2015
Connected Mathematics Project 3

8th Grade - Gateway 3

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See the series overview page to confirm the review tool version used to create this report.

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Gateway Ratings Summary

Usability

Gateway 3 - Partially Meets Expectations
78%
Criterion 3.1: Use & Design
8 / 8
Criterion 3.2: Teacher Planning
8 / 8
Criterion 3.3: Assessment
9 / 10
Criterion 3.4: Differentiation
5 / 12
Criterion 3.5: Technology
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Criterion 3.1: Use & Design

8 / 8

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials meet expectations for use and design to facilitate student learning. Materials are well designed and take into account effective lesson structure and pacing. The materials meet the criterion for use and design. The underlying design of the materials makes a distinction between problems and exercises. In essence, the difference is that in solving problems students learn new mathematics, whereas in working exercises students apply what they have already learned in order to build mastery. Each problem or exercise has a purpose. The design of assignments is not haphazard; exercises do seem to be given in intentional sequences. Furthermore, the design is not distracting or chaotic, but supports students in engaging thoughtfully with the subject. Additionally, in most cases, the manipulatives and/or models accurately and consistently represent the mathematical objectives. Overall, the materials reviewed for Grade 8 meet the expectations for this criterion.

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Indicator 3a

2 / 2

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

In the teacher edition, the section before the investigation starts is divided into the following sections:

  • Launch
    • Launch Video
    • Connect Prior Knowledge
    • Present Challenge
  • Explore
    • Provide for Individual Needs
    • Planning for Summary
  • Summary
    • Orchestrate Discussion
    • Reflect on Student Learning

The underlying design of the materials does distinguish between problems and exercises meeting the expectations for this indicator.

The investigations are divided into problem sets, and they typically follow a sequence. For example, Investigation 1 could be divided into the following problems set: 1.1, 1.2, and 1.3.

At the beginning of each investigation problem set, there are some explanations and definitions given. Next, there is a set of problems to be completed within the class period. 

At the end of each investigation, there is a homework section. This section is divided into Application, Connection, and Extension sections. 

There is also a mathematical reflection section at the end of each investigation where students reflect on the mathematics content they have learned. During this section, there is also a connection between the mathematical practices used throughout the investigation.

Indicator 3b

2 / 2

Design of assignments is not haphazard: exercises are given in intentional sequences.

The design of assignments is not haphazard; exercises do seem to be given in intentional sequences meeting the expectations for this indicator.

On page 12 of A Guide to Connected Mathematics, the three phases of the CMP model are explained: launching, exploring, and summarizing. 

In the first phase, the teacher launches the problem with the whole class. Launches include connecting to prior knowledge as well as presenting the challenge of the problem.  

For the Explore phase, the nature of the problem suggests whether students work individually, in pairs, in small groups, or occasionally as a whole class to solve the problem. As students work, they gather data, share ideas, look for patterns, make conjectures, and develop problem solving strategies. 

It is during the Summarize phase that the teacher guides the students to reach the mathematical goals of the problem and to connect their new understanding to prior mathematical goals and problems in the unit. The Summarize phase begins when most students have gathered sufficient data or made sufficient progress toward solving the problem. In this phase, students present and discuss their solutions and the strategies they used to understand the problem, organize the data, and find the solution. During the discussion, the teacher helps students enhance their conceptual understanding of the mathematics in the problem and guides them in refining their strategies into more efficient, generalizable problem-solving techniques, or algorithms.

Indicator 3c

2 / 2

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

There is a variety in what students are asked to produce, meeting the expectations for this indicator.

  • Throughout various investigations and within the problem sets, students are asked to produce answers and solutions as well as to describe their answers, discuss ideas, make conjectures, explain their work and reasoning, make sketches and diagrams, justify their reasoning, and use appropriate models.
  • Sometimes only one aspect is specified, such as only requiring an answer, and other times a problem requires students to provide an answer, provide an explanation or steps, include a diagram and/or use a model.

Because problems require different responses, the type of response is intentional, like requiring models when a concept is introduced and then not requiring the same model when a more procedural method for solving similar problems is developed.

Indicator 3d

2 / 2

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The manipulatives are almost always faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models, meeting the expectations for this indicator.

On page 40 of A Guide to Connected Mathematics, an explanation is provided for the use of manipulatives in the instructional materials. Manipulatives are used only when they can help students develop understanding of the mathematical ideas.  

For example, in Filling and Wrapping, students find all the different rectangular arrangements possible for a given number of cubes. They find the surface area of each arrangement by creating a net (covering) for the arrangement that require the least and the most material to wrap. This activity sets the stage for developing the ideas of surface area and volume of rectangular prisms. 

Most of the manipulatives used in CMP are commonly available, and many schools may already have them. Included are rulers, protractors, angle rulers, cubes, square tiles, counters, spinners, and dice.

Two manipulatives are unique to CMP:

Polystrips are plastic strips that can be pieced together with brass fasteners to form polygons. They are used in Grade 7 to investigate the relationship among the side lengths of triangles and quadrilaterals. They also are useful in the Grade 8 Geometry unit, “Butterflies, Pinwheels, and Wallpaper.”  

The CMP Shapes Set is a set of polygons used in Grade 7 to explore sides, angles, and tilings.

Indicator 3e

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The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The visual design is not distracting or chaotic but supports students in engaging thoughtfully with the subject.

  • The student materials are clear and consistent between investigations within a grade level as well as across grade levels.
  • Each investigation and problem set is clearly labeled and provides consistent numbering for each investigation and problem set with both a lesson number and page number.
  • The investigations are clearly named, and the problem set examples within each investigation are labeled. The homework section includes clear labels for the Applications, Connections, and Extensions sections. 
  • At the end of each Investigation, there is a clearly labeled Mathematical Reflection section.
  • There are no distracting or extraneous pictures, or captions within lessons.

Criterion 3.2: Teacher Planning

8 / 8

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

Teacher Planning and Learning for Success with CCSSM: Materials support teacher learning and understanding of the standards. The materials reviewed meet the criterion for teacher planning and learning. The materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. Materials contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition. The strongest point is that each module begins with an overview section that gives teachers an understanding of the mathematical content in the lessons as well as where it fits in the scope of mathematics from Kindergarten through Grade 12. Overall, the material reviewed for the Grade 8 meets the expectations for this criterion.

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Indicator 3f

2 / 2

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Materials support teachers in planning and providing effective learning experiences through teacher questioning. This is a strength of the program with multiple questions for each lesson/problem throughout all of the Grade 8 units. 

  • Lesson sections that include questions are Launch, Presenting the Challenge, Explore and Summarize.
  • Additionally, teachers are prompted to reflect on student learning after the lesson in order to prepare for the next daily lesson or unit.

Indicator 3g

2 / 2

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The teacher’s edition has many suggestions on presenting the content to students. 

  • The Launch section and Connecting to Prior Knowledge are the two beginning sections of each lesson.
  • The purple book contains ideas for technology implementation for student learning.

Indicator 3h

2 / 2

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The teacher’s edition is an excellent resource for mathematics teachers to understand the mathematics of the unit and for teachers to expand their understanding of the mathematical concepts.

  • Mathematical background is included at the beginning of each unit. In the It’s in the System book, there are 13 pages of mathematical background for teachers to understand.

It’s in the System includes problems, explanations of problems, examples, and connections to CCSSM. 

Indicator 3i

2 / 2

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The teacher’s edition clearly explains the role of specific grade-level mathematics in the context of the overall mathematics for grades 6-12.

  • Standards are aligned in Grades 6-12 (curriculum is only written for these grade levels--K-5 is not an option)
  • Teacher’s editions connect the learning from previous grade levels and explain how standards build on one another throughout the program.

Indicator 3j

Narrative Only

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The teacher’s edition contains a detailed planning chart of each Problem and estimated pacing of the problems as well as assessment and mathematical reflections. 

  • The standards are aligned for each investigation throughout the book.

Indicator 3k

Narrative Only

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The materials do contain strategies for informing parents or caregivers about the mathematics program and give suggestions for how they can help support student progress and achievement.

A parent letter is available for each of the 23 units in A Guide to Connected Mathematics. These letters provide valuable information about how parents can be helpful to their children in learning mathematics. The letters can be personalized to include teacher name and contact information. The letters can be sent home with students or can be sent electronically

Indicator 3l

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Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials do contain explanations of the program's instructional approaches and identify the research-based strategies within the teaching materials.

A list on page 7 of A Guide to Connected Mathematics provides CMP’s guiding principles. Page 8 provides a rationale for a problem-centered curriculum. On page 10, a rationale is provided for depth versus spiraling. On pages 16–19, a theory and research rationale is provided for the following topics: cooperative learning and classroom discourse, teaching through problem solving, equity and motivation for learning, conceptual and procedural knowledge, formative assessment, mathematical knowledge for teaching, and teacher development and school change.

Criterion 3.3: Assessment

9 / 10

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The materials reviewed partially meet expectations for the criterion of assessment in Grade 8. The materials do not provide strategies for gathering information about students' prior knowledge within and across grade levels. Materials only sometimes provide strategies for teachers to identify and address common student errors and misconceptions. Reflections and problem sets - which could be used as formative assessment tasks - are not aligned to a specific standard or group of standards. The materials provide opportunities for ongoing review and practice. The materials offer formative and summative assessments, such as Notebook, Mathematical Reflection, Looking Back, Check Up, Partner Quiz, Unit Test, Self-Assessment, Project, and Group Work/Discussion. The materials do encourage students to monitor their own progress.

Overall, the materials reviewed for the Grade 8 partially meet the expectations for the assessment criterion.

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Indicator 3m

2 / 2

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

In the first phase of each lesson, the teacher launches the problem with the whole class.  Launches include connecting to prior knowledge. 

Indicator 3n

2 / 2

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Materials provide very few strategies for teachers to identify and address common student errors and misconceptions.

  • In the Teacher Planning section, there were occasionally “Notes” about potential student errors.

They present situations and ask students to identify mistakes or provide rationale based on common misconceptions.

Indicator 3o

2 / 2

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

  • The materials provide several opportunities for ongoing review and practice, such as Notebook, Mathematical Reflection, Looking Back, Check Up, Partner Quiz, Unit Test, Self-Assessment, Project, and Group Work/Discussion.
  • Within a unit the set of practice problems includes Application, Connection, and Extension to promote both increased understanding of a concept as well as developing procedural skill/fluency.
  • Beyond a lesson, future lessons/units typically expand practice and depth of previous learning.

Teacher Planning offers a multitude of prompts to orchestrate discussion.

Indicator 3p

Narrative Only

Materials offer ongoing formative and summative assessments:

Indicator 3p.i

2 / 2

Assessments clearly denote which standards are being emphasized.

Assessments clearly denote which standards are being emphasized.

  • Each standard is aligned to one or more lessons as noted at the beginning of each topic.
  • The Check up and Unit Test appear to be developed to fully assess a particular standard and the scoring guidelines specify which item aligns to which standard.
  • However, reflections and problem sets—which could be used as formative assessment tasks—are not aligned to a specific standard or group of standards.

Indicator 3p.ii

1 / 2

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

  • Formative assessments include scoring guidelines that provide sufficient guidance to teachers for interpreting student performance but do not include suggestions for follow-up.
  • Each Check up and Unit Test includes a scoring guideline as well as worked out solutions for correct responses but do not include suggestions for follow-up for below or above grade level students.
  • Although the rubric and scoring guidelines are complete, there are no strategies or suggestions for follow-up provided for the teachers.

Indicator 3q

Narrative Only

Materials encourage students to monitor their own progress.

The materials do encourage students to monitor their own progress. On page 57 the approach to self-assessment is explained. After every unit, students complete a self-assessment summarizing the mathematics they learned in the unit and the ideas with which they are still struggling. The self-assessment also asks students to provide examples of what they did in class to add to the learning of the mathematics. The goal of this activity is to have students reflect on their learning.

An explanation of the Mathematical Reflections questions is provided on page 83 of A Guide to Connected Mathematics. These questions provide an opportunity for the teacher and students to discuss the goals of the Investigation. After the class discussion, which can take place orally or in written form, students record their responses to the Mathematical Reflection questions to have a record of their current understandings.

Using the Mathematical Practices Reflections at the close of each Investigation allows students to name what they have done. Students demonstrate to themselves the power of the Practices.

Criterion 3.4: Differentiation

5 / 12

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The materials reviewed for Grade 8 do not meet expectations for the criterion for differentiated instruction. Materials sometimes provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners. There is no guidance to support teachers if a lesson does not work as written or if students need additional support to master the content. There is little provided that would help struggling or ELL students access the content successfully. Although there are occasionally challenge problems, there are minimal opportunities for advanced students to go beyond the mathematics provided in the classroom lessons. What is provided is not enough to guarantee that all students have content that is accessible. Overall, the materials do not meet the criterion for differentiated instruction.

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Indicator 3r

1 / 2

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners:

  • Sequencing and scaffolding are embedded into lesson development, but materials provide few strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
  • In Teacher Planning pages, there is always a section called “Orchestrating Discussion” which offers suggestions of questions at varying levels to help students understand the content.
  • There is no evidence of an explanation of how the lessons develop.

There is no guidance to support teachers if a lesson does not work as written or if students need additional support to master the content.

Indicator 3s

1 / 2

Materials provide teachers with strategies for meeting the needs of a range of learners.

Materials provide teachers with strategies for meeting the needs of a range of learners:

  • The materials provide very general strategies to help teachers make the content accessible to a range of learners.
  • There are some questioning prompts in the Teacher Planning section under “Provide for Individual Needs.” However, they do not seem geared toward a struggling learner but are simply good discussion points for all students.
  • Occasionally, “Provide for Individual Needs” includes “Going Further” which offers approximately 100 challenge problems.
  • The homework problems always offer “Extensions.”
  • There is a concern that the suggestions provided are not enough to guarantee that all students have content that is accessible.

Indicator 3t

1 / 2

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

  • A variety of solution strategies are not always encouraged.
  • Students are given opportunities to create solution paths on their own.
  • Most tasks do not allow students to use multiple entry points; they start at the same place.
  • Students do have the opportunity to solve problems using a variety of strategies, paths and/or models, though the materials sometimes undermine this concept by using tasks that explicitly state how to solve the problem or which representation to use.

Indicator 3u

0 / 2

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

Materials provide little support, accommodations, or modifications for English language learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

  • This series is clearly designed with best practices for all students in mind; however, there is nothing explicitly differentiated or modified for struggling or ELL students, other than translating assessments into Spanish.
  • What is provided is not enough to guarantee that all students have content that is accessible.

Indicator 3v

0 / 2

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Materials provide limited opportunities for advanced students to investigate mathematics content at greater depth.

  • Occasionally there are challenge problems.
  • There were minimal opportunities for advanced students to go beyond the mathematics provided in the classroom lessons.

Indicator 3w

2 / 2

Materials provide a balanced portrayal of various demographic and personal characteristics.

Materials provide a balanced portrayal of various demographic and personal characteristics.

Indicator 3x

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Materials provide opportunities for teachers to use a variety of grouping strategies.

Materials provide limited opportunities for teachers to use a variety of grouping strategies.

  • Investigations are intended for cooperative learning groups, though there are no recommendations for forming groups or any mention of why to have a student work within a certain group size.
  • Within the lessons, there are no group roles, no group expectations, etc., to help teachers enhance the involvement of every student.

Indicator 3y

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Materials encourage teachers to draw upon home language and culture to facilitate learning.

Materials do not encourage teachers to draw upon home language and culture to facilitate learning.

  • There is no evidence of teachers needing to draw upon home language and culture to facilitate learning.

Criterion 3.5: Technology

Narrative Only

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

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Indicator 3aa

Narrative Only

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are platform neutral (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices:  Accessibility was tested on Chrome, Firefox, Safari, an iPhone, and an iPad. All access was successful.

Indicator 3ab

Narrative Only

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology: There are multiple “Student Activities” in both Student Place and Teacher Place online that could easily be used as opportunities for assessment such as the Integer Product Game, Paper Pool, and the Transformation Tool. In addition, MathXL was developed to provide online readiness tests and skills assessments at the end of units as well as additional skills practice. There is also a test generator disk.

Indicator 3ac

Narrative Only

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

Materials can be easily customized for individual learners.

  1. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations: Math XL can be used to assign students practice in areas not yet mastered.
  1. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic: There is a test generator disk that is customizable. However, there are no digital lessons that can be adapted.

Indicator 3ad

Narrative Only

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.): There is no evidence of this component.

Indicator 3z

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Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the MP: There is an extensive list of Digital Math Tools available in both Student Place and Teacher Place online including tools such as 3D Geometry, Algebra Tiles, Fraction Shapes, and Pattern Blocks. In addition, within the text, there are prompts to the student to go online to watch a video that demonstrates a point or application. And teachers have a disk that includes a Launch Video for each lesson.