High School - Gateway 3

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

Each set of materials is divided into five modules, which are then divided into topics, and the topics are divided into lessons and activities. The structure for the topics are as follows: Connections, Getting Started, Activities, Talk the Talk, and Assignment. Each part of the structure has a specific goal. For instance, Connections, located on the topic page, directs students’ instruction through the use of a question or questions. In Getting Started, students access prior knowledge needed to be successful in the activities. Activities provide a variety of teaching strategies to encourage learning, and, in Talk the Talks, students reflect on the lesson by applying mathematics to real world problems. Finally, Assignment is separated into Write, Remember, Practice, Stretch, and Review. In each section, students use concepts learned to solve problems. The MATHia Software provides additional exercises incorporating a variety of instructional tools, such as: explore, animations, classification, problem-solving, and worked examples.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

According to the Carnegie Learning Math Solutions Traditional textbook, “Each lesson of the High School Math solution has the same structure. This consistency allows both you and your students to track your progress through each lesson.” The sequence of every lesson is divided into categories called Engage, Develop, and Demonstrate.

• Engage establishes “Mathematical Goals to focus Learning.” This section includes the following: a warm-up to activate prior knowledge, the learning goals, connections to previous learning, and a Getting Started page meant for students to solve/think/share and notice other’s work/thinking, usually for a non-routine problem.
• Develop involves “aligning Teaching to Learning.” It includes lesson activities which allow students to “build a deep understanding of mathematics through a variety of activities.”
• Demonstrate includes Talk the Talk in which students have “an opportunity to reflect on the main ideas of the lesson.”

After completing the categories within the lesson, students practice their learning in assignments, which includes: Write, Remember, Practice, Stretch, and Review. Students have additional practice with “Learn Individually” lessons using the MATHia software, or if the technology is not accessible, students use the Skills Practice workbooks.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet the expectation that there is a variety in what students are asked to produce. Students produce a variety of products in digital and written form.

Examples of variety in how students present the mathematics include:

• In Algebra 1, Module 2, Topic 2, Lesson 2, Activity 2.2, students analyze work to explain errors in students' work.
• In Geometry, Module 1, Topic 1, Lesson 3, Getting Started, students determine the slope of a line, create a segment parallel to a line utilizing patty paper, and describe the movements needed to complete the segment.
• In Algebra 2, Module 3, Topic 2, Performance Task, students examine a scenario involving administering the correct dose of medications. Students write functions, sketch a graph with technology, and interpret information.

The materials include performance tasks which usually involve real world scenarios and applying mathematics. Students produce tables, graphs, and charts to model data and solutions using different representations. The assignments provide numerous opportunities for students to write, explain, determine values, describe, and graph on a coordinate plane.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and are appropriately connected to written methods.

Virtual manipulatives are embedded within the MATHia Software for each section and include graphing tools, geometry tools for transformation, and the ability to manipulate a graph. Within the instructional materials, physical manipulatives for lessons are listed in the lesson overview. For example, in the Teacher Lesson Plans for Algebra 1, Module 5, Topic 1, the material needed for a specific portion of the lesson is patty paper.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. 

Each section has extensive facilitation notes provided in the Teacher’s Edition. There are suggested questions that teachers may use to aid in instruction and mathematical development. There are also suggestions as to what teachers should look for as they observe/examine student work. Within each lesson, there are questions that ask students to explain or justify their answers. For example, in Geometry, Module 3, Topic 1, Lesson 3, teachers are given the following questions to ask while students complete questions 4-7:

• “Do the ratio and scale factor represent the same thing?”
• “If all pairs of corresponding angles in the image and pre-image are congruent, is the shape of the image always the same as the shape of the pre-image?”

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for containing 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. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide detailed information regarding the instructional design of the series, lesson structure, assignment structure, problem types, thought bubbles to promote student self-reflection, mathematical habits of mind, academic glossary, modeling process, and assessments. Detailed information regarding content alignment within a given course complement a general overview of standards addressed within that course.

The Facilitation Notes embedded within every lesson provide ample and useful annotations and suggestions regarding an overview of the mathematical concepts addressed, standards addressed, essential ideas, pacing, what to look for from students, questions to ask, grouping strategies, common student misconceptions, differentiation strategies, and a summary statement of the mathematical ideas addressed.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide guidance for using the MATHia software and collecting data reports generated by MATHia. The table of contents explicitly identifies certain MATHia workspaces that correlate to specific modules in the given course.

The instructional materials for Carnegie Learning Math Solutions Traditional series meet expectations for containing a teacher’s edition 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, Approaching Infinity (Algebra 2, Module 2, Topic 3, Lesson 2), teachers view suggestions for implementing the lesson. 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. 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 instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for containing a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series.

• 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 why the module is named, the mathematics included 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 across grades and courses. Knowledge required from prior chapters and/or grades is identified in this section.
• The Module Overview of each section does not identify specific standards related to each of the connections. For example, in Geometry, Module 3, the prior connections states, “Students have extensive experience with ratios and proportional reasoning in middle school and in previous courses” but does not state the specific standard. In Algebra II, Module 1, Topic 1, the Family Guide explains that students have previously studied linear, exponential, and quadratic functions and also explains that this topic will prepare students to study more complex function  throughout the rest of Algebra II. The Teacher's Edition lists the use of patterns in the course as a way to promote mathematical thinking for the series.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for providing 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 the standards which 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. This provides teachers with information about students' learning gaps as they work through the instructional materials.
• In the Module Overview, there is a connection to student’s prior learning. This explains to the teacher what students should know or be able to do based on previous learning. At the beginning of each module, there are teacher notes that indicate “Where We Have Been.” This small explanation frames student’s previous learning and knowledge in the context of the new lesson. At the beginning of each lesson is a Warm-Up and a Getting Started portion so that students begin to connect previous learning to new learning. These exercises  allow instructors to gather information related to prior knowledge.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

The Topic Guide regularly includes Misconceptions with suggestions for teachers to identify and address common student errors and misconceptions. For example, in Geometry, Module 1, Topic 1, Lesson 4, “Students may assume all equiangular polygons are also equilateral polygons...” 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….”

The MATHia software provides a solution pathway for 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 the Carnegie Learning Math Solutions Traditional series meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Examples include:

• Each assignment at the end of the Lesson has review questions included.
• The workbook has the answers to the odd exercises for immediate feedback.
• Within each assignment, there is practice for the current material as well as practice for past material. 
• There are Topic Performance Tasks with rubrics to provide feedback based on knowledge for the topic.
• Warm Ups and Getting Started tasks provide additional practice for students as they start to expand their learning.
• The MATHia software includes “Hints” which students can select when reviewing and practicing concepts and skills. Just-in-Time Hints automatically appear when a student makes a common error, and On-Demand Hints are provided when a student asks for a hint while working on a problem. Step-by-Step Hints demonstrate how to use the tools in a lesson by providing step-by-step guidance through a sample problem.

The instructional materials reviewed for Carnegie Learning Math Solutions Traditional series meet expectations 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 Math Solutions Traditional series meet expectations for assessments providing sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

• Expected answers and outcomes are provided for each assignment/assessment, and some guidance is provided to aid teachers in the interpretation of student performance. 
• Guidance is provided about formative assessments, and strategies teachers can use to address student misunderstandings are included in lesson facilitation notes.
• Performance Tasks include detailed scoring rubrics.
• 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.
• The materials 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 the Carnegie Learning Math Solutions Traditional series meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Scaffolding is evident throughout the series and teachers have support for different types of learners. Every activity has Differentiation Strategies included in the Facilitation notes that provide teachers detailed instructions about the organization of the topic, entry point for the students, how to see students are demonstrating understanding, and the importance of learning the topic as well as how the topic provides expertise in the mathematical standards.

The topic overview provides a detailed chart explaining the scope and sequence of the topic, and it is Learning Together. The chart provides the number of days needed for the lesson, aligned standards, and highlights for the lesson.

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

MATHia differentiates the learning experience for every learner, adapting the amount of support based on the students' answers and path through each problem. This level of support is similar to a one-on-one tutoring experience, where the software is adapting based on everything the student is doing.

The differentiation strategy notes for teachers includes strategies for struggling students and suggestions for all students. The Questions to Ask, and other facilitation strategies are intended for all students. The varied activities throughout a topic address different types of learners. For example, in Geometry, Module 1, Topic 3, Lesson 5, “What is the difference between reflectional symmetry and rotational symmetry? How do you determine if a figure has rotational symmetry?” The questions are intended for all students, not a specific subgroup. On the same page, there is a differentiation strategy stating, “To extend the lesson, encourage students to investigate how to describe the location of the center of rotation for each figure in Question 9.”

The instructional materials for the Carnegie Learning Math Solutions Traditional series meet expectations for embedding tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations. 

In each module, students complete series of consecutive problems that provide different entry points for students to demonstrate their understanding and skills. For example, Module 2, Topic 1, Lesson 1, Activity 3, scaffolds problems within the student problem set to help students have an entry point and to allow teachers to determine where students may be having difficulty. Facilitation notes at the beginning of the chapter have a section labeled “What is the entry point for students.” Teachers are given instructions such as, “As students work, look for: different translations of triangle ABC. Some students may move a single vertex of the triangle to the origin while others may move vertices to the axes. Share different strategies to accomplish the task.”

There are also problems when the materials provide two possible solution paths for a problem and students choose their path, for example, Geometry, Module 1, Topic 1, Lesson 5, Problem 3.

The instructional materials for the Carnegie Learning Math Solutions Traditional series 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.

Throughout the series, there are suggestions for ELL students within the teacher’s materials. For example in Algebra 1, Module 3, Topic 2, Lesson 4, the ELL Tip states, “Students may not understand the term intuitively in the introduction. Discuss the problem-solving strategies that students automatically used to solve the problem and how they relate to the modeling process.” ELL support is evident in all of the teachers guidance, and there are ELL Tips that address other strategies to help ELL students, such as honoring the use of students’ native languages, building relevant background, simplifying sentences, and modifying vocabulary.

Suggestions for supporting other special populations (i.e. struggling students and advanced students) are included in the Teacher’s Implementation Guide in the Facilitation Notes for each activity. Some of these suggestions for other special populations are general, and some are specific to the content of the lessons in which they are found.