Thomas B. Fordham Institute - Advancing Educational Excellence

The State of State Math Standards 2005

January 5, 2005

by David Klein, Bastiaan J. Braams, Thomas Parker, William Quirk, Wilfried Schmid, W. Stephen Wilson, Chester E. Finn, Jr., Justin Torres, Lawrence Braden, Ralph A. Raimi

States still have far to go in setting rigorous, high quality expectations for K-12 math instruction. Although a majority have replaced or revised their math standards since 2000, many have failed to make substantial improvements. The review was led by David Klein, Professor of Mathematics at California State University-Northridge, and evaluates the content, writing quality, and clarity of K-12 math standards in each state. Klein and his team attribute many of  the shortcomings to overuse and wrong applications of manipulatives and calculators; wrong-headed guidance from the National Council of Teachers of Mathematics; and lack of true mathematics competence among those writing the standards.

Contents

California

Review: Mathematics Framework for California Public Schools, 2000 Revised Edition provides standards for each of the grades K-7 and for the courses and topics: Algebra I; geometry; Algebra II; Trigonometry; Mathematical Analysis; Linear Algebra; Probability and Statistics; Advanced Placement Probability and Statistics; and Calculus. At the time of this writing, the revised edition of the Framework was the latest available, but additional revisions are in progress, including new appendices addressing algebra readiness and intervention programs.

California’s standards are excellent in every respect. The language is crystal clear, important topics are given priority, and key connections between different skills and tasks are explicitly addressed. Computational skills, problem-solving, and mathematical reasoning are unambiguously supported and integrated throughout the standards. For example, the fifth-grade standards addressing fraction multiplication and division proceed logically and clearly:

Understand the concept of multiplication and division of fractions.

Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.
Sample problems follow the latter standard. Procedural skill, conceptual understanding, and problem-solving are all required here. Another illustration is this Measurement and Geometry standard for fifth grade:

2005 State Report Card
California
Clarity: 3.83 A
Content: 3.94 A
Reason: 3.83 A
Negative Qualities: 3.92 A
Weighted Score: 3.89 Final Grade: A
2000 Grade: A
1998 Grade: A

Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram).

Sample problems immediately follow in the Framework, and a fourth-grade Measurement and Geometry standard carefully lays the groundwork for the above standard:

Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.
Top-Notch

The elementary grade standards require memorization of the basic number facts and facility with the standard algorithms of arithmetic, including the important long division algorithm. Standards calling for facility with the standard algorithms of arithmetic also ask for understanding of why the algorithms “work,” as in this fourth-grade Number Sense standard:

Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multi-digit number by a two-digit number and for dividing a multi-digit number by a one-digit number; use relationships between them to simplify computations and to check results.

 The K-7 standards build the prerequisites for secondary algebra and geometry systematically and coherently. California aims to place students in Algebra I, or an integrated math course, by eighth grade, but the Framework acknowledges on page 199 that this ambitious program is not always appropriate:

One purpose of a seventh grade assessment, as described previously, is to determine the extent to which students are mastering prealgebraic concepts and procedures. Another is to identify those students who lack the foundational skills needed to succeed in eighth grade algebra and need further instruction and time to master those skills. This additional instruction may be provided through tutoring, summer school, or an eighth grade prealgebra course leading to algebra in the ninth grade.

 California’s Framework clearly and appropriately addresses the role of technology. Chapter 9, “The Use of Technology,” provides clear guidance on calculator and computer usage that other states would do well to emulate. A section entitled “The Use of Calculators” begins,

The Mathematics Content Standards for California Public Schools was prepared with the belief that there is a body of mathematical knowledge—independent of technology—that every student in Kindergarten through grade twelve ought to know and know well.
Indeed, technology is not mentioned in the Mathematics Content Standards until grade six. More important, the STAR assessment program—carefully formulated to be in line with the standards—does not allow the use of calculators all through Kindergarten to grade eleven.

The Framework, however, does encourage the use of calculators in specific, appropriate circumstances:

It should not be assumed that caution on the use of calculators is incompatible with the explicit endorsement of their use when there is a clear reason for such an endorsement. Once students are ready to use calculators to their advantage, calculators can provide a very useful tool not only for solving problems in various contexts but also for broadening students’ mathematical horizons. One of the most striking examples of how calculators can be appropriately used to help solve problems is the seventh grade topic of compound interest.
 A Few Flaws

The K-7 standards are not without shortcomings. The standards, pitched at an internationally competitive level, place stiff demands on students that exceed those of most states, and the Framework does not elaborate sufficiently on how best to help students who fall behind. Probability and statistics are overemphasized, although not as much as with most other states. For example, these sixth-grade standards stray too far in the direction of social science and away from mathematics:

Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population.

Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.

Identify data that represent sampling errors and explain why the sample (and the display) might be biased.

California’s K-7 mathematics standards are demanding enough without the inclusion of such diversions as data collection.

 The section, “Grade-Level Considerations, Grade Four: Areas of Emphasis” has an egregious error that should be corrected, along with supporting material in Appendix A to which the passage refers. On page 135, the paragraph labeled “Fractions equal to one” includes this statement:

When the class is working on equivalent fraction problems, the teacher should prompt the students on how to find the equivalent fraction or the missing number in the equivalent fraction. The students find the fraction of one that they can use to multiply or divide by to determine the equivalent fraction.
Fourth-grade students cannot use multiplication and division of fractions to find equivalent fractions because multiplication and division of fractions are not introduced until fifth grade. Moreover, equivalence of fractions is fundamental to the arithmetic of rational numbers. The concept of equivalence of fractions must be firmly established, using only whole number operations, before multiplication and division of fractions can be defined and explained. However, equivalence of fractions is correctly addressed by the third-and fourthgrade standards themselves.

 A Model for States

The Framework identifies the high school content intended for all students as Algebra I, Geometry, and Algebra II (although it does allow integrated math courses covering the same topics). The content standards for the more advanced courses are listed by topic (rather than as courses) with the intention that those standards may be collected and combined in a variety of different possible ways. As the document explains:

To allow local educational agencies and teachers flexibility in teaching the material, the standards for grades eight through twelve do not mandate that a particular discipline be initiated and completed in a single grade. . . . Many of the more advanced subjects are not taught in every middle school or high school. Moreover, schools and districts have different ways of combining the subject matter in these various disciplines. For example, many schools combine some trigonometry, mathematical analysis, and linear algebra to form a precalculus course. Some districts prefer offering trigonometry content with Algebra II.

The Algebra I, Geometry, and Algebra II standards are exemplary. In Algebra I, students “know the quadratic formula and are familiar with its proof by completing the square.” Geometry students prove major theorems including the Pythagorean Theorem. The standards for the more advanced courses are demanding, and can prepare motivated students for university studies and scientific careers.

California’s Framework is not perfect. But it comes as close to perfection as any set of mathematics standards in the country, and should be a valuable model for other states.

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