# Developing Strategic Competence through Modeling Mathematical Ideas

**Text Box 1.1 A Math Happening 1a: The Handshake Problem**

At a neighborhood party, there were a total of 25 guests. All 25 guests shook hands to introduce themselves. How many distinct handshakes happened at this party? Show the different ways you can represent your thinking.

The problem above is one of the classic problem-solving tasks called the “handshake problem.” It is one that we always start within any workshop. One reason is not only because it allows us to introduce ourselves to other members in the room but also because it illustrates one of the most important teaching practices (NCTM, 2013) that we focus on throughout this book, which is *Implementing tasks that promote reasoning and problem solving across grade levels.*

Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. (p. 3)

The handshake problem is one of the best examples of a task that has multiple entry points and varied solution strategies.

In this book, we explore how students and teachers at all levels can engage and challenge in an inquiry-based learning environment through interactive modeling approaches in mathematics. Such opportunities with rich mathematical tasks require the use of higher-level critical-thinking strategies and self-monitoring problemsolving skills. This book presents approaches that will engage the reader in the modeling of real-world problems across the curriculum and discusses teaching and learning opportunities that will benefit students and teachers. In addition, we focus on the learning progression across grades 3-8 by presenting rich problem-solving tasks that model mathematics across the learning progression that aligns to the standards.

With the recent release of the Common Core State Standards (CCSS) (NGACBP & CCSSO, 2010), researchers and mathematics educators are looking at the standards to help teachers unpack the learning progressions that is crucial in building mathematical understanding and guiding the sequence of mathematical concepts. Learning progression extends “previous learning while avoiding repetition and large gaps” (Hunt Institute, 2012, p. 8) and guides “a path through a conceptual corridor in which there are predictable obstacles and landmarks” (Confrey, 2012, p. 4).

Understanding the learning progression is important for teachers because it serves as the critical markers for analyzing student learning and tailoring their teaching sequence. This essential instructional practice leads to the vital professional development that focused on vertical articulation around essential math concepts and worthwhile tasks. In each chapter, we use the learning progressions to map out student learning outcomes as they advance through different grade levels and provide standards for formative and summative assessments.