4th Grade - Gateway 2

The Grade 4 Expressions instructional materials partially meet the expectations for developing the conceptual understanding of key mathematical concepts. This program consistently devotes instructional time to the use of models and mathematical language to explore and develop understanding of grade level concepts. However, additional focus on 4.NBT and 4.NF is needed to fully develop conceptual understanding of these important concepts.

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

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

The Grade 4 Expressions instructional materials partially meet expectations for attention to procedural skill and fluency. The program materials give some attention to the individual standards that set an expectation of procedural skill and fluency (4.NBT.B.4), but this attention is not sustained throughout the year.

• A component of the Expressions program is a “Fluency Plan for helping students achieve fluency with the CCSSM that are suggested for each grade” (TE, page xxvi). The Grade 4 program attends to both procedural fluency with addition and subtraction (4.NBT.B.4) and intervention for those students that still need practice with multiplication facts. The plan includes Diagnostic Quizzes, Practice Sheets, Quick Practices, and Fluency Checks.
• Grade 4 materials promote the use of strategies based on place value and properties of operations, building on students’ learning from Grade 3. Lessons engage students in both pure and applied mathematics exercises to develop multi-digit procedural fluency with addition and subtraction. Students explore various methods for adding and subtracting multi-digit numbers, with time and attention given to sharing and analyzing work with peers.

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

The Grade 4 program instructional materials partially meet the expectations for giving attention to all three aspects of rigor, both in individual lessons and in units of study. The rigor aspects are treated separately and together over the course of the year, depending on the content and lesson activities. While there is some balance of the three components, there tends to be a heavy emphasis on procedural skill and fluency in some areas, and this leads to an under-emphasis on conceptual understanding.

• Each lesson generally devotes some time to conceptual understanding and application. Using models and emphasizing mathematics talk are two important components that illustrate a daily focus on developing conceptual understanding. Each lesson includes an “Anytime Problem” that is independent of the current unit of study, allowing students daily practice in applying skills and understandings to solve routine word problems. 
• The CCSSM required fluency for Grade 4 is multi-digit addition and subtraction using the standard algorithm. There is an appropriate balance of conceptual and procedural work in this area, as students are building on learning from Grades 2 and 3. This fluency is explored and practiced consistently in Unit 1, but there isn’t a strong emphasis on this skill in subsequent units. The students continue to practice this skill in the context of their work with single- and multi-step word problems with the four operations.
• In many lessons, there is a strategic overlap of the aspects of rigor. For example, in Unit 7 Lesson 2, students explore comparing fractions of different-size wholes. As students solve word problems that require comparison of fractions, they are directed to explain their reasoning and show work, allowing an opportunity for students to make connections between their conceptual understanding and their ability to apply this understanding in a situational context. A second example of this overlap is evident in Unit 2 Lesson 17, where students compare methods for multi-digit multiplication. Students use their understanding of place value and properties to look for similarities and differences in various methods of multiplication, as a way to access their conceptual understanding. This work also develops students’ procedural skill.
• The Puzzled Penguin provides opportunities throughout the year for students to analyze the Penguin’s mistakes and give written feedback to correct his thinking. The Puzzled Penguin’s errors are almost always procedural in nature.
• In the Grade 4 program materials, there are some topics that over-emphasize or jump quickly to procedural skill and fluency. The treatment of rounding begins with modeling on the MathBoard but quickly moves to oral conversations about places of numbers and rounding rules; more time with visual models like a number line may more effectively develop this understanding for students. Multi-digit division is another topic where there seems to be a strong emphasis on procedures and algorithmic skill, rather than the use of concrete and visual models.
• In addition to daily lessons/unit plans, the team analyzed the balance of rigor in Review/Tests for each unit as well.
  • Unit 1 Review/Test: Place Value and Multi-digit Addition and Subtraction—30 percent of the items are primarily conceptual (items 1–5, 20), 60 percent are primarily procedural (items 6–17), and 10 percent are primarily application (items 18–19). The high number of procedural items is appropriate, as place value and multi-digit addition and subtraction are topics students have explored since Grade 2; the Grade 4 standards call for skill and fluency in these areas.
  • Unit 2 Review/Test: Multiplication with Whole Numbers—28 percent of the items are primarily conceptual (items 1–6, 25), 52 percent are primarily procedural (items 7–19), and 20 percent are primarily application (items 20–24). A high emphasis on procedural skill is reasonable for this content; however, since Grade 4 is the first opportunity for students to explore multi-digit multiplication, a more balanced assessment of conceptual and procedural work could be appropriate.
  • Unit 3 Review/Test: Division with Whole Numbers—30 percent of the items are primarily conceptual (items 1–6), 55 percent are primarily procedural (items 7–17), and 15 percent are primarily application (items 18–20). As with Unit 2, since this is students’ first exploration of multi-digit division, a more balanced assessment of conceptual versus procedural understanding could be applied.
  • Unit 4 Review/Test: Equations and Word Problems—32 percent of the items are primarily conceptual (items 1–6, 18–19), 44 percent are primarily procedural (items 7–17), and 24 percent are primarily application (items 20–25). The numbers here are reasonable, although one might expect that a unit focusing on word problems would have a higher number of application items on the unit assessment.
  • Unit 5 Review/Test: Measurement—25 percent of the items are primarily conceptual (items 1–5), 55 percent are primarily procedural (items 6–16), and 20 percent are primarily application (items 17–20). The conversion of measurements is a procedural process, so the high percentage of procedural items on this assessment is reasonable; however, converting measurements is rarely done without a context, so a higher number of application items could be present.
  • Unit 6 Review/Test: Fraction Concepts and Operations—16 percent of the items are primarily conceptual (items 1–4), 72 percent are primarily procedural (items 5–22), and 12 percent are primarily application (items 23–25). A stronger emphasis on the conceptual understanding of this content is expected, as Grade 4 is the first time students are using operations with fractions.
  • Unit 7 Review/Test: Fractions and Decimals—20 percent of the items are primarily conceptual (items 1–3, 5–6), 56 percent are primarily procedural (items 7–20), and 24 percent are primarily application (items 4, 21–25). As with Unit 6, Grade 4 is the first time students work with decimal numbers, so one might expect a higher emphasis on the assessment of conceptual understanding.
  • Unit 8 Review/Test: Geometry—60 percent of the items are primarily conceptual (items 1–6, 9–17), 8 percent are primarily procedural (items 7–8), and 32 percent are primarily application (items 18–25). Much of this work involves recognizing attributes (parallel, perpendicular) and working with angles; any items involving identifying and labeling were considered as conceptual understanding. The balance of items seems appropriate.

The Grade 4 instructional materials partially meet expectations for attending to the full meaning of each of the eight MPs. By repeatedly aligning lessons to multiple practice standards, the grade-level materials don’t attend to the full meaning of each of the practice standards.

• The MPs are clearly visible throughout each lesson; however, the tendency to include multiple practice standards in an individual lesson does not allow for careful attention to the full meaning of the practices. For example, in Unit 2, 14 of the 19 lessons have five or more MPs tagged in a single lesson. In Unit 5, 7 of the 8 lessons have five or more MPs tagged in a single lesson. In some cases, a single question within a class discussion is tagged as developing an identified MP.
• On numerous occasions, the MPs are abbreviated or altered when included in the program materials. For example, MP3, “Construct a viable argument and critique the reasoning of others,” is simplified to “Critique the reasoning of others” or “Construct viable arguments;” MP7, “Look for and make use of structure,” is abbreviated to “Use structure;” and MP5, “Use appropriate tools strategically,” is shortened to “Use appropriate tools.” While it is reasonable that a given activity may only target one part of a specific MP, it is concerning that the rationale for abbreviating these practices is not made explicit. In addition, this shorthand notation of the MPs downplays the importance of the full meaning of these practice standards as the CCSSM authors intended them.
• MP5, “Use appropriate tools strategically” calls for students to self-select tools for a given context or situation, and to be strategic both in how they choose and use tools for a mathematical task. In a number of activities that are tagged with this MP, students are prescribed specific tools to use, rather than selecting tools themselves. In many of these cases, the teacher models the appropriate use of the tool, negating any opportunities for students to be strategic. While it is expected that students need some direct, explicit instruction in how to use mathematical tools as they are introduced, these supports should be gradually removed as student experience grows, to allow students to experience the full intent of this mathematics behavior. In the Grade 4 Expressions materials, this prescription of tools continues throughout the entire year. An example of this can be seen in Unit 6, where students explore fraction concepts. Lessons 2 and 4 prescribe the use of Class Fraction Cards for a whole group activity, where certain students are given large cards labeled with unit fractions to hold up in front of the group; Lesson 3 prescribes the use of pre-printed fraction bars; Lesson 10 engages students in solving word problems with fractions, and the teacher notes suggest that teachers tell students to use fractions strips or fraction bars to represent their thinking.
• Class discussion is the most common setting for students’ work with MP6, “Attend to precision.” This standard calls for students to use the language of mathematics and to communicate about mathematics in a clear and precise way. MP6 is tagged in each of the 150 lessons. In many instances, where MP6 is identified, questions are posed in a whole-group setting by the teacher while individual students respond, or the teacher does much of the explaining. Often the corresponding work in the Student Activity Book doesn’t follow up on these questions, so all students are not given opportunities to practice this behavior or held accountable for engaging in this practice. For example, in Unit 2 Lesson 3, materials suggest that a sole volunteer explains a solution strategy (TE, page 130). In Unit 3 Lesson 5, an activity tagged as MP6 calls for “a student who found the correct answer using the Digit-by-Digit Method to share his or her solution steps” (TE, page 305). In the part of Unit 5 Lesson 6 that is tagged with MP6, the teacher notes state: “Discuss how students can find the area of rectangles X, Y, and Z…. Ask questions to help them understand that area is measured in square units” (TE, page 491).
• The Puzzled Penguin activities are labeled as MP3, “Construct a viable argument and critique the reasoning of others.” While these activities do allow students to critique another’s work, many of the Penguin’s errors are procedural in nature and do not involve genuine meaningful mathematical reasoning. These activities might be more accurately tagged as MP6, “Attend to precision,” as they facilitate opportunities for students to consider and write about the precise nature of mathematical procedures.

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

The Grade 4 instructional materials partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.  Overall, the materials offer consistent opportunities for students to construct viable arguments, but opportunities to analyze the arguments of others are inconsistent and don’t hold students accountable for engaging in this behavior.

• The last lesson of each unit includes an “Establish a Position” activity (eight lessons total), where students are given a mathematical statement, and they must decide if the statement is true or false and justify their thinking verbally and/or in writing. Volunteers are asked to share their positions, and the other students are allowed to question the volunteer for clarification or to verify reasoning.
• Students are seldom asked to critique the reason of others. During Math Talks and Math in Action, students often share mathematical methods with the class, but they are rarely asked to critique the reasoning of those presenting content. The identification of MP3 in many of these lessons is, “Construct Viable Arguments,” as is the case in Unit 4 Lesson 7 (TE, page 403), where students compare different equations for solving a two-step word problems to see how they are alike and different, with no suggestion for or time devoted to critiquing these solution methods. Another example of this abbreviation of MP3 occurs in Unit 6 Lesson 5 (TE, page 555), which calls for the students to compare three solutions for subtracting fractions greater than one that require ungrouping/regrouping. The materials suggest students look for ways the solutions are similar, and suggests that the teacher “point out” the ungrouping, again with no time spent critiquing these strategies.
• While analyzing Unit 4, the team noted that MP3 was identified as a targeted practice in a number of lessons; however, there were no activities tagged with MP3 on TE pages 356, 359, 364, 365, 374, 375, 380, 382, 386, 396, 397 or 404.

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

The Grade 4 instructional materials partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the program’s teacher materials consistently provide opportunities for students to construct viable arguments. However, teachers using this program would need to seek out or construct additional opportunities for students to engage in critiquing the reasoning of others.

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

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

The Grade 4 instructional materials partially meet the expectations for attending to the specialized language of mathematics. While this program promotes classroom discussion and mathematics talk, there is not a strong emphasis on developing mathematical vocabulary or communicating mathematics understanding in writing.

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