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 offers teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials support students’ and teachers’ learning, respectively. 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 their 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. “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.

The teacher 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 how students are positioned in terms of status inside and outside the classroom. Teachers also guide students in understanding the problem they are asked to solve, ask questions to advance students’ thinking in productive ways, provide structure for students to share their work, orchestrate discussions so students have the opportunity to understand and take a position on the ideas of others, and synthesize the learning with the whole class at the end of activities and lessons.

Community Building

Students learn math by doing math, both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978). To support students in developing a productive disposition about mathematics and to help them engage in the mathematical practices, begin establishing norms and building a math community at the start of the school year. In a math community, all students have the opportunity to express their mathematical ideas and discuss them with others, which encourages collective learning. “In culturally responsive pedagogy, the classroom is a critical container for empowering marginalized students. It serves as a space that reflects the values of trust, partnership, and academic mindset that are at its core” (Hammond, 2015).

The materials foster conversation so that students voice their thinking around mathematical ideas, and support the teacher in making use of those ideas to meet the mathematical goals of the lessons. Additionally, the first unit in each grade level features lesson structures that build a math community, establish norms, and invite students into the mathematics, with accessible content. Each lesson offers opportunities for the teacher and students to learn more about one another, develop mathematical language, and become increasingly familiar with the curriculum routines. To maintain this community, the materials offer ideas for ongoing support to revisit and highlight the math-community norms in meaningful ways.

Model with Mathematics in K–5

In K–5, modeling with mathematics is problem solving. It is problem solving that offers opportunities for students to notice, wonder, estimate, pose problems, create representations, assess reasonableness, and continually make revisions, as needed. In the early grades, these opportunities involve various precursor modeling skills that support students in solving problems flexibly. In upper elementary, these precursor skills become various stages of the modeling process that students will experience in IM Grades 6–12. In addition to the precursor skills and the modeling stages that appear across lessons, each unit culminates with a lesson that explicitly addresses these modeling skills and stages while pulling together the mathematical work of the unit.