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: As math educators, you are probably more familiar with the organization and content of the math Standards than you are with Science for All Americans or Benchmarks for Science Literacy. Let me orient you with this Benchmarks table of contents. Where would you begin looking in Benchmarks to find mathematics ideas if you were to compare the two documents? (Take responses; chapters 2 and 9 are usual responses.)
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 set of topic areas, by a progression of grade ranges. The grade ranges themselves are also different; Standards has three—K-4, 5-8, and 9-12, whereas Benchmarks has four—K-2, 3-5, 6-8, and 9-12. K-2 is specified separately because the nature of the recommended goals for students is significantly different for early and late elementary grades.
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. Coverage 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 does not include the college-intending parts of Standards such as Discrete Mathematics, Trigonometry, and Calculus. This difference in philosophy and its implications for curricula and teaching in mathematics and science are important and can be fruitful areas for discussion by mathematics educators.
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.