ERIC ED484399: Sequenced Benchmarks for K-12 Mathematics

2004·0.13 MB·English

by  ERIC

SEQUENCED BENCHMARKS FOR
K–12 MATHEMATICS

Regional Educational Laboratory
Contract #ED-01-CO-0006
Deliverable #2004-02

Prepared by
John S. Kendall
Keri DeFrees
Jill Williams
© 2004 McREL

This document has been funded at least in part with federal funds from the U.S. Department of Education under
contract number ED-01-CO-0006. The content of this publication does not necessarily reflect the views or policies
of the Department of Education nor does mention of trade names, commercial products, or organizations imply
endorsement by the U.S. Government.
TABLE OF CONTENTS

PREFACE......................................................................................................................................iii

INTRODUCTION..........................................................................................................................1
Source Documents......................................................................................................................1
Method........................................................................................................................................2
How This Document Can Be Used.............................................................................................5

SEQUENCED BENCHMARKS
Communication
1. Communicating about Mathematics.................................................................8
Properties/Concepts of Numbers
2. Exponents/Logarithms/Roots............................................................................9
3. Fractions..........................................................................................................10
4. Integers............................................................................................................11
5. Ratio/Proportion/Percent.................................................................................12
6. Whole Numbers – Procedures........................................................................13
7. Whole Numbers – Relationship......................................................................14
Computation
8. Addition/Subtraction – Processes...................................................................15
9. Addition/Subtraction – Properties and Order of Operations...........................16
10. Multiplication/Division...................................................................................17
Measurement
11. Basic and Linear Measures.............................................................................18
12. Perimeter/Area/Volume..................................................................................19
13. Units of Measurement.....................................................................................20
14. Volume and Capacity......................................................................................21
15. Weight and Mass.............................................................................................22
Geometry
16. Lines and Angles.............................................................................................23
17. Motion Geometry............................................................................................24
18. Shapes and Figures.........................................................................................25
Data
19. Data Organization and Display – Methods.....................................................26
20. Data Organization and Display – Reading and Interpretation........................27
21. Data Collection and Sampling........................................................................28
22. Data Distribution ............................................................................................29
Probability
23. Likelihood/Chance/Certainty..........................................................................30
24. Reasoning and Predicting from Data..............................................................31

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Algebra
25. Graphs and Graphing Systems – Correspondence to Algebra........................32
26. Graphs and Graphing Systems – Solving Problems.......................................33
27. Equations and Inequalities – Representation..................................................34
28. Equations and Inequalities – Solving Problems..............................................35
29. Expressions.....................................................................................................36
30. Patterns............................................................................................................37
31 Variables.........................................................................................................38

BIBLIOGRAPHY.........................................................................................................................39

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PREFACE
This report is one in a series of reference documents designed especially to assist those
who are directly involved in the revision and improvement of content standards. It presumes a
basic understanding of the purposes for standards and the process of standards review. It is
important to note that it is intended to be a desktop reference as opposed to a practical guide.
Readers desiring more background and context for the work described here may wish to consult
A Technical Guide for Revising or Developing Standards and Benchmarks (Kendall, 2001).

This report presents mathematics benchmarks organized into instructional sequences by
topic. Each sequence is based upon an analysis of the order of content as it appears among a set
of highly rated state standards. This report does not assign content to specific grades or
recommend that content be taught at specific grades; however, each item within a sequence
includes information on the grades at which that content is found within the state standards
documents analyzed. The report is intended to inform and guide state or district curriculum
directors or others who, starting from content that is placed within broad grade bands, need to
assign specific grades to benchmarks or objectives. Alternatively, those who have benchmarks or
objectives already placed at specific grades may wish to compare the sequence of content
outlined in this report in order to confirm the commonality of that sequence. Teachers may find
the sequence of content useful for organizing instruction.

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INTRODUCTION

At the beginning of the standards movement in the early to mid 1990s, most states
delineated the content of standards as benchmarks or objectives for a range of grades, such as
K–4, 5–8, and 9–12. As states have revised their standards, usually as part of an established
review cycle, they have described them in narrower grade ranges, for example, K–2, 3–5, 6–8,
and 9–12. Some states have taken this process a step further by describing content for each level
from kindergarten through grade 8. A significant number of states, however, still do not provide
such grade-by-grade distinctions in standards.

This lack of grade-level benchmarks or objectives provides districts with some freedom
to define the local curriculum yet, for many districts, it also presents an immediate problem. In
order to implement standards, districts must find a way to translate these grade-range
benchmarks into meaningful benchmarks or objectives for day-to-day schooling at specific
grades. Unfortunately, there has been little guidance available to districts as they undertake this
process. Documents in each subject area produced by national professional organizations might
be considered the highest authority in this regard, but none provides grade-by-grade
recommendations. It seems likely that these organizations were deterred by the lack of research
that guides the placement of content at specific grade levels. In addition, many organizations
avoid the assignment of content to a grade because it is likely to be seen as overly prescriptive.
Yet, the problem for a school district remains. In most schools, content must be assigned to a
specific grade because it must be taught at a specific grade.

Although it seems likely that there will never be adequate research to support the
assignment of specific content definitively to a specific grade, we can infer that the sequence of
content in state standards documents reflects the authors’ beliefs about how the ideas and skills
that students learn in different grades and topics depend on and support one another. To assist
educators in the development of grade-by-grade benchmarks in mathematics, this report presents
information about the sequence in which knowledge and skills appear in a small set of
mathematics documents that have been highly rated by a number of national organizations. The
method employed here, and discussed in detail below, entails the analysis of exemplary
standards documents for patterns of content sequence by three individuals, and the review of the
established content sequences by two reviewers experienced in mathematics instruction. This
method has been used previously by Mid-continent Research for Education and Learning

(McREL) to identify the sequence of content in the English language arts (Kendall, Snyder, &
Flynn, 2003) and science (Kendall, DeFrees, & Richardson, 2002). Users of the previous studies
have reported that the sequencing information has been useful as a comparison with their state
standards and as a help in developing district objectives. Teachers also report consulting the
sequenced topics to help them organize instruction.

SOURCE DOCUMENTS
Three evaluation reports were used to help select the state standards documents analyzed
in this study. One report was the American Federation of Teachers’ (AFT) Making Standards
Matter (1998), which includes ratings of the state standards in terms of specificity and clarity.
Another perspective on state standards was published by the Fordham Foundation in the report
State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of
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Columbia, and Japan (Raimi & Braden, 1998). Finally, the Council for Basic Education
evaluated mathematics documents across the states in Great Expectations: Defining and
Assessing the Rigor in State Standards for Mathematics and English Language Arts (Berman &
Joftus, 1998).

Although a variety of state standards documents have been highly rated for their
mathematics standards, five state documents that were highly rated by all three organizations and
that identify objectives at each grade level from kindergarten through 8th grade were selected as
source documents for this analysis:

• Utah’s Core Curriculum Standards: Mathematics (1994)
• Mathematics Content Standards for California Public Schools (1990)
• Mathematics Standards of Learning for Virginia Public Schools (1995, June)
• Ohio’s Model Competency-Based Mathematics Program (1990, November)
• West Virginia Programs of Study: Instructional Goals and Objectives (1995,
June)

Not since these ratings appeared in the mid 1990s has there been any comparable review
of state standards documents by multiple organizations. Thus, in order to select standards
documents for this report that are widely endorsed for their quality, we were limited to the
documents reviewed in the mid 1990s, which are now some 10 to 15 years old. However, we
believe that these documents fairly represent the current state of content standards in
mathematics. This view is based on our continued familiarity with state standards over the last 10
years as we have conducted our own studies or reviews for state and district clients. For this
sequencing study, we determined that it was preferable to select standards documents that were
highly rated by multiple organizations, although somewhat dated, than to select documents that
had been highly rated by only a single organization.

METHOD
In order to track mathematics content across these state documents, each of which varies
somewhat in the content described, a uniform set of benchmarks was required as a basis for
comparison. McREL’s online standards database was used for this purpose. The database, the
online equivalent of Content Knowledge: A Compendium of Standards and Benchmarks for K–
12 Education (3rd ed.) (Kendall & Marzano, 2000), provides a synthesis of 137 standards

documents representing 14 content areas. By using the benchmarks in this document as a basis
for comparison, McREL analysts were able to track the presence or absence of mathematics
content in each of the state documents in order to find content that was addressed in common.

It should be noted that benchmarks in the Compendium were not used in the actual
sequencing of content; they were merely used as a means for tracking and organizing the content
of the five standards documents analyzed. On some occasions, in fact, benchmarks that appear in
the same grade band within the Compendium — and thus indicate no preferred sequence of
instruction — were found to have a preferred sequence of instruction based upon evidence from
the state documents analyzed in this study.

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An example can be found under the topic of “integers.” In the Compendium, the concept
of the characteristics and properties of integers, as well as the concept of the role of positive and
negative integers, appear in benchmarks at the 6–8 grade range; at this same level appears a
benchmark regarding addition, subtraction, multiplication, and division of integers. In the course
of our analysis, we found evidence that three states consistently address the first two topics in a
grade prior to addressing the skills of addition, subtraction, multiplication, and division of
integers. In order to uncover this sequence and many others like it, it was necessary to
deconstruct the more broadly written benchmarks of the Compendium into their component
parts, then combine those content elements that were found to be shared among the source
documents and could be established in a sequence.

In addition to identifying a set of benchmarks for comparison, it also was necessary to
use a set of topics to help organize this information. A topic is a level of organization that is
more specific than a standard, but more general than a benchmark. A topic names an idea that
organizes a small collection of benchmarks or objectives. In McREL’s Compendium, several
topics are commonly found within a standard, and each topic organizes two or more benchmarks
(for a description of the process of topic development and samples in mathematics and language
arts, see Kendall, 2000). The list of topics used here, organized by sub-discipline, is presented in
Exhibit 1.

The articulation of benchmarks under each topic in this report was based entirely on the
presence of that content in the state standards source documents specified earlier. Each state
document was reviewed for any sequence information it provided by topic. Sequence
information is defined as the presence of a concept or skill in a grade that is topically related to
another concept or skill at a higher or lower grade. That is, in any given document, two or more
topic-related concepts must appear separated by at least one grade to be considered informative
in the development of articulated content. Simply put, if two benchmarks addressing a given
topic appeared in the same grade, it was inferred that the authors of the document did not
consider the difference between content to be significant enough that the benchmark or objective
should be addressed in separate grades. In such a case, the state document did not contribute
information about the sequence of content.

Thus, each articulation under a topic was established by sufficient evidence from the state
standards documents. Closely related ideas beneath a topic often appeared at different grade
levels in each of the standards documents, but as long as these ideas were presented in the same
order in each of at least three documents, a sequence was established. For example, the idea that
students should understand the defining properties of three dimensional figures was found in
three of the standards documents at grades 6 or 7 (see Topic 18, Shapes and Figures). Four of the
documents also addressed, at different grades, the concept that students should be able to predict
and verify the effects of combining, subdividing, and changing basic shapes, yet each document
introduced this concept prior to the grade at which it introduced the idea that students should
understand the defining properties of three dimensional figures, thereby establishing a sequence
of information.

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Exhibit 1. Summary of Mathematics Topics
Addition/subtraction Multiplication/division
Analytic geometry Number systems
Basic and linear measures Patterns
Communicating about mathematics Perimeter/area/circumference
Data collection and sampling Perimeter/area/volume
Data distribution Permutations/combinations
Data organization and display Precision/accuracy
Decimals Problem-solving
Equations and inequalities Proof and empirical verification
Estimation Rate
Experimental probability Ratio/proportion/percent
Exponents/logarithms/roots Reasoning and predicting from data
Expressions Representing problems
Factors/multiples/primes Sequences and series
Formulating/testing hypotheses Sets
Fractions Shapes and figures
Functions Similarity and congruence
Graphs and graphing systems Solution strategies
Integers Triangles
Likelihood/chance/certainty Trigonometry
Lines and angles Units of measurement
Mathematical enterprise Uses of mathematics
Mathematical reasoning Variables
Mathematics, science, and technology Vectors
Matrices Volume/capacity
Measurement estimation Weight and mass
Motion geometry/transformations Whole numbers/place value/numeration

In other words, if the presence of sequenced content was established in any one
document, the same relative sequence ⎯ that is, the sequence from earlier to later grade ⎯ had

to be supported by at least two additional documents in order to be considered useful relative to
the articulation of content within the topic. One additional requirement was established to ensure
that the sequences were meaningful. If a sequence of content was found in the reverse order in
any other standards document, it was removed from consideration. For example, one standards
document anticipated that students should predict and verify the effect of combining,
subdividing, and changing basic shapes in the grade prior to their understanding that shapes can
be congruent or similar. Another standards document reversed the grade sequence; that is,
students were first expected to understand that shapes could be congruent or similar in the grade
before they were expected to be able to predict and verify the effect of operations on shapes.
Because of this reversal of grades, no sequence of content was established, despite evidence of
content sequence available from other states.

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The product of this analysis across the standards documents is presented in 31 topic
sequences. Each sequence represents the articulation of that aspect of a topic that was supported
by evidence from state standards documents. There are 27 topics addressed in the 31 topic
sequences because some topics involve more than one sequence to completely cover the
associated content. For example, the topic of Addition/Subtraction has one sequence that
addresses the process itself (Topic 8) and another that addresses knowing the properties and
order of operations (Topic 9).

It should be understood that each of the topics sequenced in this document has more
benchmarks associated with it, both in the Compendium and in the state standard documents,
than appear in this document. Only those benchmarks whose articulation could be supported by
the state standards documents appear here.

Similarly, there are many topics that are not articulated in this study because there was
inadequate support from the state standards documents. Of the 54 topics identified for the
mathematics standards in McREL’s standards database, approximately 57 percent could be
presented in some sequence.

The topics that could be sequenced in this report comprise 123 unique benchmarks.
Readers should be aware that the content described in 100 benchmarks of McREL’s
Compendium could not be sequenced using the methodology adopted for this study. The fact that
this content could not be sequenced of course says nothing about its relative importance. For the
benchmark content that could not be sequenced in this report, readers will need to employ other
strategies to determine grade level placement (for a discussion of such strategies, see Kendall,
2001, pp. 16–19). This report will not be useful for helping to identify all the significant content
in mathematics. For such a purpose, readers might consult the highly rated state standard
documents listed above, or consult a synthesis of these highly rated documents (see Kendall,
Snyder, Schintgen, Wahlquist, & Marzano, 1999).

HOW THIS DOCUMENT CAN BE USED

This collection of content sequenced by topic should prove useful for those districts and
schools that seek to assign their state’s grade-range content to specific grades for instruction. It is
quite likely that the topics in the pages that follow are addressed in nearly every state’s
mathematics standards. Reviewing each topic in turn, users can compare the content to their own
state standards document.

Of first interest during such a review is whether all content identified in this study for a
given topic also can be found in the state or district standards being compared. Users may likely
determine that any content not found should be added to their own standards because this content
is commonly found in highly rated documents. Once the scope of content for a topic has been
reviewed, the content should be examined for grade placement. For this work, two kinds of
information are available. First, the study indicates where content, in the form of benchmarks,
appears in sequence relative to other content in the same topic across three or more states.
Second, for each benchmark in a topic, the grade or grades at which that content is found in the
state documents is also provided.
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