understand, and be able to do; they are intended to serve as a basis for the development of curricula, assessment procedures, teacher-training programs, and supplementary instructional material.

The committee focused on two sets of standards:

  1. Principles and Standards for School Mathematics, prepared by the National Council of Teachers of Mathematics in 2000. This is an update of the first-ever set of standards, those published for mathematics in 1989.

  2. National Science Education Standards, prepared by the National Research Council in 1996.

Several sets of standards were examined, including those for geography (Geography Education Standards Project, 1994), but the mathematics and science standards offer a direct connection to spatial thinking and reasoning and they are fundamental to the process of education and to the idea of a technologically skilled workforce. (Spatial thinking is integral to geography as a discipline and, thus, figures prominently throughout the national geography standards. The first 3 of the 18 standards and all five of the sets of geographic skills are a primer in thinking spatially about geographic data; see Box 5.1).

As is the case for most education standards, these two sets of standards are organized in terms of intellectual themes with progressively more challenging standards of performance established for different grade levels along each theme. For example, the science standards are built around eight intellectual categories: unifying concepts and processes in science, science as inquiry, physical science, life science, Earth and space science, science and technology, science in personal and social perspectives, and the history and nature of science. For each category there is a content standard and “as a result of activities provided for all students in those grade levels, the content of the standard is to be understood or certain abilities are to be developed” (NRC, 1996, p. 6). In the case of the first category, there is no distinction by grade level; for the other seven categories, understanding is organized into three grade clusters: K–4, 5–8, and 9–12. The eight standards are to be used as a whole in order to achieve scientific literacy.

There are two questions about the relationship between spatial thinking and the sets of content standards: (1) Are the basic tenets of spatial thinking an explicit part of the expectations established by various standards? (2) Are spatial thinking concepts implicitly contained within the standards? To answer these questions, the committee considers the two sets of standards in sequence, beginning with mathematics because it has been in place since 1989 in its original form.

5.5.2 Spatial Thinking and the Principles and Standards for School Mathematics

The mathematics standards are built around five content standards—number and operations, algebra, geometry, measurement, and data analysis and probability—and five process standards—problem solving, reasoning and proof, communication, connections, and representation (NCTM, 2000). For each standard, there are two to four statements of what students should be able to do as a consequence of instruction, with specific expectations keyed to grade levels K–2, 3–5, 6–8, and 9–12. These expectations are summarized in Table 5.2.

In Table 5.2, only the geometry standard uses the language of spatial thinking (two and three dimensions, shapes, relationships, locations, spatial reasoning, etc.). However, in reading the detailed discussions, grade by grade, for each of the ten mathematics standards, it is clear that spatial thinking pervades and permeates the detailed articulation of what is expected of students. There are explicit references to spatial concepts in the geometry, measurement, data analysis and probability, and representation standards. Although the term spatial thinking is not used explicitly in the text of Table 5.2, the underlying concepts are implicit throughout. Thus, an understanding of spatial relations is presumed but never discussed explicitly. For many of the statements in Table 5.2,



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