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

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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.

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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.

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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).

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Then…

 

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?”

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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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Then…

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.

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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.

Combining Feet, Now Combining Dots!

When students left the summarize from our last lesson,  One is a Snail, students had two takeaways to wonder about:

  • There are different ways to make 15
  • Ten could be one crab or ten could be a spider and a person.

In Embedding Formative Assessment, Dylan Wiliam and Siobhan Leahy remind us that research evidence suggests that classroom formative assessment can have a significant impact on how much students learn. Further, attention to classroom formative assessment can produce greater gains in achievement than any other change in what teachers do.

So thinking about what we noticed and noted, I wondered:

  • About students considering different combinations equaling the same total.
  • About providing a more focused opportunity for students to consider different combinations.
  • Wanting these students to have the opportunity to:
  1. work in partnerships to encourage one to one math conversations.
  2. learn how to create a recording sheet in their journals so a copied worksheet wouldn’t be needed and they could play this game anytime.
  3. have the chance to be flexible when writing equations: 15= 10 + 5 rather than simply 10 + 5 = 15.

When planning, I used this outline provided by Solution Tree: Mathematics at Work, to help make my thinking visible for all of us.

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Prior to the launch, the classroom teachers  pre-planned partnerships and where their game playing spots should be.

Launch

Remembering yesterday’s lesson, we revisited some of the ways students combined legs to equal 15. Today, instead of combining legs to equal 15, they would play a game called Dot Addition. While playing this game, they would combine dots equalling different sums. And, instead of working alone, students would collaborate with a partner.

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Using a directions chart, I introduced the game. Choosing a student partner that could benefit from this extra rehearsal, my partner and I modeled how to set up for the game, including sectioning off the recording sheets in our journals. We played two rounds of the game.

  • Partner 1 places cards equal to the required total,
  • Partner 2  double checks the cards or helps if needed.
  • Then both partners record that same combination.

I modeled recording the combination we made with the dot cards by writing: 10 = 2 + 5 + 3. Several students were concerned that I wasn’t correct. That information is helpful to notice and note and will influence future lessons.

Every student on the perimeter leaned in as we played those first two rounds. They were excited to have their turns play. AWESOME!

With journals and pencils on the rug’s perimeter, together students prepared that first recording page. Partners were called, given one of the differentiated gameboards and a bag of dot cards, and then off they went to play!!

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Differentiated Gameboards

 

Explore

What did we notice students doing?

Here are some of their recordings.

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These partners showed more than one combination equalling the total! Showing more than one way to equal the total was a pre-planned advancing question.

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These partners recorded with dots and equations.

What did you notice?

Summarize.

When planning, for the summarize, I was going to choose two of the students’ combinations having the same sum, with one of them containing the decomposed combinations of the other. However, I wanted to honor all the amazing partner work we had just seen, while still asking students to notice combinations.

When everyone came back to the perimeter with their notebooks, I asked students to find if they had recorded a combination equalling 10. If the combination was recorded in their journals, and it wasn’t already on our chart, they read it to me and it was added to the list. Look at this !!

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From our noticing and notes from these students’ work, we can plan next steps.

Next Steps

We decided next steps were to play Dot Addition again, nudging students to consider combining their combinations of dots in flexible ways. Students connected the dot cards to our previous lesson using One is a Snail, Ten is a Crab. Two people (two sets of 2 feet) were the same as one dog (4 legs)!

During this second Dot Addition summarize, I recorded this combination from a partnership’s gameboard, and asked, “How could we know how many dots without counting each one?”

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This recording reflects their suggestions.

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Until next time!

Investigations in Number, Data, and Space. Pearson Scott Foresman, 2012.

Sayre, April Pulley., et al. One Is a Snail Ten Is a Crab: a Counting by Feet Book. Candlewick Press, 2006.

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

 

 

One is a Snail, Ten is a Crab- combinations equaling 15

After noticing and noting during the lesson using Rooster’s Off to See the World as the context, I knew:

  • I wanted to continue to encourage students to move between the concrete examples and abstract reasoning.
  • I  wanted to provide a context for a problem to be solved.
  • I wanted a task that had multiple entry points- a rich task.
  • The classroom teachers wanted visual references supporting students as they worked independently.
  • I wanted the problem to support multiple responses, encouraging the flexibility to show combinations that equaled 15.

And… I wanted students to think about 15 again, as they had during the Rooster’s Off to See the World lesson. During that lesson, the equation students wrote to determine how many animals went off to see the world was 1 + 2 + 3 + 4 + 5 = 15. One is a Snail, Ten is a Crab could offer more combination possibilities.  1 is a snail, 2 is a person, 4 is a dog, 6 is an insect, 8 is a spider, and 10 is a crab are all in this book! Additionally, this story doesn’t have an animal with 3 or 5 or 7 feet. Those numbers of feet are represented with 1 person and 1 snail (3 feet) , 1 dog and 1 snail (5), and 1 insect and 1 snail (7 feet)… combinations already built into the story!!

Additionally, this task lent itself to several possible extensions.

For example:

  • “If ___ (numbers of ) animals had 15 feet, what animals could that be?”

(4)                  (5)                    (6)

  • “What are the fewest number of animals you could combine to have 15 feet?”
  • “If you added one more animal to your total number of animals with 15 feet, how could that change your animal combination?”

Wanting to make my planning process more visible to the teachers, I used the planning template from Solution Tree’s Beyond the Core. I apologize for the fine print.

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We used the same notice and note recording sheet used with our previous Rooster’s Off to See the World lesson.

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The Launch

Fortunately, I have the big book One is a Snail, Ten is a Crab which makes it easier for students to see. We read the book together, anticipating each illustration. The book includes “1 is…snail, 2 is…person, 3, 4, 5, 6, 7, 8, 9, and 10 is…a crab!

The students’ challenge was to find combinations of animals’ legs that could equal 15! I told them that I had already tried to find ways to combine animals’ legs to equal 15, and had found eight ways!

After encouraging students to record their thinking as they found combinations equal to 15, I posted pictures, like these, as a visual reference.

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The Explore

What did we notice students doing?

Here are some of their recordings.

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The section with the star was a challenge, asking if the student could use 6 animals to make 15 feet.

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Can you see recording the snail’s foot in hops by ones and people feet in hops by twos?

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Can you see this student’s plan? She wrote the combination, and then found animals whose feet would equal the addends.

What did you notice?

 

The Summarize

 

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During the summarize, I had originally planned to choose two representations and notice what was the same about them, connecting those numbers to the context. I chose, instead, this student’s work. His recordings connected his numbers to the story, but he also showed flexibility within his combinations equalling 15. With this clear illustration, I wondered if students would notice.

In the top recording, the student’s combination equalling 15 included 1 crab, 1 snail, and 1 dog. But in the bottom combination, he decomposed the crab into 1 person’s legs and 1 insect’s legs (2 and 6), and then combined those two animals with that same snail and the dog that we saw in the top recording!

I asked students:

What is the same about these two representations?

  • They both have animals.
  • They both have snails.
  • They both have dogs.
  • They both equal 15.

What is different?

  • One has a crab and the other one doesn’t.
  • One has person and a spider and the other one doesn’t.
  • One made 10 with a crab. The other one made 10 with a spider and a person. (Yay!)

Learning about  the students during this lesson has informed my next steps… Dot Addition!

Until next time!

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

Sayre, April Pulley., et al. One Is a Snail Ten Is a Crab: a Counting by Feet Book. Candlewick Press, 2006.

What’s the same? What’s different? Compare and Connect in Pre-K

The target for this Pre-K lesson was to count how many, to know how many after counting, and to notice that equal amounts could be in arranged differently and still be the same total (conversation of number). Looking for a context for this lesson, I noticed something interesting in the illustrations for 1, 2, 3 To The Zoo by Eric Carle.

My Lesson Plan

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On each page of this picture book, the number of animals increased by one. Each animal group had to fit in same sized train cars. First 1 elephant, then 2 hippos, and 3 giraffes stood side by side in their cars. But when there were 4 lions, they had to be stacked in their car to fit- 2 on the top and 2 on the bottom! The next car had 5 bears- their arrangement was 2 on the top and 3 on the bottom.

Counting and Subitizing!!!!

After reading and discussing the animals and their arrangements through 6, students were asked to look at dot cards with different arrangements of 2, 3, 4, and 5 dots. Additionally, I had arrangements of 6 dots available if needed. Students were asked to sort cards into groups by the total number of dots on each card.

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As students sorted cards, individuals were asked,

“How many dots are on your card?”

“How do you know how many?”

“I wonder how they have the same amount when they don’t look the same?”

The Notice and Note recording sheet for teachers.

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What we saw:

  • Most students counted by touching each dot once.
  • No students only looked for cards whose arrangements were a match.
  • Many students “just knew” arrangements of 2 or 3 without counting.

Following the explore, students were then invited to gather back together on the carpet.

The Summarize

I revised the plan, asking students to consider combinations of 4 rather than 3. Most of the students already, “just knew” 3 and I hoped for more noticing than that!

Here’s what happened.

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During this share, we heard Pre-K’ers demonstrate use of:

  • Counting
  • Subitizing
  • Conservation of number
  • Algebraic thinking

Wow! And now, because of what we’ve learned about students from this lesson, their understandings and their struggles, we can strategically plan what’s next.

Megan Franke, in her foreword to Intentional Talk, reminds us that classroom conversations are crucial. Engaging in mathematical discussions in productive ways can help students see themselves as smart and competent in mathematics.

Until next time!

Rooster’s Off to See the World: Noticing and Noting

In our Early Elementary Division, I am having mini-residencies in Kindergarten and 1st grade classrooms. By spending 4 of 6 rotation days in one classroom, we are able to deepen our mathematical understanding at both the teacher and student level.
                            .
I am beginning a ‘first rotation’ with the fourth 1st Grade classroom. Through initial conferencing with the teachers, I learned that their students had been using;
  • number lines
  • part/whole models
  • tallies

Additionally, students were drawing four boxes in their math journals and recording their thinking in them with pictures, numbers, words, and equations.

The teachers and I determined that the goal for this first lesson would be to ask students to solve a contextual problem. As students solved the given problem, we would notice and note the tools and strategies used.

As we considered a higher-cognitive demand task for this lesson, I referenced a helpful checklist by Sarah Schuhl from Solution Tree’s Mathematics at Work™ Workshop.

  • Is the problem interesting to students?
  • Does it involve meaningful math?
  • Does it provide an opportunity for students to apply and extend math?
  • Is it challenging?
  • Does it support multiple strategies?
  • Will students’ interactions with the problem reveal information about their understanding?
Referencing this checklist, I wrote a task, using the context Rooster’s Off to See the World.

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This is the notice and note recording sheet teachers and I used as students worked to solve the problem.

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The Launch.

As I read the story, I stopped often, asking students to help retell it while including the sequence and numbers of animals. Students anticipated what might happen next, and even wondered how those five fish must have sounded when they were talking to the rooster!

When it was time to answer the question, “How many animals went off to see the world?” and “How do you know?”, students went to tables with paper, pencils (plain and colored), two-colored counters, and workspaces (plain on one side and with two ten frames on the other).

The Explore.

What did we notice student doing?

Here are some of their recordings.

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Several students wrote this “sun” view with 15 in the middle and numbers around the sides. I wondered, “How did they know how many circles to draw?”

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Several students wrote equations that equaled 15.

I wondered, “How did you decide to write those equations? How did they help you know how many animals went off to see the world?”

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What did you notice?

The Summarize.

In the original plan, the summarize was to share and connect students’ representations. However, as we noticed and noted what students actually did as they solved the problem, I thought about the idea of contextualizing and decontextualizing. As Mike Flynn says, we need to give students lots of practice interpreting contextualized problems so they see the operations as actions that exist in the real world. Mike suggests one way we can do that is to facilitate conversations with students that encourage them to move between concrete examples and abstract reasoning.

This was one of those opportunities!!!

I wondered, “If I post student recordings using pictures, numbers, and words on a chart, could students verbalize connections among the recordings and to the story?

I posted the recordings, and as students noticed, I noted on the chart.

This is an excerpt from our conversation:

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T: “What does this first triangle mean?”

S: “1 rooster.”

T: “What about these two triangles?” 

S: “2 cats.”

T: “Where is that in this equation?”

S: “1 + 2 = 3.”

T: “What does the 3 mean in this next equation? (3 + 3)”

S: “The rooster and the cats.”

T: “I see 3 + 3 = 6. Where is that 3?”

S: “The 3 triangles.” 

T: “Then where is this 3 (3 + 3)”

S: “You could just take one off the 4 and use it.”

Hmmm. (Students determined it could be helpful to remember what the numbers meant in the story. So we began to label them with letters).

What are next steps?

I want to continue to encourage students to move between the concrete examples and abstract reasoning.

I  want to provide a context for the problem to be solved.

I want a task that has multiple entry points- a rich task.

I want the problem to have multiple answers so students will have different solution paths and therefore additional opportunities to show their thinking.

The classroom teachers asked to include a visual support for students to reference when left to solve whatever problem there was.

Based on today’s work, because of what we’ve learned about these mathematicians, I can plan for what’s next. And I get to come back tomorrow!

Until next time!


Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

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

Noticing and Wondering with Early Learners about Counting

To support development of students’ rational counting skills, in Big Ideas of Early Mathematics, the Early Math Collaborative identifies two Big Ideas on which teachers should focus. The first Big Idea is that counting can be used to find out how many in a collection. The second Big Idea is that counting has rules that apply to any collection- stable order, one-to-one correspondence, order irrelevance, and cardinality.

As we are reminded in Young Children’s Mathematics, students extend their emerging understanding by counting collections of objects. Watching and listening over time helps us see what students know and what they need to learn.

Our Early Learners (three year olds), are studying about harvest and fall. To provide a context and reason to count, we chose to use Ten Apples Up on Top.  Each time another apple would be added, students would represent and count to see how many apples there would be. I found a site that sings the words and shows the illustrations from the book.

Here was the plan. Because classrooms had also been exploring plane and solid figures, I included building a tower of 10 with various sizes of rectangular prisms to connect both counting collections and geometry.  The idea was to teach this lesson consecutively in two Early Learners classrooms.

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In the first classroom:

Students were provided a bowl of light and dark red unifix cubes and a black workspace. As the song Ten Apples Up On Top played, students would show those collections of apples on their workspaces and count to see how many.

  • Would students create columns of cubes, unsnapped and resting on the workspace?
  • Would students snap these easier to snap unifix cubes into stand up towers?
  • Might students simply add one cube to their workspace in a scattered group or a line as each apple was added in the song?

In that first classroom, all of the students chose to snap cubes into towers, adding one at a time as the song continued. The challenge for these students came as they were asked to count how many cubes they had in their snapped together towers.

I noticed:

  • A few students touched each cube in their stack and said a counting word for each cube.
  • Some students touched some of the cubes in the stack one by one while counting, but slipped their fingers past others- even when there might be only a few cubes in the stack to count.
  • Some students were unable to touch the cubes one to one and count.
  • When struggling students was asked to unsnap a tower and count how many, most were able to demonstrate the principle of one-to-one and cardinality by counting each cube, saying a counting word for each, and telling how many there were altogether.
  • Students were able to count how many to varying totals, because the story asked them to count collections of 1 through 10.

Immediately after this lesson, I went to the next Early Learner class. Because of what I noticed with students in the first class, I quickly decided to modify the lesson for the next one.

With this second class of Early Learners, the song and book were able to be projected on the smart board. I chose not to give students cubes and workspaces at first. Using an apple pointer (how fortunate was that!), individual students were invited to take turns counting apples on the screen.

My questions for students included:

“Can you count how many apples?” 

“How many apples are there?”

As the song played, I also asked,

How many apples do you think there will be now?”  and

literacy questions for example:

Why do you think that tiger looks grumpy?” (he didn’t have as many apples as his friends)

 

I was surprised at how well these three year olds could manage the pointer to count one to one. The rest of the students watched intently (and surprisingly) as one by one, students took a turn with the pointer. There were predictions about how many apples there would be next. BI decided not to use the cubes and workspaces at all during this lesson.

To summarize this lesson, I asked students to work together to build a tower of 10 using different solid figures (rectangular prisms) . We noticed whether these figures’ faces were rectangles or squares and how we knew before adding each block to the tower!

During both of these lessons, we noticed what students did.

Now, I needed to research those the Big Ideas on a more granular level.

In Big Ideas of Early Mathematics, explains that the one-to-one correspondence principle means a child learns to coordinate the number words with the physical movements of a finger and the eye along a line of objects.

One of the rules that apply to any collection, the order irrelevance principle, generalizes the idea behind one-to-one correspondence. It says no matter in what order the items in a collection are counted, the result is the same and students develop strategies to know which items have been counted and which have not.

So now, I wonder

  • What Big Idea(s) came into play as these three year olds counted the parts of these snapped together collections?
  • Would it have mattered to students as they counted if the snapped together cubes (apples) had been different colors? For example if they were in a pattern (red, green, yellow, red, green, yellow) instead of all red?
  • What might happen if students placed cubes separately on their workspaces and counted, then snapped them together to count again?

These students’ understandings about counting are emerging. What we notice and wonder will guide what we will do next.

I would appreciate your thoughts.