Kindergarten - Gateway 3

The instructional materials for Math Expressions Kindergarten meet expectations that materials distinguish between problems and exercises.

Materials provide the opportunity for students to learn new mathematics through problem solving activities. In a typical lesson, Activity 1 and Activity 2 develop the new math content of the lesson. Lessons are outlined according to an Inquiry Lesson Path based on four phases: Phase 1 Guided Introduction, Phase 2 Learning Unfolds, Phase 3 Knead Knowledge (practice stage), and Phase 4 Maintaining and Integrating Fluency. Students build mastery through practice problems/exercises. In a typical lesson, during Activity 2 and Activity 3, students complete problems in the Student Activity Book which provide practice with the math content. The purpose of each Activity within a unit is explained in the “Teaching the Lesson Section” found on the first page of each lesson.

Examples include but are not limited to:

• In Unit 2, Lesson 7, Teaching the Lesson Section, Activity 1, Counting Mat Activities with 5-Groups, is stated as being important because “Children build a strong foundation for understanding the concepts of equal and not equal by comparing both numerals to objects and objects to objects. Activity 2, Review Numbers, is stated as being important because “Children practice writing numbers.”
• In the Student Activity Book, Unit 1, Lesson 6, Activity 1, students observe pictures and count the number of objects in them. They practice visualizing groups of 3 and 4 objects. In Activity 2, students look at a picture of dogs, identify Puzzled Penguin’s counting error, and count the number of cats and dogs in the picture. To deepen their understanding, students draw pictures of 5 hats, 3 cats, 4 stars, and 2 cars.

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide tasks in an intentional sequence.

The design of the assignments follows a natural progression, leading to full understanding and mastery of new mathematics. Lessons follow a consistent pattern of two or three activities per lesson. Activity 1 usually focuses on the new learning. This learning is reinforced in Activity 2, and then students practice the new learning by completing Student Activity Book pages during Activity 3. Activity 3 either reinforces the new skill, or it reviews previously learned content.

Examples include but are not limited to:

• In Unit 1, Lesson 9, students build an understanding of comparing numbers using a vertical Comparing Mat and square tiles to compare the height of the columns of tiles. On Student Activity Book page 27, numbers are represented on pictures of the vertical Comparing Mat with square tiles. On the following page, students color in boxes on blank Comparing Mats to represent the numbers being compared. Numerals are shown for each of the numbers being compared, but students are not expected to write the numbers yet.
• In Unit 2, Lesson 13, students learn about triangles by exploring models of triangles. After the initial exploration, students practice identifying objects in the classroom that are triangles, and then they identify triangles in pictures in the Student Activity Book.
• In Unit 4, Lesson 21, Activity 3, students practice combining 3D shapes to make new shapes using cubes, cones, cylinders, and cube-, cone-, and cylinder-shaped objects. Next, students build their own solid shapes using clay and straws and combine the shapes to make a new shape.

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide varied opportunities for students to present their mathematical knowledge.

Examples of how students produce answers and solutions include but are not limited to:

• Coloring groups of objects
• Identifying the error and correcting it (Puzzled Penguin)
• Using counters to practice counting to 20
• Explain their thinking during Math Talk in Action Activities 
• Modeling addition with fingers
• Modeling number partners using a Break-Apart Stick and square tiles
• Acting out problems
• Model story problems with objects
• Using Secret Code Cards to represent teen numbers
• Drawing circles to represent numbers
• Writing expressions to represent the number of objects in pictures, “11 = 10 + 1”
• Using 2-D shapes
• Using sorting cards to classify objects
• Using base-ten blocks to represent teen numbers
• Class discussions
• Fluency Practice in the Student Activity Book
• Problem Solving exercises in the Student Activity Book

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide virtual and physical manipulatives that are faithful representations of the mathematical objects they represent and are connected to the written material.

Students use a variety of manipulatives including counters, Secret Code Cards, base-ten blocks, 5-Counter Strips, comparing mats, square tiles, Stair Steps, and Math Mountains. Most of these manipulatives are available virtually in the itools found in ThinkCentral. Manipulatives are often connected to written methods when appropriate. 

Examples include but are not limited to:

• Unit 3 Lesson 6, counters and Break-Apart Sticks are used to model partners of 5. These manipulatives make the decomposition easier for students to see.
• Unit 3, Lesson 8, Square Tiles and 5-Square Tiles are used to represent the ten and ones in teen numbers. This is connected to pictures found in the Student Activity Book and to equations.

The instructional materials for Math Expressions Kindergarten meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Examples of teacher support include but are not limited to:

• Questions for teachers to pose are consistently included in the lesson narrative. They are italicized, making them easily visible. 
• MathTalk in Action boxes include questions for the teacher to ask and potential student responses. For example, in Unit 3, Lesson 15, the teacher is guided to ask the questions: “What do you see is the same when you look at them (Teen Number Displays)? What do you see that is different? Does it matter what objects you use to show a number?”
• Teacher Notes are also provided at the bottom of the lesson pages and include questions to deepen students understanding of the mathematics. For example, in Unit 3, Lesson 6, the Learning Community notes provide several probes to promote student thinking and discussion: “How do you know that you need a 5-group and another one to show 6? How do you know how many extra ones you need to draw after the 5-group? What numbers do you want to use for your story? Why?”
• Teacher Notes at the bottom of lesson pages also include information about best practices to strengthen teachers questioning techniques. For example, in Unit 4, Lesson 1, the MathTalk note provides the following questions for teacher reflection: “What types of questions are you asking your students? Are you asking questions that elicit a short answer, or do you probe to learn more about children’s thinking as they tell about their work?"

The instructional materials for Math Expressions Kindergarten meet expectations that 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. Materials also, when necessary, provide teacher guidance for the use of embedded technology to support and enhance student learning.

Ample guidance is provided in the Teacher Guide for planning. The Pacing Guide provides guidance for each unit. Charts show the Learning Progression for the Content Standards Across Grades for the standards addressed in the Unit. A Planning Chart for each Unit that includes Math Activity Center Resources, Big Idea Resources, and Lesson Resources is provided. The Planning Chart also includes the standards addressed in each lesson, the digital and print resources for each lesson, and the assessments for the Unit. A table of the Standards for Mathematical Practice and the lessons where each is embedded is included. Also, a Table of the Math Content Standards and the lessons where they are taught is provided. Finally, a list of Assessment, Review, and Intervention Resources for the Unit is provided.

Examples include but are not limited to:

• Each lesson includes guidance on the focus of each Activity and why it is important. For example, in Unit 3, Lesson 6, Activity 1, Show All the Partners of 2 Through 6, is stated as important because “Using number Tiles and +/- Tiles to show expressions is preliminary work before writing them.”
• Each Activity includes an explanation of what the teacher should do or say and includes possible correct responses to questions posed by the teacher. 
• Formative Assessment and Check for Understanding questions are highlighted in the Teacher Guide.
• Math Practices are highlighted in the lesson narratives.
• A list of questions that can be used to build a Math Talk community is included at the beginning of each Unit. 
• Notes at the bottom of each page of the lesson narrative give useful suggestions for implementing the lesson, asking questions, acquiring vocabulary, and building concepts. For example, in Unit 1, Lesson 1, the Learning Community note, Building Concepts, an explanation on the use of 5-groups states, “Using the 5-group arrangements repeatedly on the Number Parade helps build recognition of small quantities less than or equal to 5.”
• Digital Resources for each lesson are highlighted on the first page of the lesson, and itools, which includes virtual manipulatives, are shown in the lesson narrative when it may be beneficial to use them. For example, in Unit 2, Lesson 9, a picture of itools Number Tiles is shown because they may be used in the lesson.

The instructional materials for Math Expressions Kindergarten meet expectations that 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.

Notes are provided at the bottom of each lesson narrative in the Teacher Edition to deepen teacher understanding of the mathematics and to improve instruction. Math Background Notes provide information about the math topic to deepen teacher’s understanding. Watch For! Notes provide information about potential misconceptions and things to watch for as students complete the lesson. What to Expect from Students notes provide information about how students might engage with the math and why the math is important. Building Concepts notes provide explanations of the math and how students learn.

Examples include but are not limited to:

• Path to Fluency Charts are provided.
• Chart of the Addition/Subtraction and Multiplication/Division problem types is provided.
• Table of the Major Work and Major Clusters of the Grade is provided.
• Table of the Common Core State Standards for Mathematical Content is provided.
• Table of the Common Core State Standards for Mathematical Practice with an explanation for each Mathematical Practice is provided.
• The Putting Research into Practice section at the beginning of each unit provides research about best practices in teaching children mathematics.
• The Math Background section, prior to each unit, includes sections that deepen teacher knowledge of the math in the unit. Examples include Learning Path in the Common Core Standards, Help Students Avoid Common Errors, Effective Practice Routines, Relate Mathematics to the Real World, and Focus on Mathematical Practices. 
• The Math Background section, prior to each unit, provides excerpts from the Progressions for the Common Core State Standards.
• The Mathematical Practices section, prior to each unit, provides information on how students will engage with the Practice Standards throughout the unit.
• A Teacher Glossary is provided.

The instructional materials for Math Expressions Kindergarten meet expectations that materials contain a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series.

A Path to Fluency: Kindergarten through Grade 6 Chart is provided and highlights the fluency requirements of each grade level, activities that target fluency, and interventions for Grades 3, 4, 5, and 6. Also, a Major Work and Major Clusters of the Grade Chart for Grades K-6 is provided. Finally, for each unit, a Learning Progressions for the Common Core State Standards Chart for the domains addressed in the unit, which includes the current, prior, and next grade level standards is provided.

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Examples include but are not limited to:

• The Assessment Guide contains a 25 question Prerequisite Skills Inventory Test, organized by Domains, and a corresponding Prerequisite Skills Inventory Test Correlation document. The correlation aligns each question with a description of the prerequisite skill addressed, as well as the DoK level of the question. This correlation document is formatted as a table so each student’s performance by question/skill can be recorded. The Prerequisite Skills Inventory Test is designed to be administered at the beginning of the school year.
• When a student completes practice opportunities and tests in the Personal Math Trainer, all of the performance data and adaptive learning information follows each student to the next grade.
• Quick Practice activities at the beginning of each lesson are designed to “provide opportunities for students to call to mind their prior understanding of a topic that has already been discussed in class or to begin to build a prerequisite skill for a topic that is to come later” (Teacher Edition page I4).
• In the Unit 4 Overview, the author discusses developing the prerequisites for Level 2 and Level 3 Methods of Adding and Subtracting. The author states, “Students are working on all the prerequisites for the Level 3 derived fact methods, such as make-a-ten. The prerequisites are: Seeing the tens in teen numbers; knowing all the partners of 10, and knowing all the partners of numbers below 10” (Teacher Edition page MB1-U4).
• Students take three progress monitoring assessments to assess grade level skills and concepts students have learned. The Beginning of Year test assesses concepts they will learn throughout the year, the Middle of Year Test shows progress made in the first half of the year, and the End of Year Test measures growth throughout the school year. 

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide support for teachers to identify and address common student errors and misconceptions.

Examples include but are not limited to:

• Common student errors are identified for each Unit Review/Test question along with a direction on how to help students. For example, on the Unit 3 Review/Test, if a student misses Question 3, the common error identified states, “Students may be confused about the objects not in the group of ten.” Teachers are directed to “Encourage children to count ten stars and then ring that group. Then have them count the remaining or extra stars.”
• The Math Background section of each Unit provides a narrative called “Help Students Avoid Common Errors”. 
• Puzzled Penguin activities highlight typical student mistakes and misconceptions by challenging students to find the Puzzled Penguin’s mistake and correct it. Teachers are provided questions in order to lead classroom conversations through a MathTalk format that revolve around the mistake and its correction, helping students understand the mathematics.
• Watch For! are teaching notes periodically found in each unit. These notes alert teachers to common misconceptions they should be on the lookout for. For example, in Unit 3, Lesson 3, the Watch For! note states, “Some children may just move the Break-Apart Stick instead of actually switching partners. Instruct them that they need to change the positions of the solid tile group and dot tile group to show they are the same 2 groups just not shown in different positions.”

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Examples include but are not limited to:

• Homework and Remembering pages provide a review of recently taught topics as well as a spiral review throughout the year. The Personal Math Trainer online platform allows students to complete homework tasks for each lesson, receive instant feedback, and  step-by-step guidance if needed.
• Unit Review/Test and Performance Tasks for each unit are found in the Student Activity Book. The author states, “You can use this Unit Review/Test as an end-of-unit review to determine if children have mastered the content of the unit. You can assess children’s knowledge with one of the forms of the Unit 1 Test in the Assessment Guide.” Teachers are provided with a Data-Driven Decision Making Table which suggests specific reteaching activities for students who incorrectly answer the correlated questions, as well as suggestions for which Standards Quiz to assign in the Personal Math Trainer which provides a personalized intervention for the student. The Performance Task includes a detailed scoring rubric which can be used to provide feedback to students.
• The Personal Math Trainer can be used for homework practice, fluency practice, standards practice, unit pre-tests with instant feedback, and step-by-step guidance when needed. Everything a student completes in the platform helps to improve the adaptive workflow (powered by Knewton Adaptivity) for the student throughout the year.  
• The Knewton Adaptivity, Homework with Daily Intervention and Enrichment can be used in multiple ways in the classroom. A 5-minute Warm-Up provides students with personalized review prior to the assignment. On-level and advanced students may receive less or no warm-up, as determined by Knewton. After the warm-up, the HMH pre-built assignment is given to students. A 10-minute personalized enrichment is provided for students who demonstrate mastery (95% or higher) on the assignment.  Enrichment shows students proximate, forward-looking concepts based on the assignment content.
• Other Formative Assessment opportunities include: daily Check Understanding tasks on select Student Activity Book pages, daily observation with anecdotal notes, observations during Math Talk conversations, and analyzing student work samples and student responses in the Student Activity Book. Portfolio suggestions are also provided at the end of each unit.

The materials reviewed for Math Expressions Kindergarten meet the expectation for offering ongoing assessments that clearly denote which standards are being emphasized.

Examples include but are not limited to:

• Every unit includes two versions of a Unit Assessment, Form A and Form B, found in the Assessment Guide. Both assessments provide PARCC and Smarter Balance question formats and a Standards Correlation Document which can be used to collect student performance data. This document also aligns each question to a DoK Level and Standard(s). 
• Each unit contains a Performance Assessment which can be found in the Assessment Guide. The standards are clearly noted for the assessment as a whole, and not by specific question. 
• There are three Benchmark Assessments (Beginning of the Year Inventory, Middle of the Year Inventory and End of Year Assessment) found in the Assessment Guide. Standards for these assessments are clearly noted on the Correlation Document and DoK Levels are noted.

The instructional materials for Math Expressions Kindergarten meet expectations that assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Examples include but are not limited to:

• Scoring Guides are provided for each Unit Performance Assessment found in the Assessment Guide. Each question is assigned a point value and a rubric is provided to determine Performance Levels 0-3 based on the number of points earned. Additionally, each Performance Level is further defined on a task-specific basis and indicates specifics about student understanding to assist teachers in interpreting student work. Sample student work for each Performance Level is also provided in the Assessment Guide. 
• Answer keys for the Unit Assessments, Form A and Form B, are located in the back of the Assessment Guide. However, no guidance or suggestions for follow-up instruction are included in the Assessment Guide. 
• The online Personal Math Trainer can be utilized to administer Beginning, Middle and End of Year Tests, Unit Assessments, and Fluency Checks. The data from these assessments is collected and analyzed, and a Personal Study Plan is prescribed through Adaptive Workflow settings (through Knewton Adaptivity) based on the data and the mastery threshold percentage established for the assessment. The primary use is for end of unit assessments, or to provide targeted students with occasional review, intervention, and re-assessment opportunities. Students must complete an initial assignment (test). Students who do not demonstrate mastery receive a Personal Study Plan, consisting of a personalized review and intervention assignment lasting 15 minutes. After completing the Personal Study Plan, the initial assignment is given again, but the numbers in the assessment are changed.
  

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Teachers guide students through an inquiry path to become mathematically proficient. The four stages of the path to learning are guided introduction, learning unfolds, knead knowledge through practice, and maintain fluency. As stated by the publisher in the Teacher’s Edition, “Within the curriculum, a series of learning progressions reflect research on students’ natural learning stages when mastering concepts such as computation and problem-solving strategies. These learning stages informed the order of concepts, the sequence of units, and the positioning of topics in Math Expressions.”

Examples include but are not limited to:

• In Unit 1, Lesson 16, students learn about and write the number 5. Prompts are given for EL students at three different levels: emerging, expanding, and bridging. The prompt states, “Help children think about the number 5. Hold up 1 hand.” Emerging: “What is this? Wiggle your fingers. What are these? How many fingers? 1 hand has 5 fingers. Have children repeat.” Expanding: “Wiggle your fingers. How many fingers are on 1 hand? Ask children to repeat. Draw 5 toes on a foot. Continue.” Bridging: “Have children count their fingers. How many fingers do you have on 1 hand? Have children tell how many toes are on 1 foot.”
• In Unit 3, Lesson 9, the Watch For! teaching notes state, “Some children may try to use shapes (like squares) that fit inside but leave empty spaces. Remind them that the entire shape must be filled with no empty spaces.”
• In Unit 5, Lesson 2, the Universal Access/Extra Help differentiated instruction note states, “You can form a group of children who need additional help and solve as many problems with them as necessary. Alternatively, choose some Student Leaders to help those who need assistance, while the rest of the class works on the problems independently.”

The instructional materials for Math Expressions Kindergarten meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Examples include but are not limited to:

• An explanation of differentiated instruction is provided in the Teacher Edition. 
• A list of intervention resources is provided for each unit in the Unit Overview Assessment.
• Math Activity Centers resources for on-level, challenge, and intervention are provided for each unit’s lessons. 
• Teaching notes for English Learners are provided for emerging, expanding, and bridging students and are provided for each unit’s lessons.
• Some lessons have Differentiated Instruction notes provided for universal access/extra help.

The instructional materials for Math Expressions Kindergarten meets expectations that materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations.

MathTalks provide “an inquiry environment that encourages constructive discussion of problem-solving methods through well-defined classroom activity structures. . . comprises four components: questioning, explaining math thinking, contributing math ideas, and taking responsibility for learning” (Teacher Edition page I3). Initially, teachers model MathTalks and then students run the MathTalk. For example, in Unit 5, Lesson 2 the MathTalk instructs teachers to “Explain that Puzzled Penguin was asked to write the numbers 14, 17, and 19. Puzzled Penguin is not sure that each number was written correctly. Help Puzzled Penguin look at the first number to see if it is correct.” The teacher is then provided prompts to help guide the student discussion.

The instructional materials for Math Expressions Kindergarten meets expectations that 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.

Examples include but are not limited to:

• Scaffolding of vocabulary is provided. For example, in Unit 5, Lesson 10, teachers are provided support to explain the word fewer for EL students. Teachers are instructed to “Write ‘fewer’ on the board. Fewer means a smaller amount of something of less. Draw a row of 3 circles above a row of 4 squares. The are fewer circles than squares. Have students repeat.” 
• Extra support is provided for EL students. For example, in Unit 5, Lesson 13, students count by ones and tens to 100. Teachers are instructed to “Write 100 and one hundred on the board. Display a hundred base ten block. Lead children in counting to one hundred as you point to the ones cube in the hundred block.” 
• Each unit lesson contains a Math Activity Center with activities and resources for students who are on-level and those needing challenge and intervention.
• Teaching notes included in some lessons provide specific guidance for teachers to support students who are emerging, expanding, and bridging language acquisition.

The instructional materials for Math Expressions Kindergarten meets expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Examples include but are not limited to:

• Math Lessons contain Differentiated Instruction Math Activity Centers. Challenge Resources specify which Activity Card will challenge advanced students.
• The online Personal Math Trainer provides personalized enrichment with learning supports.
• Challenge worksheets for each lesson are available in print and digitally and are noted on the Differentiated Instruction page for each lesson.
• Math Readers, books in the Math Activity Center, place math content in the context of stories and support higher levels of critical thinking.

The instructional materials for Math Expressions Kindergarten meets expectations that materials provide a balanced portrayal of various demographic and personal characteristics. 

Examples include but are not limited to:

• Puzzled Penguin appears throughout the unit to provide opportunities to help students avoid common errors. These errors are presented as letters to students. Students teach Puzzled Penguin the correct way and explain why the penguin is wrong. 
• Math Readers contain a variety of animals, children, and adults.