Mathematics @ NPS

Naugatuck Public Schools uses a problem-based instructional model for mathematics learning. We define problem-based learning as:

Problem-based learning means building a deep conceptual understanding through problem solving, creative thinking, critical thinking and metacognitive processes to access content. Students use knowledge to interpret, gather, identify, evaluate and present information about problems in real-world contexts in order to organize and build new knowledge and understandings. In problem-based learning, collaboration and the context of the problem drives mathematical thinking and reasoning while the teacher facilitates and scaffolds the process. In this way, students can clarify their own misconceptions and transfer what they know to problems that matter in the world.

Model of Problem-based learning

In mathematics, students will encounter daily lessons designed to facilitate problem-based learning. Mathematical discourse (i.e. number talks) is promoted and an integral component of this instructional model. Students cooperatively and collaboratively solve problems, test solutions, challenge each other’s thinking and provide justification for a solution’s reasonableness.

A model of problem-based learning entails many instructional and learning components in play at the same time, all working in concert to provide purposeful educational experiences designed around what students will learn by doing.

In Naugatuck Public Schools, students engage in mathematical discourse as a problem-solving approach and as a way to gain experience as problem solvers in a collaborative context.

This model also supports the National Council for Teachers of Mathematics’ position that procedural fluency should be taught and developed through students’ growing conceptual understanding of math.

COmponents of a Problem-Based Math Program

The district uses Illustrative Mathematics and Desmo Math as the foundation for math instruction K-8. In the programs , each lesson is composed of a Warm Up, instructional activities, a Lesson Synthesis, and a Cool-Down.

Warm Up: The Warm-up is an instructional routine that invites all students to engage in the mathematics of the lesson. The Warm-up routines offer opportunities for students to bring their personal experiences as well as their mathematical knowledge to problems and discussions. These routines place value on the voices of students as they communicate their developing ideas, ask questions, justify their responses, and critique the reasoning of others.

Instructional activities: After the Warm-up, lessons consist of a sequence of 1–3 instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class. Each activity has three phases: Launch, Student Worktime, and Activity Synthesis.

Lesson Synthesis: Each lesson includes a Lesson Synthesis that assists with ways to help students incorporate new insights gained during the activities into their big-picture understanding.

Cool-Down: The Cool-down task is given to students at the end of the lesson. Students are meant to work independently on the Cool-down for about 5 minutes and then turn it in. The Cool-down serves as a brief formative assessment to determine whether students understood the lesson. Teachers use students’ responses to the Cool-down to make adjustments to further instruction.

What happens during problem-based learning?

Mathematical discourse is promoted and an integral component of this instructional model. Students cooperatively and collaboratively solve problems, challenge each other’s thinking, and provide justification for a solution’s reasonableness.

Problem-Based Learning embodies the National Council of Teachers of Mathematics principles for quality instruction:

- Questioning: The three components of a math lesson also encourages teachers and students to engage in inquiry by developing and asking questions. Questioning strategies should encourage exploration and deeper thinking about mathematics and its connections to the world. Teachers can facilitate learning through carefully crafted questions that lead students to new learning and new understandings.
- Facilitation: Teachers guide the learning by providing students with a framework for discourse. Naugatuck Public Schools embraces a model of mathematical discourse that promotes questioning, challenging one another’s thinking, attempting multiple strategies to solve a problem, developing action plans, and accountable talk. Through discourse, students can probe one another’s thinking, justify their own ideas and approaches to problem-solving, and model potential solutions.

- Building Procedural Fluency through Conceptual Understanding: The district believes that students gain fluency with addition, subtraction, multiplication and division through practice and conceptual understanding. Fluency means that students can flexibly, efficiently and accurately solve problems. This skill is built through students’ deepening understanding of number sense, procedures, and operations.
- Intentionality: Being intentional happens on many different planes: planning for the learning, planning for your role in the learning, and planning for differentiation.
  - - Purposeful Curriculum Design: Curriculum and its accompanying resources are tools for teachers to be purposeful in how they design and execute a lesson. Competencies are the “for what?” to learning and provide a progression K-12 that ensures students have opportunities to master skills. It provides a structure to ensure that lessons are aligned to expectations and the progression of learning.
    - Planning for the learning (coherence): When planning what students will learn, it is also important to connect with how they will learn it and how will you know they learned it? By setting a goal for the learning through exploration of how standards referenced in Illustrative Mathematics connect to Competencies and Performance Indicators. When planning for the learning, ask not only what students will do, but what students will learn and understand as a result of doing.
    - Planning for your role in the learning: A guiding question when planning for learning is, “What will I be doing?” Be intentional about your role in student learning and place students at the center of that learning. How will you guide the learning? How will you facilitate the learning? How will you empower students to take ownership of their learning? What structures and processes (i.e. discourse protocols, success criteria) will you create and/or implement to ensure that students are thinking deeply and making sense of mathematical concepts for themselves?

Planning for Differentiation: Differentiating learning is more than just a few leveled activities. When you differentiate the learning, you offer students a variety of ways to experience the learning. Students benefit from multiple pathways to a single goal. How will you monitor the learning and perform checks for understanding? What tools can you use to ensure all students are meeting high expectations and learning at deep levels? Differentiation goes beyond what students will do; it captures how different experiences can deepen the learning in different ways for all students. According to Carol Tomlinson, differentiating instruction means accommodating the different ways in which students learn. It requires active planning for student differences so that all students achieve the same deep levels of learning.

These are some of the components of high-quality instruction in Naugatuck Public Schools. For more information, please refer to the National Council for Teachers of Mathematics’ Principles to Actions: Ensuring Mathematical Success for All. All teachers of mathematics in Naugatuck have a copy of this seminal text.

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