Comparison of PASS to Common Core in Mathematics - Ze'ev Worman

Comparison of Oklahomas 2009 Mathematics Standards with Common Cores Mathematics Standards

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Comparison of Common Cores Mathematics Standards with Oklahomas

2009 Priority Academic Student Skills (PASS) for Mathematics Zeev Wurman

Visiting Scholar, Hoover Institution

April 2014

Executive Summary This comparison between PASS and Common Core mathematics was done in response to a request of

State Representative Jason Nelson. The comparison was performed based on the Thomas B. Fordham

Institute 2005 criteria and indicates that the two sets of standards very close in terms of their quality

and expectations, with a slight edge to the Oklahoma standards over Common Core. The report also

draws on additional sources to validate these findings. Based on the totality of the evidence, the report

concludes that there seems to have been no educational purpose for Oklahoma to switch to the

Common Core in 2010 because the existing Oklahoma standards were on par with or slightly superior to

the Common Core.

Table 1 summarizes my comparison results for both sets of the mathematics standards, offering a closer

look at the minor differences between them. The 2010 Fordham comparison between the two sets of

standards, which used a newly revised set of criteria, found similar results, albeit with the order

reversed. Further, a recent 2012 study (Schmidt and Houang, 2012)1 found Oklahoma mathematics

standards to be among the top ten states in terms of their alignment with Common Core, further

indicating that any differences in the quality of standards are rather small.

Table 1: Points per Section and Totals for Both Math Standards Sets

CC OK

Language Clarity: Clarity, Definiteness, Testability 3.00 3.83

Content: Elementary, Middle, High School 3.33 3.33

Reason 4.00 2.50

Negative Qualities: False Doctrines, Inflation 2.75 4.00

Total Score 3.28 (B+)

3.40 (A-)

While both sets of standards get similar total scores, both could be improved. PASS strengths are its

clear and relatively jargon-free language and its strong mathematical content. Minor revisions to PASS,

particularly in the area of clarifying expectations of fluency with elementary arithmetic operations,

1 William H. Schmidt and Richard T. Houang, Curricular Coherence and the Common Core State Standards for

Mathematics, Educational Researcher v41 p. 294 (2012)


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would significantly improve their quality. No such easy fix exists for the Common Core because its

weaknesses are spread throughout the standards. Furthermore, correcting them is beyond Oklahomas

power because their copyright belongs to Washington, D.C., organizations. Consequently, this report

recommends that Oklahoma returns to its own mathematics PASS standards and consider their revision

over time.

Introduction This comparison between Oklahomas 2009 Priority Academic Student Skills (PASS) in mathematics and

Common Core mathematics was done in response to a request of State Representative Jason Nelson.

The purpose is to assist legislators and the public to evaluate whether the replacement of PASS by the

Common Core standards is likely to improve or weaken Oklahomas public education.

The Thomas B. Fordham Institute performed a comparison of Common Core and PASS in 2010 as a part

of its State of the State Standards project. Unfortunately, Fordham had modified its evaluation criteria

at that time, thereby breaking the trend line used by Fordham since 1998. For this comparison I used the

original Fordham 2005 evaluation criteria.2

The Rubrics The evaluation is based on four rubrics: Clarity, Content, Reason, and Negative Qualities. Each rubric is

scored on a scale of 0 to 4 and totaled to a weighted average, with Content double the weight (40%) of

the other three rubrics (each 20%). The following description of the rubrics is drawn from the 2005

Fordham report:

Clarity refers to the success the document has in achieving its own purpose, i.e., making clear to teachers, test developers, textbooks authors, and parents what the state desires. Clarity refers to more than the prose, however. The clarity grade is the average of three separate sub-categories:

1. Clarity of the language: The words and sentences themselves must be understandable, syntactically unambiguous, and without needless jargon.

2. Definiteness of the prescriptions given: What the language says should be mathematically and pedagogically definite, leaving no doubt of what the inner and outer boundaries are, of what is being asked of the student or teacher.

3. Testability of the lessons as described: The statement or demand, even if understandable and completely defined, might yet ask for results impossible to test in the school environment. We assign a positive value to testability.

Content, the second criterion, is plain enough in intent. Mainly, it is a matter of what might be called

subject coverage, i.e., whether the topics offered and the performance demanded at each level are

sufficient and suitable. To the degree we can determine it from the standards documents, we ask, is the

state asking K-12 students to learn the correct skills, in the best order and at the proper speed? For this

report, the content score comprises 40 percent of the total grade for any state.

2 David Klein et al., The State of State MATH Standards, 2005. Thomas B. Fordham Foundation and Institute.


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Here we separate the curriculum into three parts (albeit with fuzzy edges): Primary, Middle, and

Secondary. It is common for states to offer more than one 9-12 curriculum, but also to print standards

describing only the common curriculum, often the one intended for a universal graduation exam,

usually in grade 11.

Content gives rise to three criteria:

1. Primary school content (K-5, approximately)

2. Middle school content (or 6-8, approximately)

3. Secondary school content (or 9-12, approximately).

Reason. Civilized people have always recognized mathematics as an integral part of their cultural

heritage. Mathematics is the oldest and most universal part of our culture. In fact, we share it with all

the world, and it has its roots in the most ancient of times and the most distant of lands.

Therefore, in judging standards documents for school mathematics, we look to the topics as listed in

the content criteria not only for their sufficiency, clarity, and relevance, but also for whether their

statement includes or implies that they are to be taught with the explicit inclusion of information on

their standing within the overall structures of mathematical reason.

We therefore look at the standards documents as a whole to determine how well the subject matter is

presented in an order, wording, or context that can only be satisfied by including due attention to this

most essential feature of all mathematics.

Negative Qualities. This fourth criterion looks for the presence of unfortunate features of the

document that contradict its intent or would cause its reader to deviate from what otherwise good,

clear advice the document contains. We call one form of it False Doctrine. The second form is called

Inflation because it offends the reader with useless verbiage, conveying no useful information. Scores

for Negative Qualities are assigned a positive value; that is, a high score indicates the lack of such

qualities.

Under False Doctrine, which can be either curricular or pedagogical, is whatever text contained in the

standards we judge to be injurious to the correct transmission of mathematical information. While in

general we expect standards to leave pedagogical decisions to teachers (as most standards documents

do), so that pedagogy is not ordinarily something we rate in this study, some standards contain

pedagogical advice that we believe undermines what the document otherwise recommends.

Under the other negative rubric, Inflation, we speak more of prose than content. Evidence of

mathematical ignorance on the part of the authors is a negative feature, whether or not the document

shows the effect of this ignorance in its actual prescriptions, or contains outright mathematical error.

Repetitiousness, bureaucratic jargon, or other evils of prose style that might cause potential readers to

stop reading or paying attention, can render the document less effective than it should be, even if its

clarity is not literally affected.


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The most common symptom of irrelevancy, or evidence of ignorance or inattention, is bloated prose,

the making of pretentious yet empty pronouncements. Bad writing in this sense is a notable defect in

the collection of standards we have studied.

We thus distinguish two essentially different failures subsumed by this description of pitfalls, two

Negative Qualities that might injure a standards document in ways not classifiable under the headings of

Clarity and Content: Inflation (in the writing), which is impossible to make use of; and False Doctrine,

which can be used but shouldnt.

Standards Analysis We now compare the two sets of standards along these different rubrics. The results are as follows:

Clarity: Language Clarity

CC Rating: 3.5

Generally Common Core is clearly written, yet in many places it reverts to education

jargon such as ill-defined strategies and properties. High school standards are

phrased in generic ways that make it particularly difficult to understand their scope and

depth. Further, the standards lack clear prioritization at each grade and course level.

OK Rating: 4.0

Oklahoma standards, almost completely bereft of education jargon, stand out in their concision and clarity. Their clear prioritization of content at each grade and course level is superior. Directly embedded references to the test items clarify any possible ambiguities.

Clarity: Definiteness

CC Rating: 3.5

In elementary and middle grades Common Core standards tend to have well-defined

limits and scope. The high school standards are frequently lacking specificity and limits.

OK Rating: 4.0

Throughout all grades the standards tend to be well-defined in their scope, with examples providing additional clarity. The embedded references to OCCT item specifications and sample items remove any remaining ambiguities.

Clarity: Testability

CC Rating: 2.0

A large number of Common Core standards are infused with pedagogies that are difficult

to test. For example, grade 3 standards expect students to explain why the fractions are

equivalent, e.g., by using a visual fraction model and to compare two fractions with the

same numerator or the same denominator by reasoning about their size; this presents


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close to insurmountable difficulties to the test maker who must assess not only students

content knowledge but also the students thought processes.

OK Rating: 3.5

Standards are overwhelmingly written in terms of what students can do rather than how they go about doing it. Nevertheless, a few standards do impose difficult to test constraints such as students using variety of strategies to solve application problems.

Clarity: Total

CC Average Rating: 3.00

OK Average Rating: 3.83

Content: Primary School

CC Rating: 3.5

Primary school content is largely there but some content is absent. Fluency with division

of integers is deferred beyond the National Math Advisory Panel recommendation to

grade 6. Use of the standard algorithms is avoided until the grade when full fluency is

expected. Geometry content such as area of a triangle, or sum of triangle angles, is

deferred to middle school grades.

OK Rating: 3.5

Primary school content is largely there but some content is absent. Fluency with arithmetic is expected in proper grades, yet the standards are not explicit in expecting the use of the standard algorithms. Percents are mentioned in early grades much before students actually study them in grade 6.

Content: Middle School

CC Rating: 3.5

Prime numbers are poorly handled and prime decomposition is absent. Missing

conversion of fractional representations

OK Rating: 3.0

Prime numbers are poorly handled and prime decomposition is absent. Irrational numbers are absent.

Content: High School

CC Rating: 3.0

Course content is scattered and needs additional elaboration. Weak Geometry content,


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insisting on experimental pedagogy. Weak Algebra 2 content. Absence of majority of the

optional content necessary for STEM.

OK Rating: 3.5

High school course content is generally well defined. Algebra 2 includes conic sections

and logarithms. Absence of optional content necessary for STEM.

Content: Total

CC Rating: 3.33

OK Rating: 3.33

Reason: Total

CC Rating: 4

Multiple standards stress understanding across all grade levels. Poorly developed

estimation skills in primary grades.

OK Rating: 2.5

Reasoning and coherence are few and far between in primary grades, except through generic process standards. Estimation skills in elementary grades are well developed. High school standards demand explicit reasoning only in geometry, although cohesion is clearly present through topic clustering within high school courses.

Negative Qualities: False Doctrines

CC Rating: 3

Heavy and sustained stress on multiple approaches and multiple strategies. Heavy

overuse of aids such as visual fractions particularly in the higher primary and middle

grades.

OK Rating: 4

No major False Doctrine appear in Oklahomas old standards.

Negative Qualities: Inflation

CC Rating: 2.5

Wordy standards with a significant amount of jargon.

OK Rating: 4

Concise language with little jargon.


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Negative Qualities: Total

CC Rating: 2.75

OK Rating: 4

Summary: Totals for Both Math Standards Sets

CC OK

Language Clarity: Clarity, Definiteness, Testability 3.00 3.83

Content: Elementary, Middle, High School 3.33 3.33

Reasoning 4.00 2.50

Negative Qualities: False Doctrines, Inflation 2.75 4.00

Total Score 3.28 (B+)

3.40 (A-)

It is quite clear from the summary table that the standards are comparable in quality, although each set

has slightly different strengths and weaknesses, and there is little objective educational reason to select

one over the other

Independent Confirmation The validity of our analysis is confirmed by two independent sources.

First is the 2010 Fordham review that used a new set of criteria, in which Content and Rigor account for

70% of the overalls core and Clarity and Specificity account for only 30%. The 2010 review (attached as

an appendix) offers less evaluation precision than the 2005 process and somewhat different stresses,

yet it found them comparable overall, with Oklahoma scoring B+ to Common Cores A-. Common Core

lost points on Clarity and Specificity, while Oklahoma PASS lost points on Content and Rigor. Highlights

of the 2010 Fordham review follow.

Common Core:

Overview: The expectations are generally well written and presented and cover much mathematical content with both depth and rigor. But, though the content is generally sound, the

standards are not particularly easy to read, and require careful attention on the part of the reader.

The development of arithmetic in elementary school is a primary focus of these standards and that

content is thoroughly covered. The often-difficult subject of fractions is developed rigorously,

with clear and careful guidance. The high school content is often excellent, though the

presentation is disjointed and mathematical coherence suffers. In addition, the geometry standards

represent a significant departure from traditional axiomatic Euclidean geometry and no

replacement foundation is established.


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Clarity and Specificity: ... Many standards are clear and specific Though the standards are not succinct, which detracts from the ease of reading, careful reading reveals that they are generally

both literate and mathematically correcta rare combination in standards Unfortunately, despite the inclusion of examples, some standards are not specific enough to determine the intent,

and they are subject to quite a bit of interpretation on the part of the reader The high school standards, in particular, are often too broadly stated to interpret In addition, in high school, the presentation is not always coherent. 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.

Content and Rigor: The standards have many strong features and cover a lot of rich mathematics they are generally mathematically sound, and the content is usually presented coherently. Arithmetic is well covered Fractions are developed rigorously and with a great deal of specificity The high school material, despite its sometimes incoherent presentation, is often strong. The coverage of linear equations, which begins in eighth grade, includes some

rigorous standards High school geometry has very good coverage of content, and proofs are included throughout the standards. There is, however, no obvious foundation for geometry, in

part because axioms and postulates are never mentioned. Instead, the standards approach

geometry through transformations. Unfortunately, it takes a good deal of work in Euclidean

geometry (based on axioms) to work with transformations.

Oklahoma PASS:

Overview: Oklahomas standards are generally strong. They are well written, and K-8 grades are introduced with a section that focuses and clarifies the standards by providing explicit guidance

on priorities. The standards are not rigorous enough in places, however, and some important

content is missing.

Clarity and Specificity: The standards are generally clear and easy to read. They make frequent

and excellent use of examples to clarify the meaning of the statements The clarity is also greatly enhanced by the inclusion of the major concepts, explained above, which specify the

topics that should be taught in depth. These provide the standards with focus and are clear and

explicit. Taken together, these earn Oklahoma a score of three points out of three for Clarity and

Specificity.

Content and Rigor: In grades K-8, Oklahoma has set priorities in an exemplary way Some of the development of arithmetic is very strong There are some problems with the development of arithmetic. The major concepts clearly state that fluency with whole number addition,

subtraction, multiplication, and division is required. However A rigorous treatment of computational fluency requires the standard algorithms, but the standards never specify that

students know them and are able to compute with them The development of the arithmetic of fractions similarly fails to specify standard methods for computation and instead requires a

variety of strategies. In addition, standards are provided for only three high school courses and some STEM-ready material is missing, particularly trigonometry beyond the basic definitions

Oklahomas standards cover most of the essential content well, and they set priorities beautifully. There are some weaknesses in the areas of arithmetic, the study of rates, and the

inclusion of STEM-ready material. These shortcomings result in a Content and Rigor score of

five points out of seven.

As should be clear by now, both sets of standards are quite strong, yet both have some weaknesses that

ought to be addressed.


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A second independent verification of the strength and coherence of Oklahoma PASS appears in a 2012

study by William Schmidt and Richard Houang. In that study, among other things, the authors analyze

state standards and compute their congruence with Common Core in an effort to correlate that similarity

to NAEP achievement. What is important for our purpose is the following table from their paper, which

shows Oklahoma PASS to be in the top ten state standards most aligned with the Common Core.

In summary, it is clear that Oklahoma PASS standards are on par with the Common Core. One can argue

whether they are slightly weaker or stronger in specific areas, yet all those who seriously studied both

agree they are of similar strength. The big difference is that Oklahoma can modify and improve its PASS,

but it cannot modify the Common Core.

Recommendation:

Oklahoma should return to its PASS standards in mathematics and embark on a deliberate minor revision

to strengthen them, and supplement them with optional STEM-oriented high school courses. In the

interim, Oklahoma can continue to use its existing tests aligned with PASS. Finally, Oklahoma should

seek approval from its public colleges and universities to certify the PASS standards as college-ready.

Given the strong high school content already present this should be a relatively easy task.

Zeev Wurman is visiting scholar with the Hoover Institution at Stanford University. Between 2007 and

2009 Wurman served as a senior policy adviser with the Office of Planning, Evaluation and Policy

Development at the U.S. Department of Education in Washington, D.C. Wurman analyzed the

mathematics drafts Common Core for the Pioneer Institute and for the State of California. He served on

the California Academic Content Standards Commission that evaluated the suitability of the Common

Core standards for California. He has published numerous professional and opinion articles about

education and about the Common Core.

THE STATE OF STATE STANDARDSAND THE COMMON COREIN 2010 264

OklahomaMathematics

DOCUMENTS REVIEWED

Priority Academic Student Skills: Math Content Standards. Spring 2009. Accessed from: http://sde.state.ok.us/Curriculum/PASS/Subject/math.pdf

OverviewOklahomas standards are generally strong. They are well written, and K-8 grades are introduced with a section that focuses and clarifies the standards by providing explicit guidance on priorities. The standards are not rigorous enough in places, however, and some important content is missing.

General OrganizationOklahoma organizes its K-8 standards into five content standards that are common across grade levels: Algebraic Rea-soning, Number Sense and Operations, Geometry, Measurement, and Data Analysis. Each strand is then divided into grade-specific standards.

In addition, Oklahoma introduces its K-8 standards with three major concepts, which are the three most important topics students must master in each grade. For example:

Develop quick recall of multiplication facts and related division facts (fact families) and fluency with whole number multiplication.

Develop an understanding of decimals and their connection to fractions.

Develop an understanding of area and acquire strategies for finding area of two-dimensional shapes (grade 4)

The high school standards are organized similarly, with two important differences. First, the content is divided into three courses, rather than five content strands. Second, each course is introduced with a list of major concepts (which should be taught in depth) and maintenance concepts (which have been taught previously and are prerequisites).

Clarity and SpecificityThe standards are generally clear and easy to read. They make frequent and excellent use of examples to clarify the meaning of the statements. For example, the parenthetical examples in this standard serve to make it clear exactly what students are supposed to be able to do:

Identify, describe, and analyze functional relationships (linear and nonlinear) between two variables (e.g., as the value of x increases on a table, do the values of y increase or decrease, identify a positive rate of change on a graph and compare it to a negative rate of change) (grade 7)

Similarly, the example further clarifies this standard:

Write and solve one-step equations with one variable using number sense, the properties of operations, and the properties of equality (e.g., -2x+4=-2) (grade 7)

GRADEClarity and Specificity: 3/3Content and Rigor: 5/7

Total State Score: 8/10

(Common Core Grade: A-)B+

Appendix: Fordham 2010 Review of PASS Mathematics


Oklahoma Mathematics

The clarity is also greatly enhanced by the inclusion of the major concepts, explained above, which specify the top-ics that should be taught in depth. These provide the standards with focus and are clear and explicit. Taken together, these earn Oklahoma a score of three points out of three for Clarity and Specificity. (See Common Grading Metric, Appendix A.)

Content and RigorContent Priorities

In grades K-8, Oklahoma has set priorities in an exemplary way. The major concepts introducing each grade are stated as the major goals for the year and specified as concepts that should be taught in depth. They are explicit and clear. For example, major concepts for the fourth grade are:

Develop quick recall of multiplication facts and related division facts (fact families) and fluency with whole number multiplication (grade 4)

Develop an understanding of decimals and their connection to fractions (grade 4)

Develop an understanding of area and acquire strategies for finding area of two- dimensional shapes (grade 4)

These effectively and appropriately set priorities. Standards on less important topics, such as tessellations, will not be misinterpreted as important content.

In each grade, 1-6, two out of three of the major concepts deal with numbers and computations, giving mastery of arith-metic appropriate priority.

Content Strengths

Some of the development of arithmetic is very strong. For example, the following standard explicitly requires memoriza-tion of basic facts:

Demonstrate fluency (memorize and apply) with basic multiplication facts up to 10 x 10 and the associated division facts (e.g., 5 x 6 = 30 and 30 6 = 5) (grade 3)

Other strengths include explicit mention of common denominators and the rigor of the high school Geometry course.

Content Weaknesses

There are some problems with the development of arithmetic. The major concepts clearly state that fluency with whole number addition, subtraction, multiplication, and division is required. However, the standards themselves do not ad-equately support such fluency. A rigorous treatment of computational fluency requires the standard algorithms, but the standards never specify that students know them and are able to compute with them. For example, the capstone stan-dard for multiplication, which has fluency with multiplication as a major concept, is:

Estimate and find the product of up to three-digit by three-digit using a variety of strategies to solve application problems (grade 4)

As the capstone standard for multiplication, this lacks the rigor required for true fluency with multiplication. Worse, by allowing students to use a variety of strategies, rather than requiring mastery of the standard algorithms, this standard may actually undermine such fluency by allowing students to rely on inefficient techniques.

The development of the arithmetic of fractions similarly fails to specify standard methods for computation and instead requires a variety of strategies.

There are some other weaknesses in the standards. Calculators, while not prevalent until high school, are a suggested material beginning in first grade. The inverse nature of addition and subtraction and of multiplication and division are not mentioned. Other missing content includes work with rates and rational numbers as repeating decimals (though this is mentioned in the glossary).

In high school, the standards for the Algebra courses become noticeably less clear, and there is a tendency to rely on graphing calculators. This is illustrated by the following standard:


Oklahoma Mathematics

Graph a quadratic function and identify the x- and y-intercepts and maximum or minimum value, using various methods and tools which may include a graphing calculator (Algebra II)

In addition, standards are provided for only three high school courses and some STEM-ready material is missing, par-ticularly trigonometry beyond the basic definitions. However, the standards state explicitly that students planning to continue their mathematics education should study additional advanced mathematics topics such as trigonometry

Oklahomas standards cover most of the essential content well, and they set priorities beautifully. There are some weak-nesses in the areas of arithmetic, the study of rates, and the inclusion of STEM-ready material. These shortcomings result in a Content and Rigor score of five points out of seven. (See Common Grading Metric, Appendix A.)

The Bottom LineOklahomas standards are generally clear and well presented. Standards are briefly stated and frequently include ex-amples, making them easier to read and follow than Common Core. In addition, the high school content is organized so that standards addressing specific topics, such as quadratic functions, are grouped together in a mathematically coher-ent way. The organization of the Common Core is more difficult to navigate, in part because standards dealing with related topics sometimes appear separately rather than together.

While Oklahomas standards provide well-organized high school courses, they are missing some of the advanced content for high school that is covered in Common Core. In addition, the coverage of arithmetic displays some serious weaknesses. Common Core explicitly requires standard methods and procedures, and the inclusion of these important details would enhance Oklahomas standards.

Comparison of PASS to Common Core in Mathematics - Ze'ev Worman

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