Introducing Rekenreks in Pre-K

A chapter in Beyond Answers by Mike Flynn, is devoted to Math Practice 5: Use Appropriate Tools Strategically.  In that chapter, Mike discusses five distinct categories of tools: supplies, manipulatives, representational tools, digital tools, and mathematical tools. As Mike explains, sometimes we have appropriate tools, but we lack real strategies and techniques to use them well.

Rekenreks are an available and often appropriate tool in classrooms, but a tool rarely chosen by students I am teaching in kindergarten and first grade.

I wanted to plan an explicit introduction to rekenreks that included using them to think flexibly about numbers. But how should that lesson be structured?

I started with a context. In the book Bunk Bed and Apple Boxes by Catherine Twomey Fosnot, the author connects the rekenreks two rows of beads to bunk beds. In The Sleepover, Aunt Kate invites friends for a sleepover. The guests arrange themselves on the two beds. When Aunt Kate brings treats for her guests, she places treat cups matching her guests’ arrangements on the bunk beds.  But each time she leaves to rearrange the treats, the friends move around on the beds, making different arrangements and confusing Aunt Kate.

Rather than read The Sleepover aloud, each class acted out the story. Creating bunk beds with tape on the whole group carpet, six friends arranged, and later rearranged themselves on the beds, with some students sitting on the top bunk and some on the bottom. An additional student in the class played the part of Aunt Kate (or Uncle Charlie). Here is the lesson plan written for Pre-K.

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Here’s Aunt Kate in one classroom coming back to the bunk beds and realizing the arrangement has changed!

Next, students used the rekenreks to retell the story!

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Top- beads on resting side Bottom- 4 and 2

Before putting the rekenreks into students’ hands, I introduced this “new to them” mathematical tool. Red and green stickers were positioned at the top. The red one showed students where beads would rest. To be ready to begin a math challenge, all  the beads rested on the red side. The green sticker marked the go side. When showing math thinking, the beads slid to the green side. To help me (and their teachers) see their “good math thinking”, rekenreks rested on the rug in front of them as they worked.

After students retold the story with the rekenreks and teachers noticed and noted, we played Guess My Way. I secretly arranged my rekenrek into an arrangement of 6 beads. Students guessed my way.

Student: “I have 4 on the top and 2 on the bottom.”

Me: That makes 6! But that’s not my way…

At the end of our lessons, students have free time with the materials. During free time for this lesson, all the students, in each classroom, chose to play Guess My Way!

Pre-K students playing Guess My Way during free time!


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  • Cooperation among characters on the bunk beds (who should move from one bed to another)?
  • Playing Guess My Way was amazing! When students were guessing what my way was, students found many many combinations. During free time following the lesson, all the students voluntarily broke into small groups and spontaneously continued playing the game!
  • During that free time play, students asked each other, “What number do you want to play?” demonstrating an understanding of whole and parts.
  • The notice and note post-its made from the lesson plan represented the different student responses during the lesson.PostIts.jpg
  • This lesson provides an anchor chart for future problems: “Remember when we used rekenreks to show the people on the bunk beds?”


Mathematizing Seven Little Rabbits- part 2

A chapter in Big Ideas in Early Mathematics from the Erikson Institute is titled, Every Operation Tells a Story. Through Seven Little Rabbits, students demonstrated that “a quantity (whole) can be decomposed into equal or unequal parts and the parts can be composed to from the whole.”

I read Seven Little Rabbits and students joined me in the repeated lines.

“Seven little rabbits

Walkin’ down the road

Walkin’ down the road.

Seven little rabbits

Walkin’ down the road

To call on old friend toad.”



Some of the rabbits are going to nap at Toad’s house. Some of them are going to walk down the road. The green tape separated rabbits that were above ground from those below ground.



7 puppies are napping! How                                                  many will be walkin’ down the                                            road?



Students use designed workspaces to retell and model

the story.


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My Takeaways

I am pleased that students were able to pretend the puppies were rabbits.

Using the blanket and tape to separate the above ground and below ground settings in the story seemed clear to students. Whew!

Sometimes during read alouds, students are the actors representing stories. Having concerns about seven three and four-year olds moving about into different groups, wondering if the audience would be able to see the combinations, prompted my use of the stuffed animals. Students were able to see the combinations and name or count or predict because of these tools.

Choosing a set of tiles in only one color for each student’s story telling cup was a good decision. During this lesson, the location of tiles guided student responses. Some rabbits were on the road, some were below ground.  Using only one color made representing the story more manageable for these students. I gave them 8 tiles in their cups to use for modeling the story. If I had chosen to use two colors, they would have needed at least 14 tiles!

The stickies used for teachers to notice and note were appropriate for the range of student responses. A few students needed to count how many rabbits were in one group or another, but others could subitize, and still other students were able to predict how many rabbits would be in the combinations coming next in the story.

And…some students were able to explain how they knew how many would be in the next groups: that when one group got one more rabbit, the other group gave up one. Wow!

Call to action for myself:

I will continue to mathematize read alouds. These young mathematicians are continuing to move along the Counting and Cardinality progression.
















Mathematizing Seven Little Rabbits for Early Learners (three year olds)

“Young students come to understand quantities by having lots of experiences with counting. As they work to count concrete objects, they begin to solidify their understanding that numbers represent a specific value. Students cannot reason abstractly and quantitatively if they have weak number sense, so a big part of working toward this practice standard is helping them deepen their understanding of quantities,” Beyond Answers: Exploring Mathematical Practices with Young Children” by Mike Flynn.

Part of my joyful work at my school is planning lessons for our Early Learners (three year olds). The goal for this lesson provides one of those “lots of experiences” to support mathematical understanding.

early learners rotation 16.png

The lesson begins with reading Seven Little Rabbits by John Becker and Barbara Cooney.  I want students to see the combinations, count how many rabbits in each group, and have the opportunity to notice how those groups change as the story unfolds. Sometimes  students act stories and poems during our math time. But with this lesson, I worry that the combinations won’t be clear with seven student rabbits hopping and sleeping in a small rug space. I don’t have stuffed rabbits to copy the text, so I am going to ask students to use pug puppies and their imaginations to represent the story as we read. Dividing the rug into two parts, one side will be above ground, and the other side will be down in the hole when the rabbits visit Mole. Additionally, on the Mole side, there will be a blanket to tuck sleepy rabbits in as one and then another and another decide to take a nap.

Each time another bunny is tucked in, students will be asked, “how many rabbits are down in the hole? How many rabbits are walkin’ down the road? How many rabbits are there in all?

After using the rabbits (pugs) to tell the story, each student will have a workspace and tiles to retell the story. Wanting a distinction between the above ground and below ground rabbits, I have decided to create a workspace. I have drawn the road on the top half of the space and a hole on the bottom half. Because I want students to notice the decomposed parts of seven, I have also chosen to use only one color for the tiles.

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As I teach the lesson, teachers use stickies to “notice and note” what students do and say as referenced in Five Practices for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein.




More to come after I teach the lesson.



One is a Snail, Ten is a Crab-Teen Numbers part 3!

It is extremely valuable to have the opportunity to teach the same lesson in four different classrooms. In the third kindergarten class, we read the book. The crab and spider’s legs were chosen to be combined and represented on the ten frames. And then, the experience changed!

There were many student approaches for showing 18 legs on the ten frames.

  • Putting 20 counters and taking away 2
  • Counting every leg in the pictures by ones, then counting that many counters on the workspace
  • Counting on from 10 on the workspace, then doing the same on the workspace
  • Seeing 10 crab legs, putting 10 counters; seeing 8 spider legs, placing 8 counters
  • “Just knowing” it’s 18 and placing 18.
  • Putting 9 on one ten frame and 9 on the other.

I decided to select, sequence, and summarize strategies during this explore!

I shared:

  • knowing 18 is 10 and 8
  • counting on from 10 (10, 11, 12, 13, 14, 15, 1, 17, 18)
  • counting each leg.

Although “counting all” is a less efficient strategy , I chose to share it third. The student that counted all the legs in the picture and then counted all the counters on her workspace again  wanted to feel sure she was correct. Her strategy and her perseverance were honored. In visible Learning for Mathematics, Grades K-12, the authors **** John Hattie quote.


This class didn’t have time during the lesson to play the matching game or estimate how many shells. But the explore and conversations during the share were invaluable.

In the final classroom, students and I followed the plan. But after 14 legs were represented on the ten frames, this conversation happened.


Me: “I wonder, is 14 closer to 10 or closer to 20?”

R: “20.” 

Me: “How do you know?”

R: “Because 14 is 6 away from 20, and 3 away from 10.”

Me: “Can you show me how you know 14 is 6 away from 20?”

R: (pointed to the 6 empty squares).

Me: “Can you show me how you know 14 is 3 away from 10?”


R: (First he pointed to the counters. He skipped the 14th one and instead pointed to  the 13th, 12th, and 11th and counted 1, 2, 3).

Me: “Could you move some counters and show me how you can get to 10?”

R: (moved 4 counters and felt some disequilibrium between what he had first said by pointing and what he saw by moving counters). That formative assessment opportunity informs my next plan.


Context mattered. In the previous lesson, we talked about teen numbers with Ten Apple Up on Top. Although I know combinations of apples and combinations of feet don’t have to only be represented as 10 ones + some more ones, these contexts gave an anchor chart feel to the idea that teen numbers are 10 ones and some more ones.This lesson is building conceptual understanding.

Asking the questions, Is (14) closer to 20, or closer to 10? to the whole group rather than partners was advantageous.  Everyone considered the question at the same time. Some students counted on their fingers, no one looked at the number line (like I anticipated they might),  and many used their ten frames to think about the question. We were able to get a quick read on student approaches before the partner work. Mike Flynn reminds us in Beyond Answers that representations help all students see the mathematics. During this lesson, their ten frames provided that representation.

In the classes that played the matching game, the organization of it, more than the questioning, provided opportunities to notice and note student understanding. Playing the game again, asking the questions from the plan, would be a meaningful follow-up.



One is a Snail, Ten is a Crab- Teen Numbers part 2

One is a Snail, Ten is a Crab part 2

What happened?

While reading the book…


As I was reading the book to students, I realized the format of those first pages.

Page 1, 1 is a snail.

Page 2- 2 is a person.

Turn the page, 3 is a snail and a person.

In that moment, remembering how difficult it has been for first graders to share their solutions with words, pictures, numbers, and equations, I decided to take advantage of this book’s format for these kindergarteners.

Page 4-4 is a dog.

Page 5?- I asked students to predict with words and numbers how the total number of feet would be represented on the next page!

Me to the class: Can you tell me with words how you could make five feet?”

Student: One snail and one dog”.

Me: Can someone tell me how you could make five feet using numbers?”

Student: “4 and 1 = 5″.

Making the words and numbers connection to this text wasn’t part of the plan, but I was thankful it popped into my head.

Legs, Workspaces and Counters…


Students helped me choose a page to combine with the crab’s ten legs. Then they showed the total number of legs on their ten frames.


In the moment, I modified the plan by questioning the whole group. Since I am not their every day teacher, I felt I needed a quick formative assessment about the reasonableness of the questions.

Here is one of the exchanges:

Me: “How many legs do you have on your workspace?”

T: “14.”

Me: “How do you know it’s 14?”

T: “Because I have 10 and 4 more and I know that’s 14.”

Me: “Is 14 closer to 10 or to 20?”

T: “10.”

Me: “How do you know?”

T: “It takes 6 more to get to 20 and it only takes 4 to get to 10. So 10 is closer.”

Me: (wow)

Partner Work…


This part of the lesson turned out to be more about organizing and finding matches than questioning. I’m thankful I asked the questions during the launch.

Some of the students’ ideas organizing ideas included:IMG_0762

Putting all the counting number cards in order from 11 to 20 before finding matches, Placing the counting numbers on the workspace in descending order before finding matches; and Sorting types of cards together- all the number cards together, ten frame cards together, phrase cards together before finding matches.

An Extension

Noticing a student easily matching these teen representations, I wondered if recording equations would be a good extension for him  As a quick formative assessment,  I showed him a numeral card (14) and asked him what equation we could write to show

FullSizeRender[3].jpghow many there were. He wasn’t sure. So, I asked him to build the (14) on the two ten frames. After building the (14), I asked him what equation he might write to equal the total number of legs he saw on the frames. He immediately said 10 and 4 is 14. Using the demonstration cards, I challenged him to build one teen number at a time using the counters and ten frames, and then record an equation to show how many.

FullSizeRender[4].jpgFor the lesson summary, I gathered students back to the circle and gave a quick view of a pile of shells. I asked each student around the circle to guess how many shells might be in the pile. Placing them on the ten frames, we talked about how many there were- with a number and with words.

I wish I had asked each student to record his or her guess. But each guess was reasonable and a teen number!

I teach the lesson in two more K classes tomorrow.

To be continued…

One is a Snail, Ten is a Crab: Teen Numbers.

My goal is to plan a lesson for K classes with the target goal , I can compose and decompose numbers from 11-19 into ten ones and some further ones, and I can understand that teen numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

I am choosing One is a Snail, Ten is a Crab as the context for the lesson. This book combines different numbers of legs for readers to count. Since the crab with ten legs is one of the animals in the book, I think that the crab’s legs (10) plus the legs of another animal would create teen number of legs to represent and count.

kindergarten rotation 17.png

Each student will get a workspace with two ten frames,and  cup of counters. Since our last workspace had two side by side ten frames using tiles as manipulatives, to encourage flexibility,  students will use a workspace with ten frames under each other and counters. With photos of the crab’s feet and either the person’s, dog’s, insect’s, or spider’s feet, students will create combinations of two animals’ feet for students to build on their workspaces and know how many feet altogether.

The teachers and I will notice how students approach the task on stickies.

Do students:

  • Just place counters on the ten frames without the need to count?
  • Count by ones as they place each counter on the ten frames?
  • Fill in the first ten frame “just knowing” it’s ten?
  • Count on from ten?

Next, partners will have cards representing teen numbers to match. A match could be a card with 11, a card with one group of ten and 1 one, and a card with two ten frames with one ten frame filled in and one dot on the second ten frame.

Using large replicas of the playing cards, I will demonstrate finding “matches” for the game.

Partners will share one plate of cards with teen number and a workspace (a laminated, open file folder) on which they will place their matches. These cards are part of a unit from the Georgia Standards of Excellence Curriculum Frameworks. As partners manage cards and work to find matches, I will notice helpful strategies to organize and match and share them with the group.

As seen on the plan, I will offer questions teachers can ask partners as they work to find teen numbers represented as pictures, words, and numbers.

If extensions are needed, dry erase markers and erasers will be available for students to record expressions (10 + 4) or equations (14 = 10 + 4) for each matching set,

During the summary, students gather into a whole group and will be asked to quickly look at a pile of seashells (all sizes) and estimate how many seashells there are in all.

As students make their predictions, I will notice:

  • Are students willing to take a risk?
  • Are their guesses reasonable?

Using the ten frame workspaces, we will place the seashells on the ten frames, first stopping after the first ten frame is filled to ask, “How many seashells are there so far.” And then asking “How many seashells they think there might be in all?”.

I am teaching this lesson in four different kindergarten classrooms and I will share those experiences.

To be continued…


PD planning: #Mathematizing Read Alouds


How might we deepen our understanding of numeracy using children’s literature? What if we mathematize our read aloud books to use them in math as well as reading and writing workshop?

Have you read Love Monster and the last Chocolate from Rachel Bright?


Jill Gough and I planned the following professional learning session to build common understanding and language as we expand our knowledge of teaching numeracy through literature.  Each Early Learners, Pre-K, and Kindergarten math teacher participated in 2.5-hours of professional learning over the course of the day.


To set the purpose and intentions for our work together we shared the following:



My lesson plan for Love Monster and the last Chocolate is shown below:


After reading the story, we asked teacher-learners what they wondered and what they wanted to know more about.  After settling on a wondering, we asked our teacher-learners to use pages from the book to anticipate how their young learners might answer their questions.

After participating in a gallery walk to see each other’s methods, strategies, and representations, we summarized the ways children might tackle this task. We decided we were looking for

  • counts each one
  • counts to tell how many
  • counts out a particular quantity
  • keeps track of an unorganized pile
  • one-to-one correspondence
  • subitizing
  • comparing

When we are intentional about anticipating how learners may answer, we are more prepared to ask advancing and assessing questions as well as pushing and probing questions to deepen a child’s understanding.

If a ship without a rudder is, by definition, rudderless, then formative assessment without a learning progression often becomes plan-less. (Popham,  Kindle Locations 355-356)

Here’s the Kindergarten learning progression for I can compare groups to 10.

Level 4:
I can compare two numbers between 1 and 10 presented as written numerals.

Level 3:
I can identify whether the number of objects (1-10) in one group is greater than, less than, or equal to the number of objects in another group by using matching and counting strategies.

Level 2:
I can use matching strategies to make an equivalent set.

Level 1:
I can visually compare and use the use the comparing words greater than/less than, more than/fewer than, or equal to (or the same as).

Here’s the Pre-K  learning progression for I can keep track of an unorganized pile.

Level 4:
I can keep track of more than 12 objects.

Level 3:

I can easily keep track of objects I’m counting up to 12.

Level 2:

I can easily keep track of objects I’m counting up to 8.

Level 1:
I can begin to keep track of objects in a pile but may need to recount.

How might we team to increase our own understanding, flexibility, visualization, and assessment skills?

Teachers were then asked to move into vertical teams to mathematize one of the following books by reading, wondering, planning, anticipating, and connecting to their learning progressions and trajectories.

During the final part of our time together, they returned to their base-classroom teams to share their books and plans.

After the session, Jill received this note:

Hi Jill – I /we really loved today. Would you want to come and read the Chocolate Monster book to our kids and then we could all do the math activities we did as teachers? We have math most days at 11:00, but we could really do it when you have time. We usually read the actual book, but I loved today having the book read from the Kindle (and you had awesome expression!).

Thanks again for today – LOVED it.

How might we continue to plan PD that is purposeful, actionable, and implementable?

Cross posted on Experiments in Learning by Doing.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L. (2016-09-16). Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Norris, Kit; Schuhl, Sarah (2016-02-16). Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K-5 Learners (Kindle Locations 4113-4115). Solution Tree Press. Kindle Edition.

Popham, W. James. Transformative Assessment in Action: An Inside Look at Applying the Process (Kindle Locations 355-356). Association for Supervision & Curriculum Development. Kindle Edition.