I’m Stuck

I am a writer and I’m stuck. It’s obvious.


But I’m not helpless.


I can read other’s people’s work.

I can practice.

And I can get feedback.


I must set the goal. I must decide what kind of catch I need to work on and then practice it- from different angles.


And Odell needed a quarterback to help him practice.

Someone to pass to him.

Someone who knew the placements of the ball that would sharpen his skills.


I need that, too.


Building Confidence (Izzy’s 3-day Journey from Self-Doubt to being ‘on fire’

Izzy surreptitiously motions for me to come to her table. Smiling at me she asks,

“Miss Becky, am I still on fire?”

Oh how far we have come in 3 days! As part of our previously discussed ‘nudging exercise’ to encourage children to find more than one way to solve problems (see previous post, The Championship Game: Georgia vs Alabama!), Izzy has gone from confusion to confidence!

I spent three days with Izzy and her classmates solving football-themed problems and nudging them toward thinking outside the box and promoting full use of math tools and strategies.

Day 1: Discovery

During the Day 1 lesson, I discovered that Izzy, and several of her classmates were clearly struggling. They weren’t quite sure how to approach the problem, how a tool might help them make sense of it, or how to show their thinking so that a reader might understand.

After observing these struggles, I decided rather than summarizing Day 1 by sharing student work as planned, I would instead take extra time to look at their recordings after school and strategically choose two of them to begin our second day’s lesson.

Day 2: Becoming a Mathematician…

Learning is not an event but is a process. It does not follow a linear pattern but rather has stops and starts and ‘there is no way to [suggest] a specific order or scaffold of methods.’ 

John Hattie, Visible Learning for Mathematics: What Works Best to Optimize Student Learning

Although I had planned to end yesterday’s lesson with this summarize, I began our Day 2 lesson with these two examples (see below) to show varying approaches to illustrate solving the problem.

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Share 1: 

With the first share, using a 100 grid, a student demonstrated how he hopped in sets of 7 or 3 (for touchdowns and field goals) and recorded with equations after each score.

Share 2: 

For the second share, I chose a student recording using stacks of cubes representing  (a less abstract representation of those points scored) scores in groups of 7 and 3. Additionally, this representation illustrated using a “make a ten” strategy, combining smaller groups (7 and 3) toward scoring a total of 34 points.

Before Izzy and her classmates left that plenary to explore their new football challenge, I asked,

“I wonder…what tool will you try to use today (cubes, counters, 100 grid, ten frames, cups)?”

Izzy decided to try using the number grid.

Checking in with her on Day 2, Izzy confidently explained,

“I am using the number grid. And when I scored a touchdown, I hopped 7 hops and I landed on 7. Then I wrote 0 + 7 = 7. Every time I scored, I hopped that many, and I wrote a math sentence.”

-Izzy, Age 6


Previously on day 1, Izzy had tried multiple tools and had timidly attempted to record something without fully grasping the concept. However, today, she found a tool and recording technique that made sense to her. After recording touchdowns and field goals for this Day 2 challenge, and with time to spare, Izzy asked, ” Can I go back to yesterday’s score and finish?” I responded with an enthusiastic, ‘Absolutely!’

As mentioned before, math learning is not always linear or predictable, and this day was no exception!

Although I don’t remember it, this must be the first time I said…

“Izzy, you are on fire!”

Day 3 and Beyond: A mathematician is born! 

Izzy, the mathematician, needed tools, the chance to choose one that made sense, and encouragement. But what Izzy remembers today is that she believes that she can understand and be understood. Likely her math journey will continue to be non-linear. But,  I believe we all can recognize that she is on fire!




Changing One Number to Another… with Snowflakes

“Children who understand the concepts of more and less understand the relative size of numbers, and the particular differences between them.” “When changing one number to another, children begin with a number and then add on counters or take away counters in order to end up with a specified amount.”- Kathy Richardson.

In Pre-K, our students are strong counters. Additionally, teachers are asking students to make equivalent sets via the counting jar lesson. Students are comparing collections, and being asked which collection has more or fewer objects in it. And some students answer the question, “How many more or fewer  are in one collection than in another?”

But the words of Kathy Richardson kept reminding us about the idea of changing numbers; that 4 isn’t just 4 and 6 isn’t just 6 but 4 is inside 6.

Thinking about the lesson ‘Grow and Shrink’, included in Developing Number Concepts: Counting, Comparing, and Pattern, but with a winter context, this became our lesson for Pre-K… and Early Learners!

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Included with the plan was an illustrated note and note recording sheet helping us think developmentally about how students might approach the task.

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Oops!  Even before I got to the classroom, I modified the plan.

We began with the poem and discussed how more snow can fall and there can be more of it, or how sometimes snow can melt (in the poem, it melts in your hand).

Here’s the modification.

It made better sense to ask students to be snowflakes before giving them the tools to represent them. Down went 10 pieces of tape in two rows of 5 in the center of the carpet (for Early Learners we started with 5 pieces),and, boom, students became snowflakes! These snowflakes would either melt, or be joined by additional ‘snowflakes’, depending on the number of dots showing on the die.

Additionally, wanting these mathematicians to be able to play this snowflake game independently, we needed to model how to manage those dice! So… with a large sized die and bowl, I demonstrated covering the top of the bowl with my hand, shaking three times, “Shake, shake, shake!” and counting the dots on the face of the die looking up at me.


Following this ‘students as snowflakes’ experience, our mathematicians used ten dot workspaces and a bowl of counters/snowflakes to change the snow totals. Students became the dice rollers, taking turns using an applesauce cup and regular sized die, placing  a small hand over the top of the cup, and saying, “Shake, shake, shake!” three times!

Side bar…

I have the honor of co-teaching in Pre-K and Early Learner classrooms. After one lesson in a Pre-K class, I went straight away to try it out with Early Learners. Although these 3 year olds have not had as many experiences as the Pre-K students (this is their first year at our school), as Mike Flynn encourages, these Early Learners have had and must continue to have many many opportunities to count and think about numbers, quantities, and their relationships.

So why not try it?

Therefore… notice and noting happened with Pre-K students and these Early Learners (pictured below!). As you read what we noticed, can you decide in which box each would be noted?


This student took off some counters and is counting to see if he has two counters now.


She is thinking, I have 2 counters, and I need to change it to 5. What do I need to do?


He had 4 counters. He is putting some more on and then he will count to see if that was enough to change it to 6.


As students worked, we asked,

  • “If you are changing the number of snowflakes from (this) to (this), would you need to put more on or take some off?”


  • “When you change from (this) to (this), will you have more snowflakes? or fewer?”

And sometimes, we asked,

  • “How many more snowflakes would you need to (add or take away) to change the number of snowflakes?” And the Early Learner shown above said, “2!”

Sometimes, all we have to do is ask!





The Pre-K teacher  in this class created a workspace and snowflakes students could move. It was a helpful tool for her during their second lesson’s launch, but also during that lesson’s summarize.

For this summarize, thinking about the size of numbers and the concepts of more and less, I took advantage of the snowflake context, and asked a comparing question!

“Claire had 2 snowflakes and Walker had 6 snowflakes. Claire, could you show us your 2 snowflakes? And Walker, could you show us your 6 snowflakes?”

Did you notice how Claire and Walker lined them up so neatly on the workspace???? Wow! I anticipated they might have to match them to answer. Nope.

Then I asked, “Who has more snowflakes, Claire or Walker?”

“Walker, because we you count, 6 comes after 2.” Amazing!

Then I asked, “How many more snowflakes does Walker have than Claire?”

In the photo, this mathematician is deciding what she thinks.

Isn’t that pondering face amazing? Actually, we are both pondering!

What did we learn?

When students had 4 and needed to change to 6, we noticed and noted…

  •  Adding 6 more to the workspace.
  • Counted the 4 counters by 1s and continuing to count by 1s adding the needed counters. (1, 2, 3, 4, 5, 6)
  • Had 4 counters, didn’t recount them, and counted 2 more on. (5, 6)
  • Had 4, needed to change to 6, just put some more on, then counted to see if they actually had 6. (1, 2, 3, 4, 5, 6,7, 8) Then said, “Oh I’ll take off the extras and then have 6”)
  • Had 7, needed 4, isolated the 4, and took the rest off. (1, 2, 3, 4, 5, 6, 7) I see these 4, I’ll take off the extras”)
  • Answer the question, “What did you have to do to change the snowflakes from 4 to 6? “”Put 2 more.”

We didn’t notice any students counting back to change a larger amount to a smaller one.

What fabulous, rich data!

What’s next?

  • After noticing and noting during the whole group, all of the classrooms wanted to follow up with chances to observe students in small groups.
  • Knowing that as we continued to notice and note we would want to use that data to meet students where they were, we thought about ways to differentiate. Initial considerations include:
    • numeral dice rather than dotted dice.
    • dice having higher numbers – for example 3, 4, 5, 6, 7, 8.
    • using the blank side of the workspace rather than the 10 dotted side.
    • using cards rather than dice.
    • continued thoughtfulness about the range of numbers and their differences we offer students. A student using a 1 – 6 dice, changing that total by 1 or 2, might “just know” how to change it. But that same student, using 1 – 6 and asked to change the total from 2 to 6 (a difference of 4) might not “just know”. That student might still need practice with 1 – 6 dice, discovering a strategy for a bigger difference.


Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Stenhouse Publishers, 2017.

Richardson, Kathy. Counting, Comparing and Pattern. Seymour, 1999.

Richardson, Kathy. How Children Learn Number Concepts: a Guide to the Critical Learning Phases. Math Perspectives, 2012.



Surface Knowledge, Deeper Understanding, and First Graders

In this first grade classroom, their teachers noticed that students knew strategies, but struggled to use them flexibly in their work.

John Hattie defines learning as, “the process of developing sufficient surface knowledge to then move to deeper understanding such that one can appropriately transfer this learning to new tasks and situations.” Visible Learning in Mathematics page 35.

Where in this process of learning would we find these first graders?

We began with a “what if” context and an “I wonder”question. “What if” the final score of the College National Championship Game (about to be played in our city) was Georgia 34 – Alabama 27? “What if” Georgia and Alabama only scored their points with touchdowns/extra points and field goals? And “I wonder” what combinations of TDs and FGs could equal those teams’ final scores?

The Plan



Day 1

The Number String

This number string was planned with combining touchdowns and field goals in mind. With these non-contextual equations, would students add doubles, count on, decompose numbers, use derived facts, or make tens to solve them?

Yes they did!

As students shared their strategies, I decided to name and label them on the chart.

Here is the class recording for the string.

1st football number talk.jpg

So now, would students connect these strategies to the upcoming challenge? What strategies might students use during the football challenge and how might those strategies help them be strategic?

  • Would they just keep adding scores until they got to the desired total?
  • Would they start with the total (34) and decompose?
  • If they added scores and the total was greater or less than 34, what would they do next?

The Launch

After completing the number string,  we teachers modified our “notice and note” strategies list to include those we had just seen students use!

So exciting!!!!!!!!

At each table were bins with tools (100 grids, counters, cubes, ten frames, and learning progression cards encouraging students to show their work so the reader could understand without asking a question.  Before leaving the large group to head off to work, each student identified a tool with which they would begin the task.

The Explore

This is what we saw.

Students and Tools

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This student is using the grid to count points. He tried different combinations until he reached the total score. I’m wondering why his first cube was placed on 53.

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Can you tell which color cube marks when a field goal or a touchdown was scored?

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Here, each cup represented a score and in each cup were either 7 or 3 counters. You can see the learning progression card (in blue trim) encouraging students to show their work so that a reader can understand without asking a question.

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This student had begun the task using a grid to count scores and points. But then, she decided to try a different tool. Yay! With cubes, she first built the total points scored and then…

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…decomposed them as she thought about TDs and FGs! Do you see that the last stack only has 2 cubes? I wonder what she will do next?

The Summarize

Although the original plan asked students to determine scoring for both team scores (34 and 27 points), students only had time to think about combinations equaling Georgia’s score of 34 points during this session.

During our summarize, I wanted to share the use of different tools (100 grid, cups and counters, and equations) and some of the strategies we had noticed during the number talk (counting on, decomposing numbers, doubles, and making a 10)

1st football share.jpg

What we learned:

  • That actually naming the strategies gave students the vocabulary (building their toolboxes) to talk about them.
  • That the lesson was messy.
  • That although there were initial concerns about the difficulty of the task, it proved accessible to all students.
  • That for some students, color was an important and useful factor (for example, having 2-colored counters, marking TDs and FGs with different colored cubes).
  • That we wanted to know how they decided what scores to add together and also how their tools helped them decide those combinations.
  • That we wanted to challenge students to explore ways to score 27 points. Having learned about tools and strategies used by individuals, we had ideas where to meet each of them and nudge during this next challenge.

What happened next?

Day 2

Revisiting yesterday’s number string. we listed their strategies at the top of a new recording chart – counting all, counting on, doubles, making a ten, using what you know, decomposing numbers). As strategies for today’s number talk were used, students looked to see if they were similar to yesterday’s list.

Next, students were asked to think about Alabama’s fictional score of 27 points and what combination of touchdowns and field goals could equal that total.

Before sending students off to work, we wondered, “Would they choose to work with the same tool they had used the day before? Would they be influenced by yesterday’s share and choose a different tool? So we asked them! Before they went off to work, we asked students to think about how they wanted to begin and to name that initial tool!

What we saw, noticed, and wondered:

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“E”‘s tool of choice was 2-colored counters. She used the red side for her TDs and her FGs were yellow. It’s interesting that in her recording, she didn’t add the numbers from left to right in the order she arranged her counters. Instead of 7 + 3 + 7 + 3 + 7, she added all the 7s and then the 3s. I wonder why…

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We noticed that “S” labeled his numbers, showed how he combined numbers to equal 27, and recorded his answer to the question of how many touchdowns and field goals equaled  27 points- 3TDs and 2FGs. I’m curious if he used a tool to know what scoring was needed to equal 27 points. Did he think about a number line as he added scores, because part of his recording looks like a number line. I wonder…

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“A” showed how she combined the scores to equal 27 points. She actually used a grid to count on touchdowns and field goals. We can’t see that grid in her recording, but she told us with words. She also explained how many TDs and FGs equaled 27 points. 

I asked her if she might level up by trying to find another way of scoring those 27 points. She shows us her thoughts on the left page of her journal if all the scores were field goals.

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What we’ve learned… and what’s next?:

  • The importance of tools and strategies  (surface learning), but also the importance of providing opportunities for students to apply them (deep learning).
  • What we’ve noticed and noted about individual students helps us plan purposeful and effective next steps lessons.
  • It’s exciting for us and students to continue to learn more about each other as mathematicians.
  • And, that using this learning progressions could be a useful next step.

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Gough, Jill L. “#Learning Progressions: SMP.” Experiments in Learning by Doing, 1 Apr. 2017, jplgough.blog/112lu-learning-progressions-smp/.

Hattie, John. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning. Corwin Mathematics, 2017.


The Championship Game: Georgia vs Alabama!

During collaboration with two first grade teachers, goals for their students were to:

  • solve more story problems (situations)
  • consider using combinations as a strategy rather than simply counting on
  • build number sense through flexible thinking
  • and for us to provide low floor/high ceiling tasks that include all students.

Because we are a school in Atlanta, I was of course drawn to Monday’s game as a context for a problem to solve! After gathering stats on the two teams, and considering which ones could be useful for this first grade lesson, this is became my plan.

In this post, I’m sharing the lesson, the stats I plan to share, and my anticipations of student responses. Although this is very short notice, I would be grateful for any feedback about the lesson or the anticipations. I will teach the lesson Monday, of course :), and soon after, write a post sharing its outcome.

*Side note: On the stats page, I wrote, for example,  “Total Points Georgia 508 to 220”. 220 is the total number of points scored by their opponents. Being concerned about how this data would look visually, I am choosing to explain that detail to students during the lesson.

Thank you ahead of time for your thoughts.



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Shapes: Singing, Creating, and Building with Early Learners (3-year olds)

That shapes can be defined and classified by their attributes is one of the big math ideas from the book, Big Ideas of Early Mathematics. We are gratefully reminded that we want math in children’s eyes, ears, hands, and feet. We want to provide opportunities for our students to explore and define shapes. And we must look for evidence that they see the attributes that differentiate types of shapes.

Here was one opportunity to immerse our Early Learners in Geometry.

The Plan

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Our Notice and Note Recording Sheet


The Launch

We sang!

The Square (sung to the tune of “You are My Sunshine”)

“I am a square, a lovely square

I have four sides, they are the same

I have four corners, four lovely corners

I am a square, that is my name.”


Make a Rectangle (to the tune of “The Eensy, Weensy Spider”)

“A long line at the bottom,

A long line at the top.

A short line to connect each side,

A rectangle you’ve got!

A short line at the bottom

A short line at the top

A long line to connect each side

A rectangle you’ve got!”

We made squares and rectangles with our bodies and stretchy bandages!

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This friend is counting the long sides and the short sides of a rectangle!















The Explore

We used plastic snap together pieces to make squares, and then make rectangles.





The Summarize

Modifying the originally planned summarize, we invited students to make whatever shapes they wished. As they played, we asked questions about their creations:

  • “What is the name of your shape? How do you know it’s a (triangle)? (describing attributes or it looks like a ____)”
  • “Does your shape look like your friend’s shape?  What makes them look the same?”

And as always, we noticed and noted to inform next steps!

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Was math in students’ eyes, ears, hands, and feet during this geometry lesson?

Yes, yes, yes, and yes!

Brownell, Jeanine O’Nan, et al. Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know. Pearson, 2014.

Peas and Carrots with Pre-K: Developmentally Appropriate?

In Pre-K, we wondered whether this peas and carrots task would be developmentally appropriate.

Using In Big Ideas of Early Mathematics as a resource, a path to build an understanding of part/whole relationships was revealed:

  • Perceptual subitizing- students rapidly identify a collection’s total of three or less without having to count.
  • Conceptual subitizing- using two steps, students recognize the small parts, and combine them without counting.
  • Composing and decomposing numbers- students decompose a quantity (whole)  into equal or unequal parts; the parts can be composed to form the whole (which utilizes conceptual subitizing!).

During multiple opportunities to notice and note, we knew that many of our Pre-K students:

  • Knew the counting words.
  • Counted objects using one-to-one.
  • Organized a group of objects to count them.
  • Knew how many after counting.

Additionally, students had had some perceptual and conceptual subitizing experiences through number talks and dot card sorts.

This was the Original Plan using a format from Solution Tree.

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What Happened

While teaching the Peas and Carrots lesson using this plan, students didn’t seem connected to the story using cubes to represent the peas and carrots.  In Kindergarten, prretending to cook them using those cubes, scooping out 5 of with a clear ladle, and having cooking pots and plates on which to serve them, felt like a supportive context for the learning intention. It didn’t feel that way this class of Pre-K’ers.

Knowing that students can use objects, fingers, drawings, sounds, acting out, and verbal explanations when representing situations, during the next class’s lesson, the plan was modified.

The New Launch

With the class seated on the perimeter of the rug, some students were chosen to be peas or carrots and wore either a green or orange headband. All of these ‘veggies’ gathered in the middle of the rug to get cooked. With help from a counting assistant, (we would be serving 5 veggies in all), some peas and some carrots were invited to sit on the stage (the plate).

peas and carrots on stage 2.JPG



I said, “We need to remember how many peas and how many carrots we have on these plates.”

“How many carrots do we have on the stage?”

“What could you draw on this plate to remember that many carrots?”

That first child suggested writing the number! (Surprising!)

I asked, “Could you draw 3 somethings to also show how many carrots you have?”

peas and carrots first plate.JPG


Following that same protocol: 5 different ‘peas’ and ‘carrots’ were invited to the stage and other students were asked to record.

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After acting out the situation twice, we transitioned to using cubes to represent the story. With the help of three students, 5 peas and carrots were served, the number of carrots were recorded, and finally, a student recorded the amount of peas.

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The Explore

With bowls, workspaces, cubes, and recording sheets, students served their own peas and carrots and recorded their combinations.

peas and carrots recording 1.JPG

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What happened?

Some students needed scaffolding to:

  • Have a total of 5 cubes on their workspaces.
  • Include some peas and some carrots in the collection.
  • Record the numbers of peas and the carrots together on the same recording sheet plate.

Most students:

  • Were comfortable recording peas and carrots with pictures on their recording sheets

Some students responded to advancing questions:

  • What do you notice about this plate (2 Ps and 3 Cs) and this plate (2 Cs and 3 Ps)
  • If you have 4 peas, how many carrots would you have to make 5 in all?
  • Without clearing your workspace, could you change how many peas and carrots you have, to make 5 altogether? (for example, substituting one green cube for an orange cube and knowing the arrangement is different without changing the total).

The Summarize

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Looking at these two plates of peas and carrots, these Pre-K students were asked…

What do you notice that is the same?”

  • “They both have 3.”
  • “They both have 2.”
  • “The peas are on this side and the carrots are on that side.”

Me, “What do you notice that is different?”

  • “The 2s are on different sides.”
  • “The 3s are on different sides.”
  • “The peas have different numbers.”
  • “The carrots have different number.”

What did we learn about our students?

I thought about those pathways of conceptual understanding.

Perceptual subitizing. When looking at the total of 5, some students just knew 2 peas, or 3 carrots without counting, but then needed to count to answer how many vegetables there were altogether.

Conceptual subitizing. Some students ‘just knew’ the 2 peas and 3 carrots without counting and were able to combine them to ‘just know’ there were 5 altogether.

Composing and decomposing numbers. Some students created and recorded different combinations of 5 peas and carrots, either by clearing the space each time and building new combinations, or by substituting one color for another to change the combinations without changing the total.

Reflecting on this plan for these students.

Was this a high level task? (adapting from (Taking Action Implementing Effective Mathematics Teaching Practices)

  • There were multiple entry points.
  • There were connections to conceptual ideas.
  • Ideas were represented in multiple ways.
  • Cognitive effort from students was required

But additionally, our intention for a task is not to help the students complete it- it is to help them learn. (Embedding Formative Assessment).

Were there times we had to help too much? 

As reflection continues, your thoughts are appreciated.

Brownell, Jeanine O’Nan, et al. Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know. Pearson, 2014.

Huinker, DeAnn, and Victoria Bill. Implementing Effective Mathematics Teaching Practices in Kindergarten-Grade 5. National Council of Teachers of Mathematics, 2017.

Kanold, Timothy D., and Sarah School. Mathematics at Work™. Solution Tree Press, 2017.

Wiliam, Dylan, and Siobhan Leahy. Embedding Formative Assessment: Practical Techniques for K-12 Classrooms. LearningSciencesInternational, 2015.