enVision MATH 2.0 Texas Kindergarten

Evaluation for 2.1
Materials dedicate most attention to the development of the primary focal area for the grade-level.

The materials concentrate on the development of the primary focal areas for the grade level, which are whole numbers, addition and subtraction, two-dimensional shapes, and three-dimensional solids. They spend the majority of concept development time on the primary focal areas for the grade level as outlined in the TEKS, embed strategic and integrated instruction in conceptual understanding, and provide practice opportunities to meet mastery throughout the instructional materials.

Evidence includes but is not limited to: 

The materials devote 12 of the 17 topics to the focal areas as outlined in the kindergarten TEKS. Topics 1–6 and 11 focus on whole numbers, Topics 7–8 and 10 center around solving problems with addition and subtraction, and Topics 12–13 address identifying and creating two-dimensional shapes and three-dimensional solids. 

The design of the materials develops the systematic teaching and learning of math concepts. Introductory components, including “Focus on the Process Standards,” “Formulating a Plan,” and “Focus on Content Information,” provide context before each lesson. Each topic contains a lesson overview, which includes a math breakdown, essential understanding, vocabulary, and materials for each lesson. The instructional material provides scaffolding by reviewing basic concepts required prior to beginning each topic in a “Review What You Know” section. The lessons themselves contain various sections, such as “Math Anytime,” “Problem Based Learning,” “Visual Learning,” and “Assess and Differentiate.”

Kindergarten materials provide various practice opportunities in multiple settings, including center activities, online games, “Problem Solving Reading Mats,” and “Math and Science Project” pages. Each lesson provides a mixture of videos, paper-and-pencil worksheets, and games, as well as opportunities for group, partner, and individual work. Each lesson reinforces focal area skills through activities such as the “Daily TEKS Review,” the “Daily Challenge,” and intervention activities. The materials build upon concepts taught across previous weeks and units to increase rigor and ensure students master the full intent of the concept.

Evaluation for 2.2
Materials strategically develop students’ conceptual understanding by following a progression of learning from concrete to representational to abstract (CRA) as is appropriate for the grade-level and content.

The materials provide activities across the CRA continuum; student learning progresses from more concrete to more abstract as the instructional materials progress throughout the year. The materials introduce increasingly rigorous practice with models, manipulatives, and representations, as the materials spiral skills. Although the materials’ teacher professional development resources discuss ways to teach the topics, they do not explicitly support teachers with instructional suggestions to help students progress through the CRA continuum.

Evidence includes but is not limited to: 

The materials include a variety of concrete models and manipulatives, pictorial representations, and abstract representations to introduce and practice mathematical concepts. Concrete materials are included in the “Center Games Manipulatives Kit”; they include number tiles, number cubes, and red and blue squares. Online “Math Tools” are also available for students to use when completing activities. The curriculum starts exclusively with pictures and introduces increasingly rigorous practice with models, manipulatives, and representations, as the materials spiral skills. In Topic 1, Lesson 1, the worksheet has students color in bars to represent the number of items they see on the page. As the year progresses, students are tracing and then writing more numbers. In Topic 6, Lesson 5, students trace and then write numbers up to 30. The abstract level of the continuum is briefly addressed in Topic 7 for addition, Topic 8 for subtraction, and Topic 10, where both are reviewed.

Materials provide teacher guidance in professional development videos at the beginning of each topic. Additionally, teacher resources and info documents are available to click on under each lesson. Each lesson provides teachers with “Essential Understanding” and “Materials” sections to review important information. Although this provides the student with progression along the CRA continuum, it is not specified to the teacher that this is what is being done. The materials provide guidance for teachers to help students move through the phases of the CRA continuum. For example, the students may begin the study of identifying whether a problem situation shows addition or subtraction and match the problem to a number sentence using seven connecting cubes of two different colors (concrete). The materials then move to write number sentences for related addition and subtraction facts with totals of six or seven (representation). The materials guide teachers to move back to concrete practice with students who struggle with the transition to representation by showing students four blue cubes and three green cubes, telling a story about these cubes, and asking, “How can you choose which operation to use to solve this problem?” Though this helps students, the materials do not intentionally support teachers with instructional suggestions to help students to progress through the CRA continuum.

Progressions from concrete to pictorial to abstract can occur across lessons and within lessons. The main focus of a specific lesson might be concrete, pictorial, abstract, or a combination of those. The Essential Understanding and Math Background on the first page of each lesson in the Teacher’s Edition highlight the focus of the lesson and make connections to progression across the grade.

A “Solve & Share” begins the lesson, and students can solve this problem in any way they choose. They use concrete materials at times (e.g., counters, snap cubes, place-value blocks, fraction tiles) and pictorial representations at times (e.g., number lines, arrays, grids, area models, bar diagrams). When students share and discuss their solutions, rich conversations deepen conceptual understanding of connections between concrete, pictorial, and abstract. The “Visual Learning Bridge” then provides instruction that connects students’ work in Solve & Share to new ideas taught in the lesson. The Visual Learning Bridge at times shows pictures of concrete materials, drawings of concrete materials, or diagrams that are representations of mathematical concepts. These representations are connected to abstract numbers, symbols, and procedures. The Teacher’s Edition provides sample student work and guiding questions to facilitate this as part of a classroom conversation.

Evaluation for 2.3
Materials support coherence and connections between and within content at the grade-level and across grade levels.

The materials include many areas to support students’ coherence and connections between and within content at and across the grade level. They support teachers in understanding the horizontal and vertical alignment guiding the development of concepts. The teacher guidance documents explain an increase in depth, breadth, and complexity to prepare students for the next year’s work.

Evidence includes but is not limited to: 

The materials show vertical content knowledge by referring back to previous work at the beginning of every lesson. For example, Topics 1 and 2 introduce and review the whole numbers 0 to 5 and then practice how to compare the numbers. Topic 2 starts with a “Review What You Know” section, which reviews the introduction of numbers 0 to 5 taught in Topic 1. This vertical progression continues with Topic 3, which starts with a Review What You Know section that reviews what was taught in Topic 2. Topics 3 and 4 introduce and review the whole numbers 6 to 10 and then practice how to compare the numbers. These identification and comparison skills are used again in Topics 5 and 6, where students practice their numbers up to 20 and up to 30, respectively. Also, at the beginning of each topic, there is an “Interactive Math Story” that activates prior knowledge. The materials contain tasks that direct teachers to build on students’ prior knowledge before presenting a new concept or problem aligned with a grade-level focal area. For example, as students begin Topic 14, the materials activate prior knowledge of comparing sizes; the teacher holds up two pieces of chalk and two pencils of different lengths and asks, “Which one is longer than the other? Which one is shorter?”

The materials include tasks and problems that intentionally connect two or more concepts as appropriate for the grade level and demonstrate that mathematics is used in many real-world situations. In the kindergarten materials, there are 16 “Math and Science Projects” to help students recognize and apply mathematics outside of the mathematics curriculum. The Topic 7 Math and Science Project shows students a picture of a mother cat with her kittens and asks them to explore the difference between living and non-living. Then, students draw a picture of a cat with her five kittens and tell a joining story about how many cats there are in all. The Topic 10 Math and Science Project has students explore rocks and then write addition and subtraction sentences about the rocks. The materials also provide opportunities to apply mathematical concepts to everyday situations. For example, in Topic 13, Lesson 1, there is a math and science activity that shows students how objects around them are made up of the three-dimensional objects they have been learning about. For example, a ball is a sphere, a soda can is a cylinder, and a block is a cube.

Materials include tasks that require students to understand how mathematical ideas interconnect and build upon one another and provide opportunities for students to examine relationships and patterns within and across concepts. For example, in Topic 9, Lesson 5, students count the values and add different coins to determine how much money they have and which items they could buy with that amount of money. The materials include tasks that require students to understand how mathematical ideas interconnect and build on one another to produce a coherent whole. For example, in Topic 10, Lesson 3, fact families are used to emphasize the inverse relationship between addition and subtraction. Another example of mathematical interconnectedness is in Topic 15, Lesson 5, where students count several designated objects, record the amounts on separate graphs, and then use the graphs to answer questions such as “Which object was the greatest? Which object had the least? Were the objects equal in number?”

The instructional materials include teacher supports that help teachers understand how concepts build over time and how the skills covered in each unit progress vertically and horizontally. The “Topic Planner” section for each topic gives an overview of the TEKS and Mathematical Process Standards covered in each lesson, as well as the “Essential Understanding” for students. The “Skills Trace” section in the “Content Guide” shows how TEKS build from kindergarten through 2nd grade and beyond. A “Scope and Sequence,” also in the Content Guide, shows when students are introduced to certain content, when they practice it, and when they are expected to apply it. Materials build students’ vertical content knowledge by referencing or showing how concepts progress in rigor. The three-step lesson format is highly dependent on the teacher modeling or using questions effectively to promote student discourse and connect previous learning to the current objective. Materials reference familiar models and strategies to facilitate rigor and concept development. The materials include tasks and problems that intentionally connect concepts in the “Solve and Share” problems and the “Visual Learning Bridge.” The Student Edition and center activities use story problems to help students discuss and apply math to real-world problems.

Evaluation for 2.4
Materials are built around quality tasks that address content at the appropriate level of rigor and complexity.

The materials include quality tasks that address content at the appropriate level of rigor and complexity. The materials provide guidance for the teachers on how to appropriately revise content to be relevant to their specific students, their backgrounds, and their interests. The materials provide teachers with possible student responses or strategies to practice questions and tasks. The materials provide teachers with common misconceptions of student responses and strategies. The materials provide teacher guidance on preparing for and facilitating strong student discourse grounded in the quality tasks and concepts.

Evidence includes but is not limited to: 

The materials guide students through the CRA tools, models, and understandings with increasing depth and complexity while also increasing the rigor within and throughout the topics. The materials introduce numbers in Kindergarten, starting with the numbers 0 to 5 in Topic 1, the numbers 6 to 10 in Topic 3, numbers up to 20 in Topic 5, numbers up to 30 in Topic 6, and then counting to 100 in Topic 11. Activities regularly start at the concrete level, using manipulatives, and then advance to pictorial and abstract levels of knowledge. For example, in Topic 7, students start by joining groups of counters; then, they progress to counting pictures of two different things to join together; by the end of the lesson, students are writing the numbers to show how many they get when they join the numbers together. In Topic 11, Lesson 2, students investigate and look for patterns on a hundreds chart. Students initially focus on counting by tens and eventually see that the design on the hundreds chart is a mathematical pattern. In Topic 14, Lessons 1–6, students begin by using words to describe attributes. This prepares them for the next lesson, where they use tools such as cubes to measure length and then move to a different tool such as containers to measure capacity and weight.

The materials provide guidance for the teachers on how to appropriately revise content to be relevant to their specific students, their backgrounds, and their interests. The materials provide teachers with possible student responses or strategies to practice questions and tasks, but they do not describe which ones are the most appropriate for the task based on grade-level expectations. The materials provide teachers with common misconceptions of student responses and strategies. The materials provide teacher guidance on preparing for and facilitating strong student discourse grounded in the quality tasks and concepts.  

The materials include explanations of the mathematical concepts and goals behind each task and serve to build teacher content knowledge in the “Essential Understanding” and “Math Background” sections at the beginning of each lesson. The lesson overview also includes TEKS, vocabulary, and ELPS instruction to use with a specific part of the lesson. In Topic 9, Lesson 2, on the nickel, the Essential Understanding section informs teachers that the value of some coins can be represented as a combination of other coins; specific coins each have a unique value, and the size of the coin does not indicate its value. 

The instructional materials include “Interactive Math Stories” as well as “Math and Science Projects” within each topic, which allow students to demonstrate mastery of math concepts set in real-world contexts. The Math and Science Project for Topic 5 asks students to discuss with other students living organisms versus non-living organisms. Students then draw a group of 10 living organisms, coloring them one color, and a group of 20 non-living organisms, coloring this group a second color. The students write the numbers that tell how many are in each group. In Topic 14, Lesson 1, “Solve and Share” allows students to practice word problems with realistic stories. For example, “There are two tools for measuring. What can you measure with the cup? What can you measure with the cube train? Draw an object you can measure with each tool.” Topic 15, Lesson 3, has a “Math Science Connection” activity that discusses how scientists use real graphs to sort data. In Topic 16, Lesson 1, students develop an understanding of how coins and money relate to the real world. Students understand that people earn money by working, that money is used to pay for something, and that people can earn money by working in many different ways. 

The materials provide teachers with possible student responses and misconceptions to practice questions and tasks. The “Solve and Shares” provide examples of student responses; some are found in the “Guiding Questions” and “Prevent Misconceptions” sections. In Topic 2, Lesson 1, the teacher asks, “How does showing each group with connecting cubes help you?” A sample answer is, “I can line up the cubes. The group that has a cube without a match has more.” “Error Intervention” strategies are provided in some lessons; for example, in Topic 11, Lesson 1, the strategy states that if students do not understand why they are writing numbers, the teacher can remind them that the chart shows the numbers in order from 1 to 100. Students write the number that comes just before or just after another number. 

Every lesson follows the same three-step structure. The first step is called “Problem-Based Learning,” which engages students in the content with the authentic Solve and Share problem. The Teacher’s Edition (TE) includes student work samples and questions to help students think deeply about the problem and analyze each other’s work. The second step is the “Visual Learning Bridge” (VLB), which supports the development of conceptual understanding using interactive features of Problem-Based Learning tasks and the step-by-step “Visual Learning” activity. Error analysis is included in many lessons. There are print and digital resources for both the students and teachers to support this step in the lesson. The materials develop problem-based learning and provide the appropriate level of rigor (conceptual understanding, procedural fluency, or application) as identified in the TEKS.

The materials provide questions for teachers to use to support discourse and a set of discussion questions that can be used to facilitate discourse without limiting student responses. In Topic 3, Lesson 2, the provided questions are, “What are you asked to find?” and “How can you use the mat to help you solve this problem?” Each lesson also offers a “Share and Discuss Solutions” section that encourages students to talk about the strategies they used to solve the given Solve and Share problem. The Topic 11, Lesson 1 Solve and Share handout gives teacher guidance for building and understanding. The sample questions and answers are “What Are you asked to do?” (Find the number that comes just after 30) “What tool do you have to solve the problem?” (Hundreds chart).

Evaluation for 2.5
Materials include cohesive, year-long plan for students to develop fluency in an integrated way.

The materials include a year-long plan for building fluency. There are scaffolds and extension opportunities to aid and advance fluency content. The materials provide multiple areas of teacher guidance for teaching fluency and integrate fluency at appropriate times, with purpose, as students progress in conceptual understanding.

Evidence includes but is not limited to: 

The materials provide strategic discussion opportunities around conceptual understanding. For example, in Topic 2, Lesson 1, the students determine if there are more tables or chairs represented by different colors of stacking cubes. Teachers ask, “How can you show how many are in each group?” and “How does showing each group with connecting cubes help you?” The students have counters to help them; some students come up and explain their strategy to solve the problem. This daily repetition of discussion in ever-increasing rigor helps students to build fluency in the associated number facts. The instructional materials include guidance and support for teachers for conducting fluency practice and point out how this practice helps students with concept development. For example, in Topic 3, Lesson 8, the “Math Background” section states: “Knowing equivalent ways to represent numbers is an important skill in developing computational fluency and number sense.” The materials explain the conceptual underpinnings of the program that provide conceptual cohesion across lessons, topics, and grades, as well as across TEKS and reporting categories. They connect “Essential Understandings” that occur within and across lessons and explain the connection between the fluency within the lesson and the focus of the lesson. For example, students count and write numbers from 1 to 100 in Topic 11, Lesson 1. Using a hundred chart, the teacher asks, “If you wanted to just count up numbers one at a time, would you go across like the numbers in blue or down like the numbers in yellow?” Additional questions to support student understanding of the concept behind the strategic fluency practice include “How can you use numbers 61 and 63 to find out what number is missing? What number belongs between 61 and 63? What number comes just after 84?”

The materials include teacher guidance and support for conducting fluency as appropriate for the concept development and grade. They include guidance for teachers on the structure and design of the fluency practice within the program, including the connections between concept development and fluency. The materials provide the teacher with information to give students regarding the definition of fluency as well as why it is important. Topic 3, Lesson 8 provides information in the “Math Background” section of the lesson overview for the teacher: “Research says that knowing equivalent ways to represent numbers is an important skill in developing computational fluency and number sense.” The lesson then guides students in multiple ways to represent 10, including using objects, counters, and a ten-frame. The teacher asks, “What are the ways you have already made 10? If you show four yellow bugs for one part, how many red bugs will be in the other part?” Students discuss their strategies in daily “Share and Discuss” activities.

The materials integrate fluency at appropriate times and with purpose as students progress in conceptual understanding. Two ways the materials do this is through providing multiple strategies for students to use to complete their grade-level tasks and providing students with multiple opportunities to practice and master solving grade-level tasks by applying their understanding of number relationships and strategies. For instance, students work on numbers 1 to 5 in Topic 1 and numbers 6 to 10 in Topic 3; in Topic 5, they begin learning numbers to 20. The materials include an “Interactive Math Story” at the beginning of each topic that allows students to activate their prior knowledge. In Topic 5, the “Activate Prior Knowledge” section states: “In this story, we will count some of Matt’s favorite things, and you will meet his friend Gina and also count her favorite things. Write the numbers from 10 to 20 on the board. Count them aloud with the students.” In another example, in Topic 7, Lessons 1 through 3 build on knowledge about different joining strategies. Lesson 1 introduces how to join together two groups to determine how many in all. Lesson 2 has students write the number in one group and the other group and tell what the total is. Finally, in Lesson 3, students use connecting cubes to demonstrate the joining and figure out a missing addend in a joining story. 

The materials provide guidance that recognizes the need for differentiated supports to include extension and scaffolded fluency activities. The materials in Topic 3, Lesson 1 include an extension for early finishers. Students look at the counters on the five-frames and count on to 6 by starting at 5. They point to the next counter to say 6. They do the same with the five-frame and count on to 7. Topic 5, Lesson 5 uses a worksheet to determine if students need differentiated support drawing pictures by connecting dots. If the students are having difficulty grasping this concept, intervention activities allow them to practice by finger tracing dots that are already connected and correcting dot-to-dot pictures that are done incorrectly. They then have another task to practice with more connect-the-dot pictures. Students ready for extension activities have several center games they can play; number tiles are randomly placed in squares on a game board, and students finger trace how they would connect the numbers 0 to 9. Additionally, online math games help students practice the same skills to higher numbers. Topic 12, Lesson 4 includes extensions for students mastering how triangles and circles look different. Students complete an activity and answer the questions “Does the size of a shape change what type of shape it is? If no, explain; why?” The materials also provide a math diagnosis and intervention system for teachers to use with students.

Evaluation for 2.6
Materials support students in the development and use of mathematical language.

The materials support students in developing and strengthening their mathematical vocabulary throughout the lessons and activities, as well as provide information for teachers to scaffold and support students’ use of this vocabulary in context.

Evidence includes but is not limited to: 

The materials build math vocabulary using vocabulary cards, a glossary, and an online animated glossary in the Student Edition. Math vocabulary is also embedded within the lessons to give students opportunities to develop and strengthen their use of mathematical vocabulary. The materials provide mathematical vocabulary cards and “Interactive Math Stories” at the beginning of each topic. Many of the daily lessons also provide problem-solving reading mats. Teacher-guided lessons give the students opportunities to hear the words used in the proper context. Daily “Share and Discuss Solutions” sections encourage students to use the vocabulary they learn. In Topic 4, the students have vocabulary cards for greater and less. The students circle the number on the front that matches the word on the back — either the greater number or the lesser number. The students then listen and respond in a discussion about an Interactive Math Story on “greater than” and “less than.” Afterward, students listen to a problem and share and discuss how they might solve it.

Within and across lessons, materials provide embedded scaffolding suggestions to support students’ development and use of academic vocabulary in context. The materials include a glossary and an animated glossary with motion and sound. Vocabulary terms are embedded in many sources, including the reteach lessons, online games, and center games. The Teacher Edition has a “Language of Math” section, which explains the vocabulary cards, glossary, and online animated glossary. The lessons are all formatted in a similar way; they include vocabulary cards for each topic and daily opportunities to discuss solutions to assigned problems with partners and with the class. Materials provide teachers with recommended questions to ask to draw out student responses that would use the appropriate terminology. For example, Topic 12 starts with vocabulary cards with words such as same, different, and sort. Pairs receive one square and three triangle attribute blocks. The teacher asks the students to sort the blocks into groups that are the same and different. Students share their work by repeating the words the teacher uses so they can become familiar with the words. Initially, the shapes are just referred to as objects; the actual names of the shapes are scaffolded in over several lessons.

Evaluation for 2.7
Materials contain opportunities for students to apply mathematical knowledge and skills to solve problems in new and varied contexts, including problems arising in everyday life, society, and the workplace.

The materials include opportunities for students to work on real-world problems using previously learned knowledge and skills. The materials also provide students opportunities to analyze data through real-world contexts. 

Evidence includes but is not limited to: 

The instructional materials contain grade-level appropriate opportunities for students to solve real-world problems in a variety of contexts. As students practice the problem-solving process in activities such as “Solve and Share,” “Math and Science Projects,” independent practice, and reteach lessons, they are given real-world information or situations in which to apply their math knowledge and skills. In Topic 1, students use counters as tools to help count real-world objects shown in the pictures. Topic 1’s “Interactive Math Story” tells about a mama bird and her eggs. Students count the eggs in the nest in each picture. Additionally, students must refer to previous knowledge to develop strategies to solve problems. There are many opportunities for students to solve real-world problems in a wide variety of situations, as each topic begins with “Math Science Activities” and Interactive Math Stories. For example, Topic 3 starts with activating prior knowledge by practicing numbers to five. Next, an Interactive Math Story introduces students to numbers six to ten with shells, crabs, and other things found on a beach. The extension activity for the story encourages students to draw pictures of something they might find at a beach based on a number the teacher provides. The Math and Science Activity at the beginning of Topic 3 talks about how cows come in different colors. Students color different colored cows, count the number of cows by color, and write that number on their papers. In Topic 3, Lesson 3, students count the number of beach balls in all. In Topic 10, students use pictures of real-world objects, such as polar bears or counting cubes, to help solve addition or subtraction problems. Topic 10, Lesson 2 has a Solve and Share section that shows students green ducks and yellow ducks at a pond. They solve to find how many ducks in all and also write the corresponding subtraction sentence. The program contains opportunities for students to solve real-world problems from a variety of contexts. The materials provide multiple opportunities for students to make sense of open-ended, real-world contexts involving mathematics. For example, in Topic 3, Lesson 9, students count the number of fish they see. In Topic 16, Lesson 1, students learn that people work to make or earn money. In Topic 8, Lesson 7, students learn to subtract using doggie biscuits. The program requires students to integrate knowledge and skills to make sense of a context and develop an efficient and successful solution strategy. The materials provide opportunities for students to apply their knowledge and skills from previous units in “Problem-Solving Tasks.” For example, in Topic 4, Lesson 6, the Problem-Solving Task asks the students, “How many frogs do you see?” to assess or analyze the problem. Students create a successful and efficient solution strategy or plan, integrating knowledge and skills; solve and justify their answer; and check for reasonableness.

The materials provide opportunities for students to analyze data from another content area (e.g., science). In Topic 5, Lesson 6, students begin to use a ten-frame format to help count and subitize numbers. The ten-frame is a helpful graph format used throughout the curriculum for the next several years. Each topic begins with an Interactive Math Story and Math and Science Activities that are of interest to the students. Topic 6’s Interactive Math Story is about a child’s outing with the child’s dad. Students find the missing number of the bus the characters take, view a sign to see which number in line they are for retrieving their lunch, count on to find the seat numbers for the theatre, and count on to find their apartment number when they return home. In the Topic 6 Math and Science Project, students view a picture of different colored train cars and determine the location of objects in relation to one another (e.g., which colored train car comes before the white car and which colored train car comes after the white car). In another example, Topic 10’s Math and Science Project asks students to observe, describe, compare, and sort rocks by size, shape, color, and texture. In Topic 11, students use hundreds charts throughout the lessons to find patterns in numbers and to find the number that comes before or after a specific number. Topic 15 begins with an Interactive Math Story that discusses voting and shows students how to put data into a chart. For example, “How many students want to go to the zoo?” or “How many students want to go to the beach?” Students draw pictures of something they might need wherever the class consensus shows they are going. The topic continues with a Math Science Activity about favorite pets; students make a graph with three different pets and then survey ten classmates to fill in their graph. Graphing continues in Math Science Activities in Topic 15, Lessons 1 and 3. The Topic 16, Lesson 1 “Math Background” section states: “In previous lessons, students have learned the names and values of coins. In this topic, students are developing an understanding of how these coins and money relate to the real world. Students should have an understanding that people earn money by working. Money is used to pay for something. You can earn money by working in many different ways. Students will have varying degrees of understanding based on their personal or family experiences.”

Evaluation for 2.8
Materials are supported by research on how student develop mathematical understandings.

The materials include some research to aid teachers in understanding mathematical concepts and the validity of the publisher’s approach to the lesson. There is little evidence that cited research is current, academic, relevant to skill development in mathematics, and applicable to Texas-specific context and demographics. Materials do not provide a bibliography.

Evidence includes but is not limited to: 

In the program overview, the materials do describe the design of the program as “Problem-Based Learning,” where students must think critically about a real-world math problem, evaluate options, collaborate, and present solutions, followed by “Visual Learning” to solidify the underlying math concepts. The Topic 5, Lesson 6 “Math Background” reads, “Research says students who are mentally and actively engaged in adjusting and constructing sets to compare quantities exhibit developing understanding of number relations” (Kami, 1985).

The “Lesson Overview,” Math Background, and “Essential Understanding” sections within the “Teacher Edition” provide general guidance for instruction to help enrich an educator’s understanding of the mathematical concepts and the validity of the publisher’s recommended approach, but they do not cite the supporting research. Twelve of the lessons provide specific, research-based guidance for instruction that enriches educator understanding of the mathematical concepts and validity of the publisher’s recommended approach. Topic 1, Lesson 1 cites Kato et al. (2001) in a discussion about young children and their relationships with numbers. Kato states that engaging students in thinking about quantities helps build these mental relationships. Topic 1, Lesson 4 cites Van de Walle (2004), who explains that students need to develop two different skills in order to count: First, they need to memorize the correct number words in order; then, they need to relate those words to the objects given, with each object only getting one word. Topic 3, Lesson 2 cites Fisher (1990), explaining that kindergarteners who learn part-part-whole representations showed improved understanding of basic number concepts, addition, subtraction, problem-solving, and place value. Topic 3, Lesson 4 cites Van de Walle (2004), explaining that students come to school with different understandings about numbers and that we need to build on those understandings to develop their numerical relationships. Topic 3, Lesson 8 cites R. Charles (2005), explaining how decomposing and composing numbers ready children for addition and subtraction. Topic 5, Lesson 3 cites Miller et al. (1995), explaining how the lack of pattern in the numbers 11 through 20 makes it difficult for children to memorize these numbers. Topic 5, Lesson 6 cites Kamii (1985), explaining how students who adjust and construct sets to compare quantities have a better understanding of number relationships. Topic 5, Lesson 9 again cites Van de Walle (2004), explaining how students learn that one and two more than or less than is important for all numbers. Topic 9, Lesson 1 cites Moniighan-Nourot, Scales, and Van Hoorn (1987), explaining how using coins in everyday contexts helps children understand them better. Topic 11, Lesson 3 cites Fuson and Briars (1990), who explain that relative sizes of concrete representations help students think about bigger numbers. Topic 15, Lesson 1 cites Friel, Curio, and Bright (2001), explaining that graph comprehension is based on knowledge of the components of the graph, ability to make comparisons and do computations for the information on the graph, and facility in relating information back to its context. Topic 17, Lesson 9 cites Van Hiele (1986), explaining how, at this age, children are starting to understand that a square is not just a square because it looks like it, but that it has four equal sides and four square corners. Research is mentioned three other times with no author given.

The cited research is academic and relevant to skill development in mathematics, though not current. Twelve sources are quoted, with the most current being R. Charles (2005) and the oldest being Kamii (1985). It is not possible to determine if the research is academically vetted or demographically aligned with the Texas-specific context. The materials do not contain a bibliography to cite research. 

The author team and well-known mathematicians bring an impressive level of experience as classroom teachers, teacher educators, researchers, and authors. They have written numerous professional articles based on their research and observations, and their contributions to the program are an implementation of successful teaching methods. The program offers an instructional model that is based on a research foundation and has proven efficacy shown by statistically significant advantages in independent, scientific research done with randomized controlled trials. enVisionmath2.0 meets ESSA’s “Promising” evidence criteria.

Evaluation for 3.A.1
Materials develop student ability to use and apply a problem-solving model.

The materials provide guidance to help the students develop their ability to use and apply the problem-solving model throughout the lessons. Student texts pose insightful questions that help them evaluate their approach to the problems. The teacher text provides numerous questions for each problem to guide the students through solving and thinking about possible methods to solve each problem. 

Evidence includes but is not limited to: 

The materials support the development and practice of a consistent problem-solving model across the topics. This model, consisting of four parts (“Analyze, Plan, Solve and Justify, and Evaluate”), is used in the problem-solving sections of Topics 1–16 in the kindergarten curriculum. Topic 1, Lesson 10, begins the introduction to problem-based learning by showing students how to use counters to help plan, solve, and analyze the math problem. Students begin by analyzing, and the materials ask, “What do you see? How can you find out how many birds there are on the branch?” The students move on to the next part, the plan, and the materials ask, “How can you use these counters to help you count the number of birds?” Next, students solve and justify. Materials ask, “A counter was put on each bird. How many counters are there? How many birds are there?” Finally, they are introduced to the evaluation part to check for reasonableness. Materials ask, “Look at the number 3. How can you check this answer?” In Topic 8, Lesson 8, students continue to use counters matched to the number of items in a picture to work through a subtraction problem. At the end of some lessons, a problem-solving reading activity provides a page of math problems to solve using data on a mat.

The materials include prompts for students to apply the problem-solving model taught within the materials. The Teacher Edition provides prompts and guiding questions for teachers to give or ask students as they help guide the students through the lessons. Topic 4, Lesson 1 has several guiding questions: “What do you think Emily is doing? How many chicks are there? How do you know? Does a set with seven chicks have more than or fewer than a set of 10? How do you know?” A student worksheet on mixed problem-solving in Topic 16 Lesson 5 shows the problem-solving steps of analyze, plan, solve and justify, and evaluate written in red letters at the top of each step of an example problem. To guide students in using the problem-solving model, teachers ask provided prompts, such as “How many coins did Emily earn?” and “How do you know?” 

The materials provide prompts for students to reflect on their approach. In Topic 4, Lesson 6, the problem-solving guided practice leads students through the problem-solving model with assistance from the teacher. Teachers give students provided prompts to help formulate a plan, including “How many frogs do you see? How many would there be with 1 more frog? How could we find out?” Under the Plan section, the question prompt is, “How can we use Emily’s counters to find out how many frogs?” For the Solve and Justify section, the question provided is, “What does Emily do to show 1 more frog?” In the Evaluate section, the question “What number should Emily write to show how many frogs in all?” helps students connect the ideas formulated when solving the problem. In Topic 5, Lesson 10, the teacher builds understanding by asking questions such as “What are you asked to find? What do you already know about numbers to help you solve the problem?”

The materials provide teacher prompts and questions in each lesson to use when guiding students to reflect on problem-solving. For example, in Topic 6, Lesson 6, teachers ask, “What are you asked to find and complete? What do you already know about the problem?” The teachers can give hints as needed, including “Do the numbers get bigger or smaller each time? How can we find how much bigger the numbers are getting each time?” The students are given an opportunity to share and discuss solutions and finally summarize and generalize. In Topic 8, Lesson 6, teachers ask provided guiding questions to build understanding, including “What are you asked to find? What tools do you have to solve the problem?” There is a guidance section for the teacher to give hints as needed; for example, “How can you use counters to show the problem? What does the minus sign mean?” Topic 8, Lesson 8, provides another example of prompting. Teacher prompts encourage students to analyze the given problem; for example, “What kind of story is this, a joining or a take-away? Do you add or subtract when you want to find how many there are in all?” 

Evaluation for 3.A.2
Materials require students to select appropriate tools for the task, concept development and grade.

The materials provide opportunities for students to select and use real objects, manipulatives, representations, and algorithms as appropriate for the stage of concept development, grade, and task. The materials also provide opportunities for students to select and use technology (e.g., calculator, graphing program, virtual tools) as appropriate for the concept development and grade; they provide teacher guidance on tools that are appropriate and efficient for the task. There is not, however, any specific teacher guidance about the reasons for using tools or which tools would be the best for any specific problems. 

Evidence includes but is not limited to: 

The materials provide students with opportunities to learn to use grade-appropriate tools for solving. Throughout the lessons, materials provide pictorial models, and students also have opportunities to use counting cubes, counters, etc. Topic 1 uses counters and number cards; Topic 2 introduces connecting cubes; Topic 3 uses a ten-frame. In Topic 6, Lesson 1, students use a double ten-frame mat to represent the number 30. In another example, in Topic 14, Lesson 1, students use real-world objects to solve problems about height. 

The materials provide students with opportunities to learn to use grade-appropriate technology for solving tasks and understanding concepts. For example, at the end of Topic 6, Lesson 3, students complete a digital math tools activity to reinforce the lesson content; they use a suite of digital math tools, with virtual manipulatives that allow students to use representations. Some of the virtual online tools include counters, place value blocks, data and graphs, a number line, an input-output machine, number charts, a pan balance, and strip diagrams. The assessments all have a “listen” option for those students who need help reading. In Topic 9, Lesson 6, students play a game called “Amazing Savings 1,” where they move around a maze as a mouse collecting slices of cheese and then spend them to open gates to earn more cheese.

Students have opportunities to use a variety of tools throughout the lessons, particularly for the “Solve & Share” problems where digital tools are always available. Through this and also with the discussion and sharing of solutions, students are gaining experience in deciding which tools work best for different situations and seeing the different ways tools can be used. They will see, for example, how the same tool can be used differently to reach a correct solution or why they find a particular tool more suitable than another. Student sample work is provided in the Teacher’s Edition and can encourage further discussion as needed.

The materials use many tools and recommend specific tools for use in each lesson. However, they do not specifically explain which tools are appropriate and efficient for a task nor which tool might be more appropriate than any other tool. 

Evaluation for 3.A.3
Materials require students to select appropriate strategies for the work, concept development, and grade.

The materials include teacher prompts on student usage of appropriate techniques (mental math, estimation, number sense, generalization, or abstraction) to solve problems. There is evidence that the materials also support teachers in understanding the appropriate strategies that could be applied and how to guide students to more efficient strategies. They also provide opportunities for students to use multiple strategies to solve problems. 

Evidence includes but is not limited to: 

At the kindergarten (early reader) level, the materials have the teacher prompt the students with an appropriate technique for solving a task. In Topic 4, Lesson 6, in the “Build Understanding” section, teachers prompt, “What are you asked to find? What tools do you have to solve the problem?” The materials then support students to select techniques appropriate for the grade-level task; teachers ask, “What should you do with your counters? How many counters will you need?” In another example, in Topic 5, Lesson 9, the “Solve and Share” has a “Summarize and Generalize” section in the Teacher Edition. It reads, “There are several different ways to find 1 more or 1 less than a number. Counting forward or backward mentally is a good strategy and one that we will practice later in the lesson.” Topic 7, Lesson 4, provides another example of this. In the “Build Understanding” section, the teacher asks, “What do you already know about joining that will help you solve this problem?” The “Share and Discuss Solutions” section of this same lesson allows students to explain their technique to their classmates. The teacher then explains that a plus sign can be used to represent joining. The lesson continues by having students use the word and as well as the plus sign in joining problems. This model allows the students to select their own technique for solving the problem. In Topic 10, Lesson 6, the last three exercises in the independent practice direct the teacher to instruct the students to “use objects, drawings, or a number sentence to solve the problem.”

The materials support teachers in understanding the appropriate strategies students can apply to solve math problems and how to guide students to more efficient strategies. The “Focus on Content” and “Focus on Process” sections in the “Math Background” help to support teachers in understanding which strategies are appropriate for specific math problems. In Topic 4, materials explain two process standards students use during the topic: check for reasonableness and communicate. For check for reasonableness, materials explain, “Checking to see if an answer makes sense is part of the problem-solving process.” They also explain how using counters can help a student show the problems and check if their answer is reasonable by determining if their answer is more than the number of counters they started with. The communicate section explains, “A ten-frame makes it easier for students to justify that their answer is correct and can explain that one more frame is filled in a ten-frame showing a number that is one (or two) more.” Another example of this is in Topic 8, Lesson 1, where students use counters to represent bugs; they then take away the number of counters explained in a separating story. This model is the first step in learning subtraction. As the topic progresses, the students draw their own bugs, and then they use numbers and the subtraction sign in a number sentence to demonstrate the separating story. Materials explain these steps to the teacher to show how students can become more efficient in their strategies. The Math Background for this lesson explains to teachers that students must have practice developing and using language such as “How many are left?” before understanding operations of subtraction and the minus symbol. In Topic 8, Lesson 7, students explain the solution of the separating story by using manipulatives, words, drawing pictures, or writing subtraction number sentences. This strategy prepares them for the next part of the lesson, where they explain solutions to separating problems. In Topic 11, materials guide teachers to help students count to 100 by first using a hundreds chart and then looking for patterns, focusing on counting by tens. The materials have students build up to counting big numbers and adding ones. Finally, after learning to recognize patterns of numbers, students can correlate the pattern (e.g., 21, 22, 23) with the words (twenty-one, twenty-two, twenty-three).

The kindergarten materials provide opportunities to solve problems using multiple appropriate strategies. Some of the strategies students use as they progress through the units include making drawings (beginning in Topic 1), using counters (beginning in Topic 1), using ten-frames (beginning in Topic 3), and writing corresponding number sentences (beginning in Topic 7). In Topic 4, Lesson 1, students use a ten-frame. In Topic 5, Lesson 8, students explain how they know that the number they circled is greater than the other. Students may use a variety of strategies, such as comparing the numbers to ten, counting, visually seeing which set has more counters, or using another way. In Topic 5, Lesson 9, students learn how to use mental math to count on or count back using counters or double ten-frames to determine one more or one less. In Topic 11, Lesson 5, students can count on; count back; count by tens; or use counters, cubes, or hundreds charts to figure out missing numbers in a series that follow a pattern.

Evaluation for 3.A.4
Materials develop student’s self-efficacy and mathematical identity by providing opportunities to share strategies and approach to tasks.

The materials include many opportunities for students to see themselves as mathematical thinkers who can learn from solving problems, make sense of mathematics, and productively struggle in their efforts to come to acceptable solutions. Materials support students in understanding that there can be multiple ways to solve problems and complete tasks. Materials support and guide teachers in facilitating the sharing of students’ approaches to problem-solving.

Evidence includes but is not limited to: 

The materials support students to see themselves as mathematical thinkers who can learn from solving problems, make sense of mathematics, and productively struggle. The formatting of the lessons ensures that all students participate in solving problems and making sense of mathematics. Each lesson starts with a “Solve and Share” problem that students work on by themselves or sometimes with a partner. The teacher then chooses some of the students to explain their solution to the class. The Solve and Share for Topic 7, Lesson 5, asks students to determine that the plus sign is needed to show the joining of two groups. A guiding question helps foster the students’ thinking: “What tools do you have to help you solve the problem?” In Topic 7, Lesson 7, pairs of students receive three connecting cubes of one color and two of another. They learn that their teacher made three nametags in the morning and two in the afternoon. Teachers ask, “How does their teacher know that she has five nametags?” Students need to explain their answers and show their work. This problem allows students to choose a method that helps them to make sense of the problem best. By working on this with a partner, they can productively struggle through numerous ways of thinking to solve the problem ultimately. Also, in Topic 8, Lesson 7, students demonstrate their understanding of subtraction by explaining their solution in a variety of ways. In Topic 10, Lesson 4, students share their solutions and discuss them with the teacher. Teachers ask questions such as, “What kind of story is this, an addition or a subtraction story? Why? How do the cubes, and addition sentences, show the story?” These questions allow all students to participate. There are also tasks designed to support students in productive struggle as they make sense of the problem and solve it. The materials include images of individuals students can relate to, who are working on math problems and are successful in mathematics; this helps students understand that everyone can be a mathematical thinker. 

The materials support students in understanding that there can be multiple ways to solve problems and complete tasks. In Topic 1, Lesson 3, students work to communicate the number 2 in pictures, manipulatives, and symbols. Students learn multiple ways to solve problems. In Topic 10, Lesson 3, the teacher gives students nine connecting cubes, six of one color and three of another. They must create a subtraction story with the cubes and then write the number sentence. In Topic 10, Lesson 7, the teacher explains that addition and subtraction can both be solved with pictures, objects, or number sentences. In Topic 14, students learn to compare the length of objects by using materials available to them (crayons, pencils, cubes). 

The materials provide many problems that encourage divergent solution strategies. The Solve and Share problems in the Student Edition introduce a lesson by giving students problems that have embedded important math ideas. Students can solve the problem any way they choose. Topic 4, Lesson 6 guides students to formulate a plan and connect ideas; teachers ask, “How many frogs do you see? How many would there be with one more frog? How could we find out? Emily uses counters to count the frogs. How can we use Emily’s counters to find out how many frogs? What does Emily do to show one more frog? What number should Emily write to show how many frogs in all?” In Topic 6, Lesson 4, Exercise 3, teachers explain that, since both numbers fill up the ten-frames, students can just count those outside the frames to see which number has more. For Exercise 4, teachers remind students that the number with the most filled ten-frames is the largest. For Exercise 7, teachers remind students to say the numbers clearly. Finally, for Exercise 9, the teacher reminds students that they need to compare the tens before the ones. These prompts help the teacher to monitor students while they develop solution strategies and also sequence the discussion of student strategies. Another example of this is in Topic 13, Lesson 5, where students sort a variety of two- and three-dimensional shapes and justify their strategies. The shapes could be sorted by color, dimension, or 2D to 3D equivalent. 

Evaluation for 3.B.1
Materials prompt students to effectively communicate mathematical ideas, reasoning, and their implications using multiple representations.

Materials provide students the opportunity to communicate mathematical ideas and reasoning while solving problems using multiple representations, including writing and the use of mathematical vocabulary. They also prompt teachers to encourage students to use correct mathematical terminology in oral and written communication and use a variety of representations in solving and explaining problems. 

Evidence includes but is not limited to: 

The materials provide students with opportunities to communicate mathematical ideas and to solve problems using multiple representations as appropriate for the tasks. The “Solve and Share” section found at the beginning of the lessons provides many opportunities to communicate by giving students problems in which important math ideas are embedded. In Topic 2, Lesson 3, students compare cubes to find out which group has fewer; they use concrete manipulatives such as connecting cubes. In Topic 3, Lesson 5, the students use two colors of counters to show the number 8 in different combinations of each color. They then explain their solutions to the class. In Topic 5, Lesson 1, the materials direct teachers to begin by discussing students’ solutions and to discuss how counting 12 counters show there are 12 toys. The materials also prompt teachers to ask questions, giving the students the opportunity to communicate, such as “How can you tell how many trucks there are? Why do you think this number is in this box?” In Topic 7, Lesson 1, students show how they determine the number of kids sitting on the bench using counters. In the center games for Topic 7, Lesson 5, students draw number tiles out of a paper bag to make an addition sentence. For each number drawn, students count out that specific number of red squares to put on the plate on the worksheet. Students count the total number of squares to find the sum and also air write the math sentence. In Topic 8, Lesson 7, students use counters, drawing, or number sentences to solve a missing-part problem. They then explain their work to the class. The use of representations to show thinking and communicate with others is almost always used in the Solve and Share problems. The Topic 1, Lesson 4 Solve and Share problem has students use counters to show the number 5, and then a crayon to circle the colored box that shows 5. They then explain their solution to the class. In the Topic 16 “Art” center, students draw two pictures: one of how they can receive money as a gift and one of how they can earn money.

Materials provide ways for students to communicate mathematical ideas and reasoning in multiple representations, including writing and using mathematical vocabulary. Topics 1–16 each include vocabulary cards; the vocabulary words on these cards are used and referred to often within the topic and in subsequent topics. There are prompts for teachers to help students use the correct vocabulary throughout the materials. For example, in Topic 4, Lesson 2, the teacher shows students how to know the difference between greater than and less than. After students watch the “Visual Learning” that introduces greater than and less than, the class has a guided lesson. During this lesson, the teacher asks, “Which number is greater, 3 or 7? Which number is less, 3 or 7?” In the problem-solving reading activity guide for Topic 5, students count the number of birds in each exercise and write the correct number in the corresponding blank. Afterward, the teacher asks provided prompts, such as “How many wings do you see? Can you count the feathers on any of the birds?” In Topic 8, Lesson 6, students use counters to model a subtraction sentence and then write that sentence. Teacher prompts to support students’ writing include “How can you use counters to show this problem? What does the minus sign mean?” In Topic 12, Lesson 7, the materials provide teachers with prompts and possible answers, such as “When making shapes, what are the numbers of sides needed to make a triangle, square, and rectangle?” (A triangle has three sides, and a square and rectangle each have four sides.) 

Evaluation for 3.B.2
Materials provide opportunities to discuss mathematical ideas to develop and strengthen content knowledge and skills.

The materials provide opportunities for students to engage in mathematical discourse in partners, in small groups, and whole class. The materials integrate discussion throughout to support students’ development of content knowledge and skills. Additionally, materials guide teachers in structuring and facilitating discussions as appropriate for the concept and grade level.

Evidence includes but is not limited to: 

The materials intentionally provide opportunities for students to engage in mathematical discussions in a variety of different groupings. The “Solve and Share” problems provide opportunities for students to share their problem-solving in a whole group setting, while the center games provide opportunities for students to share mathematical discussions as they play games with one another and are prompted to explain how they solved the math questions. For example, in Topic 6, Lesson 1, materials prompt, “Jada has 23 granola bars. How can you use counters to show how many granola bars Jada has?” The students begin by building understanding through questions prompted by teachers: “What are you asked to find? What do you already know?” Finally, students are asked to share and discuss their solutions with one another and the entire class. In Topic 7, Lesson 3, the Solve and Share problem at the beginning of the lesson has students working with two colors of connecting cubes to represent boats on the water and then writing the numbers in an “and-is” statement to show joining. Some students share their solutions with the class. The whole class then does guided practice together, using pictures of connecting cubes and boats to show joining sentences. After the teacher assesses their independent practice, students either work with the teacher in a small group for intervention practice, drawing pictures to do joining sentences, or in pairs, working on activity centers. In Topic 16, Lesson 3, the teacher asks students to describe what is happening in the pictures without using their assigned word. In groups, students decide which pictures, if any, depict income or a gift and explain why. 

The materials include opportunities for discussion at the beginning, middle, and end of every lesson. The lessons begin with a Solve and Share section that introduces the concept in the lesson and allows for discussion as students try their different strategies to solve the problem. For each Solve and Share problem, there are “Build Understanding” and “Give Hints as Needed” sections. During the lessons, students may participate in problem-solving reading activities, which allow them to share their learning further as they solve more math problems related to the lesson. The “Center Games” and “Concept Review” sections wrap up the lessons and provide additional opportunities for students to share their mathematical ideas in discussion. In Topic 7, Lesson 6, students begin with the Solve and Share problem: “Daniel sees 2 rabbits under a bush. He sees 5 other rabbits eating grass. How many rabbits are there in all? What number sentence can you write to solve the problem?” Students solve the problem and share solutions under the “Share and Discuss” section. In the middle of the lesson, students answer questions and discuss: “What do you see in the box? What joining story can you make up about the rabbits?” Then, they complete guided questions. At the end of the lesson, students work on an intervention activity where they also discuss questions posed to them. In another example, Topic 7, Lesson 7 starts with pairs of students using three cubes of one color and two of another to represent nametags their teacher made before and after lunch. They work together to explain how she knows she has five nametags. Some of the pairs then discuss their solutions with the class. In the middle of this lesson on explaining addition, during guided practice, the whole class discusses the topic. Then, after independent practice, students either discuss the topic further with a small group directed by the teacher or while doing a reading mat with a small group. 

The materials offer guidance for teachers on how to structure a discussion that is appropriate for the grade level. In the “Program Overview,” there are tips for facilitating problem-based learning: “Set expectations to make sure students know you expect them to do the thinking. Foster communication by having students share their thinking with a partner, small group, or the whole class. Be encouraging to show that you value student’s thinking even when they struggle. Use the language of the process standards during discussions.” The Teacher Edition provides guiding questions with the Solve and Shares; it also provides guiding questions within the problem-solving reading activity guides, which include questions to ask the students to discuss with their peers. Questions can be found within the center games, which teachers can use for guided small groups. An example of this is found in Topic 8, Lesson 8. Students use tools to add or subtract. The teacher prompts the students by asking, “What are you asked to find? What tools do you have to solve the problem? What kind of story is this, a joining or a take away? How can you use counters to help solve this problem?”

Evaluation for 3.B.3
Materials require students to justify mathematical ideas using multiple representations and precise mathematical language.

Materials provide opportunities for students to construct and present arguments that justify mathematical ideas using multiple representations. They also assist teachers in facilitating students to construct arguments using grade-level-appropriate mathematical ideas. 

Evidence includes but is not limited to: 

The materials provide students with opportunities to construct arguments to justify their mathematical ideas using multiple representations. The “Solve and Share” for Topic 5, Lesson 6 asks students to look at pictures of the two trays and circle the tray that has more pieces of fruit. Students also draw an X on the tray with fewer pieces of fruit. Once this is complete, students show how they know their answer is correct. Students can use any strategy to count the pieces of fruit and also to prove their answer. In Topic 8, Lesson 2, the teacher tells students a number story about separating and asks, “How does acting it out help you solve this subtraction problem?” In Topic 8, Lesson 7, students use counters, drawings, and number sentences to solve and justify a subtraction problem to the class. In the Topic 13 Lesson 5 Solve and Share, students analyze given information and formulate a plan to determine and justify a solution to the problem. In Topic 14, Lesson 3, the materials provide tasks for students to compare length and justify their answers using cubes.

The material assists teachers in facilitating students to construct arguments. For example, in Topic 5, Lesson 7, there are prompts for teachers to facilitate students’ arguments. Teachers ask, “What does the double ten-frame show? Does the double ten-frame show more than, less than, or the same as the number on the card? How do you know?” The Topic 5 Lesson 9 Solve and Share has students using counters, ten-frames, mental math, count on, and count back to find a number that is one more or one less. Provided teacher prompts help students to justify their answers. The teacher asks, “What do you know? What tools could you use to find the answer? How does counting help you solve this problem?” The Teacher Edition for Topic 6, Lesson 6 provides the following guided questions to assist students when constructing arguments: “How can we find which set has more? What does the double ten-frame show? Is the number of apples greater than, less than, or the same as the number of bananas? How do you know?” In Topic 14, Lesson 3, the materials provide questions to help elicit different types of responses from students as they present their arguments: “Is the cube train longer than the pencil? How do you know? How many cubes long is the cube train? Martha checks to be sure her cube train is longer than the pencil and writes down the number of cubes she used. What number does she write?”

Evaluation for 4.1
Materials include developmentally appropriate diagnostic tools (e.g. formative and summative progress monitoring) and guidance for teachers and administrators to monitor progress.

The instructional materials include a variety of diagnostic tools that are developmentally appropriate (e.g., observational, anecdotal, formal). They include some guidance in each lesson for how to grade and use the individual practice assignment to place students in appropriate groups (intervention, on level, or advanced) for the remainder of the lesson. They include diagnostic tools to measure all content and process skills for the grade level, as outlined in the TEKS and Mathematical Process Standards. No evidence shows the materials include tools for students to track their own progress and growth.

Evidence includes but is not limited to: 

The materials provide a variety of diagnostic tools that are developmentally appropriate. These tools are formal and informal and grade-level appropriate. Materials provide two grade-level diagnostic test forms: Form A and Form B. There is also an online placement test at the beginning of the curriculum. This test has an option to have questions read to the students for struggling readers. The test provides teachers with a list of what students need more work on and what they have mastered. Materials also include digital assessments that are taken online and auto-scored. Assessments can be customized online and allow the teacher to upload a district assessment. Online quick checks may also be used; they include five questions for students to answer based on specific skills. Data from these quick checks are provided in an online teacher report, allowing teachers to assess student mastery of the skill. Ongoing assessments are given during a lesson throughout various sections, such as questions to use with “Visual Learning Bridge,” “Visual Learning Animation Plus,” “Do You Understand,” and guided practice. Independent practice worksheets in each lesson can also be used for diagnostic purposes. The Teacher Edition explains how to grade the sheets and sort students into groups that need intervention, are on level, or are advanced. For example, in Topic 11, Lesson 1, materials instruct the teacher to use the independent practice as a quick check. Students who score 0–3 points go to interventions; 4 points mean students are on level; and 5 points are equivalent to advanced. Follow-on activities are provided to challenge each group on the topic of the lesson. The materials also come with a “Math Diagnosis and Intervention System” (MDIS), which provides additional lessons and face-to-face checks to ensure students have grasped the concept. Further teacher support for the formal and informal assessments in the program is provided at the publisher’s website, which explains how to use assessment data to inform instruction. Support includes tutorials and downloadable resources for placement tests, topic assessments, performance tasks, lesson assessments, practice, cumulative assessments, state test preparation, and MDIS.

The materials provide some guidance for the administration of the diagnostic tools. The MDIS includes a “Teacher’s Guide for K–3,” which provides an individual record form and a class record form. The overview of this guide briefly details four areas. For assessment, it explains that an “Entry Level Assessment Form A” is given for a student entering a grade; “Form B” is used as a diagnostic test to check performance after providing instruction or intervention. For diagnosis, teachers use the “Class Record Form”; the MDIS gives a brief explanation of how to use the form to make placement decisions. “Intervention” lessons can be used for the content taught during the year. For monitoring, there is an “Individual Record Form” to help record student progress. Further in the MDIS, there are in-depth details explaining these four areas and instructions for how to use the system. The materials also include tips or recommendations to support consistent and accurate administration of the diagnostic tools. For example, in Topic 2, the materials provide explicit instructions to administer the topic test. The directions include “Have students mark the best answer. Which shows more cups than plates? Which shows two more colored dots than the group at the top? Which shows one fewer dog than bones? Which shows the same number of baseballs as footballs?” The independent practice section is used as a diagnostic check to determine mastery of the concept taught in that lesson. In Topic 8, Lesson 5, the teacher’s manual explains how to grade the quick check and how the number of correctly answered questions correlates with the category the student is placed in for the remainder of the lesson (intervention 0–3, on level 4, or advanced 5). 

The materials do not include any tools to assist students in tracking their own progress and growth. The materials include diagnostic tools to measure all content and process skills for the grade level, as outlined in the grade-level TEKS. The placement test at the start of the year identifies what TEKS students already know. The instructional materials provide a “Texas Assessment Resources for Teacher’s Guide.” This guide contains performance tasks pages for students to complete and includes 4-point scoring rubrics that outlines the four levels of achievement for students’ understanding of the concepts and skills in that topic, as well as answer keys. The questions in the performance tasks are matched to specific TEKS and include several open-ended questions for each topic. Class and individual record forms are available in the teacher’s guide of the MDIS to track student mastery of skills. Materials also provide TEKS correlations from prior grade levels to assist teachers in developing plans for additional support. 

Evaluation for 4.2
Materials include guidance for teachers to analyze and respond to data from diagnostic tools.

The materials support teachers with guidance and direction to respond to individual students’ needs in all areas of mathematics based on developmentally-appropriate student data. Additionally, diagnostic tools provide assistance in planning instruction and differentiation through a variety of activities to respond to student data. The materials provide a variety of resources and teacher guidance on how to leverage different activities to respond to student data. Materials provide guidance for administrators to support teachers in analyzing student data. 

Evidence includes but is not limited to: 

Materials support teachers with guidance and direction to respond to individual students’ needs in all areas of mathematics, based on measures of student progress appropriate to the developmental level. The materials include recommendations for responding to student needs within mathematics based on data from developmentally appropriate assessments. For example, at the end of each lesson, there is an “Assess and Differentiate” section. The teachers use the quick check on the previous page to prescribe differentiated instruction. The material includes learning support for students who need an intervention, students who are on level, and students who are advanced in that content area. For example, based on the students’ data from the quick check, students either do the reteach activity to build understanding or the on-level and advanced activity centers. For example, in Topic 7, Lesson 3, students who score three or fewer points on the quick check receive additional intervention. They work in a small group to create joining rhymes about sea creatures, draw pictures and determine how many they have in all, and then work independently to do more joining sentences based on pictures on the worksheet. Additional worksheets for remediation are available if the student needs additional practice after intervention time. Students who are on-level or advanced move on to center games at their level. The “Texas Assessment Resources” guide explains that comprehensive test reports are available to use with each of the end-of-course tests provided within the Texas Assessment Resource guide. In the report, each item tested is referenced by specific TEKS. The items tested are also referenced to the “Math Diagnosis and Intervention System 2.0” (MDIS). When teachers see an item on the end-of-course assessment a student has not mastered, the teacher can identify lessons that give students additional reviews and practice through the corresponding MDIS listed on the report. For example, if a student missed question number 19 on the practice test, which asks the student to compose and decompose numbers up to 10 with objects and pictures, the “Practice Test Report” shows that this question tested TEKS K.2I and that the teacher can use MDIS lessons B9 and B10 to have the student review and practice this skill. Lesson B9 includes exercises on making 6 and 7. Lesson B10 includes exercises on making 8 and 9. In the end-of-course tests, there is a comprehensive test report that provides information for each student once an assessment has been taken. Each item in this assessment is referenced to a TEKS standard and to the MDIS for review. It allows the teacher to identify whether a student has reached proficiency on each specific concept assessed within each test. It also provides lesson references to provide additional reviews and practice for students who have not yet reached proficiency.

The diagnostic tools provide meaningful data to teachers to help them plan instruction and to differentiate within their instruction. For example, the materials contain the “Item Analysis for Diagnosis and Intervention” charts in the Teacher Edition and auto-generated reports for online assessments. The Texas Assessment Resources under the “Teacher Resources” tab explain how to read the datasheet printed out for the online practice end-of-course exam. This datasheet identifies for teachers the areas that need additional instruction prior to state testing. The Texas Assessment Resources guide and the MDIS list the corresponding TEKS. This list helps the teacher understand which lessons/activities to use for a student who needs additional practice to master a specific skill. For example, in the Practice Test Report, in the Texas Assessment Resource guide, a teacher is informed to use the MDIS lessons A12 for a student who did not answer test questions 1 and 20 correctly; these questions cover TEKS K.2A. Materials provide online teacher reports based on the online activities students complete; they allow teachers to see individual and class views of progress. There are TEKS reports to show mastery of individual TEKS. Assignment reports show the status of resources that have been assigned online. Assessment reports show performance on items in the online assessments. Teachers can use these reports to identify areas students need to review and practice. The lessons and activities within the instructional materials list the corresponding TEKS, which help guide a teacher to know which activities to use. If a student needs more practice on specific TEKS, the teacher can also find additional activities related to the TEKS within the center activities. If a student has mastered a skill, a teacher can look in the “Scope and Sequence” within the “Content Guide” to know the related higher-level skill the students work on next so that the student can begin working toward those skills. A “Leveled Assignment Guide,” provided in the Teacher Edition, provides intervention, on-level, and advanced assignments. There is also a quick check at the end of each lesson. The materials include guidance to support teachers in understanding the results of diagnostic tools and provide teachers support for planning instruction and differentiation based on data gathered from the diagnostic tools. The MDIS references lessons with additional reviews and practice to provide to students, especially for those students who have not yet reached proficiency.

The materials provide a variety of resources and teacher guidance on how to leverage different activities to respond to student data. The material includes an item analysis for diagnosis and intervention for each test. Each item or problem is correlated with a grade-level TEKS, and most are given an intervention system with additional support for students who need it. Online quick checks provide information to the teacher to choose intervention, on-level, or advanced activities for students based on the student data. For example, there is an online quick check for comparing capacities; it includes five questions. At the end of the quick check, it tells the students any areas they need more practice with and also provides the data to the teacher so that the teacher may assign activities based on the data. In Topic 11, Lesson 1, the teacher uses hundreds charts and riddles to review the concept with those who are still struggling. The teacher might say, “I’m a number that comes after 82 but before 84. Who am I?” There are a variety of resources a teacher can choose from when assigning activities, including “Visual Learning Animation Plus,” online math games, digital math tools activities, “Reteach to Build Understanding” activities, center games, and intervention lessons in the MDIS. The materials provide intervention that can be auto-assigned in the online tool. Customizable digital intervention is available through the use of online resources, eText pages, printable PDFs, animations, online tools, and math games. 

The materials offer guidance for administrators to analyze and respond to data from diagnostic tools, including the following, available on mysavvastraining.com:

• Progress Monitoring and Reporting
• Realize Reports Administrator Guide
• Realize Reports: Getting Started for Administrators
• Realize Administrator Assessments Recorded Webinar
• Realize Reports FAQ
• Realize: Data and Security FAQs

Evaluation for 4.3
Materials include frequent, integrated formative assessment opportunities.

The instructional materials include ways to accurately measure student progress in a routine and systematic manner. The frequency of progress monitoring is appropriate for the age and content skill.

Evidence includes but is not limited to: 

The instructional materials provide routine and structural progress monitoring opportunities to track and measure student growth. The materials include suggested timelines for checking students’ progress. For example, the materials include an online placement test at the start of the year. At the start of the topic, there is a review of what students know. During the lesson, students are assessed by a “Do You Understand?” section and through guided practice. “Texas Assessment Resources” provide summative testing at the end of each topic, a benchmark test for every four topics, and two end-of-course practice exams to ensure students are prepared for state testing. These tests are all aligned with grade-level TEKS and specified program goals. The tests include a grade-appropriate question format. Materials also include digital assessments that are taken online and auto-scored. Teachers can track the data on the online assessments or activities they assign, and the program shows the data on the standards students have mastered and haven’t mastered. The data can be sorted by class results by assignment or by class mastery by standard. TEKS reports show which TEKS have been mastered. Links to resources for practice in any needed areas are also provided with this data. For example, when a student completes a quick check, they are assigned an activity based on their results. The activity may be an intervention assignment, an on-level assignment, or an advanced-level assignment. The instructional materials also allow assessments to be customized. Teachers can customize assessments or upload district-created or teacher-created assessments for data tracking. The online data tracking of the instructional materials allows teachers to monitor student progress closely and to know when extra support is needed. The usage data lets teachers know how much time students are spending in the online course.

The materials include appropriate and frequent assessments that reflect student learning at their current age. Materials guide teachers to administer progress monitoring assessments regularly, allowing students to demonstrate their learning as appropriate for the age and content skill. Formal assessments are available at the end of each topic. The materials suggest assessments for the start of the year, at the start of a topic, during a lesson, at the end of a lesson, at the end of every four topics, and at the end of the year. The end-of-course exams provide a report for each student, which is linked to the “Math Diagnostic and Intervention System 2.0.” This system provides students who have yet to achieve mastery with additional reviews and practice. The materials include suggestions to support more frequent monitoring of students demonstrating difficulty in order to support instructional interventions and response to intervention. The materials provide an ongoing assessment during the lesson through questions in the “Visual Learning Bridge,” in the “Visual Learning Animation Plus,” in the Do You Understand? section, and through guided practice. Informal assessments are also conducted during each lesson with quick checks, anecdotal checks, and observational assessments. The material includes learning support for students who need an intervention, students on level, and for students who are advanced in that content area. For example, based on the students’ data from the quick check, students either do the reteach to build understanding or the on-level and advanced activity centers. 

Evaluation for 5.1
Materials include guidance, scaffolds, supports, and extensions that maximizes student learning potential.

Throughout each lesson, the materials provide support to meet the diverse learning needs of all students. There are enrichment questions asked throughout each lesson in the “Extend Your Thinking” and “Daily Challenge” sections. There are differentiation tools after each topic, and teachers can base every four topics on the results of the online topic benchmark tests. The materials provide recommended targeted instruction and activities for students who have mastered the content. 

Evidence includes but is not limited to: 

The “Math Diagnosis and Intervention System,” listed in the Table of Contents, includes guidance for differentiating support for students who are struggling to master content. The “Diagnostic Tests” within this system include two diagnostic tests (Form A and Form B) to assist the teacher in identifying areas in which students need additional practice. Multiple booklets provide intervention activities based on the students’ needs. The materials provide guidance for scaffolding instruction and differentiating activities based on targeted areas. An example can be found in the Topic 1 “Review What You Know,” which helps students develop and practice prerequisite skills. Teachers find support in the “Teacher Resources” section throughout the book. In Topic 1, Lesson 6, the “Errors Interventions” section states: “If students have difficulty representing the number of objects in the picture, then have students cover each object in the picture with a counter and move the counters one by one to the box on the right. Then have students count to check that the numbers are the same.” Each lesson contains a reteach worksheet to provide additional practice for students who struggle to master the content. In Topic 4, Lesson 3, for example, the worksheet has directions to assist the teacher, but it could be completed by the student individually, as no reading is required to complete the work. The end of each topic also provides a reteach activity to assist struggling students.

Topics 1–16 include a section titled “Today’s Challenge,” which provides extension activities in the form of various math problems for students who have mastered the content of that specific lesson. For example, the Topic 4 Today’s Challenge provides five different challenge questions/activities. On the first day, it gives a factoid that explains to students that many of our clothes are made of cotton and that West Texas grows more cotton than almost anywhere else in the country. Days 2–6 use a picture of a field of cotton plants divided into six columns. Day 2 asks students to identify the column with enough cotton to finish filling in a ten-frame. Day 3 asks students to identify which two columns together have 10 cotton plants. Day 4 asks how much cotton would be in a column if one more plant was planted. Day 5 adds two more plants to a column and then asks how many plants are in that column plus another column. On Day 6, students write their own problems. Another component developed for students who have mastered the content is the “Extend for Early Finishers,” located in the Teacher Edition. For example, in Topic 2, Lesson 2, Extend for Early Finishers directs teachers to show three cubes in a row and have students count them. Then, teachers say, “How many cubes would I have if I put two more cubes in the row?” Questions for Extend for Early Finishers can be found throughout each topic. 

The instructional material provides center activities for Topics 1–16 to allow students to continue to explore and apply their learning in new ways. A wide variety of activities, online games, and center games provide for concept practice for all students in each topic.

There are lessons and support materials for struggling, on-level, and advanced students during and after each lesson. Intervention activities within the lesson reinforce the same model and problem-solving strategy used in the main lesson. Activities for on-level and advanced students provide extension within the same topic and encourage application to real-world tasks and discussion between peers.

A “Differentiated Instruction” page near the beginning of a topic shows these resources organized by Ongoing Intervention during the core lesson (RTI 1), Strategic Intervention at the end of the lesson (RTI 2) and intensive intervention, or more instruction for struggling students and enrichment for advanced students, as needed (RTI 3). These are also shown as they apply to specific lessons in each “Topic Planner.”

The website can auto-assign differentiation based on the results of online Topic Tests and online Benchmark Tests. These include both students who struggle to master content and those who have mastered the content. Examples include “Visual Learning Animation Plus,” “Online Math Game,” “Digital Math Tools Activity,” “Reteach to Build Understanding Master,” “Center Games Master,” and lessons from MDIS 2.0.

Evaluation for 5.2
Materials provide a variety of instructional methods that appeal to a variety of learning interests and needs.

The materials provide a variety of instructional methods. The guidance and support to teachers to help them meet the diverse learning needs of all students, specifically addressing teaching approaches, instructional strategies, and flexible settings utilized to support the mastery of content. The materials support multiple types of practices (e.g., guided, independent, collaborative) and provide guidance and structures to achieve effective implementation. 

Evidence includes but is not limited to: 

The materials incorporate a variety of instructional approaches to engage students in the mastery of content. In each topic, there are components, such as visual learning, games, online activities, independent work, guided work, and lessons, that can be used for small group or whole group. There are manipulatives to use for concrete practice, technological options for independent practice, cross-curricular stories and related science topics to demonstrate how math applies to everyday life, visual representations, videos, and symbolic abstractions that can be taught to large or small groups. About half of the lessons include the center games, which include on-level and advanced-level resource material. The whole program is interactive and can be used on any device at any time. As an example, Topic 4 provides the students with an “Interactive Math Story” Jake’s Garden. As the story is read aloud, students participate by pointing and counting the number of rows; they complete the assignment by circling the correct number of flowers of the correct color and then draw two more flowers.

The material guides teachers in selecting appropriate teaching strategies. The Teacher Edition (TE) includes guidance for the teacher on how to work through a “Solve and Share” problem. It provides six steps for this guidance: “Pose the Solve-and-Share Problem,” “Build Understanding,” “Give Hints as Needed,” “Share and Discuss Solutions,” “Summarize and Generalize,” and “Extension for Early Finishers.” Within these steps, materials provide questions for the teacher to ask the students to assist in their thinking while they are analyzing the math problem. The “Realize Reader” online TE provides instructors with step-by-step formatting for lessons. It provides suggestions for how to deliver the lesson; how to assess student learning; and how to provide follow-on instruction for those who are struggling, those who are on level, and those who are above level. The materials support the use of a variety of instructional strategies to support delivery. In the TE, the “Math Background” section for each topic includes related TEKS, “Big Ideas,” “Essential Understandings,” a “Focus on the Process,” and a “Focus on the Content.” Materials support language with visual support, manipulatives and online tools, and Solve and Share as well as an in-depth understanding of the latest TEKS. For example, in Topic 6, Lesson 2, students receive hands-on materials. They use 30 counters to organize and communicate the number of cherries on a tray. Students also share and discuss finding various ways cherries can be counted without changing the number of cherries. 

Materials support flexible grouping. A placement test can be given at the start of the year to check students’ content knowledge. Students can be placed in groups based on their content mastery for each topic and lesson. Any of the lessons can be done with an individual, small group, or large group, and the text also provides examples of activities to do with struggling learners. The Solve and Share sections in each topic provide a routine activity for whole group or small group instruction. For example, in Topic 9, Lesson 1, the Solve and Share shows a little boy and a beach blanket. The teacher says, “Alex finds seven cents at the beach. Write how much money he found. Draw the coins he could have found.” The teacher’s guide provides questions to pose to the students. The center games for the same lesson allow students to play with a partner.

The framework of each lesson starts with a guided activity followed by independent work that is assessed for content mastery. Students who are on level or have mastered the content have centers or collaborative opportunities to practice the content or expand on their learning. Students who struggle are provided with intervention work with the teacher to reteach the topic. The materials walk the teacher through the process step-by-step. 

Evaluation for 5.3
Materials include supports for English Learners (EL) to meet grade-level learning expectations.

The instructional materials include accommodations for linguistics commensurate with various levels of English language proficiency (Beginning, Intermediate, Advanced, and Advanced High). They also include an “ELPS Toolkit,” which offers research-based scaffolds as an intentional and natural part of the lesson. They encourage the strategic use of students’ first language as a means to develop linguistic, affective, cognitive, and academic skills in English (e.g., to enhance vocabulary development).

Evidence includes but is not limited to: 

The materials include accommodations for linguistics for various English proficiency levels. The instructional materials provide instruction in one or more ELPS for English Learners (ELs) at Beginning, Intermediate, Advanced, and Advanced High levels of English proficiency within the lessons and through additional activities in the ELPS Toolkit. Materials include visual learning in math instruction, such as “Visual Learning Animation Plus,” “Visual Learning Bridge,” animated glossary, and visual learning in exercises. At the beginning of each lesson, materials provide guidance to the teacher for which ELPS are addressed in the lesson. ELPS are used with specified parts of the lessons such as “Solve and Share,” Visual Learning Bridge, and “Do You Understand?” Materials provide activities for students to complete with the teacher based on their individual English language proficiency level. For example, in Topic 4, Lesson 1, the ELPS focus on speaking (3d) and learning strategies (1b). Beginning ELs receive their own ten-frame and some counters. Students place the counters on their ten-frame. The teacher puts five counters on her ten-frame and asks each student to tell if his/her group has more than or fewer than the group of five. Intermediate ELs compare their number with the teacher’s group of five using the sentence stems “A group of ... has fewer than five”; “A group of ... has more than five”; “... is greater”; or “... is less.” Advanced ELs compare their group with the group of five using more than and fewer than in their responses. Advanced and Advanced High ELs find a student who has more or fewer counters than they do and use the words more and fewer. Topic 6 includes prompts for a variety of proficiency levels. Beginning ELs point to each cube as they aloud and use the sentence stems “There is/are … cube(s) in this ten-frame. There are ... cubes in all.” Intermediate ELs ask their partners, “How many cubes are there in this ten-frame? How many cubes are there in all?” Students respond using there is/are in their answers. Advanced ELs pair up. Student A chooses a number between 0 and 30 and fills the ten-frame(s) with that many cubes. Student B describes how many cubes there are in all and explains how he or she knows. For Advanced High ELs, Student A chooses a number between 0 and 30 and instructs Student B to show the number with the cubes and ten-frames. Student B explains the total and how he or she knew the number. In Topic 7, Lesson 3, the ELPS are focused on teaching ELs the concept of parts and wholes. ELs receive connecting cubes; they are told that there are three cubes in one part and two cubes in another part; they must find the whole. Beginning ELs show the problem with their cubes, and then fill out the sentence stems that say “...and…is…. We joined…and…to get…in all.” Intermediate ELs also show the parts with their cubes. Teachers ask them, “What are the parts? How many are there in all?” They respond using complete sentences. Advanced ELs work in pairs. Student A instructs student B to make two parts and then join them into a whole; students then switch roles. Advanced High ELs are similar to the Advanced ELs, but they tell their partner a number story, and the partner must use the cubes to demonstrate the story. These exercises are interactive, playful, and allow opportunities for repetition of the desired terms.

The material includes suggestions for scaffolds to support students learning English. The material includes routine scaffolds through teaching academic vocabulary, Visual Learning Bridges, and connecting new information to prior experiences and learning using the review of what students know. The materials include resources and support materials that make scaffolding intentional and natural in the lessons. The ELPS Toolkit emphasizes seven specific instructional strategies. These strategies include the following: modeling thinking out loud, partner talk, providing a word list, providing sentence stems, rephrasing, suggesting a sequence, and using repetition. Each lesson has a Visual Learning Bridge and an animated glossary. Lessons allow students to collaborate with others often as they discuss the learning. The materials frontload the lesson by activating prior knowledge. For example, in Topic 1, the materials provide an “Interactive Math Story”; students find out how many eggs there are, activating prior knowledge. The materials include resources and support materials that make scaffolding intentional and natural in the lessons and provide modeling. For example, in Topic 1, teachers model counting three counters with the students. They put the counters on the table and point to each counter as it is counted. They demonstrate with sentence stems: “I have ... counters. I counted ….” The lessons are planned with the use of manipulatives and hands-on experiences in small groups; this allows students to practice concepts and new English terms in less threatening environments. An example is in Topic 5, Lesson 7, where students practice the use of double ten-frames. The students are broken into their capability groups and given 20 counters and two ten-frames. The teacher shows two ten-frames. The top is full, and the bottom only has five counters in it. Beginning ELs read aloud the number of counters in each ten-frame. Intermediate ELs are paired up. Each student must show 13 in their ten-frames; they then check each other’s work. Advanced ELs are also working in pairs. They each get assigned a number between 11 and 20 to show in their frames; they then check each other’s work. Advanced High ELs also work with partners. Each student picks a number between 11 and 20; they write that number by their frame and then switch papers; the partner must put the correct number of counters in the frame to equal the number their partner gave them. Materials include suggestions for small group instructions that focus primarily on language development. For example, Topic 5, Lesson 7 in the ELPS, suggests that intermediate students be put into pairs. Then, “Ask each partner to show 13 on his or her frames and write their numbers. Partners will read and compare each other’s work.”

The materials encourage strategic use of the student’s first language to develop linguistic, affective, cognitive, and academic skills in English and include examples of how to use students’ first language as the foundation for developing skills in English. The materials include accessible resources, such as an ELPS Toolkit, that share strategies that are effective with ELs for teachers to use. The Toolkit provides a chart with mathematical “thinking words” in English, Spanish, Chinese, Vietnamese, and Hmong. Students are encouraged to discuss mathematical concepts with a partner who speaks their language. If this is not available, the materials encourage the use of an online translator, dictionary, or pictures that students can point to in order to communicate their thoughts. The use of cognates is also discussed, as many words are similar enough to determine meaning. The toolkit gives an example of the Spanish words división, hexágano, ángulo, triángulo, álgebra, circunferencia, and cubo, which are all very similar to the English words for the same things.

Evaluation for 6.1
Materials include year-long plans with practice and review opportunities that support instruction.

The materials build students’ concept development by including a cohesive year-long plan that is vertically aligned with focal areas in the TEKS. There is also evidence that the material provides review and practice of mathematical knowledge and skills throughout the curriculum.

Evidence includes but is not limited to: 

The materials include a cohesive, year-long plan to build students’ concept development and consider how to vertically align instruction that builds year to year. The content plan is cohesively designed to build upon students’ current level of understanding with clear connections within and between lessons and grade levels. A pacing guide is included in the “enVisionMATH Texas 2.0 and Texas” guide in the teacher overview. This guide assumes one lesson per day, which adds up to 110 days, with an additional 10 days for the “Step Up to Grade 1” lessons. This schedule allows additional time for differentiation, review, local and state testing, and other requirements. A correlation chart can be found in the teacher overview “enVisionMath Texas 2.0 Correlations” guide. This reference shows where each TEKS is taught in the curriculum. Within the “Content Guide,” a “Big Ideas in Mathematics” section lists which mathematical concepts are detailed for grades K–5. The instructional materials state: “Big Ideas are the conceptual underpinnings of the program and the glue that provides conceptual cohesion across lessons, topics, and grades, as well as across TEKS and reporting categories. Big Ideas connect Essential Understandings that occur within and across lessons. Math Background at the start of each topic shows the Big Ideas and Essential Understandings for the topic.” For example, the fifth Big Idea is “Comparison and Relationships.” The Big Ideas chart shows that this Big Idea can be found in kindergarten Topics 2, 4, 5, 6, 14, and 15. It can be found in grade 1 Topics 4, 10, 13, and 15. For grade 2, this Big Idea can be found in Topics 4, 10, 12, and 15. Vocabulary terms used for the fifth Big Idea, such as order, more (than), fewer (than), and number sentence, are consistent across the grade levels. The Content Guide also includes a “Scope and Sequence” section that charts the mathematical concepts across grades K–5 and the grade level in which a concept is introduced, practiced, and applied. For instance, “Number and Operations, using concrete/pictorial models, strip diagrams, number lines” is shown to be introduced in grade K, practiced in grades K, 1, 2, 3, and 4, and applied in grades 4 and 5. The materials include a vertical alignment chart, “Skills Trace,” that shows how topics align, both directly and indirectly, to topics outlined for students in preceding and subsequent topics. For example, in kindergarten, Topic 9, under “Looking Back TEKS 1.4A,” the materials state that students identify the value of a dime. In grade 2, Topic 10, the materials state that with TEKS 2.5A, students put a set of coins in order from the one with the greatest value to the one with the least value and then count on to find the total value. 

The material provides review and practice through the curriculum. Every topic starts with a “Review What You Know” section, and each lesson starts with a review practice sheet to ensure students have the foundation necessary on which to build new skills. Each lesson then has a “Quick Check” to review the lesson and ensure that students learned the required concepts in the lesson. The material includes various components, such as ongoing “Daily TEKS Review,” “Solve and Share,” “Independent Practice,” “Guided Practice,” and “Homework” practice, for each lesson. The “Math Tool Activities” can cover more than one lesson concept. The Math Tools Activity can be used with the first three lessons in Topic 6. “Center Games” provide opportunities for students to use and practice newly taught skills as well as previously taught skills. In the Center Games for Topic 6, Lesson 3, students use counters to show various numbers from 15 to 30. Once they show the number, they either color in the appropriate number of shapes or label the amount with the appropriate number. Skills Trace shows how the materials build upon previously taught content: how the TEKS covered in a topic build upon TEKS covered in a previous topic and also how they are addressed or built upon in a future topic. Topic 8, “Understanding Subtraction,” shows how TEKS covered in Topics 2 and 4 are built upon in Topic 8 and built upon further in Topic 10. “Today’s Challenge Online” has problems that apply to any content taught before the topic.

Evaluation for 6.2
Materials include implementation support for teachers and administrators.

The materials provide a TEKS-aligned scope and sequencing that builds and connects across grade levels. Resources and guidance are included to help administrators support teachers in implementing the materials as intended. The materials include a school year’s worth of math instruction, including realistic pacing guidance and routines. 

Evidence includes but is not limited to: 

The material includes a “Content Guide” with the “Big Ideas in Mathematics,” “Texas Focal Points,” “Skills Trace” for each kindergarten topic, and a “Scope and Sequence” that supports the teacher in understanding the vertical alignment. This Scope and Sequence shows clear alignment through the “TEKS Correlation” document. The Scope and Sequence outlines which essential knowledge and skills are taught. The chart lists the concepts and skills and the grade level in which they are introduced, practiced, and applied. The “Numbers and Operations” section of the Scope and Sequence details comparing and ordering whole numbers. It charts how “one and two more” and “one and two less” are introduced in kindergarten and practiced in kindergarten and first grade. “Ten more and ten less” are introduced in first grade and practiced in grades 1 and 2. “One hundred more and one hundred less” are introduced and practiced in grade 2. Comparing whole numbers is introduced in kindergarten, practiced in kindergarten through grade 3, and applied in grades 4 and 5. Using comparison symbols <, >, = is introduced in grade 1, practiced in grades 1 through 3, and applied in grades 4 and 5. Ordering whole numbers is introduced in kindergarten, practiced in kindergarten through grade 4, and applied in grade 4. Comparing and ordering on number lines is introduced in grade 1, practiced in grades 1–4, and applied in grade 4. Skills Trace details the order in which the essential knowledge and skills are presented and revisited. It describes how the essential knowledge and skills build and connect across grade levels. Each lesson includes a “Lesson Overview,” which includes TEKS; “Essential Understanding”; vocabulary; materials; and “Math Background” to use with a specific part of the lesson provided prior to learning a topic. A “Big Idea” chart shows the topics where each conceptual underpinning is taught from kindergarten to fifth grade. 

The materials support the teacher by providing teacher prompts, topic planners, Math Backgrounds (“Focus on Process” and “Focus on Content”), differentiation for all three “RtI” levels, and a “Language of Math” section for each topic. These areas provide the teacher with required supplies, TEKS, ELPS, and an understanding of the cognitive requirements for students to master the content in the topic. They provide suggestions for how to assess student progress, how to address struggling students, and how to challenge those who are at grade level or who have achieved mastery. The materials provide resources that include class sets of individually packaged manipulatives for each student and a pocket chart in which to store them. At the beginning of each lesson, a “Materials List” shows which manipulatives are used. In addition, materials include many online resources, such as assessments, online tools, online games, and review materials. Most of the materials are organized in a repetitive and logical manner that is consistent throughout the topics. The “User Guide” guides teachers on how to use each page in the student’s workbook. There is the option on “Realize” for users to quickly add resources to create their own Table of Contents (playlist). This playlist is then accessed from the “My Library” tab.

The materials include resources and guidance to help administrators support teachers in implementing the materials as intended. For example, the materials contain a TEKS-aligned scope and sequence outlining the essential knowledge and skills that are taught in the program and the order in which they are presented, as well as a “Skills Trace” to show how knowledge and skills build and connect across grade levels. The “TEKS Correlation” and “Texas Focal Points” indicate mathematics content to emphasize at each grade level. For each focal point, there is a group of related TEKS. The materials provide tools to support teachers in recognizing best instructional practices and arrangements in an elementary math classroom. The materials include guidance to support teachers in understanding developmentally appropriate mathematical practices in elementary classrooms, including the use of small groups and guidance for implementation of the lessons. The teacher’s guide gives much direction and details on how to implement the instructional materials. For each topic, there is a “Topic Planner” section, a Math Background section, a section for “Differentiated Instruction,” and a section for the “Language of Math,” which all help to give the teacher guidance and support in implementing the instructional materials. The Math Background lists and details the TEKS covered during that topic, along with an explanation of the Essential Understandings that are addressed. Detailed information for Focus on Process and Focus on Content is also provided. The Topic 1 Math Background lists and details TEKS K.2B; the Big Idea that matches this TEKS is also detailed; Essential Understandings are provided for Lessons 3, 7, and 9 in Topic 1, in which this TEKS is taught; the Focus on Process provides detailed information for the two process standards featured in Topic 1, “Connect and Formulate a Plan”; the Focus on Content provides information on number arrangements and ways to count. Professional development videos and research-based information is provided within the topics and in the “Content Guide” in the teacher overview section. Administrators can use this information to understand developmentally appropriate practices. The data section of the curriculum provides class and individual charts and tracking sheets to show content mastery or intervention material for those still struggling. 

The material includes lessons and activities for a full year of instruction and realistic pacing guidance for each topic and lesson. The topics allow for reasonable implementation throughout a school year, and the activities and routines in each topic can be completed within the length of the year. For example, the material includes pacing for 16 topics that assumes one lesson per day, which is a total of 110 days. Additional time may be spent on review, remediation, differentiation, and assessment as needed. The “Step Up to Grade 1” includes an additional 10 days. In the “enVisionMATH 2.0 and Texas” guide, a year-long “Pacing Guide” is located on page 17. This guide proposes 10 days for Topic 1, 10 days for Topic 5, five days for Topic 11, and six days for Topic 14. The guide proposes 110 days in all for Topics 1–16 and 10 days for Topic 17. The lessons are all formatted in a similar manner and provide spiraling work that keeps content fresh in students’ minds. Additional time is scheduled at the end of each lesson so the teacher has an opportunity to work with those needing intervention while providing lesson-specific activities, at an appropriate level of challenge, for those on level and those who have achieved mastery. 

Evaluation for 6.3
Materials provide implementation guidance to meet variability in programmatic design and scheduling considerations.

The instructional materials are sequenced and spiraled in an order that ensures students develop prerequisite skills prior to scaffolding on higher-order concepts. The materials provide guidance for strategic implementation without disrupting the sequence of content that must be taught in a specific order following a developmental progression. Evidence shows that materials are designed in a way that allows LEAs the ability to incorporate the curriculum into district, campus, and teacher programmatic design and scheduling considerations. 

Evidence includes but is not limited to: 

The materials include strategic guidance for implementation to ensure that content is taught in an order consistent with the developmental progression of mathematics. The materials provide a suggested sequence of units that considers the interconnections between the development of conceptual understanding and procedural fluency; information can be found in the “envisionMATH Texas 2.0 Content Guide.” The curriculum is designed so that teachers can import additional items for use in lessons in accordance with individual, grade-level, school, or even district requirements. The lessons can be customized, and the plans can be organized by day, week, or month. District-created content or teacher content can be uploaded. Topics can be resequenced to match district-level curriculum guides or district scope and sequence preference. The materials provide a “Skills Trace” for each kindergarten topic and a “Scope and Sequence” guide to help ensure the correct sequence of the mathematical concepts is taught. The Skills Trace lists the TEKS that prepared the students for the current topic, the TEKS that the current topic covers, and the TEKS covered in future topics that build upon the current topic’s TEKS. For example, Topic 11 addresses TEKS K.5 as students count to 100; the “Looking Back” column shows this being covered in Topic 6 when students counted to 30. In the “Looking Ahead” column, TEKS 1.5A is addressed in grade 1, Topic 7, where students use a hundred chart to count forward from any number. The topics are organized in sequential order. In kindergarten, Topic 1 addresses numbers 0 to 5, Topic 3 addresses numbers 6 to 10, Topic 5 addresses numbers to 20, and Topic 6 addresses numbers to 30. 

The materials are designed in a way that they can be easily implemented in a variety of ways. Within the materials, it is easy to navigate, assign resources, search, customize, organize plans by day, week, or month, assess, and analyze data. Each topic is flexible enough that it can be expanded with “Math Science Activities,” “Interactive Math Stories,” “Today’s Challenges,” “Daily TEKS Reviews,” “Interventions,” “Centers,” and online math games. It is also possible to shorten in-class instruction and assign quick checks, homework, and assessments in the online program. The curriculum provides support for English Learners, struggling students, students who are on level, and students who have achieved mastery. The lessons can easily be presented to a full class or a small group both in person or online. The instructional materials are designed in a way to allow the ability to incorporate the curriculum into the district, campus, and teacher programmatic design and scheduling considerations. The materials provide the ability for the topics to be rearranged online so that LEAs can organize the topics as needed to match their curriculum or district scope and sequence preference. Online, to the right of the topics, the “Rearrange” button allows teachers to click and drag the topics into any order they choose. The ability to create content is also available. A teacher can upload a file, add a link, or build a test. 

Evaluation for 6.4
Materials provide guidance on fostering connections between home and school.

The materials support the development of relationships between teachers and families. They also specify activities for use at home to support students’ learning of appropriate mathematical skills. 

Evidence includes but is not limited to: 

The instructional materials support the relationships between teachers and families. The materials include “Home-School Connections” pages that encourage the development of strong relationships between teachers and families by giving families an overview of the content in the topic. The Topic 4 Home-School Connection explains how students use ten-frames to compare numbers in groups. It suggests families draw six stars on a paper and asks students to draw one or two more or one or two less. The student picks a number between seven and ten. The parent draws one or two more or less. They can continue this way, taking turns. Another activity suggests writing numbers one through ten on pieces of paper or index cards and having the student practice putting them in order. Students then pick a number, and the parent finds the number that is one or two more or less. The Topic 12 Home-School Connection shares how students are learning about geometry and specifically how they are learning to identify shapes as the same or different based on the attributes of shape and to organize them based on this attribute. Materials share information about the different types of attributes the students are learning and provide activities for students to find shapes around their home and neighborhood as well as to draw different shapes they have learned about. Student progress reports are located in “Teaching Tools” under the “Teacher Resources” tab on the right side of the “Table of Contents.” This form is intended to be sent home at the end of each topic. It reviews the student’s progress on the topic and has a portion at the bottom that parents/guardians are to sign and return to the teacher. This response ensures that parents are aware of how their students are progressing in math. The materials also include leveled homework for each lesson on which there is a “Home Connection” section. The materials provide suggestions and examples of exemplary family engagement practices. For example, in Topic 13, Lesson 1, the Home Connection reads, “Your child learned to sort objects by different attributes, such as shapes and color. The Home Activity suggests parents set out a variety of simple objects, such as marbles in different colors and sizes, and ask your child to sort the objects into groups in at least two different ways. Ask your child to tell what is alike about all of the objects in each group. For example, your child may sort all of the large marbles into a group and then sort all of the blue marbles into a group.”

The materials include online access to resources that parents can use at home to work with their children on specific skills. Some “Math and Science Projects” included in each topic also provide some activities for students to specifically complete at home. One example is the Math and Science Project for Topic 2, which directs students to discuss with family and friends how seeds grow into plants and if they have ever planted a seed that has grown into a plant. Students then make a poster showing more seeds than growing plants and compare the number of seeds versus growing plants. Online materials include resources that are easy to use and are related to current skills. There are printable versions of worksheets, an online animated glossary, and online manipulatives and content-specific games for students to practice math skills. On each homework sheet, there is a “Math Tools and Math Games” callout that has a link to a specific math tools activity or math game to use with the lesson. For example, the Topic 3, Lesson 1, Home Connection informs the parents that their child counted six or seven objects and drew counters to show each number. The “Home Activity” suggests that parents have their child count groups of six objects. The child can then draw pictures of six objects and repeat using the number seven. Home-School Connections pages give families an overview of the content in each topic. The Home-School Connection at the beginning of Topic 15 explains how students are learning to use real graphs and picture graphs. The sheet suggests that families should encourage students to think of questions to ask family members about their favorite color, TV show, or food, and create graphs with their answers. It also suggests parents help children cut out ten red and ten blue squares (or two other colors) and then have them find items of those colors around the house and put one square down on the table for each item of that color they find. They can then compare which color had more items, which had less, and how much more or less. The curriculum is available online, and teachers can assign specific activities for students to complete at home, such as assessments, online games, or practice activities. There are also online manipulatives that students can use to help them solve problems when they are at home and do not have school manipulatives to use. Home support materials are readily available in English and Spanish, and there are suggestions at the beginning of each topic for real-world ways to practice skills being learned in class. 

Evaluation for 6.5
The visual design of student and teacher materials (whether in print or digital) is neither distracting nor chaotic.

The materials are structured in a way to facilitate ease of instructional support to teachers for planning and implementing lessons that contribute to student learning. The pictures and graphics are supportive of student learning and include appropriate use of white space and design that supports and does not distract from student learning. 

Evidence includes but is not limited to: 

The materials are designed to support students’ learning. The teacher’s guide is clear and is designed in a way that teachers can locate important information. The materials consistently include a place for instructional support to aid teachers in planning and implementing lessons. For example, the “Topic Planner” at the beginning of each topic provides an overview of the lessons. It gives the lessons, pages, TEKS covered, ELPS, “Essential Understanding,” and the materials needed and where to find them. The teacher’s guide includes instructional support with information that is easily identified throughout the lessons. For example, there are callouts and notes on the side of each page that have guidance questions for teachers to check for students’ understanding. The visuals and graphics that are included are concise and user friendly. The materials adhere to the “User Interface Design” guidelines. For example, the font is clear and easy to read. Items with photographs and colorful pictures do not distract from the text on the page or interfere with learning. Also, under “Visibility” in system status guidelines, the materials allow for the user to immediately enlarge images when clicked upon. Materials meet the “Aesthetic and Minimalist Design” guidelines; icons provide easy access to the “Table of Contents,” “Resources,” “Standards,” “eTexts,” and “Tools.” The design of the student instructional materials is consistent from topic to topic; lessons begin with the “Visual Learning Bridge,” then move into the “Guided Practice,” and end with “Independent Practice.” The student book pages are easy to follow; the print is appropriate, and the pages are not crowded. Any tables, charts, and visuals included are clear and concise. The characters and illustrations are age-appropriate and adequately display the mathematical concepts being taught without being overly distracting.

The same theme of the grade-level robot is used throughout and shows consistency. The graphics used in online games are aesthetically pleasing and engaging for the students without being overwhelming. The interactive “Math Tools,” such as the base-ten blocks and the pan balance, are straightforward to use and allow for easy user control and freedom. New vocabulary and concepts are introduced with pictures and words to help students visualize what they are learning. Page designs are simple, with clear and easy-to-read information and plenty of white space for student work. Online games, assessments, and resources are intuitive, with opportunities to go back and review work before submitting it. The graphics and pictures on the pages are colored and go along with the learning. The publisher provides some text resources like pictures, books, and charts. Items with photographs and colorful pictures do not distract from the text on the page or interfere with learning.

Evaluation for 6.6
If present, technology or online components included are appropriate for grade level students and provide support for learning.

The technological components align to the curriculum’s scope and approach to mathematics skill progression; they support the materials’ progression of math content and skills. The materials provide a full suite of online curricular components to reach all students, including math games, science activities, and problem-solving reading activities that promote reasoning and application. Every part of every lesson can be assigned to students to perform at home. This access includes videos, stories, assessments, reviews, and games. Students can even participate online in “Solve and Share” using the “DrawPad,” where students can write their solutions during whole-class discussion. At the beginning of each topic, there is a “Today’s Challenge” and an “Animated Math Story.” The “Begin Topic” folder contains vocabulary cards that may be downloaded or printed. It also contains a “Review What You Know” activity. For each topic’s lesson, there is an available “ACTIVe-book” activity. “Visual Learning Animation Plus” includes interactives to build understanding through classroom conversations. An online “Quick Check” assesses student progress. Many lessons include online interactive games to give students additional practice in the lesson’s concepts. These items can be found within each lesson of each of the topics. For every four topics, the instructional materials include an online “Benchmark Test”; there is also an online “End of Year Test” available.

The materials include technology to enhance student participation, colorful interactive math manipulatives, and “Math Games” to help motivate and enhance learning. For example, there is an opportunity for a differentiated assignment after every four topics. Students are assigned remediation or enrichment tasks, which can include Visual Learning Animation Plus, an online Math Game, or a “Digital Math Tools Activity.” The Visual Learning activity provides animated math problems for students to answer via discussion, the drag and drop tool, and text box tools. In the Topic 9, Lesson 3 Visual Learning activity, students are shown a nickel and five pennies and are asked what coins they see. The video reviews the value of a penny and a dime and discusses how students can count on to find the value of all the coins (ten cents). The video then discusses that a dime is worth ten cents. The material provides teachers with appropriate and sufficient guidance on how to use technology. For example, the teacher guide has sidebars within each lesson that reference what technology can be accessed and where to find it. Online manipulatives and games are engaging and allow students to review new skills. The materials provide intuitive ways for teachers to assign online activities to students and make online learning feasible. The teacher’s “User’s Guide” provides information for the different online activities available for student use. The online component features many online tutorials and quick-start guides to help teachers jumpstart their “enVisionMath Texas 2.0” training.