Singapore Math: Dimensions Math (2013-2015)

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Report Overview

Summary of Alignment & Usability: Singapore Math: Dimensions Math | Math

Math 6-8

The instructional materials for Dimensions Math Grades 6-8 do not meet expectations for alignment to the CCSSM. In Gateway 1, the instructional materials for Grades 6-8 do not meet the expectations for focus. For coherence, the instructional materials for Grades 6-7 are partially coherent and consistent with the Standards. The instructional materials for Grades 6-7 contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. The instructional materials for Grade 8 are not coherent and consistent with the Standards. The materials for Grade 6 were reviewed for rigor and the mathematical practices in Gateway 2. The instructional materials for Grade 6 meet the expectations for rigor and balance, but they do not meet the expectations for practice-content connections. Since the materials do not meet expectations for alignment to the CCSSM, they were not reviewed for usability in Gateway 3.

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

Does Not Meet Expectations

Gateway 3

Usability

Not Rated

Report for 7th Grade

Alignment Summary

The instructional materials for Dimensions Math Grade 7 do not meet expectations for alignment to the CCSSM. In Gateway 1, the instructional materials do not meet the expectations for focus as they assess above-grade-level standards and devote less than 65% of instructional time to the major work of the grade. For coherence, the instructional materials are partially coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. Since the materials do not meet the expectations for focus and coherence in Gateway 1, they were not reviewed for rigor and the mathematical practices in Gateway 2 or usability in Gateway 3.

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 Dimensions Math Grade 7 do not meet expectations for focus and coherence in Gateway 1. For focus, the instructional materials do not meet the expectations for assessing grade-level standards, and the amount of time devoted to the major work of the grade is less 65 percent. For coherence, the instructional materials are partially coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade.

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 instructional materials reviewed for Dimensions Math Grade 7 do not meet expectations for not assessing topics before the grade level in which the topic should be introduced. The instructional materials include assessment items that align to standards above this grade level.

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 instructional materials reviewed for Dimensions Math Grade 7 do not meet expectations for assessing grade-level content. The FAQ page on the website for Singapore Math states, “There are currently no tests, but the workbook could be used as a test bank.” In Dimensions Math workbooks 7A and 7B, above grade-level items are present and could not be modified or omitted without a significant impact on the underlying structure of the instructional materials. For example:

Students evaluate expressions involving square and cube roots (8.EE.2, N-RN.2). For example, in Workbook 7B page 4, problem 19b.iii states, “Find the cube root of $(15\times12,600/7)$ .” There are problems similar to this in Workbook 7A on pages 4, 8, 10, 11, 15, 16 and 17.
Students use factorial notation, which does not align to standards from Grades 6-8. For example, in Workbook 7A page 19, problem 25 part a states, “Find the value of the following: 5!, 8!, 10!, 100!/98!” Part b states, “What is the relation between n! And (n-1)!?”
Students evaluate exponential expressions by applying the laws of exponents (8.EE.1). For example, in Workbook 7A page 2, problem 11 part e states, “ $(3^2\times 3^5)^2$ .”
Students calculate the rate of change for linear functions (8.EE.B). For example, in Workbook 7B page 11, problem 9 states, “Find the slopes of the lines in the following diagram.” Problems 18-20 on page 13 are similar.
Students describe subsets of a sample space (S-CP.1). For example, in Workbook 7B, page 57, problem 3 part a states, “Find the set A’ in each of the following cases. a) E = {1, 3, 5, 8, 9, 12, 14} and A = {3, 5, 9, 12}.” Parts b through f are similar, and the remainder of problems 1-20 on pages 57-60 are also similar.

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.

The instructional materials reviewed for Dimensions Math Grade 7 do not meet expectations for devoting the large majority of class time to the major work of the grade. The instructional materials spend less than 65% of instructional time 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 Dimensions Math Grade 7 do not meet expectations for spending a majority of instructional time on major work of the grade.

The approximate number of chapters devoted to major work of the grade (including assessments and supporting work connected to the major work) is 8.5 out of 17, which is approximately 51 percent.
The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 36.5 out of 69, which is approximately 53 percent.
The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 67 out of 125, which is approximately 54 percent.
In Grade 7, two of the 17 chapters do not address major work or supporting work connected to major work, and there are five chapters that have less than 50 percent addressing major work (including supporting work connected to major work).

A lesson-level analysis (which includes lessons and sublessons) is most representative of the instructional materials because it addresses the amount of class time students are engaged in major work throughout the school year. As a result, approximately 53 percent of the instructional materials focus on major work of the grade.

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 Dimensions Math Grade 7 partially meet expectations for being coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. The instructional materials do not include an amount of content that is viable for one year, do not attend to the full intent of some standards, and do not give all students extensive work with grade-level problems.

Indicator 1C

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

The instructional materials reviewed for Dimensions Math Grade 7 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting standards/clusters are connected to the major standards/clusters of the grade with one exception. Connections are not explicitly stated except for the connection of ratios to scale drawings as described in the Teaching Notes and Solutions, Book B, page 6.

Examples of supporting work that engage students in the major work of the grade include:

In Lesson 8.2, students write an equation (major standard 7.EE.4) to solve for a missing angle (supporting standard 7.G.5).
In Chapters 12 and 13, students use geometric formulas to solve problems (supporting standard 7.G.6), including setting up and solving equations for unknown measures (major standard 7.EE.4).
In Lesson 12.3 and Chapter 14, students use proportional relationships (major standard 7.RP.2) involving scale drawings and maps with scale factors (supporting standard 7.G.1).
In Lesson 16.2, students conduct a probability experiment and calculate probabilities (supporting standard 7.SP.7) as percentages (major standard 7.RP.3).
In Chapter 13, students use rational numbers (major standard 7.NS.3) to find surface area and volume of figures (supporting standard 7.G.6).
In Chapter 17, students calculate simple and compound probabilities (supporting standard 7.SP.8) using rational numbers (major standard 7.NS.3).

The missed connection between supporting work and major work includes:

In Chapter 16, a connection to major work is missed when students do not use proportional reasoning (7.RP.2) to make predictions about populations (7.SP.8).

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 for Dimensions Math Grade 7 do not meet expectations that the amount of content designated for one grade level is viable for one year.

As designed, the instructional materials can be completed in 125 days. The total days were computed in the following manner:

Each lesson was counted as one day of instruction.
A “lesson” with subsections (i.e., 1a, 1b, 1c) counted as three lessons or three days.
A practice day was added for each chapter.

The suggested amount of time for the materials is not viable for one school year, and/or the expectations for teachers and students are unreasonable. Significant modifications would be necessary for the materials to be viable for one school year. In addition, there are several lessons that are off-grade level, which, if not completed, would reduce the number of days provided in the materials.

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 for Dimensions Math Grade 7 partially meet expectations for being consistent with the progressions in the standards. In general, materials follow the progression of grade-level standards, though they don’t always meet the full intent of the standards. In addition, lessons utilize standards from prior grade levels, though these are not always explicitly identified in the materials.

Examples where standards from prior or future grades are utilized but not identified include:

In Chapter 1, students represent the prime factorization of numbers in exponential notation (6.EE.1), find the GCF and LCM of pairs of numbers (6.NS.4), and find the square roots and cube roots of small numbers (8.EE.2). All work is presented as grade-level work.
In Chapter 2, students are introduced to negative integers and absolute value (6.NS.5-7). This material is presented as grade-level material. In Lesson 2.7, students round numbers to specific decimal places (5.NBT.4), but this is not identified as below grade-level content.
In Lesson 2.6, students encounter irrational numbers, which aligns to 8.NS.A. This lesson is used as an extension lesson, but the content is not identified as above grade level.
In Lesson 8.3, students encounter perpendicular and angle bisectors, which aligns to G-CO.C. The content is not identified as above grade level.

The “Notes on Teaching” in Teaching Notes and Solutions provide some direction for teachers to explicitly relate the content to prior learning:

In Real Numbers (Book A, page 3), “Students have learned whole numbers and fractions in their earlier grades. In this chapter, the concept of numbers is extended from whole number to integers…”
In Factors and Multiples (Book A, page 1), “Students have learned factors and multiples in their earlier grades.”
In Percentages (Book A, page 9), “In the earlier grades students have learned the meaning of percentage and the conversion between decimal, fraction, and percentage. In this chapter, they will learn to apply percentage to solve more daily life problems.”
In Perimeter and Areas of Plane Figures (Book B, page 4), “The idea of perimeter and area has been studied in elementary schools. This chapter extends the idea to find the perimeters and areas of plane figures…”
Student Workbook 7A, page 134 states, “We have learned the idea of a ratio in the previous grade. Let us recall its meaning.” and provides the definition for ratio and notation.

The instructional materials do not attend to the full intent of some standards. Examples include:

In Lesson 5.1, students solve linear equations in one variable but not ones that arise from word problems (7.EE.4a). However, there are no opportunities to “compare an algebraic solution to an arithmetic solution (by) identifying the sequence of the operations used in each approach.”
In Lesson 5.4, students solve linear inequalities (7.EE.4b) but do not graph the solution sets.
In Lesson 10.2, students graph linear relationships, but none of those represent proportional relationships (7.RP.2d).

The materials do not give all students extensive work with grade-level problems for some standards. Examples include:

In Lesson 13.2, although students determine side lengths, surface area, and volume of a figure shown (7.G.6), there is one real-world problem in the Math@Work section and one real-world problem in the BrainWorks section.
In Chapter 14, students “reproduce a scale drawing at a different scale” (7.G.1) in one problem.
In Lesson 14.3, the instructional materials identify k in the equation y = kx as a constant. Students calculate y/x to “decide whether two quantities are in a proportional relationship” (7.RP.2a), but the materials do not use the term “constant of proportionality” (7.RP.2b).
For 7.NS.1b, there is one problem that illustrates a + -a = 0.

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 for Dimensions Math Grade 7 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards.

The materials include learning objectives that are visibly shaped by CCSSM cluster headings, and there are correlations between Dimensions Math Grade 7 learning objectives and CCSSM cluster headings. Examples include:

In Chapter 5, learning objectives are shaped by 7.EE.B, “Solve real-life and mathematical problems using numerical and algebraic expressions and equations.” Examples of learning objectives shaped by the cluster include: “use letters to represent numbers of variables; interpret algebraic notations; evaluate algebraic expressions and formulas; and express real-life situations in algebraic terms.”
In Chapter 2, some learning objectives are shaped by 7.NS.A, “Apply and extend previous understandings of operations with fractions.” For example, learning objectives “identify integers, rational numbers, and real numbers and perform the four operations on them.”

The materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade. Examples include:

In Lesson 5.1, students solve linear equations (7.EE.B) that include rational numbers (7.NS.A).
In Lessons 4.1 and 4.3, students rewrite expressions (7.EE.A) that include rational numbers (7.NS.A).
In Lesson 6.1, students express equivalent ratios (7.RP.A) involving rational numbers (7.NS.A), and in Lesson 6.2, students calculate rates involving rational numbers.
In Lesson 7.2, students solve linear equations (7.EE.B) that include percentages (7.RP.A).
In Chapter 15, students use data identified from random sampling (7.SP.A) to draw informal comparative inferences about two populations (7.SP.B).

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

2013-2015