Student work is just the best. It is the one thing that will always motivate me to write! So, let’s kick this post off with a great work example from 3rd grade. In this task, students are asked to locate 1 on a number line labeled with 0 and 1/3.

When I look at student work, I typically think about 3 things: 

1. I look for evidence of what the student understands. In this case, it also involves making some assumptions because we can only know so much based on written work.
2. I think about question(s) I would want to ask the student. Again, because we can only know so much from written work, there can be so many hidden understandings that can only be found by talking with the student.
3. I think about next steps, in particular, how I might  create a connected, grade-level onramp experience that might be helpful for the student.  

Before reading on, what do you see in the work? Take a minute to jot down your thoughts and I would love to hear them in the comments at the end! 

Here are some ideas I think the student understands based on the work:

• Since the denominators increase from 1 to 5, there is evidence of an understanding that whole numbers increase as they move to the right on the number line. 
• Since the unit fractions are all between 0 and 1, there is evidence the student knows that unit fractions are less than 1 and can be in many denominations, especially since 5ths are not typically in the work of earlier grades.
• Since the fractions are spaced out fairly equally on the number line, there is evidence the student understands there should be equal intervals between numbers on a number line. 

These are great understandings on which to build for a problem like this. So, what is up with those denominators increasing to the right?

At first, I thought the student was counting the denominator like whole numbers and thinking fractions are always written with a 1 above the fraction line, so it was just ‘put there’ but had no meaning. But then, in the same set of student work I saw this cool-down. 

This student did a similar thing with the increasing denominators, but it was the sketch in the corner that caught my eye. This student is showing the number of equal parts in a whole for halves and  fourths. This made me wonder if the denominator did have a fractional meaning to them and not just a counting number in a sequence.

If the student understands the denominator as a number that represents the number of equal parts in a whole, then it makes total sense that they would label them in increasing order. In that way of thinking, the denominator is like a whole number in their mind and whole numbers increase to the right on the number line. For example halves are 2 parts, thirds are 3 parts, fourths are 4 parts,…etc. So the more parts, the larger the fraction in their eyes. They see the fraction as a numerator and denominator, 2 numbers with different meanings, rather than 1 number located on the number line.

As I looked back at the progression, I realized how intricate this move is from K–2 understanding of fractions as a quantity to grade 3 understanding of a fractions as numbers.

Using the Standards as a reference, in grade 2 students use fraction language to describe quantities such as “half of a rectangle” or “a fourth of a circle.” That half and fourth are quantities that can be a different size each time depending on the whole because it is a fraction of a shape. At the same grade level, within their measurement work, students are locating whole numbers on a number line.

In grade 3, students learn fraction notation to represent those quantities and later merge that understanding with the early number line work to understand a fraction as a number on the number line. That number has a specific location and does not vary in the way the quantities did in earlier grades. It is a super intricate move in fraction understanding which makes it easy to understand why students have brilliant moments like the one in the cool-downs above.

So, let’s revisit the student work.

To learn more about their thinking, I would love to ask the following questions: 

1. How did you use the ⅓ to locate the other fractions?
   • In this question, I am curious if the student split the distance from 0-⅓ into 3 parts (bc of the 3 in thirds) and then did the same on the other side of ⅓ and that is how they determined to stop at ⅕ and then put 1 – three intervals on each side of the ⅓. 
2. What did you pay attention to when you put the other fractions on the number line?
   • Want to hear about the equal spacing of the fractions. 
3. Why did you stop at ⅕? Where would you put ⅙?
   • This could be answered in question 1, but if not, I am so curious about ⅙. 
4. [draw a number line labeled with 0–1] Where is ½?
   • Do they know ½ is in the middle of 0–1?
5. [draw an identical number line below the first one] Where is ⅓? 
   • Do they know it is 3 equal intervals between 0–1? And, if the number lines are the same length, I want to see if they can reason about ½ being larger than ⅓.

Now, the fun part of thinking about next steps. How can we connect what students understand about equal-sized parts to understanding a fraction as a number on the number line? It seems like activities that connect the number of parts in a whole to the size of one part would be helpful here. Right now, I think the students understand 1/4 as 4 parts and maybe 3/4 as 3 out of 4 parts. But, when moving to understanding a fraction as a number we want them to understand 1/4 as the size of 1 part of a whole partitioned into 4 equal parts and 3/4 as 3 pieces the size of 1/4.

There are many ways to think about this but below is a sequence of learning that I think would be nice.

• To connect to the 2.G.A.3 work, we could:
  • Have several same-sized rectangles drawn one above another. Ask students to divide the top one into halves the middle one into thirds, and the bottom one into fourths. (grade 2 work)
  • Ask them how they would name each piece and what they notice about the size of the pieces. (grade 2 work)
  • Ask them how they would record the size of each piece. If they use fraction notation that is grade 3 work.
• Transition to creating fraction strips for halves, thirds, and fourths. Label the fractional pieces of the strips and discuss the size of the pieces and the number of pieces in the whole. (The fraction strips from Marilyn’s Do the Math kits made the WORLD of difference in my 5th graders understanding of fractions!)
• Place the fraction strips above a number line labeled with 0 and 1.
  • Ask students where 1/2, 1/3, and 1/4 would be located on the number line based on what they know about the length of the fraction strips.

• Ask where they think non-unit fractions might go? Where is 2/3? Where is 2/4? Where is 3/4?
• Ask for different ways they could label 1.

Then, I would have them revisit the cool-down and see what they do! There are also more of my favorite number line tasks from IM here. Try some out and dig into the student thinking. It is always such an amazing learning experience!

Over the past months, like many people, I have been thinking a lot about what it looks like to reimagine a curriculum that was intended for use in an in-person setting for digital implementation. Alongside an amazing working group of teachers and coaches, we have gone piece by piece through the IM K-5 curriculum components and discussed the purposes, opportunities, and challenges of each when transitioning to a digital setting, whether it be synchronous or asynchronous. Discussions around the student experience and student engagement were challenging, but really pushed us to think deeply about the activity purposes, questions, and ways in which to design opportunities to learn more about each student.

Together, we are in the process of creating ‘storyboards‘ for warm-up routines and generalizing processes for planning activities, but the piece that I really cannot get off my mind is assessment. While I think every activity is an assessment opportunity, I am thinking in particular about the cool-downs and checkpoints in the curriculum. These pieces were designed as in-person formative check-ins on student understanding after a lesson and section. Now, in some cases, they will be completed asynchronously. While it seems like one of the easier pieces to recreate digitally with the platform being the biggest decision, there are nuanced constraints that limit what a teacher might learn about student understanding when completed asynchronously. In the classroom, a teacher can observe students as they work and is available to answer questions. The sheer nature of sending home an assessment digitally can raise a lot of different, yet related questions.

Family members or caregivers might wonder:

• Is this graded?
• What if they cannot do it independently?
• How much should I help and if I don’t help, will they get a bad grade?
• If I do help them, how do I know if the work is aligned to how they are learning?

Teachers might wonder:

• Should I grade this?
• Did the student do it independently?
• If the student had help, how much?
• If the student had help, what did they know and where did they struggle?

I think these questions stem from two very caring places: family members want their child to learn and be successful and teachers want to know what students understand so they are able to best support them as learning continues throughout the unit.
I have been thinking about these questions alongside the results from Learning Heroes most recent family survey (webinar/slides), in particular, the areas outlined in yellow.

I wonder if this year is an opportunity to reframe how we approach assessment, reframe how we engage families in discussions around assessment, and break the typical lens families might have of assessment being the ‘final grade.’ How could reframing assessment as ‘finding out what students understand and are able to do’ change the way in which: families approach assessments at home, teachers respond to what they receive from home, and shape ongoing communication between families and teachers throughout the year?
Here are few ideas I keep tossing around to support this communication:

• At the beginning of the year, when meeting with families, let them know that they are not expected to teach their child the math content but instead encouraged to continually ask students questions (provided for them) to find out what they know, such as:
  • Retell the problem to me in your own words.
  • What do you notice? What questions do you have?
  • How do you know?
  • Where do you see the problem in your picture or diagram?
  • Explain how you solved the problem?
• Reinforce learning as an ongoing journey that, in order to best support students, teachers need to know what they understand and are able to do. Explain that the activities and assessments are ways in which to do this, so every conversation during the year will focus first on what students know.
• Create a schedule to meet with family members or caregivers to check in with some pre-arranged topics to discuss that focus on what they are seeing or hearing:
  • What do you see your child understanding and able to do?
  • When, if ever, do they get stuck?
  • What can I do to help?
  • (Consider communication types families view as most effective on page 24.)
• On each assessment, put 2 checkboxes at the end for students to choose from (no repercussions for whichever is chosen – also established in the opening family meeting):
  • ___ I did this all by myself
  • ___ I did this with some help.
  • [text box for the student or whoever helped] What was the sticky point?
• Provide opportunities for students to retake problems they didn’t get correct as the unit progresses. For example, if a student gets 2/3 problems correct on the Section A checkpoint, put the one they missed on the Section B checkpoint after the next week’s instruction. This communicates to families that learning happens across time.

All of this is so challenging with no easy solutions, but I would love to engage in discussions that push us to reimagine learning at a distance that includes strengthening teacher<>family relationships and moves beyond creating digital worksheets of in-person materials.

(Originally written for IM: https://illustrativemathematics.blog/2019/05/07/designing-coherent-learning-experiences-k-12/)

One challenge in curriculum design is considering all we know and believe to be true about math teaching and learning and translating that into realistic and actionable pieces for teachers and students. Our recent post about the K–5 curriculum focused around our belief that each and every student should be seen as a unique person with unique knowledge and needs. And while that post centered on elementary materials, to truly design around this belief we must look past K–5 to consider each student’s unique K–12 mathematical journey. A journey that, for most students, looks very different as they move from elementary to middle to high school.

While it is not surprising that the math content is different across grades, there is something really important and powerful in curriculum design that can make a student’s mathematical journey much more connected and accessible than it currently is. As we write the final component of our complete K–12 OER math curriculum, we continually focus on the consistent structures that support the focused, coherent mathematics within, and across, grade levels.

Why consistent structures for students?

Consistent problem-based curriculum structures can help remove some of the guesswork and anxiety about how math class is going to look and feel each year. Imagine a student entering any grade level knowing they will be given the opportunity to grapple with mathematical ideas, both independently and with their peers, before being expected to apply the understandings on their own. The problem-based atmosphere supports students in understanding their important role as a mathematical thinker and over time students get used to generating the mathematical knowledge as a community of learners.

Within our problem-based design is a consistent lesson structure:

This means every day, no matter what grade they are in, students know they have an accessible invitation to the mathematics during the warm-up, are given opportunities to build ways of thinking and doing the mathematics, discuss the ideas of the lesson as a whole group during the synthesis, and show what they know on the cool-down. This predictable flow allows students to focus on the math of the lesson because they know there will be opportunities to discuss ideas, ask questions, and consolidate their learning.

Another important consistent structure within the curriculum are the instructional routines. Instructional routines provide an opportunity for all students to engage in, and contribute to, mathematical conversations. The predictable routine for the interactions between teachers and students allows each student in the classroom to focus on the mathematics rather than the actions that need to be taken. And the more experience students have with the routine gives them greater ability to focus on the mathematical content.

Why consistent structures for teachers?

Within the classroom, consistent structures give teachers time to focus on what’s most important—the students. As a teacher, there is something really nice about not having to worry about “what comes next” in a lesson or “how the activity runs,” just as it is for students. When all of those things are kept consistent, it offers teachers the time to listen for the mathematical understandings students have and think more deeply about the classroom experience: What questions are being asked? How are they are asked? With a consistent structure, there are more opportunities to think about whose ideas are being shared, and the ways in which these ideas are represented.

Outside of the classrooms, it would be amazing to have the opportunity to collaborate K–12 in professional learning. I would love to have a world where a district did not always split into K–5, 6–8, and 9–12 math professional development sessions, but instead had a room of K–12 teachers discussing the coherent progression of mathematical ideas, representations, or routines in a curriculum.

For example

Below is a series of warm-up routines from the IM math curriculum by grade. While there are some really cool math connections that show up in this series, I would like you to imagine a student who experienced the structure of these routines since kindergarten.

• What might they know about the structure of the routine by the time they are in Algebra 1?
• How might what they know about the structure help them attend to the mathematics more by the time they reach Algebra 1?
• How might what they know about the structure help them be seen as a unique person with unique knowledge by the time they reach Algebra 1?

Kindergarten

Grade 2

Grade 3

Grade 6

Algebra 1

Next Steps

For more info on the upcoming K-5 curriculum, you can visit the IM site here. We would love to read your reflections to the questions above! Please respond in the comments.