learning with deep conceptual understanding or, more simply, learning with understanding. Learning with understanding is strongly advocated by leading mathematics and science educators and researchers for all students, and also is reflected in the national goals and standards for mathematics and science curricula and teaching (American Association for Advancement of Science [AAAS], 1989, 1993; National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000; NRC, 1996). The committee sees as the goal for advanced study in mathematics and science an even deeper level of conceptual understanding and integration than would typically be expected in introductory courses.

Guidance on how to achieve learning with understanding is grounded in seven research-based principles of human learning that are presented below (see Box 6-1).1 In Chapter 7, these principles are used as the framework for the design of curricula, instruction, and assessments for advanced study—three facets of classroom activity that, when skillfully orchestrated by the teacher, jointly promote learning with understanding. These principles also serve as the foundation for the design of professional development, for it, too, is a form of advanced learning.

The design principles for curriculum, instruction, assessment, and professional development provide one of the organizing frameworks of the committee’s analysis of the AP and IB programs (see Chapters 8 and 9, this volume). While it could be argued that all components of the educational system (e.g., preservice training and leadership) should be included (and we believe they should), our analysis was limited to these four facets. Although this framework was developed to assess current programs of advanced study, it also can serve as a guide or framework for those involved in developing, implementing, or evaluating new educational programs.


Principle 1: Principled Conceptual Knowledge

Learning with understanding is facilitated when new and existing knowledge is structured around the major concepts and principles of the discipline.

Highly proficient performance in any subject domain requires knowledge that is both accessible and usable. A rich body of content knowledge about a subject area is a necessary component of the ability to think and


The research on which these principles are based has been summarized in How People Learn: Mind, Brain, Experience and School (Expanded Edition) (NRC, 2000b).

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