Illustrative Mathematics is a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials are designed to support students’ and teachers’ learning. This document defines the principles that guide IM’s approach to mathematics teaching and learning. It then outlines how each component of the curriculum supports teaching and learning, based on these principles.

All Students are Capable Learners of Mathematics

With unique knowledge and needs, every student enters the mathematics learning community as a capable learner of meaningful mathematics. Mathematics instruction that supports students in viewing themselves as capable and competent leverages and builds upon the funds of knowledge they bring to the classroom. Instruction is grounded in equitable structures and practices that provide all students with access to grade-level content and provide teachers with necessary guidance to listen to, learn from, and support each student. The curriculum materials include classroom structures that support students in taking risks, engaging in mathematical discourse, productively struggling through problems, and participating in ways that make their ideas visible. It is through these classroom structures that teachers have daily opportunities to learn about and leverage students’ understandings and experiences, and to position each student as a capable learner of mathematics.

Learning Mathematics by Doing Mathematics

Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or getting told what to do. “Doing mathematics” means learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, constructing arguments and critiquing the reasoning of others, modeling with mathematics, using appropriate tools strategically, attending to precision in the use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers, with worthwhile ideas and perspectives, and to cultivate positive attitudes and beliefs about mathematics.

“Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving” (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the problem solvers learning the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them. 

Principles for Mathematics Teaching and Learning

Active learning is best: Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and students practice, then students will learn how to do mathematics.

Decades of research tells us that the traditional model of instruction is flawed. Traditional instructional methods may get short-term results, with procedural skills, but students tend to forget the procedural skills and do not develop problem-solving skills, deep conceptual understanding, or a mental framework for how ideas fit together. They also don’t develop strategies for tackling non-routine problems, including a propensity for engaging in productive struggle to make sense of problems and persevere in solving them.

In a problem-based instructional framework, teachers play a critical role in mediating student learning, but that role looks different than simply showing, telling, and correcting. The teacher, in fact, has many roles in this framework: listener, facilitator, questioner, synthesizer, and more. In all these roles, teachers listen to and make use of students’ thinking, are mindful about who participates, and continuously monitor students’ positions in terms of status inside and outside the classroom. 

The teacher’s role is multifaceted:

1. Ensure students understand the context and what is asked.
2. Ask questions to advance students’ thinking in productive ways.
3. Help students share their work and understand others’ work, through orchestrating productive discussions.
4. Synthesize the learning with students at the end of activities and lessons.

Teachers should build on what students know: New mathematical ideas are built on what students already know about mathematics and the world. As they learn new ideas, students need to make connections between them (NRC, 2001). In order to do this, teachers need to understand what knowledge students bring to the classroom and monitor what they do and do not understand as they are learning. Teachers must themselves know how the mathematical ideas connect in order to mediate students’ learning.

Good instruction starts with explicit learning goals: Learning goals are clear not only to teachers, but also to students, and they influence the activities in which students participate. Without a clear understanding of the learning objectives, activities in the classroom, implemented haphazardly, have little impact on advancing students’ understanding. Strategic negotiation of whole-class discussion during an Activity Synthesis is crucial to making the intended learning goals explicit. Have a clear idea of the destination for the day, the week, the month, and the year, and select and sequence instructional activities (or use well-sequenced materials) that will get the class to their successive destinations. To arrive at a destination, know the address and plan a route, because driving around aimlessly leads nowhere.

Different learning goals require different instructional moves: The kind of instruction that is appropriate at any given time depends on the learning goals of a particular lesson.

Lesson and activities have one or more student learning goals:

• Explore new topics of study and access the mathematics.
• Study new concepts and procedures in depth.
• Integrate and connect representations, concepts, and procedures.
• Work toward mastery.
• Apply mathematics.

Lessons are designed on the basis of the intended learning outcomes. To facilitate these goals, have a toolbox of instructional moves to use as appropriate.

Intentional planning: Different learning goals require different instructional moves. Plan instruction appropriately. While a high-quality curriculum does reduce the burden to create or curate lessons and tasks, it does not reduce the need to spend deliberate time planning them. Instead, planning time can shift to high-leverage practices: reading and understanding the curriculum materials; identifying connections to prior and upcoming work; diagnosing students’ readiness to do the work; leveraging instructional routines to address students’ individual needs and differentiate instruction; anticipating students’ responses that will move the learning forward; planning questions and prompts to help students attend to, make sense of, and learn from each other’s work; planning supports and extensions to give the greatest access to the main mathematical goals; figuring out timing, pacing, and opportunities for practice; preparing necessary supplies; and providing feedback on students’ work.

Each and every student should have access to the mathematical work: With proper structures, accommodations, and supports, all students can learn mathematics. Teachers’ instructional toolboxes should include knowledge of and skill in implementing supports for different learners.