High School - Gateway 3
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Usability
Gateway 3 - Meets Expectations | 83% |
|---|---|
Criterion 3.1: Use & Design | 7 / 8 |
Criterion 3.2: Teacher Planning | 6 / 8 |
Criterion 3.3: Assessment | 8 / 10 |
Criterion 3.4: Differentiation | 9 / 10 |
Criterion 3.5: Technology Use |
Criterion 3.1: Use & Design
Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for being well designed and taking into account effective lesson structure and pacing. In the instructional materials, the underlying design distinguishes between problems and exercises, the design of assignments is not haphazard, and there is variety in how students are asked to present the mathematics.
Indicator 3a
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.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations that the underlying design of the materials distinguish between problems and exercises. The materials distinguish between problems and exercises within each lesson. Most problems or exercises have a purpose. Each lesson starts with Explore & Reason, Model & Discuss, or Critique & Explain to introduce the new concept for that lesson. Then, there are several examples that guide students through the learning of the content. The lessons end with exercises where students use what they have learned to develop procedural skills, application, and conceptual understanding as appropriate.
Indicator 3b
Design of assignments is not haphazard: exercises are given in intentional sequences.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences. Exercises within student assignments are intentionally sequenced to build understanding and knowledge. There is a natural progression within student assignments leading to full understanding of new mathematics. Within each set of exercises, students are brought back to the essential question for understanding. Students progress in the lesson by starting with problems that focus on understanding, move to practice exercises which are more procedural, and complete application problems.
Indicator 3c
There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations that there is variety in how students are asked to present the mathematics. Students compute numerical answers while also providing diagrams and graphs. There are problems in each lesson that ask students a question about the mathematics and require them to explain their thinking. Each lesson includes at least one error analysis problem where students must find, describe, and correct an error in mathematical work. Examples include:
- In Mathematics I, lesson 8-5, students construct an argument on the possibility of a figure having rotational symmetry and no reflectional symmetry. Students explain and give examples to construct arguments.
- In Mathematics II, lesson 3-2, students graph quadratic functions using vertex form. In Problem 40 students are presented with two ordered pairs identifying the path a soccer ball travels. Students determine the quadratic function in vertex form, defend possible solutions that can not be determined, and explain why. Students generate a realistic graph using technology to explore undetermined values as well as find values that generate a realistic graph. Students also explain how key features of the graph correspond to the given situation.
- In Mathematics III, lesson 5-1, Problems 3 and 11, students analyze an error and explain or correct it. In problem 5, students identify the different parts of an exponential function. In problem 30, students create an exponential function to model a radioactive isotope and make a prediction.
Indicator 3d
Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations that manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent, and when appropriate are connected to written methods. Manipulatives are not consistently connected to written methods, when appropriate. Examples include:
- The materials occasionally direct students to use manipulatives within the materials, but the materials do not provide directions for the use of virtual manipulatives such as Desmos. For example, there were no Desmos screenshots or other supports offered in the materials.
- Algebra tiles are used in Mathematics I, but there was no evidence of their use in Mathematics II or Mathematics III to make connections across the courses.
- The Digital Math Tools include a graphing calculator and geometry tools to explore transformations, evaluate equations, and plot tables of data. The materials state, “much more is always available to students and teachers at PearsonRealize.com.”
Indicator 3e
The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
Criterion 3.2: Teacher Planning
Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for supporting teacher learning and understanding of the Standards. The instructional materials support teachers in planning and providing effective learning experiences by providing quality questions, and the teacher edition contains ample and useful annotations and suggestions on how to present the content in the student edition and in ancillary materials. The instructional materials rarely explain the role of the specific mathematics standards in the context of the overall series, and the teacher edition partially includes explanations and examples of the course-level mathematics specifically for teachers so that they can improve their own knowledge of the subject.
Indicator 3f
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. For each topic throughout the series, teachers are provided a math background section, specific to the focus, coherence, rigor, and mathematical practices addressed, and a topic planner. Within the math background section, teachers are given a clear and concise explanation of what students will be covering, what concepts students should already know, and where the concepts lead. Each lesson begins with an Explore & Reason, Model & Discuss, or Critique & Explain question in the student edition and teachers are provided an assortment of questions to ask their students to encourage discourse, conceptual understanding, support for productive struggle, and differentiation. For example, in Mathematics II, lesson 7-3, teachers are given the following to ask students: “Draw 5 new points on your map. How can you tell which middle school each point is closer to?” In addition, “What do you notice about the points that are the same distance from each middle school?”
Indicator 3g
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 instructional materials reviewed for enVision Integrated Mathematics meet expectations for containing ample and useful annotations and suggestions on how to present the content in the student edition and in ancillary materials. The materials provide an overview at the beginning of each topic explaining the overarching ideas. This overview is broken up among conceptual understanding, procedural skills, and applications.
Each lesson also includes useful annotations such as a lesson overview that contains student objectives, connections to previous and future content, common errors, vocabulary, and guiding questions with sample student answers. There is also a section with suggestions for advanced students, struggling students, and English Language Learners.
The teacher’s edition contains an abundance of teaching supports for both planning and in-class instruction. Within the side margins, teachers find highlights on effective teaching practices, essential questions, probing questions, habits of mind questions, additional examples, differentiated instruction supports for English Language Learners, advanced and struggling students, and common errors.
Examples from the teachers edition that show useful annotations and suggestions include:
- In Mathematics I, lesson 8-2, the student edition describes how to write a translation rule. The teacher’s edition suggests two questions to ask students to help them think about and make sense of these directions.
- In Mathematics III, lesson 3-3, the teacher’s edition provides a question to ask students at the beginning of the Explore & Reason activity and specifies that this question is intended for the whole class. The teacher’s edition provides one more question to ask students as they are completing the activity, specifying that the additional question should be asked to small groups. There are also two questions that the teacher could use to extend the thinking of early finishers, followed by one more question to summarize the activity for the whole group. The teacher’s edition also indicates that this activity could be done using an online tool instead of paper and gives a picture of what this tool looks like.
Indicator 3h
Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.
The instructional materials provide narrative explanations for answers and solutions in the Teacher’s Edition and in the Answer and Solutions Application.The Teacher’s Edition includes answers for “Do You Understand? And Do You Know How,” and Practice and Problem Solving tasks which include explanations that build teacher understanding of the mathematical content. In the Answer and Solutions Application, teachers are also given narrative explanations for answers and solutions. Examples include:
- In Mathematics I, Teacher Edition, Lesson 4-5, Problem 12, teachers are provided with the following: “Answers may vary. Sample: A system of two linear inequalities is similar to a system of two linear equations because their solutions are determined by where the graphs of the inequalities or equations intersect or overlap. They are different because a system of linear equations has infinitely many solutions when the two equations in the system are equivalent. A system of linear inequalities can have infinitely many solutions even when the inequalities are not equivalent.
- In Mathematics I, Lesson 9-5, Try It!, the Answer and Solution Application states, “Yes, there are two cases. If the congruent legs are included between the congruent acute angle and the right angles, the triangles are congruent by ASA. If the congruent legs are not included between the congruent acute angles and the right angles, the triangles are congruent by AAS.”
- In Mathematics II, Lesson 5-2, Answers and Solution Application, Practice and Problem Solving, Problem 11, “You can compare the zeros of the graph to the solutions you calculated and, because completing the square rearranges the equation to vertex form, you can also compare the vertices.”
The instructional materials make connections in the Topic Overview, Math Background and Coherence between prior knowledge, the lesson, and future content, but do not provide support for building teachers’ understanding of more advanced mathematical concepts. Examples include:
- In Mathematics I, Topic 5, Topic Overview, Math Background, Coherence, Looking Ahead connects the content of this topic to additional topics in Mathematics II and III: “Quadratic Functions Students will extend modeling with exponential functions to model using quadratic functions. They will compare models using linear, exponential, and quadratic functions. Students will apply what they learned about rational exponents to solve quadratic equations containing radicals. Radical Functions Students will apply their knowledge of radical exponents to solve radical equations. Students will also extend their knowledge of the effects of constants on functions as they transform radical functions. Rational Exponents Students will build upon their skills of writing radicals and creating, solving, and graphing exponential functions, when writing, solving, and graphing radical functions.”
- In Mathematics II, Topic 8, Topic Overview, Math Background, Coherence, Looking Ahead identifies Topic 11 and states: “Students will re-examine some Properties learned in Topic 8 and prove these properties using Coordinate Geometry. Topic 12 Students will use the polygon interior Angle-Sum Theorem when studying inscribed angles and polygons. Trigonometry In Mathematics III, students will use polygon angle sums to generalize a formula for the side length of regular polygons.”
- In Mathematics III, Topic 7, Topic Overview, Math Background, Coherence, Looking Ahead connects the content to “Precalculus Trigonometric Identities Students verify and apply trigonometric identities and the sum and differences of formulas. In Precalculus, they will verify and evaluate functions involving Multiple-Angle Formulas, Product-to-Sum Formulas, and Sum-to-Product Formulas.”
Indicator 3i
Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
The instructional materials reviewed for enVision Integrated Mathematics partially 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 materials rarely explain the role of the specific mathematics standards in the context of the overall series. In the beginning of each topic, the teacher’s edition provides a page that discusses 3 types of connections: how the topic connects with what students have learned earlier in the course or in previous courses, how different concepts are connected throughout the topic, and how that topic is connected to what students will learn later in the course or in future courses. For example, in Mathematics II, Topic 5, Quadratic Equations and Complex Numbers, there is an explanation of how this topic connects to Mathematics I work on functions as well as linear systems in Topic 5. There is also an explanation of how this topic will connect later in Mathematics III to polynomial functions. These descriptions do not reference specific mathematical standards.
Indicator 3j
Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
Indicator 3k
Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
Indicator 3l
Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
Criterion 3.3: Assessment
Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for offering teachers resources and tools to collect ongoing data about student progress. The instructional materials provide strategies for gathering information about students' prior knowledge, support for teachers to identify and address common student errors and misconceptions, and clearly denote which standards are being emphasized on assessments. Ongoing review and practice is available in the digital materials but not in the print materials, and the materials do not include guidance for teachers to interpret student performance.
Indicator 3m
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing strategies for gathering information about students’ prior knowledge within and across courses. There is a pre-assessment that addresses prior knowledge that is often not on course level. These online pre-assessments are editable. Answer keys are provided along with a list of the prior standards associated with each item on the assessment. Each topic in the series includes a topic readiness assessment, also found online, that provides the same features as the pre-assessment.
Indicator 3n
Materials provide support for teachers to identify and address common student errors and misconceptions.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing support for teachers to identify and address common student errors and misconceptions. The materials highlight common student errors and/or misconceptions for teachers. The materials also provide strategies to teachers for addressing common student errors and/or misconceptions. In the Teacher’s Editions there are red boxes titled “Common Error” that describe common errors and misconceptions. There are at least two different descriptions of common errors in each lesson. Examples include:
- In Mathematics I, lesson 4-2, students solve systems of equations using the substitution method. In example 2, students compare the graphical representation to the substitution method. The materials note that students may incorrectly simplify when removing parentheses and solving for the variable using the substitution method. Teachers are prompted to remind students to use the Distributive Property and guide them through a review of this property and how it applies to this context.
- In Mathematics II, lesson 9-2, example 3, the teacher’s edition notes that students may have difficulty identifying a reflection as a similarity transformation. The teacher’s edition suggests that teachers have students trace the figures and mark congruent angles while considering the rigid motion that maps figures with different orientations.
- In Mathematics III, lesson 2-4, example 2, the teacher’s edition notes that students may multiply the zero of the divisor, 1, by -5 and place that answer in the wrong location. The teacher’s edition suggests that teachers have students use long division to do the same problem and consider the relationships between the values in long division and synthetic division.
Indicator 3o
Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials do not provide support for ongoing review and practice for students in learning concepts. The materials also do not provide support for teachers to provide feedback.
The problems in each lesson address the content taught in that lesson. There is no ongoing review and practice built into the materials. However, the online materials allow teachers to assign a 10-question mixed review for each section. This is not included in the print materials, only in the online materials. Students are given immediate feedback on whether each answer is correct or not. If the answer is incorrect, there is an explanation of the content, and students can try again to get the correct answer. After two or three attempts, the correct solution is shown, and students can choose to try a similar question or move on to the next question. These problems mainly address procedural skills. An example of a conceptual question from this digital source can be found in the topic review for Mathematics III, Topic 4. One question states, “A relation has one element in its domain and two elements in its range. Is the relation a function? Is the inverse a function? Explain.” Students select a multiple choice answer. There is no support for teachers in grading these assessments, as the feedback is provided by the digital device when answers are incorrect. There is no support for teachers in using this information as students progress through these digital online reviews.
Indicator 3p
Materials offer ongoing assessments:
Indicator 3p.i
Assessments clearly denote which standards are being emphasized.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for assessments clearly denoting which standards are being emphasized. There are lesson quizzes at the end of each lesson, topic assessments and performance assessments at the end of each topic, four benchmark tests throughout the year, a mid-year assessment, and an end-of-course assessment for each course. Each of these assessments include an answer key and the standards being assessed for each item of the assessment. The benchmark tests, mid-year assessments, and end-of-course assessments are found online, not in the print materials.
Indicator 3p.ii
Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for offering ongoing assessments that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The materials do not include guidance for teachers to interpret student performance. The materials provide some suggestions for follow-up after students complete some assessments. Each lesson quiz provides suggestions of differentiated assignments based on scores from the quiz. These assignments include: a print or digital assignment called Reteach to Build Understanding, an Additional Practice worksheet, an Enrichment worksheet, and a Vocabulary worksheet. There is no guidance for interpreting student performance, and there is no follow-up suggestions for any of the other assessments.
Indicator 3q
Materials encourage students to monitor their own progress.
Criterion 3.4: Differentiation
Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide teachers with strategies for meeting the needs of a range of learners, tasks with multiple entry-points that can be solved using a variety of solution strategies or representations, support, accommodations, and modifications for English Language Learners and other special populations, and support for advanced students to investigate mathematics content at greater depth.
Indicator 3r
Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide some strategies to scaffold lessons, but there are no general statements about sequencing provided. The lessons do include common misconceptions for the teacher to point out, as well as a section that provides instructions on how to assist struggling students and advanced students; however, these sections only contain additional questions and explanations. They do not contain any information on how to sequence the lesson for any learner.
Indicator 3s
Materials provide teachers with strategies for meeting the needs of a range of learners.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing teachers with strategies for meeting the needs of a range of learners. Some general statements for the teacher about meeting the needs of all learners are included. There is also a section in the teacher’s edition with ideas and guiding questions to support struggling students with each lesson. The questions in this section are typically questions that help reinforce concepts students need to be successful in the lesson or questions that help build a student’s conceptual understanding of the lesson. There is also a section in the teacher’s edition for advanced students, which contains more complex problems for more advanced students to complete.
For example, in Mathematics III, Topic 7-2, Teacher’s Edition, advanced students deepen their understanding as they explore the right triangle case of the Law of Cosines. Struggling students review how to use the Law of Cosines based on the abbreviations for triangle congruence.
Indicator 3t
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. The materials provide teachers with guidance on helping students solve problems with multiple entry points. Some of the Mathematical Modeling in 3-Acts and STEM projects give students multiple entry points to a problem as well as allow students to try a variety of methods to solve the problem. Examples include:
- In Mathematics I, Topic 3, Mathematical Modeling in 3-Acts, students determine whether it would be faster to check out in a regular line or an express line at a grocery store. Students are given information about a hypothetical situation involving a certain number of customers in each line as well as how many items each customer has. There are various ways students could approach this problem.
- In Mathematics II, Topic 5, STEM project, students design a ballpark and determine what it would take to hit a home run at the park. Students can choose every dimension for their ballpark and determine an appropriate quadratic equation to fit their design.
The teacher’s edition embeds Mathematical Modeling in 3-Acts into the pacing guide, but it does not include the STEM projects into this guide. There are no suggestions of where to use the STEM projects or how much time to allow for them.
Indicator 3u
Materials provide 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).
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in the learning of mathematics. Within each topic in the teacher’s edition, the series provides guidance for beginning, intermediate, and advanced ELL students. Students are also given access to read and listen for English and Spanish definitions. Each lesson provides a “Concept Summary” which includes definitions of the introduced vocabulary. Students are provided multiple representations of the concepts addressed within the lesson.
There are strategies for special populations to practice vocabulary as seen in Mathematics III, Lesson 2-7, Teacher’s Edition, page 109A, ELL supports at the bottom of the page provide guidance on vocabulary practice for Beginning, Intermediate, and Advanced speakers. Each practice section begins with “Do you understand”, from which teachers can modify pacing for special populations if needed.
Indicator 3v
Materials provide support for advanced students to investigate mathematics content at greater depth.
The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing support for advanced students to investigate mathematics content at a greater depth. The materials provide multiple opportunities for advanced learners to investigate the course-level mathematics at a greater depth. There are no instances of advanced students doing more problems than their classmates. Each topic begins with a “Topic Readiness Assessment” which, based on students’ performance, assigns a study plan tailored to students’ specific needs, including advanced students. Each lesson provides teachers with additional problems for advanced students. Some examples of enrichment for advanced learners include:
- In Mathematics II, Lesson 9-1, page 417 in the Teacher’s Edition, an extension is provided of a problem on dilations.
- In Mathematics III, Lesson 2-2, page 71 in the Teacher’s Edition, an extension is provided for advanced students to extend their work with polynomial functions.
Teachers are also provided an assignment guide for advanced students for each lesson.
Indicator 3w
Materials provide a balanced portrayal of various demographic and personal characteristics.
Indicator 3x
Materials provide opportunities for teachers to use a variety of grouping strategies.
Indicator 3y
Materials encourage teachers to draw upon home language and culture to facilitate learning.
Criterion 3.5: Technology Use
Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
The instructional materials reviewed for enVision Integrated Mathematics integrate technology in ways that engage students in the Mathematical Practices and are web-based and compatible for multiple internet browsers. The instructional materials also include opportunities to assess student mathematical understanding and knowledge of procedural skills using technology, to personalize learning for all students, and to easily customize for local use. The instructional materials do not include or reference technology that provides opportunities for teachers and/or students to collaborate with each other.
Indicator 3aa
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 Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
Indicator 3ab
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
Indicator 3ac
Materials can be easily customized for individual learners.
Indicator 3ac.i
Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
Indicator 3ac.ii
Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
Indicator 3ad
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.).
Indicator 3z
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.