What Kind of Bigger? -Early Learners and Balances

Sung to the tune: “Lazy Mary Will You Get Up”

Which one is heavy and which is light,

which is light,

which is light?

Which one is heavy and which is light,

The balance will help us measure.

“Making direct comparisons in the preschool classroom is critical preparation for later, more sophisticated indirect measurement activities and builds the conviction that to be accurate measurement must be fair.” – Big Ideas of Early Mathematics

Lesson Plan

 

 

 

Notice and Note

With The Balancing Act as context, these Early Learners (3 year olds) wondered what would happen next as animals got on one side or the other of a see saw Singing the Weight Song, using their arms, and using a homemade balance, they predicted and then represented what happened!

What we noticed:

• Predicting first with their bodies became students’ connector for applying language to their comparing results. Singing and moving their bodies to the Weight Song brought language to the results and explained ‘what kind of bigger’.
• The format of the book invoked some misconceptions. Many students made predictions about heavier and lighter based solely on the number of animals on each side of the balance, and expecting a balanced scale only when there were equal numbers of animals on each side.

How we adjusted the lesson:

• After reading A Balancing Act and noticing misconceptions, we strategically asked students to compare:
  • 1 wooden solid figure to 1 foam solid figure (same size, different mass).
  • 1 bunny to 1 plastic egg, (1:1 and the bunny was heavier).
  • 1 bunny to three plastic eggs (1:3 and the bunny was STILL heavier).
  • 4 plastic frogs to 4 plastic insects (close in size, same number on each side of the balance).

What are next steps?

• Time for students to independently explore.
• Asking “I wonder” and “What if” for example:
  • What if you compared 2 objects that are the same shape and size like two different balls.
  • I wonder if an object that is smaller than another one could be heavier?
  • I wonder if we could balance only one object on one side and many objects on the other side?
• Conversations about what they have noticed using comparison language.”Which side is lighter?heavier? How do you know?”
• Can we compare two objects another way? What one is longer? Shorter? Slower? Faster?

 

Pre-K 3 Little Kittens Have Lost Their Mittens

“Young students come to understand quantities by having lots of experience with counting.” Mike Flynn /Beyond Answers

There is a lot of counting going on in our Pre-K classrooms! In this lesson, we counted, but also we chose to lean in to algebraic thinking as well!

Pre-K had been studying nursery rhymes. With that them in mind, we decided to mathematize 3 Little Kittens Have Lost Their Mittens.

As the Pre-K team and I talked through the idea, we wondered how students would understand that although each kitten has four feet, each kitten would only have two mittens. Additionally, we wanted to take advantage of the literary connection by asking students to retell the story. Finally, we became concerned about whether our goals- literary and mathematical- could be accomplished during one 40 minute lesson.

With all of those discussions in mind, this became our plan.

Three Little Kittens Have Lost Their Mittens- How many mittens would we have if…?

Math Goals:

Count to tell how many.

• one to one
• counts by ones
• organizes to count

Notices patterns:

• one cat / two mittens; two cats / four mittens…

Math Practice:

I can make sense of problems and persevere in solving them.

I can use appropriate tools strategically. (drawing, objects, numbers…)

Materials:

• The nursery rhyme- Three Little Kittens Have Lost Their Mittens
• The video
• Tools with which to represent/solve the problem (“pretend” mittens for our Pre-K kitten actors to wear, cut out ‘kittens’ and ‘mittens’, beans to represent mittens.
• Photos (screenshots from the video) to support retelling and sequencing the nursery rhyme.
• Chart paper.
• Recording paper for students on day 2.

“The three little kittens, they lost their mittens,

And they began to cry.

“Oh mother dear, we sadly fear,

That we have lost our mittens.”

“What! Lost your mittens, you naughty kittens!

Then you shall have no pie.”

The three little kittens, they found their mittens,

And they began to cry,

“Oh, mother dear, see here, see here,

For we have found our mittens.”

“Put on your mittens, you silly kittens,

And you shall have some pie.”

“Purr, purr, purr,

Oh, let us have some pie.”

The three little kittens, put on their mittens,

And soon ate up the pie,

“Oh, mother dear, we greatly fear,

That we have soiled our mittens.”

“What! soiled your mittens, you naughty kittens!”

Then they began to sigh,

“Meow, meow, meow,”

Then they began to sigh.

The three little kittens, they washed their mittens,

And hung them out to dry,

Oh, mother dear, do you not hear,

That we have washed our mittens?”

“What! Washed your mittens, then you’re good kittens,

But I smell a rat close by.”

“Meow, meow, meow,

We smell a rat close by.”

Notice and Notes:

• I can model the story.
• I can organize my collection of mittens to count.
• I can count and tell how many.
• I can record my thinking.
• I can extend the nursery rhyme beyond 3 kittens and determine the total number of mittens the kittens had (with the number of kittens determined by the teacher)
• I can explain my thinking: “If I know how many mittens 3 kittens have, I know how many mittens 4 kittens have because…”
• I can notice/describe a pattern (1 kitten= 2 mittens; 2 kittens=4mittens…

Launch

Day 1:

• Listen to the nursery rhyme video.
• Students help place picture cards in order to retell the story. The teacher could place the first card. If needed, offer two choices for a difficult the next card e.g., Did the kittens wash their mittens next or did mother cat say, “Then you shall have no pie.”
• Retell the story once again with students playing the parts of the kittens and the mother cat. With help from the audience, reference the picture cards to retell.
• When the Pre-K kittens hang up their mittens to dry, we can see two and two and two mittens hanging on the line (two mittens per kitten).
  • What do you notice about the kittens and their mittens? (they were different colors, each kitten had two mittens…)
• Retell the story one final time using manipulatives. We want to see if students might be confused by the two-dimensional kittens having two beans each for mittens. They didn’t!
• We ended with our T-chart. What do you notice?

K                                                       MM

KK                                                    MMMM

KKK                                                 MMMMMM

These are the sequenced cards. Those bags were our pretend mittens. They fit on the kittens’ hands and were way easier to get on and off. You can see them hanging on the line.

              

      

Day 2

• I wonder…”What if there were more than 3 kittens who lost their mittens? Could we figure out how many mittens were lost then?”
• “Let’s imagine that (5) little kittens have lost their mittens and they began to cry. Meow, meow, meow, meow, and they began to cry.”
• How many mittens were lost?
  • Could you show us how many?Could you show how you know?
  • Could you draw a picture of the kittens and their mittens?

Explore:

• Small groups have charts showing how many kittens they will think about who have lost their mittens.
• Students will have a collection of tools (recording sheets, workspaces, pencils, crayons, kittens and beans) available in each group.

Possible questions:

• Tell me about what you’re doing/thinking.
• How many mittens did one kitten have?
• What are you trying to find out?
• What did you do first?

Recording questions:

• Could you draw a picture to show the mittens? Think about how you could draw that.
• Could you write the number/numbers to show how many there are altogether?
• Can you help us remember what the numbers mean? (with pictures or the beginning letter because we knew they had already studied the m and the K!)

What if there were 4 kittens? What if there were 5?

Can you see the labels? 12 mittens and 6 kittens!

What we learned.

• Two days were needed for this experience! Good decision.
• This was a multi-entry task. Everyone had a place at the problem solving table.
• These Pre-K students are becoming more more and comfortable being asked, “What do you notice?” and “What’s the same and what’s different?”
• Anticipating possible struggles and/or stumbling blocks as a Pre-K teacher team helped make this task accessible and meaningful for all of the students.
• I thought students would model the kittens and mittens, push the beans together, and then reorganize them to count. I never saw that happen!
• There were some students who “just knew” without counting how many mittens the next number of kittens would have, because they saw a pattern!
• Students were comfortable being asked to label their numbers. It just seemed to be a reasonable request to them. How lovely to nudge that habit as Pre-Kers!

 

 

 

 

 

Feast for 10- Mathematizing, Deep Learning, and Learning Progressions

“Deep learning is a period when students consolidate their understanding and apply and extend some surface learning knowledge to support deeper conceptual understanding.” Visible Learning for Mathematics.

At Trinity School, Jill Gough, the Director of Teaching and Learning, leads a series of weekly professional development sessions called Embolden Your Inner Mathematician. Inspired by a session called, Sheep Won’t Sleep #Mathematizing Read Alouds – implement tasks that promote reasoning and problem solving, I planned a lesson with first graders mathematizing Feast for 10.

The Plan- Feast for 10

Big Ideas:

• Solving problems in adding and subtracting contexts.
• Developing a repertoire of strategies for addition and subtraction.
• Making strategies for adding and subtracting explicit.

Math Practices

• Look for and make use of structure (to their advantage!).
• Use appropriate tools strategically.

Materials:

• Feast for 10
• Math journals
• Tools/manipulatives

Notice and Note:

• I can represent/record food types and quantities by listening to a story.
• I can determine the total amount of food bought:
  • I can count all.
  • I can count on.
  • I can strategically compose/decompose numbers.
  • I can strategically compose/decompose numbers in more than one way.

Introduction and Warm-up.

Strategic goals for this number talk:

• to use magnetic ten frames and counters as physical tools that visually represent composing and decomposing numbers.
• to connect the students’ strategies used to determine ‘how many’ to a learning progression (Did you count all? Count on? Did you pull numbers apart/put them together strategically?)
• to think aloud with students about how their visual actions/strategies might be recorded with numbers and equations.

We use physical and thinking tools to help us solve problems. In this number talk, the physical tools are double ten frames and counters. The thinking tools might be counting on, making a 10, doubles??? Let’s see what helps us know how many dots we see in all.

Notice and note during the number talk: counting all, counting on, decomposing and composing, making ten, using doubles, other.

The Launch:

• Students gather with math journals and pencil.
• Students listen to the story and when food is being purchased for the Thanksgiving meal, each student will choose their own way to record each type of food and the quantity.
• After the story is read, each student’s task is to determine how much food was purchased for the meal.
• Learning Progression:
  • 4. I can compose and decompose numbers strategically in more than one way.
  • 3. I can compose and decompose numbers strategically to solve problems.
  • 2. I can count on and or back to solve problems.
  • 1. I can count all to solve problems.
• Before beginning the task, ask, what tools might help them work through this task.
  • Physical tools: Cubes, counters, double ten frames, ten frame, fingers, pictures, number line…)
  • Thinking tools: counting all, counting on, making doubles, making 10, facts we know.

The Explore:

If there are struggles by students as they begin the task, we must remember that the goal of unpacking the problem is to support sense making around the context, not to help students come up with a strategy or an answer.

• “What did you know after listening to the story?”
• “What do you know? What are you trying to find out?”
• “I see… can you tell me what you have in your recording?” (e.g., How many of each food from the grocery)
• Ask: “Can you show what you are doing while you are still solving the problem?” Tell them what you notice them doing. (“I see you drawing all of the …. “”I can see you counting all of them one at a time.” “I can see you thinking about a ten frame and putting food on it to help you count.” “I can see you finding food groups that you can combine to make a ten or to make a double”…)
• “Is there a tool that you could try to help you think about the problem?”
• “What could you do first?”
•  

As they work, we might ask:

• “Can you tell me how you solved it?”
• “What did you do? Tell me about your strategy.”
• “Can you tell me how you…”
• “Try to write it so someone can tell exactly how you combined them.”“How did you use that (tool) to…”
•  

We must consider and remember:

• Does a student’s representation match the actions taken and the ways in which the quantities were organized?
• These recordings offer a window into students’ mathematical thinking. They become artifacts supporting class discussions.

The Summarize:

Choose two representations.

• Survey students to tell what tool, physical or thinking that helped them solve the problem. List their suggestions and tally.
• Ask someone to share their way to solve.
• Ask others, “Is your way the same or different? Thumbs up if your way is like ______’s. Did anyone solve it differently than _____’s? David, (you’ve already scouted out David’s way and would like it to be shared), do you think you did it differently? Can you tell us what you did?

Anticipations:

 

What we saw students record as they listened…

Some unanticipated surprises.

• Students discovered the importance of labeling their data so that they would know what the numbers represented. Initially, some students were using numbers from their data more than once- simply combining numbers rather than solving the task.
• A few students had difficulty accessing their data to solve the task because of the way they had recorded it. In one classroom, food cards representing their data in number talk format were offered on Day 2.

Artifacts from students determining how much food was purchased.

This student started the task using the larger quantities of food (‘the top 5 highest numbers’). She counted those amounts by ones and then wrote equations to tell us what she combined. When she was ready to consider the smaller quantities (‘the lower numbers’), she leveled up and strategically combined them to make a 10. THEN, she was again strategic and combined both totals!

 

This next student showed his thinking more than one way!

• On the first page he shows us how he combined amounts to make tens. He showed us how many tens and ones he made in all on his tens/ones chart.
• On the second page he wrote to us about how sometimes he combined two addends to make ten and other times he decomposed and composed to make ten. His labeling made it easy for us to make sense of his thinking!

 

This student struggled to show her thinking on paper. But when she saw these picture cards, she comfortably used them as references and revealed deep thinking. She decomposed the 6 bunches of greens three times to make combine with other addends to make tens! What a fabulous reminder of how showing your thinking might not always be on paper. Thank- you!

Reflections from the evidence:

*A magnetic board showing the images for the number talk was extremely helpful!

• Using red discs on the top ten frame/yellow on the bottom ten frame and physically moving them to demonstrate student strategies helped them see how combinations were decomposed and composed.

*Using this learning progression, encouraged by the strategies outlined in Cognitively Guided Instruction (CGI), were a just right fit for this task.

• The learning progression helped to bring everyone to the problem solving table by providing a starting point for all. This learning progression nudged students to deepen their learning by helping them understand what their next steps could look like.
• Using the word “strategically” in the learning progressions proved to be powerful . Students knew how to break numbers apart, but the word “strategically” influenced their decisions about what and when to break apart and what to combine.

*We needed at least two days for this lesson. Deep learning takes time!

*During the lesson:

• Students needed more opportunities to talk to each other:
  • About how they were recording their data (numbers, pictures, initials, words, ten-frame pictures…) as the story was being read
  • About the strategies they were trying to use to determine the total amount of food (making tens, counting by ones, writing equations, drawing pictures, adding, subtracting, a combination of strategies…

*During the summarize:

• The whole class, and I, needed more time during the summarize to compare representations.

What’s next?

In Principles to Actions: Ensuring Mathematical Success for All, Mathematics Teaching Practice 8 states, “I can elicit and use evidence of student thinking.”

Referencing these learning progressions, #LL2LU by @jgough and @jwilson828:

The lesson elicited multiple student responses, but we needed more time. We wanted students to make their thinking visible to each other as well as to us and have the opportunity to make connections among those representations.  And we needed the time to notice and note student thinking. Their conversatons and representations (our evidence of their thinking) will drive our instructional decisions.

My ‘next steps’ task for these students will include more time to elicit and use student thinking while solving a task that mathematizes Hershel and the Hanukkah Goblins.

To be continued…

Conceptual Learning Progressions in Kindergarten

During a workshop training at NCSM this spring, a group of math coaches from Howard County, Maryland shared their work with conceptual learning progressions.

“This framework creates conditions for teachers and coaches to partner and think deeply about math content, how students might learn the content, and how teachers can assess student learning of the content in effective and efficient ways.”

– Howard County Public School System

I wondered…

“Could we at Trinity School continue to build our conceptual learning progressions with these protocols?” 

“Could we deepen our math content?”

“And approach solving math tasks using both a teacher and a student lens?”

“And… do it collaboratively, in teams?”

We agreed to give it a go.

So… this school year, in kindergarten and first grade:

• We will use the Course Blueprints from Illustrative Math to provide a mathematics roadmap for the year.
• Prior to beginning a new topic, we will have extended grade level team collaborations.

During pre-planning, our Kindergarten team came together to think about Numbers to 5.

Our Agenda

Kindergarten Math Planning Session- Topic 1, Numbers to 5.

Resources:

Standards and I Can document

Illustrative Math: blueprints

Howard County Mathematics

Lessons

Kindergarten “I Can” statements.

 

Goals What will we accomplish in our time together?		Outcomes Skill/understanding/strategy we will take away
We will explore Topic 1 – Numbers to 5 including the standards and learning progressions.		We will create a map of what students will be learning and doing.
We will solve counting tasks.		We will uncover students’ strategies, knows, and dos, and add to our map.
We will consider an order for the tasks, thinking about what we are asking students to do and what order makes sense.		We will relate these tasks to our goals for the unit (and “I Can” statements) and consider a conceptual learning progression for counting.
We will investigate Routines and Resource Bank on the Howard County mathematics site.		We will add ideas for instructional routines, including “Count Around the Circle”.

12:00	5 min	Welcome, Goals, and Outcomes
12:05	10 min	Review Topic 1 and standards (map)
12:15	20 min	Solve tasks What do students have to be able to do/understand to solve the task?
12:35	10 min	Discuss: In what order would you teach these lessons? What is your thinking? (small group)
12:45	5 min	Share with large group & Brainstorm a conceptual learning progression
12:50	10 min	Explore Routines and Resource Bank (Howard County site)

Reflection:

I like:

I wish:

I wonder:

One of the Task Cards

Samples of Teachers’ Feedback

• “I liked thinking about the sequence of sub-skills within the larger objective of “count to five.”
• “I liked thinking through the steps that students needed to know in order to solve the associated problems.”
• “I wish there was more time to fully unpack and discuss our findings.”
• “I like that we were able to work through tasks from the perspective of our students and how we were able to think about the ‘I can’ statements to go with tasks, what might be hard for our students, what questions we might ask them, etc. It was nice to be able to think about topic 1 together before we start and all be on the same page.”
• “I wish we could have time to look at tasks like these and anticipate before each topic together.”

My Takeaways

• It was hard to decide whether to offer only task cards or to give teachers the complete lessons (which were linked in the agenda). Our time together is limited I worried but also wondered:
  • Is having only the task enough as we think about conceptual understanding?
  • Are pedagogical considerations being supported enough through that choice?
• As teachers thought about what students have to know to solve problems and what they have to be able to do to solve them, what was helpful? The tools? The standards? Their “I Can” statements? Hard copies or electronic versions?
• Having more scheduled time seems necessary to discuss solutions, knows, and dos as a team (and the feedback seemed to agree).
• Time to think about sub-skills and enabling bodies of knowledge will help build out our conceptual map.

The First Grade Team collaboration is NEXT!

 

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.

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!

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

Launch

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.

Explore

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?”

and

• “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!

 

 

Summarize

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.

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

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.

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

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.

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…

…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)

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:

“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…

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…

“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.

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.

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.

 

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.

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).

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?”

 

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

 

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.

The Explore

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

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

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.

 

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.

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!!

Differentiated Gameboards

 

Explore

What did we notice students doing?

Here are some of their recordings.

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.

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 !!

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?”

This recording reflects their suggestions.

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.

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

 

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.

 

The Explore

What did we notice students doing?

Here are some of their recordings.

 

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

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

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

 

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.