# The Nuances of Understanding a Fraction as a Number

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.

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!

# Thinking Through Asynchronous Assessments

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:

• 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:

• 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.

# Coherent Learning Experiences K-12

(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

Algebra 1

### Next Steps

I cannot count how many times classroom norms has been a topic of my conversations in the past month. From creating and facilitating professional learning to thinking about how a curriculum can offer support in this area, I find myself obsessively thinking about ways in which norms might support both students and adults in their learning.

If you asked me a year ago about the norms in my classroom, I would have felt pretty good about how the list hung proudly on my classroom wall, was collaboratively established by students, and appeared to be in place during their math activities.

However, like the majority of my teaching life, the more I learn, the more I realize how much there is still left to learn. In this particular case, it is norms in a classroom.

I think most people would agree that establishing norms is important. Norms can encourage students to work collaboratively and productively in a classroom, elicit use of the Mathematical Practices and help students see learning mathematics as more than just doing problems on a piece of paper.  But, how often do we create norms in our classroom only to complain a month or two later that students aren’t thinking about any of them when working together and we struggle with how to refocus students to keep in mind those things they said were important at the beginning of the year? I know I have been there and looking back, wonder how I could have done that better.

While I think good curriculum tasks, lesson structures, and relationships I had with students helped me a lot in encouraging students to be mindful of the norms in the classroom, I don’t think I put an equal amount of effort into maintaining norms as I did establishing them. With that, I wonder what it even looks and sounds like to maintain them?

To me, maintaining norms is about moving from a poster on a wall to a living and breathing culture in the classroom. But, what things can a teacher do to make the norms not only a list, but a part of their classroom math community?

Of course, as the journey begins on writing the IM K-5 Math curriculum, I am also wondering how a curriculum can support teachers in establishing and maintaining classroom norms in a meaningful way. Even more specifically, what could this look like in Kindergarten when we have the opportunity to influence the way students view learning mathematics?

As I think through these questions, I would love to hear how you think about norms in your math classroom. What things can we do as teachers to support students in thinking more about what it means to learn and do mathematics? How could a curriculum, especially in Kindergarten, help teachers in this process?

# Purposeful Warm-up Routines

As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading

# Small Change, Big Impact.

After looking at the Standards, Learning Progressions, and discussing the 3rd grade fraction work, all of us on the thread agreed it was an appropriate question for 4th grade but then started to question whether we would give this problem as is, or adapt it. I really appreciate these conversations because they move us beyond ‘this problem sucks, don’t do it’ or ‘do this problem instead, it’s more fun’ to thinking about a realistic thought process teachers can use when working with a text that may not be aligned to the standards or lessons that may not best meet the needs of their students.

While all of the adaptations we discussed sounded similar, I couldn’t help but wonder how each impacted so many classroom things in different ways. What may seem like a small change can easily impact the amount of time it takes in class, students’ approaches to the problem, what we learn about student thinking, and the follow-up question we could potentially ask.

There is no right or wrong adaptation here, but I wanted to sketch out how each change to the problem impacts teaching and learning.

Change 1: Remove the context and only give the expression 1 – 26.

This would be the quickest way to do this problem in class. With this change, I would be curious to see how students think about 1 in the expression.

Do they record it as 66? Do they draw a diagram? If so, what does it look like? How is it partitioned? Are the pieces removed in the diagram or is there work off to the side? A potential follow-up could be to ask students to write a context to match the expression. I think in their contexts I would have the opportunity to see how they thought about 1 in a different way.

Change 2: Remove the expression and only give the context.

I find anytime there is a context it takes a bit longer because of the time to read and reread the problem so this change would take a bit more class time than the first. This change would give students more access to the problem and I could possibly learn how they make sense of a context, but I wonder what I would learn about their fractional thinking. Since the context pushes students to think about 1 as the whole pizza and also tells them that there are 6 equal pieces, the diagram, partitioning, and denominator are practically done for them.

Because of this, I may not learn if they know 1 is equivalent to 6and may not find out how they represent fractions in a diagram because I imagine most would draw a circle. Since they could do the removal on the circle, I also wonder if I would learn much about how they saw this problem as an expression so I would add that as my follow-up question.

Change 3: Remove the expression and the numbers from the context.

This one ​would definitely take an entire class period as a numberless word problem and probably the longest to plan. Because it takes the longest to plan and implement I really have to think about what I learn above and beyond the two changes previously mentioned if I were to do it this way.

I imagine the scenario could sound like this:

“Sam ordered a pizza cut into equal pieces. He ate some of the pizza and put the rest away for later.” or “Sam ordered a pizza. He ate some of the pizza and put the rest away for later.”

When I do a numberless problem, my goals are to give students access to the problem and see how they make sense of a context without the numbers prompting them to feel like they have to do something. I have to plan for how I craftily find the appropriate time to let them notice and wonder and plan questions that elicit the subtraction from 1 that I hope to see. I also like to give students a chance to choose their own numbers for problems like these in order to see how they think about the reasonableness of numbers, which adds more time. The hardest part here is getting to the fraction work because I think students could stay in whole numbers as they talk about number of pieces. I can hear them wondering how many pieces it is cut into and how many he ate – neither of which guarantees fractions. So, while this has the potential to get at everything Change 1 and 2 do, a teacher must weigh how much the making sense of context portion meets the needs of his or her students.

All of this for one problem and I haven’t even discussed the two most important questions we need to ask ourselves before even making these changes – what understandings are students building on? and what understandings are students building towards?

There is much to think about in planning that is often hard to think about all of the implications of one tiny change to a task, however, thinking about how each of these changes impacts teaching and learning is the fun and exciting part of the work!

After the post, Brian had another change that he posted on Twitter! I wanted to capture it here so it is not lost in the crazy Twitter feed:

# Looking for Patterns in a Number Talk

I love when I read a blog post in which I can relate to how the teacher felt, learn from both the teacher and student thinking, want to hear what happens next, and leave with questions circling around in my head. This happened when I read Marilyn’s recent post. I really appreciated how her recount of the lesson demonstrated the importance of number choice and the honest way we all have felt when we made a decision during a lesson that we wished we hadn’t. It was really interesting to think about how changing the divisor from 4 to 5 changed what students experienced. I cannot wait to hear what they do when they try the original problem and see the remainder as 1 in the balloon context, 25¢ in the money context, 1/4 in the cookie context and .25 on the calculator. Awesome discussions could happen there!

I left this post still thinking about the math talk at the launch of the lesson. I was going to tweet about it, but because it seemed long and I have many questions of my own I want to play around with and revisit, I decided to put it here. I loved the connectedness of the number talk to the division task, and wondered how the recording of those strings could impact division patterns and structure students may see in future lessons. I started playing around with it in my journal in terms of how we think about recording a choral count.

• How many problems in each row?
• Does horizontal vs vertical recording impact what we see?
• What might students notice about the remainders in each row? column?
• What might students notice about the change in dividends in each row? column?

Not that I would launch the 4 Problems task with this following string, but I wondered what it would look like to change the divisor and what students might see here:

I think the number of problems with remainders at the bottom of the list versus the top is really interesting.

THEN, I started wondering about ways we could record the remainder and how that may impact how students interpret it? Not sure how this would work in terms of launch and facilitation, but I like thinking about the pink writing here.

Recording is one of those things I get so intrigued by and cannot wait to revisit this post, play around with patterns that could be elicited in different ways and think about tasks in which these talks could be connected.

Thank you, as always, Marilyn for sharing your work – you continue to be such an inspiration! My only hope is one day I can be in the room for one of your lessons!

# Fraction Division and Complex Fractions

It is posts like Lisa’s most recent one that make me long for more collaboration K-12. I have to admit, when I saw her Twitter post with the words pre-calculus and simplifying complex fractions, my inclination was to skim right by because I would not understand the post anyway. Literally, my only recollection of simplifying complex fractions like the one at the beginning of her post is through a set of procedures I was explicitly taught step by step. However, when I looked at the accompanying image that showed fraction division, I was curious how my understandings of fraction division connected to her pre-calculus work.

I loved reading Lisa’s process of making the math accessible for her students because I am sure many would have felt like I did if shown the CPM opener from the very beginning. It is that same process of thinking about what students know and how we can build on it that made me get out my journal and start sketching out connections I was making as I read. In no time, my journal was full of problems, diagrams, concepts, questions and every tab on my computer referenced the progressions, standards, references linked in Lisa’s post, and a blank email to Kate and Ashli to jot down my questions for them about the math. Talk about a wonderful rabbit hole to be going down.

The more I read and reread this post, the more I think it could lead to many more posts connecting how students are introduced to ideas in elementary school, the impact it has on later work, and the questions I have as I go. My questions revolve around not only the math, but also how these mathematical ideas build, how our representations impact student understandings, and how there are times when a problems lends itself to one way of thinking versus another.

During my first read, two things I wondered were:

1. How does the way the fractions are written impact the way I think about them?
2. What happens when I have two ways of thinking about fractions and two ways of thinking about division?

How does the recording of the fractions impact the way I think about them?

As the post progressed from an image of a complex fraction to one of fraction division, I felt like Lisa must have felt, wondering what students may know about the complex fraction and why they may struggle. My initial thought was they may not understand that a complex fraction is even division. This may not be the case for most, however based on what I remember from high school, I saw complex fractions as one thing I did operations on. As an elementary school teacher, it seems similar to the difference between seeing a fraction as a number (introduced in 3rd grade) versus seeing fraction as division (introduced in 5th grade).  As I looked at CPM’s complex fraction and how it was written, I only thought about it as multiplying the numerator by the reciprocal of the denominator because of how I was taught. However, when I looked at the fraction division problem written horizontally, I found myself attending more to each fraction as a number, using what I know about division to find the quotient. Less intimidating to me solely because of the way it was written on the paper.  I wonder if this compares a bit to how we record computation problems horizontally versus stacked during number talks to encourage thinking about a problem versus always relying on the algorithm?

I know the fraction division problem means the same thing written either way, but how they are written impacts my thinking a lot. From an elementary perspective where we spend so much time attending to developing understanding of fraction as a number, I am not inclined to really think about what it means to divide the two terms when written as a complex fraction. To that end, I wonder if the opening problem written one way versus another evokes a different meaning for some students?

Knowing that there are things to be learned in between the problems listed below, but in terms of seeing the complex fraction as division where I think about the individual pieces as things in their own right, is one possibly a small transition to the other for me or students like me?

Lisa – I would love to hear more about the transition prompt between the fraction division problems the students were solving and the CPM problem. I think that is a really important piece of what you did so beautifully in this lesson.

What happens when we have two ways of thinking about fractions and two ways of thinking about division?

I think about fractions in the two ways I mentioned above: as a number and as division.

I think about division in two ways: how many groups? and how many in each group?

First, fractions: In 3rd grade, students learn a fraction is a number in which the numerator indicates the number of pieces and the denominator (as the denominator of a unit fraction) represents the size of the piece. For example, we say 3/is 3 pieces the size of 1/4. This understanding and associated language are so beautiful when students use it to compare fractions and create equivalent fractions. In my 5th grade class, my students were comfortable using complex fractions such as1/2 / when talking about 1/6 because they were thinking ½ a piece the size of ⅓ is . No division, just reasoning about the pieces and their size. When comparing 4/9 to 5/7, students would use the reasoning that four and a half ninths and three and a half sevenths are equivalent to a half so 5/7 is more than a half and 4/9 is less than a half. I saw a glimpse into how that thinking was not helpful when they asked what happened when there is a fraction in the denominator. This is where understanding fraction as division would have been more helpful.

In 5th grade students also learn about fractions as division. In terms of sharing situations, they learn that 5 things shared by 3 people results in each person getting 5/3 of the things or 5 divided by 3. In these situations, thinking about 5 pieces the size of 1/3 is not particularly helpful in solving, but division is. However, when it comes back to interpreting the solution, 5 pieces the size of 1/3 is needed.

Questions I am thinking about at this point:

• How does the complex fraction in the post relate to either or both of these ways to think about fractions?
• How does the way we represent fraction division relate to one or both of these ways to think about fractions?

Now, division: In 3rd grade, students learn division in two contexts: how many in groups and how many in each group.  In 5th grade, students use those understandings to divide whole numbers by unit fractions and unit fractions by whole numbers. Those two meanings of division carry into middle school to divide fractions by fractions and conceptually understand the reason we multiply by the reciprocal.

After reading Kristin and Bill’s series of posts on fraction division, I am now constantly thinking about how the context (interpretation) for division impacts the way students represent and solve a problem. I know changing the way I think about the division context changes how I represent the problem as well as how I operate with the reciprocal.

Questions I am thinking about at this point:

• Does one context of division connect more closely with the CPM complex fraction problem?
• Does the visual fraction model of the the division problem impact the way students approach the complex fraction problem?
• Is an array representing both fractions being divided helpful in this complex fraction?
• Is one bar model representing both fractions on one helpful in this complex fraction?
• Is one way of representing it more helpful than the other?

Obviously, I have a lot to read about how a problem such as the one Lisa posed progresses after middle school but after seeing the division of fraction problem,  I am even more intrigued to see how these ideas progress from the time they are introduced. I am so curious when certain ways of thinking are more helpful than others and how we can construct learning experiences that help all students have access to the mathematics in a lesson in the way Lisa did.

# CCSSMashup – Fractions

I never tire of conversations about the 3rd – 5th grade fraction progression because after each one, I leave with the desire to reread the Standards and Progressions with a new lens.

A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag’s typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don’t mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.

The conversation focused on this standard:

With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.

I expect a student would mark off the 1/4’s starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?

Is the student counting?

Is the student doing both?

How does adding and counting look or sound the same in this scenario? different?

This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.

In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:

Don’t get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn’t seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.

When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:

However, because the 3rd grade work is on a number line and the arrangement and order does matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:

So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:

With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.

In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:

The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.

The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .

There is also a great “Meaning of Fractions” video on the Illustrative Mathematics site that explains this idea with visuals.

There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.

I look forward to hearing your favorite #CCSSMashup!

# Explicit Planning vs Explicit Teaching

Planning is like…..

How would you finish that sentence?

As a facilitator, I use this sentence starter to open Illustrative Mathematics’ 5 Practices Professional Learning. To be completely honest, when I designed the PD I was a little hesitant of using it because I was nervous it was opening a can of worms within the first 5 minutes of the day.  I am, however, always surprised with all of the beautiful analogies participants share and feel challenged each time to come up with something new and better than the one I used in previous sessions. When I first delivered this PD, I started with analogies like a marathon or really hard workout – something that is exhausting, a lot of work, but ends with something I take pride in. While these analogies were accurate representations of how hard I think lesson planning truly is, I was continually unhappy with where students’ ideas fit into my analogy.

Planning is like putting together a puzzle.

When sharing my reasoning with participants for the first time, I included a lot of beautiful words around mathematical connections but in the middle somewhere I used the phrase “making connections explicit” in relation to the puzzle pieces and saw an immediate reaction from a few people in the room. Of course, I had to pause and ask, “Was it the word explicit?” – answered by many nods in the room.

For a long time, the word explicit in relation to teaching held a negative, cringe-worthy connotation for me as well. If ever asked to paint a picture of what explicit teaching looks like in the math classroom, I would describe scenarios in which a teacher is either at the board telling students how to solve a problem or showing a struggling student how to solve a problem because they are stuck or “taking the long way there.” To me, being explicit meant telling students a way to do something in math class – typically in the form of a procedure.

Through teaching a problem-based curriculum [Investigations], designing and implementing math routines such as number talks, and reading Principles to Actions5 Practices and , I realized that I was guilty of making mathematical ideas explicit every day in my classroom, but not in the way that made me cringe.

I was explicitly planning, not explicitly teaching.

To me, those two phrases indicate a big difference in how I think about structuring a lesson. I have found when teaching a problem-based curriculum, it is easy for ideas to be left hanging and important connections missed, forcing me to explicitly teach an idea to ensure students “get it” before they leave the class period without any understanding of the mathematical goal for the day. Many days, I would find myself frustrated because students would completely miss the point of the lesson, however now I realize this was because I was expecting them to read my mind of what I wanted them to take away from the problem. On the flip side of that coin, however, not teaching a problem-based curriculum and explicitly teaching students how to do the math in each lesson is not an option (and is a topic that could be its own blogpost).

This is exactly why I find the 5 Practices framework invaluable in planning. The framework forces me to continuously think about the mathematical goal, choose an activity that supports that goal, plan questions for students toward the goal, and sequence student work in a way that creates a productive, purposeful discussion toward an explicit mathematical idea. I have learned so much using this framework over and over again in planning for my 5th grade class, collaborating with other teachers and coaching teachers across different grade levels.

Explicit planning is how I would describe the new, open education resource (OER) by Illustrative Mathematics. As a part of the writing team, I explicitly planned warm-ups such as number talks and notice and wonder activities to elicit specific mathematical ideas that play a purposeful role in the coherent plan of the lesson and unit. But not only are the warm-ups explicitly planned, but each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While they may not all arrive at that landing in the same way, the problems and discussions are structured to ensure students do not leave the work of the day without any idea of what they were working toward.

While I would love to think my blog posts paint a clear picture of explicit planning, I am not that naive. So, what does explicit planning look like in a 5 Practices Framework?

This lesson from Grade 7, Unit 2, Lesson 2 from Illustrative Mathematics’ Middle School Curriculum is one of many in the curriculum. (All images are screenshots from the online curriculum that is linked at the bottom of the post)

Practice 0: Choosing a Mathematical Goal and Appropriate Task

Lesson Learning Goals

With the goals in mind, the lesson begins with a notice and wonder warm-up that engages students in thinking about tables, followed by two activities that build on those ideas and support the mathematical goals. While both activities demonstrate explicit planning, I am focusing on one for the sake of space.

Practice 1 and 2: Anticipating & Monitoring

Activity Narrative

Practice 3 & 4: Selecting & Sequencing

Activity Synthesis

Practice 5: Connecting

Activity Synthesis

So..Planning is like putting together a puzzle. It is hard, takes time, and is sometimes difficult to figure out where to start. We know all of the pieces connect in the end, but making a plan for all of those pieces to connect takes an understanding of the final picture – the goal. There will be missteps along the way and some parts will take longer than others, but we know it is important to carefully connect each piece to another as one missing piece will leave unconnected ideas and the final picture unfinished. As you work alone, the way the pieces connect to form the final picture may not always be obvious, but as others help us see the pieces in different ways during the process, connections become explicitly clear and the final picture is something in which you can take a lot of pride.

The ‘others’ in my teaching journey have helped me see a difference between explicit teaching and explicit planning. Through explicit planning I have seen the importance in understanding the mathematical goal in a way that enables me to structure activities and lessons that enable students to make important mathematical connections through their own work and discussions. It is so exciting to see IM’s curriculum be a model for how I think about explicit planning in such a coherent, purposeful progression.