Design Principles

It is our intent to create 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, student thinking, and their own teaching practice. In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.

All Students are Capable Learners of Mathematics

All students, each with unique knowledge and needs, enter the mathematics learning community as capable learners of meaningful mathematics. Mathematics instruction that supports students in viewing themselves as capable and competent must leverage and build upon the funds of knowledge they bring to the classroom. In order to do this, instruction must be 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 will have daily opportunities to learn about and leverage their students’ understandings and experiences and how 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 being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, 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 ones doing the problem solving to learn 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 must listen to and make use of student thinking, be mindful about who participates, and continuously be aware of how students are positioned in terms of status inside and outside the classroom. Teachers also guide students in understanding the problem they are being 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.

Coherent Progression

To support students in making connections to prior understandings and upcoming grade-level work, it is important for teachers 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. Each activity and lesson is part 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.

Each 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 teachers the opportunity to observe students’ prior understandings.
  • Each lesson starts with a warm-up to activate prior knowledge and 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 ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned.
  • Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.

Balancing Rigor

There are three aspects of rigor 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 are developed together and are therefore interconnected in the materials in ways that support student 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-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.

Community Building

Students learn math by doing math both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978) within this collective learning. To support students in developing a productive disposition about mathematics and to help them engage in the mathematical practices, it is important for teachers to start off the school year establishing norms and building a mathematical community. In a mathematical 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 the teacher is supported to make use of those ideas to meet the mathematical goals of the lessons. Additionally, the first unit in each grade level provides lesson structures that establish a mathematical 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 provide ideas for ongoing support to revisit and highlight the mathematical community norms in meaningful ways.

Instructional Routines

Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature. They are “enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” (Kazemi, Franke, & Lampert, 2009)

In the materials, we intentionally chose to use a small set of instructional routines to ensure they are used frequently enough to become truly routine. The focused number of routines benefits teachers as well as students. Consistently using a small set of carefully chosen routines is just one way that we attempt to lower the cognitive load for teachers. Teachers are free to focus the energy that would be used on structuring an activity on other things, such as student thinking and how mathematical ideas are playing out.

Throughout the curriculum, routines are introduced in a purposeful way to build a collective understanding of their structure, and are selected for activities based on their alignment with the unit, lesson, or activity learning goals. While each routine serves a different specific purpose, they all have the general purpose of supporting students in accessing the mathematics and they all require students to think and communicate mathematically. The Instructional Routines section of the teacher course guide gives more details on the specific routines used in the curriculum.

To help teachers identify when a particular routine appears in the curriculum, each activity is tagged with the name of the routine so teachers are able to search for upcoming opportunities to try out or focus on a particular instructional routine. Professional learning for the curriculum materials includes video of the routines in classrooms so teachers understand what the routines look like when they are enacted. Teachers also have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the routines.

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. The Instructional Routines section of the teacher course guide describes the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) and points teachers to the book for further reading. In all lessons, teachers are supported in the practices of anticipating, monitoring, and selecting student work to share during whole-group discussions. In lessons in which there are opportunities for students to make connections between representations, strategies, concepts, and procedures, the lesson and activity narratives provide support for teachers to also use the practices of sequencing and connecting, and the lesson is tagged so teachers can easily identify these opportunities. Teachers have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the 5 Practices.

Task Complexity

Mathematical tasks can be 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, teachers can look to warm-ups and activity launches for built-in preparation, and to teacher-facing narratives for further guidance.

In addition to tasks that provide access to the mathematics for all students, the materials provide guidance for teachers on how to ensure that during the tasks, all students are provided the opportunity to engage in the mathematical practices. More details are given below about teacher reflection questions, and other fields in the lesson plans help teachers assure that all students not only have access to the mathematics, but the opportunity to truly engage in the mathematics.

Purposeful Representations

In the materials, mathematical representations are used for two main purposes: to help students develop an understanding of mathematical concepts and procedures or to help them solve problems. For example, in grade 3, equal-groups drawings are used to introduce students to the concept of multiplication. Later on, students make equal-groups drawings to find the total number of objects in situations involving equal groups.

Curriculum representations and the grade levels at which they are used are determined by their usefulness for particular mathematical learning goals. Representations that are more concrete are introduced before those that are more abstract. For example, in kindergarten, students begin by counting and moving objects before they represent these objects in 5- and 10-frames to lay the foundation for understanding the base-ten system. In later grades, these familiar representations are extended so that as students encounter larger numbers, they are able to use place-value diagrams and more symbolic methods, such as equations, to represent their understanding. The teacher course guide makes explicit the selection of a representation when appropriate, so that teachers understand the reasoning behind certain representation choices in the materials.

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 supports the development of procedural fluency.

Teacher Learning Through Curriculum Materials

In order for all students to have access to mathematical learning opportunities, teachers must first believe that every student can learn mathematics. It is the responsibility of teachers to provide equitable instruction and position all students in a way that supports learning. Getting better at teaching requires teachers to plan at the course, unit, and lesson level, and to reflect on, and improve on, each day’s instruction. Equitable instruction requires teachers to develop both their knowledge of mathematics and the socio-cultural contexts of the students they teach in order to deepen learning for all students. The materials are designed to support teachers in both of these areas.

This is how 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 provide teachers with 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 provide 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 the lived experiences and students’ 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 that provide opportunities for students to see themselves in the activities or learn more about others’ cultures and experiences. In places where there are opportunities to adapt a context to be more relevant for students, we have provided suggested prompts to elicit these ideas.

There are two sections within each lesson plan that support teachers in learning more about what each student knows and that provide guidance on ways in which to respond to students’ understandings and ideas.

  • Advancing Student Thinking
    This section offers look-fors and questions to support students as they engage in an activity. Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). As teachers monitor during the course of an activity, they gain insight into what students know and are able to do. Based on these insights, the advancing student thinking section provides teachers questions that advance student understanding of mathematical concepts, strategies, or connections between representations.
     
  • Response to Student Thinking
    Most lessons end with a cool-down to formatively assess student thinking in relation to the learning goal of that day’s lesson. The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, next-day support or prior-unit support, based on the anticipated student response. The purpose of this guidance is to allow teachers to continue teaching grade-level content with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lessons to recommending a section from a prior unit with activities that directly connect to the concepts in the lesson.

Teacher Reflection Questions
To encourage teachers to reflect on the teaching and learning in their classroom, each lesson includes a teacher-directed reflection question on the mathematical work or pedagogical practices of the lesson. The questions are drawn from four categories: mathematical content, pedagogy, student thinking, or beliefs and positioning. The questions are designed to be used by individuals, grade-level teams, coaches, or anyone who supports teachers.

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). The beliefs and positioning questions support teachers in identifying and acknowledging their own mindsets and beliefs. These questions prompt teachers to reflect on, and challenge, the assumptions they make—about mathematics, learners of mathematics, and the communication of mathematics in their classrooms.

Professional Learning Communities (PLCs)
Teaching mathematics requires continual learning. Teachers must be adept at moment-to-moment decision making in order to engage students in rich discussions of mathematical content (O’Connor & Snow, 2018). We believe this learning should be embedded within a teacher’s daily work and be a collective experience within professional learning communities. To support teachers and coaches in this collective work, each unit section has an activity identified as a PLC activity. This activity either highlights an important mathematical idea in the unit or has complex facilitation that would benefit from teachers planning and rehearsing the activity together. We have also included a structure for the learning community included in the Professional Learning Community section of the Course Guide.

Model with Mathematics K–5

In K–5, modeling with mathematics is problem solving. It is problem solving that provides 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 being flexible about the way they solve problems. In upper elementary, these precursor skills become various stages of the modeling process that students will experience in grades 6–12. In addition to the precursor skills and 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.

References

  • Gay, G. (2010). Culturally responsive teaching: Theory, research, and practice. New York: Teachers College Press.
  • Hammond, Z. (2015). Culturally responsive teaching and the brain: Promoting authentic engagement and rigor among culturally and linguistically diverse students. Thousand Oaks, CA: Corwin Press.
  • Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H, Alwyn, O., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher 25(4), 12–21. doi.org/10.3102/0013189X025004012
  • Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 1).
  • National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.
  • O’Connor, C., & Snow, C. (2018). Classroom discourse: What do we need to know for research and for practice? In M. Schober, A. Britt, & D. Rapp (Eds.), The Routledge handbook of discourse processes (2nd ed.) (pp. 315–342). London: Routledge.
  • Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.
  • Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.
  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.