6th Grade - Gateway 3

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

The course has five modules with each module broken into topics. Each topic has a set of three to six lessons/activities. Each lesson consists of several sections, which may include Warm Up, Getting Started, Activities, Talk the Talk, and an Assignment. The Warm Up and Getting Started sections activate students’ prior knowledge and engage students in non-routine problem solving. The Activities develop students' understanding of concepts by exploring problems through both individual and whole group instruction. The students demonstrate their understanding of concepts by applying their knowledge to real-world problems in the Talk the Talk section. The Assignment includes five mini-sections that reinforce understanding of the new mathematical concept. Each lesson has a coordinating practice set called Skills Practice with exercises for students to solve using their new learning. MATHia (online) provides additional personalized exercises for students to show their understanding of the activity/lesson.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

Lessons follow a consistent format that intentionally sequences assignments:

• “Warm Up” - exercises that activate students’ prior knowledge.
• “Getting Started/Engage” - students solve/think/share and notice others' work/thinking, usually for a non-routine problem.
• “Develop/Activities” - new learning takes place; students explore problems that engage them with examples and explanations of the targeted skill in a whole-class setting. Each Activity includes verbiage describing how the new knowledge relates to previous understanding.
• “Demonstrate/Talk the Talk” - students reflect on and connect what was learned.
• “Assignment” - five sections that review the lesson: Write - reviewing rules or vocabulary, Remember - summary of one or two key points, Practice - problems related to the activities, Stretch - an extension, and Review - looping in previous skills.

Students practice with “Learn Individually” lessons using the MATHia software or, if technology is not accessible, students use the Skills Practice workbooks.

Overall, each topic is sequenced to begin with prior knowledge and build upon that knowledge to develop conceptual understanding and procedural skill.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that there is a variety in what students are asked to produce. Students are asked to produce a variety of products in both digital and written form.

Some of these products include:

• Multiple representations including expressions and equations, models, arrays, number lines, etc.
• In Module 4, Topic 2, Lesson 2.2, students are given parts of quadrilaterals on a coordinate plane and use their knowledge of polygons and horizontal or vertical distances to complete the missing parts, then determine the area of the polygons.
• Justification of their thinking and others', critiquing others’ work, explaining why answers given are correct.
• Writing, reviewing, practicing, and stretching activities at the end of each lesson, for example, writing their own situations to model a given expression. (Module 1, Topic 2, Lesson 3.5)

Finally, each module includes a real-world connection where students produce solutions in a variety of ways to demonstrate their mathematical knowledge, such as plotting band member locations on a coordinate plane.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and are appropriately connected to written methods.

Manipulatives are embedded in activities and the MATHia Independent Digital Lessons. Number lines, patty paper, equivalency cards, etc. are used throughout the year in connection to the mathematics being presented and are faithful representations. For example, students cut, fold, and tape a cube net and answer questions about the net, then make conjectures on different nets based on creating this one. (Module 1, Topic 3, Lesson 3, Getting Started: Breaking Down a Cube)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

In the Teacher Edition, facilitator notes for each activity include questions for the teacher to guide students' mathematical development and to elicit students' understanding. The material indicates that questions provided are intended to provoke thinking and provide facilitation through the mathematical practices as well as getting the students to think through their work. The Note provided on page FM-21 of the Teacher’s Implementation Guide Volume 1 reads, “When you are facilitating each lesson, listen carefully and value diversity of thought, redirect students’ questions with guiding questions, provide additional support with those struggling with a task, and hold students accountable for an end product. When students share their work, make your expectations clear, require that students defend and talk about their solutions, and monitor student progress by checking for understanding.”

Each lesson guide in the Teacher Edition provides quality questions to help guide students' mathematical development.

For example:

• “Did you calculate the area of both figures, or did you take a shortcut? If you took a shortcut, what was it?”
• “If you didn’t have a rectangle with the same dimensions as a parallelogram, how could you determine the area of the parallelogram?”
• “Which formula did you learn first, the area of a parallelogram or the area of a rectangle?”
• “Explain how knowing the area formula for a rectangle helped you derive the area formula for a parallelogram.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that the materials contain a teacher 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 such as how to use and read data in the MATHia software.

In the Lesson Resources, the teacher guide provides information including a lesson overview, lesson structure and pacing facilitation notes, questions to ask, connections to standards, a materials list, essential ideas, facilitation notes, what to look for when students are working, and a summary of the lesson.

As part of the blended learning approach, there is Learning Individually with MATHia software. There is ample support for students and teachers to engage with this software such as the Getting Started guide, a table of contents, an RTI table of contents, and MATHia system requirements.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for containing a teacher edition (in print or clearly distinguished/accessible as a teacher edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Within MyPL, teachers can view instructional videos that provide adult-level explanations and examples for teachers to enhance their own knowledge of the content. The instructional videos address textbook lessons, MATHia, mathematical content, and classroom strategies. For example, in the video, March MADness - Mean Absolute Deviation (Course 1, Module 5, Topic 2, Lesson 3), teachers view suggestions for implementing the lesson. Course 1 contains 51 lesson videos. MyPL also includes 33 videos addressing mathematical content that are not lesson-specific, and the advanced mathematics concepts addressed by the videos include, but are not limited to: ellipses, hyperbolas, and discontinuities and asymptotes of rational functions. The Teacher’s Implementation Guide for each course provides detailed information regarding how mathematical content fits into the series overall, and the materials include module overviews that describe the mathematics of the module and how the content is connected to prior and future learning.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials contain a teacher edition (in print or clearly distinguished/accessible as a teacher 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 12.

• The Module Overview includes information for the teacher with explanations that build the teacher’s understanding of how the lesson content fits into the curriculum. It tells what mathematics is in the module and how the module connects to prior and future learning.
• Each Topic Overview provides information on the mathematical content in the lessons as well as where it fits in the scope of mathematics from Kindergarten through Grade 12. Knowledge required from prior chapters and/or grades is explicitly called out in this section.
• The Topic Overview also connects each lesson to standards from a previous grade. These previous grade standards are embedded into each lesson through the Warm Up and Getting Started sections as well as the Activity sections.
• The Topic Overview describes the entry point or prior experience with the mathematical concept for students, why what is being learned is important, and how the activities in the topic promote student expertise in the MPs.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

• There is a pretest for every topic in each module that addresses standards that will be taught. The post-test for the topic is the same test.
• The Topic Overview provides a list of Prerequisite Skills needed for the topic, which creates an indirect opportunity for teachers to gather information about students’ prior knowledge although there is no direct guidance provided to the teacher about how to use the information.
• The MATHia software is used as an assessment and progress monitoring tool, providing personalized data about where a student stands on various skills.
• In every assignment in the textbook, there is a Review section. Students practice two questions from the previous lesson, two questions from the previous topic, and two questions that address the fluency standards outlined in the Standards. This provides teachers information about students' learning gaps as they work through the instructional materials.

While there are opportunities to collect information about students’ prior knowledge, the materials do not provide strategies about how to utilize the information in the classroom.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies for teachers to identify and address common student errors and misconceptions.

• In the Topic Guide, lessons regularly have a section titled “Misconceptions” with suggestions for teachers to identify and address common student errors and misconceptions.
• Example: “Students may get confused over the varied uses of the term scale—such as scaling up, scaling down, scale on a number line, even bathroom scale. Take the time to explain how all the uses of the term are related.” (M2-37J)
• Teachers are encouraged to engage students in mathematical conversations to address student errors and misconceptions with phrases such as, “Remind the students…, Discuss with students…, Point out that….”
• MATHia software provides a solution pathway to common student misconceptions. “Like a human tutor, MATHia re-phrases questions, re-directs the student, and hones in on the parts of the problem that are proving difficult for the student. Hints are customized to address the individual student, understanding that there are often multiple ways to do the math correctly.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that 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:

• Students practice with “Learn Individually” lessons using the MATHia software or, if technology is not accessible, the Skills Practice workbooks. In the Skills Practice book, odd number answers are provided, so students know if they’re solving problems correctly; and in the MATHia software, feedback is continually given for both correct and incorrect answers.
• The MATHia software includes “Hints” which students can select while reviewing and practicing skills. There are three types of “Hints”:
• Just-in-Time Hints automatically appear when a student makes a common error.
• On-Demand Hints are hints that a student can ask for at any time while working on a problem.
• Step-by-Step demonstrates how to use the tools in a lesson by guiding step-by-step through a sample math problem.
• Each lesson ends with Talk the Talk, a few questions that capture the learning of all of the activities the students have engaged in with the lesson.
• Each lesson also has a short review section that provides a spiral review of previous concepts.
• Standardized Practice Test that the teacher can use at any time to review and practice concepts and skills learned throughout the course.
• Prior to each lesson there is a Warm-Up that reviews previous topics.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for assessments clearly denoting which standards are being emphasized. The series offers several types of assessments, print and digital:

• MATHia provides information for each student based on standards.
• Performance tasks clearly note which standards are being assessed. 
• The student-facing versions of the Pretest, Post test, and the End of Topic Test do not denote which standards are being emphasized.
• The digital overview contains assessments and an assessment overview document. The document contains each assessment as well as which standard is assessed for each individual problem.
• The Carnegie Edulastic Assessments Suite displays standards for each problem within each assessment provided. These standards are not student-facing.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. Materials include some guidance for teachers to interpret student performance. Answer keys are provided for all assessments. Performance Tasks include a detailed scoring rubric for teachers to use when interpreting student performance; however, no other assessment provides guidance for teachers about scoring student performance. MATHia reports provide teachers with detailed information about student performance in relation to progress on standards and suggestions on the skills that require additional support. Teachers can monitor students working in MATHia and view in-the-moment guidance that indicates to teachers which students need additional support. The materials also offer teachers an APSLE (Adaptive Personalized Learning Score) report which is a predictor for year-end summative assessments. Videos within MyPL explain this report in more detail while outlining the research and models behind the report.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The materials include a detailed Scope and Sequence of the course, including pacing. The lesson summary and the essential ideas provide further information on sequencing of the lessons. There is a chart in the Teacher’s Implementation Guide that includes a table with a column entitled, “Connections to Prior Learning,” which enhances the opportunity to scaffold instruction by identifying prerequisite skills that students should have.

All lessons include instructional notes and classroom strategies that provide teachers with key math concepts, sample questions, differentiation strategies, discussion questions, possible misconceptions, what to look for from students, and summary points providing structure for the teacher in making content accessible to all learners.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 partially meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

A primary strategy for meeting the needs of all learners in this program is MATHia software. MATHia provides differentiation, providing varying levels of support based on how students solve the problems. The software adapts to the specific responses and “tutors” each student to meet their needs.

Most lessons provide “Differentiation strategies,” “Questions to ask,” and a “Misconception” section. Most of the suggestions and the questions included in the “Questions to ask” section are intended for all students rather than geared toward helping students who struggle or challenging students ready to go deeper. For example, in Module 4, Topic 2, Lesson 3.1, Questions to ask: “What quantities are being compared? Which quantity depends on the other? How did you decide which quadrants you will need in order to graph the data? How did the signs of the numbers in the table help you decide which quadrants are needed in order to graph the data? How did you determine the weight differentials in Question 2?”

However, in the “Differentiation strategies” section, suggestions are limited but more specific. For example, in Module 1, Topic 2, Lesson 3.1: “For students who struggle, reduce the number of diagrams for which they must create fact families.”

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

• Each Topic Overview includes a section called, “What is the entry point for students?” For example, in Module 1, Topic 1, the Factors and Area overview states: “Students enter Grade 6 with a conceptual understanding of area and fluency in computing the perimeter and area of rectangles. They have used tiling to relate area to multiplication and addition, and they have used informal statements of the number properties. Students have also used area models to represent multiplication. Factors and Area draws on this prior knowledge to formalize the Distributive Property, to reinforce connections between area models and multiplication, and to determine the area of new shapes.”
• Some application tasks, particularly the Performance Task, allow for multiple solution strategies or representations. For example, in Module 5, Topic 1, “The Statistical Process,” students create a histogram for each display given and then describe how they selected the size of the bin. Students make a conclusion about the amount of sleep that children, teenagers, and adults get.
• Some assessment questions allow for multiple entry points. For example, in Module 3, Topic 3, Post-test question 5, students write a story that matches a graph shown, allowing answers to vary.
• Lesson activities provide limited opportunities for students to create their own solution paths since strategies are often provided.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for suggesting 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).

ELL Tips are specifically cited throughout the materials. For example, “ELL Tip: In the first sentence, there are several words that may be challenging for English Language Learners (analyze, situation, corresponding). Have students read the paragraph independently and circle words that they do not know. Then have students find people in the class that have different circles. The students should help each other define words that they have circled. If numerous students are circling the same words, define these as a class.” (M3-197)

Additional differentiation strategies included in the materials are often general, such as providing additional examples, using manipulatives, or using a graphic organizer.

Some suggestions are specific to the lesson, but don’t necessarily further knowledge, such as in Module 2, Topic 2, Lesson 1.3: “Differentiation strategy: To support students who struggle, provide a hint sheet to support their learning while they play the game with peers who may think faster than they do."

There are differentiation suggestions that do not include a rationale as to how they would provide support. For example, in Module 2, Topic 1, Lesson 1.1: “For students who struggle, provide another example to compare/contrast. For example, scores for two games are 99-100 and 1-2. Which game was probably 'closer'?” It is unclear how this suggestion would support a student's development of additive and multiplicative reasoning.

However, there are numerous examples that do support accommodations for special populations such as:

• “For students who struggle with base and height, it may be helpful to have them cut out figures. That way, they can turn the paper to see different sides can be the base. They can then use folds to determine the height.” (M1-15E)
• “For students who struggle, transferring measurements from Question 2 into the diagram may be confusing. You may want to have pre-labeled diagrams available in those cases.” (M1-29D)
• “For students who struggle, using the terms greatest and least may sound counter-intuitive compared to their answers. Stress that they should concentrate on key terms. In GCF, factor is the key term, and common factors cannot be larger than both values. In LCM, multiple is the key term and multiples may be larger than both values.” (M1-39G)
• “Questions 1 through 4 and Questions 5 through 6 require the same knowledge; however, Questions 5 through 6 are less scaffolded and different information is provided. Assign Questions 1 through 4 to students who struggle and assign Questions 5 through 6 to students who need a challenge.” (M4-73H)
• “For students who struggle, the use of inequality symbols often adds more confusion. You may want to adjust the directions so that students circle the larger number or label the values on a number line.” (M1-71G)
• “To support students who struggle, it may be difficult to draw tape diagrams with rectangles of the same size. For this reason, you may want to provide a rectangle stencil or pre-drawn tape diagrams.” (M2-37E)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for providing a balanced portrayal of various demographic and personal characteristics.

• No examples of bias were found.
• Pictures, names, and situations present a variety of ethnicities and interests.
• The text is black and white with green as the only color. The people are gray with black hair, but still appear to represent many ethnicities.
• Problems include a wide span of international settings, as well as situations in urban, suburban, and rural settings.
• There is a wide variety of names in the problems, from James, Ben, and Haley to Keirstin, Miguel, and Miko, representing a variety of cultures.