The Natick Middle School math department is piloting Illustrative Math in grades 5-8 from February 2023 to April 2023. We have picked units that align with the Massachusetts standards that fit the curriculum path we are currently on.
The IM Math certified curriculum is rigorous, problem-based, and fully aligned to the standards, with coherence across grade bands. In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time.
More about Illustrative Math
Yet research shows that students who believe that hard work is more important than innate talent learn more mathematics.1
We want students to believe anyone can do mathematics and that persevering at mathematics will result in understanding and success.
In the words of the NRC report Adding It Up, we want students to develop a “productive disposition—[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”2
Conceptual understanding: Students need to understand the why behind the how in mathematics. Concepts build on experience with concrete contexts. Students should access these concepts from a number of perspectives in order to see math as more than a set of disconnected procedures.
Procedural fluency: We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.
Application: Application means applying mathematical or statistical concepts and skills to a novel mathematical or real-world context.
These three aspects of mathematical proficiency are interconnected: procedural fluency is supported by understanding, and deep understanding often requires procedural fluency. In order to be successful in applying mathematics, students must both understand and be able to do the mathematics.
In a mathematics class, students should not just learn about mathematics, they should do mathematics.
This can be defined as 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.
How teachers should teach depends on what we want students to learn.
To understand what teachers need to know and be able to do, we need to understand how students develop the different (but intertwined) strands of mathematical proficiency, and what kind of instructional moves support that development.
Intentional planning: Because different learning goals require different instructional moves, teachers need to be able to plan their instruction appropriately. While a high-quality curriculum does reduce the burden for teachers to create or curate lessons and tasks, it does not reduce the need to spend deliberate time planning lessons and tasks. Instead, teachers’ planning time can shift to high-leverage practices (practices that teachers without a high-quality curriculum often report wishing they had more time for): reading and understanding the high-quality curriculum materials; identifying connections to prior and upcoming work; diagnosing students' readiness to do the work; leveraging instructional routines to address different student needs and differentiate instruction; anticipating student responses that will be important to move the learning forward; planning questions and prompts that will help students attend to, make sense of, and learn from each other's work; planning supports and extensions to give as many students as possible access to the main mathematical goals; figuring out timing, pacing, and opportunities for practice; preparing necessary supplies; and the never-ending task of giving feedback on student work.
Establishing norms: Norms around doing math together and sharing understandings play an important role in the success of a problem-based curriculum. For example, students must feel safe taking risks, listen to each other, disagree respectfully, and honor equal air time when working together in groups. Establishing norms helps teachers cultivate a community of learners where making thinking visible is both expected and valued.
Building a shared understanding of a small set of instructional routines: Instructional routines allow the students and teacher to become familiar with the classroom choreography and what they are expected to do. This means that they can pay less attention to what they are supposed to do and more attention to the mathematics to be learned. Routines can provide a structure that helps strengthen students’ skills in communicating their mathematical ideas.
Using high quality curriculum: A growing body of evidence suggests that using a high-quality, coherent curriculum can have a significant impact on student learning.5 Creating a coherent, effective instructional sequence from the ground up takes significant time, effort, and expertise. Teaching is already a full-time job, and adding curriculum development on top of that means teachers are overloaded or shortchanging their students.
Ongoing formative assessment: Teachers should know what mathematics their students come into the classroom already understanding, and use that information to plan their lessons. As students work on problems, teachers should ask questions to better understand students’ thinking, and use expected student responses and potential misconceptions to build on students’ mathematical understanding during the lesson. Teachers should monitor what their students have learned at the end of the lesson and use this information to provide feedback and plan further instruction.