Principles for Organizing Mathematical Content
Contexts before Formality
Students should encounter plenty of examples of a mathematical or statistical idea in various real-world and mathematical contexts before that idea is named and studied as an object in its own right. It’s not just about the contexts in which the math idea is presented. It’s also about giving students the space and time to interact or play with the idea in intuitive ways and to talk about it in terms that make sense to them (rather than in formal terms)
This is a problem-based curriculum where students should have opportunities to see what they can figure out before having things explained or being told what to do. We purposefully designed the tasks in our curriculum to provide students with these opportunities and teachers with the guidance to facilitate them. The teacher’s role is to ensure students understand the context and what is being asked, ask questions to advance students’ thinking in productive ways, help students share their work and understand others’ work through orchestrating productive discussions, and synthesize the learning with students at the end of activities and lessons.
Math is Learned by Doing Math
Students should spend time in math class doing mathematics, which can be defined as engaging in the mathematical practices: making sense of problems, decontextualizing and recontextualizing and checking for reasonableness, 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. We designed our materials with the goal that all students are doing mathematics most of the time. We chose a problem-based approach for this reason, and we designed tasks and instructional routines so all students are actively engaged.
We attend diligently to alignment to grade-level content standards. When content below grade level is included as an on-ramp to new content, this is indicated so teachers can make an informed choice about the extent to which they will use it in their classes. Extension activities are offered for any student who is ready to go deeper into the content, as an alternative to offering work above grade level. We also attend closely to alerting the teacher to opportunities for students to engage in standards for mathematical practice, as well as designing many activities that are particularly conducive to practice standards.
Often, mathematical representations are used as tools for problem solving. Representations are good tools for problem solving when they are linked to developing understanding, but we also believe strongly that representations are also productive tools for understanding. Our materials make heavy and thoughtful use of representations (in both ways) throughout. First of all, where appropriate, representations introduced go from more-concrete to more-abstract. For example, in grade 6, equivalent ratios are first represented as groups of two associated recognizable objects, then as groups of squares, then the association is represented using a double number line, then a table of values. In grade 7, these familiar representations are extended so that students encounter proportional relationships represented with equations and graphs. Any representation introduced is connected to symbolic methods. Students are encouraged to use representations that make sense to them, but also connect their existing understanding to more efficient methods.
Balancing the Aspects of Rigor
These materials carefully and deliberately engage students in all three aspects of rigor: conceptual understanding, procedural fluency, and applications. For conceptual understanding, we always help students understand the why behind the how. Concepts are built gradually upon experience with concrete contexts. We take procedural fluency to mean solving problems expected by the standards with speed, accuracy, and flexibility. Practice is built into tasks whose purpose is to give students the opportunity to practice applying a concept or using a procedure, as well as warm-ups that revisit prior skills or prime students to use a skill in a new way in that day’s lesson. Additionally, a practice problem set is provided for each day’s lessons that asks students to practice new material from that day and review materials from earlier lessons and units. Application means applying mathematical or statistical concepts and skills to a novel situation. Most units end with a culminating lesson that gives students an opportunity to do just that. Each grade level ends in a culminating unit that gives students opportunities to apply several concepts and skills from throughout the year to a novel situation.
Purposeful Tasks and Lessons
Instructional units are organized with a flow: invitation to the mathematics, deep study to develop concepts, and consolidating ideas and applying them to new contexts. Ongoing practice and work toward fluency is continuous. Every lesson has a purpose within this instructional unit. Nested inside the instructional units, lessons are laid out with an organization: warm-up, one or more instructional activities, synthesis, cool-down. Each activity serves a purpose within the lesson. Even deeper in the lesson, activities hold a routine structure: launch, student work time, synthesis. Each component has a purpose within the activity.
Purposeful Digital Components (for Middle School)
Many activities have a digital alternative to the paper-and-pencil activity. The purpose of replacing an activity with a digital version is to help kids see many examples of a relationship in a short amount of time, to provide a manipulative to use in lieu of a physical artifact (like tangrams or pattern blocks), to shorten the amount of time it takes students to create a representation (like a double number line diagram), to illustrate a concept dynamically, or offer a way for students to practice many iterations of a skill with error checking. Digital versions are only developed when they offer a meaningful alternative to the paper-and-pencil version to achieve the learning goals of the lesson.
Digital Components Statement for High School
We expect high school students to become fluent users of digital tools to solve problems and produce representations to support their own understanding. In most cases, instead of being given a pre-made applet to explore, students have access to a suite of linked applications, such as graphing tools, synthetic and analytic geometry tools, and spreadsheets. Students (and teachers) are taught how to use the tools, but not always told when to use them, and student choice in problem-solving approaches is valued. When appropriate, pre-made applets may be included to allow for students to practice many iterations of a skill with error checking, to shorten the amount of time it takes students to create a representation, or to help students see many examples of a relationship in a short amount of time. Digital versions are developed when they are required by the standard being addressed, or when they offer a meaningful alternative to the paper and pencil version to achieve the learning goals of the lesson.
Principles for Designing Teacher Supports
Easing the Teacher’s Workload When Possible
These materials are tested in real classrooms and designed for real classroom use. Many decisions were made about the design of the materials with the purpose of reducing teacher prep time and class time demands on the teacher’s attention. For example, many activities follow one of a small set of instructional routines. The purpose of using routines is so that students and the teacher become familiar with the classroom choreography and what they are expected to do, so that less attention can be paid to “what am I supposed to do?” and more attention can be paid to the mathematics.
Supporting Teacher Learning
Every decision made about the flow of the mathematics is explicitly described through lesson narratives and activity narratives, including whether the decision is required by the standards or a choice made by the authors. Whenever there is a connection to the mathematical work of a prior or future lesson, unit, or grade, it is pointed out. The mathematical purpose of every lesson and activity is made clear. Careful study of the teacher-facing portions of lesson plans is intended to help teachers better understand the mathematics they are teaching and the instructional moves baked in that are designed to help students learn.
Tools for Heterogeneous Classrooms
These materials can be used in heterogeneous classrooms. Careful attention is paid to giving every student access to the mathematics by not assuming familiarity with contexts and giving teachers tools to recognize shortcomings in understanding below grade-level work. Built-in supports are included for teachers of students with disabilities and students with emerging English language proficiency. Extension activities are offered for any student who is ready to go deeper into grade-level content.