Kindergarten - 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 Kindergarten meet the expectation for gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency, 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 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 Kindergarten meet the expectations 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. The materials reviewed in Kindergarten for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Kindergarten this includes K.OA.5, which asks students to add and subtract within 5. Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around five minutes for each lesson in Kindergarten. The three aspects are balanced within the lessons and modules. Overall, the Kindergarten materials meet the criteria for rigor and balance.
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 Kindergarten 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.
- In module 1, specific lessons are devoted to the understanding of zero.
- In module 4, significant time is spent developing an understanding of addition and subtraction using a variety of settings. For example, students are asked, “How did we use the number bond to show putting things together?”
- In module 5, students work with models to develop conceptual understanding of the teen numbers.
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 Kindergarten for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Kindergarten this includes K.OA.A.5 "Add and subtract within 5."
- Within the distribution of instructional minutes the schedule allows for 10-20 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 within the "how to implement" documents, "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 section called "Curriculum Overview Sequence."
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 Kindergarten 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 five minutes for each lesson in Kindergarten.
- If the focus standard of the lesson includes language requiring application, the application problem will become the major portion of the lesson.
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 Kindergarten for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always combined, nor are they always separate.
- The distribution-of-minutes chart and the structure of the lessons 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 Kindergarten partially meet this criterion. MPs are often identified and often used to enrich mathematical content. There are missed opportunities for identifying MPs, and some instances where they are misidentified. 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, there are lessons where the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Kindergarten attend to the standards' emphasis on mathematical reasoning. Overall, the materials partially meet the criterion 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, however, it only partially meets expectations. 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 6 describes MP7, one of the focus standards in this module: "Students make use of their understanding of a shape's attributes to build three-dimensional from two-dimensional shapes."
- MPs are listed in the margins of the teacher notes, mostly in the "Concept Development" and the "Student Debrief" portions of some lessons.
- While reviewers appreciate that the MPs are not over-identified or used in contrived situations, there are many missed opportunities for identifying MPs as a way 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).
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The materials only partially meet the expectations. While they often attend to the full meaning of each practice, there are instances where the students are not using the practice as written. For example, there are many lessons where the tools are chosen for the students or the modeling expected is a simple representation.
- Students are using 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).
- The overview of module 1 describes MP4-related activity: "Students model decompositions of three objects as math drawings and addition equations (page v)." There is no mention of a contextual situation.
- MP4 ("Model with Mathematics") is irregularly applied. There is ambiguity over whether "model" means to draw a picture 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 Kindergarten meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.
- Students are prompted within problem sets, interviews and application problems to explain, describe, critique and justify.
- In module 1 the debrief section suggests using the frame "we have zero_______ in our classroom" to talk about items in or not in the classroom.
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 Kindergarten 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, when working on teen numbers and place value, students are asked, "How does finding parts help you to understand big numbers better?"
- In module 4, students are asked if there were many different answers today or if there were different ways to solve to get an answer.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Kindergarten 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, students are asked to be explicit about the attributes of shapes.
- In module 4 the debrief section includes a question focused on language: "How did we use our math words to talk about what happened in the song?"