Alignment to College and Career Ready Standards: Overall Summary

The Grade 5 Expressions instructional materials partially meet the requirements for alignment to the CCSSM.  These materials meet expectations for Gateway 1. The lessons and assessments focus appropriately on grade-level content and the major work of the grade. The materials are also coherent, following the progression of the standards and connecting the mathematics within the grade level. The Grade 5 materials partially meet expectations for Gateway 2. All three of the aspects of rigor are present and attended to in the materials, although not fully. All eight MPs are included in a way that connects logically to the mathematical content. However, the program materials have a tendency to include multiple practice standards within a lesson, while only exploring one aspect of these multi-faceted standards. This limits teachers and students to investigating these eight practices at only a surface level. The program materials set up opportunities for students to engage in mathematical reasoning as they discuss concepts and construct arguments. There are missed opportunities for critiquing the reasoning of others and supporting mathematics language development through writing. 

See Rating Scale
Understanding Gateways

Alignment

|

Partially Meets Expectations

Gateway 1:

Focus & Coherence

0
7
12
14
13
12-14
Meets Expectations
8-11
Partially Meets Expectations
0-7
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
10
16
18
11
16-18
Meets Expectations
11-15
Partially Meets Expectations
0-10
Does Not Meet Expectations

Usability

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Not Rated

Not Rated

Gateway 3:

Usability

0
22
31
38
0
31-38
Meets Expectations
23-30
Partially Meets Expectations
0-22
Does Not Meet Expectations

Gateway One

Focus & Coherence

Meets Expectations

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-
Gateway One Details

The instructional materials reviewed for Grade 5 meet the expectations for Gateway 1. As a whole, students are assessed on prior and/or current grade-level topics. The instructional materials spend the majority of the time on the major clusters of the grade. This includes all clusters within 5.NBT and 5.NF, as well as 5.MD.C. The instructional materials reviewed for Grade 5 meet the expectations for coherence. The materials use supporting content as a way to continue exploring the major work of the grade. For example, students use their data collection for line plots to continue their work with fractions. The materials include a full program of study that is viable content for a school year, including 150 days of lessons and assessments. This set of materials is consistent with the Mathematical progression of learning set forth in the standards with a few exceptions. Support offered to help struggling students continue to work on grade level problems is inconsistent. Some include suggestions for lowering expectations to previous grade levels, instead of supporting students with current grade-level work. For teachers, minimal connections are made to learning from previous grades within the lessons. These instructional materials are visibly shaped by the cluster headings. Connections are made between domains and clusters within the grade level. For instance, materials make connections between line plots and their work on fraction operations. Generally, the Grade 5 materials support coherence, spend the majority of time on major work, and assess grade-level skills and content. The materials do not fully support all students with grade-level work and do not make explicit connections to prior knowledge. Overall, the Grade 5 instructional materials meet the requirements of Gateway 1.

Criterion 1a

Materials do not assess topics before the grade level in which the topic should be introduced.
2/2
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Criterion Rating Details

The instructional materials reviewed for Grade 5 meet the expectations for Focus within assessment. The unit review/test items assess prior and/or current grade-level content as written.

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.
2/2
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-
Indicator Rating Details

The instructional materials reviewed for Grade 5 meet the expectations for Focus within assessment. The unit Review/Test items included in the Student Activity Books assess prior and/or current grade-level content as written.

 Points of discussion:

  • In Unit 6, item 10 includes a 5-digit dividend ($90,000), which is beyond the 4-digits called for in 5.NBT.B.6. However, because students could solve this word problem with multiple approaches and draw on their work with understanding place value and the base-ten number system (5.NBT.A), this item is Mathematically reasonable.
  • For assessment items aligned to standards that include the possible use of varying strategies (i.e. 5.NBT.B.6-7, 5.NF.A.2, 5.NF.B.3,6-7), the student directions do not explicitly call for the use of any particular strategies.

*Evidence updated 10/27/15

Criterion 1b

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.
4/4
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Criterion Rating Details

The instructional materials reviewed for Grade 5 meet the expectation for this criterion by spending the large majority of the time on the major clusters of the grade. This includes all clusters within 5.NBT, 5.NF and cluster C within 5.MD.

Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.
4/4
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-
Indicator Rating Details

The instructional materials reviewed for Grade 5 meet the expectation for this indicator by spending the majority of the time on the major clusters of the grade. This includes all clusters within 5.NBT, 5.NF and cluster C within 5.MD.

  • More than 80% of the instructional time is spent on the major work of the grade level.
  • Units 1, 2, 3, 4, 5 and 6 focus exclusively on major work.
  • Unit 7 includes additional/supporting work.
  • Unit 8 includes both major and additional/supporting work.

Criterion 1c - 1f

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
7/8
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Criterion Rating Details

The instructional materials reviewed for Grade 5 meet the expectations for coherence. The materials use supporting content as a way to continue work with the major work of the grade. For example, students use their data collection for line plot to continue their work on fractions. The materials include a full program of study that is viable content for a school year, including 150 days of lessons and assessments. This set of materials is consistent with the mathematical progression of learning set forth in the standards with a few exceptions. Support offered to help struggling students continue to work on grade level problems is inconsistent. Some include suggestions for lowering expectations to previous grade levels instead of supporting students with current grade level work. Minimal connections are made for the teacher within the lessons to prior knowledge from previous grades. These instructional materials are visibly shaped by the cluster headings. Connections are made between domains and clusters within the grade level. For instance, materials make connections between line plots and their work on fraction operations. Overall, the Grade 5 materials support coherence and are consistent with the progressions in the standards.

Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
2/2
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-
Indicator Rating Details

The instructional materials reviewed for Grade 5 meet the expectations for their use of supporting content as a way to enhance coherence. For Grade 5, reviewers focused on the use of data and conversion of measurements as methods for supporting operations with whole numbers and fractions.

  • Unit 8 lessons on 5.MD.A support the work students are doing in 5.NBT. 1, 2, 5 and 6.
  • There are missed opportunities in unit 7 for making connections between 5.OA.1, 2 and 3 and the 5.NF and 5.NBT domains.
  • There are missed opportunities in unit 8 to highlight 5.NBT.2 within the 5.MD.1 lessons.
  • Unit 1 connects line plot data with fraction operations.
  • Unit 2 connects conversions to computations with decimals.

Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
2/2
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-
Indicator Rating Details

The instructional materials reviewed for Grade 5 meet the expectation for this indicator by providing a viable level of content for one school year.

  • Materials provide for 150 days of instruction, quizzes, fluency checks and formal assessment.
  • Most lessons are appropriate in length for Grade 5.
  • Some lessons may take longer than indicated.

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.
1/2
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Indicator Rating Details

The instructional materials reviewed for Grade 5 mathematical progressions partially meet the expectation for this indicator.

  • Learning progressions for CCSSM described at the beginning of each unit. This includes explicit connections to the mathematics of the unit.
  • Students are not given adequate opportunities to reason about division with whole numbers.
  • Students are not given adequate opportunities to use concrete models or drawings in their work with decimal operations.
  • Students are given extensive time to work on grade level problems.
  • Support offered to help struggling students continue to work on grade level problems is inconsistent. Some include suggestions for lowering expectations to previous grade levels instead of supporting students with current grade level work. This was found in units 1 and 4. This was not identified as previous grade level work.
  • Minimal connections are made for the teacher within the lessons to prior knowledge from previous grades.
  • A progression chart including Grade 4 and Grade 6 content is included in the planning pages.

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.
2/2
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Indicator Rating Details

The instructional materials reviewed for Grade 5 foster coherence through connections in the grade.

  • Lesson objectives include language shaped by the cluster headings.
  • For example, students are asked to write and evaluate expressions that contain variables.
  • Students are asked to graph ordered pairs and use them to represent and solve real world problems. This is similar to the cluster heading 5.G.A: Graph points on the coordinate plane to solve real-world and mathematical problems.
  • Connections are made in unit 3 between addition of fractions and multiplication of fractions.
  • Connections are made in unit 4 between area and multi-digit multiplication.
  • Connections are made in unit 7 between numerical patterns and graphing ordered pairs.
  • Connections are made between 5.NF.B.4 (fraction operations) and 5.NBT.B.5 (whole number operations) in units 3, 6 and 8.
  • Connections are made between line plots and fractions in units 1, 3 and 8.

Gateway Two

Rigor & Mathematical Practices

Partially Meets Expectations

+
-
Gateway Two Details

The Grade 5 Expressions instructional materials partially meet the expectations for Gateway 2: Rigor and Mathematical Practices. All three of the aspects of rigor are present and attended to in the materials, although not fully. All eight MPs are included in a way that connects logically to the mathematical content. However, the program materials have a tendency to include multiple practice standards within a lesson, while only exploring one aspect of these multi-faceted standards. This limits teachers and students to investigating these eight practices at only a surface level. The program materials set up opportunities for students to engage in mathematical reasoning as they discuss concepts and construct arguments. There are missed opportunities for critiquing the reasoning of others and supporting mathematics language development through writing.

 

*Evidence updated 10/27/15

Criterion 2a - 2d

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.
5/8
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-
Criterion Rating Details

The Expressions instructional materials for Grade 5 partially meet the expectations for Rigor and Balance called for in the CCSSM. These materials give attention to all three aspects of rigor, both in individual lessons and in units of study. The rigor aspects are treated separately and together as appropriate, depending on the content and lesson activities, and in some cases there is a strategic overlap in these aspects to help students make meaningful connections and develop a deeper understanding of Grade 5 content. However, while each of the aspects of rigor is present, none of them are particularly well-developed. Much of students’ conceptual understanding develops through class discussions and mathematics talk, which happen in a whole-group setting with little follow-up during independent student work. There is a focus on multi-digit multiplication, which is the CCSSM-required fluency in Grade 5, but practice does not occur throughout the year.  While students are given multiple opportunities to tackle word problems in a variety of contexts, there were not enough lessons that consistently applied the major work of the grade through independent problem solving.

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.
1/2
+
-
Indicator Rating Details

The Grade 5 Expressions instructional materials partially meet expectations for developing the conceptual understanding of key mathematical concepts. This program consistently devotes instructional time to the use of models and mathematical language to explore and develop understanding of grade level concepts. However, additional focus on 5.NBT.A, 5.NBT.B and 5.NF.B is needed to fully develop conceptual understanding of these important concepts.

  • A strong component of the Expressions curriculum is the “Math Talk” featured in each lesson. Students have daily opportunities to engage in math talk, allowing them to develop their understanding of concepts through speaking and listening.
  • Math drawings and visual models are used in various contexts to support students’ understanding, including the key areas of multi-digit multiplication and division with whole and decimal numbers and with fractions. These models are often utilized during whole class, teacher-directed discussion, with some follow-up opportunities for students to demonstrate understanding independently.
  • There are strong examples of building conceptual understanding in lessons. For example, Unit 1 Lesson 4 calls for students to practice comparing fractions with unlike denominators through reasoning about their size or relationship to a benchmark fraction, before students move into lessons finding common denominators. Another example occurs when students are introduced to volume concepts in Unit 8 Lesson 9, and they use centimeter cubes to fill nets of rectangular prisms.
  • The program includes correspondences across mathematical representations, in order to deepen students’ understanding of concepts. Unit 1 Lesson 2 connects fraction bars and number line models to equations for finding equivalent fractions. Unit 2 Lesson 1 relates decimals to fractions with visual models and money. Unit 3 Lesson 3 uses an area model and number line model to multiply fractions. Unit 3 Lesson 10 relates dividing by a unit fraction to the inverse operation of multiplication, with a focus on mathematical reasoning.
  • The program misses some opportunities to develop students’ understanding of concepts at a deeper level. In studying volume, for example, the instructional sequence jumps from filling rectangular prisms to counting layers to multiplying dimensions to the formula for calculating volume within three lessons (Unit 8 Lessons 9-11). Also, in Unit 2 Lesson 2 students explore decimals with a place value chart and using Secret Code cards; they then practice identifying and writing expanded form, but the values are always in the traditional order, leading some students to simply recognize a pattern of digits, not actually understand the value of each one.
  • The program lacks consistent opportunities for students to develop and explain their mathematical understanding in written form. An analysis of Unit 2 shows that only 12 items in the 10 lessons call for students to construct a written explanation of their mathematical understanding of addition and subtraction with decimals. An analysis of Unit 6 shows that only 8 items in 11 lessons call for students to construct a written explanation of their mathematical understanding of operations and word problems. Teachers using this program may wish to supplement with additional lessons and/or activities that prompt students to write about mathematics.

The grade level materials sometimes introduce multiple ideas in a single lesson, which may not allow students time to explore and develop a deep understanding of these ideas.  For example, Unit 3 Lesson 4 explores multiplying a fraction by a fraction; students model the multiplication with fraction bars, use area models, write an algebraic rule, and solve word problems. A second example of this occurs in Unit 7 Lesson 5, where the focus is the coordinate plane. Activities within this lesson include: Introduce the Coordinate Plane, Read Points, Discuss Plotting Points, Plot Points, Discuss Distance, Horizontal and Vertical Distance, and What’s the Error?

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.
1/2
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-
Indicator Rating Details

The Grade 5 Expressions instructional materials partially meet expectations for attention to procedural skill and fluency.  The program materials give some attention to the individual standards that set an expectation of procedural skill and fluency (5.NBT.B.5), but this attention is not sustained throughout the year.

  • A component of the Expressions program is a “Fluency Plan for helping students achieve fluency with the CCSSM that are suggested for each grade” (TE, page xxvi). The Grade 5 program attends to both procedural fluency with multi-digit multiplication of whole numbers (5.NBT.B.5), and intervention for those students that still need practice with multiplication facts. The plan includes Diagnostic Quizzes, Practice Sheets, Quick Practices, and Fluency Checks.
  • Grade 5 materials promote the use of strategies based on place value and properties of operations, building on students’ learning from Grade 4. Lessons engage students in both pure and applied mathematics exercises to develop multi-digit procedural fluency with the four operations. Students explore various methods for multiplication and division of multi-digit whole and decimal numbers, with time and attention given to sharing and analyzing work with peers.

Instruction targeting multi-digit multiplication fluency occurs in Unit 4 (in 6 of 12 lessons), Unit 5 (in 1 of 11 lessons), and Unit 6 (in 6 of 11 lessons), although some of these lessons only include an item or two that relate to the targeted fluency. It is questionable whether students would develop fluency with this procedural skill, given these limited opportunities for practice.

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
2/2
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Indicator Rating Details

The Grade 5 Expressions instructional materials partially meet expectations for attention to applications. The program materials are designed in a way to allow teachers and students to spend time working with applications of the mathematics, although these opportunities are sometimes overly simplistic and inconsistent in appearance.

  • The program presents single- and multi-step word problems for students to solve, in all units and in connection with a variety of grade level content. The program includes attention to the different problem types for addition/subtraction and multiplication/division (TE, page 473FF-473JJ), as outlined in the CCSSM.
  • Each lesson includes an “Anytime Problem” that is independent of the current unit of study, allowing students daily practice in applying skills and understandings to solve routine word problems.
  • Unit 6 focuses on Operations and Word Problems. Lessons include instruction on situation versus solution equations for all four operations, additive and multiplicative comparison problems, and writing and evaluating equations and expressions related to word problems. Exercises in these lessons include whole numbers, fractions, and decimals. The scope of the numbers used in these problems vary widely, sometimes using only single-digit numbers.
  • NF.A.2, 5.NF.B.6, and 5.NF.B.7.C are Grade 5 standards that explicitly target application, calling for students to represent and solve word problems with fractions using the four operations, and assess the reasonableness of their answers. Lessons that target these standards occur in Units 1, 3, and 6; these lessons appropriately connect the use of visual models, estimation and reasoning to support student thinking and check answers. Assessments include minimal opportunities, however, to assess these standards.
  • Each of the eight units in Grade 5 concludes with a lesson focused on connecting mathematics to the real world. These lessons prompt students to apply mathematics in a variety of contexts, including building birdhouses, the solar system, insects, and currency.

While this program devotes an adequate amount of time to solving and discussing word problems, most of these problems are routine and predictable. Teachers using this program should supplement with additional opportunities to engage students with non-routine problem situations.

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.
1/2
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Indicator Rating Details

The Grade 5 instructional materials partially meet the expectations for giving attention to all three aspects of rigor, both in individual lessons and in units of study. The rigor aspects are treated separately and together over the course of the year, depending on the content and lesson activities. While there is some balance of the three components, there tends to be a heavy emphasis on procedural skill and fluency in some areas, possibly at the expense of conceptual understanding.

  • Each lesson generally devotes some time to conceptual understanding and application. Using models and emphasizing mathematics talk are two important components that illustrate a daily focus on developing conceptual understanding. Each lesson includes an “Anytime Problem” that is independent of the current unit of study, allowing students daily practice in applying skills and understandings to solve routine word problems.
  • The CCSSM-required fluency for Grade 5 is multi-digit multiplication with whole numbers using the standard algorithm.There is an appropriate balance of conceptual and procedural work in this area, as students are building on learning from previous grades. This fluency is practiced in Units 4, 5, and 6, in addition to opportunities for students to practice this skill in the context of their work with single- and multi-step word problems with the four operations.
  • In many lessons, there is a strategic overlap of the aspects of rigor. For example, in Unit 3 Lesson 1, students solve comparison problems using comparison bars (tape diagrams) to model the given situations. This allows students to make sense of the problems conceptually, so they can apply skill with number and operations to complete the work. A second example of this overlap is evident in Unit 8 Lesson 12, where students explore volume in real-world situations. Students must apply their understanding of the concept of volume to given situations; in doing this, they are also practicing their procedural skill and fluency when using operations on whole and fractional numbers.
  • The Puzzled Penguin provides opportunities throughout the year for students to analyze the Penguin’s mistakes and give written feedback to correct his thinking. The Puzzled Penguin’s errors are almost always procedural in nature.
  • In addition to daily lessons/unit plans, the team analyzed the balance of rigor in Review/Tests for each unit as well.
    • Unit 1 Review/Test: Addition and Subtraction with Fractions—25 percent of the items are primarily conceptual (items 1–5 ), 60 percent are primarily procedural (items 6–17), and 15 percent are primarily application (items 18–20). The high number of procedural items is appropriate, as addition and subtraction of fractions was introduced and practiced in Grade 4.
    • Unit 2 Review/Test: Addition and Subtraction with Decimals—28 percent of the items are primarily conceptual (items 1–5, 10–11), 60 percent are primarily procedural (items 6–9, 12–22), and 12 percent are primarily application (items 23–25). Work with addition and subtraction with decimals is new to Grade 5; 5.NBT.B.7 calls for this operational work to include concrete models or drawings and strategies based on properties of operations, place value, and the relationship between operations. A more balanced assessment of conceptual and procedural work could be appropriate.
    • Unit 3 Review/Test: Multiplication and Division with Fractions—40 percent of the items are primarily conceptual (items 1–8), 45 percent are primarily procedural (items 9–17), and 15 percent are primarily application (items 18–20). As noted for Unit 2, operational work with decimals in Grade 5 calls for a conceptual emphasis; the balance of procedural and conceptual items on this assessment reflects the emphasis called for in the CCSSM.
    • Unit 4 Review/Test: Multiplication with Whole Numbers and Decimals—20 percent of the items are primarily conceptual (items 1–4, 25), 72 percent are primarily procedural (items 5–22), and 8 percent are primarily application (items 23–24). By Grade 5, students should be developing fluency with multi-digit multiplication with whole numbers; however, operational work with decimals in Grade 5 calls for a conceptual emphasis. The number of procedural and conceptual items on this assessment should be more balanced to reflect this.
    • Unit 5 Review/Test: Division with Whole Numbers and Decimals—16 percent of the items are primarily conceptual (items 1–4), 76 percent are primarily procedural (items 5–23), and 8 percent are primarily application (items 24–25). Division work with whole numbers and decimal numbers in Grade 5 should have a conceptual emphasis as called for in CCSSM. The number of procedural and conceptual items on this assessment should be more balanced to reflect this.
    • Unit 6 Review/Test: Operations and Problem Solving—40 percent of the items are primarily conceptual (items 1–4), and 60 percent are primarily application (items 5–10). A high number of application problems is appropriate for a unit on problem solving. Procedural work is embedded within the problem solving, which illustrates an authentic approach to mathematics application.
    • Unit 7 Review/Test: Algebra, Patterns, and Coordinate Graphs—40 percent of the items are primarily conceptual (items 1–7, 17), 45 percent are primarily procedural (items 8–16), and 15 percent are primarily application (items 18–20).

Unit 8 Review/Test: Measurement and Data—16 percent of the items are primarily conceptual (items 1–3, 16), 48 percent are primarily procedural (items 4–15), and 36 percent are primarily application (items 17–25). The conversion of measurements is a procedural process, so the high percentage of procedural items on this assessment is reasonable; the similar number of application items appropriately reflects that measurement always involves a context.

Criterion 2e - 2g.iii

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
6/10
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Criterion Rating Details

The Grade 5 Expressions instructional materials partially meet the expectations for meaningfully connecting the MPs. This program does a solid job of weaving in all eight of the MPs in an intentional way, to support students’ learning of content. However, the program’s tendency to include multiple practices in a lesson and only explore one aspect of these multi-faceted standards limits teachers and students to investigate some of the practices at only a surface level, therefore not allowing students to engage fully in these behaviors. Students using this program as designed have limited opportunities to critique the reasoning of others and develop mathematical communication skills.  Overall, the Grade 5 program materials somewhat support teachers and students in rigorous instruction that includes the connection of MP and the CCSSM.

Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
2/2
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Indicator Rating Details

The Grade 5 Expressions instructional materials meet expectations for identifying the MPs and using them to enrich the mathematics content. Overall, the MPs are an intentional part of the design of this program, as evidenced by the inclusion of practice standards in every lesson.

  • The MPs align and connect with the content of daily lessons, rather than being included as stand-alone topics.
  • The MPs are clearly identified and elaborated for teachers in multiple places: Introduction, Unit Planning, Research & Math Background/Getting Ready to Teach Unit, the introductory page of each lesson, and within daily lesson guidelines.
  • Within the Introduction of the program, an overview of the “Problem Solving Process” links each part of the problem-solving routine to a mathematical practice (TE, page xxvii). This problem-solving process is integrated routinely within each unit. The Introduction also includes a table (TE, page xviii) that relates the eight MPs to the learning path of the “Math Talk Community,” a key element of this instructional program.
  • In the “Research & Math Background” section at the beginning of each unit, the MPs are not simply listed but elaborated—two specific examples excerpted from lessons illustrate how each practice is integrated with and used to develop students’ understanding within each respective unit. For example, in Unit 1 students engage in MP6, “Attend to precision” when they explain their process to change a fraction to a mixed number, and when they explain in their own words how they found a common denominator for a group of fractions (TE, page 1W). In Unit 7, students engage in MP8, “Look for and express regularity in repeated reasoning,” when they recognize that evaluating any expression involves two steps: replacing the variable with a given number; and using the order of operations to simplify the expression (TE, page 557W). Students also engage in MP8 when they generalize that an x-coordinate is a horizontal distance from the origin and a y-coordinate is a vertical distance from the origin.
  • Within lessons, the MPs are identified in the teacher and student dialogue as they connect to specific activities; however, these sections don’t include explicit content-practice connections.
  • The final lesson in each unit presents a real-world application of mathematics content, allowing students a practical opportunity to engage in these mathematical behaviors. For example, Unit 7 Lesson 7 gives students an opportunity to explore the arrangement of stars in constellations, and connect these arrangements to their understanding of the coordinate plane.

It should be noted that while the MPs are clearly identified in the teacher materials and purposefully used to support teaching and learning in this program, there is little time or attention spent discussing these mathematical habits of mind explicitly with students.

Indicator 2f

Materials carefully attend to the full meaning of each practice standard
1/2
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Indicator Rating Details

The Grade 5 instructional materials partially meet expectations for attending to the full meaning of each of the eight MPs. By repeatedly aligning lessons to multiple practice standards, the grade level materials don’t attend to the full meaning of each of the practice standards.

  • The MPs are clearly visible throughout each lesson; however, the tendency to include multiple practice standards in an individual lesson does not allow for careful attention to the full meaning of the practices. For example, in Unit 3, 11 of the 14 lessons have four or more MPs tagged in a single lesson. In Unit 8, 12 of the 17 lessons have four or more MPs tagged in a single lesson. In some cases, a single question within a class discussion is tagged as developing an identified practice standard.
  • The MPs are routinely abbreviated or altered when included in the program materials. For example, MP3, “Construct a viable argument and critique the reasoning of others is simplified,” to “Critique the reasoning of others” or “Construct viable arguments.” MP7, “Look for and make use of structure,” is abbreviated to “Use structure;” and MP5, “Use appropriate tools strategically” is shortened to “Use appropriate tools.” While it is reasonable that a given activity may only target one part of a specific MP, it is concerning that the rationale for abbreviating these practices is not made explicit. In addition, this shorthand notation of the MPs downplays the importance of the full meaning of these practice standards as the CCSSM authors intended them.
  • MP5, “Use appropriate tools strategically,” calls for students to self-select tools for a given context or situation, and to be strategic both in how they choose and use tools for a mathematical task. In a number of activities that are tagged with this mathematical practice, students are prescribed specific tools to use, rather than selecting tools themselves. In many of these cases, the teacher models the appropriate use of the tool, negating any opportunities for students to be strategic.
  • Class discussion is the most common setting for students’ work with MP6, “Attend to precision.” This practice standard calls for students to use the language of mathematics and to communicate about mathematics in a clear and precise way. MP6 is tagged in each of the 150 lessons. This seems excessive. In many instances where MP6 is identified, questions are posed in a whole-group setting by the teacher while individual students respond, or the teacher does much of the explaining. Often the corresponding work in the Student Activity Book doesn’t follow up on these questions, so all students are not given opportunities to practice this behavior or held accountable for engaging in this practice.
  • The Puzzled Penguin activities are labeled as MP3, “Construct a viable argument and critique the reasoning of others.” While these activities do allow students to critique another’s work, many of the Penguin’s errors are procedural in nature and do not involve genuine meaningful mathematical reasoning. These activities might be more accurately tagged as MP6, “Attend to precision,” as they facilitate opportunities for students to consider and write about the precise nature of mathematical procedures.

The final lesson of each unit lists all eight Standards for MP as targets for a one-day lesson. One day does not allow for adequate exploration and development of any one of the practice standards, and almost certainly not all eight.

Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
0/0

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.
1/2
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Indicator Rating Details

The Grade 5 instructional materials partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the materials offer consistent opportunities for students to construct viable arguments, but opportunities to analyze the arguments of others are inconsistent and don’t hold students accountable for engaging in this behavior.

  • The last lesson of each unit includes an “Establish a Position” activity (eight lessons total), where students are given a mathematical statement, and they must decide if the statement is true or false and justify their thinking verbally and/or in writing. Volunteers are asked to share their positions, and the other students are allowed to question the volunteer for clarification or to verify reasoning.
  • Students are seldom asked to critique the reason of others. During Math Talks and Math in Action, students often share mathematical methods with the class, but they are rarely asked to critique the reasoning of those presenting content. The identification of MP3 in many of these lessons is to “Construct Viable Arguments,” as is the case in Unit 3 Lesson 5 (TE, page 220), where students compare different methods for multiplying a fraction with a whole number, with no suggestion for or time devoted to critiquing these solution methods.

The Math Expressions program uses the Puzzled Penguin to give students consistent opportunities to “Analyze and correct errors, explaining why the reasoning was flawed” (TE, page 1Z).  These activities occur multiple times in each unit, to allow students to engage in mathematical critique in connection with varied content. However, students are informed that the work contains an error, rather than analyzing and determining this for themselves. Also, the Penguin’s mistakes are generally procedural in nature, which may lead students to critique the procedural skill rather than the underlying mathematical understandings.

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.
1/2
+
-
Indicator Rating Details

The Grade 5 instructional materials partially meet expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the program’s teacher materials consistently provide opportunities to support students in constructing viable arguments. However, teachers using this program would need to seek out or construct additional opportunities for students to engage in critiquing the reasoning of others.

  • Math Talk is an integral component of this program, as stated in the introduction to the program materials: “A significant part of the collaborative classroom culture is the frequent exchange of mathematical ideas and problem-solving strategies, or Math Talk” (TE, page xx). The teacher materials include directives, prompts, and/or questions the teacher can use to support students in constructing viable arguments, in some cases including scaffolded dialogue with expected answers.
  • Math Talk discussions occur mainly in a whole group format—the discussions are generally teacher-led and follow a question-and-answer format. Little direction is given for teachers to engage students in this work independently, beyond the Puzzled Penguin activities.

During discussions, there is little guidance for teachers in how to promote or scaffold productive discourse if needed. Teacher commentary around discussions includes many instances of “be sure to…,” “make sure students understand that…,” “make sure students conclude that…,” but no explanation as to how to “make sure” students do these things.

Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.
1/2
+
-
Indicator Rating Details

The Grade 5 instructional materials partially meet the expectations for attending to the specialized language of mathematics. While this program promotes classroom discussion and mathematics talk, there is not a strong emphasis on developing mathematical vocabulary or communicating mathematics understanding in writing.

  • Many lessons have vocabulary terms listed in the Teacher Edition and the Student Activity Book. The vocabulary words include both general mathematical terms (i.e., additive, multiplicative, expression, Order of Operations) and terms that are specific to this textbook program (i.e., Place Value Sections Method, comparison bars, New Groups Below).
  • Each assessment begins with a Vocabulary section that targets general mathematical terms from the unit.
  • The instructional materials do not guarantee individual students the opportunity to attend to the specialized language of mathematics. Math Talk is an integral component of this program, as stated in the Introduction to the program materials: “A significant part of the collaborative classroom culture is the frequent exchange of mathematical ideas and problem-solving strategies, or Math Talk” (TE, page xx). The teacher materials include directives, prompts, and/or guiding questions the teacher can use to support students in constructing viable arguments and communicating their mathematical thinking. Math Talk discussions occur mainly in a whole group format—they are generally teacher-led and follow a question-and-answer format. This format allows only some students to verbalize their thinking making it easy for others to limit their participation or get overlooked.

Gateway Three

Usability

Not Rated

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
0/8

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.
0/2

Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
0/2

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.
0/2

Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
0/2

Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
0/0

Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
0/8

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
0/2

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.
0/2

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.
0/2

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.
0/2

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).
0/0

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.
0/0

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.
0/0

Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
0/10

Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
0/2

Indicator 3n

Materials provide strategies for teachers to identify and address common student errors and misconceptions.
0/2

Indicator 3o

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
0/2

Indicator 3p

Materials offer ongoing formative and summative assessments:
0/0

Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
0/2

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.
0/2

Indicator 3q

Materials encourage students to monitor their own progress.
0/0

Criterion 3r - 3y

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
0/12

Indicator 3r

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
0/2

Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
0/2

Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
0/2

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).
0/2

Indicator 3v

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
0/2

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
0/2

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
0/0

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
0/0

Criterion 3aa - 3z

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
0/0

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.
0/0

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
0/0

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.
0/0

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.).
0/0

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.
0/0

Additional Publication Details

Report Published Date: Wed Feb 11 00:00:00 UTC 2015

Report Edition: 2013

About Publishers Responses

All publishers are invited to provide an orientation to the educator-led team that will be reviewing their materials. The review teams also can ask publishers clarifying questions about their programs throughout the review process.

Once a review is complete, publishers have the opportunity to post a 1,500-word response to the educator report and a 1,500-word document that includes any background information or research on the instructional materials.

Educator-Led Review Teams

Each report found on EdReports.org represents hundreds of hours of work by educator reviewers. Working in teams of 4-5, reviewers use educator-developed review tools, evidence guides, and key documents to thoroughly examine their sets of materials.

After receiving over 25 hours of training on the EdReports.org review tool and process, teams meet weekly over the course of several months to share evidence, come to consensus on scoring, and write the evidence that ultimately is shared on the website.

All team members look at every grade and indicator, ensuring that the entire team considers the program in full. The team lead and calibrator also meet in cross-team PLCs to ensure that the tool is being applied consistently among review teams. Final reports are the result of multiple educators analyzing every page, calibrating all findings, and reaching a unified conclusion.

Math K-8 Rubric and Evidence Guides

The K-8 review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The K-8 Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

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