Plan B During a Kindergarten Lesson

In chapter 8 of Taking Action: Implementing Effective Mathematics Teaching Practices, the authors remind us that we can use “evidence of student reasoning and understanding (or misunderstanding) to inform teacher actions during instruction as well as to assess students’ progress in learning”.

When planning with a kindergarten class, we began by looking at their data. That data told us that these students were organizing collections greater than 10 and tell how many after counting them. Most students were creating equivalent sets. We decided to plan a subitizing lesson. During the lesson, students would participate in a number talk with arrangements of dots and later be introduced to rekenreks using combinations and totals of 5.

As the number talk began, students struggled to answer both “How many do you see?” and “How do you see them?” I was doing all the talking talking talking! So, I stopped talking and suspended the number talk. I chose to move to the rekenrek part of the lesson.

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Here is the rekenrek plan.

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Below, from an earlier blog post, is an explanation of the rekenrek introduction.

 

“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, the 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).

Next, students used the rekenreks to retell the story!

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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.”

For this kindergarten class’s lesson, our story had 5 friends on the bunk beds. Making arrangements using 5 beads, students used their rekenreks to retell the story. Then they used their mathematical tool to play “Guess My Way”. I made a combination of 5 on my rekenrek and and students had to guess the way I had arranged my beads. The original plan would have concluded with students guessing each others’ “way” while further exploring their rekenreks.

However, responding to the information gathered during the number talk, I made the instructional decision to connect their rekenrek combinations to recordings. Students were asked to show 5 in different ways on their rekenreks and then choose their favorite “way” to share with the class. I would record some of their “ways” for all of us to see. And as I recorded, I asked students to “see what they noticed”.

Here is the recording.

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What did students notice?

Mine is the same! I have 1 on the top and 4 on the bottom like that one!”

This one has 3 on the top and 2 on the bottom and this one has 2 on the top and 3 on the bottom.”

These two boxes have the same- 4 on the top and 1 on the bottom.”

Oh! I can’t have 5 on the top and 1 on the bottom. Then it wouldn’t be 5 altogether!”

 

Takeaways:

  • We (teachers) must listen for understandings but also misunderstandings during a lesson.
  • Even when we think we have accurately anticipated student responses, based on data, we might be surprised.
  • Understanding learning progressions and developmental learning trajectories are helpful when planning, anticipating, and responding during instruction.
  • When I’m doing all the talking, I’m the one learning. During the number talk, I had to stop, and look for a plan B.
  • This time, Plan B was offering an opportunity, during the lesson, for students to connect acting out (with the bunk bed story) and constructing combinations (with the rekenreks) to the recordings that represented those combinations.

 

 

 

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Eliciting Evidence of Student Understanding- Place Value in First Grade

“Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.” – from the NCTM’s Mathematics Teaching Practices.

In first grade, teachers are planning a unit on place value. What mathematical understandings do students need? How can we discover what they already understand?

When in kindergarten, students think about teen numbers as 10 ones and some more ones. And in first grade, students begin thinking of a group of ten as one unit and as a collection. They must understand that ten is both one ten and ten ones.

The mathematical standard for understanding place value states: Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

a. 10 can be thought of as a bundle of ten ones- called a “ten.”

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

In Beyond Answers, Mike Flynn reminds us that “as primary students develop their understanding of the base ten number system, they should have plenty of opportunities to contextualize the base ten structure to give it meaning. He suggests that any familiar context in which objects are grouped in tens and ones will help students make sense of this numerical structure. And once they understand that structure, they can begin to develop problem-solving strategies based on place value.

Looking to introduce a context for ten and wanting a task with several entry paths to notice and note place value understandings, I created a bare bones task. Introducing the context of markers coming in packages of ten, I would ask students to help me solve a problem.

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The Plan

 

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A notice and note recording sheet

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A different notice and note format

 

 

 

 

 

 

 

I planned a story with 27 markers. I wanted students to consider a total greater than 19, without it being too large for students to represent with with pictures or manipulatives. In bins at tables, students had snap cubes, ten frames, markers, pencils, and clipboards as available tools.

Here is some of the evidence of student thinking elicited through this task.

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“B” and “C” both counted out 27 and then made groups of ten. Both students also showed how many boxes of ten markers they could make. Can you see how?

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“L” counts out the 27 by writing the numerals, and creates a box of ten. She seems unsure about how many would be in the next box of ten.

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“E” showed two packages of markers counting to 10 for each box. She kept a running count of the total number of markers, counting by 10s. After some adjustments, she showed the 7 leftover markers at the bottom of her page. When I asked her about the yellow highlighted numbers, she said she wanted me to know how many more markers I would need to make another box of ten markers. Notice her equation at the top of the page, “27 + 3 = 30!

 

 

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“A” drew the markers and labeled how many in each box plus the leftovers. And “A” added a sentence and equations. Did you notice his flexibility when writing them? While he was explaining his thinking, he apologized for two backwards letters. 🙂

 

 

 

These next two students began the task by starting with 27 taking away the 7 extra markers.

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“B” first changed the context to cookies! After he took 7 away from 27, he said he subtracted 20 – 10 = 10 and 10. Notice he just X’d off the extras!

 

 

 

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“A” also took 7 away. Notice her sign”Arts and Crafts” where the newly packaged markers will go. I wondered when she said there would be 2 boxes if she “just knew” there were 2 tens from looking at the number 27. So I asked her how she knew there would be 2 boxes. She said, “because 10 + 10 is 20. Wondering what would happen if she considered a different total, I wrote the number 47 on the back of the page and asked how many boxes we could make using 47 markers. Without much hesitation, she said, “4”. I asked how she knew and she said because “20 + 10 + 10”.

 

Finally, here are “D”‘s recordings. She reminds us that looking at recordings is only part of the evidence we gather to uncover mathematical understanding.

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This is how she explained her work.

“I had 27 sharpies. I made 1 box of 10. Then I made a box of 5 and another box of 5. That could make another box of 10.

I asked, “Did you have any markers leftover?” She wasn’t sure. I asked her to think about it and I’d ask her later. When I did, she said, “7. I had 7 leftovers.” Do you see her thinking in her recording?

During the summarize in another class,

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S1, “It keeps getting bigger.”

S2, “They all have tens.”

S3, “I see one 10 and there’s a 1 (pointing to the 1 in 17). I see two 10s and there’s a 2 (pointing to the 2 in 27), three tens and a 3, and four 10s and a 4.”

S4, (revoicing S3), “There’s a 10, and there’s a 1”.

Me, “Is that a 1? There are 10 markers. So what is this” (pointing to the 1)?

Ss, “That’s 1 box of markers!”

Me, “So this is two 10s, so what does the 2 mean?”

Ss, “Two boxes of markers!.”

Boom!

A connection was noticed between ten ones and one ten. The context using markers and packages helped students view those quantities and their relationship. There will be many more opportunities for students to investigate these relationships. The evidence elicited from this experience will inform those instructional decisions. And “once (students) understand the structure, they can begin to develop problem-solving strategies based on place-value.”

Boom again!

 

 

 

Using Resources Flexibly- Graham Fletcher’s Geometric Subitizing Cards with Three Year Olds

 

When planning a geometry formative assessment for our Early Learners (three year olds), Math Practice 5, using appropriate tools strategically, seemed especially important. Research tells us that as students understand shapes, they match, recognize, identify, and describe them in different sizes and orientations. Understanding math content is important but because we want students to have conceptual understanding of that content, the math practices are important, too. As Mike Flynn reminds us in Beyond Answers, the math practices identify behaviors, or practices, that students enact in order to understand and become proficient in mathematics.

But what tools could I strategically use as I planned this common formative assessment lesson?

The target skills for this lesson were to notice and note students matching, recognizing, naming, and even describing shapes.  What strategic tools would be helpful for this formative assessment lesson? Knowing that I wanted students’ hands on shapes, and not wanting color and size to limit responses, I considered using Graham Fletchers’s subitizing cards. These cards are black shaded arrangements of shapes. Could I be flexible and use these subitizing cards as a tool for this non-subitizing lesson?

I decided to enlarge Graham’s subitizing cards picturing various arrangements using only circles, squares, and triangles. Using attribute blocks, I gathered those three shapes in different sizes, colors, and thicknesses.  Each student would have a bin of shapes and a black workspace.

Here is the plan for these Early Learners.

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This formative assessment lesson would be my first time teaching math with these students. I would begin with Shapes, Shapes, Shapes, by Tana Hoban. Showing only a few of the pages in this big book, I would ask, “What shapes do you see? Can you show me where they are? Could you find a (triangle)? How did you know it was a (triangle)?” Hearing students’ use of geometric language would inform subitizing card choices during the next part of the plan.

During this first part of the lesson, various students recognized and/or named and/or described the shapes. Wow! We can never be sure what students know unless we ask!

Beginning with one shape on per card, and moving to cards with combinations of two shapes, I asked students to show me the shapes on their workspace.

As they worked, we asked,

“What shapes did you use?”

“How do you know that’s a (square)?

“How many (circles) did you use?”

“What shape is beside the (triangle)?”

As they worked, we noticed, 

With one look, could they remember what to show on their workspaces?

Did they look back and forth to be sure of the arrangments?

Did size or color matter?

Did they place shapes beside each other?

Did their arrangements match left to right with the subitizing card?

 

When it was time to clear workspaces and be ready to consider the next subitizing card, I swiped a xylophone. That would always be our signal to “clear your workspace”.

Here are some photos of students replicating the subitizing cards.

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Partners thinking together.
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Notice the different sizes and colors  used to represent the subitizing card.
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Mathematicians at work and teachers noticing and noting. There were unexpected conversations about position words, too (above, below, beside).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This subitizing card provided some interesting conversation.

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After showing this card, some students thought they couldn’t match it because they didn’t have a “diamond”. This student said she just put these shapes down, and then she turned one. In the photo, the student going to show a friend how she did it!IMG_2353.JPG

 

 

 

 

 

 

 

 

After recreating the subitizing cards, students were challenged to create pictures using only triangles and later, to use only squares. Here are some results!

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Reflections

The target goal for the lesson was to notice and note students’ abilities to match, recognize, name, and/or describe circles, squares, and triangles. There were opportunities with this plan to recognize all of that. Using the subitizing cards promoted the flexibility of responses for which I had hoped. The cards and materials felt “just right” manageability wise. Whew!

There are many more experiences and opportunities for these three year old mathematicians. as they think about shapes and positions. This open-ended response lesson gave a first glimpse of what our students know and understand about shapes as we continue to nurture and support our math community of learners.

 

Beginning a unit about Measurement & Data with a 3- Act task

“…evidence suggests that attention to classroom formative assessment can produce greater gains in achievement than any other change in what teachers do”, Embedding Formative Assessment.

As the school year begins, I am having the opportunity to collaborate with several grade level teams at my school. Our initial work begins with common formative assessments. Embedding Formative Assessment, reminds us that we, as teachers, must be clear what we want our students to learn because if not, we won’t know what evidence to collect. And having that evidence tells us how we can help students.

First grade has decided to begin with a study of measurement and data. The standard explains that students will organize, represent, and interpret data for up to three categories and ask and answer questions about the total number of data points, how many in each category, and how many more/less there are in one category than another. Using the operations and algebraic thinking standard “represent and solve problems involving addition and subtraction”, students can represent and solve the measurement and data questions using objects, drawings, and equations.

When deciding how to begin this formative assessment, I discovered a post from Graham Fletcher, Beginning a Unit with a 3-Act Task.  I began the search for one.   I looked for a 3-act task asking students to represent and solve a problem and compare quantities. Graham wrote a task using measurement and comparison called Lil’ Sister. In this task, sisters’ heights were compared. Their heights were measured with cubes snapped into towers. How much shorter one sister was than another was illustrated and labeled by count in Act 3. of the task.

For our formative assessment, I decided to write a 3-act task modeled after Graham’s task, but using  quantities less than 20. Students would be asked to represent and solve how many less there would be of one quantity than another, but also the total number of objects (peppers). I’m going to live the task with a first grade class tomorrow and reflect on my decisions.

Here is the task I will share with first grade mathematicians.

Act 1: 

Look at the picture.

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  1. What stories come to mind?
  2. How many fewer yellow peppers do we have than green peppers?

Act 2:

What info do you need to solve the problem?

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Act 3

How many fewer yellow peppers than green peppers?

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(I chose to show comparison and ten frames and numerals as possible entry points or connections during the share).

I will share what we notice and note in another post and my reflections!

 

The Right Number of Elephants- Formative Assessment in Kindergarten- teaching the lesson

In a previous post linked here, I shared a plan for a formative assessment lesson in kindergarten. Today we taught the lesson. Here’s what happened…

During the read aloud, The Right Number of Elephants, it sounded like this.

  • “Let’s count their noses!”
  • “Oh no! These elephants are in a bunch on the page. We can’t tear them off and place them on the table. How can we count them?” S1– “Count them in rows. 1-2-3″ S2– “And 3 are on the next row. So 3 and 3 makes 6!”
  • “Let’s double check and see if we still count 10. If we count starting here (from right to left instead of left to right), do you think we’ll still have 10 elephants?”
  • “There are 7 elephants in all. If I see 1 elephant on this page (and then turn the book toward me), how many elephants do you think you’ll see on the other page?” S3“6!” (yes, she knew. I only had to ask).

As students came to the table to count collections, some reflections were:

  • The teacher recording sheets were helpful “look fors”, but teachers wished for additional room for anecdotal notes . We considered using formatted labels to later peel and place on students’ recordings. Here’s an adjustment idea.IMG_0672.JPG
  • Some of the collections were challenging for students when they were asked to sort them into groups. Items in our collections included buttons, bugs, frogs, sea animals, wooden cubes, and seashells. Each individual item, not including the wooden cubes, had unique attributes. Would they sort by color? or kind? or size?
  • Several students struggled to record their collections on paper.
  • The free explore materials and procedures (partners sharing a bin, workspaces for each partner, partners on the rug, clean up a bin and wait for everyone to be ready and then switch materials) were successful in engaging students as they waited for their turn at a table.

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What did we notice?

  • Numeral reversals.
  • Difficulty recording quantities.
  • Several strategies for counting (collections not greater than 11) for example, counting one by one as each item is removed from the bin, lining up to count, making rows, touching or scooting each item…)
  • Noticing attributes in the collections.
  • Identifying one subset greater or less than another and some able to explain how they knew.
  • Identifying how many more in one set than another using different strategies: for example, a visual check, or matching, or counting. One student removed the extras of one of two sets, making it equal to the other, and then knew the extras removed told him how many more that first set had than the other!!! (again, all we had to do was ask).

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With this set-up and rotations we were able to talk with all the students as they counted and recorded. They were engaged for 75 minutes!

Additionally, we were able to have a gathering of all students back at the rug for a summary. I chose two students’ work.

One student was able, after initial trepidation, to show what she knew on paper. She wasn’t sure how to write the numbers, but she went ahead and drew figures to represent the objects!
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The second shared work showed how a student  realized she had made a mistake when counting, by double checking (Yay!), was willing to mark out her original recording (she had recorded with a crayon), and write again!

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Both shares demonstrated making sense and persevering!

So, what’s next?

Provide:

  • Reasons to write numerals.
  • Opportunities to count and record.
  • Opportunities to subitize (for example, number talks)
  • Problems to solve, definitely including- “If there were (4) elephants, how many trunks would we count? How many legs would we count?” Woo hoo!

“…evidence suggests that attention to classroom formative assessment can produce greater gains in achievement than any other change in what teachers do”, Embedding Formative Assessment.

Now our students can be the direct beneficiary of the data we have collected – Embedding Formative Assessment.

A 3-Act Task on a shoestring! M&M Jam

3-Act Tasks are lesson structures designed to specifically engage students in modeling with math. Through Dan Meyer’s blog and Graham Fletcher’s 3-Act Task lessons , I have learned about and found amazing resources.

When our teacher math committee participated in a school-wide “recruiter” fair for new teacher members, the committee wanted to share a quick math lesson with our teammates. Thinking about the structure of 3-act tasks, this was our plan.

Act 1The central conflict is introduced. Typically this is shown in a quick video. For us, it was a small mason jar full of M&Ms.

Act 2We must overcome obstacles, look for resources, and/or develop new tools to solve the conflict. For us, we offered a resource… provided only upon request.


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Additionally, teachers were asked to think of an estimate that was too high (but close) and an estimate that was too low (but also close). During a 3-act task, students could be asked to write these estimates down or record them on a number line.

Each teacher recorded his or her guess on a slip of paper and put a tally to show in what range their guess fell.

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Act 3Resolve the conflict. Set up a sequel/extension. In a 3-Act Task with students, the resolution is often revealed in a video. For us, we emailed the result of our count to teachers.

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With the range of teachers at our school, from Early Learners (three year olds) through sixth grade, this task wasn’t focused on one particular standard or big idea. But in the classroom, we would begin with what we want students to understand (standard/big idea) before choosing a task.

  • What standards/ big ideas could be addressed in your classroom with this 3-act task on a shoestring?
  • What resources for students would be appropriate for those chosen standards?

Your thoughts are welcomed!

 

 

The Right Number of Elephants: Formative Assessment in Kindergarten

“…evidence suggests that attention to classroom formative assessment can produce greater gains in achievement than any other change in what teachers do”, Embedding Formative Assessment.

As Early Learner, Pre-K, Kindergarten, and First Grade teams at Trinity begin the year, they are considering common formative assessment lessons. Using pacing guides and learning progressions, and with the hope of a literature connection, I planned some beginning of the year, common formative assessment lessons.

In this post, I am sharing the kindergarten lesson.

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I began the plan by defining the content targets for the lesson. Included in this plan: knowing number names and the counting sequence, counting to tell the number of objects, and comparing two numbers. To provide developmental details of those learning targets, notes are included in the notice and note section of the plan.

Next, I thought about what questions we could ask students to uncover their understandings. Since students are going to be asked to count collections, I chose The Right Number of Elephants, as the context for the lesson. In this book’s illustrations, there are elephants on each page or two page spread, starting with 10 elephants on the first page and ending with only 1 elephant. These elephants are shown in different arrangements. For example, 8 elephants stand in a nice straight line making counting efficient and accurate. However, 6 elephants on another page are in a big bunch. How can we be certain we’ve counted every one of them? What strategy would be helpful? And when there are 5 elephants spread across two pages, 2 elephants are on one page and 3 are on the other!

Following the read aloud, each student will be asked to count the objects in a collection. The questions included on the planning sheet ask students more than just “how many” there are on their workspaces.

Can students reason abstractly and quantitatively (For example, when thinking flexibly about these collections, do they notice that the total could be sorted and decomposed into smaller combinations?)

Can students make mathematical arguments? (For example, asking, “How do you know you counted them all?” “How do you know there are more shells than buttons?”)

Are students modeling with mathematics? (For example, Do students represent their collections by attempting to draw pictures that look just like the objects or an abstraction like circles pretending to be shells…numbers?)

When writing the plan, I did not include an explicit summarize. I plan to respond in the moment as I get to know this class of mathematicians.

For this lesson, these additional materials are needed:

Collections to count…

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“Clipboards” used by students to record their collections
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Teacher recording sheets to “notice and note”

 

 

 

 

 

 

 

 

 

Kindergarten teachers and I will follow this plan in their classroom tomorrow. In my next post, I will share what happened!