8th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 77% |
|---|---|
Criterion 2.1: Rigor | 7 / 8 |
Criterion 2.2: Math Practices | 7 / 10 |
The instructional materials for the Grade 8 partially meet the quality expectations for rigor and MPs. The instructional material meets the expectations for the criterion of rigor and balance by reflecting the balances of all three aspects of rigor throughout the lessons and helping students meet the standards rigorous expectations. Within the concept development sections of each lesson, the mathematical topic is developed through understanding as indicated by the standards and cluster headings. Procedural skill and fluency is a focus throughout the material. It is most evident in Module 3 which is devoted to 8.EE.7 and 8.EE.8.B. Application of the mathematical concepts is evident in real-world problems in the beginnings of lessons and in guided and independent practice.
The instructional materials for Grade 8 partially meet the expectations for the practice-content connections criteria. The MPs are identified and often used to enrich mathematical content. Materials sometimes attend to the full meaning of each practice standard. There are many places where students are prompted to construct viable arguments and analyze the work of others. However, there are many places where the label does not match the problem or the problem covers more than one practice, but only one is listed. Materials are very limited in assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Also, only some materials actually attend to the specialized language of mathematics. Overall, the instructional materials partially meet the quality expectations for Gateway 2 in rigor and the mathematical practices.
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 8 meet the quality expectations for these criteria by reflecting the balances of all three aspects of rigor throughout the lessons and helping students meet the standards' rigorous expectations.
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 8, procedural skill and fluency is most evident in module 3 which is devoted to 8.EE.7 and 8.EE.8.B. There are places that practice fluency throughout the material. Application of the mathematical concepts is evident in real-world problems in the beginnings of lessons and in guided and independent practice. In the instructional materials, the three aspects are balanced within the lessons and modules.
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 8 for this indicator meet the expectations by attending to conceptual understanding within the lesson.
- In module 3, there is conceptual development of the connection between proportional relationships, lines and linear equations. There are guided practice activities that have the students use whiteboards to draw graphs of the independent and dependent variables from a table to see the line they make when the points are connected.
- In module 3 (8.EE.B), modules 4 and 6 (8.F.A) and modules 9-11 (8.G.A), there were ample opportunities for students to develop these concepts. All modules had opportunities to answer thought provoking questions.
- Connections and further conceptual development continue in module 4, when the module introduces non-proportional relationships and connects them to y=mx+b. Another great place to help build understanding is the "Unpacking the Standards" material, which shows exemplars of what it looks like when the standard is completed correctly.
- In module 9, conceptual understanding is taught by using models and tracings to demonstrate that figures are congruent after translations occur.
- In module 10, students will connect similarity to proportionality when learning about dilations.
- In module 12, students again will engage with whiteboards, use manipulatives and watch interactive models to demonstrate the Pythagorean Theorem.
Each module has an evaluation page that states the concept and skill and how knowledge develops along with the mathematical practice. Throughout the unit and modules, teachers are directed to key ideas for which students should develop a deep understanding. Items in the teacher edition that will aid in conceptual understanding in every lesson or module include:
- "Unpacking the Standards"
- "Reading Startup/Visualize Vocabulary/Active Reading"
- Questioning strategies
- Engagement with whiteboards or other manipulatives
- "Avoid Common Errors"
- "Focus on Critical Thinking"
- Blue heading boxes that provide different tips
- Many questions on guided practice and independent practice that have students explain their answers.
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 8 for this indicator meet the expectations by attending to fluency and procedural work within the lessons.
- All units spend time on the procedural skills of the major work of the grade.
- Students must use the skills they have acquired in Grades 6 and 7 and use them to solve equations, systems of equations, slope, functions and the Pythagorean Theorem.
- Module 3 (solving equations and systems of equations) is devoted to 8.EE. Although additional examples may be required for some students to attain fluency, solving equations is done elsewhere in the book, thus addressing these throughout the year.
There are places that practice fluency throughout the textbook:
- At the beginning of each module, there is an activity called "Are You Ready?" Its primary focus is skill prerequisites for the lessons to come.
- Most guided practice pages have a few skill practice problems.
- Module quizzes called "Ready to Go On?"
- Mixed review of assessment readiness.
- Study guide review.
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 8 for this indicator partially meet the expectations for being designed so that teachers and students spend sufficient time working on engaging the applications of the mathematics,
- Several problems ask the student to apply their knowledge to real-world situations.
- In module 12, page 379 of the student edition asks the student to read and analyze the question about a painter using a ladder. Students must apply the Pythagorean Theorem and come up with an answer to the nearest tenth and then use the answer to connect to two additional parts of the problem.
Places where application is evident:
- Each lesson begins with "Motivate the Lesson," a question that puts the mathematics concept into a real-world situation and is meant to engage the student in the learning.
- The examples at the beginning of the lesson are often real-world problems and tie the learning to context.
- A section titled "Guided and Independent Practice" has numerous problems that have students apply their knowledge.
Areas in need of improvement:
- Very few problems are multistep; those that are break the problem down for the students.
- The performance tasks are lengthy questions but do not require students to do more than answer similar question to that on the independent practice and are often can be solved in one step.
- Many problems are simply "word problems" that use an algorithm to solve.
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 8 for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always combined together nor are they always separate.
- Conceptual understanding, fluency and procedural work, and application are practiced in most lessons.
- There is a blend that naturally happens for conceptually understanding equations, systems, and functions since this understanding also needs integer and rational number fluency skills.
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 8 partially meet the criteria of meaningfully connecting the Standards for Mathematical Content and the standards for Mathematical Practice. In the instructional material, the MPs are identified and often used to enrich mathematical content. Materials sometimes attend to the full meaning of each practice standard. There are many places where students are prompted to construct viable arguments and analyze the work of others. However, when looking beyond the labels of the practices there are many places where the label does not match the problem or the problem covers more than one practice but only one is listed. Materials are very limited in assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Also, only some materials actually attend to the specialized language of mathematics. Overall, the materials partially meet the expectations for 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 materials reviewed for Grade 8 meet the expectation for identifying the Standards for Mathematical Practice (MPs) and using them to enrich mathematics content.
- The front of the teacher edition has the practices listed and where to find them being utilized.
- The teacher edition describes what students should be able to do for each practice and gives examples from the book where they are to demonstrate the skill.
- Each module in the teacher manual has a section called "Professional Development," which tells how the MP is integrated (an example is found in module 7, lesson 7.1 on page 197).
- In the section called "Explore and Explain," found in the lessons in the teacher edition (see page 197), the MP focus is listed.
- On the evaluation page of the teacher edition (see module 7 on page 201 for an example), the MP is again listed, along with its associated Depth of Knowledge and the exercise to which it is connected.
- Additional references to the MPs can be found in the teacher edition's index.
- Charts at the end of each lesson in the teacher edition show what mathematical practices are covered in the independent practice questions. An example of this is on page 233, for lesson 8.1.
- Throughout the teacher edition, there are general references to MPs, but specific MPs are not designated by number. An example of this would be on pages 83 and 85 in lesson 3.3 in the teacher edition.
- Assessment readiness questions have also been analyzed based on the practices.
- Unit performance tasks include the use of the mathematical practices and identify the standards.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The Grade 8 materials meet the criteria of attending to the full meaning of each Standard for Mathematical Practice.
- Locations of MPs are listed in the teacher edition. In looking through each page where MP6, MP7 and MP8 are identified, the practices sometimes occurred only in the questioning strategies as opposed to the independent practice, where the students would have to employ the practices on their own.
- Following the listed pages on CC6-CC9, the activity or problem(s) do correspond with the stated practices. MPs 1, 2, 3, 4 and 8 are accurately listed in all occasions, while MP5 is named correctly 14 of 15 times, MP6 is correctly presented 8 of 11 times and MP7 is correctly presented 20 of 21 times.
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 8 partially meet the expectation for appropriately prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards.
- There are places where students are prompted to construct viable arguments and analyze the work of others. When looking beyond the labels of the practices there places where the practice is listed but not evidence of being used.
Places where evidence for MP3 is found:
- In the student edition on page 20, 100 and 146, MP3 is evident because it asks the student to explain an error or relate data in order to explain why or why not a statement is true.
- In the student book, students consider this question: "If irrational numbers can never be precisely represented in decimal form. Why is this?" Students must use and apply previous knowledge in stating or arguing that this is true or false and provide evidence to back it up.
- P.CC.13 describes MP3 and then shows examples of where it is evident in the book. It can be found in each lesson as an "essential question check-in" and in the independent practice section under such question headings as "Critique Reasoning," "Error Analysis," "Justify Reasoning" and "Communicate Mathematical Ideas."
- At the end of each lesson the question analysis lists the questions that are connected to MP3.
Problems with the evidence.
- MP3 is not evident in in the "Assessment Readiness" section or in performance tasks in the units
- There is only one question in Unit 6 that claims to address MP3.
- Four of the six performance tasks had MP3 listed, but at no point were students asked to critique the work of others. When they were asked to explain, it was to explain how they solved it, not to explain how they know they are correct or to justify.
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 8 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others.
- On page 100 in the student book there are exercises that relate to MP3. MP3 does match with exercises 10, 11, and 12, which ask students to analyze relationships, communicate mathematical ideas and critique reasoning, respectively.
- On page146 in the Student book there are exercises that relate to MP3. They are listed as 14, 16, 17 and 18 and they ask students to Make a Conjecture, Critique Reasoning and Explain the Error.
All lessons include many "why/how/explain" questions. These are in the questioning strategies for the teachers, as well as in the independent practice sections for students.
- Each lesson has a whiteboard activity. This is electronic and does not come with the teacher's textbook. Of the several that were looked at only one had langue that could engage students in discussion.
- There is a section called "Extend the Math." It only describes part of the activity as the rest is online. Only one of the 22 activities had students engaged in looking at each other's work and discussing.
- In the lessons is a section called, "Talk about It." These are questions to which the students respond. There is not any instruction on how the students should respond or any guidance for how to engage the students in meaningful conversation.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Grade 8 partially meet the expectation for attending to the specialized language of mathematics.
- Vocabulary is taught with pages titled "Reading Start-Up" and "Unpacking the Standards." The former is a review of previous vocabulary and the latter covers current and includes a real-life example. The vocabulary is highlighted when introduced.
- "Math Talk" is throughout the book. The publisher considers this a formative assessment. Students could discuss, or write their response.
- The study guide review includes a re-cap of the vocabulary.
- The "H.O.T." questions expect students to respond and explain using precise language.
- Attending to the specific language of mathematics is not evident in "Assessment Readiness" for any unit even though two units list MP6.
- Attending to the specific language of mathematics was evident in five of the six performance tasks, units 1, 2, 4, 5 and 6.
- In the teacher edition on page CC9 it states that MP6 is evident on pages 56, 106 and 143. In the teacher guide on the "Evaluate" page (page 55) MP6 is there and listed as questions 28 and 30. Students are to describe and identify and/or write about scientific notation and the relationships to adding and subtracting to scientific notation.
- On page 143 of the teacher edition, it states MP6 should be evident; however, MP6 is not identified on any of the questions.
- In the teacher edition on page CC9 it states that MP6 is evident on page 13. However, on page 13 the "reflect" question (1) asks the student only to look at the accompanying data and number line to solve the problem; there is no specific language or vocabulary.
- Vocabulary is not evident in "Assessment Readiness" for any unit.
- Vocabulary is evident in only two of the six performance tasks, units 1 and 3.