Opportunities for communication, in particular classroom discourse, are foundational to the problem-based structure of the IM K–5 Math curriculum. The National Council of Teachers of Mathematics’s Principles and Standards for School Mathematics (NCTM, 2000) states, “Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically.” Opportunities for each of these areas are intentionally embedded directly into the curriculum materials through the student task structures and supported by the accompanying teacher directions.

One highly visible form of discourse is student discussion during the course of a lesson. Another, not as highly visible form of discourse is writing. While this is often only seen as the written responses in a student workbook, journal writing can provide an additional opportunity to support each student in their learning of mathematics.

Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify their ideas and develop a deeper understanding of the mathematics at hand.

To encourage the use of journal-writing in math class, we have provided a list of journal prompts that can be used at any point in time during a unit and across the year. These prompts are divided into two overarching categories: Reflections on Content and Reflection on Beliefs and Feelings.

Reflections on content focus on the students’ learning or specific learning objectives in each lesson. We first ask students to reflect on the mathematical content because, in general, the act of writing requires a deliberate analysis that encourages an explicit association between current and new knowledge that becomes part of a deliberate web of meaning (Vygotsky, 1987). For example, when students are asked to write about ways in which the math they learned in class that day was connected to something they knew from an earlier unit or grade, they are explicitly connecting their prior and new understandings.

John Dewey asserted that students make sense of the world through metacognition, making connections between their lived experiences and knowledge base, and argued that education should provide students with opportunities to make connections between school and their lived experiences in the world. This belief, alongside one of Ladson-Billings’ principles of CRT that states teachers must help students effectively connect their culturally- and community-based knowledge to the learning experiences taking place in the classroom, supports the need for students to continually reflect not only on the mathematics, but on their own beliefs and experiences as well. Reflections on beliefs are more metacognitive and focus on students’ feelings, mindset, and thinking around using mathematics. Writing about these things promotes metacognitive frameworks that extend students’ reflection and analysis (Pugalee, 2001, 2004). For example, as students describe something they found challenging during a lesson, they have the chance to reflect on the factors that made it a challenge.

Since the prompts, regardless of the category, can be used at any point during the year, they live in the Course Guide. We imagine these prompts could be used in a variety of ways. In the early grades, they might be used as discussion prompts between partners or students may be asked to respond to a prompt in the form of a drawing or example from their work of the day. In later grades, students can establish a math journal at the beginning of the year and record their reflections at the beginning, in the middle, or at the end of lesson, depending on the prompt. For schools or districts who require homework, the prompts may serve as a nice way for students to reflect on their learning of the day or ask questions they may not have asked during the class period.

Journal writing not only encourages explicit connections between current and new knowledge and promotes metacognitive frameworks to extend ideas, but it also provides opportunities for teachers to learn more about each student’s identity and math experiences. We believe that writing in mathematics can offer a means for teachers to forge connections with students who typically drift or run rapidly away from mathematics and offer students the opportunity to continually relate mathematical ideas to their own lives (Baxter, Woodward, and Olson, 2005). Writing prompts and journaling work well because students who may not advocate well for themselves when they are struggling get their voices heard in a different way, and thus their needs met (Miller, 1991).

It is our hope that through the use of these questions and prompts, students will communicate to learn mathematics as well as learn to communicate mathematically.

Reflecting on Content and Practices

• What math did you learn and do today that connected to something you knew from an earlier unit or grade?
• Describe a time you used the math you learned today outside of school. 
• How did your thinking change about something in math today?
• How did any predictions you made in class today work out? Why do you think that happened? 
• What questions do you still have about the math today? What new questions do you have? 
• Describe the way you solved a problem in class today that you are proud of.
• Where do you see the math you did in class today outside of school? (MP4)
• What math tool did you find most helpful today? Why? (MP5)
• What patterns did you notice in the mathematics today? Why did that pattern happen? (MP7, MP8)
• Starter prompts:
  • The most important thing I learned today is . . .
  • Today, I struggled and worked through a problem when . . . (MP1)
  • I could use what I learned today in math in my life when I . . .
  • At the end of this unit, I want to be able to . . .
  • I knew one of my answers was right today when . . .
  • Another strategy I could have used to solve a problem today is . . .
  • The most important thing to remember when doing the problems like we did today is . . .

Reflecting on Learning and Feelings about Math

• Describe something you really understand well after today’s lesson or describe something that was confusing or challenging.
• In math class, it’s important to be able to explain your thinking. Describe a time when you were able to explain your ideas to other people in your class. (MP3)
• In math class, it’s important to listen to other people’s ideas. Describe a time when you learned something by listening carefully to someone in your class. (MP3)
• What does it mean to be good at math?
• Describe a time when you asked a question about math you were working on. How did asking a question help you?
• If you could change anything about math class, what would it be? Why?
• Starter prompts
  • I learned from a mistake today in math when . . .
  • When it comes to math, I find it difficult to . . .
  • I love math because . . .
  • I felt heard during class today when . . .
  • I felt my ideas were valued during class today when . . .
  • The most helpful thing that happened today was . . .

References

• Baxter, J. A., Woodward, J., & Olson, D. (2005). Writing in mathematics: An alternative form of communication for academically low-achieving students. Learning Disabilities Research & Practice, 20(2), 119–135.
• Baxter, J. A., Woodward, J., Olson, D. & Robyns, J. (2002). Blueprint for writing in middle school mathematics. Mathematics Teaching in the Middle School, 8 (1), 52–56. 
• Liedtke, W. W. & Sales, J. (2001). Writing tasks that succeed. Mathematics Teaching in the Middle School, 6 (6), 350–55.
• Miller, L. (1991). Writing to learn mathematics. The Mathematics Teacher, 84(7), 516-521.
• Pugalee, D.K. (2001), Writing, mathematics and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, 101(5), 236–245.
• Pugalee, D. (2004). A comparison of verbal and written descriptions of students’ problem solving processes. Educational Studies in Mathematics, 55, 27-47.
• Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber, & A. S. Carton (Eds.), The collected works of L. S. Vygotsky: Vol. 1. Problems of general psychology (pp. 39-285). New York: Plenum Press.

As stated in our design principles, within a problem-based curriculum “students learn mathematics by doing mathematics.” Given the nature of math classrooms, however, students come with differing math identities, which means some students are more prone to see themselves as doers of mathematics than others. Furthermore, apparent inequities in math instruction suggest that some students have opportunities to bring their voice into the classroom, and others do not. In order to extend the invitation to all students to do mathematics, we must work to explicitly develop the math learning community.

Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms that expect students to communicate their mathematical thinking to their peers and teacher, both orally and in writing, using the language of mathematics—positively affect participation and engagement among all students (Principles to Action, NCTM).

To support teachers to develop math learning communities in their classrooms, the first six lessons of each course embed structures to collectively identify what it looks like and sounds like to do math together, create, and reflect on classroom norms that support those actions.

Beyond the first six days, teachers should revisit these norms at least once a week to sustain the math learning community. Consistently returning to these ideas shows students that we value the math learning community as much as we value the math content. Students should also be provided with opportunities to reflect on the norms by stating which ones are the most challenging for them and why. Teacher reflection questions periodically remind teachers of points in a unit where it may be helpful to reflect on these norms.

Additional teaching moves can be used to support the development of math learning communities throughout the school year. The section below highlights teaching moves, put forth by Phil Daro and the SERP Institute, that are intended to support students’ engagement in the mathematical practices. A solid math learning community exists when all students display these observable actions, called student vital actions. 

Teaching Moves to Support Math Community

student vital actions	teacher moves
All students participate.	Assign rotating roles, and provide routines for collaboration so that every student is actively engaged in each task, and has experience in all roles over time. When students are confused, ask them to show where they got lost or ask a question that can help them move forward (more than “I don’t get it” or “How do you do it?”).  Check to see if there are recognizable patterns between participation and prior achievement or social groups (for example, EL, race/ethnicity, or gender).
Students say a second sentence.	Ask and encourage students to ask:  “Can you tell me more about that?” “Why do you think that?” “What changed and what stayed the same?” “Is that an answer that makes sense for this problem? How do you know?” “How did you get that answer? Why did you (reference student work)?” “Is it always true? Sometimes true?”
Students talk about each other’s thinking.	Show and discuss work generated by students when working with mathematics concepts. Questions that may be used to prompt students:  “Did anyone approach the problem a different way?” “How is your thinking different from theirs?” “What does their way of thinking help you understand?” “Do you think their method would work with this kind of problem? Why or why not?” Try only responding to questions from groups when no one in the group can answer the question and everyone in the group can ask it.
Students revise their thinking.	If a student is presenting an explanation, play the role of not understanding and say, “Could you help me make sense of your thinking? Could you revise your explanation?” Have a student quote a classmate’s statement that inspired them to revise.  Have students confer in small groups after whole-class presentations to revise and refine their way of thinking.
Students engage and persevere.	Ask a student who has given a wrong answer additional questions to explore their thinking. Demonstrate curiosity about that thinking. Have students share their thinking and attempts even when they have not found a viable solution.  When some groups are “finished” earlier than others, ask them to analyze their work and seek places to revise their explanation so more students will understand it, or look for an alternative approach.
Students use general and discipline-specific academic language.	Before beginning small group work, give students sentence frames and probing questions that feature important terms. Accept students’ everyday way of talking as a starting point for joining the math conversation.  Teachers can refer to student statements using some student language while strategically incorporating more precise academic language with the addition of a key word or phrase.
English learners produce language.	For everyday words that have precise mathematical meaning, provide multiple contexts where the word is useful and have students explain what it refers to in that context. Ask them to use the word to make connections between the different representations. Encourage students to use language to construct meaning from representations with prompts such as:  “Explain where you see (length, ten, oranges) in the (figure, equation, table). How do you know it represents the same thing?”  Every student speaks, listens, reads, and writes.

For more details and a full list of teaching moves, visit the SERP Institute site: https://www.serpinstitute.org/5x8-card/vital-student-actions

Teaching mathematics is complex work. It requires teachers to plan lessons that offer each student access, elicit students’ ideas during these lessons, find ways in which to respond to those ideas, and build a classroom community where students feel known, heard, and seen. Teachers must always be flexible and timely in decision-making in order to engage students in rich mathematical discussions. Within each decision lies the opportunity to orient students to one another’s ideas and the mathematical goal, and position each student as a competent learner and doer of mathematics. One of the biggest challenges to learning from the work of teaching is that the majority, if not all, of a teacher’s learning, planning, and decision-making happens in isolation. 

Professional learning communities (PLCs), are spaces in which teachers can work together around planning and teaching. PLCs include any time teachers or coaches work collaboratively in recurring cycles of collective inquiry and action research to achieve better results for the students they serve. Professional learning communities operate under the assumption that the key to improved learning for students is continuous job-embedded learning for educators (DuFour, R., Dufour, R., Eaker, R., & Many, T, 2006). 

To support teacher collaboration around planning and teaching, we have identified an activity in every unit section as a PLC activity. This activity was chosen because it is either a key mathematical idea of the section or requires a more complex facilitation. We also organized a structure for teachers to use as they work together in professional learning communities.

The suggested structure is categorized as pre-, during-, and post-lesson to offer teachers the opportunities to experiment with instruction during both planning and the classroom enactment by collectively discussing instructional decisions in the moment (Gibbons, Kazemi, Hintz, & Hartmann, 2017). These suggestions are meant to provide guidance for a professional learning community of teachers and coaches that meet to plan for upcoming lessons. While using all of the suggestions in the given structure is ideal, they are flexible enough to adapt to fit any teacher’s given schedule and context. 

Suggested before a professional learning community meeting

• Read the upcoming lesson that is the focus of the meeting.
• Review student cool-downs from previous lessons.
• Discuss: 
  • current student understandings
  • ways in which these understandings build toward the PLC activity

Suggested during a professional learning community meeting 

• Do the math of the PLC activity individually.
• Read the CCSS and learning goal addressed by the activity.
• Discuss how the standard and learning goal are reflected in what the activity is asking students to do. Think about: 
  • Are students conceptually explaining a new, or developing, understanding?
  • Are students making connections between a conceptual understanding and a procedure or process? 
• Based on students’ previous lesson cool-downs, discuss 1–2 ways students might complete the activity. 
• Discuss: 
  • How might student responses reflect the CCSS and lesson learning goal?
  • What unfinished learning might students have?
• Based on these discussions, make a plan for: 
  • look-fors as you monitor students during their work time
  • questions to ask that assess and advance student thinking 
  • the sharing of work and student discussion during the activity synthesis

Suggested after a professional learning community meeting 

• Record observations as students work.
• Review student cool-downs in relation to the learning goal of the lesson.

References

• DuFour, R., DuFour R., Eaker, R., & Many, T. (2006). Learning by doing: A handbook for professional learning communities at work. Bloomington, IN: Solution Tree. 
• Gibbons, L. K., Kazemi, E., Hintz, A., Hartmann, E. (2017). Teacher time out: Educators learning together in and through practice. Journal of Mathematics Educational Leadership, 18(2), 28–46.

“The power of a representation can . . . be described as its capacity, in the hands of a learner, to connect matters that, on the surface, seem quite separate. This is especially crucial in mathematics” (Bruner, 1966).

Mathematical representations can be used for two main purposes: to help students develop an understanding of mathematical concepts and procedures or to help them solve problems. The materials make thoughtful use of representations in both ways. 

Curriculum representations and the grade levels at which they are used are determined by their usefulness for particular mathematical learning goals. Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent. Over time, they will see and understand more efficient methods of representing and solving problems, which support the development of procedural fluency.  

In general, more concrete representations are introduced before those that are more abstract. There are a couple of key progressions of representations that occur across grade bands in different domains.

These progressions, as well as the descriptions below, can be helpful in providing support for students who have unfinished learning and would benefit from more concrete representations to make sense of mathematical concepts.

 

Two-color Counters (K–1)

Counters of one color are used frequently to represent quantities in the early grades. Students use the two-color counters to support their work in comparing, counting, combining, and decomposing quantities. In later grades, the counters can be used to visually represent properties of operations.

Connecting Cubes (K–5)

Like counters, cubes can be used in the early grades for comparing, counting, combining, and decomposing numbers. In later grades, they are used to represent multiplication and division, and in grade 5, to study volume. Teachers of grade 5 should use cubes that connect on multiple sides to develop understanding of volume.

5-frame and 10-frame (K–2)

5- and 10-frames provide students with a way of seeing the numbers 5 and 10 as units and also combinations that make these units. Because we use a base-ten number system, it is critical for students to have a robust mental representation of the numbers 5 and 10. Students learn that when the frame is full of ten individual counters, we have what we call a ten, and when we cannot fill another full ten, the “extra” counters are ones, supporting a foundational understanding of the base-ten number system. The use of multiple 10-frames supports students in extending the base-ten number system to larger numbers.

Connecting Cubes in Towers of 10 (1–2)

Cubes that are in towers of 10 support students in using place value structure for adding, subtracting, and comparing numbers. Connecting cubes have the advantage that students can physically compose and decompose numbers, unlike place value blocks or Cuisenaire rods. The cubes are a helpful physical representation as students begin to unitize. For example, students can understand that 10 of the single cubes are the same as 1 ten and 10 of the tens are the same as 1 hundred.

Base-ten Blocks (2–5)

Base-ten blocks are used after students have had the physical experience of composing and decomposing towers of 10 cubes. The blocks offer students a way to physically represent concepts of place value and operations of whole numbers and decimals. Because the blocks cannot be broken apart, as the connecting cube towers can, students must focus on the unit. As students regroup, or trade, the blocks, they are able to develop a visual representation of the algorithms. The size relationships among the place value blocks and the continuous nature of the larger blocks allow students to investigate number concepts more deeply. The blocks are used to represent whole numbers and, in grades 4 and 5, decimals, by defining different size blocks as the whole.

Base-ten Diagram (1–5)

Base-ten diagrams offer students a way to represent base-ten blocks after they no longer need concrete representations. Although individual units might be shown, the advantage of place value diagrams is that they can serve as a “quick sketch” of representing numbers and operations.

 

Tape Diagram (2–5)

Tape diagrams, resembling a segment of tape, are primarily used to represent the operations of adding, subtracting, multiplying, and dividing. Students use them first with whole numbers and later with fractions and decimal numbers to emphasize the idea that the meaning and properties of operations are true as the number system expands. They can help students represent problems, visualize relationships between quantities, and solve mathematical problems.

Number Line Diagram (2–5)

Number line diagrams are used to represent and compare numbers, and can also be used to represent operations. Understanding of number line diagrams is built on students’ grade 2 experience with rulers. Students begin by working with number lines with tick marks to represent the whole numbers. Then, they work with number lines where tick marks correspond to multiples of 10, 100, or 1,000 to develop an understanding of place value and relative magnitude. In later grades, students understand that there are numbers between the whole numbers. They extend their work with whole number operations on the number line to include fractions and decimals. 

Fraction Strips (3–4)

Fraction strips are rectangular pieces of paper or cardboard used to represent different parts of the same whole. They help students concretely visualize and explore fraction relationships. As students partition the same whole into different-size parts, they develop a sense for the relative size of fractions and for equivalence. Experience with fraction strips facilitates students’ understanding of fractions on the number line.

Array (2–3)

An array is an arrangement of objects or images in rows and columns that can be used to represent multiplication and division. Each column must contain the same number of objects as the other columns, and each row must have the same number of objects as the other rows.

Inch Tiles (2–4)

Inch tiles offer students a way to create physical representations of flat figures that have a certain area and to cover a flat figure with square units to determine its area. Students organize inch tiles into rows and columns to connect the area of rectangles to multiplication and division.

Area Diagram (3–5)

An area diagram is a rectangular diagram that can be used to represent multiplication and division of whole numbers, fractions, and decimals. The area diagram may be overlaid with a grid to show individual units. As students move from working with an area diagram overlaid with a grid to one without, they move from a more concrete understanding of area to a more abstract one. In an area diagram without a grid, the unit squares are not explicitly represented, which makes this diagram useful when working with larger numbers or fractions and making connections to the distributive property and algorithms. 

As the numbers in products become larger, area diagrams are difficult to read if the ones, tens, and eventually hundreds are shown accurately. This diagram shows a way to visualize the product \(53 \times 31\).

It shows how to decompose the product into 4 parts, represented in the diagram as smaller rectangles. The size of each smaller rectangle in the diagram does not represent its actual size since the segment labeled 30 is not 30 times as long as the segment labeled 1. Even though the small rectangles do not have the correct relative size, the diagram can still be used to correctly decompose the product \(53 \times 31\),

\(\displaystyle 53 \times 31 = (50 \times 30) + (50 \times 1) +(3 \times 30) + (3 \times 1)\)

The diagram helps visualize geometrically why the equation is true.

References

• Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.