a great way to get everyone up and moving at 8:30 in the morning, but it also started something they weren’t going to be aware of immediately. One thing I did very intentionally throughout the two days was embed FREE resources from my online PLC, the Math Twitter Blogosphere (MTBoS). Unbeknownst to everyone, one of my primary goals for the course is to connect them with this inspiring community. And what better way to entice them than by taking these two days to show off some of the rich resources this community creates and shares freely?

Community Circle

After our getting-to-know-you activity, we moved into a community circle. Regina set the tone by talking about why our district is excited about and invested in this course. Then everyone went around to introduce themselves to the group and talk a bit about why they chose to apply for the course. Their reasons varied, but there were some overriding themes. For many of us in the group, math is not a subject we loved as a kid. In fact, several folks went so far as to say they hated it growing up. On the bright side, these same folks want their students to have better experiences with math than they did. Everyone agreed that math is a rich subject, and they want their students to experience and appreciate that richness.

Their stories during the community circle provided a nice segue into our next activity. We asked the participants to reflect on their own experiences learning math. They had to choose three images that came to mind that symbolize what math was like to them as a student and sketch them on a blank sheet of paper. When everyone was finished, we did a gallery walk.

IMG_9587 IMG_9586 IMG_9585 IMG_9584 IMG_9583 IMG_9582 IMG_9581 IMG_9580 IMG_9579

There were a few recurring themes here as well. Many pictures showed formulas with variables. People said that they remembered being told to use these formulas because they would “work” but they never understood what they meant or why they were using them. Many pictures also showed numerous worksheets, indicating that math was more about quantity of problems than quality of reasoning or understanding. For those that said they disliked math as a child, we talked about when that started happening, and the group was split over it being Algebra or Geometry.

By the way, I’m sharing a lot of the negative experiences, mostly because I felt like I was hearing those most, but I do have to say that there were some voices of folks who did like math as a kid or they grew to like it as they got into higher grades. So negative stories were definitely not universal, which was encouraging.

After debriefing these experiences, we watched Tracy Zager’s talk from Shadow Con 2015. This was basically a small teacher-led mini-conference in the “shadow” of NCTM Boston (hence the name). All of the talks given at Shadow Con are available on the website, along with a facilitator’s guide if you’re interested in utilizing any of the videos in your own PD. Two of the videos really struck a chord with me and ended up becoming the inspiration for our two course goals.

Tracy’s video is called Breaking the Cycle. Here’s a short synopsis. I could write a whole blog post about this video and my thoughts on it, but really you should take 15 minutes and watch it for yourself. It’s powerful stuff.

The majority of elementary school teachers had negative experiences as math students, and many continue to dislike or avoid mathematics as adults. We’ll look at how we can better understand and support our colleagues, so they can reframe their personal relationships with math and teach better than they were taught.

We watched the video, debriefed, and then I shared our first goal for Math Rocks: Relationships.


We want our participants to focus on building relationships this year with:

  • their teammates,
  • their administrators,
  • me and Regina,
  • with their students, and
  • with other educators.

We also want them to build their relationship and their students’ relationships with mathematics.

To help them start working on this goal, we took Tracy’s call to action from the end of the video. Each participant chose a word from a word cloud that shows how mathematician’s describe math. Over the course of the next month, as they attend PD and prepare for the start of the school year, their mission is to plan for math instruction with that word as an inspiration and guide. We’ll revisit how this went when we meet back in September.


And then it was time for lunch. Whew! We crammed a lot in that morning.

After lunch we did a little math courtesy of Mary Bourrassa’s Which One Doesn’t Belong? If you’re unfamiliar with this site, students are presented an image of four things. They have to answer one question, “Which one doesn’t belong?” The fun part is that you can justify a reason why each one doesn’t belong. Here’s the one we did as a group:

Everyone had to pick one picture that doesn’t belong and go stand in a corner with other people who chose the same picture. Once they were grouped, they discussed with one another to see if their justifications were the same, and then we shared out as a group. Here are some of their reasonings:

  • The quarters don’t belong because they equal a whole dollar. The value of each of the other three pictures equals part of a dollar (4 cents, 5 cents, 40 cents).
  • The quarters don’t belong because the word you say for their value (one dollar, one hundred cents) doesn’t start with “f” like in the other three pictures (four, five, and forty cents).
  • The pennies don’t belong because they are not the same color as the other coins.
  • The pennies don’t belong because they are the only coin where the heads face right instead of left.
  • The nickel doesn’t belong because there is only one.
  • The dimes don’t belong because they are the only one where the tails side is showing.
  • The dimes don’t belong because the value of a dime has a 0 in the ones place. All the other coins have some number of ones in the ones place (5 ones in 25, 1 one in 1, 5 ones in 5).

Like Estimation 180, this activity was included intentionally because this is yet another FREE resource created by the MTBoS (pronounced “mit-boss”). It’s actually inspired by another FREE resource created by someone in the MTBoS, the Building Better Shapes Book by Christopher Danielson.

After talking about money, we prepared to watch Kristin Gray’s talk from Shadow Con. Hers is called Be Genuinely Curious, and you should take a few minutes to watch it for yourself:

When students enter our classroom, we ask them to be genuinely curious about the material they are learning each day: curious about numbers and their properties, about mathematical relationships, about why various patterns emerge, but do we, as teachers, bring that same curiosity to our classes? Through our own curiosities, we can gain a deeper understanding of our content and learn to follow the lead of our students in building productive, engaging and safe mathematical learning experiences. As teachers, if we are as genuinely curious about our work each day as we hope the students are about theirs, awesome things happen!

Again, we watched the video, debriefed, and then I shared our second goal for Math Rocks: Curiosity.


We want participants to use their time in this course to get curious about mathematics, about teaching, and about their students. We also want them to find ways to spark their students’ curiosity about mathematics.

When you’re curious about something, you need resources to help you resolve your curiosities. I didn’t want the folks in this course to feel like we were going to leave them hanging. That’s when I formally introduced the MTBoS.


I told them the story of how I joined the MTBoS back in August 2012. (On a side note, it’s hard to believe I’m approaching my third anniversary of being part of this amazing community of educators!) This is a community that prides itself on freely sharing and supporting one another. If the educators in Math Rocks really want to take their math teaching to the next level, getting connected to a network like the MTBoS is the way to go.

One of the amazing things the MTBoS has done to help new members join and get started is to create Explore MTBoS. Periodically, the group kicks off an initiative to help new members start blogs and Twitter accounts. Unfortunately, there isn’t an initiative starting up right when Math Rocks is starting, so I started one up myself. I created a blog where I tailored the existing missions from Explore MTBoS to guide our group as they become members of this online PLC. We did the first two missions to wrap up the first day of Math Rocks. Each person had to make a blog and create a Twitter account.

I’ll admit, I was super stoked about this, but I’ll be honest that I threw more than a few people way out of their comfort zone that afternoon. Despite that, they still made their accounts, wrote their first blog posts, and sent out their first tweets. I am so proud of them for taking these steps, and I am eager to see where it leads from here.

That wraps up Day 1, our first 6 hours together. I’ll share Day 2 in another post.

Intentional Talk Book Study, Chapter 2

I don’t know that I would ever recommend reading Intentional Talk cover to cover. I know that’s exactly what I’m doing this summer, but I can see why it might backfire with regards to changing teacher practice. Across 6 chapters, roughly 112 pages, the book goes in depth into 6 different ways to plan for and structure classroom conversation. The material is so rich, I can see teachers feeling overwhelmed attempting to put all of it into practice in a meaningful way.

In fact, that happened this past year at an elementary school in my district. The faculty did a book study of Intentional Talk, but by the end of the year there was little evidence that teachers were leading varied and intentional discussions. They liked what they read, sure, but trying to plan for and implement all these discussion types got pushed to the side because of numerous other demands on their time. They suffered from the problem of biting off more than they could chew.

This is a defeatist way to start a post, but the reason I’m leading with these thoughts is because I’ve been thinking about how to share this book with teachers and begin to help them successfully incorporate these ideas into their practice. My answer is to start with chapter 2.

Chapter 1 is excellent, don’t get me wrong. It provides rationale for why well-planned discussions are important, but honestly that chapter is only a must read for someone who craves background information or a reason to pursue this work. If you’re reading the book, you’ve probably already decided you want to learn more about facilitating classroom conversations. You aren’t looking for convincing. In that case, you probably want to jump right in and learn something new. So my advice is to skip chapter 1 (for now, at least) and move straight to chapter 2.

And then stop.

Seriously. Quit reading the book.

Take the time to apply what you learn in this one chapter. Believe me, there’s plenty to sustain you for a while! Take time to establish your classroom norms. They are critically important to creating the safe, respectful environment your students need before you start tackling the other discussion types. Take time to teach students the talk moves. Students need practice in learning what to say and how to say it. Let them practice. Let yourself practice! The talk moves are probably going to be new to you, too. Practice using them until you feel comfortable with them. That will take the pressure off when you do finally decide to tackle one of the targeted discussion types.

How long should you wait before picking the book up again? I have no idea. But chances are you’ll know when you’re ready. It might take a couple weeks, a month, or it might take a full semester, but at some point you’re going to notice that your students have gotten really good at sharing and discussing their strategies for solving problems. You’re going to realize that you know the talk moves like the back of your hand. That will be the time to branch out and ask yourself the question, “What other kinds of conversations could we be having?”

At this point, I might recommend reading pages 1-5 in chapter 1 to reconnect with the book and the principles that guide it. When you’re done, consult the table at the bottom of page 3 that provides a brief summary of the goals of each discussion type. Read through those and think about which type might support your students where they are currently in their math learning. After you choose one of the targeted discussion types, read the corresponding chapter and then stop. Quit reading the book again, and take the time to practice the new discussion type while continuing to do open strategy sharing.

Assuming you’re like most teachers, you’re going to teach more than one school year. There’s no reason you have to learn and master all these discussion types in one school year. The most important thing you can do is start by creating a classroom culture where students’ ideas are valued by you and their peers. Where students feel safe taking risks in sharing their ideas because they know everyone in the class is there to support each other in making sense of mathematics. Chapter 2 will help you with this goal. The rest of the book is great, but it can wait. No rush.

Intentional Talk Book Study, Chapter 1

“Math discussions aren’t just about show-and-tell: stand up, sit down, clap, clap, clap.”

Designing and implementing quality mathematical discussions takes effort. It’s not as easy as just having students get up and share their answers to a problem. But don’t let that turn you away from working to improve your practice! This is just chapter 1, after all, and there’s still so much to read and learn. In this chapter the authors lay out four principles that should guide our classroom discussions:

  1. Each discussion should have a goal. This means thinking in advance what it is mathematically you want students to get out of the discussion.
  2. Be explicit! Students likely don’t come to you with the skills needed to participate in classroom discussions. The teacher’s role is to help the students learn what they should be sharing and how they should be sharing it.
  3. Students should be talking and responding to one another. It’s easy in a classroom “discussion” for all comments from students to be directed at you, the teacher. Instead the discussion should be a conversation amongst all the students around a particular mathematical idea.
  4. Students must believe that they can make sense of math, and their ideas are valuable, even when they aren’t fully correct. There’s a lot of learning to be found in mistakes, and we need to value those as much as correct answers. Getting students to share means they have to be willing to take risks, and as teachers our job is to make our students feel safe to do so.

In addition to presenting these guiding principles, this chapter also differentiates two types of classroom discussion: open strategy sharing and targeted discussion. Open strategy sharing is what many teachers already do to some degree in their classrooms. This is when you let students share their answers and solutions to a problem. The goal is to get a variety of responses out in the open. 

However, sometimes you have a particular mathematical goal you want to focus on, such as having the students justify why a particular strategy works. That’s when you would use a targeted discussion instead. These types of classroom discussion are much more nuanced and planning for each one is different. This is why open strategy sharing only has one chapter in the book while 5 chapters are focused on the different types of targeted discussion:

  • Compare and Connect – comparing similarities and differences among strategies
  • Why? Let’s Justify – justifying why a certain strategy works
  • What’s Best and Why? – determining the best (most efficient) strategy in a particular situation
  • Define and Clarify – defining and discussing how to use models, tools, notation, etc. appropriately
  • Troubleshoot and Revise – determining which strategy produces a correct solution or figuring out what went wrong with a particular strategy

The chapter includes three vignettes to help illustrate the differences between open strategy sharing and targeted discussion. I love how the authors insert comments about the intentional decision-making the teacher did throughout each conversation. It shows early on in the book that the teacher isn’t being herded into some lock-step approach. Rather, at every moment you have the power to guide and steer the conversation based on the needs of your students.

One thing that really stood out to me that I didn’t catch the first time I read this chapter is that it’s okay to stop a conversation and come back to it later. I know as a teacher I often let conversations run so long that I wouldn’t get to other things I had planned. In my mind the conversation was so great, it was okay that we were cutting into our next subject by 10-15 minutes. I think this speaks to how I wasn’t planning my discussions in advance. I just let them happen and let them run their course for as long as they were interesting. As the vignettes in this chapter show, however, important mathematical topics can be discussed over several class periods instead of trying to cram it all in to one sitting.

The other thing that stood out to me was how these discussions have the power to give a voice to all our students, not just the high achievers or the outspoken ones. Creating a sense of community where all ideas are valued and respected allows all children the opportunity to be heard and to demonstrate what they understand about math. As the authors say in the book, there are many different ways to be smart in mathematics:

  • making connections across ideas
  • representing problems
  • working with models
  • figuring out faulty solutions
  • finding patterns
  • making conjectures
  • persisting with challenging problems
  • working through errors
  • searching for efficient solutions

How much more exciting to look for and honor these skills in our students rather than seeking out just correct answers! Just think of what valuing these skills tells students about what it means to learn and do mathematics.