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.

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.

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.

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.

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.

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.