4th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Meets Expectations | 88% |
|---|---|
Criterion 2.1: Rigor | 8 / 8 |
Criterion 2.2: Math Practices | 8 / 10 |
The materials reviewed for Grade 4 meet the expectations for gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency, and application. These three aspects are balanced within the lessons. The materials partially meet the expectations for the connections between the MP and the mathematical content. There are missed opportunities for identifying MPs and some instances where they are misidentified. The materials do attend to the mathematical reasoning that is embedded in the standards.
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 materials reviewed for Grade 4 meet the expectation for this criterion by providing a balance of all three aspects of rigor throughout the lessons. Within the concept development sections of each lesson the mathematical topic is developed through understanding as indicated by the standards and cluster headings. In Grade 4 fluency and procedural work includes 4.NBT.B.4 which asks students to add and subtract within 1,000,000. Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 3-10 minutes for each lesson.
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 in Grade 4 for this indicator meet the expectations by attending to conceptual understanding within the lessons.
- Within the concept development sections of each lesson, the mathematical topic is developed through understanding as indicated by the standards and cluster headings.
- Significant time is spent developing understanding fractions, place value and operations.
- Although some sample scripts offered to the teachers show procedural methods, the methodology, instructions and guiding questions are conceptual.
- Students spend time in module 1 working on rounding using their understanding of place value.
- Module 5 spends a significant amount of time developing an understanding of fraction addition and subtraction using equivalence.
- In module 5 students are guided to use their understanding of multiplication to generate equivalent fractions.
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 in Grade 4 for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Grade 4 this includes 4.NBT.B.4, which focuses on adding and subtracting within 1,000,000.
- Within the distribution of instructional minutes the schedule allows for 10-15 minutes per day to practice fluency. This varies according to the timeline of the school year and the focus mathematics in the module.
- Module 1 spends a significant amount of time on fluency for addition and subtraction with whole numbers.
- As described in "How to Implement A Story of Units," "Fluency is usually first-by beginning class with animated, adrenaline-rich fluency, students are more alert when presented with the Concept Development and Application Problems."
- Attention is paid to the use of the words "fluency" and "fluent" within the standards.
- Required fluencies are listed within the curriculum overview sequence.
- Lessons include mental strategies, problem sets, homework assignments and sprints.
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 in Grade 4 for this indicator meet the expectations by attending to application within the lessons.
- Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 3-10 minutes for each lesson in Grade 4.
- If the focus standard of the lesson includes language requiring application, the application problem will become the major portion of the lesson.
- Contextual word problems are used with a variety of problem types that increase in difficulty throughout the year. These problems focus on a variety of operations.
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 materials reviewed in Grade 4 for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always treated together nor are they always treated separately.
- The structure of the lessons and the distribution-of-minutes charts show a balance of the three aspects of rigor.
- Application problems often call for fluency and procedural skills.
- Fluency work and application problems are used to develop conceptual understanding.
- Conceptual problems often involve procedures.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The materials reviewed for Grade 4 partially meet this criteria. The Standards for Mathematical Practice are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MP, however, and in some instances they are misidentified. In module 1, students are directed to use a place-value chart and not given an opportunity to solve an open problem or make meaning of problem and solve. This is incorrectly identified as MP1. The materials often attend to the full meaning of each practice. However there are instances where the students are not using the practice as written. There is little explicit reference to modeling (MP4) and lessons identifying this practice incorrectly. There are lessons where the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 4 attend to the standards' emphasis on mathematical reasoning. For example, in module 5, students are asked to share their solution paths with their partners. Then they are asked, "Why is it necessary to decompose the total into ones and a fraction before subtracting? How does that relate to a subtraction problem such as 74- 28?" Students are prompted within problem sets, and application problems to explain, describe, critique, and justify. Each lesson includes a debrief section with questions for the teacher to use in facilitating classroom discussion about the mathematical content. Overall, the materials partially meet the criteria for practice-content connections.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The Standards for Mathematical Practice (MPs) are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MPs and some instances where they are misidentified.
- MPs are listed at the beginning of each module with a description of the explicit connection to the mathematics of the module.
- Module 4 describes an activity tied to MP3 ("Construct Viable Arguments and Critique the Reasoning of Others") as follows: "Knowing and using the relationships between adjacent and vertical angles, students construct an argument for identifying the angle measures of all four angles generated by two intersecting lines when given the measure of one angle. Students explore the concepts of parallelism and perpendicularity on different types of grids with activities that require justifying whether or not completing specific tasks is possible on different grids."
- MPs are listed in the margins of the teacher notes, mostly in the concept development portion and the student debrief of some lessons.
- In module 5, MP7 is correctly marked in a problem in which the understanding of 6x2 as repeated addition that can be displayed on a number line is linked to what 6x1/2 must mean and how it can then equal 3x2/2.
- While reviewers appreciate that MPs are not over identified or used in contrived situations, there are missed opportunities for identifying them in order to enrich the content in these lessons.
- The debrief section of the lessons offers an opportunity to highlight, for both teachers and students, how they might reason abstractly and quantitatively (MP2) and construct arguments and critique the reasoning of others (MP3).
- There is little explicit reference to modeling (MP4), and some lessons identify this practice incorrectly.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The materials often attend to the full meaning of each practice; however, there are instances where the students are not using the practice as written. For example, in many lessons the tools are chosen for the students or the modeling expected is a simple representation.
- Students are using the MPs when engaging with the content as designed, fully meeting Publisher's Criteria #9.
- Throughout the lessons the debrief section includes opportunities to construct viable arguments and critique the reasoning of others (MP3).
- In module 3, students practice MP4 to solve a real-world problem with modeling.
- Many lessons list MP without attending to the full meaning of the standard. For example, in module 1, students are directed to use a place-value chart and not given an opportunity to solve an open problem or to make meaning of a problem and solve. This is incorrectly identified as MP1. Another example in module 1 is that students are given place-value disks to use in solving the problem and this is listed as MP5, not attending to the full meaning which includes strategically choosing a tool.
- MP4 ("Model with Mathematics") is irregularly applied. There is ambiguity over whether "model" means to draw a picture representing the problem or whether it means to create a mathematical representation in a real-world context.
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 materials reviewed for Grade 4 meet the requirement of this indicator by attending to the Standards' emphasis on mathematical reasoning.
- Students are prompted within problem sets and application problems to explain, describe, critique and justify.
- In module 3, students are asked to "assess the reasonableness of your answer."
- In module 5, students are asked to share their solution paths with their partners. Then they are asked, "Why is it necessary to decompose the total into ones and a fraction before subtracting? How does that relate to a subtraction problem such as 74-28?"
- In module 5, students are asked on a problem set to "Explain the reasoning you used when determining whether 11/8 or 15/12 is greater.
- In module 5, students are asked to look at the work of their classmates in order to analyze the solution paths of others.
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 materials reviewed for Grade 4 meet the requirement of this indicator by attending to the standards' emphasis on mathematical reasoning.
- Each lesson includes a debrief section with questions for the teacher to use in facilitating classroom discussion about the mathematical content. For example, "Why is a vertical number line a good tool to use for rounding?"
- In a module 5, the teacher is prompted to ask students to explain why they might have different answers and the reasoning they used for each.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Grade 4 meet the requirement of this indicator by attending to the Standards' emphasis on mathematical reasoning.
- Each module lists terminology for the module including "new or recently introduced terms" and "familiar terms and symbols."
- In module 6 students are asked to attend to precise mathematical language in their work with classifying angles.