**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 many other chapters, in particular Chapter
1: The Nature of Science, Chapter 11: Common Themes, and Chapter 12: Habits
of Mind.

In most cases, each bulleted statement under a Standard is followed
by benchmarks that relate to it. But sometimes the same benchmark (or set
of benchmarks) is related to more than one of the statements within the
same Standard. In these cases, rather than repeating the benchmarks under
the separate Standard statements, the statements may be grouped together.
Some benchmarks relate to two or more *different* Standards, thereby
appearing multiple times in the comparison. (Had the comparison been organized
by benchmarks, of course, the opposite effect would have occurred.)

Benchmarks often integrates the meaning and understanding of computations, 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 benchmarks are more specific than the standards they relate to, offering examples for, commentary on, or expansion of standards. Yet some benchmarks are more general than standards, providing an overall interpretation of the nature and processes of mathematics and science rather than specific procedures, concepts, or strategies.

*Standards* is organized in three grade level groupings, K-4, 5-8,
and 9-12, and by topic areas within each grade grouping. Benchmarks is
organized by topic and by four grade level groupings within each topic,
K-2, 3-5, 6-8, and 9-12. In order to make the comparison most accessible
to mathematics educators, the comparison uses the organizational scheme
of *Standards*. Since the grade levels overlap, benchmarks for K-2
and 3-5 are included under K-4 standards and benchmarks for grade levels
3-5 and 6-8 are included under the grade 5-8 standards. This approach to
organizing the comparison sometimes results in taking individual benchmarks
out of context and making them vulnerable to misinterpretation. In a few
cases, this approach may make it appear that a particular standard has
either no similar benchmark or that the related benchmark does not treat
the topic comprehensively. The reader is encouraged to refer to the full
*Benchmarks* document in order to obtain the most accurate and comprehensive
interpretation of benchmarks that relate to a particular standard.

The *set* of benchmarks that follows such a grouping should be
understood as being relevant to the *set* of grouped Standard statements.
If a benchmark relates to more than one Standard, a footnote indicates
which other Standards are also relevant.

**Curriculum Standards for Grades K-4**

The coverage of the K-4 standards by *Benchmarks* is perhaps the
most comprehensive. Although both K-2 and 3-5 benchmarks have been use
in the comparison, the level of knowledge seems to be an appropriate match.
K-4 teachers should find the comparison especially useful, since mathematics
and science are often integrated at this level. The first four standards
are clearly important ones for this integrated approach. *Benchmarks*
provides useful comprehensive and in-depth interpretations of these standards.

Conceptual understanding of operations is illustrated in *Benchmarks*
through its emphasis on being able to explain computations and judge their
reasonableness. At this grade level, modeling appears informally in both
*Benchmarks* and *Standards*, developing the notion of concrete
representations of mathematical ideas. In *Benchmarks*, students are
expected to describe and compare many properties of things and ideas.

The concepts and uses of number in *Benchmarks* is primarily focused
on quantities rather than on number in the abstract. *Benchmarks*
also connects measurement, computation, estimation, and other uses of numbers
more extensively than does *Standards*. In measurement, *Benchmarks*
mentions specific meanings of length, area, volume, weight and time as
well as measuring dry and liquid materials. The ideas of patterns and relationships
is mainly drawn from real world examples in *Benchmarks*, rather than
being an explicit area of study as in *Standards*.