**Detailed Comparison Format**

Readers will notice that mathematical elements in *Benchmarks*
are not confined to the chapters labeled mathematics -- that is, to Chapter
2: The Nature of Mathematics and Chapter 9: The Mathematical World. References
are also made to benchmarks in other chapters, in particular Chapter 1:
The Nature of Science, Chapter 11: Common Themes, and Chapter 12: Habits
of Mind. Finally, a few mathematically related benchmarks appear in Chapter
4: The Physical Setting and Chapter 8: The Designed World. In Chapters
4 and 8, the benchmarks present applications of mathematics, rather than
statements about what mathematics should be learned.

Each statement under a Benchmark is followed by standards that relate to it. Some standards relate to two or more different benchmarks, thereby appearing multiple times in the comparison. (In the comparison organized by standards, the opposite effect occurs.) There are many benchmarks that are not directly related to mathematics. In this document, only those benchmarks which are judged to have mathematically-related content have been included.

*Benchmarks* often integrates the meaning and understanding of
ideas, connecting them with applications in familiar contexts. Also, *Benchmarks*
provides more concrete examples of how a student's number sense and facility
with computations should be demonstrated. Often, standards are more general
than the benchmarks they relate to, offering a broad perspective on the
benchmarks.

*Benchmarks* is organized by topic and by four grade level groupings
within each topic, K-2, 3-5, 6-8, and 9-12. *Standards* is organized
in three grade level groupings, K-4, 5-8, and 9-12, and by topic areas
within each grade grouping. In order to make the comparison most accessible
to science educators, the comparison uses the organizational scheme of
*Benchmarks*. Since the grade levels overlap, standards for K-4 are
included under K-2 benchmarks; standards for grade levels K-4 and 3-5 are
included under the grade 3-5 benchmarks; and standards for grades 5-8 are
included under grades 6-8 benchmarks. In some cases, even higher grade
level standards are included under a benchmark, since benchmarks are focused
on what students should know at the end of secondary school whereas standards
tend to aim at what students should study at particular grade levels. This
approach to organizing the comparison sometimes results in taking individual
standards out of context and making them vulnerable to misinterpretation.
The reader is encouraged to refer to the full *Standards* document
in order to obtain the most accurate and comprehensive interpretation of
benchmarks that relate to a particular standard.