4th Grade - Gateway 2
Back to 4th Grade Overview
Note on review tool versions
See the series overview page to confirm the review tool version used to create this report.
- Our current review tool version is 2.0. Learn more
- Reports conducted using earlier review tools (v1.0 and v1.5) contain valuable insights but may not fully align with our current instructional priorities. Read our guide to using earlier reports and review tools
Loading navigation...
Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 77% |
|---|---|
Criterion 2.1: Rigor | 7 / 8 |
Criterion 2.2: Math Practices | 7 / 10 |
The instructional materials reviewed for Grade 4 partially meet the expectations for rigor and the mathematical practices. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. However, the materials partially meet the expectations for mathematical practices as they do not attend to the full meaning for each of the MPs and rarely prompt, or have the teachers prompt, students to analyze the arguments of others.
Criterion 2.1: Rigor
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.
The instructional materials reviewed for Grade 4 partially meet the expectations for rigor and the mathematical practices. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. However, the materials partially meet the expectations for mathematical practices as they do not attend to the full meaning for each of the MPs and rarely prompt, or have the teachers prompt, students to analyze the arguments of others.
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.
The materials reviewed for Grade 4 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. In the instructional materials visual representations, verbal explanations, and written equations are used to develop conceptual understanding.
- In Unit 6 Session 1.3 students use visual fraction models to generate equivalent fractions (4.NF.1). While working, students share their ideas with another student. Students also complete Student Activity Book pages 348 and 349 which require them to find equivalent fractions using fraction models and then provide a written explanation of how they know that the fractions are equivalent.
- In Unit 6 Session 2.3 student view Fraction Cards and are asked to determine which fraction is greater (4.NF.2). Students are told to use the picture on the cards and other knowledge that they have about fractions to help them decide. Teachers are told to listen for the use of “landmark fractions” in the student explanations. Students continue comparing fractions playing the game Capture Fractions and record equivalent fractions on Student Activity Book page 371.
- In Unit 5 Sessions 3.1 and 3.2 students create and hang a 10,000 chart in order to recognize that a digit in one place represents ten times what it represents in the place to its right (4.NBT.1).
- In Unit 1 Session 1.1 students draw rectangular arrays, write problems to represent the arrays, and calculate how many items are in an array (4.NBT.5). For Daily Practice, students are given real-world pictures of arrays and are asked to find the total and dimensions and draw the arrays.
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.
The materials reviewed for Grade 4 meet the expectations for giving attention throughout the year to individual standards that set an expectation for procedural skill and fluency. The materials include opportunities to practice and review in order to build procedural skill and fluency. Students are provided Daily Practice in every session and Homework in many sessions. The instructional materials provide direct instruction regarding the standard algorithm, 4.NBT.4, and additional practice is provided.
Standard 4.NBT.4 requires students to fluently add and subtract multi-digit whole numbers using the standard algorithm.
- In Unit 4, nine Ten-Minute Math activities address 4.NBT.4. The Ten-Minute Math activities are titled “Broken Calculator” and are found in Sessions 1.1, 1.2, 1.3, 1.4, 1.5, 3.1, 3.2, 3.3, and 3.4. These activities are presented to students without any explicit instruction on the standard algorithm, and many of these numbers are less than 1,000 or use 0 in the tens and ones place. For example, in Session 1.1 students are asked to create five expressions that equal 1,800. They can use either subtraction or addition and the 1 and 8 keys are broken. In Session 1.2 students create five expressions that 2,600 using addition and subtraction, and 2 and 6 broken. These Ten-Minute Math activities are not aligned to 4.NBT.4.
- In Unit 5, ten sessions- 1.4, 1.5, 1.6, 2.4, 2.5, 2.6, 2.7, 3.4, 3.5, and 3.6- provide the formal instruction for 4.NBT.4. Session 1.4 is the first time that students are formally taught the standard algorithm. In the opening activity students discuss and explain how two different algorithms work. The Discussion section provides support for the teacher to explicitly model the standard US algorithm. Practice is provided in student activity book pages 278-279. Session 1.5 provides more practice but no further review or instruction for using the standard algorithm. Session 1.6 provides support to review the steps with students. Session 2.4 includes an opening activity that provides an example of subtracting by place. This section explains how the standard US algorithm works step-by-step and includes Student Activity Book pages 298, 299, and 300. In Session 2.5 students practice and talk through the steps of subtraction problems, and in Session 2.6 students continue to subtract using the standard algorithm and practice problems including items with a story context. In Session 2.7 students subtract a number from a number with a zero in the tens place. Students talk through the steps of the problem without the teacher modeling or providing instruction. Sections 3.4, 3.5, and 3.6 provide practice with the standard algorithm.
- In Unit 6 students are provided with five formal opportunities to practice 4.NBT.4. In the Daily Practice portion of Sessions 1.1, 1.3, 1.4, 1.5, and 2.5 students are provided opportunities to add or subtract using the standard algorithm.
- In Unit 7 students are provided with another formal opportunity to practice 4.NBT.4 in the Daily Practice portion of Session 1.3.
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
The materials reviewed for Grade 4 meet the expectations for teachers and students spending sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.
Practice with application of the major work in fractions is limited to Unit 6. Standard 4.NF.3d, solve real-world problems involving addition and subtraction of fractions referring to the same whole and having like denominators, is found within Sessions 3.1, 3.2, 3.4, 4.2, 4.3, and 4.4. Standard 4.NF.4c, solve word problems involving multiplication of a fraction by a whole number, is in Sessions 4.1, 4.2, 4.3, and 4.4. For example, in Session 4.1 (4.NF.3d) students are required to solve word problems like “Jake bought three kinds of pizza for a party. Each pizza was the same size. By the end of the party, ¾ of the pizza was eaten. How much pizza was eaten in all?”
Practice for 4.OA.3 is found throughout five units of instruction. Through the first four units, materials focus on one-step problems, scaffolded two-step problems, or two-step problems with inverse operations. In Unit 8, Analyzing Patterns and Rules, the full intent of the standard is reached. In this Unit, students are generating tables from situations and then generating and/or using rules from those tables to solve word problems. However, often the activities are still very scaffolded.
Much of Unit 2 provides real-world application of 4.MD.A and 4.MD.B. Sessions include “How Many Raisins in a Box,” “How Tall are Fourth Graders,” and “How Tall are First Graders.” Students collect data from the real-world and use that data to create representations, compare data, and analyze data.
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.
The instructional materials reviewed for Grade 4 partially meet the expectations for balance of the three aspects of rigor within a grade. Although the instructional materials meet expectations for each aspect of rigor, these aspects of rigor are often addressed in separate parts of the Sessions. Materials targeting application are often scaffolded, detracting from the balance of rigor. Overall, the three aspects of rigor are most commonly treated separately.
In general, conceptual understanding, procedural skill and fluency, and application are adequately addressed in the Sessions; however, for the most part they are addressed in separate sections of the instructional materials. Conceptual understanding is typically addressed in the Discussion and Math Workshop portions of the Sessions. Procedural skill and fluency is typically introduced in separate Sessions and then practiced in the Daily Practice portion of sessions. Application consists of routine word problems in the instructional materials. As a result, all aspects of rigor are almost always treated separately within the curriculum including within and during Sessions, Practice, and Homework.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for Grade 4 partially meet the expectations for practice-content connections. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialize language of mathematics. However, the materials do not always attend to the full meaning of each MP, and there are few opportunities for students to analyze the arguments of others either through prompts from the materials or from their teachers.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials for Grade 4 meet the expectations for identifying the Standards for Mathematical Practice (MPs) and using them to enrich the mathematical content. The MPs are clearly identified in Implementing Investigations on page 44 and can also be found in each unit. The instructional materials highlight two MPs in every unit. During the sessions, Math Practice Notes dialogue boxes are given to provide tips to the teacher on how to engage students in the MPs. Additionally, Math Practice Notes are provided for the MPs that are not highlighted so that students continue to work on the practices all year.
The Introduction and Overview of each unit includes a “Mathematical Practices in this Unit” section. This section of each unit highlights the two MPs that are the focus of the unit. The MPs are described and examples from the unit are provided. A chart showing where Mathematical Practice Notes occur and when the MP is assessed is also included in this section.
- The Unit 2 “Mathematical Practices in this Unit” is found on pages 8-11. This unit focuses on MP2 and MP7. An example of MP2 from Session 2.2 is included.
- The Unit 8 “Mathematical Practices in this Unit” is found on pages 8-11. This unit focuses on MP4 and MP8. An example of MP4 from Session 1.2 is included, and an example of MP8 from Session 1.3 is included.
Math Practice Notes are provided in sessions alongside content. Math Practice notes are provided for the MPs highlighted within the Unit and MPs that are not the highlighted practices for the unit.
- Unit 1 Session 1.6 includes a Math Practice Note for MP7, a practice not highlighted in the unit. Students are noticing and using the properties of multiplication.
- Unit 4 Session 1.2 includes a Math Practice Note for MP6, a practice highlighted in the unit. The note discusses measurement differences.
- Unit 5 Session 3.1 includes a Math Practice Note for MP5 and MP7, practices highlighted in the unit. The note discusses how the 10,000 chart is a tool for understanding the structure of the base-10 system.
- Unit 7 Session 3.5 includes a Math Practice Note for MP8, a practice not highlighted in the unit. Students are working with problems that emphasize the relationship between multiplication and division.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 4 partially meet expectations that materials carefully attend to the full meaning of each practice standard (MP). Although the instructional materials attend to the full meaning of some of the MPs, there are some MPs for which the full meaning is not developed.
At times, the instructional materials only attend superficially to MPs. The following are examples:
- Unit 1 Session 1.1 in the Math Practice Note it lists MP5 and has students represent multiplication situations with arrays. The Math Practice Note talks specifically about what a rectangular array is and how it provides images students use to understand key properties. This activity only has students using arrays and does not allow them to choose any other tool.
- Unit 3, Session 2.2 in the Math Practice Note it lists MP4 and has students learning how to represent remainders in a division problem. In the Math Practice Note it states that while students are learning the division notations, teachers should emphasize “reading”each equation with meaning by referring to the context it is modeling. This session doesn’t allow for students to realize the mathematics present in the real-world problem, but has them notate and solve the problem and then refer back to it to find the meaning.
- Unit 3 Session 2.4 in the Math Practice Note it lists MP5 and has students play a Missing Factors game. In the Math Practice Note they state that this game is to learn how to use the rectangular array to model division. Students are not able to choose any tool other than an array in this lesson.
- Unit 6 Session 1.2 in the Math Practice Note it lists MP5 and has students finding fractional parts of a rectangle. In the Math Practice Note students are told to use a 4X6 rectangle.
At times, the instructional materials fully attend to a specific MP. The following are examples:
- Unit 6, Session 3.2 in the Math Practice Note it lists MP5 and has students adding fractions. In the Math Practice Note it states students choose which mathematical tool to use and the session supports this choice while offering suggestions such as number lines and rectangles, to represent addition of fractions. This session attends to the full meaning of MP5.
- Unit 8, Session 1.6 in the Math Practice Note it lists MP4 and has students work on a Window and Tower Activity where they can use tables and arithmetic expressions to model the number of floors and windows they would have in a tower taking into consideration if it is a single or double tower. This session allows for students to engage in real world problem situations.
Indicator 2g
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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.
The instructional materials reviewed for Grade 4 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade level mathematics.
When MP3 is referenced, students are often asked to solve and share solutions. The independent work of the student is most often about finding the solution to a problem without creating a viable argument. Students often listen to peer solutions without being asked to critique the reasoning of the other student. Much of the student engagement in the class discussion is teacher prompted without giving students the opportunity to create their own authentic inquiry in the thinking of others.
- Unit 6 Session 1.2 directs students to talk with a partner and find a way to explain whether ⅓ = ⅙ while directing teachers to “listen for explanations that focus on the relationship between thirds and sixths.” Students do not critique the reasoning of others during this activity.
- Unit 8 Session 1.8 asks students to explain how one penny jar will catch up to another and provides other students the opportunity to ask questions. However, those questions may not necessarily be a critique of the explanations originally offered.
At times, the materials prompt students to construct viable arguments and analyze the arguments of others.
- In Unit 6 Session 2.4 students work in groups to compare fraction cards to ½ by placing fraction cards between landmarks. This activity encourages group discussion with language such as “I understand your thinking, but here is my reason for placing this one (card) here.”
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.
The instructional materials reviewed for Grade 4 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.
Most of the time when MP3 is referenced, teachers are asked to have students share or explain their solutions. Teachers are also directed to have students ask questions but are not supported in focusing those questions toward critiquing the arguments of others.
- In Unit 1 Session 2.2 the teacher asks, “What strategies are you using to find the factors of 200 and 300?” The Math Practice Note connects this to critiquing the arguments of others. However, students do not critique, and no assistance is provided for the teacher to engage students in such a discussion.
- In Unit 2 Sessions 1.2 and 1.4 students are asked to construct arguments about height based on data. The teachers is assisted in supporting students with sentence stems like “Can anyone finish a sentence that starts ‘More than half our class…’ or ‘About half our class…’ The Math Practice Note focuses teacher attention on the construction of arguments based on data modeling.
- In Unit 6 Session 1.1 the teacher is instructed to provide the following directions: “Work with a partner to shade in and label these fourths and eighths on the 4 X 6 rectangles. You don't each have to do all of the fractions, but between the two of you, do all of them; and you have to agree that the fractions your partner did are correct.” This does not support the student in how to critique or express disagreement if the fractions done are incorrect.
- In Unit 6 Session 1.6 the teacher ask students to bring an activity book page to a discussion. The students are asked to show and explain how they know each pair of fractions is equivalent. The Math Practice Note states that “In this discussion, students connect the numerator and denominator with the ideas of the number of pieces and the size of the pieces, respectively, in order to begin to make arguments about equivalence.”
The materials assist teachers, at times, in engaging students in constructing viable and analyzing the argument of others.
- In Unit 2 Session 1.5 the teacher is prompted to provide the students with questions such as “What do you think of [___]’s idea?” or “How is [__] supporting what she is saying with evidence from the data?”
- In Unit 7 Session 2.2 the teacher is is given the following guided questions: “Let’s talk about Set A. The problem you want to solve is 37x52. Who wants to explain what cluster problems they used to help them solve 37x52? ...[Marisol] noticed she could multiply by 50s to get her answer, and [Emann] used what he knew about multiplying by 3 to multiply by 30. Did other people recognize multiplication facts or problems that you saw right way you could easily solve?”
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 4 meet the expectations for explicitly attending to the specialized language of mathematics.
The instructional materials provide opportunities for teachers to say mathematical terms to students during the whole group portion of the lessons. The materials use precise and accurate terminology when describing mathematics. New terminology is introduced on the summary page of the TE at the beginning of the session where it will first be used. The mathematical terminology is highlighted in italics throughout the sessions within the TE. There is also an index at the end of each unit manual in which math terms are listed for the unit.
- In Unit 2 Session 2.2 students are collecting data about shoe size. The materials prompt the teacher to state, “You’ve been collecting data that includes the fraction ½ . To think about how we might display that data using a line plot, let’s do a quick data collection.”
- In Unit 4 Session 2.1 students are identifying lines. The materials prompt the teacher to state, “There are two special kinds of lines we’re going to learn about-parallel lines and perpendicular lines.”
- In Unit 6 Session 2.5 students are comparing fractions. The materials prompt the teacher to state, “Could these two fractions be equal? Why or why not? Could thinking about equivalent fractions help? How?”