8th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 66% |
|---|---|
Criterion 2.1: Rigor | 8 / 8 |
Criterion 2.2: Math Practices | 4 / 10 |
The materials reviewed for Course 3 partially meet the expectations for Gateway 2: Rigor and Mathematical Practices.
The materials reviewed for Course 3 meet expectations for rigor and balance by providing a balance of all three aspects of rigor throughout the lessons. The Grade 8 instructional materials reflect the balances in the standards and help teachers to help their students meet rigorous expectations. They do this by helping students develop conceptual understanding, procedural skill and fluency, and application. The materials weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.
The materials reviewed for Course 3 do not meet the expectations for practice-content connections. The materials attempt to incorporate the MPs in each lesson but all instruction of the MPs happens at the beginning of the Teacher Implementation Guide and never directly links the standards to the lessons using the MP vocabulary. This makes it extremely difficult for a teacher to reliably use the materials to know when MPs are being carefully attended to. The materials incorporate questions in which students have to justify and explain their answers, but no teacher supports are given which creates a lack of lesson structures for which students would discover their own solution paths, present their arguments, and justify their conclusion. Vocabulary is presented and almost always incorporated meaningfully into the lesson.
Overall, the materials partially meet the expectations for Gateway 2 in rigor and 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 Course 3 meet expectations for rigor and balance. The Grade 8 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 weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.
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 instructional materials for Course 3 meet the expectations to develop conceptual understanding of key mathematical concepts, especially when called for in specific content standards or cluster headings. Overall the instructional materials present real-world situations and multiple visual examples as a way to develop conceptual understanding.
The materials in the Teachers Implementation Guide, student text, and student assignments were all used to gather evidence. In the Teachers Implementation Guide it states that they provide many opportunities for conceptual understanding, especially those in which the standard calls for it. Each chapter begins with an overview and in this overview is a column titled "highlights." This section summarizes what the students will be able to do after the lesson and was a guide to searching for the evidence below to justify the meets rating.
- Chapter 3, Lesson 2: Students use visual representations and linear graphs to represent a situation from a given context.
- Problem 1, page 164, gives students information about a school soccer team trip and then has follow-up questions that ask students to determine rate of speed during the trip. They are then asked create a concrete visual representation of the story based upon facts presented.
- Chapter 4: Students interpret meanings of equations and analyze intervals. The lessons have students use tables and graphs to see the relationships of the function which correlates with standards in 8.F and 8.EE.
- Chapter 6: Teaches the use of the Pythagorean Theorem (8.G). Lesson 1 has students use shapes to develop the Pythagorean Theorem. They use grid paper to show the relationship of the area of the triangles.
- Chapter 7: Students explore transformations (8.G) with shapes in the coordinate plane. Many questions require students to explain their thinking.
- Chapter 9: Students work with understanding dilation and similarity (8.G) by investigating and exploring shapes in the coordinate plane.
In the lessons listed below and others, the students are asked to explain their reasoning or explain why they believe the answer to be correct. The student assignment book allows individual students to show their understanding through the following questions in the listed lessons.
- Lesson 1.2 - Why Doesn't This Work?
- Lesson 2.1 - Patterns, Patterns, Patterns ...
- Lesson 7.4 - Mirror, Mirror
- Lesson 12.3 - Making Decisions
- Lesson 14.2 - Piling On!
An area of concern for this review team was that many of the guided lessons walk students through entire procedures and do not allow them to explore or discover concepts on their own.
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 instructional materials for Course 3 meet the expectations to give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, there are multiple opportunities for students to develop procedural skills and fluency which include various questioning strategies for students to explain procedural skills, and chances for students to apply procedural skills to new situations. Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency meeting the expectations for this indicator.
- Based on where the lesson is in the timeline of the unit, the level of difficulty varies. As expected, it starts off with a low level of difficulty, and as the students gain more practice, the difficulty increases as the unit progresses.
- Along with problems during the guided lessons, each lesson begins with a warm up that is procedural practice.
- Some of the student assignments are procedural in nature as well as each lesson having suggested pages for students to complete in the student skill practice book.
Examples of fluency practice that justify the meets rating are:
- Chapter 1, Lesson 1.1, when the students start by having to write the steps they follow to solve multi-step equations with one variable then they move onto just solving the problems.
- Chapter 5, Lesson 5.1, students practice writing fractions as decimals.
- Chapter 8, Lesson 8.2, students practice determining what transformation is present.
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 instructional materials for Course 3 meet the expectation 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. Overall, the materials have multiple opportunities for real-world application.
There are a variety of single and multi-step contextual application problems found in the student text, students' assignment book and student skills practice.
- Chapter 2, Lesson 2.6 has students calculate the cost for orders of T-shirts for various given values from a competitor with a different cost value, determine the amount of shirts that can be purchased, and create a table and graph to represent the situation.
- Chapter 3, Lesson 3.3 asks the students to determine the rate of change based on a soccer tournament.
- Chapter 11, Lesson 11.3 asks the students to apply what they know about rate of change and linear equations to solve real-world problems such as how much money a person will make if they work at an hourly rate. It also asks them to apply what they know about functions and graphing to demonstrate a variety of scenarios for a person to earn a certain amount of money.
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 instructional materials reviewed for Course 3 meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separate. Overall, the majority of the lessons focus on procedural skills and fluency, but do balance that out with conceptual understanding and application problems. The material's weakest section, but still enough quality representations to receive the "meets" rating, is on conceptual understanding. Students are able to work in groups to develop understanding, but then sometimes the narrative, in the text, scaffolds the work in such a way that the students are just walking through that understanding step by step.
The student text, along with the wrap around Teacher Implementation Guide help guide the educator and student to the level of rigor needed to prepare the student for upcoming mathematics. An example of this is in Chapter 3, Analyzing Linear Equations.
- The chapter begins with introducing students to finding two points on a line and then determining the rate of change.
- Delving deeper into this skill, students are then asked, on page 143 problem 1, to determine the unit rate of four different cars using the same graph.
- They are then asked to describe what the steepness of each line implies regarding their individual unit rate.
- Then each lesson ends with a complex problem which delves deeper into student understanding of the standard.
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 Course 3 do not meet the expectations for practice-content connections. The materials attempt to incorporate the MPs in each lesson. However, all instruction of the MPs happens at the beginning of the Teacher Implementation Guide and never directly links the standards to the lessons using the MP vocabulary. This makes it extremely difficult for a teacher to reliably use the materials to know when MPs are being carefully attended to. The materials incorporate questions in which students have to justify and explain their answers, but no teacher supports are given which creates a lack of lesson structures for which students would discover their own solution paths, present their arguments, and justify their conclusion. Vocabulary is presented and almost always incorporated meaningfully into the lesson.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The Instructional materials reviewed for Course 3 partially meet the expectation for identifying and using MPs to enrich mathematics content within and throughout each grade. While each practice is represented in this book, they are not often used in a way that would promote or enrich the mathematics content, are not user-friendly and are under-identified in many of the units because they are not specifically stated.
- MPs are described on pages FM-34 through FM-42 in the Teacher's Implementation Guide, along with a description of what it looks like in the student text.
- On pages FM-45 through FM-55, there are more examples of how the practices are implemented throughout the series. This section also defines symbols that clue teachers and students that they should discuss to understand, think for yourself, work with your partner, and share with the class. It also defines that a "thumbs up" means a worked example is correct and a "thumbs down" means a worked example is incorrect.
- Specifically, FM-45 through FM-49 attempt to define the academic terms analyze, explain your reasoning, represent, estimate, and describe. When these verbs appear in the series, they are suppose to correlate with the MPs listed. There is no further identification of MPs in the Teacher's Implementation Guide.
- All eight MPs are evident throughout the materials, but it was very difficult to find them since they are not specifically marked and not all practices had identifying academic terms to label them.
- The MPs could be used to enrich the mathematical content, but one would have to keep referencing the only guide to using them in the first few pages of the Teachers Implementation Guide, in order to have a better understanding of what practices to emphasize and how to use the problems to enrich them. Since all of the chapters have an overview section, this would be a place to identify the practices for each lesson and further encourage the use of these practices to enrich the mathematics content.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Course 3 do not meet the expectations for carefully attending to the full meaning of each practice standard. The publisher rarely attends to the full meaning of the practice standard and when they do, it is cumbersome to use since they are not specifically called out.
- The lessons are set up for students to work in groups and then the teacher is to guide them through discussion of their findings. The graphs, tables, equations, etc., are usually created for the students, which doesn't allow them to have to model with mathematics.
- There are sequential questions to lead the students through the process, so they are not having to make sense of the problem, or persevere, because the text book does it for them. With this guidance on how to complete the problem, there is usually only one way to solve the problem, not allowing for multiple entry points.
- Students are not given the opportunity to choose tools to help them with the mathematics, the tools are provided.
- Almost all of the lessons are designed for group work. Students are rarely asked to work independently, and given opportunity to compare.
- There are a variety of questions for teachers to pose during each lesson, however there are no MPs indicated and no sample student responses given to aid a teacher who is unsure what MP is involved and how students may be thinking as related to that MP.
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 Course 3 partially meet the expectation for appropriately prompting students to construct viable arguments and analyze the arguments of others. Each chapter provides opportunities for students to construct viable arguments and places to critique worked examples that are sometimes correct and sometimes wrong, however, each time they let the students know which argument is correct and which is incorrect by putting a thumbs up or a thumbs down with the problem. This directs students thinking and doesn't force them to go through the thought process of finding viable arguments to critique the reasoning of others.
- The questions following 'thumbs up' are usually comparing how two different people solved it correctly, but differently, which does allow for a student to then describe why they are both right.
- For 'thumbs down' situations, students are asked to find the errors that were made.
- In the Teacher's Implementation Guide, page FM-45, there are icons that direct student questioning that provide norms for what students are to do when they see the icons throughout the text.
- In the Teacher's Implementation Guide, page FM-48 students learn specifically how to explain their reasoning which is identified as SMP3.
- In the Teacher's Implementation Guide, pages FM-52 to FM-55, there is a "Who's correct" option that, if used, better allows students to form their own opinions and arguments for the work done.
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 Course 3 do not meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, there is not enough guidance in the teacher materials to direct teachers on questioning strategies, setting up scenarios where students experiment with mathematics and based on those experiments construct and present ideas, examples of higher level questions and suggested activities that lead students to construct viable arguments and analyze the arguments of others.
In the Teacher's Implementation Guide, page FM-37 and pages FM-43 through FM-55 do provide teachers with instruction on how to get students to construct viable arguments and critique the work of others, while each lesson also has a "Share Phase" section in the margins that poses questions teachers are supposed to ask for discussion; however, many of the questions being asked are closed ended questions and do not promote discourse. Also, there are no suggestions for how students should share or report out their thinking, and the lessons are written in the same format all the way through the series which does not promote students to think about mathematics in different ways.
As the wrap around teachers edition was reviewed, the publisher did not specifically address potential teacher moves regarding constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics. The vast majority of the time, the areas that addressed MPs were merely closed ended questions added to the practice section of the lessons. Teachers are not given any specific examples on how to address this practice in their daily lessons.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Course 3 meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit.
- In the Teacher Implementation Guide, page FM-40, there is an explanation of MP6.
- It states that students should communicate clearly and use clear definitions in discussions and writing.
- It emphasizes that students should label things to clarify their work.
- This section also states that the answers provided in the Teacher's Implementation Guide are exemplars of student responses and model precision appropriately.
- Along with information about a vocabulary section in the skill practice for each lesson. The book also states that each lesson provides opportunities for students to communicate precisely in writing and when sharing their solutions.
- Each chapter has key terms listed in Lesson 1. The words are then defined somewhere in the lesson and written in bold font.
- The terms are not in bold or referenced after that first lesson.
- The terms are listed again in the chapter summary, but it does not define or use them in any way.
- Lesson 1 in the student skills practice book, has students work with the vocabulary and the answers in the teacher's guide provide the precise answers written in the language of mathematics.