Coherent Progression

To support students in making connections to prior understandings and upcoming grade-level work, it is important to understand the progressions in the materials. Grade-level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors.

The basic architecture of the materials supports all learners through a coherent progression of the mathematics, based both on the standards and on research-based learning trajectories. Activities and lessons are parts of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense.

Every unit, lesson, and activity has the same overarching design structure: The learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.

The overarching design structure at each level is as follows:

Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer the opportunity to observe students’ prior understandings.
Each lesson starts with a Warm-up to activate students’ prior knowledge and to set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The Lesson Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned.
Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.

Purposeful Representations

“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 serve two main purposes: to help students develop an understanding of mathematical concepts and procedures, and to help them solve problems. For example, in IM Grade 6, students first use tape diagrams to make sense of two interpretations of division—finding the number of equal-size groups, and finding the amount in each group. Later, they use tape diagrams to solve division problems that involve fractions.

Several considerations guide the choice of representations, the timing of their introduction, and the sequence in which they are presented, such as the extent to which they:

Serve the mathematical learning goals.
Help to build conceptual understanding.
Support a coherent progression of mathematical ideas.
Are relevant across cases and contexts.

The principle of “concrete before abstract” also guides the use of representations. This principle comes into play in two ways:

First, more concrete representations are introduced before those that are more abstract. For example, to develop an understanding of ratios, students first use physical objects to represent quantities related by a ratio. Next, they create discrete diagrams to represent such quantities, followed by double-number-line diagrams and tables of equivalent ratios. This thinking eventually bridges into representing non-proportional linear relationships and the multiple ways to represent functions that starts in IM Grade 8 and continues throughout high school. Each representation is less concrete, more abstract, and offers greater flexibility and efficiency than the one before it.
Second, a representation can show concrete concepts and numbers in earlier courses, and abstract ideas and quantities in later courses. For example, starting in IM Grade 3, students use rectangular diagrams to represent the multiplication of whole numbers. In IM Grade 6, they use such diagrams to represent the product of multi-digit decimals. In these early uses, students relate the discrete factors and product to concrete attributes of the rectangle: its side lengths and area. Later, students use rectangles to represent the multiplication of expressions that include variables, such as and . The same diagram is now used in a more abstract sense: to represent and organize decomposable factors and their partial products. In high school, students use the same diagram to reason about the multiplication and division of binomials and higher-degree polynomials.

Across lessons, units, and courses, students are encouraged to use representations that make sense to them, and to make connections—between representations, as well as between representations and the concepts and procedures they show. Over time, students learn to recognize and use efficient methods of representing and solving problems, which in turn supports fluency.

Applying Mathematics

Students make connections to real-world contexts throughout the materials. Carefully chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts. The first unit on geometry is such an example.

Developing Conceptual Understanding and Procedural Fluency

Three aspects of rigor are essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects, developed together, are interconnected in the materials in ways that support students’ understanding.

Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations. Warm-up routines, practice problems, centers, and other built-in activities help students develop procedural fluency, which develops over time.

Each unit begins with Check Your Readiness, a diagnostic assessment to gauge what students know about both prerequisite and upcoming concepts and skills. Adjustments are then made accordingly. The initial lesson in a unit activates prior knowledge and provides an easy entry point to new concepts, so that students at different levels of both mathematical and English language proficiency engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift toward procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.

Task Complexity

Mathematical tasks are complex in different ways, with the source of complexity varying based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities, without losing the intended mathematics, look to the Warm-up routine and activity Launch for built-in preparation, and to teacher-facing narratives for further guidance.

In addition to tasks that provide all students access to the mathematics, the materials include guidance on how to ensure that all students engage in the mathematical practices during the tasks. Teacher-reflection questions and other fields in the lesson plans help assure that all students have not only access to the mathematics, but also the opportunity to truly engage in the mathematics.

Tasks in IM lessons serve a variety of purposes:

Provide experience with a new context. Activities that give all students experience with a new context ensure that they are ready to make sense of the concrete before encountering the abstract.
Introduce a new concept and associated language. Activities that introduce a new concept and associated language build on what students already know, and ask them to notice or describe something new.
Introduce a new representation. Activities often present a new representation of a familiar idea and ask students to interpret it. Where appropriate, new representations connect to or extend from familiar representations. Give students clear instructions on how to create a representation such as a tool for understanding or solving problems. For subsequent activities and lessons, provide opportunities for students to practice using these representations and to choose which representation to use for a particular problem.
Formalize a definition of a term for an idea previously encountered informally. Activities that formalize a definition offer a general definition of a concept that students have encountered through examples.
Identify and resolve common mistakes and misconceptions. Activities that give students a chance to identify and resolve common mistakes and misconceptions usually present a piece of incorrect work and ask students to identify and explain the error. Students deepen their understanding of key mathematical concepts as they analyze and critique the reasoning of others.
Practice using mathematical language. Activities that provide students an opportunity to practice using mathematical language focus on that task as the primary learning goal. They give students a reason to use mathematical language to communicate. In these activities, students frequently use the Information Gap instructional routine.
Work toward mastery of a concept or procedure. Activities in which students work toward mastery are included for topics with which students typically need additional time. Often these activities are marked as optional and can be safely skipped because no new mathematics is covered.
Provide an opportunity to apply mathematics to an open-ended problem such as modeling. Activities that provide an opportunity to apply mathematics to an open-ended problem, such as modeling, most often are found toward the end of a unit. Their purpose is to give students experience using mathematics to reason about a problem or a situation that one might encounter naturally outside of a mathematics classroom.

A note about standards alignments: There are three kinds of alignments to standards in these materials: building on, addressing, and building toward. Oftentimes a particular standard requires weeks, months, or years to achieve, and in many cases depends on work in prior grade levels. When an activity reflects the work of prior grades but is used to bridge to a grade-level standard, alignments are indicated as “building on.” When an activity lays the foundation for a grade-level standard but doesn’t reach the level of the standard, the alignment is indicated as “building toward.” When a task focuses on the grade-level work, the alignment is indicated as “addressing.”

Teacher Learning through Curriculum

For all students to have access to mathematical learning opportunities, it is important for teachers to believe that every student can learn mathematics. And it is the responsibility of teachers to provide equitable instruction and to position all students in a way that supports learning. Getting better at teaching requires planning at the course level, the unit level, and the lesson level, and reflecting on and improving each day’s instruction. Equitable instruction requires development of teachers’ knowledge of mathematics and the socio-cultural contexts of the students in order to deepen the learning for all. Key components of the materials support teachers in understanding the mathematics they are teaching, the students they are teaching, or in some cases, both.

Unit, Lesson, and Activity Narratives

The narratives included in the materials afford teachers a deeper understanding of the mathematics and its progression within the materials.

Authentic Use of Contexts and Suggested Launch Adaptations

The use of authentic contexts and adaptations offers students opportunities to bring their own experiences to the lesson activities and see themselves in the materials and mathematics. When academic knowledge and skills are taught within students’ lived experiences and frames of reference, “they are more personally meaningful, have higher interest appeal, and are learned more easily and thoroughly” (Gay, 2010). By design, lessons include contexts in which students see themselves in the activities or learn more about others’ cultures and experiences.

Building on Student Thinking

Certain activities within each lesson plan include ways to provide guidance, based on students’ understandings and ideas. Building on Student Thinking offers look-fors and questions to support students as they engage in an activity. Effective teaching requires supporting students as they work on challenging tasks, without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). Monitor during the course of an activity to gain insight into what students know and are able to do. Based on these insights, the Building on Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.

Teacher Reflection Questions

To encourage reflection on the classroom teaching and learning, each section includes three teacher-directed reflection questions on the mathematical work or pedagogical practices of the lesson. The questions are drawn from three themes:

Mathematical Content and Student Thinking questions invite consideration of both the mathematics of the activities and how students reasoned about the activities. These questions may invite making a connection to a concept or a representation students encountered previously, as well as reflecting on how students access the content during a lesson through their representations and communications with each other.
Pedagogy questions invite consideration of instructional decisions throughout planning, teaching, and assessing a unit. These decisions encompass both the implementation of a problem-based curriculum, using available resources, as well as creating a mathematical classroom community.
Access and Equity questions invite reflection on teachers’ beliefs about students and how those beliefs impact students’ learning and access to the learning. These questions also ask how classroom structures and instructional decisions impact students’ beliefs about their own and each others’ mathematical capabilities.

Due to the overlapping nature of these themes, a question listed as aligning with a specific theme may align with more than one.

The questions are designed to be used by individuals, grade-level teams, coaches, and anyone who supports teachers. Choose to focus on one theme over the course of the year, one theme each unit, the question of greatest interest in each section, or some other combination to best suit your professional goals.

To ensure that all students have access to an equitable mathematics program, educators need to identify, acknowledge, and discuss the mindsets and beliefs they have about students’ abilities (NCTM, 2014). Spanning the three themes are beliefs-and-positioning questions that support the identification and acknowledgment of teachers’ mindsets and views. These questions prompt reflection and challenge the assumptions teachers make—about mathematics, learners of mathematics, and the communication of mathematics in their classrooms.

Using the 5 Practices for Orchestrating Productive Discussions

Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipate, monitor, select, sequence, and make connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons in which students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting as well, and the lesson is tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, Compare and Connect supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices.

Why is the curriculum designed this way?