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

Indiana

Reviewed: Indiana Mathematics Academic Standards, Approved September 2000, subsequently updated. Indiana provides standards for each of the grades K-8, and for each of the secondary courses, Algebra I, Geometry, Algebra II, Integrated Math I, II, and III, Pre-Calculus, Probability and Statistics, and Calculus.

2005 State Report Card
Indiana
Clarity: 3.67 A
Content: 3.83 A
Reason: 4.00 A
Negative Qualities: 3.75 A
Weighted Score: 3.82 Final Grade: A
2000 Grade: C
1998 Grade: C


Indiana’s 2000 revision of its standards was a remarkable success, vaulting it from the middle rank of states to near the top of the pack. These standards have many admirable features. The writing is generally clear and the content is excellent and well organized. Mathematical reasoning is implicitly or explicitly required in many of the content standards. One particparticularly commendable feature, that other states would do well to emulate, is that the standards for grades K-3 do not have a probability and statistics strand.

The elementary grade standards require mastery of the basic number facts, mental calculation, and skill with the standard algorithms for addition and subtraction. Facility with the standard algorithms for multiplication and division is required in the case of single-digit divisors for division and a single-digit factor for multiplication as indicated by these fourth-grade standards:

Use a standard algorithm to multiply numbers up to 100 by numbers up to 10, using relevant properties of the number system. Example: 67 x 3 = ?

Use a standard algorithm to divide numbers up to 100 by numbers up to 10 without remainders, using relevant properties of the number system. Example: 69 ÷3 = ?

A shortcoming is that the fifth-grade standard for multiplication and division of whole numbers in general leaves students free to choose their own methods:

Solve problems involving multiplication and division of any whole numbers. Example: 2,867 ÷ 34 = ? Explain your method.

The absence of any requirement to learn the long division algorithm for whole numbers in general slightly undermines the foundations for the understanding of irrational numbers in later grades. Computations with decimals are required in sixth grade, but with no specified methods:

Multiply and divide decimals. Example: 3.265 x 0.96 = ?, 56.79 ÷ 2.4 = ?

 The development of fractions is fast-paced. By third grade students are expected to:

Show equivalent fractions using equal parts. Example: Draw pictures to show that 3/5 , 6/10 , and 9/10 are equivalent fractions.

 This may even be overly ambitious, since memorization of all of the multiplication facts is not expected until the following year in fourth grade. Third-graders also add and subtract fractions with the same denominator.

Fifth-graders multiply and divide fractions and add and subtract mixed numbers and decimals.An example of the commendable attention given to reasoning by the Indiana standards is illustrated in this fifth-grade standard: Use models to show an understanding of multiplication and division of fractions. Example:

Draw a rectangle 5 squares wide and 3 squares high. Shade 4/5 of the rectangle, starting from the left. Shade 2/3 of the rectangle, starting from the top. Look at the fraction of the squares that you have double-shaded and use that to show how to multiply 4/5 by 2/3.

Minor Complaints

The treatment of areas in the lower grades is one of the few defects of the lower elementary grade standards. A legitimate (though redundant) fourth grade standard is, “Know and use formulas for finding the areas of rectangles and squares,” but in grades 3 and 2 respectively, one finds:

Estimate or find the area of shapes by covering them with squares. Example: How many square tiles do we need to cover this desk?

Estimate area and use a given object to measure the area of other objects.

Example: Make a class estimate of the number of sheets of notebook paper that would be needed to cover the classroom door. Then use measurements to compute the area of the door.

The concept of area should be developed more carefully than indicated in this last example especially. Sheets of notebook paper are not square and the area of the door, calculated by multiplying length times width, is not the number of notebook sheets needed to cover it. Area should be introduced initially for rectangles with positive whole number sides and then determined exactly. Only after that should students be expected to estimate areas, especially when the exact area is not a whole number of square units.

The middle school grade standards and secondary course standards are for the most part well crafted and complete. However, examples that accompany them leave room for improvement, as illustrated in these two consecutive Algebra I standards:

Understand the concept of a function, decide if a given relation is a function, and link equations to functions. Example: Use either paper or a spreadsheet to generate a list of values for x and y in y = x2. Based on your data, make a conjecture about whether or not this relation is a function. Explain your reasoning.

Find the domain and range of a relation. Example: Based on the list of values from the last example, what are the domain and range of y = x2?

Spreadsheets have no legitimate role to play in deciding whether y = x2 is a function and what its natural domain and range are.

A Plethora of Probability

The Data Analysis and Probability strand that runs from fourth grade to eighth grade, while better than analogous strands for many other states, is nevertheless overblown. For example, in eighth grade, students are expected to:

Represent two-variable data with a scatterplot on the coordinate plane and describe how the data points are distributed. If the pattern appears to be linear, draw a line that appears to best fit the data and write the equation of that line.

 To develop the topic of lines of best fit properly is college-level mathematics, and to do it in other ways is not mathematics. Moreover, some of the data analysis standards stray too far from mathematics in the direction of social science, such as this eighth-grade standard:

Identify claims based on statistical data and, in simple cases, evaluate the reasonableness of the claims. Design a study to investigate the claim. Example: A study shows that teenagers who use a certain brand of toothpaste have fewer cavities than those using other brands. Describe how you can test this claim in your school.

A few of the standards are poorly stated, such as this eighth-grade example:
Understand that computations with an irrational number and a rational number (other than zero) produce an irrational number. Example: Tell whether the product of 7 and ¹ is rational or irrational. Explain how you know that your answer is correct.
or this standard for Integrated Math:

Know and use the relationship sin2 x + cos2 x = 1. Example: Show that, in a right triangle, sin2 x + cos2 x = 1 is an example of the Pythagorean Theorem.

In the above standard, the phrase “in a right triangle” is out of place. In a similar vein, the glossary needs careful editing (e.g., “prime number” and “composite number” are not correctly defined).

Despite these minor flaws, Indiana’s excellent mathematics standards are among the best in the nation.

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