Carnegie Learning Math Series (2011)

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Report Overview

Summary of Alignment & Usability: Carnegie Learning Math Series | Math

Math 6-8

The instructional materials reviewed for Grades 6-7 do not meet the expectations for alignment to the CCSSM. The materials reviewed for Grade 8 partially meet expectations for alignment. The materials devote insufficient time to the major work for grades 6 and 7 and do not meet the expectations for coherence. These grades were not reviewed for Gateway 2 because the materials do not meet expectations for focus and coherence in Gateway 1,. The grade 8 materials partially meet the expectations for Gateway 1 in focus and coherence and were reviewed for Gateway 2. For Gateway 2, the instructional materials meet the expectations for rigor and balance but do not meet the expectations for practice-content connections.

6th Grade

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Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Does Not Meet Expectations

Gateway 3

Usability

Not Rated

7th Grade

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Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Does Not Meet Expectations

Gateway 3

Usability

Not Rated

8th Grade

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Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Partially Meets Expectations

Gateway 3

Usability

Not Rated

Report for 7th Grade

Alignment Summary

The instructional materials reviewed for Grade 7 do not meet the expectations for alignment to the CCSSM. In Gateway 1, the instructional materials do not meet the expectations for focus on major work due to devoting an insufficient amount of time to the major work of the grade. The materials also do not meet the expectations for coherence. They partially meet the expectations for each of the following indicators within coherence: supporting content enhances focus and coherence simultaneously; being consistent with the progressions in the Standards; and fostering coherence through connections at a single grade. The materials do not meet expectations for having an amount of content that is viable for one school year. Since the materials do not meet expectations for focus and coherence in Gateway 1, they were not reviewed for evidence of rigor and the mathematical practices in Gateway 2.

7th Grade

Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Alignment (Gateway 1 & 2)

Does Not Meet Expectations

Gateway 3

Usability

Usability (Gateway 3)

Not Rated

Overview of Gateway 1

Focus & Coherence

The instructional materials reviewed for Course 2 do not meet the expectations for focus and coherence with the CCSSM. For focus, the instructional materials do not meet the criteria for the time devoted to the major work of the grade with only 56.0 percent of the days allocated in the timeline aligning to the major work. For coherence, supporting work is sometimes connected to the focus of the grade with some missed opportunities for natural connections to be made. The amount of content for one grade level is not viable for one school year and this team does not believe it will foster coherence between the grades. Content from prior or future grades is clearly identified, but materials that relate grade level concepts to prior knowledge from earlier grades is not explicit. Overall, the materials are shaped by the CCSSM and do incorporate some natural connections, but not enough to prepare a student for upcoming grades. The material also lacks some consistency for grade-to-grade progressions, and content that is not on grade-level or supports on grade-level learning is not explicit.

Gateway 1

v1.0

Gateway 1

Focus & Coherence

Criterion 1.1: Focus

Criterion 1.2: Coherence

Criterion 1.3: Coherence

Does Not Meet Expectations

Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The post, chapter, and standardized assessments that are included in the Teacher's Resources and Assessments were reviewed for Course 2 and found to meet the expectations for instructional material that assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades is sometimes introduced, but students should not be held accountable on assessments for those future expectations. If the future grade content was removed, it would not change the underlying structure of the assessments. Overall, the instructional material in the summative assessment items reviewed in Course 2 addressed the grade-level content in a challenging and effective manner with most units having little or no above grade level standards addressed.

Indicator 1A

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The post, chapter, and standardized assessments that are included in the Teacher's Resources and Assessments were reviewed for Course 2 and found to meet the expectations for instructional material that assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades is sometimes introduced, but students should not be held accountable on assessments for those future expectations. If the future grade content was removed, it would not change the underlying structure of the assessments. Overall, the instructional material in the summative assessment items reviewed in Course 2 addressed the grade-level content with most units having little or no above grade-level standards addressed.

Quality on grade-level examples are:

Chapter 2, End of Chapter Test. Question 6 uses a real-world scenario to assess 7.RP.A.2.C by representing proportional relationships by equations and asking a student to explain their reasoning.
Chapter 10, End of Chapter Test. Question 9 asks students to evaluate some claims made about triangles and if the claims are false, to give examples that disprove the claim. This on grade-level example fits 7.G.A.2 very well and allows for a student to work with the MPs as well.

The following items are above grade and should not be assessed, but they can be removed without drastically changing the materials:

Chapter 6, End of Chapter Test, Question 17b. The question asks the students to graph two expressions to see if they are equivalent. It has the students using a graphing calculator when the Grade 7standards expect students to apply the properties of operations to determine equivalence between expressions.
Chapter 9, Post Test, Question 11. Students are asked to construct a perpendicular line through a given point not on the line. This is a high school geometry standard.

Criterion 1.1: Focus

Indicator 1A

Criterion 1.2: Coherence

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

Students and teachers using the materials as designed will not devote a majority of class time in Grade 7 to the major work of this grade. The Teacher's Implementation Guide Volume 1 and 2 reviewed for Course 2 do not meet the expectations for majority of class time on the major clusters of the grade. For example, based on the pacing (one period = 50 minutes), 51 days out of 91 days have 56.0 percent of the time spent directly on the major work of the grade.

Indicator 1B

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Course 2 do not meet the expectations for spending the majority of class time on the major cluster of each grade. A chapter overview was found at the beginning of each chapter. This included the standards being taught in the lesson and a suggested pacing guide. Overall the instructional materials do not meet the criteria outlined in the CCSS publisher guidelines for the majority of class time on the major clusters of each grade.

To determine the three perspectives we evaluated: 1) the number of chapters devoted to major work, 2) the number of lessons devoted to major work, and 3) the number of days devoted to major work. It was decided that the number of days devoted to major work is the most reflective for this indicator because it specifically addresses the amount of class time spent on concepts and our conclusion was drawn through this data.

Evidence was determined from the Contents Pages FM-6 through FM-56 and the number of days suggested in each chapter overview found in the the Teacher Implementation Guide written by the publisher.

Chapters – 10 out of 17 chapters, or approximately 58.8 percent of time is spent on major work.
Lessons – 47 out of 81 lessons, approximately 58.0 percent of time is spent on major work.
Days – 51 out of 91 days, or approximately 56.0 percent of time is spent on major work.

The major work of the grade is:

7.RP.A- Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.NS.A- Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.EE.A- Use properties of operations to generate equivalent expressions.
7.EE.B- Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

All numbers are based on number of days on major work listed above.

Modules and Chapters that contain these Standards are:

Module 1 (Focus on Ratios and Proportional Reasoning): Chapter 1- 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 (6 days); Chapter 2- 2.1, 2.2, 2.3, 2.4,2.5, 2.6, 2.7 (9 days).
Module 2 (Focus on Ratio work with Percents and Integer Operations): Chapter 3- 3.1, 3.2, 3.3, 3.4, 3.5 (5 days); Chapter 4- 4.1, 4.2, 4.3, 4.4, 4.5 (5 days): Chapter 5- 5.1, 5.2, 5.3, 5.4, 5.5 (5 days).
Module 3 (Focus on Expressions and Equations): Chapter 6-6.1, 6.2, 6.3, 6.4, 6.5 (5 days); Chapter 7- 7.1, 7.2, 7.3, 7.4, 7.5 (6 days); Chapter 8- 8.1, 8.2, 8.3, 8.4, 8.5, 8.6 (7 days).
Module 4 (Focus on Expressions and Equations through Geometry): Chapter 10- 10.1 (1 day).
Module 5 (Focus on Expressions and Equations through Geometry): Chapter 12- 12.3, 12.4 (2 days).

Criterion 1.2: Coherence

Indicator 1B

Criterion 1.3: Coherence

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

The instructional materials reviewed for Course 2 partially meet the expectations for being coherent and consistent with the standards. Supporting work is sometimes connected to the focus of the grade with some missed opportunities for natural connections to be made. The amount of content for one grade level is not viable for one school year, and the materials do not foster coherence between the grades. Content from prior or future grades is clearly identified, and materials that relate grade level concepts to prior knowledge from earlier grades is not explicit. Overall, the materials are shaped by the CCSSM and incorporate some natural connections that will prepare a student for upcoming grades. However, the material does lack some consistency for grade-to-grade progressions, and content that is not on grade level or supports on grade level learning is not explicit.

Indicator 1C

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Course 2 partially meet the expectations for the non-major content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. In some cases, the non-major work enhances the major work of the grade level, while other areas could be stronger.

Non-major Clusters of the Grade 7 are:

7.SP.A - Use random sampling to draw inferences about a population.
7.SP.C - Investigate chance processes and develop, use, and evaluate probability models.
7.G.A - Draw, construct and describe geometrical figures and describe the relationships between them.
7.G.B - Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.SP.B - Draw informal comparative inferences about two populations.

Evidence of non-major content enhancing focus and coherence and supporting a partially meets score are:

7.RP.2 is supported by 7.G.1 in lesson 2, lesson 3 and lesson 4 of Chapter 11 and on pages 553-580 in the Teacher's Implementation Guide, volume 2.
- Ratios and proportions are also used with scale drawings in Chapter 11's pre-test, post-test, end of the chapter test and standardized test practice. Tests are found on pages 1205-1220 in the Teacher's Resources and Assessment book, volume 2.
7.G.2 and 7.G.5 support 7.EE.B in:
- Chapter 9, lesson 9.3, has students solve equations that are created by using the concepts of complements, supplements, midpoints, perpendiculars, and perpendicular bisectors (Teacher Implementation Guide, page 485).

Examples of missed opportunities:

Chapter 16, lesson 3, there are proportions used to answer questions about the data. Students could have been asked to make inferences using ratios and proportions in more of the lessons and questions.
Chapters 15, 16 and 17 on statistics and probability are treated separately without connecting the standards to the major work of the grade.

Though there are some connections to major work, the majority of non-major clusters are taught in isolation and are only very loosely, if at all, support the major work of this grade level. Several times, Course 2 misses important opportunities to engage students in the major work of Grade 7, even at spots that offer a natural opportunity to do so such as in lesson 3 of Chapter 16.

Indicator 1D

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Course 2 do not meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. Without including any assessment days, there are approximately 91 days of lessons in the materials. There needs to be additional material, other than assessment days, to ensure a students' grasp of all major work at this grade level. Overall, the amount of content that is designated for this grade level is short of the amount of material needed to make it truly viable for one school year.

According to the pacing guide, each period is 50 minutes in length and there is a suggested 91 days of lessons.
When pre-tests, mid-chapter tests and post-test assessments are also included in the pacing, this would add an additional 51 days. If all assessments are given during the course of the year, one extra day per assessment, the total would be 142 days.

The guiding focus taken for this indicator for our team was, "Will the non-major and major work of this material be enough to prepare a student for the next grade level?" With the amount of days, many of those days not focusing on major work, and the non-major work days not often supporting the major work of the grade, it will require the teacher to make significant modifications to prepare the student for the next grade level and supports this indicator receiving a 'does not meet' rating.

Indicator 1E

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials reviewed for Course 2 partially meet the expectations for the material to be consistent with the progressions in the standards. Content from prior grade standards is clearly identified, however above grade level standards are not clearly marked as such. There is ample practice for students to engage deeply with with the problems related to the Grade 7 standards; however there are no connections explicitly made to prior or future content in the Teacher Implementation Guide or the student text.

Some examples of areas where identification of standards from lower grades is beneficial and supports a partially meets rating:

Lower grade level material is clearly identified in the grade level outline found in the Teacher Implementation Guide on page FM-28. They are also identified and explained in the same resource at the beginning of each lesson.
Chapter 6, pages 293-338, titled "Numerical and Algebraic Expressions and Equations" starts with standard 6.EE.6 (page 295A) before moving into Grade 7 standards.

The instructional materials reviewed for Grade 7 partially meet the expectation of giving all students extensive work with grade-level problems. Overall, the materials do not consistently give students of varying abilities extensive work with grade-level problems.

Some examples of giving all students extensive work with grade-level problems, but not of varying abilities, and supports a partially meets rating:

There is ample practice for each standard. Every lesson has guided practice with a script for the teacher to follow. This portion has the students conceptually developing the skill being taught and are given practice problems as well. Along with the guided practice are assignments. The number of assignments and number of problems varies per lesson. In addition there are skill practice pages to accompany each lesson as well. The number of skill pages also varies with each lesson.
The Teacher Implementation Guide does not list any lessons or ideas for differentiated instruction except when it talks about the Mathia Software product. No differentiated or extension lessons in the Student Text, Students Skills Practice book, or the Student Assignment book were found by the reviewers.

The instructional materials reviewed for Grade 7 do not meet the expectation of relating grade-level concepts explicitly to prior knowledge from earlier grades. Overall, no support materials were found that relate grade-level concepts explicitly to prior knowledge from earlier grades.

The Teacher Implementation Guide is a wrap around of the student text. In the margins of the Teacher Implementation Guide, the authors have reworded the question asked in the student text but do not seem to add anything to the instruction. The margins also have steps for the teachers to follow, ways to groups students (i.e., have students complete questions 2 and 3 with a partner. Then share the responses with a class, page 5) and guiding questions to ask students. However, it does not clearly make connections between previous knowledge and new concepts. There are not any indicators that knowledge is being extended.

Indicator 1F

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for Course 2 partially meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings.

The Chapter titles are clearly labeled and aligned to the standards without a need for much interpretation.
- Chapter 1 - Ratios and Proportions (7.RP.A)
- Chapter 2 - Proportional Reasoning (7.RP.A)
- Chapter 7 - Solving Linear Equations and Inequalities (7.EE)
- Chapter 8 - The Power of Algebraic Reasoning (7.EE)

The instructional materials do include some problems and activities that serve to connect two or more clusters in a domain. They include a few problems and activities that connect two or more domains in a grade, in cases where these connections are natural and important. However, overall the materials only partially foster coherence through connections in Course 2.

For the majority of the work, most standards were taught and covered within one unit out of the entire series and not aligned with any other concept throughout the year.
There are no connections identified by publisher. However, there are connections within the Grade 7 standards they are just not noted or stated by the publisher.

An example of where connections were made and supports a partially meets rating is:

Chapter 6, "Numerical and Algebraic Expressions and Equations" lesson 2 connects 7.NS and 7.EE.A by having students use the Distributive Property of Multiplication and Division over Addition and Subtraction to simplify algebraic expressions. They use the property to also develop an understanding of combining coefficients of like terms and calculating the product of two numbers.

Criterion 1.3: Coherence

Overview of Gateway 2

Rigor & Mathematical Practices

Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One

Gateway 2

v1.0

Gateway 2

Rigor & Mathematical Practices

Criterion 2.1: Rigor

Criterion 2.2: Math Practices

Not Rated

Criterion 2.1: Rigor

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

Indicator 2A

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Indicator 2B

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Indicator 2C

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

Indicator 2D

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

Indicator 2E

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

Indicator 2F

Materials carefully attend to the full meaning of each practice standard

Indicator 2G

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

Indicator 2G.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

Indicator 2G.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

Indicator 2G.iii

Materials explicitly attend to the specialized language of mathematics.

Criterion 2.2: Math Practices

Overview of Gateway 3

Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two

Gateway 3

v1.0

Gateway 3

Usability

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Criterion 3.5: Technology

Not Rated

Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

Indicator 3A

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

Indicator 3B

Design of assignments is not haphazard: exercises are given in intentional sequences.

Indicator 3C

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

Indicator 3D

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Indicator 3E

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

Indicator 3F

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Indicator 3G

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

Indicator 3H

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Indicator 3I

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Indicator 3J

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

Indicator 3K

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Indicator 3L

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

Indicator 3M

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

Indicator 3N

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Indicator 3O

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Indicator 3P

Materials offer ongoing formative and summative assessments:

Indicator 3P.i

Assessments clearly denote which standards are being emphasized.

Indicator 3P.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3Q

Materials encourage students to monitor their own progress.

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

Indicator 3R

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Indicator 3S

Materials provide teachers with strategies for meeting the needs of a range of learners.

Indicator 3T

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Indicator 3U

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

Indicator 3V

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Indicator 3W

Materials provide a balanced portrayal of various demographic and personal characteristics.

Indicator 3X

Materials provide opportunities for teachers to use a variety of grouping strategies.

Indicator 3Y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

Criterion 3.4: Differentiation

Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

Indicator 3AA

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

Indicator 3AB

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

Indicator 3AC

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

Indicator 3AD

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

Indicator 3Z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Criterion 3.5: Technology

Downloadable Resources

2011