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

Massachusetts

Reviewed: Massachusetts Mathematics Curriculum Framework, November 2000; Supplement to the Massachusetts Mathematics Curriculum Framework, May 2004. The Framework provides standards for two-year grade spans from PreK-K to 11-12, and for the courses Algebra I, Geometry, Algebra II, and Pre-Calculus. In addition, the 2004 Supplement gives standards for the individual grades 3, 5, and 7 for the purpose of annual testing required by the No Child Left Behind Act. Each grade span includes extra standards under the heading, “Exploratory Concepts and Skills.” These enrichment topics are not assessed by the state at the grade levels in which they appear, but some of them are also listed in later grade-level standards.

2005 State Report Card
Massachusetts
Clarity: 3.67 A
Content: 3.67 A
Reason: 2.00 C
Negative Qualities: 3.50 A
Weighted Score: 3.30 Final Grade: A
2000 Grade: D
1998 Grade: F

Massachusetts did its students a tremendous service in 2000 by jettisoning its old standards and substituting these clear, well-organized documents. They outline a solid and coherent program for mathematics education.

The elementary grade standards are particularly strong. They require memorization of the basic number facts and facility with the standard algorithms of arithmetic. Students are expected to compute with and solve word problems involving fractions, decimals, and percents by the end of sixth grade. Rational number arithmetic and the field properties are thoroughly developed in the middle grades, with algebra and more advanced topics addressed by the high school standards.

Mixed Guidance on Technology

 Technology plays a mixed role in these standards. A section of the Framework, “Guiding Principle III: Technology,” begins with the declaration,“Technology is an essential tool in a mathematics education.” The opening sentence of the final paragraph of the section is, “Technology changes what mathematics is to be learned and when and how it is learned.” Both of these sweeping assertions overstate the importance of technology for K-12 mathematics.

On the other hand, this section also includes an important and refreshing caveat:

Elementary students should learn how to perform thoroughly the basic arithmetic operations independent of the use of a calculator. Although the use of a graphing calculator can help middle and secondary students to visualize properties of functions and their graphs, graphing calculators should be used to enhance their understanding and skills rather than replace them.

The Massachusetts standards deal admirably with technology in the elementary grades, but offer little guidance for its proper use in the higher grades. For example, Exploratory Concepts and Skills for grades 9 and 10 includes the suggested project, “Explore higher powers and roots using technology.” Several standards include the ambiguous statement “use technology as appropriate,” such as the following:

7.P.6 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate. AI.P.11 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate. AII.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, and logarithmic functions; expressions involving the absolute values; and simple rational expressions.

 Considering the diversity of teachers’ opinions on the use of technology, the Framework would be improved if it clarified its directive to “use technology appropriately.”

Inconsistent Reasoning

Standard AI.P.11 cited above illustrates the lack of specificity found in some of the higher grade-level standards. What “everyday problems” are intended here? What is a “reciprocal function”? And what are “tabular methods”? In a similar vein, a seventh-grade standard calls upon students to “solve linear equations using tables, graphs, models, and algebraic methods.” How can linear equations be solved using tables? The appropriate methods for solving a linear equation in seventh grade are algebraic. If the solution of simultaneous linear equations is intended here, then graphical methods also play an important role, but this standard does not specify whether one or more linear equations are to be solved.

Mathematical reasoning is prominently featured. All Massachusetts standards are prefaced with the phrase, “Students engage in problem solving, communicating, reasoning, connecting, and representing as they: . . .”But the standards also go beyond this perfunctory exhortation. The following two standards for algebra and geometry respectively illustrate the incorporation of mathematical reasoning in the Massachusetts standards:

Use properties of the real number system to judge the validity of equations and inequalities, to prove or disprove statements, and to justify every step in a sequential argument.

Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive.

 The Framework also requires a comprehensive treatment of methods for finding the roots of quadratic polynomials in the algebra standards:

Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods.

 We interpret this to mean that students are expected to know how to derive the quadratic formula by completing the square, and to understand that the roots of a quadratic polynomial are given by the quadratic formula. However, we would prefer a clearer statement such as:

Find the roots of quadratic polynomials (with real roots) by factoring, by completing the square, and by using the quadratic formula. Derive the quadratic formula by completing the square, and prove that the roots of a quadratic polynomial are given by the quadratic formula.

 There is no standard that explicitly requires students to see or understand a proof of the Pythagorean Theorem. The closest the Framework comes to this requirement is the following eighth-grade standard:

Demonstrate an understanding of the Pythagorean Theorem. Apply the theorem to the solution of problems.

What does it mean to “demonstrate an understanding of the Pythagorean Theorem”? Does it mean to understand the statement of the theorem? Or to understand a geometric interpretation of the theorem in terms of areas? Or, perhaps, even a proof? One can only guess.

Opportunities for the incorporation of mathematical reasoning are missed in standards that address topics in area, volume, and perimeter. Consider the following geometry standard:

Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area.

There is no requirement in any of the Massachusetts standards for students to understand how to derive or deduce any formula for area, perimeter, or volume of any geometric figure or solid. Surely, students can be expected to find the lateral surface area of prisms without being “[g]iven the formula.”

Minor Problems

 There are other problems with the Bay State’s standards. As elsewhere, data analysis, statistics, and probability standards are overemphasized throughout the standards. This starts in pre-K and Kindergarten, where students are expected to construct bar graphs. In grades 1 and 2, students

decide which outcomes of experiments are most likely
It makes no sense to teach probability to students before they have reasonable facility with fractions, since probabilities are, by definition, numbers between zero and one.

Data analysis, statistics, and probability standards are also inappropriately included among the standards for Algebra I and Algebra II. Among these standards is:

Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use technology when appropriate.

 To develop the topic of lines of best fit properly is college-level mathematics, and to do it in other ways is not mathematics. Manipulation of polynomials is too restrictive:
Add, subtract, and multiply polynomials. Divide polynomials by monomials.

 The four basic arithmetic operations should be performed with rational functions, not just with polynomials (or monomials). Requiring division by binomials would at least support a theorem addressed in the Pre- Calculus standards:

Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients.

 In spite of these shortcomings, the Massachusetts math standards are among the best in the nation.

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