and difficult for students may in fact have intuitive or experiential underpinnings, and it is important to discover these and use them for formalizing students’ thinking.

Principle #2: Building Conceptual Understanding, Procedural Fluency, and Connected Knowledge

The focus of Principle 2 is on simultaneously developing conceptual understanding and procedural fluency, and helping students connect and organize knowledge in its various forms. Students can develop surface facility with the notations, words, and methods of a domain of study (e.g., functions) without having a foundation of understanding. For students to understand such mathematical formalisms, we must help them connect these formalisms with other forms of knowledge, including everyday experience, concrete examples, and visual representations. Such connections form a conceptual framework that holds mathematical knowledge together and facilitates its retrieval and application.

As described previously, we want students to understand the core concept of a fuctional relationship: that the value of one variable is dependent on the value of another. And we want them to understand that the relationship between two variables can be expressed in a variety of ways—in words, equations, graphs, tables—all of which have the same meaning or use the same “rule” for the relationship. Ultimately, we want students’ conceptual understanding to be sufficiently secure, and their facility with representing functions in a variety of ways and solving for unknown variables sufficiently fluid, that they can tackle sophisticated problems with confidence. To this end, we need an instructional plan that deliberately builds and secures that knowledge. Good teaching requires not only a solid understanding of the content domain, but also specific knowledge of student development of these conceptual understandings and procedural competencies. The developmental model of function learning that provides the foundation for our instructional approach encompasses four levels—0 to 3—as summarized in Table 8-1. Each level describes what students can typically do at a given developmental stage. The instructional program is then designed to build those competences.

Level 0

Level 0 characterizes the kinds of numeric/symbolic and spatial understandings students typically bring to learning functions. Initially, the numeric and spatial understandings are separate. The initial numeric understanding is one whereby students can iteratively compute within a single string of whole numbers. That is, given a string of positive, whole numbers such as 0,



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