**Estimated Time:** 30 minutes.

**Example of Use: **Sample 2.5 Day Workshop
Agenda.

**Sample Presentation**
**Presenter: ***The purpose of this presentation is to show the
extent of agreement and to identify differences between *Benchmarks*
and the *Curriculum and Evaluation Standards for School Mathematics.

*Both* Benchmarks for Science Literacy* and the *Curriculum
and Evaluation Standards for* *School Mathematics* describe what
American students are expected to know and be able to do at various grade
levels during their K-12 education. *Benchmarks* includes mathematics
as an important area of learning both in its own right and for its relationship
to science and technology. (In Project 2061 usage, "science" includes natural
and social sciences, mathematics, technology, and the relations among them.)*

*Although there are some significant differences in format, content,
and philosophy between the two documents, there is considerable overlap
in the substance of their recommendations. Their comparison provides important
information* *for educators,* *especially those involved in K-12
mathematics and science. Today we will compare the purpose, domains, grade
levels, and organization of the two documents, as well as their substance.
The examples we will examine are taken from a comparison on the CD-ROM
disk, *Resources for Science Literacy: Professional Development.

TRANSPARENCY*: Benchmarks
for Science Literacy* - Table of Contents.

**Presenter :**

HANDOUT: *Benchmarks*
Table of Contents.

**Presenter:** *When we use numbers and letters, like 1B or 2C,
in speaking about *Benchmarks*, we are simply referring to its chapters
and sections, as you can see in this handout; for example, 2C: chapter
2, The Nature of Mathematics; section C, Mathematical Inquiry.*

TRANSPARENCY: *Benchmarks*
Chapters 11 and 12.

**Presenter:** *Although you would expect to find mathematical
content in *Benchmarks* chapters 2 and 9, you will find it in several
other chapters as well; for example, in Chapter 11: Common Themes, section
C, Constancy and Change; or in Chapter 12, Habits of Mind, section B, Computation
and Estimation.*

TRANSPARENCY: Comparison
of *Benchmarks* and *NCTM Standards*.

**Presenter:** *The purposes of *Benchmarks* and the NCTM
*Standards* are somewhat different. *Benchmarks* addresses the
question "What knowledge and skills in mathematics should characterize
all adults and what are the stages along the way?" The NCTM *Standards*
document more directly addresses the question "What should K-12 students
do in school mathematics?" This difference in purpose is reflected in the
different ways *Benchmarks* and *Standards* statements are expressed¾
most benchmarks are stated in terms of what students should know and retain
(whenever they learn it) whereas most standards are stated in terms of
what students should do as mathematics students. However, it is clear in
*Benchmarks* that the way students become literate adults is through
doing and investigating while they are students. Another difference in
purpose is that *Benchmarks *describes the floor of expectations,
whereas *Standards* describes the whole mathematics curriculum for
average and college-intending students as well.*

*Domain: *Benchmarks*, like *Science for All Americans,*
includes science, mathematics, and technology, and their interconnections.
The *Standards* deal exclusively with mathematics. A closer examination
reveals considerable similarity and some differences in the mathematical
topics addressed and in how they are treated.*

*The difference in emphasis between the two documents shows up in
their opposite organizations. Standards is organized first by three grade
ranges—and then, within each grade range, by a somewhat variable set of
topic areas. *Benchmarks* is organized first by topic area and then,
within this uniform*

TRANSPARENCY: *NCTM Standards*
1-4 Comparison (K-12).

**Presenter: ***The first four standards—Mathematics as Problem
Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical
Connections—are the same in each grade-level group—K-4, 5-8, and 9-12—as
shown in the left *column*. Listed on the right are related *Benchmark*
sections. For example, for Standard 2, Mathematics as Communication, we
find *Benchmarks* 1B; that is, chapter 1: The Nature of Science, section
B, Scientific Inquiry. As noted earlier, the mathematical ideas in*
Benchmarks* are not confined to the chapters labeled mathematics¾
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. *

TRANSPARENCY: *NCTM*
K-4 Comparison (Standards 5-13).

**Presenter:** *Unlike Standards 1 through 4, Standards 5 through
13 differ from one grade range to the next. Here are Standards 5-13 for
grade-level K-4.* C*overage of standards by benchmarks is fairly complete
for the K-4 grade-level. The different organization of ideas between *Benchmarks*
and *Standards* continues in the other nine standards for grades K-4.
*(Point to Standard 5.)

*Standard 5: Estimation includes an idea that appears in five *Benchmarks*
sections; contrariwise, *Benchmarks* section 9C: Uncertainty, includes
ideas relevant to five different standards. *(Point to Standards
5, 8, 9, 10, and 11.)

TRANSPARENCY: *NCTM Standard*
3: Mathematics as Reasoning Comparison (K-4).

**Presenter:** *This example, using Standard 3: Mathematics as
Reasoning, shows a number of K-2 and 3-5 benchmarks with similar expectations.
Note that in many cases benchmarks are more specific and can be used to
clarify the more general statements in *Standards.* *

TRANSPARENCY: *NCTM Standard*
11: Statistics and Probability Comparison (K-4).

**Presenter:** *In this example from Standard 11: Statistics and
Probability, you can see considerably more substance in *Benchmarks*.*

TRANSPARENCY: *NCTM*
5-8 Summary Comparison (*Standards* 5-13).

**Presenter:** *These are standards 5 through 13 for grade-level
5-8. Note that the topical areas for the 5-8 standards differ from the
K-4 topical areas. Corresponding *Benchmarks* are shown in the right
column.*

*Without pointing to details, we can make some general statements
about how math ideas are treated somewhat differently in *Standards*
and *Benchmarks*. Generally, *Benchmarks* pays less attention
to abstract concepts. For example, the coverage of numbers and their relationships
in *Benchmarks* is more or less integrated with measurement and computation.
(Most of the grade 5-8 measurement in *Standards* is covered by grades
3-5 in *Benchmarks*.) *Benchmarks* treats algebra, patterns,
and functions in terms of patterns of change in the real world. *

TRANSPARENCY: *NCTM*
9-12 Summary Comparison (*Standards* 5-14).

**Presenter:** *The 12th-grade comparison of *Standards* and
*Benchmarks* reveals the most significant differences between the
two documents, both in emphasis and in the amount of mathematical content.
The 12th-grade benchmarks represent the original intent of Project 2061
in *Science for* *All Americans*, which was to describe what
all students should know at the end of high school. Therefore,*
Benchmarks

TRANSPARENCY: *NCTM Standards*
(for All) with No Comparable Benchmarks.

**Presenter:** *This* *list* *shows standards*,*
mostly 9-12, recommended by NCTM for all students, that are not found in
*Benchmarks*.*

*At present, the mathematics content of *Benchmarks* is not widely
known in the mathematics education community. As *Standards* and other
reforms are implemented in mathematics, it is important to look carefully
at *Benchmarks*, especially in the light of renewed attention to applications
of mathematics and integration with science and other subjects. A detailed
comparison of the *Benchmarks* and *Standards*, at the level
of detail of the examples we have seen here, is available on *Resources
for Science Literacy: Professional Development*. *