The Project
2061 Analysis Procedure for Mathematics Curriculum Materials
Introduction
Deciding which curriculum
materials to use is one of the most important professional judgments
that educators make. Textbook adoption committees make recommendations
that influence instruction for years to come, and the daily
decisions that teachers make about which teaching units or chapters
to use and how to use them largely determine what and how students
will be expected to learn.
Such important decisions
require a valid and reliable method for evaluating the quality
of curriculum materials. Even an indepth review of the topics
covered by a textbook or a teaching unit may not be sufficient
to determine whether the material will actually help students
learn that content. What is needed is a manageable process for
examining curriculum materials that gets below the surface by
focusing intensely on the appropriateness of content and the
utility of instructional design.
With funding from
the National Science Foundation and in collaboration with hundreds
of K12 teachers, curriculum specialists, teacher educators,
scientists, and materials developers, Project 2061 of the American
Association for the Advancement of Science (AAAS) has been developing
a process for analyzing curriculum materials. Field tests suggest
that Project 2061's curriculumanalysis procedure will not only
serve the materials adoption needs of the schools but also help
teachers revise existing materials to increase their effectiveness,
guide developers in the creation of new materials, and contribute
to the professional development of those who use it.
Specific
Learning Goals Are Key
Until recently, there
was nothing against which to judge appropriateness of content
and utility of instructional design. Now, as a result of the
standardsbased reform movement in education, these judgments
can be made with a high degree of confidence. In mathematics,
for example, the appearance of Science for All Americans
(AAAS, 1989), Curriculum and Evaluation Standards for School
Mathematics (NCTM, 1989), Benchmarks for Science Literacy
(AAAS, 1993), and more recently Principles and Standards
for School Mathematics (NCTM, 2000) has made it possible
to make more thoughtful decisions about curriculum materials
than ever before.
Although the Project
2061 curriculumanalysis procedure was developed using the learning
goals in Benchmarks and the mathematics and science standards,
subsequent work has indicated that some state education frameworks
also can be used. Indeed, the process would seem to apply to
any K12 school subject for which specific learning goals have
been agreed upon. These goals must be explicit statements of
what knowledge and skills students are expected to learn, and
they must be precise. Vague statements such as "students
should understand fractions" are not adequate. Instead,
consider this following NCTM Standards and corresponding Benchmarks:
NCTM
Standards Used in the Analysis and AAAS Benchmarks Comparisons
NCTM
Standard
Representing
Functions
 Select appropriate
representations (numerical, graphical, verbal, and symbolic)
for the functions and relations embedded in quantitative
situations, convert flexibly among representations,
interpret representations, and use them to interpret
the situations represented.

AAAS
Benchmarks
 9B68#3:
Graphs can show a variety of possible relationships
between two variables. As one variable increases uniformly,
the other may do one of the following: increase or decrease
steadily, increase or decrease faster and faster, get
closer and closer to some limiting value, reach some
intermediate maximum or minimum, alternately increase
and decrease indefinitely, increase or decrease in steps,
or do something different from any of these.
 9B912#4:
Tables, graphs, and symbols are alternative ways of
representing data and relationships that can be translated
from one to another.
 12D68#1:
Organize information in simple tables and graphs and
identify relationships they reveal.
 12D68#2:
Read simple tables and graphs produced by others and
describe in words what they show.
 12C912#2:
Use computers for producing tables and graphs and for
making spreadsheet calculations.
 12D912#1:
Make and interpret scale drawings.

Modeling
with Functions
 Model a
wide range of phenomena with a variety of functions
including linear, quadratic, exponential, rational,
trigonometric, and recursively defined functions and
recognize that a particular type of function can model
many different situations.

 11B68#2:
Mathematical models can be displayed on a computer and
then modified to see what happens.
 11B68#3:
Different models can be used to represent the same thing.
What kind of a model to use and how complex it should
be depends
on its purpose. The usefulness of a model may be limited
if
it is too simple or if it is needlessly complicated.
Choosing a useful
model is one of the instances in which intuition and
creativity
come into play in science, mathematics, and engineering.
 9B68#3:
Graphs can show a variety of possible relationships
between two variables. As one variable increases uniformly,
the other may do one of the following: increase or decrease
steadily, increase or decrease faster and faster, get
closer and closer to some limiting value, reach some
intermediate maximum or minimum, alternately increase
and decrease indefinitely, increase or decrease in steps,
or do something different from any of these.
 11B912#1:
The basic idea of mathematical modeling is to find a
mathematical relationship that behaves in the same ways
as the objects or processes under investigation. A mathematical
model may give insight about how something really works
or may fit observations very well without any intuitive
meaning.
 11B912#2:
Computers have greatly improved the power and use of
mathematical models by performing computations that
are very long, very complicated, or repetitive. Therefore
computers can show the consequences of applying complex
rules or of changing the rules. The graphic capabilities
of computers make them useful in the design and testing
of devices and structures and in the simulation of complicated
processes.

Representing
Variable Quantities
 Represent
situations that involve variable quantities with expressions,
equations, inequalities, and systems of equations using
a variety of equivalent forms.

 9B912#2:
Symbolic statements can be manipulated by rules of mathematical
logic to produce other statements of the same relationship,
which may show some interesting aspect more clearly.
Symbolic statements can be combined to look for values
of variables that will satisfy all of them at the same
time.
 9B912#4:
Tables, graphs, and symbols are alternative ways of
representing data and relationships that can be translated
from one to another.
 11C912#4:
Graphs and equations are useful (and often equivalent)
ways for depicting and analyzing patterns of change.

Operating
with Symbols and Equations
 Become fluent
in generating equivalent expressions for simple algebraic
expressions and in solving linear equations and inequalities.
 Develop
fluency operating on polynomials, vectors, and matrices
using byhand operations for the simple cases and using
technology for more complex cases.

 12B68#6:
Insert instructions into computer spreadsheet cells
to program arithmetic calculations.
 9B912#2:
Symbolic statements can be manipulated by rules of mathematical
logic to produce other statements of the same relationship,
which may show some interesting aspect more clearly.
Symbolic statements can be combined to look for values
of variables that will satisfy all of them at the same
time.
 9B912#5:
When a relationship is represented in symbols, numbers
can be substituted for all but one of the symbols and
the possible value of the remaining symbol computed.
Sometimes the relationship may be satisfied by one value,
sometimes more than one, and sometimes maybe not at
all.
 12B912#2:
Find answers to problems by substituting numerical values
in simple algebraic formulas and judge whether the answer
is reasonable by reviewing the process and checking
against typical values.
 12B912#3:
Make up and write out simple algorithms for solving
problems that take several steps.
 12C912#2:
Use computers for producing tables and graphs and for
making spreadsheet calculations.

The
Project 2061 CurriculumAnalysis Procedure
At its simplest level,
the Project 2061 curriculumanalysis procedure involves the
following five steps:
 Identify specific
learning goals to serve as the intellectual basis for the
analysis. This is done before beginning to examine any curriculum
materials. The source for appropriate goals can be national
standards or documents such as those mentioned above, state
or local standards and curriculum frameworks, or sources like
them. To be useful, the goals must be precise in describing
the knowledge or skills they intend students to have. If the
set of goals is large, a representative sample of them should
be selected for purposes of analysis.
 Make a preliminary
inspection of the curriculum materials to see whether they
are likely to address the targeted learning goals. If there
appears to be little or no correspondence, the materials can
be rejected without further analysis. If the outlook is more
positive, go on to a content analysis.
 Analyze the curriculum
materials for alignment between content and the selected learning
goals. The purpose here is to determine, citing evidence from
the materials, whether the content in the material matches
specific learning goals, not just whether the topic headings
are similar. At the topic level, alignment is never difficult,
since most topics, variables, equations, and so forth, lack
specificity making them easy to match. If the results of this
analysis are positive, then reviewers can take the next step.
 Analyze the curriculum
materials for alignment between instruction and the selected
learning goals. This involves estimating the degree to which
the materials (including their accompanying teacher's guides)
reflect what is known generally about student learning and
effective teaching and, more important, the degree to which
they support student learning of the specific knowledge and
skills for which a content match has been found. Again, evidence
from the materials must be shown.
 Summarize the
relationship between the curriculum materials being evaluated
and the selected learning goals. The summary can take the
form of a profile of the selected goals in terms of the content
and instruction criteria, or a profile of the criteria in
terms of the selected goals. In either case, a statement of
strengths and weaknesses should be included. With this information
in hand, reviewers can make more knowledgeable adoption decisions
and suggest ways for improving the examined materials.
In addition to its
careful focus on matching content and instruction to very specific
learning goals, the Project 2061 procedure has other features
that set it apart. For example, its emphasis on collecting explicit
evidence (citing page numbers and other references) of a material's
alignment with learning goals adds rigor and reliability to
decisions about curriculum materials. Similarly, the Project
2061 procedure calls for a team approach to the analytical task,
thus providing opportunities for reviewers to defend their own
judgments about materials and to question those of other reviewers.
These and other characteristics help make participation in the
analytical process itself a powerful professional development
experience.
The
Procedure in Detail
To provide a better
sense of how the procedure works, the following describes in
more detail each step in the procedure. The description pays
particular attention to the various criteria used to evaluate
the instructional effectiveness of materials.
Identify specific
learning goals to serve as the intellectual basis for the analysis.
After reviewers have agreed upon a set of learning goals as
a framework for the analysis, the task is then to choose specific
learning goals that will serve as the focus of further study.
When evaluating standalone
curriculum units that cover a relatively short period of time,
it might be possible and worthwhile to analyze all of the learning
goals that appear to be targeted by the material. However, in
the evaluation of yearlong courses or multiyear programs, this
becomes impractical. Therefore, a crucial step in the analysis
procedure is the sampling of a few learning goals that will
lead to valid and reliable generalizations about the material.
Sampling of standards
should be representative of the whole set of goals specified
in the framework or standards being applied and should reflect
the reviewers' needs. For example, if the review committee's
task is to select a course in high school algebra that is aligned
with a state mathematics framework or NCTM Standards,
it might identify a sample of learning goals from important
topic areas (e.g., number systems, equations, graphs, functions)
and include learning goals that reflect different types of knowledge
(e.g., skills, conceptual understanding, problem solving). When
examining elementary or middle school mathematics materials,
one would probably want to broaden the range of learning goal
statements examined to include important strands in mathematics
(e.g., number, geometry, algebra, and statistics).
Make a preliminary
inspection of the curriculum materials to see whether they are
likely to address the targeted learning goals. Once learning
goal statements have been selected, the next step is to make
a first pass at the materials to identify those whose content
appears to correspond reasonably well to the learning goals.
Materials that do not meet these initial criteria are not analyzed
further.
Reviewers then examine
materials on the shortened list more carefully to locate and
record places where each selected learning goal seems to be
targeted (e.g., particular readings, experiments, discussion
questions). If several sightings are found for some or all of
the sample learning goals in the material, then these sightings
will be looked at more carefully in subsequent steps of the
analysis. If, on the other hand, sightings cannot be found for
a significant number of the sample learning goals, then the
material is dropped from the list.
Analyze the curriculum
materials for alignment between content and the selected learning
goals. This analysis is a more rigorous examination of the
link between the subject material and the selected learning
goals and involves giving precise attention to both ends of
the match; the precise meaning of the learning goal on one end,
and the precise intention of the material on the other.
With respect to each
of the sampled learning goals, the material is examined using
such questions as:
 Does the content
called for in the material address the substance of a specific
learning goal or only the learning goal's general "topic"?
 Does the content
reflect the level of sophistication of the specific learning
goal, or are the activities more appropriate for targeting
learning goals at an earlier or later grade level?
 Does the content
address all parts of a specific learning goal or only some?
(While it is not necessary that any particular unit would
address all of the ideas in a learning goal or standard, the
K12 curriculum as a whole should do so. The purpose of this
question is to provide an account of precisely what ideas
are treated.)
In addition, an attempt
is made to estimate the degree of overlap between the material's
content and the set of learning goals of interest. Thus, this
step in the analysis is designed to answer questions regarding
the material's inclusion of content that is not required for
reaching mathematics literacy and the extent to which the material
distinguishes between essential and nonessential content. (While
distinguishing content essential for literacy from nonessential
content in material might seem to be a luxury, it assists teachers
in determining the range of students for which the material
can be used. Identifying the nonessential material makes it
easier for the teacher to direct better students to enrichment
activities and allows students themselves to avoid overload
from ideas that go beyond what is vital.)
Analyze the curriculum
materials for alignment between instruction and the selected
learning goals. The purpose here is to estimate how well
the material addresses targeted learning goals from the perspective
of what is known about student learning and effective teaching.
The criteria for making the judgments in the instructional analysis
are derived from research on learning and teaching and on the
craft knowledge of experienced educators. In the context of
mathematics literacy, these are summarized in Chapter
13, "Effective Learning and Teaching," of Science
for All Americans and in Chapter
15, "The Research Base," of Benchmarks for
Science Literacy. From these and other
sources, seven criteria categories (shown below) have been
identified to serve as a basis for the instructional analysis.
Category
I: Identifying a Sense of Purpose
Part of planning
a coherent curriculum involves deciding on its purposes and
on what learning experiences will likely contribute to achieving
those purposes. But while coherence from the designers’
point of view is important, it may be inadequate to give students
the same sense of what they are doing and why. This category
includes criteria to determine whether the material attempts
to make its purposes clear and meaningful to the student and
genuinely relates lessons to the unit purpose.
I.1 Conveying Unit Purpose: Does the material
convey an overall sense of purpose and direction that
is understandable and motivating to students? 
Clarification:
This criterion involves examining whether the material begins
with (or early on presents) an overarching question or problem
to be addressed by the unit (e.g., How can a graph help to make
predictions?), or a representation of what will be learned (e.g.,
a concept map of the main ideas that will be explored), or otherwise
identifies a purpose for the unit or chapter for the students
(e.g., a clear statement of objectives, using known terms).
The problem, question, representation or purpose provided by
the material should be explicit and comprehensible by the students,
and it should be plausible that it would be interesting and/or
motivating to them. A material that begins with abstractions
or phenomena outside students’ range of perception, understanding,
or knowledge does not adequately meet the criterion. However,
a material that starts with an unfamiliar but highly interesting
phenomenon that is likely to motivate students may meet the
criterion.
Providing students
with a sense of purpose for a whole unit or chapter is not always
possible (for example, there may not be a single question or
problem that is broad enough to foreshadow all learning goals
in the unit) or even desirable (for example, providing a purpose
on a large scale can lead to a complex sequence of activities
that is too demanding on the memory of younger students). In
such cases, it may be sufficient for the material to frame sections
within a unit rather than the whole unit or chapter.
Indicators of
meeting Criterion I.1:
1. The purpose is presented to students explicitly (or implicitly
through a problem, question, or representation).
2. The purpose is likely to be comprehensible to students.
3. The purpose is likely to be interesting and/or motivating
to students.
4. Students are given an opportunity to think about and discuss
the purpose.
5. Most activities or lessons are consistent with the stated
purpose.
6. The material returns to the stated purpose at the end of
the unit or chapter.
Scoring Scheme:
High: The material meets all six indicators.
Medium: The material meets indicator 1, along with three of
the other five indicators.
Low: The material meets indicator 1, along with one of the other
five indicators.
None: The material does not meet any of the indicators.
I.2 Conveying Lesson Purpose: Does the material
convey the purpose of each activity or lesson and its
relationship to others? 
Clarification:
The question is whether the purpose of individual activities
or lessons (as opposed to the whole unit) is made apparent to
the students and whether there are logical transitions and connections
between activities or lessons. If a classroom visitor asked
students what they were doing and why, is there reason to think
they would know?
The purpose of individual
activities or lessons could be brought out through the text,
teacher comments (suggested in the material), and/or student
responses to questions. For example, the purpose of gathering
data about the heights of students in the class (to learn about
dispersion of data) might be brought out by text explanation,
teacher explanation, or by the students coming up with a description
of the spread of data.
Indicators of
meeting Criterion I.2:
1. The material conveys or prompts teachers to convey the
purpose of each activity or lesson to students.
2. Each activity encourages each student to think about the
purpose of the activity or lesson.
3. The material conveys or prompts teachers to convey to students
how each activity or lesson relates to the other activities.
4. The material periodically engages students in thinking about
what they have learned so far and what they need to learn/do
next.
Scoring Scheme:
High: The material meets all four indicators.
Medium: The material meets three of the four indicators.
Low: The material meets one of the four indicators.
None: The material does not meet any of the indicators.
I.3 Justifying Sequence of Activities: Does
the material involve students in a logical or strategic
sequence of activities (versus a collection of activities)
that build toward understanding of the ideas in the
unit or chapter purpose? 
Clarification:
The issue here is whether there is a logical or strategic
sequence of activities in the material and whether this logic
or strategy is made explicit to the teacher or just inferred
by the reviewer. A rationale or implicit reason for the sequence
of activities should be clear, providing the teacher and students
with a sense of making progress toward the purpose of the unit
or chapter.
Indicators of
meeting Criterion I.3:
1. The material provides a rationale for the overall sequence
of activities or lessons.
2. If no rationale for the overall sequence of activities or
lessons is provided, the reviewer can identify one.
3. The sequence of activities reflects the stated or inferred
rationale or purpose.
Scoring Scheme:
High: The material meets indicators 1 and 3.
Medium: The material meets indicators 2 and 3.
Low: The material meets indicator 1 or 2.
None: The material does not meet any of the indicators.
Category
II: Building on Student Ideas about Mathematics
Fostering better
understanding in students requires taking time to attend to
the ideas they already have, both ideas that are incorrect and
ideas that can serve as a foundation for subsequent learning.
Such attention requires that teachers be informed about prerequisite
ideas/skills needed for understanding a standard and what their
students’ initial ideas are—in particular, the ideas
that may interfere with learning mathematics. Moreover, teachers
can help address students’ ideas if they know what is likely
to work. This category examines whether the material contains
specific suggestions for identifying and addressing student
ideas.
II.1 Specifying Prerequisite Knowledge. Does
the material specify and address prerequisite knowledge/skills
that are necessary to the learning of the standards? 
Clarification:
This criterion refers to (a) prerequisites to concepts or
skills in the standards examined, and (b) prerequisites to activities
used in the material to teach the concepts or skills in the
standards examined.
(a) Understanding
the ideas in standards often requires that students first understand
some other "prerequisite" concepts or skills. For
example, knowing what a prime factor is is prerequisite to learning
how to find the common denominator of fractions.
(b) In addition to
prerequisites to specific ideas in standards, additional prerequisites
may arise from the specific activities used to teach them. For
example, consider the benchmark: "Spreading data on a number
line helps to see what the extremes are, where they pile up,
and where the gaps are. A summary of the data includes where
the middle is and how much spread is around it." There
are no benchmarks that are prerequisite to the ideas in this
benchmark. However, as the students work through a curriculum
material, they may be engaged in investigations that target
this benchmark and involve measurement. In these cases, the
material should not take for granted that students will have
developed measuring skills. Teachers should be alerted to this
prerequisite (measurement skills) and encouraged to support
their students in developing measurement skills.
Responding to the
"prerequisites" criterion involves (a) making a list
of prerequisite concepts and/or skills, (b) examining whether
the material alerts to any prerequisite ideas on the list and
if so which ones, (c) examining whether the material has in
fact adequately addressed the prerequisites in the same or earlier
units, and (d) examining whether the material helps students
make connections between standards and their prerequisites.
While a standalone unit should not be faulted for not addressing
prerequisite ideas or skills, it should be expected to make
connections between standards and their prerequisites.
Indicators of
meeting Criterion II.1:
1. The material makes explicit what the specific prerequisite
ideas or skills are, if any exist.
2. The material addresses the identified prerequisites in the
same unit or in earlier units.
3. The material makes connections between standards and their
prerequisites (even if the prerequisites are addressed elsewhere).
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets two out of the three indicators.
Low: The material meets one of the three indicators.
None: The material does not meet any of the indicators.
II.2 Alerting Teacher to Student Ideas. Does
the material alert teachers to commonly held student
ideas (both troublesome and helpful) such as those described
in Benchmarks for Science Literacy Chapter 15:
The Research Base? 
Clarification:
Researchers have identified ideas that students have in
several content areas. The issue here is whether the material
informs teachers about students’ commonly held ideas in
the topic areas the material addresses. This information can
help teachers (a) understand better their own students’
ideas, (b) decide what ideas to build on and what changes to
promote, or (c) if the material is already designed in ways
that build on or attempt to change students’ commonly held
ideas, to better understand the rationale and purpose behind
designed strategies and activities.
Responding to this
question involves examining (1) whether there is research on
commonly held student ideas in the topic area/s that the material
addresses, (2) whether the material alerts teachers to such
ideas, and (3) whether the material accurately represents research
findings. Summaries of research on students' ideas in mathematics
(such as those included in Benchmarks Chapter 15: The
Research Base or the NCTM Research Ideas for the Classroom
series) will be helpful to reviewers who will want to know what
ideas students typically have about the topics that the curriculum
material they are examining addresses. If there is no research
on student ideas in the topic area/s that the material addresses,
the material should not be faulted for not addressing this criterion.
Indicators of
meeting Criterion II.2:
1. The material lists, conveys, or identifies specific commonly
held ideas that are relevant to the standard (rather than just
to relevant difficult topics).
2. The material clarifies/explains commonly held ideas.
3. The material explains or refers to commonly held ideas in
an accurate way.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
II.3 Assisting Teacher in Identifying Ideas.
Does the material include suggestions for teachers to
find out what their students think about familiar situations
related to the standards before the mathematical ideas
are introduced? 
Clarification:
Teachers need guidance in identifying students’ ideas
in these unresearched areas. But even in areas in which there
is research on commonly held student ideas, teachers need help
in identifying what proportion of their own students hold these
ideas as well as other more idiosyncratic ideas.
Responding to this
criterion involves examining not only whether the material encourages
teachers to find out students’ ideas but also whether it
provides specific suggestions for how to do so. Suggestions
may include providing (a) tasks in which students make predictions
and give their own descriptions and explanations of concepts
or skills; (b) tasks in which students are asked to represent
their understandings in drawings; (c) tasks in which students
are asked to interpret information (for example the solution
to an algorithm related to ideas in the standards), discuss
connections with related topics, or discuss alternative solutions,
or justifications; or (d) tasks which ask students about the
meaning of specific terms and/or probe for understanding of
important relationships between concepts.
Responding to the
criterion also involves examining the quality of the suggestions
provided. Tasks should not focus exclusively on identifying
students’ meaning for terms. While including such tasks
is useful, it is important to look for tasks in which students
make predictions and/or give explanations of concepts or procedures.
It is important that tasks make sense to students who have never
studied the topic and include questions posed in ways meaningful
to the students who are not familiar with the mathematical vocabulary.
It is also important that the material encourages teachers to
use probing questions to clarify what students mean or to get
more information about students’ thinking.
Indicators of
meeting Criterion II.3:
1. The material includes specific questions or tasks to
assist the teacher in identifying the ideas students have before
they study the standards.
2. The questions or tasks are likely to be comprehensible by
students before they become familiar with the concepts, procedures,
or vocabulary.
3. The material includes questions or tasks that ask students
to make predictions or give explanations of concepts or procedures
(vs. focus primarily on identifying students’ meaning for
terms).
4. The material suggests how teachers can use questions or tasks
to understand students' thinking and level of understanding.
Scoring Scheme:
High: The material meets all four indicators.
Medium: The material meets indicators 13.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
II.4 Addressing Misconceptions. Does the material
explicitly address commonly held student ideas? 
Clarification:
The issue here is whether the material includes questions
or activities that address students’ commonly held ideas
(both concepts or skills that are incorrect and those that can
serve as a foundation for subsequent learning). For example,
the material may include experiences that help students change
their ideas by providing activities that challenge students’
predictions or explanations, or prompt students to react to
commonly held misconceptions and resolve differences between
these misconceptions and the correct ideas. Alternatively, the
material may include experiences that extend common student
concepts or skills that have limited scope. Pointing out misconceptions
and telling students that they should avoid them does not adequately
address this criterion. Serious difficulties, either with concepts
or with skills, are not generally successfully addressed by
telling students they are wrong and providing them with the
"right answer."
In addition to providing
specific suggestions to teachers about how to address commonly
held student ideas reported in the research literature, materials
can be helpful by including suggestions to teachers about how
to take into account their own students’ ideas. Addressing
this aspect of the criterion may involve suggesting general
strategies that teachers can use to build on or change students’
ideas, and providing examples of how these strategies can be
implemented in the classroom. For example, teachers can be encouraged
to probe students’ ideas further, juxtapose them with other
students’ ideas, encourage students to compare their ideas
on a topic before and after instruction on the topic, etc.
Indicators of
meeting Criterion II.4:
1. The material explicitly addresses commonly held ideas
related to the standards (if there is research on these ideas).
2. The material includes questions, tasks, or activities that
are likely to help students progress from their initial ideas,
for example, by:
(a) explicitly
challenging students’ ideas, for example, by comparing
their predictions to what actually happens.
(b) prompting students
to contrast commonly held ideas and the correct concept or
procedure, and resolve differences between them.
(c) extending correct
commonly held ideas that have limited scope.
3. The material suggests
general strategies for addressing student ideas related to the
standards.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets any one indicator.
None: The material does not meet any of the indicators.
Category
III: Engaging Students in Mathematics
Much of the point
of mathematics is finding patterns and modeling ideas and relationships
in terms of a small number of generalizations or ideas. For
students to appreciate the power of mathematics, they need to
have a sense of the range and complexity of ideas and applications
that mathematics can explain or model. "Students need to
get acquainted with the things around them—including devices,
organisms, materials, shapes, and numbers—and to observe
them, collect them, handle them, describe them, become puzzled
by them, ask questions about them, argue about them, and then
try to find answers to their questions." (Science for
All Americans, p. 201)
III.1 Providing Variety of Contexts. Does the
material provide experiences with mathematics in multiple,
different contexts? 
Clarification:
Mathematicians and others construct and use mathematical
knowledge to describe, explain, predict, and design realworld
objects, systems, or events as well as abstract relationships.
Therefore, mathematical ideas need to be connected to meaningful
problems, situations, and the real world. The question is whether
the material provides a sufficient number of problems, experiences,
or events in a variety of contexts to support the ideas put
forth in the standards. The material can provide experiences
with problems, situations, systems, or events directly through
handson activities or demonstrations (firsthand experiences)
or indirectly, through the use of text, graphs, diagrams, computer
screens, videos, pictures, models, etc.
Indicators of
meeting Criterion III.1:
1. The experiences with objects, applications, and materials
are "right on target" in addressing the content of
the standards.
2. The material provides an appropriate variety of experiences
with objects, applications, and materials.
Scoring scheme:
High: The material meets indicators 1 and 2.
Medium: The material meets indicator 1.
Low: The material meets indicator 2.
None: The material does not meet any of the indicators.
III.2 Providing Firsthand Experiences.
Does the material include activities that promote firsthand
experiences with the standards ideas, when practical?

Clarification:
Students can learn more readily about things that are tangible
and accessible to their senses; thus students, especially younger
ones, will benefit most from firsthand experiences with the
objects, problems, or events to which the mathematical knowledge
in a standard refers. Providing students with some firsthand
experiences (e.g., handson activities, problem solving, or
measuring) is important, provided such experiences are practical.
When such experiences are not practical (for example, providing
firsthand experiences with measuring the height of a mountain),
students can encounter objects and events indirectly, through
the use of videos, pictures, models, etc.
However, it is neither
necessary nor optimal that all experiences provided are firsthand.
(For example, once students have had some firsthand experience
with flipping coins to find probabilities, providing them with
examples of other events that have finite outcomes would likely
be adequate.) If all experiences provided to students were firsthand,
it would limit the number of examples that could be provided
(see previous criterion, Providing Variety of Contexts). Moreover,
students should not be asked to reason only about ideas they
see firsthand, when in real life they will also encounter problems
indirectly.
Indicators of
meeting Criterion III.2:
1. The activities, whether firsthand or not, provide experiences
(e.g., text, pictures, video) that give students meaningful
connections of the concept or skill to their knowledge.
2. An appropriate number of experiences with ideas are firsthand
experiences.
(The number of firsthand experiences that are appropriate depends
on the age level of the students and the difficulty of the standard.)
3. The firsthand experiences are efficient when compared to
other firsthand experiences that could be used. (Efficiency
of an experience is judged by the time and cost of the experience
in relation to its value.)
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicator 1, along with indicator
2 or 3.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
Category
IV: Developing Mathematical Ideas
Science for All
Americans includes in its definition of mathematics literacy
a number of important yet quite abstract ideas—e.g., symbolic
representation, patterns and relationships, summarizing data.
Such ideas cannot be readily discovered in the real world; the
ideas themselves were developed over many hundreds of years
as a result of considerable discussion and debate about the
existence and logic of laws of mathematics and proofs of theorems.
Mathematics literacy requires that students see the link between
concepts and skills, see mathematics itself as logical and useful,
and become skillful at using mathematics. This category includes
criteria to determine whether the material expresses and develops
ideas in ways that are accessible and intelligible to students,
and to demonstrate the usefulness of the concepts and skills
in varied contexts.
IV.1 Justifying Importance of Standards Ideas.
Does the material suggest ways to help students develop
a sense of the importance and validity of mathematical
concepts or procedures? 
Clarification:
This criterion highlights the importance of including some
instances in the curriculum where an argument is developed in
support of the concepts, skills, or strategies presented. There
are both logical and psychological reasons for expecting a material
to provide students with a sense of why ideas make sense and
of why mathematicians are interested in them.
First, an understanding
of the link between hypotheses and argument is itself a literacy
goal. Science for All Americans includes in its definition
of science literacy a basic knowledge of the nature of mathematics—both
its logic and creativity—and its central role in human
endeavor. Learning about the nature of mathematics can be studied
in the context of learning about nearly any discipline, including
students’ own discoveries.
Given the time it
takes to properly develop an argument for ideas and the increased
level of sophistication required for understanding both the
evidence and the arguments, there is a limited number of ideas
for which an evidencebased argument is required for literacy.
However, it is possible that some concepts or procedures themselves
are sufficiently difficult for students to understand—e.g.,
long division, infinite decimals—that a case needs to be
made for students to find them plausible. The case might involve
examining whether the concept fits well with other concepts,
explains several relationships, and predicts new observations,
and how it compares to other explanations of the same observations.
If such a case is likely to be too difficult for most students
to understand, then students should at least be informed that
they are being asked to take an idea on faith. When a material
does not attempt to make a case, reviewers should comment on
(a) whether or not a case ought to have been made and, if so,
why and (b) whether or not the material makes explicit that
a case is not being built.
Indicators of
meeting Criterion IV.1:
1. The material builds a case for the mathematical importance
of the standard.
2. The material builds a case for the validity of the mathematical
ideas.
3. The material builds a case for the standard that is likely
to be comprehensible to students.
4. The material engages students in considering a case for the
validity and importance of the standard's concepts or skills.
Scoring Scheme:
High: The material meets all four indicators.
Medium: The material meets indicator 1 or 2, along with indicator
3 or 4.
Low: The material meets indicator 1or 2.
None: The material does not meet any of the indicators.
IV.2 Introducing Terms and Procedures. Does
the material introduce terms and procedures only in
conjunction with experience with them and only as needed
to facilitate thinking and promote effective communication? 
Clarification:
Understanding, rather than simply memorizing vocabulary
or algorithms, should be the main purpose of mathematics teaching.
In mathematics, many terms refer to concepts. For students to
understand a concept, they should be able to describe its properties,
give examples and nonexamples of it, and eventually give a
definition. Algorithms are important in themselves, as well
as providing efficient ways to solve problems. Students should
have opportunities to apply the concepts or procedures in problems
and reasoning.
Indicators of
meeting Criterion IV.2:
1. The material limits the use of terms and procedures.
2. The material introduces mathematical vocabulary or procedures
in conjunction with experiences, rather than having students
simply memorize definitions or procedures.
3. The material provides appropriate examples or meaningful
applications of the terms or procedures.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicator 1, along with indicator
2 or 3.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
IV.3 Representing Ideas Accurately. Does the
material include accurate and comprehensible representations
of mathematical concepts, procedures, and relationships?

Clarification:
This question highlights the importance of using accurate
representations to make (abstract) ideas intelligible to all
students. Different representations highlight different aspects
of an idea and provide a variety of opportunities for the idea
to connect to other students’ ideas and become embedded
in a student’s knowledge system. Possible modes of representation
include drawings, diagrams, graphs, images, analogies and metaphors,
models and simulations, and roleplaying. Representations need
to be clear so that students can understand fairly quickly what
ideas are being represented and how. In addition, because representations
typically highlight only some aspects of an idea, care must
be taken that they represent the real thing as accurately as
possible (or they involve students in considering which aspects
of the real thing are represented by the model and which are
not).
Indicators of
meeting Criterion IV.3:
1. The material includes accurate representations.
2. The material includes comprehensible representations (depending
on the students’ grade level and the difficulty of the
standards).
3. The material includes an appropriate number and variety of
accurate and comprehensible representations.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
IV.4 Connecting Standards Ideas. Does the material
explicitly draw attention to appropriate connections
among standards ideas? 
Clarification:
This criterion emphasizes connections among standards ideas,
concepts, and skills (rather than connections between activities,
which are examined in Category I). Some kinds of connections
can be classified as belonging to one of the following general
types:
(a) One concept
or skill may be an instance of a generalization (e.g., the
sum of the angles of a rectangle illustrates the general sum
of the angles of a polygon).
(b) One concept
or skill might be analogous to another idea (e.g., adding
rational fractions is like adding rational expressions in
that they both require a common denominator).
(c) A concept or
skill may show up in several fields or contexts (e.g., the
number pi can be the ratio of the circumference to diameter,
the radian measure of a semicircle, or the sum of a series).
Other connections
are more unique to particular ideas (e.g., linking slope of
a line to the tangent of a curve).
Responding to this
criterion involves looking to see whether any general or unique
connections are essential, which requires identifying what such
connections might be. Growth of Understanding maps provide a
rich source of potential connections. A set of maps on about
50 topics is available in Atlas of Science Literacy.
When a map is not available for a topic, the "AlsoSee"
box in Benchmarks for Science Literacy may be helpful
in identifying conceptual connections among ideas. Reading Science
for All Americans may also be helpful in identifying conceptual
connections, since several important connections among ideas
are made in the prose.
Indicators of
meeting Criterion IV.4:
1. The material notes connections among specific standards
ideas (rather than just among general topics).
2. The material adequately explains or develops the identified
connections.
3. The material engages students in making and/or explaining
the identified connections.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets one of the three indicators.
None: The material does not meet any of the indicators.
IV.5 Demonstrating/Modeling Procedures. Does
the material demonstrate/model (or include suggestions
for teachers on how to demonstrate/model) skills or
the use of knowledge? 
Clarification:
Among the ways literate adults use their knowledge and skills
are to describe and explain phenomena, to solve practical problems,
and to consider alternative positions on issues. Hence students
should learn to use their knowledge and skills in these ways.
In order for students to know the type and level of performance
expected for a skill or an application of conceptual knowledge,
they need to see examples. This is particularly important for
complex behaviors such as explaining how to solve problems,
developing a generalization, argument, or proof, or carrying
out a complex procedure or algorithm. Demonstrating or modeling
a skill involves (a) an expert's demonstrating or modeling the
skill, (b) providing running commentary about important aspects
to note about the performance or demonstration, and (c) providing
criteria for judging a good performance. Demonstrating or modeling
how knowledge might be used, for example, to solve problems
or construct a proof, is similar.
Responding to this
criterion involves examining whether (a) demonstrating/modeling
is carried out by the text or other accompanying materials (e.g.,
software, video), or (b) the material includes suggestions to
teachers about how to go about demonstrating/modeling skills
or use of knowledge in their classrooms.
Indicators of
meeting Criterion IV.5:
1. The material demonstrates (or instructs teachers how
to demonstrate) the expected procedure or performance.
2. The demonstration is clear and comprehensible.
3. The material provides commentary that points to particular
aspects of the demonstration and/or provides justifications
or explanations for the steps taken.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
IV.6 Providing Practice. Does the material
provide tasks or questions for students to practice
skills or use knowledge in a variety of situations?

Clarification:
An important part of learning mathematics consists of giving
students many opportunities to use their skill or knowledge,
in particular giving them opportunities to practice using mathematical
knowledge and skills in describing objects, relationships, and
events, solving problems, and applying knowledge in new situations
or contexts. Moreover, literacy means that people will be able
to draw upon and use their understanding of mathematics when
they encounter events that do not come with labels such as "algebra,"
"geometry, " or "statistics" but in political
arguments, discussions of literature, and walks on the beach.
Therefore, students will need practice in making connections
to new situations. Providing students with opportunities to
practice only calculating answers to predictable exercises does
not adequately address this criterion.
Indicators of
meeting Criterion IV.6:
1. The material includes appropriate practice exercises
and tasks on the standards.
2. The material provides an appropriate number of practice exercises
and tasks.
3. The material includes a variety of contexts, including everyday
tasks and contexts and novel as well as familiar practice tasks.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicator 1, along with either 2
or 3.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
Category
V: Promoting Student Thinking about Mathematics
No matter how clearly
materials may present ideas, students (like all people) will
devise their own meaning, which may or may not correspond to
targeted learning goals. Students need to make their ideas and
reasoning explicit, hold them up to scrutiny, and recast them
as needed. This category includes criteria for whether the material
suggests how to help students express, think about, and reshape
their ideas to make better sense of the world.
V.1 Encouraging Students to Explain Their Reasoning.
Does the material routinely include suggestions for
having each student express, clarify, justify, and represent
his/her ideas and how to get feedback from peers and
the teacher? 
Clarification:
It is important to provide opportunities for students’
thinking to become overt to themselves, the teacher, and other
students. By stating (clarifying, justifying, and representing)
their ideas, in writing, drawing, or speaking, students become
more aware of what they think. This may stimulate making explicit
connections between their ideas and the ideas presented by the
text or the teacher, and/or questioning of their ideas (if relevant).
Exchange of ideas in small groups or a large group discussion
may make students aware of the range of ideas that exist and
may provoke students to reconsider their own ideas in light
of others. Feedback from the teacher or other peers is necessary
to help students understand their mistakes and improve the quality
of their descriptions, explanations, or designs.
Responding to the
first part of the criterion involves examining whether the material
prompts (or encourages teachers to prompt) students to express
their ideas either orally or in writing. It also involves examining
whether the material has opportunities for each student to express
his or her ideas. Responding to the second part of the criterion
includes examining whether the material includes specific suggestions
to help the teacher provide explicit feedback, includes text
that directly provides students with feedback on the adequacy
of their ideas, or provides teachers with strategies they can
use to ensure that each student in the class receives feedback.
Indicators of
meeting Criterion.V.1:
1. The material encourages students to express their ideas
about the standards.
2. The material encourages students not only to express but
also to clarify, justify, interpret, or represent standards
ideas.
3. The material provides (or includes suggestions to help the
teacher to provide) explicit feedback to students about their
ideas.
4. The material includes suggestions to the students or teacher
on how to use student responses to diagnose errors or difficulties,
address errors or difficulties, or further develop students’
ideas about the standards.
Scoring Scheme:
High: The material meets all four indicators.
Medium: The material meets indicator 1, along with two of the
other three indicators.
Low: The material meets at least one of the indicators.
None: The material does not meet any of the indicators.
V.2 Guiding Interpretation and Reasoning. Does
the material include tasks and/or question sequences
that guide student interpretation and reasoning about
standards concepts, skills, and relationships? 
Clarification:
Experiences with handson materials, problems, and examples
of mathematical ideas are useful but not sufficient. Students
need time, opportunities, and guidance to make sense of these
experiences. If the students are turned loose to do exercises
or problems on their own, very little happens except for a small
number of students. The activities need to be guided with sequences
of questions that lead students to make relevant generalizations
and understand relationships. Similarly, students need time,
opportunities, and guidance to make sense of things they read
and ideas they are introduced to.
Responding to this
criterion involves examining whether the material includes (in
the teacher’s guide or student books) specific, carefully
chosen and sequenced tasks or questions that are likely to support
students’ thinking about exercises, problems, and investigations.
Good tasks and questions frame important issues, help students
relate their previous experiences to the mathematical ideas,
anticipate common student difficulties or misconceptions, and
focus on important generalizations and procedures.
Indicators of
meeting Criterion V.2:
1. The material includes specific and relevant tasks and/or
questions for activities related to the standards.
2. The material includes connected sequences (rather than only
collections) of questions or tasks.
3. The questions or tasks guide student interpretation and reasoning
through approaches such as:
(a) framing, introducing,
or developing important ideas,
(b) helping students to relate their own experiences to mathematical
ideas, or
(c) anticipating or eliciting common difficulties or student
misconceptions.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets two of the three indicators.
Low: The material meets one of the three indicators.
None: The material does not meet any of the indicators.
V.3 Encouraging Students to Think about What They’ve
Learned. Does the material suggest ways to have
students check their own progress? 
Clarification:
This criterion highlights the importance of having students
look back at the progress of their thinking and learning. Monitoring
one’s understanding and realizing which ideas one does
not understand can shift some of the responsibility for learning
to the students and may elicit their attempts to understand
as a result.
Responding to this
criterion involves examining whether the material includes questions
or tasks that prompt students to monitor their understanding,
or includes suggestions to teachers on how to encourage students
to do so. For example, What was confusing to you today? How
does the new knowledge compare with what you used to think?
What do you think you understand and where do you need to work
more? Encouraging students to monitor their understanding should
also include (when appropriate) questions on how and why students
changed their ideas. For example, Did you change any of your
ideas today? What evidence convinced you to do so?
Indicators of
meeting Criterion V.3:
1. The material engages (or provides specific suggestions
for teachers to engage) students in monitoring their progress
toward the standards.
2. The material asks students to think about how their ideas
have developed or changed.
3. The material gives students an opportunity to revise their
initial ideas about the standards based on what they have learned.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets two of the three indicators.
Low: The material meets one of the three indicators.
None: The material does not meet any of the indicators.
Category
VI: Assessing Student Progress in Mathematics
Assessment provides
information to students about what is important and to teachers
about what has been learned. Just as important, assessment provides
information to both the students and the teacher about adjustments
that should be made in learning and instruction. Because assessment
is so important to the teachinglearning process, it must match
the mathematics learning goal of the curriculum materials. Further,
assessments must address the range of skills, applications,
and contexts that reflect what students are expected to learn.
All of this is possible only if assessment takes place throughout
instruction, not only at the end of a chapter or unit.
VI.1 Aligning Assessment. Are assessment items
or tasks included that match the ideas, concepts, or
skills of the standards? 
Clarification:
This criterion highlights the necessity of including assessment
items for each standard that is important in the material. To
judge whether the items provided match the standard, the same
procedure should be followed as in examining the content match
between an activity and a standard. That is, examine whether
the assessment item addresses the substance rather than only
the topic of the standard, the level of sophistication of the
standard, and what part of the standard is assessed.
Indicators of
meeting Criterion VI.1:
1. The material provides
at least one assessment task that addresses the specific ideas
of the standard (assessment item should not be answerable by
reading comprehension, general intelligence, or testwiseness
alone).
2. The assessment items that do address the standard require
no other, more sophisticated, ideas.
3. If the material provides a test that is given to the students,
an appropriate number of assessment items are contentmatched
to the standard. If the material provides a bank of assessment
items that teachers select from, an appropriate proportion of
assessment items are contentmatched to the standard.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets indicators 1 and 2.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
VI.2 Assessing through Applications. Does the
material include assessment tasks that require application
of standards ideas, concepts, or skills and avoid allowing
students a trivial way out, like using a formula or
repeating a memorized term or rule without understanding? 
Clarification:
Rather than checking whether students have memorized certain
bits of information, assessment needs to test students’
mathematical understanding, reasoning, and the application of
knowledge. In addition, it needs to include tasks that engage
students in activities similar to those they will engage in
in their lives outside the classroom. Literate persons use mathematical
knowledge to describe, explain, and predict realworld phenomena,
solve a practical problem, or discuss issues. Accordingly, assessment
tasks need to engage students in descriptions, explanations,
predictions, design, and discussion of issues. This, however,
does not necessarily dictate the format that the assessment
should include. For example, assessments of students’ use
of knowledge to explain a concept could include a "multiple
choice" or "constructed response" format.
Indicators of
meeting Criterion VI.2:
1. The material provides assessment tasks that focus on
application of standards ideas.
2. The material includes assessment tasks that are familiar
as well as tasks that are novel or nonroutine.
3. If the material provides a test that is given to the students,
an appropriate number of assessment items focus on application.
If the material provides a bank of assessment items that teachers
select from, an appropriate proportion of items or tasks focus
on application.
Scoring Scheme:
High: The material meets all three indicators.
Medium: The material meets two of the three indicators.
Low: The material meets one of the three indicators.
None: The material does not meet any of the indicators.
VI.3 Using Embedded Assessment. Are some assessments
embedded in the curriculum along the way, with advice
to teachers as to how they might use the results to
choose or modify activities? 
Clarification:
This criterion highlights the need for assessment to be
in the service of instruction to guide teaching and learning.
The criterion requires that materials include assessments that
can be used as diagnostic or formative instruments, which help
determine learners’ needs, rather than largely as instruments
for grading students at the end of a unit or chapter.
Responding to this
question involves examining whether the material (a) provides
assessment tasks only at the end of a unit of study to help
grade student achievement, or also along the way to help monitor
student progress, (b) encourages and provides guidance to teachers
about how to probe beyond students’ first response to clarify
and further understand student answers, and (c) encourages teachers
to use the information from the assessments to make instructional
decisions about what ideas need to be addressed by further activities
with the whole group or smaller groups of students.
Indicators of
meeting Criterion VI.3:
1. The material uses embedded assessment as a part of the
instructional strategy or design.
2. The material includes assessments that provide opportunities,
encouragement, or guidance for students on how to further understand
standards ideas.
3. The material includes suggestions or guidance for teachers
on how to probe students’ understanding of standards ideas.
4. The material provides specific suggestions to teachers about
how to use the information from the embedded assessments to
make instructional decisions about what ideas need to be addressed
by further activities.
Scoring Scheme:
High: The material meets all four indicators.
Medium: The material meets indicator 1, along with two of the
other three indicators.
Low: The material meets indicator 1.
None: The material does not meet any of the indicators.
Category
VII. Enhancing the Mathematics Learning Environment
Several other important
considerations are involved in the selection of curriculum materials—for
example, the help they provide teachers in encouraging student
curiosity and creating a classroom community where all can succeed,
or attractiveness. These can influence student learning or even
whether the materials are used appropriately by the teacher
and students. The criteria listed in this category provide reviewers
with the opportunity to comment on these and other important
features.
VII.1 Providing Teacher Content Support. Does
the material help teachers improve their understanding
of mathematics and its applications? 
Clarification:
The issue here is whether the material includes a "content
background" section or other features that help teachers
develop their understanding of the mathematical knowledge addressed
in the material. Responding to this question involves commenting
on the quality of the support, not merely on whether such support
is included. Just providing teachers with a list of resources
that may enhance their understanding of mathematics does not
adequately address this question. A minimum requirement is that
such lists are annotated to describe the resources and specify
what can be learned from them.
Indicators of
meeting Criterion VII.1:
1. The material provides content information or recommends
resources for improving specific skills or understanding of
particular ideas.
2. The material provides content information that is in the
form that is useful and appropriate for teachers, no matter
what their background knowledge.
3. The material indicates how the ideas or skills are relevant
and important to teaching the material to students.
VII.2 Establishing a Challenging Classroom.
Does the material help teachers to create a classroom
environment that welcomes student curiosity, rewards
creativity, encourages a spirit of healthy questioning,
and avoids rigidity? 
Clarification:
Responding to this criterion involves examining whether
teachers are given guidance to (a) encourage students to raise
questions about the material being studied and suggest productive
ways for finding answers, (b) use activities in which students’
creativity and imagination will pay off, (c) respect and value
students’ ideas, and (d) avoid conveying the impression
that they themselves or the textbooks are absolute authorities
whose conclusions are always correct. In addition, the criterion
involves examining whether the materials give a vision of what
the curriculum might look like in action (i.e., teacher hints
and suggestions, dialogue boxes, vignettes, or video clips that
show desirable student teacher interactions).
Indicators of
meeting Criterion VII.2:
1. The material provides opportunities for students to express
their curiosity or creativity.
2. The material provides occasions for students to take risks
and ask questions.
3. The material suggests how to encourage students to weigh
and challenge their own and others’ ideas.
4. The material avoids sending a message that mathematics consists
only of rules and single correct answers.
VII.3 Supporting All Students. Does the material
help teachers to create a classroom that encourages
high expectations for all students, enables all students
to experience success, and provides all students a feeling
of belonging in the mathematics classroom? 
Clarification:
Several pedagogical criteria presented in previous categories
highlight the need for materials to incorporate principles of
teaching and learning that are likely to promote mathematics
understanding for all students. This question highlights the
importance of reviewing curriculum materials for features that
might distract or impede the progress of females, minorities,
students whose first language is not English, students with
disabilities, or others from the intended work. Further, the
criterion requires that materials provide specific suggestions
and resources for encouraging all students to be able to learn
mathematics and express their competence and performances during
instruction and in assessment tasks.
Indicators of
meeting Criterion VII.3:
1. The material avoids stereotypes or language that might
be offensive to a particular group.
2. The material illustrates the contribution or participation
of women, minorities, and persons with disabilities to mathematicsrelated
fields.
3. The material suggests alternative formats for students to
develop or express their mathematics knowledge during instruction
and assessment.
4. The material includes specific suggestions on how teachers
can modify activities for students with special needs, interests,
or abilities.
