Alignment to College and Career Ready Standards: Overall Summary

The Grade 4 Expressions instructional materials partially meet the expectations 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 4 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

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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 4 meet the expectation for Gateway 1 focus and coherence. The majority of assessed content aligns to prior and/or current grade-level standards, and the majority of instructional time throughout the year is devoted to the major work of the grade. The instructional materials reviewed for Grade 4 meet the expectation for coherence. The materials use supporting content as a way to continue work with the major work of the grade. The materials include a full program of study that is viable content for a school year including 150 days of lessons and assessments. More should be done to support struggling students with grade level skills and understandings. These instructional materials are visibly shaped by the cluster headings in the standards. Connections are made between domains and clusters within the grade level. For instance, fraction work is connected to data and measurement lessons. Overall the Grade 4 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 4 meet the expectations for focus within assessment. The majority of unit review/test items assess content from prior and/or current grade levels. There are instances of assessment items that do not align to grade-level expectations, but these items could be easily revised or removed by classroom teachers. Overall, the instructional materials meet the expectation for this indicator.

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 4 meet the expectations for focus within assessment. The majority of unit review/test items included in the student activity books appropriately assess prior and/or current grade-level content, with few exceptions.

Points of discussion:

  • Unit 1 review/test:
    • Items 14-17 assess procedural skill with multi-digit addition and subtraction, which is a CCSSM-required fluency by the end of Grade 4 (4.NBT.B.4).
  • Unit 4 review/test:
  • Items 7-8 involve evaluating expressions with symbols. The listed test objective is “evaluate expressions and solve equations with parentheses” which the program connects to 4.NBT.B.4-6. Parentheses are not specifically mentioned in the CCSSM until 5.OA.A.1; however, these items could be considered Mathematically reasonable or omitted.
  • Unit 5 review/test:
    • Item 18 is a word problem involving finding a missing side length when given a rectangular area. If students used division to solve this problem, it would call for the use of a 2-digit divisor, which is beyond the scope of Grade 4. However, because this is a word problem with a measurement context and could be solved in a number of ways, it is an appropriate item.
  • Units 6 and 7 review/tests:
    • Unit 6 items 6, 9, 13, 15, 16, 20, 21 and unit 7 items 8, 16 include fractions with denominators outside the limits listed in the footnote for 4.NF, including sevenths, ninths, and fortieths. It is reasonable, and should be encouraged, that Grade 4 students explore fractions beyond these denominators during instruction to avoid limiting their thinking, and these above grade-level items could easily be omitted from assessments or edited.

*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 4 meet the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes cluster 4.OA.A, all clusters in 4.NBT and all clusters in 4.NF.

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 4 meet the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes cluster 4.OA.A, all clusters in 4.NBT and all clusters in 4.NF.

  • While some lessons include multiple standards, a large majority of the lessons are explicitly focused on major work.
  • Units 1, 2, 3, 4, 6 and 7 devote all lessons to major work.
  • Units 5 and 8 are additional and supporting work.
  • Students work on place value, operations and application problem throughout the year.
  • More than 70% of the lessons and assessments are focused on major 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 4 meet the expectations for coherence. The materials use supporting content as a way to continue work aligned to the major work of the grade. 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 in the standards. Connections are made between domains and clusters within the grade level. For example, work with fractions is connected to data and measurement lessons. Overall, the Grade 4 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 4 meet the expectations for their use of supporting content as a way to enhance coherence. For Grade 4, reviewers focused on the use of data, factors and multiples, and measurement and conversion of measurements as methods for supporting operations with whole numbers and fractions.

  • Unit 4 connects 4.OA and 4.NBT.
  • Units 5 and 7 make connections between data and work with fractions.
  • Unit 6 uses fractions to continue work on line plots.
  • Unit 5 makes connections between operations, place value and measurement conversions.
  • Unit 5 also connects multiplication and division understanding to area and perimeter computations.

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 4 meet the expectations 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 4.
  • 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 4 are consistent with the mathematical progressions in the standards, partially meeting the expectations for this indicator.

  • Learning progressions for CCSSM are described at the beginning of each unit. This includes explicit connections to the mathematics of the unit.
  • Lessons include fractions with denominators outside grade level expectations, so do not follow the progression in the standards.
  • 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 support includes suggestions for lowering expectations to previous grade levels instead of supporting students with current grade level work. This was found in units 1, 4 and 6. 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 3 and Grade 5 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 4 foster coherence through connections at the grade level.

  • Examples were found within the "Lesson Focus" sections of language from the cluster headings.
  • In unit 2, students are expected to use their understanding of place value and to solve real world problems.
  • Connections are made in units 2, 4, 6 and 7 between 4.MD, 4.NF and 4.NBT.
  • Connections are made in unit 8 between 4.NF and 4.MD using addition of fractions and angle measurement.

Gateway Two

Rigor & Mathematical Practices

Partially Meets Expectations

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Gateway Two Details

The Grade 4 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 4 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 4 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 often happen in a whole-group setting with little follow-up during independent student work. There is a minimal focus on multi-digit addition and subtraction, which is the CCSSM required fluency in Grade 4. 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
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Indicator Rating Details

The Grade 4 Expressions instructional materials partially meet the 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 4.NBT and 4.NF 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 mathematics talk, allowing them to develop their understanding of concepts through speaking and listening.
  • Mathematics drawings and visual models are used in various contexts to support students’ understanding, including the key areas of multi-digit multiplication and fractions. These models are often utilized during whole class, teacher-directed discussion, with some follow-up opportunities for students to demonstrate understanding independently.
  • The program lacks opportunities for students to develop and explain their mathematical understanding in written form. An analysis of Unit 2 shows that only 28 items in the 19 lessons call for students to construct a written explanation of their mathematical understanding of multiplication with multi-digit numbers. An analysis of Unit 6 shows that only 13 items in 10 lessons call for students to construct a written explanation of their mathematical understanding of fraction concepts and operations. 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 does not allow students time to explore and develop a deep understanding of these ideas. For example, Unit 2 Lesson 6 explores using place value to multiply multi-digit numbers; this lesson introduces both the Place Value Sections Method and the Expanded Notation Method and calls for students to compare and connect visual models for these two methods. This comparison of methods may be more meaningful if students have a more solid understanding of each method. A second example of this occurs in Unit 6 Lesson 2, where the focus is finding pairs of fractions that add to one. Activities within this lesson include: Fifths that Add to One, Sixths that Add to One, Find the Unknown Addend, Build with Unit Fractions, Comparison Notation, Discuss and Compare Unit Fractions, Compare and Order Unit Fractions, and What’s the Error?

The program includes some missed opportunities to develop students’ understanding of concepts at a deeper level. For example, Unit 1 Lesson 3 builds on students’ knowledge of rounding from Grade 3, beginning with MathBoard modeling of rounding numbers to the nearest 10. However, this lesson quickly moves from the visual representation to questions like “Which place tells us the way to round?” and “Which place is increased if the ones tell us to round up?” (TE, page 21). Also, in Unit 3’s treatment of division with whole numbers, students are moved through this content very quickly. The third lesson of this unit calls for students to work with four-digit dividends, and the fifth lesson calls for them to relate three different methods of division. Teachers using this program may wish to slow down and spend more time with each of the presented methods.

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 4 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 (4.NBT.B.4), 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 4 program attends to both procedural fluency with addition and subtraction (4.NBT.B.4) 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 4 materials promote the use of strategies based on place value and properties of operations, building on students’ learning from Grade 3. Lessons engage students in both pure and applied mathematics exercises to develop multi-digit procedural fluency with addition and subtraction. Students explore various methods for adding and subtracting multi-digit numbers, with time and attention given to sharing and analyzing work with peers.

Unit 1 is the only place in the Student Activity Book where students work toward proficiency with 4.NBT.B.4, which calls for “Students [to] fluently add and subtract multi-digit numbers within 1,000,000 using the standard algorithm” (K-5, NBT Progressions, page 14), with activities in 5 of the 14 lessons. The team could not find other practice activities embedded in daily instructional materials. Also, most of the practice items are with four- and five-digit numbers.  It is questionable whether students would make meaningful progress toward addition and subtraction fluency, with these limited opportunities for practice that occur so early in the year.

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 4 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, pages 355HH, 355KK), 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.
  • OA.A.3 is the most explicit Grade 4 application standard, calling for students to represent and solve multi-step word problems using the four operations, and assess the reasonableness of their answers. Lessons that target this standard occur in the first four units of the program; these lessons appropriately connect the use of estimation and mental math to support student thinking and check answers, including lessons that focus on interpreting remainders in division word problems.
  • Unit 4 focuses on Equations and Word Problems. Lessons include instruction on situation versus solution equations, additive and multiplicative comparison problems, and identifying starting points in multi-step problems. It should be noted that for addition/subtraction word problems, most of the numbers are three digits or less, and one- or two-digit numbers for multiplication/division problems. Teachers using this program will want to expose Grade 4 students to problem solving with greater numbers to challenge students.
  • Each of the eight units in Grade 4 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 amusement parks, interactive games, engineering, and seasons.

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 4 program 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, and this leads to an under-emphasis on 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 4 is multi-digit addition and subtraction using the standard algorithm. There is an appropriate balance of conceptual and procedural work in this area, as students are building on learning from Grades 2 and 3. This fluency is explored and practiced consistently in Unit 1, but there isn’t a strong emphasis on this skill in subsequent units. The students continue 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 7 Lesson 2, students explore comparing fractions of different-size wholes. As students solve word problems that require comparison of fractions, they are directed to explain their reasoning and show work, allowing an opportunity for students to make connections between their conceptual understanding and their ability to apply this understanding in a situational context. A second example of this overlap is evident in Unit 2 Lesson 17, where students compare methods for multi-digit multiplication. Students use their understanding of place value and properties to look for similarities and differences in various methods of multiplication, as a way to access their conceptual understanding. This work also develops students’ procedural skill.
  • 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 the Grade 4 program materials, there are some topics that over-emphasize or jump quickly to procedural skill and fluency. The treatment of rounding begins with modeling on the MathBoard but quickly moves to oral conversations about places of numbers and rounding rules; more time with visual models like a number line may more effectively develop this understanding for students. Multi-digit division is another topic where there seems to be a strong emphasis on procedures and algorithmic skill, rather than the use of concrete and visual models.
  • 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: Place Value and Multi-digit Addition and Subtraction—30 percent of the items are primarily conceptual (items 1–5, 20), 60 percent are primarily procedural (items 6–17), and 10 percent are primarily application (items 18–19). The high number of procedural items is appropriate, as place value and multi-digit addition and subtraction are topics students have explored since Grade 2; the Grade 4 standards call for skill and fluency in these areas.
    • Unit 2 Review/Test: Multiplication with Whole Numbers—28 percent of the items are primarily conceptual (items 1–6, 25), 52 percent are primarily procedural (items 7–19), and 20 percent are primarily application (items 20–24). A high emphasis on procedural skill is reasonable for this content; however, since Grade 4 is the first opportunity for students to explore multi-digit multiplication, a more balanced assessment of conceptual and procedural work could be appropriate.
    • Unit 3 Review/Test: Division with Whole Numbers—30 percent of the items are primarily conceptual (items 1–6), 55 percent are primarily procedural (items 7–17), and 15 percent are primarily application (items 18–20). As with Unit 2, since this is students’ first exploration of multi-digit division, a more balanced assessment of conceptual versus procedural understanding could be applied.
    • Unit 4 Review/Test: Equations and Word Problems—32 percent of the items are primarily conceptual (items 1–6, 18–19), 44 percent are primarily procedural (items 7–17), and 24 percent are primarily application (items 20–25). The numbers here are reasonable, although one might expect that a unit focusing on word problems would have a higher number of application items on the unit assessment.
    • Unit 5 Review/Test: Measurement—25 percent of the items are primarily conceptual (items 1–5), 55 percent are primarily procedural (items 6–16), and 20 percent are primarily application (items 17–20). The conversion of measurements is a procedural process, so the high percentage of procedural items on this assessment is reasonable; however, converting measurements is rarely done without a context, so a higher number of application items could be present.
    • Unit 6 Review/Test: Fraction Concepts and Operations—16 percent of the items are primarily conceptual (items 1–4), 72 percent are primarily procedural (items 5–22), and 12 percent are primarily application (items 23–25). A stronger emphasis on the conceptual understanding of this content is expected, as Grade 4 is the first time students are using operations with fractions.
    • Unit 7 Review/Test: Fractions and Decimals—20 percent of the items are primarily conceptual (items 1–3, 5–6), 56 percent are primarily procedural (items 7–20), and 24 percent are primarily application (items 4, 21–25). As with Unit 6, Grade 4 is the first time students work with decimal numbers, so one might expect a higher emphasis on the assessment of conceptual understanding.
    • Unit 8 Review/Test: Geometry—60 percent of the items are primarily conceptual (items 1–6, 9–17), 8 percent are primarily procedural (items 7–8), and 32 percent are primarily application (items 18–25). Much of this work involves recognizing attributes (parallel, perpendicular) and working with angles; any items involving identifying and labeling were considered as conceptual understanding. The balance of items seems appropriate.

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 4 Expressions instructional materials partially meet the expectations for meaningfully connecting the CCSSM and 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 attending to the full meaning of these standards. Students using this program as designed have limited opportunities to critique the reasoning of others and develop mathematical communication skills. Overall, the Grade 4 program materials somewhat support teachers and students in rigorous instruction that includes the connection of mathematical practice and content standards.

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 instructional materials reviewed meet the expectations for identifying the MPs and using them to enrich the mathematics content. The MPs are clearly identified and used to enrich mathematics content and learning throughout the Grade 4 program materials. It is clear that 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 mathematical practices 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—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 3 students engage in MP3, “Construct a viable argument and critique the reasoning of others,” when they discuss and compare the methods used to complete multi-digit division equations (TE, page 267R). In Unit 6, students engage in MP2, “Reason abstractly and quantitatively,” when they discuss how to represent mixed numbers using paper fraction strips (TE, page 513R). Students answer teacher-posed questions that prompt them to reason about how their knowledge of unit fractions supports and informs their representations.
  • Within lessons, the MPs are identified in the teacher and student dialogue as they connect to specific activities; however, these sections don’t always include explicit content-practice connections. For example, in Unit 3 Lesson 6 the materials cite the use of MP8, “Use Repeated Reasoning,” as the teacher explains to students how making comparisons without numerical information is a useful skill that can be applied to various problem solving situations. More detailed information is necessary to support teachers in understanding how this example illustrates MP8. Similarly, Activity 2 in Unit 7 Lesson 4 (TE, page 622) cites the use of MP1, “Make Sense of Problems,” as students look for patterns in the multiplication table and explore how they can use these patterns to find equivalent fractions. It is unclear to teachers how this group activity connects to MP1.
  • 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 1 Lesson 14 gives students an opportunity to see connections between mathematics and engineering, as they analyze and represent data about the length of different bridges in the world. Unit 5 Lesson 8 presents students with a gardening context, in order to facilitate connections between this practice and students’ understanding of perimeter and area.

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 4 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 2, 14 of the 19 lessons have five or more MPs tagged in a single lesson. In Unit 5, 7 of the 8 lessons have five or more MPs tagged in a single lesson. In some cases, a single question within a class discussion is tagged as developing an identified MP.
  • On numerous occasions, the MPs are 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 MP, 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. While it is expected that students need some direct, explicit instruction in how to use mathematical tools as they are introduced, these supports should be gradually removed as student experience grows, to allow students to experience the full intent of this mathematics behavior. In the Grade 4 Expressions materials, this prescription of tools continues throughout the entire year. An example of this can be seen in Unit 6, where students explore fraction concepts. Lessons 2 and 4 prescribe the use of Class Fraction Cards for a whole group activity, where certain students are given large cards labeled with unit fractions to hold up in front of the group; Lesson 3 prescribes the use of pre-printed fraction bars; Lesson 10 engages students in solving word problems with fractions, and the teacher notes suggest that teachers tell students to use fractions strips or fraction bars to represent their thinking.
  • Class discussion is the most common setting for students’ work with MP6, “Attend to precision.” This 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. 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. For example, in Unit 2 Lesson 3, materials suggest that a sole volunteer explains a solution strategy (TE, page 130). In Unit 3 Lesson 5, an activity tagged as MP6 calls for “a student who found the correct answer using the Digit-by-Digit Method to share his or her solution steps” (TE, page 305). In the part of Unit 5 Lesson 6 that is tagged with MP6, the teacher notes state: “Discuss how students can find the area of rectangles X, Y, and Z…. Ask questions to help them understand that area is measured in square units” (TE, page 491).
  • 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 MPs 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
+
-
Indicator Rating Details

The Grade 4 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, “Construct Viable Arguments,” as is the case in Unit 4 Lesson 7 (TE, page 403), where students compare different equations for solving a two-step word problems to see how they are alike and different, with no suggestion for or time devoted to critiquing these solution methods. Another example of this abbreviation of MP3 occurs in Unit 6 Lesson 5 (TE, page 555), which calls for the students to compare three solutions for subtracting fractions greater than one that require ungrouping/regrouping. The materials suggest students look for ways the solutions are similar, and suggests that the teacher “point out” the ungrouping, again with no time spent critiquing these strategies.
  • While analyzing Unit 4, the team noted that MP3 was identified as a targeted practice in a number of lessons; however, there were no activities tagged with MP3 on TE pages 356, 359, 364, 365, 374, 375, 380, 382, 386, 396, 397 or 404.

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 1BB). 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 4 instructional materials partially meet the 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 for students to construct 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…,” or “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 4 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., estimate, rounding, partial products) and terms that are specific to this textbook program (i.e., Place Value Sections Method, comparison bars, Algebraic Notation Method).
  • 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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