Mathematics

End-of-high-school standards should thoroughly cover the arithmetic of rational numbers, including decimals, and should also cover rates, ratios, proportions, and percentages, as well as problem solving that utilizes this content. The ACT standards cover these important topics well under the “basic operations and applications” strand, which includes the following:

Solve routine one-step arithmetic problems (using whole numbers, fractions, and decimals).

Solve routine two-step or three-step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off, and computing with a given average.

Solve word problems containing several rates, proportions, or percentages.

The measurement strand covers the necessary content about lengths, perimeters, and areas.

The “properties of plane figures” strand addresses geometry, covering angles, parallel lines, and properties of triangles. Coordinates and the Pythagorean theorem are also included.

The standards contain absolute value, slope, solving linear equations, and the arithmetic of polynomials, including factoring. Logarithms, basic trigonometry, and quadratic equations are there, too, as are basic probability and statistics.

Missing from the standards, however, is any mention of the following: geometric constructions, definitions and proofs, reciprocals, any of the forms of linear and quadratic functions, equations, and rational expressions and exponential functions. Furthermore, a complete analysis of quadratic equations and functions (and their graphs) is not explicitly addressed. Combinations and permutations are absent unless they fall under “apply counting techniques.”

Yet although some content is missing, this lapse is mostly due to lack of detail, rather than lack of substance. For example, the standards do specify that students should master “counting techniques” and the most common “counting techniques” are combinations and permutations. Therefore, the standards certainly imply that students should learn this critical content. Similarly, most standards associated with linear and quadratic equations and functions are focused on solving equations, graphing functions, and working on problems that require them. They do not go into detail about the forms these equations can take and the necessity to be able to move between forms algebraically. The closest they come for quadratics, for example, is this direction:

Recognize special characteristics of parabolas and circles (e.g., the vertex of a parabola and the center of radius of a circle).

This is too non-specific to be of much help. In particular, it does not give the necessary guidance for finding the maximum or minimum of a quadratic function.

That said, while these missing details are important, they are not as important as the core content that is present—both explicitly and implicitly. It would be difficult to say that 20 percent of the mathematics was missing, and so the Content and Rigor score must earn a six out of seven.

The structure of the ACT standards is nicely done. The standards comprise only two pages, proving that quality end-of-high-school standards do not need to be lengthy. They are broken up into eight content strands; each strand has six levels of skills, knowledge, or understanding.

The organizational structure of ACT’s standards allows them to set priorities. It is easy to see what is most important or considered the deepest level of understanding. For example, one thread that uses all six levels within a strand (Expressions, Equations, & Inequalities) is the following:

Solve equations in the form x+a=b, where a and b are whole numbers or decimals.

Solve one-step equations having integer or decimal answers.

Solve routine first-degree equations.

Solve real-world problems using first-degree equations.

Find solutions to systems of linear equations.

Write equations and inequalities that require planning, manipulating, and/or solving.

This is a well-developed sequence of items requiring higher and higher levels of understanding. Furthermore, each item is straightforward, clear, and well written. Not all threads run through all six levels, but connected threads similar to these are frequent in the standards.

Few standards could comfortably be eliminated. Most of them are straightforward, simple and clear. This is a rare example of an unclear standard:

Compute a probability when the event and/or sample space are not given or obvious.

There are very few vague “process” standards about mathematical thinking; below is one exception:

Draw conclusions based on a set of conditions.

This one might be intended to cover proofs in geometry, but it is not explicit.

In short, the standards are well organized, clearly written, set priorities adequately, and contain no excesses that hide important content. The standards easily merit a three out of three for Clarity and Specificity.