4th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 83% |
|---|---|
Criterion 2.1: Rigor | 8 / 8 |
Criterion 2.2: Math Practices | 7 / 10 |
The instructional materials reviewed for Grade 4 partially meet the expectations for rigor and mathematical practices. The instructional materials meet the expectations for the criterion on rigor and balance with a perfect score, but they only partially meet the expectations of the criterion on practice-content connections because they do not fully attend to the meaning of each mathematical practice standard. Overall, the instructional materials are strong in regards to rigor, identifying mathematical practices and the language of mathematics, but need improvements in consistently attending to the full meaning of practice standards where they are identified.
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 meet expectations for rigor and balance. The instructional materials give appropriate attention to conceptual understanding, procedural skill and fluency, and application, and the materials address these three aspects with balance, but always treating them separately and not always together. Overall, the instructional materials reflect the balances in the CCSSM, which helps students meet rigorous expectations by developing conceptual understanding, procedural skill and fluency, and application.
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 instructional materials reviewed for Grade 4 meet the expectations for giving attention to conceptual understanding. Overall, the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
- In each chapter and lesson in Go Math, the sections that cover conceptual understandings are identified. For instance, chapter 1, lesson 1, page 5A indicates that conceptual understanding problems will be covered in the "share and show" sections of the lesson. In this section, students use a model to find the place value of an underlined digit and compare the value of underlined digits.
- Of the 103 lessons in the grade, 71 are primarily conceptual in nature and match the standards calling for conceptual understanding. For example, in chapter 2, lesson 3, the cluster heading for the objective is called "Use Place Value Understanding and Properties of Operations to Perform Multi-digit Arithmetic"; the lesson has students using drawings and number lines to build understanding.
- The beginning of each chapter has a page heading called "Teaching for Depth." For example, see chapter 6, teacher's manual, page 325E, which specifically identifies what students should have conceptual understanding of in that particular lesson.
- Another example of "teaching for depth" is chapter 6, lesson 2, teacher's manual, page 333A.
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 instructional materials reviewed for Grade 4 meet the expectations for giving attention to procedural skill and fluency. Overall, the materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
- Each lesson in Go Math identifies where it will focus on procedural skills and fluency. For example in chapter 1, lesson 1, this is identified on page 5a as being in the "on your own" section of the lesson.
- Each lesson has daily routines and a fluency builder. For instance, in chapter 1, lesson 1, the problem of the day is a word problem where students must find the difference in length between two rivers. For fluency, there are eight problems where students have to write the number value of ones, tens, and hundreds (example: 5 tens=50).
- Of the 103 lessons in the grade, 27 focus primarily on procedural skill and fluency in nature and match the standards calling for procedural skill and fluency. For example, in chapter 4, lesson 3, the objective is to "solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted."
- Another example of a fluency builder can be found in chapter 5, lesson 1, page 279B.
- Each chapter has "practice and homework" section to help with fluency. For example, chapter 5, lesson 1, pages 283-84 of the teacher edition.
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 instructional materials reviewed for Grade 4 meet the expectations for giving attention to applications. Overall, the 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.
- In each lesson and chapter in Go Math, the application problems are identified. In chapter 1, lesson 1, page 5a, the application problems are identified as being part of the "go deeper and think smarter" problems, which are 13-17.
- Of the grade's 103 lessons, five are primarily application in nature and match the standards calling for real-world application. For example, chapter 13, lesson 1, the objective is "apply the area and perimeter formulas for rectangles in real world and mathematical problems," and students are using picture frames and gardens to solve area problems.
- In Go Math, each lesson has a section called "Think Smarter and Go Deeper"; it is meant to give students practice with application while engaging them in the major work of the grade. The majority of these sections in Grade 4 are application problems. For example, in chapter 4, lesson 8, questions 14-16 on page 244 are "think smarter and go deeper" problems.
- Chapter 2, lesson 11, teacher's manual, page 127 is another example of using "think smarter and go deeper" as an application problem.
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 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together nor are they always treated separately within the materials, and there is a balance of the three aspects of rigor within the grade.
- In each lesson and chapter of Go Math, the three elements of rigor are identified. At times, they are covered separately, and sometimes they overlap. For instance, question 13, in chapter 1, lesson 1, is identified as being a procedural skill and fluency problem as well as one that addresses application.
- Each chapter specifically outlines how rigor is addressed and balanced. For example, see chapter 3, lesson 4, teacher's manual, page 163A.
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. The materials meet expectations for identifying the practice standards and explicitly attending to the specialized language of mathematics as addressed in indicators 2E and 2G.iii, respectively. However, the materials only partially meet the expectations for attending to the full meaning of each practice standard and engaging students in mathematical reasoning as addressed in indicators 2F, 2G.i, and 2G.ii. Overall, the materials do not attend to the full meaning of every practice standards, especially MP3, and therefore only partially meet the practice-content connections criteria.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials reviewed for Grade 4 meet the expectations for identifying the Standards for Mathematical Practice (MP) and using them to enrich the mathematical content. Overall, the instructional materials do not over-identify or under-identify the MPs, and the MPs are used within and throughout the grade.
- The identified MPs are found in the "lesson at a glance" portion of each lesson. For example, in chapter 1, lesson 1, this can be found on page 5A.
- MP3 is addressed in the "math talk" section of chapter 9, lesson 6.
- In chapter 9, lesson 6, mathematical practices 2, 6, 7, and 8 are addressed.
- The practices are related to grade-level work in the "teaching for depth" section, located in the teacher's manual of each chapter. For example, see chapter 5, teacher's manual, page 277C.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 4 partially meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the instructional materials carefully attend to the full meaning of some of the practice standards but not for all of them.
- Materials sometimes attend to the full meaning of the practice standard. For example, in chapter 3, lesson 5, page 171 of the teacher edition, MP7 is referenced along with questions to ask the students. The questions are structured to help students see and make use of structure.
- An example of not meeting the full standard is found in chapter 4, lesson 5, page 221 of the teacher edition. It lists MP2, but it provides questions with right and wrong answers that do not allow students to reason.
- The full meaning of MP5 is for students to consider all available tools to solve a math problem. In some instances, the Grade 4 materials give students the tool to use for solving the problem, which means not having them choose strategically. For example, in chapter 7, lesson 3, teacher's manual, page 398, students are told to use a number line to solve a problem.
- In chapter 2, lesson 3, teacher's manual, page 77, students are asked to communicate about products that end in zero, which is not MP5 even though it is listed as such.
- In chapter 4, lesson 1, teacher's manual, page 198, the box titled "Math Talk" is identified as MP5 and asks students to explain their thinking, which again would not fall under this practice.
- Chapter 10, lesson 2, teacher's manual, page 555, activity 1 lists MP5, however this is not what would be considered using appropriate tools strategically.
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 detailed in the content standards. Overall, the materials consistently allow students to construct viable arguments, but they do not consistently prompt them to analyze other students' arguments.
- The materials do not consistently prompt students to construct viable arguments and analyze the arguments of others. There are several places where MP3 is identified on the student materials, but the material does not always meet the intent of the standard.
- For example, chapter 6, lesson 6, page 359 of the student edition lists MP3 for the students, but gives no directions for having students create an argument or engage with other students.
- Chapter 7, lesson 3, page 397 of the student edition again MP3 is listed but gives no directions for students to construct or critique arguments.
- In chapter 4, lesson 9, students are to analyze the explanation and work of two other students and explain how they would solve on their own.
- Chapter 10, lesson 5, teacher's manual, page 575 has students explain how to tell if a shape has line symmetry, but they do not have to analyze the argument of another student.
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. Overall, the materials consistently assist teachers in having students construct viable arguments, but they do not consistently assist teachers in having students analyze other students' arguments.
- There are several places where MP3 is identified, however the instructional materials do not meet the meaning of the standard.
- Chapter 10, lesson 5, page 575 of the teacher edition lists MP3, but instead of having students construct an argument and critique the reasoning of others, it has them give a right or wrong question.
- Chapter 11, lesson 1, page 601A of the teacher edition lists MP3, but in the lesson there is no guidance for teachers to implement the practice or prompt students.
- Chapter 2, lesson 2 and chapter 10, lesson 2 lists MP3, but the practice is not included in the lessons. Chapter 4, lesson 9 has students analyze the explanations and work of two other students and explain how they would solve on their own. Exemplar responses are included for each question and response.
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. Overall, the materials for students and teachers have multiple ways for students to engage with the vocabulary of mathematics used throughout the materials.
- Each chapter begins with vocabulary builders and vocabulary games. An example can be found in chapter 12 on pages 640-640B.
- The lessons pay attention to using correct vocabulary. For example, the questions for teachers to ask use correct vocabulary. This can be seen in chapter 12, lesson 1, page 642 of the teacher edition where the questions for teachers to ask use correct terms such as "centimeters," "millimeters," "mass" and "grams."
- The instructional materials provide teachers with strategies to help students: using semantic maps to analyze relationships, introducing the new words for the chapter, making vocabulary cards and writing prompts to help students learn the vocabulary.
- The instructional materials have recommendations and examples for how to use a mathematics journal.
- The beginning of each chapter includes a page called "Developing Math Language." For example, see chapter 12, page 639H of the teacher's manual.