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.

Students as Capable Learners of Mathematics

Students enter the mathematics learning community as capable learners of meaningful mathematics, each with unique knowledge and needs. Mathematics instruction that supports students in viewing themselves as capable and competent must be grounded in equitable structures and practices that provide students with access to grade-level content and provide teachers with necessary guidance to listen, learn, 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 their ideas are visible. Through these structures, teachers have daily opportunities to learn about their students’ understandings and experiences and ways in which to position each student as a capable learner of mathematics.

Learning Mathematics by Doing Mathematics

In a mathematics learning community, students learn mathematics by doing mathematics. 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. This is in contrast to students either watching someone else engage in the mathematical practices or taking notes about mathematics that has already been done. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives. 

Problem-based Lesson Structure

“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 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 how students share their work, orchestrate discussions so students have the opportunity to understand and take a position on others’ ideas, and synthesize the learning with students at the end of activities and lessons.

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 interconnected in ways that support student understanding. For example, in order to be successful in applying mathematics, students must both conceptually understand and procedurally be able to do the mathematics. (Principles to Action, NCTM, pg. 42)

The materials support the development of the three aspects of rigor by offering students opportunities to access new mathematics, engage in rigorous tasks, and connect new representations and mathematical language to prior learning. This positions them to retrieve and apply their knowledge to novel problems, which often require both conceptual understanding and procedural fluency. Specific grade-level expectations for procedural fluency are supported by the warm-ups, centers, and practice problems. Continual opportunities for students to apply their understandings to mathematical situations give them practice with new material and a review of concepts, skills, and applications from earlier lessons and units. 

Coherent Progression

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 across units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense. 

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 flow of the mathematics, 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.

Each unit, lesson, and activity has the same overarching design structure: an invitation to the mathematics, 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 in offering students access to the mathematics, as it builds on prior knowledge and encourages the use of 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.

Community Building 

To support students in developing a productive disposition about mathematics and engage in the mathematical practices, it is important for teachers from the start of the school year to build a mathematical community that allows all students to express their mathematical ideas and discuss them with others. Students learn math by doing math both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978) within this collective learning. 

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 which 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, materials provide ideas for ongoing support to revisit and highlight the mathematical community norms in meaningful ways. 

Instructional Routines

Instructional routines create structures so that all students can engage and contribute to mathematical conversations. Instructional routines have a predictable structure and flow. 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 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. 

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. 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. The teacher course guide explains the purpose and use of the instructional routines 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 the success in a problem-based classroom. 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 curriculum activities, careful attention is given to the complexity of contexts, numbers, and required computation with few assumptions as to students’ familiarity with contexts and representations. Activities are designed to focus students’ productive struggle on specific mathematical ideas without layering additional complexities that distract from the intended learning goal. To help students gain familiarity with complexities such as contexts or representations, there are intentional warm-ups and activity launches with teacher-facing narratives that provide guidance on adapting the context of instructional tasks for student relevance without losing the intended mathematics of the task.

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. Specifically, teacher reflection questions ask teachers to think carefully about who is participating and why. Teachers are supported in considering the assumptions they make about their students’ understanding and mathematical ideas so they can work to leverage all student thinking in their classroom. 

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 third grade, 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. 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. 

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.

Equitable Teaching Structures and Practices

For each and every student to have access to mathematical learning opportunities, teachers must first believe that each and every student can learn mathematics with appropriate instruction and understand it is their responsibility to position students in a way that supports that learning. Equitable instruction leverages the teacher’s knowledge of mathematics and the socio-cultural contexts of the students in the classroom to deepen learning for all students. 

While the problem-based lesson structure and the mathematical community aspects of the curriculum support equity and access, there are five additional ways in which the materials support teachers in equitable teaching practices: 

  • authentic use of contexts 
  • suggested launch adaptations
  • advancing student thinking questions
  • response to student thinking 
  • teacher reflection questions 

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, unit 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 open the space more for students to bring in themselves and their experiences, 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
    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. 
  • Responding to Student Thinking
    Each lesson ends with a cool-down problem to formatively assess student thinking in relation to the learning goal of that day’s lesson. If students demonstrate unfinished learning on the cool-down problem, the materials offer guidance on next steps. 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 recommended center activities from prior grade level units. 

Teacher Reflection Questions
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 PtA, 64). To support teachers in identifying and acknowledging their own mindsets and beliefs, we have created a set of teacher reflection questions within the category we call Beliefs and Positioning. 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. 

Teacher Learning Through Curriculum Materials

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. Educative materials that support teacher learning from teaching must leverage and enhance the teacher’s knowledge of mathematics and pedagogical practices, their students, and the socio-cultural contexts of each student in the classroom. There are three key components of the curriculum materials that support teachers in this learning: Unit, Lesson, and Activity Narratives, Teacher Reflection Questions, and Professional Learning Community (PLC) activities and structure. 

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. 

Teacher Reflection Question 
To encourage teachers to reflect on the teaching and learning in their classroom, each lesson concludes with 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. 

Professional Learning Communities (PLC)
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 a 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

  • 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
  • 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. Teachers College Press.