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:

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.

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.

 

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!

 

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.

 

Introducing Rekenreks in Pre-K

A chapter in Beyond Answers by Mike Flynn, is devoted to Math Practice 5: Use Appropriate Tools Strategically.  In that chapter, Mike discusses five distinct categories of tools: supplies, manipulatives, representational tools, digital tools, and mathematical tools. As Mike explains, sometimes we have appropriate tools, but we lack real strategies and techniques to use them well.

Rekenreks are an available and often appropriate tool in classrooms, but a tool rarely chosen by students I am teaching in kindergarten and first grade.

I wanted to plan an explicit introduction to rekenreks that included using them to think flexibly about numbers. But how should that lesson be structured?

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, each 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). Here is the lesson plan written for Pre-K.

Here’s Aunt Kate in one classroom coming back to the bunk beds and realizing the arrangement has changed!

Next, students used the rekenreks to retell the story!

Top- beads on resting side Bottom- 4 and 2

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.

After students retold the story with the rekenreks and teachers noticed and noted, we played Guess My Way. I secretly arranged my rekenrek into an arrangement of 6 beads. Students guessed my way.

Student: “I have 4 on the top and 2 on the bottom.”

Me: That makes 6! But that’s not my way…

At the end of our lessons, students have free time with the materials. During free time for this lesson, all the students, in each classroom, chose to play Guess My Way!

Pre-K students playing Guess My Way during free time!

        

Takeaways

• Cooperation among characters on the bunk beds (who should move from one bed to another)?
• Playing Guess My Way was amazing! When students were guessing what my way was, students found many many combinations. During free time following the lesson, all the students voluntarily broke into small groups and spontaneously continued playing the game!
• During that free time play, students asked each other, “What number do you want to play?” demonstrating an understanding of whole and parts.
• The notice and note post-its made from the lesson plan represented the different student responses during the lesson.
• This lesson provides an anchor chart for future problems: “Remember when we used rekenreks to show the people on the bunk beds?”

 

Planning rich mathematics lessons

There is so much to think about when planning and teaching mathematics lessons. At many schools, there is an adopted text that dictates what teachers teach and when they teach it. At our school, the adopted mathematics textbook provides a resource for lessons, but that text is not the mathematics curriculum. This can feel freeing but also daunting. How can teachers be certain they are planning rich mathematics lessons when they aren’t following a lesson from an adopted text?”

As a math specialist at the school I have wondered, “Could a lesson-planning template be helpful to teachers when they plan their own math lessons? And what would that template look like? In order to plan a mathematically rich task, I wanted that template to include:

• A reference to national standards
• A context
• A connection to recently revised progress reports and “I Can” statements
• A way to provide mathematical progressions (developmental sequences) for the target skill
• An outline for teaching the lesson – (launch, explore, summarize)
• Anticipation student responses
• Connections with the 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein–Anticipate, Monitor, Select, Sequence, and Connect.

Gratefully, I collaborated with Jill Gough about that template. Jill is the director of Teaching and Learning at Trinity School in Atlanta and is my colleague. In addition to her mathematics expertise, she is masterful at listening to ideas and illustrating them through her sketches.

With her help, this is our latest iteration of a mathematics lesson-planning template.

This is one of the completed templates for Pre-K.

And this is an example of a completed template for a First Grade lesson.

 

I believe the important part of this math-planning template is understanding the mathematics content necessary to effectively plan with it.

Mathematics content knowledge is needed to anticipate student responses for a lesson. Anticipating responses helps us be strategic in providing entry points for the lesson and meeting students where they are during the lessons.

For example, during a lesson asking students to physically act out, and later use objects to model How Many Feet in the Bed?, these were anticipated students responses:

• Counting the feet by ones to tell how many.
• Counting the feet by twos to tell how many.
• Starting the count over to know” how many feet in the bed” after another character gets in.
• Counting on from the last count when another character gets in the bed to tell ‘how many feet in the bed”.
• ‘Just knowing’ “how many feet in the bed” without counting.
• Struggling to count or to tell how many in all as the number of feet in the bed increases.

Content knowledge is also necessary as we monitor what our students do during the lessons.

For example, during that How Many Feet in the Bed? lesson, it is important to notice and note:

Can students model the story with objects?

Are students keeping track of an unorganized pile of objects?

Does a student demonstrate one to one?

Do students know how many after counting?

Can students count on?

Where students are in their mathematical understanding is demonstrated in what they are doing and saying. And what they do and say helps us know what to ask, when to nudge, and what should happen in the next lesson.

Creating this template reminds me of planning for guided reading. When planning for a guided reading group, I wouldn’t simply ask students to read and then tomorrow pick another book and ask them to read again. I would choose a focus for the lesson: decoding, fluency, or comprehension (the target). Then I would choose a proper supporting text to match the focus (context). During the lesson, I would notice what students know how to do as they read. I would also notice what they struggle to do and what strategies they use to help themselves when they struggle. I would realize what tools students don’t yet have that can help them with future texts (notice and note). I would know how to notice all of this and make instructional decisions because I know what tools students need to become readers. There is a path. Deepening our content knowledge helps us understand the path in mathematics.

So, this math lesson-planning template is a beginning to planning rich tasks. The template helps us think about the content and the path of learning for the lessons. And it is helping us notice what students do.

What’s next:

It is crucial for me to create efficient and useful meeting agendas as grade level teams and I continue to build our collaborative math communities. During 30-minute meetings once per rotation, we have shared our notes, noticed trends across classrooms, talked about content, and discussed future lessons.

The lesson planning template should evolve to explicitly include the math practices. Routines for Reasoning by Grace Kelemanik, Amy Lucenta, and Susan Janssen Creighton states that a math practice goal is a thinking goal. Those math practices describe how students will reason mathematically about a problem.

So this journey continues…