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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:

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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.

Wondering About Classroom Norms

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.

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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.

The other day, Jamie tweeted about this example problem from a 4th grade program:

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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.

I thought about:

  • 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?

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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:

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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.

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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?

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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?

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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:

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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 adding?

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:

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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:

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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:

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

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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!