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This is Fordham’s review of the state’s standards in place prior to adopting the Common Core. To return to our review of the common core standards (which this state has adopted), click here.
Arizonaâ€™s standards are generally strong. They are well presented and often include sample problems to enhance clarity. But a serious weakness in these standards is their coverage of arithmetic, which is neither prioritized nor appropriately culminated.
Arizonaâ€™s standards are divided into five content strands, such as â€œGeometry and Measurement.â€ Each strand is subdivided into topics called â€œMajor Concepts,â€ and then further separated into grade-level â€œperformance objectives.â€ These performance objectives are what we refer to as standards in this analysis.
One of the five strands, â€œStructure and Logic,â€ is actually a process strand that is primarily focused on skills, such as problem-solving and reasoning. The skills outlined in this strand are meant to be integrated across all content strands.
The high school standards follow the same organization but combine grades 9-10 and 11-12.
Clarity & Specificity
The standards are well presented: They are generally concise, comprehensible, and easy to read. The â€œExplanations and Examplesâ€ are often quite specific and serve to clarify the standards. The use of sample problems is an excellent feature, demonstrating for the reader exactly what kinds of problems students are expected to be able to do. For example, the following fifth-grade standard is broadly stated and the intent is subject to interpretation:
Use ratios and unit rates to model, describe and extend problems in context (grade 5)
But the explanatory material for this standard includes sample problems, which helps reveal what a student is expected to know:
If you can travel 20 miles in 4 hours on a bicycle, what is the unit rate (the distance you can travel in 1 hour)? (grade 5)
While the standards are generally clear, the explanatory material is not always specific enough to provide sufficient clarification. For example, consider the third-grade standard:
Demonstrate fluency of multiplication and division facts through 10 (grade 3)
It is not clear if fluency means fluency with instant recall or fluency with computation. The distinction is important, as students who do not have instant recall will be at a serious disadvantage as they continue learning multiplication. The additional explanatory material could have served to clarify the intent of the standard, but it is, unfortunately, equally unclear:
Fact fluency includes working with facts flexibly, accurately, and efficiently. This means that students have quick recall using strategies that are efficient (grade 3)
It is not clear from this if students are required to memorize basic facts. The second sentence suggests memory by the use of the word â€œrecall.â€ However, the need to use â€œstrategies that are efficientâ€ in order to achieve â€œquick recallâ€ is confusing and undermines any assumption of memorization.
Generally, the standards are clear, and the use of examples is an excellent feature that usually serves to clarify any ambiguity in the statements. Arizona receives three points out of three for Clarity and Specificity. (See Common Grading Metric.)
Content & Rigor
Arizona does not provide any guidance as to priorities. Worse, each grade has many standards, some of which are not important from a mathematical perspective. For example, from grade 2 through high school, one of the Major Concepts is â€œVertex-Edge Graphsâ€; many standards are devoted to this topic, such as in third grade:
Solve conflict problems by constructing and coloring vertex-edge graphs (grade 3)
This atypical and unimportant content is apparently equally weighted with crucial content such as mastery of arithmetic. More generally, Arizona fails to prioritize arithmeticâ€”only one-third of the elementary school standards are devoted to it.
The standards are often very strong. They cover some of the basic properties of arithmetic well, including commutativity, associativity, and distributivity. They also explicitly address the inverse relationship of addition and subtraction and of multiplication and division. The geometry standards include the development of formulas for areas, and the development of fractions is covered in some depth, including the use of the number line.
The high school standards cover many topics with both depth and rigor. Much of the STEM-ready content is covered.
The development of arithmetic is Arizona standardsâ€™ main weakness. There are many good culminating standards for arithmetic, fluency is mentioned in the explanatory material, and sample problems demonstrate student arithmetic proficiencies. However, the development of arithmetic is not adequately specific. To illuminate this shortcoming, the discussion below traces the development of whole-number multiplication.
As discussed above, instant recall of basic multiplication facts is not explicit. There are some desirable standards on multiplication, such as the fifth-grade capstone standard for whole-number multiplication:
Multiply multi-digit whole numbers (grade 5)
A rigorous treatment of this standard requires fluency with the standard algorithm. However, the explanatory material does not specify any methods. The preceding fourth-grade standard on multiplication is:
Use multiple strategies to multiply whole numbers: two-digit by two-digit and multi-digit by one digit (grade 4)
This standard could appropriately lead to mastery of the standard algorithm. However, the explanatory material for this standard includes four separate ways to multiply whole numbers, none of which is the standard algorithm. This suggests both a lack of exposure to the standard algorithm and a lack of expectation that a student must learn it.
The development of fraction arithmetic is problematic as well. Some standards ask that students manipulate fractions, but methodology is not specified. Common denominators are not mentioned in the standards, though they are mentioned in the explanatory material.
Arizonaâ€™s standards have strong high school content, but do not properly develop or prioritize arithmetic. These â€œcritical shortcomingsâ€ result in a Content and Rigor score of four points out of seven. (See Common Grading Metric.)
THE BOTTOM LINE
With their grade of B, Arizonaâ€™s mathematics standards are decent. Still, those developed by the Common Core State Standards Initiative earn an impressive A-minus, and thus are superior to what the Grand Canyon State has in place today.
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