Alignment to College and Career Ready Standards: Overall Summary

The Grade 3 Expressions instructional materials do not meet the requirements for alignment to the CCSSM. These materials partially meet expectations for Gateway 1. The lessons and assessment focus appropriately on the major work of the grade. However, these materials do not fully foster coherence; the program is not consistent with the mathematical progression of learning set forth in the standards, including only a brief nine-lesson unit on fractions at the very end of the year. The Grade 3 program materials partially meet expectations for Gateway 2. The program includes all three aspects of rigor although conceptual understanding of multiplication and division concepts, as well as fraction concepts, is underdeveloped. The MPs are an intentional aspect of this program; however, these standards are not attended to fully within lessons and units and only partially support students’ development of mathematical reasoning.

See Rating Scale
Understanding Gateways

Alignment

|

Partially Meets Expectations

Gateway 1:

Focus & Coherence

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

Gateway 2:

Rigor & Mathematical Practices

0
10
16
18
12
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

Partially Meets Expectations

+
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Gateway One Details

The instructional materials reviewed for Grade 3 partially meet the expectations for Gateway 1 focus and coherence. Some examples were found of above grade-level content being assessed, but the occurrence of these items was infrequent, and the items could be easily revised or removed. The instructional materials do spend the majority of the time on the major clusters of the grade, spending much more time on multiplication and division than on fractions. Major work includes all clusters within 3.OA and 3.NF, and clusters A and C within 3.MD. The fraction domain is not well-developed in this grade-level program. The instructional materials reviewed for Grade 3 partially meet the expectations for coherence. The materials often use supporting content as additional opportunities to engage in the major work of the grade. For example, partitioning shapes within the geometry domain is used to support students’ understanding of 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 only partially consistent with the Mathematical progression of learning set forth in the standards, as it only includes 16 days of learning around fraction concepts. This grade-level program is visibly shaped by the cluster headings in the standards. Connections are made between domains and clusters within the grade level. Overall, the Grade 3 materials partially meet the expectations for 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 3 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 3 meet the expectations for focus within assessment. The majority of unit review/test items included in the student activity books assess content from prior and/or current grade levels.

  • In units 1 and 2, one of the test objectives is “recall basic multiplications and divisions with 0, 1, 2, 3, 4, 5, 9, and 10.” 3.OA.C.7 calls for students to “know from memory all products of two one-digit numbers by the end of Grade 3” and this program assesses fact recall after two units of instruction and then not again on any other unit review/tests.
  • Unit 2 review/test:
    • Item 25 is a word problem that includes the language “twice the number of…” This sounds like multiplicative comparison work (4.OA.A.1-2), but this is Mathematically reasonable for Grade 3 because it is used in the context of a word problem.
  • Unit 3 review/test:
    • Items 18-19 have large numbers in word problems with multiplicative contexts; however, problems could be solved with additive thinking. It should be noted that the lessons related to these items didn’t use such large numbers.
  • Unit 4 review/test
  • Items 7-17 don’t allow for much workspace, which may discourage students from using strategies based on place value, properties of operations, and the relationship between multiplication and division, as explicitly called for in 3.NBT.A.2. This could easily be rectified by reformatting the items.
  • Unit 7 Review/Test:
    • In item 6, the figures are already shaded and divided. Also, the equation ¼ + ¼ + ¼ = ¾ is a Grade 4 expectation (4.NF.B.3b), but this equation is Mathematically reasonable based upon 3.NF.A.1and 3.NF.A.2.
    • Standard 3.NF.A.3.D calls for comparing fractions with like numerators/denominators, and item 18 has neither. However, the standard also calls for comparing by reasoning, and Grade 3 students should be able to reason that 2/2 = 3/3 because they both name 1 whole (3.NF.A.3.C).
    • Standard 3.NF.A.3.D calls for comparing fractions with like numerators/denominators, and item 20 has neither. However, the students are given visual models to support their reasoning, so the item is appropriate for Grade 3.

*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 3 meets the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes all clusters within the 3.OA, all clusters in 3.NF and cluster A and C from 3.MD. The fraction domain is not well developed in these lessons.

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 3 meet the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes all clusters within the 3.OA, all clusters in 3.NF and cluster A and C from 3.MD.

  • Units 1, 2 and 7 are completely focused on major work.
  • Units 3, 5 and 6 spend most of the lesson/assessment time on major work.
  • Unit 4 is not major work.
  • These materials spend a great deal of time on multiplication and division, but not enough time is spent on fractions. Only 9 lessons in one unit at the end of the year is focused on fractions.

Criterion 1c - 1f

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
5/8
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-
Criterion Rating Details

The instructional materials reviewed for Grade 3 partially meet the expectation for coherence. The materials often use supporting content as a way to continue work with the major work of the grade. For example, partitioning shapes within the geometry domain is used to support 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 not consistent with the mathematical progression of learning set forth in the standards, including only including a few lessons on fractions. These instructional materials are visibly shaped by the cluster headings in the standards. Connections are made between domains and clusters within the grade level. Overall, the Grade 3 materials partially support coherence and are not 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.
1/2
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-
Indicator Rating Details

The instructional materials reviewed for Grade 3 partially meet the expectations for their use of supporting content as a way to enhance coherence. For Grade 3, reviewers focused on the use of data and shapes as methods for supporting operations and fractions.

  • Unit 3 contains the only example of using a scaled bar graph as a connection to multiplication.
  • Unit 3 misses opportunities to connect data to the 3.OA domain.
  • Measurement data in fractions of an inch are not connected to the fraction work of the grade level. It is instead taught as a stand-alone concept.

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 3 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 3.
  • 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.
0/2
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-
Indicator Rating Details

The instructional materials reviewed for Grade 3 are not consistent with the mathematical progressions in the standards and do not meet the expectation for this indicator.

  • The problem types included in the application problems show an increasing level of difficulty in multiplication.
  • Much of the school year is devoted to multiplication and division.
  • Fractional measurement is included in lessons before working with fractions on a number line.
  • Unit 6 requires fraction language (one-sixth) before fractions are taught in unit 7.
  • Only 9 lessons on Grade 3 fractions are contained at the end of the year. This does not allow for a complete progression of learning for Grade 3 fractions nor does it offer extensive work on grade level problems in fractions.
  • Support offered to help struggling students while still engaging with 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 4, 5 and 6. This work 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 2 and Grade 4 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 3 foster coherence through grade level connections.

  • Some lesson and assessment objectives include language shaped by the cluster headings in the standards.
  • Connections are made between 3.OA and 3.MD.
  • Connections are made between 3.G and 3.NF.A.1.

Gateway Two

Rigor & Mathematical Practices

Partially Meets Expectations

+
-
Gateway Two Details

The Grade 3 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, and all eight MPs are included in a way that connects logically to the mathematical content. There is an under-development of conceptual understanding with multiplication and fraction concepts. 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.
6/8
+
-
Criterion Rating Details

The Expressions instructional materials for Grade 3 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 both separately and together throughout the program as dictated by lesson content and activities. In many cases there is a strategic overlap of these aspects to help students make meaningful connections and develop a deeper understanding of Grade 3 content. However, while the program materials in Grade 3 are strong in developing procedural skill and fluency and providing opportunities for practical applications, the attention to conceptual understanding is inconsistent in depth, frequency, and quality. A Grade 3 student using this program may not emerge with a solid understanding of the concepts of multiplication/division or fractions as numbers, two highly important foundational topics for future mathematical learning.

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 3 Expressions instructional materials partially meet expectations for developing conceptual understanding of key mathematical concepts.  This program consistently devotes instructional time to the use of models and mathematical language to introduce and develop understanding of grade level concepts.  However, with the concepts of multiplication, division, and fractions, all key concepts introduced in Grade 3, not enough time and attention is dedicated to developing a conceptual understanding of these foundational concepts.

  • A strong component of the Expressions Grade 3 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. For example, in Unit 1 Lesson 7 the teacher and students have a discussion about the equation they are writing where they link each number and symbol they write to its meaning in the context of the problem.
  • Mathematics drawings and visual models are used in various contexts to support students’ understanding, including the key areas of multiplication/division and fractions. However, these models are often utilized during whole class, teacher-directed discussion without 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 1 shows that only 12 items in the 19 lessons call for students to construct a written explanation of their mathematical understanding of multiplication concepts. An analysis of Unit 7 shows that only 12 items in 9 lessons call for students to construct a written explanation of their mathematical understanding of fraction concepts. 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 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 1 Lesson 15 explores multiplication and addition properties, including the associative property of addition/multiplication, zero property, division rule for 1, identity property of addition/multiplication, and commutative property of addition/multiplication, all within that singular lesson. A second example of this occurs in Unit 7 Lesson 3 where the focus is to locate fractions on a number line. Students explore strategies for partitioning number lines, using unit fractions to locate non-unit fractions, whole numbers as fractions, and fractions with different denominators on the same number line. These are complex concepts that warrant more time for first-time fractions learners.
  • The program materials introduce abstract ideas, including symbols and equations, before developing a solid understanding of multiplication concepts. An example of this occurs in the first lesson of the program. Unit 1 Lesson 1 introduces the multiplication symbol, and the terms factor and product and calls for students to write equations for facts of 5 as part of their initial exposure to the concept of multiplication.
  • As students learn different multiplication facts, the program quickly encourages the use of shortcuts as strategies. For example, as students are introduced to multiplying by 5 in Unit 1 Lesson 1, they count by 5 and discuss how all multiples of 5 have a 0 or 5 in the ones place. There is little time or attention given to concretely looking at equal groups of five objects. Another shortcut is introduced in Unit 1 Lesson 8, as students are introduced to 9s multiplication. The lesson explicitly teaches a Quick 9s strategy, which is simply a trick with fingers that has no connection to the concept of equal groups.
  • The Grade 3 multiplication lessons do not allow adequate time for students to explore the meaning of multiplication; instead, the materials lend themselves to the idea that multiplication is a series of facts to be practiced and remembered. This is apparent in an analysis of Units 1 and 2, which mark the first time students explore multiplication: 21 of the 34 lessons deal with fluency strategies, and most lessons focus on facts in isolation (Multiply and Divide with 5s, Building Fluency with 2s, 5s, 9s, and 10s, Practice with 6s, 7s, 8s).
  • Program materials provide minimal opportunities for concrete exploration of multiplication concepts. Three of the nineteen lessons in Unit 1 and one of the fifteen lessons in Unit 2 call for the use of concrete materials. Teachers using this program will want to supplement with more concrete learning experiences for students to begin exploring multiplication concepts.
  • There is an imbalance between multiplication and division concepts. While the curriculum relates multiplication to division early to develop a connected understanding of these operations, division is not developed fully. There are examples in lessons where students look at various representations (equal group models, repeated addition, arrays) and write a multiplication sentence; however, there are missed opportunities to use visual representations where students were asked to write a division sentence. Division is introduced only in word problems or equations related to multiplication problems.
  • Several lessons include opportunities for students to look at a visual representation of multiplication and write an equation. However, there are few opportunities for students to create a representation of equal groups as a way to interpret a multiplication or division expression, as suggested by 3.OA.A.1 and 3.OA.A.2. Being able to move back and forth between visual representations, equations, and contexts is an indication of solid conceptual understanding.
  • The Grade 3 program’s treatment of area as it relates to multiplication is solid. These lessons are strong examples of conceptual development for multiplication and division. Students concretely tile to build rectangles and use this work to find more efficient methods of finding a total, which eventually extends to exploring the distributive property. These lessons were well scaffolded across several units, beginning with Unit 1 Lesson 11, making the connection between area and multiplication early; later in the year, when students are more comfortable with multiplication and division concepts and better able to apply this understanding in different contexts, area is more fully explored.

Grade 3 is the first time students have formal instruction around fractions. The Expressions lessons use concrete and visual models to begin to develop students’ conceptual understanding of fractions as numbers (examples: concrete paper shapes, numbers lines and fraction bars). However, the unit devoted to exploring fractions is only nine lessons, which is not enough for such an important foundational concept in mathematics.

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.
2/2
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Indicator Rating Details

The Grade 3 Expressions instructional materials meet expectations for attention to procedural skill and fluency.    The program materials give adequate attention throughout the year to the individual standards that set an expectation of procedural skill and fluency: 3.OA.C.7 and 3.NBT.A.2.

  • A central component of the Grade 3 program is a “Path to Fluency.” The materials outline seven different routines that are designed to develop multiplication fluency (3.OA.C.7). There is commentary (TE page xxvi and 1EE–1MM) to help teachers understand both how to execute the routines and why each routine is important to students’ development. Students engage in these practice routines daily, beginning with the first lesson of the program. 
  • The program incorporates opportunities for repeated practice in the introductory multiplication and division units and, over the course of the year, to develop multiplication fact fluency. These opportunities include fluency routines, games, Quick Practices, Practice Charts, Daily Study Plans, Study Sheets, Check Sheets, Dashes, Strategy Cards, Diagnostic Tests, Games, and Fluency Checks.
  • The program devotes entire lessons to the development of multiplication fact fluency (Unit 1 Lessons 6, 9, 14, 17 and 18; and Unit 2 Lessons 8, 13 and 14), as well as lessons where fluency is practiced and developed through the context of word problems (Unit 1 Lesson 16 and Unit 2 Lessons 2, 4, 9, 10 and 11).
  • Instructional materials facilitate the exploration of opportunistic and efficient strategies for multiplication. Multiplication facts are introduced in an order that allows students to grasp the easiest patterns of multiples first. As students become fluent with these easier facts, they use them to derive more difficult facts. Students also engage in mathematics talk to discuss patterns with multiples and in the multiplication table that help reinforce their facts. Additionally, students solve problem and then compare solution methods. While these discussions do not go so far as to compare efficiency of solution paths, a teacher could easily modify the discussions to help promote fluency.
  • The Grade 3 program’s “Path to Fluency” includes developing multi-digit addition and subtraction skills (3.NBT.A.2). Students explore and discuss various addition and subtraction methods for numbers within 1000; lessons engage students in both pure and applied mathematics exercises to develop multi-digit procedural fluency.

Both fact fluency and procedural fluency with multi-digit numbers are assessed through formative fluency checks and summative Unit Review/Tests.

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 3 Expressions instructional materials meet expectations for attention to applications. The program materials are designed in a way to allow teachers and students to spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of the grade.  Overall, this grade level program allows students opportunities to apply mathematical skills and understandings in various ways.

  • The program consistently presents single- and multi-step word problems for students to solve in all seven units and in connection with a variety of grade level content.
  • OA.D.8 is the most explicit Grade 3 application standard, calling for students to represent and solve two-step word problems using the four operations. The Grade 3 materials strategically scaffold application opportunities from single- to multi-step to help students develop proficiency in solving problems. For example, when students are initially introduced to multiplication and division concepts in Unit 1 and 2, they work with single-step word problems. As students develop skill and fluency with multiplication and division, they extend their work to two-step problems with these operations.
  • Unit 5 is entirely devoted to solving single- and multi-step word problems with students exploring each of the different one-step problem types and the equations that accompany them. Students are asked to compare and contrast word problems to equations. Students are also introduced to problem-solving tools and mathematics drawings that help them make sense of and solve particular problem types. The “Research & Background” section for Unit 5 walks teachers through the different problem types, with commentary on how to support students as they write both situation and solution equations, as well as how to foster student dialogue about multiple solution paths.
  • Each of the seven units in Grade 3 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 hobbies, sports, recipes, news, and gardening.

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 3 instructional materials partially meet expectations for balancing the three aspects of rigor. These grade-level 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. However, the balance is heavy toward procedural skill and fluency in Grade 3, particularly with multiplication concepts. 

  • Each lesson generally devotes some time to each aspect of rigor. Using models and emphasizing mathematics talk are two other important components that illustrate a daily focus on developing conceptual understanding. There is an opportunity to practice fluency included in almost every lesson, as described in the “Path to Fluency.” Each lesson also 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.
  • In many lessons, there is a strategic overlap of the aspects of rigor, using word problems and real world situations to explore concepts and develop fluency. For example, in Unit 4 Lesson 12, students explore the use of ungrouping to subtract. As students solve word problems involving ungrouping within 1000, they are directed to show their work numerically and with proof drawings, allowing an opportunity for students to make connections between the conceptual and procedural nature of this skill. A second example of this overlap is evident in Unit 6 Lesson 9, where students are applying their understanding of the concepts of area and perimeter to solve word problems. In addition to solving the problem, students need to recognize if the problem relates to area or perimeter, and they must draw a diagram to represent each situation.
  • Grade 3 marks the introduction to multiplication for elementary students. The learning seems to focus briefly on an understanding of multiplication and then moves quickly to practice and recall of multiplication facts without allowing for deep exploration of multiplication concepts. The lesson topics in Unit 1 indicate a heavier emphasis on procedural fluency than conceptual understanding: Lessons 2, 3, 4, and 11 focus on Multiplication as Equal Groups, Arrays, and Area, and the Meaning of Division.  Lessons 1, 5-10, 12-15, and 17-18 focus on multiplying with specific numbers in order to build fluency. This emphasis on fact fluency continues in Unit 2, with 9 of the 15 lessons focused on individual numbers and/or building fluency.
  • An understanding of fractions is also introduced to students for the first time in Grade 3. This is another area where there wasn’t an appropriate emphasis on conceptual understanding. Lesson 1 begins exploring the idea of unit fractions and jumps right into building fractions from unit fractions. Lessons 2-3 explore fraction bars and fractions on the number line, and by Lesson 4, students are expected to compare unit fractions. Grade 3 students would benefit from more time spent exploring the idea of a unit fraction before moving forward to use unit fractions to build and compare fractions. 
  • 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 balance of rigor in Review/Tests for each unit was analyzed as well.
    • Unit 1 Review/Test: Multiplication and Division with numbers 0–5, 9, and 10—Twenty percent of the items are primarily conceptual (items 1–5), 68 percent are primarily procedural (items 6–22), and 12 percent are primarily application (items 23–25). While all three aspects of rigor are present and assessed, the lack of conceptual items is concerning, and the amount of procedural items seems high for a first introduction to the concept of multiplication.
    • Unit 2 Review/Test: Multiplication and Division with 6–8, and Multiply with Multiples of 10—Twenty percent of the items are primarily conceptual (items 1–5), 64 percent are primarily procedural (items 6–21), and 16 percent are primarily application (items 22–25). The lower percentage of conceptual items on this assessment is more reasonable, as it is a continuation of multiplication concepts and is building on students’ understanding of these concepts to develop multiplication fluency.
    • Unit 3 Review/Test: Measurement, Time, and Graphs—Twenty-five percent of the items are primarily conceptual (items 1–5 ), 40 percent are primarily procedural (items 6–13), and 35 percent are primarily application (items 14–20). The balance of rigor in these items is appropriate for the content, as time and measurement are skill-based and lend themselves to applications.
    • Unit 4 Review/Test: Multi-digit Addition and Subtraction Review/Test—Thirty 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). A higher number of procedural items is appropriate for multi-digit addition and subtraction in Grade 3, as students are moving toward procedural skill with standard algorithms.
    • Unit 5 Review/Test: Write Equations to Solve Word Problems—Thirty percent of the items are primarily conceptual (items 1–3), and 70 percent are primarily application (items 4–10). With a focus on solving word problems, a strong emphasis on application is reasonable for this assessment. Although none of these items have a primary focus on fluency, the use of procedural skill is embedded within the work students do to solve the various word problems.
    • Unit 6 Review/Test: Polygons, Perimeter, and Area—60 percent of the items are primarily conceptual (items 1–12), and 40 percent are primarily application (items 13–20). A strong emphasis on conceptual understanding is reasonable in this case since students are studying the concepts of area, perimeter, and attributes of polygons.

Unit 7 Review/Test: Explore Fractions—Eighty percent of the items are primarily conceptual (items 1–10, 13–18), 10 percent are primarily procedural (items 11–12), and 10 percent are primarily application (items 19–20). A heavy emphasis on conceptual understanding is expected and appropriate for a unit on fractions since Grade 3 is the first time this content is being explored.

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 3 Expressions instructional materials partially meet the expectations for meaningfully connecting the CCSSM. This program does a solid job of weaving in all eight of the math practices 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 sometimes limits teachers and students to investigating these eight 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 3 materials somewhat support teachers and students in rigorous instruction that includes the connection of mathematics 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 MPs are clearly identified and used to enrich mathematics content and learning throughout the Grade 3 instructional 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. It is important to note 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.

  • The MPs align and connect with the content of daily lessons rather than being included as stand-alone topics.
  • The MPs are clearly identified for teachers in multiple places such as in the Introduction, Unit Planning, Research & Math Background/Getting Ready to Teach Unit, the introductory page of each lesson, and within daily lessons.
  • 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 on—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 2 students engage in MP6 when they describe a strategy for finding the product 7 x 7, and when describe patterns they see in count-bys and equations (TE p. 173X). In Unit 5, students engage in MP2 when they recognize similarities and differences between two solution methods/diagrams. Students also engage in MP2 when they compare two numbers by reasoning about the value of the digits.
  • 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. For example, Activity 1 in Unit 5 Lesson 4 (TE page 584) cites the use of “MP5 Use Appropriate Tools” as students compare numbers to the thousands, but the commentary doesn’t explicitly state what tool is being used. Is it the MathBoard or place value drawings? Similarly, the “Path to Fluency” activity in Unit 2 Lesson 6 (TE page 224) cites “MP2 Reason Abstractly and Quantitatively” as students practice solving multiplication and then division equations with square numbers. The materials do not state the reasoning that students should be engaging in to demonstrate this habit of mind.

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 3 Lesson 14 gives students an opportunity to explore connections between mathematics and sports as they collect, record, and organize broad jump data. Unit 7 Lesson 9 engages students in paper folding in order to facilitate connections between this art and students’ understanding of fractions.

Indicator 2f

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

The Grade 3 instructional materials attend to each of the eight MPs multiple times throughout the year. 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 1, 18 of the 19 lessons have four or more MPs tagged in a single lesson. In Unit 4, 16 of the 18 lessons have four or more practices tagged in a single lesson. In some cases, a single question within a class discussion is tagged although the question does relate to the identified practice standard.
  • 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 math behavior. In the Grade 3 Expressions materials, this prescription of tools continues throughout the entire year. An example of this can be seen in Unit 7 where students are introduced to fractions. Lesson 1 prescribes given paper models and drawn fraction bars; Lesson 2 directs students to draw fraction bars; Lesson 3 prescribes the use of drawn and given number lines; Lesson 4 engages students in comparing unit fractions using given fraction bars; Lesson 5 calls for students to use given paper models of fraction circles to explore and compare non-unit fractions; Lesson 6 moves to exploring equivalent fractions, using given fraction strip models and folded fraction strips; and Lesson 7 calls for students to mark and label equivalent fractions on the number line, with the note: “Ask them to be as accurate as possible and suggest that they use a ruler” (TE page 802).
  • In relation to MP5, these materials sometimes have a loose interpretation of what qualifies as a mathematical tool. While items like rulers, place value models, and fraction strips are valid and important tools for students to explore and gain experience with, items like MathBoards, Strategy Cards, and Study Sheets (copies of completed multiplication fact equations) don’t seem to warrant being categorized in the same way.
  • 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. For example, in Unit 1 Lesson 1 (TE page 7) students are never asked to communicate the concept being learned. The teacher is asked to point out specific vocabulary that is highlighted on the student worksheet. Students do not have to understand vocabulary terms to complete the worksheet. The worksheet does not have any cloze sentence where about students might have to determine the appropriate word choice. In Unit 2 Lesson 4, teachers choose two or three students to read original word problems aloud, and “Have the class discuss whether each problem is of the correct type” (TE page 206). In the part of Unit 5 Lesson 1 that is tagged with MP6 (TE page 559), the teacher notes state: “Point out to the class the various solution methods students use. Discuss the different ways students labeled their work.” Teachers using this program may need to analyze the materials and decide with activities tagged with MP6 best exemplify the intent of this practice standard.
  • 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 much 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
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Indicator Rating Details

The Grade 3 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/or don’t hold students accountable for engaging in this behavior.

  • The last lesson of each unit includes an “Establish a Position” activity (seven 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.
  • During Math Talk, students often share mathematical methods with the class; however, students are rarely pressed 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 464), where students make proof drawings and share their different methods with no suggestion for or time devoted to critiquing these solution methods. Another example of this abbreviation of MP3 occurs in Unit 6 Lesson 2 (TE page 666), which calls for the students to individually and then collectively construct a definition for a parallelogram.
  • 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 1DD). 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.

While analyzing Unit 4, the team noted that MP.3 was identified as a targeted practice in a number of lessons; however, there were no activities tagged with MP3 on TE pages 420, 425, 432, 440, 444, 448, 460, 470, 476, 478, 488, 498, 514, 518, 524, 530, 536 and 541.

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.
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Indicator Rating Details

The Grade 3 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 guiding questions the teacher can use to support students in constructing viable arguments, 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 critiquing the reasoning of others beyond the Puzzled Penguin activities.

Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.
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Indicator Rating Details

The Grade 3 instructional materials partially meet the expectations for attending to the specialized language of mathematics.

  • Many lessons have vocabulary terms listed in the Teacher Edition and the Student Activity Book. The vocabulary words include general mathematical terms (i.e., equation, variable, line plot, unit fraction).
  • Each assessment begins with a vocabulary section that targets 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

Title ISBN Edition Publisher Year
null 9780547824741 null null null
null 9780547824970 null null null

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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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