3rd 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 3 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.
*Evidence updated 10/27/15
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 3 meet expectations for rigor and balance. The instructional materials give appropriate attention to conceptual understanding, application, and procedural skill and fluency. The materials address these three aspects with balance, not 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, application, and procedural skill and fluency.
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 3 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.
- Each chapter includes a “Teaching for Depth” page. This page identifies strategies, research, mathematical practices, and professional development videos to help teachers achieve a deep understanding of content among students (for example, chapter 2, in the teacher guide on page 85E).
- The beginning of each lesson has an “About the Math” section to give teachers direction on teaching for depth and reasoning to teach each specific skill (for example, chapter 1, in the teacher guide on pages 5A and 17A).
The materials state each lesson has “Understand Concepts,” and many of them support conceptual understanding. For example, in chapter 1, lesson 2, on page 13 in the teacher edition, the “Share and Show” has students finding connections and sharing their thinking about rounding and place value. Chapter 3, lesson 1, has students using “Math Talk” to explain their thinking about making equal groups and how that leads to multiplication. In chapter 6, lesson 6, on page 334 in the teacher edition, students investigate, draw conclusions, and make connections to understand division.
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 3 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.
- The beginning of each lesson includes a “Fluency Builder” (for example, chapter 3, lesson 3, on page 151B in the teacher edition).
- Each lesson provides an “On Your Own” section, where students practice fluency and procedural skills (for example, chapter 3, lesson 2, on page 147).
- Each lesson includes a “Daily Routines” component with a problem of the day.
Many of the practice and homework pages are developing procedural skill and fluency (for example, chapter 9, lesson 5 on page 537).
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 3 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.
- Materials are designed for students to spend sufficient time working with engaging applications. Real-world applications are clearly identified in each lesson (“Think Smarter & Go Deeper”). An example can be found in chapter 4, lesson 5, in the teacher’s guide on page 217.
- Chapter 11 is focused on perimeter and area, and about half of the problems have a real-world context. For example, chapter 11, lesson 4, on pages 647 and 648 of the teacher edition, the homework pages for about half the problems have a context.
Many of the lessons have a “Problem-Solving and Applications” page where problems have a context. For example, see chapter 9, lesson 5, on page 536 of the teacher edition.
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 3 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.
- The three aspects of rigor are balanced throughout the material. These aspects are identified in each lesson under Rigor: Understanding Concepts (“Share & Show”), Procedural Skill and Fluency (“On Your Own”), and Application (“Think Smarter & Go Deeper”). An example can be found in chapter 2, lesson 2, on page 93A in the teacher edition.
- Another example can be found in chapter 1, lesson 3, on pages 19 and 20.
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 3 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 standard, especially MP 3, 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 3 meet the expectations for identifying the Standards for Mathematical Practice (MPs) 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.
- Each chapter includes a “Teaching for Depth” page that has an explanation of how the mathematical practices will be used throughout. For example, chapter 6, on page 299E in the teacher edition, states that students should explore different types of problems and contexts that pertain to division so that they can learn to make sense of problems and persevere in solving them through the use of groups, arrays, and diagrams using counters and drawings (MP 1).
- Each Lesson has a “Lesson at a Glance,” where the practices are noted for that particular lesson. For example, chapter 6, lesson 1 on page 301A in the teacher edition demonstrates how students can model with mathematics by considering a general form of an equation that pertains to dividing flowers equally among vases (MP 4).
- Each lesson has an “Explore” and “Explain” section for the teacher to prompt students to apply and work through the mathematical practices. An example of this is found in chapter 6, lesson 2, on page 309 in the teacher edition, where teachers can prompt students to write multiplication equations from division problems, which allows students to look for and make use of structure (MP 7).
- Some lessons also contain a box titled, “Mathematical Practices in Your Classroom.” For example, chapter 6, lesson 7, on page 343 in the teacher edition, is another instance where teachers can prompt students to look for and make use of structure between multiplication and division sentences (MP 7).
The MPs are identified in the teaching planning edition on pages 20 through 22. There are questions/prompts teachers can use on page 23 for each mathematical practice. Pages 24 through 29 identify each mathematical practice and the lessons that focus on each standard along with an explanation of what students learn to do.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 3 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.
- Overall, the materials attend to the full meaning of most of the MPs. One exception is in regards to MP 5, use appropriate tools strategically. In the materials, there are many instances when the MP 5 is listed, and students are given or directed to use a specific tool, which does not allow students to consider and choose the appropriate tool. Two examples of this can be found in chapter 6, lesson 4, on page 321 in the teacher edition, and in chapter 7, lesson 1, on page 365 in the teacher edition.
- There are instances where MP 5 is listed at the beginning of a lesson and within a lesson, yet there is no discussion within the teacher’s guide how MP 5 is addressed. For examples, see chapter 8, lesson 7, on pages 481 and 481A in the teacher edition, and chapter 7, lesson 4, on pages 383 and 383A in the teacher edition.
- In chapter 10, lesson 2, on page 567 in the teacher edition, MP 7 is referenced along with a question to ask the students. The question is not supporting students in looking at and make use of structure.
There are times when MPs are identified but do not appear in the lesson. For example, chapter 12, lesson 6, page 729A in the teacher edition identifies MPs 3, 6, 7, and 8, but the lesson never addresses MP 8.
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 3 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.
- Materials prompt students to construct viable arguments within grade-level mathematics. Examples can be found throughout the materials for opportunities to apply MP 3. Two examples can be found in chapter 10, lessons 4 and 5, in the teacher edition on pages 579 and 587. Lesson 10.4 asks students, “What is a different way to jump on the number line to solve the problem?” Lesson 10.5 prompts teachers to extend children’s thinking by having students explain how they could solve exercise 2 a different way.
- The “Think Smarter” problems and “Go Deeper” sections of each lesson prompt teachers to help engage students in constructing viable arguments. For example, see chapter 4, lesson 7 on page 232.
- There are several places where MP 3 is identified in the student materials, but it does not always meet the intent of the standard. For example, chapter 2, lesson 6, page 119 in the student edition, identifies MP 3 as a practice the students will engage in, but students are not required to create an argument or engage with other students.
Another example is in chapter 3, lesson 7, on page 177 in the student edition. MP 3 is listed, but no direction is given for students to construct arguments or critique arguments.
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 3 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.
- In the “Planning Guide,” page 26 is dedicated to identifying where examples of MP 3 are located throughout the materials. The “Planning Guide” also lists examples of MP 3 on page 98.
- In chapter 11, lesson 9, on page 676 in the teacher edition, the “Math Talk” box asks students, “Explain how the perimeters of example A and Example B are related. Explain how the areas are related.”
- In chapter 12, lesson 6, on page 732 in the teacher edition, the “Problem Solving/Thinking” box explains, “Exercise 9 requires students to apply what they know about rectangles to determine why the statement is incorrect. Suggest students list the characteristics of a rectangle and compare to the figure shown.”
- There are several places where MP 3 is identified, but all places do not meet the meaning of the standard. For example, chapter 4, lesson 7, on page 232 in the teacher edition, lists MP 3, but instead of having students construct an argument and critique the reasoning of others, it is asking for them to simply look at the work provided.
In chapter 11, lesson 1, on page 628 in the teacher edition, MP 3 is listed, but instead of engaging in constructing arguments or critiquing others, students simply find the perimeter of other students’ drawings.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 3 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 has a page titled “Developing Math Language.” For example, see chapter 8, on page 441H in the teacher edition.
- Each chapter includes a “Vocabulary Builder” and “Vocabulary Game” section, such as in chapter 8, on pages 442, 442A, and 442B in the teacher edition.
- Each lesson has a “Language Objective” and “Vocabulary Builder” exercise. An example can be found in chapter 8, lesson 1, on pages 443A and 443 B in the teacher edition.
- In the materials, there are routinely sections of each lesson that are connected to math language. This happens specifically in the “Math Talk” and “Vocabulary Preview” sections of each lesson.
The questions for teachers to ask use correct vocabulary. For example, in chapter 9, lesson 3, on page 519 in the teacher edition, the questions for teachers and students use correct vocabulary such as “denominator.”