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
Author Archives: mathmindsblog
Small Change, Big Impact.
The other day, Jamie tweeted about this example problem from a 4th grade program:
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 followup 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 – ^{2}⁄_{6. }
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 ^{6}⁄_{6? }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 followup 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 ^{6}⁄_{6 }and 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 followup 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:
I wonder if another change could be tweaking the problem to where you don’t tell them it was cut into six slices: Sam ordered a small pizza. He ate 2/6 and put away the rest for later. How much of the pizza did he put away for later?
— Brian Bushart (@bstockus) February 1, 2018
Oh, and still get rid of the expression, in case that wasn’t obvious. Perhaps after Ss share solutions, if no one uses 1 – 2/6, you could share it and ask how it also represents the situatuon, assuming that fits with your overarching goals.
— Brian Bushart (@bstockus) February 1, 2018
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?
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 K12. I have to admit, when I saw her Twitter post with the words precalculus 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 precalculus 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:
 How does the way the fractions are written impact the way I think about them?
 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 3/4 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 / 3 when talking about 1/6 because they were thinking ½ a piece the size of ⅓ is ⅙. 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 neargrade 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 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 35 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.
My most recent sentence was this…
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, cringeworthy 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 problembased curriculum [Investigations], designing and implementing math routines such as number talks, and reading Principles to Actions, 5 Practices and Intentional Talk , 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 problembased 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 problembased 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 warmups 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 warmups 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
With the goals in mind, the lesson begins with a notice and wonder warmup 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
Practice 3 & 4: Selecting & Sequencing
Practice 5: Connecting
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.
Link to Illustrative Mathematics 68 Math Curriculum.
Link to the 7th Grade lesson featured in this post.
Choral Count: Trusting Patterns
In the next round of 1st grade videos for my Teaching Channel Math Routines series, I am so excited to release a video of a Choral Count. When I taught 5th grade, I used Choral Counts a lot with decimals and fractions, so using them in the earlier grades was such an interesting parallel to that work.
I don’t want give away what you will see in the video, but Heidi’s recent post about trusting patterns when multiplying by 10s made me think about two particular student journal entries after the count.
In this count, students counted by 10s starting at 4 as I recorded their count and I stopped them at 154. They discussed patterns they saw in the count and afterwards, I asked them to journal about any new or extended patterns they think may happen if we continued counting. One thing I stress in their journal writing is the fact that it doesn’t have to always be written in words, we can explain mathematics with numbers as well. These two students didn’t include any description of what they noticed, but from their recording, I can assume some things about what they saw.
Through Heidi’s lens of trusting patterns, I watched this student record her extended count. Instead of writing every number in its entirety, this student wrote out all of the tens and then went back to add all of the 4s on. I asked her how she could show the pattern she noticed and she quickly went back to underline all of the tens. She trusted the pattern that the tens would continue going up by 1 and the ones place would stay the same throughout the rest of the count since we were counting by 10s. The pattern she noticed and trusted worked, would continue to work if she kept counting, and is something teachers could build on by counting by different multiple of tens.
Now, this student below invented another pattern that doesn’t keep the count in tact but brings up an interesting connection to the work above. Instead of looking at the tens going up by 1 and ones place staying the same, this student added 100s going down each column. In his work, the tens and ones stayed the same while the 100’s increased by 1 with each jump down.
The more I looked at these, I thought about how to support students in later grades who are unsure whether to trust a pattern or even why patterns work in the first place. For example, and not that counting by 7’s would be the way to go, but for Heidi’s problem of 2,000 x 7, I wonder how a choral count could support students who were struggling to explain multiplying by 10s. The student in Heidi’s example obviously has a grasp of the idea, but what about those who don’t?
Let’s see what we notice here:
7 14 21 28 35
42 49 56 63 70
77 84 91 98 105
112 119 126 133 140
Could students pull out where the multiples of 10 would show up if we kept counting?
Could they translate that pattern to equations?
Could they connect this additive thinking to multiplicative thinking?
Could they apply that same understanding to counting by another number but keeping the same structure of 5 in each row that is shown here?
All of these questions are so interesting to me and leave me wondering if we did more of this work in the earlier grades the impact it would have in later grades. Thank you Heidi for sparking such a great convo on Twitter and giving me a different lens for student work in the earlier grades!