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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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