Assessment Capable Visible Learners: My struggle to write a learning progression.

“Stated learning intentions have a priming effect on learners. They signal to the student what the purpose is for learning and prevent students from having to fall back to the lowest rung on the ladder, which is compliance. They cause students to see the relationship between the tasks they are completing and the purpose of the learning.

Success criteria empower learners to assess their own progress and not to be overly dependent on an outside agent (their teacher) to notice when they have arrived.”

Developing Assessment-Capable Visible Learners

The learning intention for one of our first grade units focused on adding and subtracting within 20. ‘Making a ten’ as a strategy would play a prominent role; students would work to make their strategies for adding and subtracting, like composing and decomposing numbers, explicit.

Our Feast for 10 lesson would ask students to determine the total number of groceries purchased for the family feast. Adding two pumpkins and three chickens and eight tomatoes and so on would provide the opportunity for students to compose and decompose numbers. I wanted to offer a learning progression to students that would make those strategic decisions visible.

First try:

               Level 4. I can determine the total number of foods bought.

          Level 3. I can strategically compose/decompose numbers.

     Level 2. I can compose/decompose numbers,

Level 1. I can count all.

  • I liked using the word “strategically”.
  • But if level 3 was the goal, it would’t make sense for the students to level up to level 4 just to get the answer. Being strategic should yield the total. So level 3 and level 4 were messy.

So…

Second try:

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There were several parts of this learning progression that troubled me.

  • Should I have two level 3s? Students might level up to level 3, but could they meet the criteria with doubles and near doubles if they didn’t make 10s to solve the task?
  • I separated the word ‘strategic’ (in level 2) from the strategies (in level 3). The goal was for students to be strategic in their use of strategies. At the same time!
  • When I tried to ‘live’ the learning progression as a student, deciding where a response would fall on the learning progression, it wasn’t always clear. How would students be able to assess their own progress when I was struggling?
  • And when Level 4 asked students to show their work so a reader understands, did that statement switch from a content goal to a process goal? Was I mixing apples and oranges?

So, yes… I needed a third try.

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This learning progression made sense to me!

The learning intention asked students to solve problems, think about making tens, and make their strategies explicit.

I felt that this iteration:

  • Cleanly and clearly brought every mathematician to the problem solving table.
  • Empowered students to assess their own progress toward the goal.
  • Demonstrated a visible connection between the lesson and the purpose for learning.

Thinking about these learning progressions with our young mathematicians is hard work. But this one DID empower learners. And that includes ME.

 

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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.

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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!)

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What if there were 4 kittens? What if there were 5?IMG_6434.jpg

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

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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.

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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:

first grade anticipate feast for 10.png

 

What we saw students record as they listened…

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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!

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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!

IMG_6566.jpg  IMG_6568 2.jpg

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!

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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:

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nctmp2a-useevidence.png

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

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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.”

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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!

 

First Graders Owning Their Learning #LL2LU

shoe my work so a reader understands

 

During a comparison lesson, with inspiration from a learning progression, three students owned their learning.

*******

Spencer

“Today I’m going to level up.”

As Spencer began to compare the number of beans in each cup, his goal was to show his thinking so that a reader could understand without having to ask him a question… Level 3.

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After using the number line to find the difference, Spencer flipped his paper over and said, “Today I’m going to level up! Changing tools, he began drawing tens and ones to represent the totals.

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Typically a quiet and independent worker, Spencer motioned me over.

“I leveled up and showed my thinking another way, but I didn’t get the same answer!”

Because this learning progression nudged Spencer to make sense of the problem another way, we were able to compare and connect his two strategies.

Asher

“I’m stuck!

Asher, Spencer’s partner, was influenced differently by the learning progression. Showing the difference between 71 beans and 90 beans, Asher chose to use a number line. Rather than hop by ones or tens and ones to find the difference, Asher hopped one big hop of 20, from 71 to 91. Knowing he only needed to count to 90, Asher didn’t know what to do about that extra hop.

Seeing his struggle, I suggested approaching the problem with a different tool. He collected tens and ones pieces, however he just couldn’t make sense of the problem with them.

With his head lowered, cheeks resting on his fisted hands Asher began to shut down.

“I’m stuck!”

At that moment, I referenced the learning progression.

“Let’s see where you are right now on the learning progression.”

Asher immediately knew he was at Level 2. He knew the answer to the problem.  How to show his thinking was where he was stuck.

Understanding his place on the learning progression lightened his mood. Asher became willing to go back to look at his work again, using that original number line, to make his own sense of the problem.

 

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Hampton

“This is the first time I have a 3!”

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As Hampton worked to determine the difference between 22 and 41, he chose to level up. Hoping to show his work so that the reader could understand without having to ask a question, he labeled the additional amount of beans in the second cup as difference (difrincis).

When reading his work, I made a point to thank Hampton for his label, appreciating how it helped me understand his thinking as he compared the two amounts.

Meanwhile, in another part of the classroom, two of Hampton’s classmates struggled to clearly illustrate their thinking. I invited Hampton to join us.

“Hampton, these friends are trying to show their thinking so that the reader can understand without having to ask a question. Could you share the detail you added in your recording?”

After Hampton shared his recording with his teammates. I added how Hampton’s label made it much easier for me to understand what the numbers in his recording meant.

As the class  gathered for the Lesson Summary, Hampton proudly declared to all,

“This is the first time I have a 3! I showed my work so the reader could understand without having to ask me a question!”

The learning progression provided the guidance Hampton needed to determine for himself what he had accomplished.

**********************************

Three students.

Three different uses of a learning progression.

Spencer was encouraged to level up.

Asher understood what he knew and what his next steps would be.

Hampton realized a goal that was met.

These students were the owners of their learning.


Gough, Jill and Kato Nims. “#LL2LU Learning Progressions.” Experiments in Learning by Doing, 30 Mar. 2018, jplgough.blog/ll2lu-learning-progressions-smp/ll2lu-learning-progressions/.

 

 

 

 

I’m Stuck

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

 

But I’m not helpless.

 

I can read other’s people’s work.

I can practice.

And I can get feedback.

 

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

https://www.youtube.com/watch?v=BY8xrmkN9w4

And Odell needed a quarterback to help him practice.

Someone to pass to him.

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

 

I need that, too.

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

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

“Miss Becky, am I still on fire?”

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

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

Day 1: Discovery

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

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

Day 2: Becoming a Mathematician…

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

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

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

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Share 1: 

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

Share 2: 

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

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

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

Izzy decided to try using the number grid.

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

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

-Izzy, Age 6

 

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

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

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

“Izzy, you are on fire!”

Day 3 and Beyond: A mathematician is born! 

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