5th 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 5 are aligned to the CCSSM. The materials are focused within assessments and spend the majority of time on the major work of the grade. The materials are also coherent, following the progression of the standards and connecting the mathematics within the grade level. The Grade 5 materials include all three aspects of rigor and there is a definitive balance between conceptual understanding, fluency and application. MPs are identified and used to enhance the mathematical content, but the materials often do not attend to the full meaning of each MP and some are misidentified.
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 5 meet the expectation for this criteria 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 5, fluency and procedural work includes 5.NBT.B.5 ("Multi-digit Multiplication"). Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 5-10 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 5 for this indicator meet the s 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.
- In module 2, students work on conceptual understanding of division and place value with decimals.
- In module 4 students work on their understanding of products.
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 5 for this indicator meet the requirements by attending to fluency and procedural work within the lessons. In Grade 5 this includes 5.NBT.B.5 ("Multi-digit Multiplication")
- 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.
- 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."
- After working with visual models and distributive property, students move to fluency with multidigit multiplication.
- 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 5 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 5-10 minutes for each lesson in Grade 5.
- 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 5 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 5 partially meet this criteria. The MPs are often identified and often used to enrich mathematics content. There are missed opportunities for identifying MPs, however, and in some instances they are misidentified. In module 5 (Multiplication and Division of Fractions and Decimal Fractions), out of 33 lessons, MPs are referenced in only 15. 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, module 6 includes lessons that are labeled with MP1 but then give students step-by-step methods for solving the problem, not attending to the full meaning of the practice. There is little explicit reference to modeling (MP4), and some lessons identify 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 5 attend to the standards' emphasis on mathematical reasoning. 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 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 6 describes an activity tied to MP2 ("Reason Abstractly and Quantitatively"): Students reason abstractly and quantitatively as they interpret the steepness and orientation of a line given by the points of a number pattern. Students attend to the meaning of the values in an ordered pair and reason about how they can be manipulated in order to create parallel, perpendicular, or intersecting lines."
- 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 4, students are asked to model with mathematics (MP4) while they are interpreting fraction as division, and this is identified in the teacher's edition.
- In module 1, students engage with structure (MP7) when representing one-thousandth and three-thousandths in standard, expanded and unit form; this is identified in the teacher's edition.
- In module 1, students reason (MP7) about place value; this is identified in the teacher's edition.
- In module 2, students are reasoning abstractly and quantitatively (MP2) within the application problem; this is identified in the teacher's edition.
- In module 5, "Multiplication and Division of Fractions and Decimal Fractions," out of 33 lessons, MPs are referred to in only 15 lessons.
- While reviewers appreciate that MPs are not over identified or used in contrived situations, there are many missed opportunities for identifying them as tools to enrich the content in these lessons.
- The debrief section of the lessons offers an opportunity to highlight, for 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 are asked to "discuss with your partner what is happening to the pieces, the units, when the numerator and denominator are getting larger" (MP7).
- In module 5, students are expected to have the "demonstrating students receive and respond to feedback and questions from peers" (MP3).
- There is ambiguity over whether "model" means to draw a picture representing the problem or whether "model" means to create a mathematical representation.
- Many lessons list MPs without attending to the full meaning of the standard. For example, module 6 includes lessons that are labeled MP1 but then give students step-by-step methods for solving the problem, not attending to the full meaning of the practice.
- In module 3, MP5 is identified without including an option for tools to be strategically chosen and used.
- In module 4 there is a suggestion that the first step in solving the problem is to model the problem, but there is little reference to the modeling process.
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 5 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 prove another (fictional) student wrong using pictures.
- In the module 2 assessment, students are asked to "explain your reasoning, including how you decided where to place the decimal point."
- In module 2, students are shown the work of another student and asked to "Explain her mistake using decimal division."
- In module 3, students are asked to share their partner's methods for solving.
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 5 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 taking 1 half of 2 halves equal to 1 half?"
- In module 2, teachers are prompted to ask during a classroom discussion, "Are both expressions acceptable?"
- In module 5, teachers are asked to "have the demonstrating students receive and respond to feedback and questions from peers."
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Grade 5 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 2 pays attention the use of the word "expression" by the students and teachers.
- In module 6, students are asked to develop a definition of a coordinate plane, using precise mathematical language.