Introduction
Have you ever heard a child talk about something that excites them and sparks their curiosity and wondered how you can bring that level of engagement and interest to your classroom environment? Has something sparked your interest and you just had to learn more about it? Do you wonder why curiosity is important and how to bring it to your lessons, especially in math class? As former teachers and current mathematics teacher educators, we wondered about upper elementary school students’ thinking and what reasoning and mathematical argumentation look like in their classrooms. We are passionate about this topic because we want to develop students’ confidence and conceptual understanding in elementary school, and we see this as building a strong foundation rooted in reasoning and curiosity. While curiosity is not an easily measurable skill on an assessment, researcher and author Susan Engel (2015) writes that it “is the linchpin of intellectual achievement. People who are curious learn more than people who are not, and people learn more when they are curious than when they are not” (p. 3). Change consultant Monica Parker (2023) writes about the value of wonder and curiosity as follows:
Curiosity supports learning and improves critical thinking. Allowing students to work with rather than against their natural curiosity also means working with and not against the neuroscience of learning. When someone’s curiosity leads them to an answer, what they learn is encoded more deeply and thus retained longer. (p. 165)
Curiosity is a powerful tool in learning, and we want to find ways to reignite it in upper grades. We know that “children enter this world as emergent mathematicians, naturally curious, and trying to make sense of their mathematical environment” (Huinker, Marshall, Rigelman, Yeh, & Barnes, 2020, p. 17), but something is lost in elementary school. In her research, Engel (2015) finds that “expressions of curiosity” (p. 88) in school classrooms were infrequent and decreased as children got older. There was variation in the data between classrooms and grade levels, but the overall average was lower for fifth graders compared to kindergartners. “There is little curiosity in grade school,” Engel (2015) reports, adding, “Observations as well as interviews and surveys suggest that though children are curious, students are not” (p. 89). How do we make sure the curious children who enter our classrooms have opportunities to be curious students in a “wonder-based” (Parker, 2023) approach? Mathematics education scholars DeAnn Huinker, Anne Marie Marshall, Nicole Rigelman, Cathery Yeh, and David Barnes (2020) address the matter as follows:
When children are curious and wonder, they ask questions, helping them to feel confident when doing mathematics. Each and every child must be afforded opportunities to not only feel confident as doers of mathematics but also to experience joy and see beauty in their mathematical discoveries. (p. 17)
In writing this book, our first goal is to support you as you integrate exploration and mathematical argumentation in grades 3–5. Mathematical argumentation is a way for curious mathematicians to explore and build a deep understanding of mathemat ics. Upper elementary school students bring so much to our classrooms and have so much desire to share what they discover within a community. Mathematical argumentation is about nurturing students as they explore, notice patterns, curiously wonder, ask questions, conjecture, and justify. We imagine you are a current or future third- through fifth-grade teacher who wants to build exploration, curiosity, and argumentation in your classroom. This book provides explicit examples of what this looks like in action. Additionally, if you are an instructional coach, coteacher, or administrator, we hope this book will be valuable as you work alongside teachers.
Our second goal is to reignite your own curiosity along with your students and potentially help you grow as a mathematician as you explore mathematics content in new ways. Exploring alongside students to understand their thinking can change how you think about mathematics too! Mathematics is a creative subject, and you can look for connections between ideas and areas within mathematics. The more you think about mathematics, the better you are able to make those connections for students and for yourself. We have included opportunities in this book for you to explore mathematics in curious ways so that you can make connections between patterns and concepts. In our own K–12 mathematics experience, we learned mathematics as procedures and algorithms, but we want you to have an opportunity to also see it as a creative and exploratory subject by nudging you to notice and wonder, conjecture, justify, and extend your ideas.
We are grateful to the K–5 teachers who explored this idea with us during professional learning. This book shares what we learned while investigating alongside teachers and students, specifically in grades 3–5. The activities, student work, and vignettes are based on our findings in the professional learning projects we have led with teachers. Most of the
teachers you’ll meet in this book were part of our professional learning projects. While we were the facilitators who brought an agenda, readings, and coaching, we all entered the experience eager to explore and learn together. Throughout the book, you’ll hear about Leslie Whitaker, a veteran teacher, and Tiffany Kane, Hannah Stalker, Emily Igarashi, Marissa Alvarez, and Christina Cho, who were all part of our ongoing professional learning projects. We’ll use their real names, but other teachers and all the students that you’ll meet have pseudonyms to keep their identities confidential. Throughout the book, you will also read Teacher Voices sections, taken from teachers’ reflective journals and used with permission.
We are so grateful for the collective learning we experienced through this project and are excited to extend the collaboration with you in this format. We look forward to continuing the conversation and exploring with you.
How Do We Get Started t oge ther?
In this book, we prepare you for success by sharing every aspect necessary to reignite and nurture math curiosity in your students. Like mathematics educational consultant Helen J. Williams (2022), we want our students to have “a broad and deep experience of mathematics that connects with their existing knowledge and understanding and helps them to flourish” (p. 8). To that end, we cover the following content in this book.
Part 1: Nurturing Your Classroom Community and Growing Your Teacher Toolbox—We begin by investigating how classroom communities, teacher tools, and instructional strategies support a foundation for curiosity and mathematical argumentation. Unpacking these ideas can be helpful for argumentation and other aspects of teaching! In chapter 1, we build a foundation for the book by unpacking examples of mathematical argumentation and offering specific definitions to guide discussion. In chapter 2, we explore essential elements of classroom environments that support argumentation. We share the importance of establishing classroom norms, using space to support sense-making and collaboration, supporting language development, and ensuring each student has a voice. In chapter 3, we identify and develop tools we find essential in argumentation, including those for planning, representing mathematical ideas, and communicating. Then, in chapter 4, we connect the environment (chapter 2) and teacher toolbox (chapter 3) as we describe instructional routines that lend themselves to argumentation.
Part 2: Growing the Layers of Argumentation— Mathematical argumentation is a way for curious mathematicians to explore and build a deep understanding of mathematics. When students engage in mathematical argumentation, they communicate their ideas, listen to those of others, and make connections. In part 2, chapters 5 through 8, we explore the layers of mathematical argumentation—noticing and wondering (chapter 5), conjecturing (chapter 6), justifying (chapter 7), and extending (chapter 8)—that establish the foundation of our work. We’ll also follow one classroom lesson throughout all four chapters to unpack each layer using the true or false instructional routine.
Part 3: Growing More Mathematical Ideas— In part 3, we tie together all the ideas from preceding parts as we explore early mathematics concepts, such as composing and decomposing operations, properties, and more! Chapter 9 describes hidden opportunities
to include mathematical argumentation within curriculum materials and tasks. Chapter 10 offers opportunities to connect with algebra. In the epilogue, we reflect on what we as educators gain with mathematical argumentation and invite the reader to consider how to continue learning and growing with others!
Throughout the book, we present classroom vignettes to help you see what mathematical argumentation looks and sounds like in third- through fifth-grade classrooms. We offer some background for each vignette and discuss alternative teacher moves or questions. When we do this, it’s not to criticize the teacher’s work in the vignette. It’s to reflect and acknowledge that there is much to think about, and teachers can take alternative pathways during meaningful discussions.
To help you grow your mathematical wonder, each chapter features a section called Your Turn, which invites you to explore related mathematics. Sometimes, this section prepares you to look at a classroom vignette with your own knowledge activated. Other times, Your Turn extends an idea, allowing you to relate better to the chapter topics as a learner. Other features in each chapter include Pause and Ponder to help you think about what you’ve read so far, as well as Questions for Further Reflection, and an Application Guide that provide prompts to help you connect what you learned in the chapter to your classroom.
We also include appendix A (page 227) to guide you through using the layers of argumentation in instructional routines we explore throughout the book, including the following.
• Choral counting
• What do you know about ?
• Always, sometimes, or never
• True or false
We also include a QR code for a video where we explain how to use color in your class records. Note that many of the figures and reproducibles in this book are available in full color online (visit go.SolutionTree.com/mathematics). Appendix B (page 241) gives you an overview of the mathematical ideas across all chapters.
We hope the features included throughout the book will help you reflect and create a vision for what mathematical argumentation will look like in your own classroom. Additionally, we hope the features in this book support book studies with the Your Turn, Pause and Ponder, Application Guide, and Questions for Further Reflection features useful for guiding group discussions.
Establishing the Foundation for Mathematical Argumentation
As we build our vision of what curiosity and argumentation look like with mathematicians in upper elementary school, let’s start by exploring a task in which third graders are building rectangles as part of a game. We’ll discuss how students’ ideas can provide opportunities for argumentation and deep conceptual understanding and look at ways that we can reignite curiosity and engagement in our classrooms while building on their own wonderings and ideas in meaningful ways. Then, we’ll share a vignette about fifth graders exploring decimals and unpack the layers of mathematical argumentation in both vignettes. Finally, you’ll have a turn to explore the connection between addition and multiplication in a task that we’ll follow for the rest of the book.
When students engage in mathematical argumentation, they look for patterns, manipulate equations, communicate their ideas, listen to those of others, and make connections as they notice and wonder, conjecture, justify, and extend. With all the different ideas and fluency strategies present in the classroom during a task and discussion, it is easy to miss the amazing mathematical argumentation that is happening. Therefore, we’ll help you see the argumentation that is already happening in classrooms and the potential for future argumentation in tasks and activities. In the third-grade classroom vignette that follows, you’ll see that many conjectures are discussed in a short amount of time. We define conjecture in this context as a statement that you think might always be true. It’s remarkable that
the students didn’t want to stop thinking about the deep conjectures and the game. They were engaged and curious about the game and the mathematics behind it.
CLASSROOM VIGNETTE: BUILDING RECTANGLES WITH THIRD GRADERS
Hannah is in her first year of teaching, and her third graders are playing a game using a fifteen-by-fifteen grid and dice. Partners are taking turns rolling a twelve-sided die and a six-sided die. Students make a rectangle with an area equal to the product of the numbers they rolled, as in the game board made by students Brian and Flora shown in figure 1.1. The goal of the game is to have the most squares in one’s color. The game Hannah bases her activity on, Rectangle Rumble (IM® K–12 Math, n.d.f), uses two six-sided dice, but hers includes one twelve-sided die and one six-sided die to prompt the grids to fill up faster so students could begin to consider how different dimensions can have the same area.
Source: IM K–12 Math, n.d.f. Task based on IM K–12 Math by Illustrative Mathematics® . This work is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0).
Figure 1.1: Brian and Flora’s game board.
As Hannah and the mathematics coach, Jill, walk around the classroom, they look for students who are running out of room on their grids to see what they might do. They plan to capitalize on this situation to prompt observations and conjectures. Jill notices Brian, who has just rolled a 6 and a 5, has nowhere to put a 6 × 5 rectangle (see figure 1.1). Other students are also running out of space on the grids, so Hannah brings everyone to the
carpet to make observations and brainstorm solutions. Hannah projects the grid onto the whiteboard so that the class can see the open spaces left on the grid.
Hannah: Brian and Flora have a problem. Brian, can you explain what is happening in your game?
Brian: I rolled a six and a five. I know the answer is thirty but I don’t have space for a six-byfive rectangle.
Hannah: I agree that the product is thirty, why can’t the rectangle fit?
Brian: We have space for a three-by-four rectangle or a three-by-ten rectangle but not six-by-five.
Violet: There are three rows of ten at the bottom.
Hannah: Can you say that again, Violet? How could that help us?
Violet: If he made a three-by-ten rectangle, it would equal thirty. See figure 1.2 for what the students’ work looks like.
6 × 5 = 30
10 x 3 = 30
1.2: Two equations with a product of 30.
Students go back to their tables to complete the first round, or they flip the paper over to start a new round using the information that the product of the dice must match the area of the rectangle but it doesn’t have to match the numbers on the dice. As the teachers walk around, they look for equations to bring back to the whole group, such as those shown in figure 1.3. 2 × 6 = 12 4 × 4 = 16
× 2 = 12 4 × 3 = 12 2 × 8 = 16
Figure 1.3: Pairs of related equations.
× 4 = 12
Hannah focuses the students’ attention on the pairs of equations, saying, “We saw some interesting pairs of equations on your game boards. Look at these pairs of equations and turn and talk about what you notice and wonder.” The students make observations about the connection between the rectangles and factors and about the relationship between pairs of factors like those in figure 1.3. We’ll share specific ideas that the students noticed and wondered in chapter 5 (page 115), but we are excited to tell you that this opportunity to notice and wonder prompted many students to begin conjecturing related to the context of the game and their observations. Brian conjectures, “We can split one of the numbers in half and double the other number” to get the same product. Poppy builds on that idea to conjecture that you can double and halve instead, noticing that if you “swap the numbers then it swaps the rule.” Going beyond the doubling and halving or halving and doubling conjectures, Violet begins conjecturing about what types of numbers would work, saying, “The numbers have to be even if you want to halve it; seven and three wouldn’t work
Figure
because they’re odd.” Diego adds that the strategy works with odd numbers but results in a decimal as one of the factors.
Not all students are thinking like Brian, Poppy, Violet, and Diego! Many are content using the dimensions on the dice rather than manipulating the numbers to squeeze the rectangles into other spaces. The students have an opportunity to think about deep mathematical ideas regarding multiplication and a strategy for finding the product, and they don’t all use the same strategies in the game. Even as the class time ends, students continue talking about their conjectures. They are playfully exploring and do not want to stop! As they continue discussing the conjecture, the students are developing the idea that the conjecture works with two even numbers and it works with an odd and even number, but it doesn’t work with two odd numbers. They also talk about the idea that with an odd and an even factor, the odd number needs to double, and the even number would need to be halved. A simple game prompted rich discussions about deep mathematical ideas!
Throughout this book, we will guide you in integrating mathematical argumentation in your classroom and support you as a learner by showing you mathematics in ways that you might not have explored before. Your curiosity is an important component to reigniting curiosity in your students. When teachers model and encourage curiosity, it makes a meaningful impact on students’ questioning and curiosity (Engel, 2015). Let’s continue to nurture that in ourselves so that we can reignite our students’ curiosity as well!
As we mentioned in the introduction (page 1), the following Your Turn, like others throughout the book, invites you to grow your mathematical wonder by taking a learner’s perspective. Let’s look again at the rectangle conjectures from Hannah’s third-grade classroom, this time with you exploring the ideas.
YOU r t U r N
CONJECTURES FOR RECTANGLES
Let’s start by looking at Brian’s conjecture. He is saying that with multiplication, we can halve one of the factors and double the other factor and get the same product. Brian has 5 × 6 as his expression based on his roll, but you can choose other factors to explore the conjecture. For example, what would happen if he rolled a 5 and 4 instead?
• Pick one of your examples and draw rectangles in the following box to represent the equations. Where is the halving in the representation and where is the doubling?
• What kind of numbers does this strategy work for?
• How could you convince someone that Brian’s conjecture is always true?
t eaching Mathematicians in Grades 3–5
You might wonder why the rectangle conjectures are so fascinating to us and why we began our journey together with it. Through our own work as elementary teachers and teacher educators, we have seen a lot going on in the minds of mathematicians in grades 3–5. Their work is brilliant and deep as they make sense of many different operations and make connections between them. They notice patterns, ask questions, and make connections between ideas with complex language and reasoning, like the students in Hannah’s class.
There are many ways that students talk in mathematics class and share their ideas. For example, we hear students discussing a solution, a procedure, a strategy, a classmate’s idea, a question, an observation, a connection to another idea, a rule, and so on. While there is value in the different ways of communicating, it is important to provide opportunities for students to develop their conceptual understanding through higher levels of discussion and mathematical argumentation at all grade levels. What does that mean? As educators, we strive to foster deep mathematical discussions within our lessons to build a strong foundation that includes well-developed conceptual understanding and communication skills. We want to nurture a strong mathematical identity and give all students a voice in the classroom, creating an environment where all students flourish.
We also want to provide opportunities beyond stating procedures and focusing on answers. For instance, in the rectangle example, Hannah encouraged students to go beyond the specific examples of the game and think about the structure of the numbers and what might always be true. This aligns with what mathematics education researchers Susan Jo Russell, Deborah Schifter, and Virginia Bastable (2011) write:
Children spend much time in mathematics solving individual problems. But the core of the discipline of mathematics is looking across multiple examples to find patterns, notice underlying structure, form conjectures around mathematical relationships, and, eventually, articulate and prove general statements. (p. 2)
We explored the idea of communicating about the “core of the discipline of mathematics” through mathematical argumentation with teachers and their students in grades 3–5. As we spent time in their classrooms, we saw many examples of students engaging as a community of learners, exploring and communicating deep mathematics much like the students in the opening vignette (page 10).
Understanding Layers of Mathematical a rgu mentation
The amazing student thinking heard in Hannah’s classroom relates back to mathematical argumentation. In the vignettes we’ll share in this book, students are communicating about the “core of the discipline of mathematics” (Russell et al., 2011, p. 2) by noticing patterns, making conjectures, justifying, and extending their ideas in new ways. Research (Rumsey, Guarino, Gildea, Cho, & Lockhart, 2019; Rumsey, Guarino, & Sperling, 2023; Rumsey & Langrall, 2016) shows that mathematical argumentation in elementary classrooms grows through the following four layers.
1. Noticing and wondering: Making observations and asking curious questions
2. Conjecturing: Making general statements about the observations
3. Justifying: Convincing others and explaining why a conjecture is true or false
4. Extending: Making connections beyond the current conjecture, and noticing and wondering with the new ideas in mind
These layers emerged in our original work with K–2 teachers and students and as we extended and explored with teachers and students in grades 3–5 in the second part of the project. The teachers found it easier to break mathematical argumentation into the four layers rather than tackling it all at once. Other researchers also find that mathematical argumentation breaks down into distinct parts. Deborah Schifter and Susan Jo Russell (2020) developed a five-phase model that separates “different points of focus in the complex process of formulating and proving such generalizations” (p.15). Those five phases are “(1) noticing patterns, (2) articulating conjectures, (3) representing with specific examples, (4) creating representation-based arguments, and (5) comparing and contrasting operations” (Schifter & Russell, 2020, p. 15). Mathematics educators Traci Higgins, Susan Jo Russell, and Deborah Schifter (2022) find “four key dimensions of student conjecturing that apply to the context of generalizing about the behavior of the arithmetic operations” (p. 188) for students in grades 2 through 5. Mathematics education researchers Jennifer Knudsen, Harriette S. Stevens, Teresa Lara-Meloy, Hee-Joon Kim, and Nicole Shechtman (2017) write that with middle school students there is a four-part model of mathematical argumentation: (1) generating cases, (2) conjecturing, (3) justifying, and (4) concluding. While the models and terminology are not the same, these examples show that there are interrelated components of argumentation. Our hope is that breaking down each of the layers we found in the elementary classrooms will support you in nurturing mathematical argumentation in your own classroom.
t E a CHE r VOICES
WHAT IS MATHEMATICAL ARGUMENTATION?
Argumentation is giving students an opportunity to develop conceptual understanding by tapping into their own ideas and curiosity. It’s about communicating precise ideas. It asks students to study a piece of work and use their natural wonderings to voice their own big ideas and conjectures about a concept. But more importantly, it asks students to stand behind their ideas and justify or revise their thinking while listening and considering the ideas of others. (Jill, elementary math coach)
By growing layers of mathematical argumentation within your classroom, you build a strong foundation for students while nurturing skills beyond the mathematics classroom. In addition, you create a classroom culture where learning is shared, everyone’s voice is valued, and students communicate clearly. Exploring, conjecturing, justifying, communicating, and considering other people’s ideas are important skills to develop, not only in mathematics, but also in all other subjects, such as English language arts and science.
We describe the four layers—(1) noticing and wondering, (2) conjecturing, (3) justifying, and (4) extending—in part 2 (page 113) so that you can see the distinctions between them and nurture them organically in your own classroom. The layers of argumentation don’t happen in isolation. Sometimes, students will flow between layers on their own; other times, you’ll need to nudge them into a new layer with questions. Harvey Mudd College professor of mathematics Francis Su (2020) writes that we can “think of exploration as a continuous cycle, passing from one phase to the other and back again” (p. 57). When we get to the fourth layer, we will have new things to notice and wonder that are built on the previous conjecture and exploration. We can modify conjectures and ideas, lingering longer on the context and continuing to explore.
Let’s look in another classroom to dig into some of the characteristics of mathematical argumentation with mathematicians in upper elementary grades.
CLASSROOM VIGNETTE: DECIMAL OBSERVATIONS WITH FIFTH GRADERS
Let’s look in another classroom to dig into some of the characteristics of mathematical argumentation with mathematicians in upper elementary grades. In Christina’s fifth-grade classroom, the students are exploring decimals. First, as a class, they engage in the routine “What do you know about ?” where students share ways they think about the number 0.01 and write everything they can about it. Figure 1.4 shows that classroom conversation on the public record , which is large chart paper where ideas are recorded and explored.
As the class compares which is bigger, 0.1 or 0.01, they compare the numbers to 1 and how many of the units it would take to get to 1. Then a student named Nick begins to think more generally about the size of the numbers.
Nick: You can add a bunch of zeros to the one and it will stay the same.
Christina: Tell me about where we could put the zeros.
Nick: On the right side.
Christina: You can add zeros and it doesn’t change. Does it work if I put two zeros? Or one-hundred zeros? You can add zeros and it doesn’t change that number. You know a lot about decimals.
Jamal: You can also add zeros before the decimal.
Maria: You can add zeros before the decimal and to the right, but if you add them in between the second zero and the one, then it will get increasingly smaller.
Christina: So, if you add a zero here, it changes it. It gets smaller if you add the zeros there. Do you think this is always going to be true?
The idea about where you can put zeros and change or not change the value of the number interests the students and sparks a lot of conversation. Their curiosity and engagement are reignited, and the excitement in the room grows. Many students want to share what they know about 0.01. Through their observations, the following two conjectures emerge, related to where you can put zeros to change the value of the number or not change the value of the number.
× 0.01 is 0.1 1 100 You need 100 to get 1. 0.01 Add zeros before, and it doesn’t change.
Zero in between changes it. It makes the number smaller. One is in the hundredths place. You can add zeros here, and it doesn’t change the value.
Not the same as 0.1 (one tenth)
0.1 is bigger than 0.01
Figure 1.4: Public record for 0.01 observations. Visit go.SolutionTree.com/mathematics for a free full-color reproducible version of this figure.
1. You can add zeros before (to the left of) the decimal point.
2. You can add zeros before the decimal and to the right, but if you add them in between the second zero and the one, then it will get smaller.
These working conjectures support the students’ understanding of place value and the size of numbers. There is room for them to make the conjectures more precise and begin thinking about how to convince someone that it is always true, but this is an amazing start to the lesson and gets the students excited to keep sharing. We’ll revisit this lesson in chapter 6 (page 135) and see how incorporating a game into the lesson pushes the students to continue thinking about this conjecture and begin to convince others that it is always true.
P a USE a ND PONDE r VIGNETTES
• What do you notice about the two classroom vignettes (pages 10 and 16) we’ve shared so far? What do they have in common?
• What are students noticing and conjecturing, and how are those ideas connected to the major work of upper elementary school?
• Where could their ideas be fleshed out and made more precise?
• What do you wish you could ask the students to learn more about their thinking?
Unpacking Layers of a rgu mentation in the Classroom Vignettes
Dropping into Hannah’s third-grade classroom (page 10) and Christina’s fifth-grade classroom (page 16), we can start to see ways that mathematicians in grades 3–5 playfully and curiously engage in mathematics, and we can see the layers of mathematical argumentation. For example, in Hannah’s class, the students noticed patterns in the multiplication expressions, looked across examples, gathered information, and conjectured that with multiplication they could halve one of the factors and double the other factor and get the same product. Students continued to use examples to convince each other, extend the ideas and modify the conjecture, even considering when the strategy would work. Christina’s fifth-grade students worked on a conjecture about decimals and where you can put zeros without changing the value. The students extended the conjecture to explore how adding zeros changes the value. The students did not only state the conjectures, they also began to justify and share with the whole class to modify their ideas and make them more precise. In these two classrooms, we’ve seen layers of mathematical argumentation already emerging! Table 1.1 shows the vignettes with examples of the four layers seen in the classroom examples.
Layer of Argumentation
Noticing and Wondering
Conjecturing
Building Rectangles With Third Graders
• Noticed patterns in multiplication expressions
• Connected the shifting rectangle dimensions to the expressions
• Compared pairs of expressions
• Conjectured about halving and doubling
• Conjectured switching the order of the rule
Decimal Observations With Fifth Graders
• Unpacked place value within the number
• Made sense of the size of numbers
• Compared 0.01 to 0.1
• Noticed how to change or not change the quantity
• Conjectured about where to put zeros to change the quantity
• Conjectured about where to put zero to keep the quantity the same
Table 1.1: Argumentation in the Classroom Vignettes
Justifying • Used the rectangle grid
• Imagined cutting and moving the rectangle grid
Extending • Considered what numbers it works for and when to use the strategy
Connecting a ddition and Multiplication
• Used examples to show changes
• Imagined numbers on a number line to show how the zero affects the location on the number line
• Considered adding more zeros to change the value of the number
This book is not just about reigniting wonder and curiosity in your students; it is about growing as teachers so you can meet students where they are. Engel (2015) writes that expressions of curiosity decline as we enter school, and it’s important that we work to reignite that in our students and ourselves. Curiously exploring is an important component of learning at all levels (Engel, 2015; Piaget, 1952). In his groundbreaking work on child development, psychologist Jean Piaget (1977) writes about how we create cognitive schemas to assimilate and accommodate new information as we learn. Learning is not about discrete facts but about growing a well-connected web of knowledge (Piaget, 1977; Van de Walle, Karp, & Bay-Williams, 2013). To help you grow your wonder and curiosity, we will explore deep mathematical connections in Your Turn: Addition and Multiplication Task, and we will revisit this task throughout the book.
YOU r t U r N
ADDITION AND MULTIPLICATION TASK
Su (2020) writes, “Doing math properly is engaging in a kind of play: having fun with ideas that emerge when you explore patterns and cultivating wonder about how things work” (p. 50). Therefore, let’s explore the relationship between addition and multiplication. The following task is for you to explore as a learner rather than to teach to your own students, although there are a lot of ideas they can unpack in this activity as well. First, set a timer and spend ten to fifteen minutes exploring what you notice and wonder about the following equations.
• 7 + 4 = 11 7 × 4 = 28
• 8 + 4 = 12 8 × 4 = 32
• 9 + 4 = 13 9 × 4 = 36
The following list includes some ideas to think about as you get started.
• Look for patterns
• Write the patterns that you notice and extend the columns to see if your predictions continue
• Organize your thoughts into a table or a chart
• Draw pictures or diagrams to make your thinking visible
• I notice . . .
• I wonder . . .
If you are reading this book as part of a book study, share your ideas with others and look for ways that your ideas are similar, different, and connected.
In figure 1.5, we provide samples of what other teachers noticed and wondered about the equations in Your Turn: Addition and Multiplication Patterns Task. Choose some of the work shown in the figure, make sense of the other teachers’ ideas, make notes and connections to what you came up with. How are some of the teachers’ ideas related to your own? Is there something you hadn’t thought of that you find interesting? How is the work organized and how is it helpful to make patterns more visible?
4 is in all equations
The sums are smaller than the products. The products are multiples of both factors.
When the first addend goes up by 1, the sum goes up by 1. 28 − 11 = 17 32 − 12 = 20 36 − 13 = 23
What is the difference between the products and sums?
Why is this increasing by 3?
Figure 1.5: Samples of teacher observations.
When the first factor goes up by 1, the product goes up by 4.
How would the equations and patterns change if 4 changes?
Visit go.SolutionTree.com/mathematics for a free full-color reproducible version of this figure.
The following list of wonderings emerged as we discussed this problem with other teachers.
• How would the equations and patterns change if the 4 changes?
• What is the difference between the product and the sum in each row? Why is the difference increasing by 3?
• How are addition and multiplication related?
• What does the 4 represent? Is there a related story problem?
• Why do the sums increase by 1 but the products increase by 4?
• What would happen if we graphed the equations or the data?
Do you have a wondering that is evolving into a conjecture? Is there an observation you could generalize into a statement that you believe to always be true? You can use the teacher work examples in figure 1.5 (page 21) to get started as you write a conjecture about the idea that interests you. Think of it as a first draft; you can always go back later to revise it to make it more precise, but get your ideas down first! Here are some ideas to get you started.
• How could you convince someone that the conjecture is true? Use objects, pictures, numbers, or words to justify why your conjecture is true.
• Are there words that could make the conjecture more precise? How can you rewrite your conjecture to make it more precise and understandable to other people?
• What else do you think might be true now that you have explored this conjecture? How does this idea extend to other numbers?
Writing a conjecture may seem like a simple task, but there is a lot to think about! Slowing down to explore a problem takes practice. We tend to be in a hurry as adults, so it can be hard to enter a task with the curiosity and wonder of our students. What new ideas did you learn through the exploration of multiplication and addition? How did it connect to your previous knowledge? Jot down your answers to the questions in figure 1.6 in the spaces provided. We will come back to this figure as we continue reflecting throughout the book.
Questions
Noticing and Wondering
• What was challenging about noticing?
• What was challenging about wondering?
• Which patterns were you drawn to?
• Which peer observations and wonderings surprised you?
• How did it feel to wonder?
Conjecturing
• Which questions helped push your wonderings into a generalized conjecture?
• What was challenging?
Justifying
• Which ways of convincing are your go-to ways?
• What was challenging about convincing someone else?
Extending
• What other ideas are you drawn to?
• Where does your brain want to go next?
• What connections can you make to other mathematical ideas?
Figure 1.6: Reflecting on your own experience with the addition and multiplication patterns task. Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Questions for Further r eflection
The following questions will help you synthesize your learning by reflecting on chapter 1.
• What inspired you within this chapter? What surprised you about the students’ ideas in the classroom vignettes?
• As you think about the addition and multiplication patterns task, how is it similar and different to your K–12 math learning?
• What did the chapter leave you curious about?
• As you reflect on the chapter, what opportunities could argumentation offer your students?
• What does argumentation offer you as a teacher?
Chapter 1 Summary
In this chapter, we shared our observations of what argumentation looks like with mathematicians in grades 3–5 through two classroom vignettes. We discussed the four layers of argumentation and reignited your curiosity through the addition and multiplication patterns task (page 20). We are glad you are beginning this journey with us. Use the “Chapter 1 Application Guide” reproducible to help you integrate this chapter’s ideas into your classroom. Next, in chapter 2 (page 27), we’ll discuss exploring a classroom community.
