Colorado

Standards are often clearly stated:

Find the value of a collection of coins and choose coins to have a given value (grade 2)
Name and locate points specified by ordered number pairs on a coordinate grid (grade 4)
Use a protractor to measure angles to the nearest degree (grade 6)
Compare and order sets of integers and rational numbers that are expressed as fractions, decimals, or percents (grade 8)

However, other standards are far too broadly stated to allow readers to interpret the intent:

Apply addition and subtraction concepts to financial decision-making (grade 2)
Use geometric properties of points and line segments to describe figures (grade 3)
Analyze various lending sources, services, and financial institutions (high school)

These standards are not specific enough to determine what students are expected to know or what kinds of problems they should be able to solve.

The high school standards in particular are often too general to be clear, and the material is often not coherent. Standards relating to a single topic, such as quadratics, may be strewn across various strands. Many topics are often included in a single standard, which makes such a standard difficult to understand. For example, in the following standard, the specific techniques mentioned do not apply to all of the topics:

Find solutions to quadratic and cubic equations and linear inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula (high school)

While the K-8 standards are often clear and easy to interpret, the high school standards are not. As a whole, the standards “do not quite provide a complete guide” to users and therefore receive a Clarity and Specificity score of two points out of three. (See Common Grading Metric.)

Content Priorities

Arithmetic is the key content priority in the early-middle grades, but it is barely prioritized in Colorado’s standards. In fact, just over one-third of the standards in the appropriate grades address the development of arithmetic. This provides an implicit indication that arithmetic is not much of a priority, which is not sufficient.

Content Strengths

The structure of arithmetic, commutativity, associativity, distributivity, and the inverse nature of addition and subtraction and of multiplication and division are all well covered.

There are some strong standards on the development of area, including:

Model area using square units (grade 4)
Determine the perimeter of polygons and area of rectangles (grade 5)
Develop and apply formulas and procedures for finding area of triangles, parallelograms, and trapezoids (grade 6)

In high school, the coverage of linear equations is also strong:

Demonstrate the relationship between all forms of linear functions using point-slope, slope-intercept, and standard form of a line (high school)

Although geometry foundations in high school are weak (see Mathematics Content-Specific Criteria in Appendix A for foundations), some standards explicitly mention proof, such as:

Know and apply properties of angles including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve problems. Justify the results using two-column proofs, paragraph proofs, flow charts, or illustrations (high school)

Content Weaknesses

The development of whole-number arithmetic is inadequate. Instant recall of number facts is not stated strongly enough, since the relevant standards can be interpreted as requiring computational fluency instead. Instant recall is an important building block for future mathematics; students who are still struggling with basic facts are not prepared to move on to the next level of mathematics.

In the continued development of arithmetic, students are expected to be able to use different methods of computing, but fluency is not required:

Use flexible methods of computing, including student-generated strategies and standard algorithms
(grade 3)

Use flexible methods of computing including standard algorithms to multiply and divide multi-digit numbers by two-digit factors or divisors (grade 5)

For addition and subtraction, the standard algorithms are given equal status with student-generated algorithms, defeating an important goal of arithmetic. For multiplication and division, it also appears that alternative algorithms are acceptable.

In the continued development of arithmetic, common denominators for fractions are not mentioned, though they appear in the peripheral material.

High school content is often weak. The coverage of linear equations is missing some essential details, including equations for parallel and perpendicular lines. The coverage of quadratics is also incomplete. Quadratics is not developed coherently, and specific mention of it is infrequent. Much of their coverage is subsumed in a single standard:

Find solutions to quadratic and cubic equations and linear inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula (high school)

Missing content includes complex roots, vertex form, and max/min problems.

While factoring is mentioned, polynomials are not, and the arithmetic of polynomials and rational functions is not covered.

Much of the STEM-ready content is also missing, including inverse trigonometric functions and polar coordinates.

Though prioritized somewhat, the development of whole-number arithmetic is not adequate. The high school material is not presented coherently and misses much essential content. These “serious problems” result in a Content and Rigor score of three points out of seven. (See Common Grading Metric.)