1st 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 1 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 MPs 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 1 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 Grade 1 attend to fluency and procedural work within the lessons. In Grade 1 this includes 1.OA.C.6 that asks students to add subtract within 10. Application problems occur in almost every lesson depending upon the lesson's focus mathematics. This is expected to last 5-7 minutes for each lesson. The three aspects are balanced within the lessons and modules. Overall, the Grade 1 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 Grade 1 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 an understanding of addition and subtraction and an understanding of place value and its impact on adding two-digit numbers and comparing larger numbers.
- In module 2, attention is paid to understanding place value by asking the students the following question: "How do you know 9 ones and 1 ten is the same as 1 ten and 9 ones?"
- In most debriefing sections students are asked to explain their thinking and justify their conclusions.
- Understanding of addition and subtraction and place value is built through contextual problems.
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 1 for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Grade 1 this includes 1.OA.C.6 "Add and Subtract within 10."
- 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."
- Lessons include sprints, dashes and flashcard activities.
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 1 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 5-7 minutes for each lesson in Grade 1.
- 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.
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 1 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 Grade 1 partially meet these criteria. MPs are often identified and often used to enrich mathematical content. There are many missed opportunities for identifying MPs and some instances where they are misidentified. Module 6 includes a lesson specifically devoted to modeling with mathematics, but it is not identified as such. 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 in which the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 1 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. For example, "What did we do today to make our counting strategy more efficient?" 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, the instructional materials only partially meet expectations because 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 MP3, one of the focus standards in this module: "Students describe and explain their strategies for adding within 40, and critique and adjust student samples to more efficiently solve addition problems."
- In module 4, the instructions read: "Repeat the process following the suggested sequence: 18+12, 28+12, 18+22, 16+23, 16+24, and 21+19. Students may choose to continue practicing adding on the tens first as in the previous exercise. Alternatively, they may start with trying to add the ones using the number bond or the arrow way, and explain their choice." This is correctly labeled as MP5, because students are expected to make and then explain their choice.
- 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 MP 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.
- Module 6 includes a lesson specifically devoted to modeling with mathematics, but it is not identified as such.
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, 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).
- In module 2, students are asked to use their fingers to point to their beads; this is incorrectly labeled as MP4.
- 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 1 meet the expectations 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 1, students are asked the following on a problem set: "Were there any problems that were just as easy using either method? Did you use a different method for any problems?"
- In module 3, students are asked to use an example from the problem set to explain their thinking.
- In module 6, students are asked to explain how knowing 3+6 can help them solve 30+60 as part of a classroom discussion.
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 1 meet the expectations 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, "What did we do today to make our counting strategy more efficient?"
- In module 1, teachers are prompted to ask, "Look at 7+1 and 6+2. Why is the total the same?"
- In module 4, teachers are prompted to ask the following: "Compare your work on problem 4 with your partner. Did you solve the same way? Do you think their way was an easier or harder way to solve? Explain why."
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Grade 1 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 5, students are asked to construct definitions of two- and three-dimensional shapes based on attributes, attending to mathematical language.