enVision MATH 2.0 Texas Grade 2

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

The materials provide overall strategic and integrated instruction. The materials emphasize the primary focal areas for grade 2, namely place value (base-10 up to 1,200), addition and subtraction with multi-digit whole numbers, measuring length, and applying two-dimensional shape and three-dimensional solid understanding. The materials also provide practice opportunities for students to master the content. 

Evidence includes but is not limited to: 

A majority of the topics in grade 2 address the focal areas. Topics 3–4 and 6–8 focus on place value; Topics 1–2, 5–8, and 15 target problem solving with addition or subtraction. Addition and subtraction are spiraled in all of the topics after they are taught. Two- and three-dimensional shapes and solids come under consideration in Topic 13, and the principles of length measurement can be found in Topic 14. The materials contain planning documents, such as scope and sequences or unit plans, that clearly state the focal areas of a week or unit; those focal areas align with the grade-level TEKS. 

Lessons are broken down into sections, such as “Math Anytime,” “Problem Based Learning,” “Visual Learning,” and “Assess and Differentiate.” Each of these sections includes several activities for students to strategically and systematically develop their content knowledge. The curriculum spirals each focal area by introducing the content and then gradually readdressing it numerous times in a slightly more rigorous manner with each repetition. For example, students learn about subtraction in Topic 2 and then again in Topics 5, 7–8, and 17.

Opportunities to practice the content occur throughout the instructional materials. 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 also reinforces focal area skills through activities such as the “Daily TEKS Review,” the “Daily Challenge,” and intervention activities. For example, a lesson on doubles and near doubles in Topic 1 has students build on their knowledge of doubles when they develop their understanding of near doubles.

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 for student learning, progressing from more concrete to more abstract as the instructional materials progress throughout the year. Students are explicitly taught how to work with models, manipulatives, and representations for concept exploration and attainment. However, the materials fail to give explicit instruction to teachers on how to use the models, manipulatives, and representations. 

Evidence includes but is not limited to: 

Concrete materials are included in the “Center Games Manipulatives Kit.” There are also online “Math Tools” for students to use when completing online activities; these include counters, shapes, place value blocks, money, data and graphs, number line, input-output machine, strip diagrams, measuring cylinders, number charts, and a pan balance. In Topic 1, the materials provide 10 connecting cubes so that students may explain how a doubles fact might help them solve a near doubles fact. In Topic 2, students use ten-frames and counters to model subtraction. In Topic 6, Lesson 1, students are asked, “What are you asked to find out? What tools do you have to help you add?” Materials support the use of models, manipulatives, and representations for concept exploration and attainment. Although number tiles are used throughout the curriculum, students progress to less concrete connecting cubes, as in Topic 14, Lesson 6, where students use connecting cubes as a random unit of measure. 

Professional development videos provide teacher guidance at the beginning of each topic. Additionally, teacher resources and information documents are available to click on under each lesson. The materials explain how to subtract a one-digit number from a two-digit number using cubes (concrete), then move to using pictorial representation to regroup 1 ten as 10 ones to subtract (pictorial). The materials guide teachers to move back to concrete practice with students who struggle with the transition to pictorial by reteaching to build understanding using cubes. Although this provides the student with a progression along the CRA continuum, it does not specify to the teacher how or why the students are progressing. 

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 and 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 the Solve and 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 help 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 meet the criteria for the indicator. They support students in building their vertical content knowledge by accessing prior knowledge and understanding of concept progression. They support coherence and connections within content at and across grade levels and 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 to previous work at the beginning of every lesson and connecting new learning to previously learned skills. For example, the “Review What You Know” section for Topic 5 reviews the concepts and skills students learned and practiced in Topic 1 and Topic 2. In addition, the “Math Background” section for Topic 9, Lesson 1, states that students use prior knowledge of addition, including skip counting, to explore the relationships between repeated addition and multiplication. Manipulatives, such as counters, are useful tools in helping students visualize and model equal groups. Students are introduced to the terminology of multiplication and the times symbol in preparation for writing multiplication sentences. As students begin Topic 14, Lesson 1, the materials activate prior knowledge by asking students what tools people might use to measure how tall a chair is and to suggest that students use footsteps to measure the width of the room. Half of the students then line up along one wall and walk heel to toe across the room as they count their footsteps. The teacher asks, “How many footsteps wide is the room?” The other half of the class then uses footsteps to measure the length of the room.

Included tasks require students to recognize and make connections among mathematical ideas. In Topic 6, students build upon their previously learned skills of making 10 when regrouping ten ones for one ten as they begin to add two-digit numbers and 1-digit numbers. They continue to use these skills as they begin to add two two-digit numbers together. Topic 9, Lesson 4, shows students how they can repeatedly subtract like-sized groups to represent division. In Topic 14, Lesson 5, a measurement task asks students to use paperclips to compare objects by length. Then, in Topic 14, Lesson 6, students compare objects by using rulers, making the connection that length can be measured using different objects. In grade 2, there are 16 “Math and Science Projects” to help students recognize and apply mathematics outside of the mathematics curriculum. The Topic 2 Math and Science Project has students look at photos of different fish or study fish in an aquarium. The Topic 9 Math and Science project has students investigate different amusement park rides and then invent one, drawing a picture. 

The materials include tasks that require students to understand how mathematical ideas interconnect and build on one another to produce a coherent whole. In Topic 5, Lesson 1, students perform mental math by breaking two-digit numbers into their place values, adding the tens first, and then adding the ones. In Topic 6, Lesson 1, students practice regrouping ones into a new group of ten by using unifix cubes to add a two-digit number with a one-digit number. In Topic 6, Lesson 3, students apply skills learned in Lesson 1 of the topic and practice adding two-digit and one-digit numbers with the traditional algorithm. These lessons on regrouping to make 10 continue to be applied when students work on Topic 4, Lesson 6, where students practice the algorithm for adding a two-digit number with another two-digit number. In another example, in Topic 10, Lesson 3, students use prior knowledge of addition, including skip counting, to explore the relationship between repeated addition and multiplication.

The “Skills Trace” section in the “Content Guide” shows how TEKS build from kindergarten through second 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. The “Topic Planner” 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. 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 and 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 provide an appropriate progression of tasks designed to engage students in the appropriate level of rigor (conceptual understanding, procedural fluency, or application) for grade-level content and skills. Topic 3, Lesson 1 begins by having students look at pictorial models of base-ten blocks to recognize how many hundreds, tens, and ones are shown; in Lesson 2, students move into writing the number shown in base-ten blocks in standard form and expanded form. As students continue to work through Topic 3, they see pictorial models of base-ten blocks with the 1,000 block added; in Lesson 5, students begin to write numbers up to 1,200 in standard form and expanded form. The understanding of place value that the students gain assists them when they begin adding and subtracting two-digit and three-digit numbers in Topic 6, Topic 7, and Topic 8. In Topic 10, Lesson 1, the “Math Background” reads: “Money provides a rich context for mathematical understanding. Counting money is a form of mental math and is related to a student’s understanding of place value and their ability to skip count by 5s, 10s, and eventually, 25s. Attributes of coins include size, color, edge, values, and heads and tails sides. Students will benefit from studying each coin’s attributes and understanding that its value is not necessarily related to its physical size. Once students understand, they can begin to find the value of coins and the value of a collection of coins.” 

Math concepts addressed in the instructional materials are explained to the teacher before students engage in the tasks. Each lesson’s “Topic Planner” section includes the TEKS, vocabulary material, ELPS instruction, and learning objectives to be addressed in that lesson. For example, the Topic 9, Lesson 1, Topic Planner explains that the lesson focuses on repeated addition and multiplication. The TEKS addressed is 2.6: Connect repeated addition and subtraction to multiplication and division situations that involve equal groupings. The mathematical process standards addressed are 2.1A, 2.1C, 2.1D, 2.1F, and 2.1G. Furthermore, the “Essential Understanding” is “Repeated addition involves joining equal groups and is one way to think about multiplication.” In Topic 9, Lesson 6, one discussion problem asks students to draw a picture and write a number sentence to solve a division problem. The teacher guide lists varying ways the students may solve the problem: Students may use objects, a picture, or a number sentence as a representation to help solve the problem and communicate mathematical ideas. The teacher then shares a student’s correct work to demonstrate how to draw a picture and write a number sentence to solve a division problem. The materials note multiple goals behind a task, emphasizing that the process is just as important for student learning as the product; teachers receive guidance to facilitate discussions on how differences in strategy relate to efficiency and how well they work for that problem type (rather than saying one strategy is better than another). Topic 11, Lesson 1 explains the Essential Understanding as: “Numbers can be classified as even or odd.” The Math Background explains: “This lesson defines even numbers as those numbers that have two equal parts. Odd numbers are numbers that do not have two equal parts. Students model the equal and unequal parts by breaking cube towers of specified numbers in two, comparing the parts, and recording the parts and whole as a number sentence. As students gain practice, they start to recognize that even numbers end in 2, 4, 6, 8, or 0, and odd numbers end in 1, 3, 5, 7, or 9. Students will also begin to recognize that doubles facts represent even numbers.”

Within each topic, the instructional materials include “Interactive Math Stories” and “Math and Science Projects” that apply to real-world contexts and allow students to demonstrate mastery of math concepts set in the real world. The Interactive Math Story for Topic 7 discusses two bunnies and the jobs they have around their home. Each page shows the bunnies completing a job and asks how much is left of the job to do. For example, one page shows the bunnies putting away books, showing 11 books in all, with seven books on the floor and four books on the shelf. Students complete the subtraction sentence 11 - 4 = 7 in their book by finding the difference. The Math and Science Project for the same topic discusses how sailors often bring fresh water when sailing. The assignment asks students to have an adult peel a potato, cut it in half, and place each half in its own separate glass of water. Salt is added to one of the glasses; students wait one day and then observe the potatoes. Students draw pictures of the results of their experiment and then describe how they use water every day. Finally, students make up and solve subtraction problems about the supply of fresh water on a ship. The Topic 8, Lesson 8 “Solve and Share” allows students to practice word problems with realistic stories, such as “The class made 547 keychains to sell. There are 283 keychains left. How many keychains has the class sold? How many of you use paper and pencil to solve? Do you need to regroup? Explain.” The Topic 11, Lesson 3 “Math Science Connection” activity discusses the phases of the moon and how to predict full moons. In Topic 16, Lesson 4, students determine how much money they would need to buy a pet turtle. 

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 and 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 provide some teacher guidance on anticipating student responses and strategies by listing possible student responses to practice questions and tasks. For example, in Topic 1, Lesson 1, the teacher edition includes questions with possible responses: “What addition sentence can you make for these 3 numbers? (6 + 2 + 3 = ?) Why do you add 5 and 6? (5 is the sum of 2 + 3. 6 is the third number being added).” In Topic 5, Lesson 1, teachers ask, “How does knowing 7 + 7 help you find 7 + 8?” In addition to the sample answers, there is also an “Error Intervention” section to guide teachers. If students do not understand the doubles plus 1 process, then teachers can have them use cubes to illustrate the doubles fact and then use one cube in a second color to change the doubles into a doubles plus 1 fact. The “Guiding Questions” and “Prevent Misconceptions” sections also provide sample answers. In Topic 7, Lesson 3, the teacher asks, “Do you need to regroup? Why or why not?” A sample answer is, “Yes; there are not enough ones to subtract from.” Topic 8, Lesson 6 provides the following misconception: “Students may not remember how the strip diagram shows the parts and the whole. Remind students that the number in the top box is the whole and that the parts are shown below.” 

The materials provide some questions for teachers to use to support discourse and sets of discussion questions that can be used to facilitate discourse without limiting student responses. In Topic 5, Lesson 2, the Solve and Share handout has teacher guidance for building understanding, including questions such as “What strategy can you use to help you solve the problem? (I can use a double fact and then add 2 more.) What should you do with the cubes? (Show how many seashells there are in all).” Similarly, in Topic 5, Lesson 3, questions help guide students through parts of the lessons. To solve 27 + 35, questions include, “Where do numbers 20 and 30 come from? Where do the numbers 7 and 5 come from? Can you add the tens of 27 to 35 first and then add on the extra ones?” In Topic 10, Lesson 2, the guided practice handout asks, “Which coins are shown here? (1 half dollar, 1 quarter, 1 dime, 1 nickel) How can you find the total amount? (Count the values of the coins).” Topic 14, Lesson 4 provides open-ended questions such as “What do you already know about area?” and “How can you measure the distance around the shapes?” 

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 Edition includes student work samples and questions to help students think deeply about the problem and to 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.

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

The materials include a cohesive, year-long plan for students to develop fluency in an integrated way. The materials provide multiple areas of guidance for the teacher for teaching fluency as well as multiple opportunities for students to practice, apply, and master the grade-level fluency expectations while building rigor throughout the year. The materials include scaffolds and supports for teachers to differentiate fluency development for all learners.

Evidence includes but is not limited to: 

The materials 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 Teacher Edition provides embedded questions throughout the lesson material for the teacher to use to help guide students. When a student is having difficulty, intervention lessons and materials help teachers reteach the concept to the student. The material introduces instructional routines at the beginning of the material that stay consistent throughout the topics. Students discuss their strategies in daily “Share and Discuss” activities. For example, in Topic 1, Lesson 2, the students solve the problem “4 + 5 = 9” and then “5 + 4 = ?” Teachers ask, “What are you asked to find?” and “Can you write another addition fact with 4, 5, and 9?” Students have counters to help them; some are asked to 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. Materials provide information for the teacher to identify when fluency is being focused on or a part of the lesson. For example, Topic 2, Lesson 5 provides the following information in the “Math Background” section: “This lesson continues to develop fluency with recall and automaticity and reinforces the use of efficient strategies.” In Topic 7, Lesson 8, the Math Background reads: “In this lesson, students construct math stories involving real-word scenarios to further develop their understanding of, and fluency with, two-digit addition and subtraction. Students utilize precise mathematical language to generate a math problem and rely on what they have learned about addition and subtraction to solve the problem.” In Topic 14, Lesson 1, the focus is on formulating a plan and reasoning. The materials provide support for conducting fluency practice with students, including clear directions for how and when to conduct fluency activities.

The materials provide a year-long plan for building fluency, connected to the concept development and expectations of the grade level. The “Skills Trace” for grade 2 gives the teacher the TEKS information to look back and look ahead for each of the topics; it shows how the current TEKS build upon the previous TEKS and support future TEKS. There is also a “Scope and Sequence” in the “Content Guide.” This document has a section called “number sense,” which is introduced in kindergarten and then practiced through grade 5. In addition, the materials provide guidance for TEKS mastery by using the placement test at the start of the year. The materials also include a quick check and topic test at the end of each lesson and topic. 

The instructional material provides opportunities for students to practice fluency activities while developing conceptual understanding at progressively higher levels. Two ways the materials do this is by providing multiple strategies for students to use to complete their grade-level tasks and by providing students with multiple opportunities to practice and master solving grade-level tasks. For instance, students work on addition strategies in Topic 1 and subtraction strategies in Topic 2 before practicing adding two-digit numbers in Topic 6 and subtracting two-digit numbers in Topic 7. Topic 8 is when students practice three-digit addition and subtraction. In Topic 9, Lesson 2, a lesson on writing multiplication stories, the materials use counters for solving. The lesson provides a “Share and Discuss Solutions” section that encourages students to share strategies they used to solve the problem. The materials also include fluency activities; for example, students explain which numbers can be shown as two equal groups of cubes in Topic 11, Lesson 1, before working on conceptual understanding of identifying odd and even numbers. 

The materials include scaffolds and supports for teachers to differentiate fluency development for all learners. For example, in Topic 3, Lesson 7, the materials provide a quick check that includes exercises for prescribing differentiation. Topic 6, Lesson 2 uses an individual worksheet to determine if students need differentiated support in adding two-digit numbers. If the students have a difficult time grasping this concept, intervention activities have students practice. They use two different colors of cubes to form cube trains of tens, count the groups, place that number in the tens place, and place any leftover cubes in the ones place. In another activity, students count drawings of cubes in tens trains before just working with the standard numbers. Students ready for extension activities have several options, including the reading mat activities that go with the lesson and online two-digit addition games. The material provides extensions for students who exhibit mastery of fluency expectations of the grade level. For example, in Topic 9, Lesson 3, there is an extension for early finishers that asks students to find other ways to share six counters equally (1 group of 6, 2 groups of 3, 6 groups of 1). The materials also include reteaching materials for students struggling to meet fluency expectations of the grade. 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 include opportunities for students to develop and strengthen mathematical vocabulary throughout the lessons and activities. Evidence also shows the materials include embedded opportunities to develop and strengthen students’ mathematical vocabulary. 

Evidence includes but is not limited to: 

Lessons provide repeated opportunities for students to listen, speak, read, and write using the mathematical vocabulary within and across lessons. Each topic introduces vocabulary using a set of vocabulary cards for the current topic and revisits vocabulary from prior topics. Students look at the word on the front of the card and then write a sentence using the word on the back of the card. “Interactive Math Stories” appear at the beginning of each topic. Many of the daily lessons use problem-solving reading mats. For example, in Topic 6, Lesson 1, students read the vocabulary word regroup and then identify if regrouping is necessary. Students also discuss strategies of how to regroup. Math vocabulary is provided at the beginning of the lessons and is also embedded within the lessons; this gives students opportunities to develop and strengthen their mathematical vocabulary. In Topic 16, students write in a journal about expenses they might have if they had a pet. Vocabulary cards for the topic encourage them to fill in blanks in sentences and write their own sentences with words such as borrow, lend, spend, and save. Afterward, students read some word problems and do the math to determine how much money was spent or saved. The first lesson has them working with partners to determine if pictures are of people spending or saving. Following that, pairs share their answers with the class. 

The materials provide scaffolding to ensure students have opportunities to use math vocabulary in context. The lessons are all formatted similarly; they include vocabulary cards for each topic and daily opportunities to discuss solutions to assigned problems with partners and the class. The vocabulary terms are embedded in many sources, such as in reteach lessons, online games, and center games. The differentiated instruction lessons continue to concentrate on the vocabulary while working on the concept development. For example, the vocabulary terms for Topic 1—doubles, near doubles, addend, and sum—are a review of terms learned in grade 1. Topic 5 reviews the vocabulary terms mental math, tens digit, next ten, and difference, which continue to be used throughout the second-grade topics. The materials build from informal language to formal language, using pictorial models on the vocabulary cards as needed, as well as continued vocabulary use throughout the guided lessons and independent lessons. The materials also include and encourage classroom routines to support language development and the use of academic vocabulary within and across lessons. For example, in Topic 3, Lesson 1, prior to the lesson, the materials provide a weekly routine of cutting out and studying the word, then completing an activity on the back. The activity asks students to complete sentences and provides a sentence frame. For extended learning, students write their own sentences using each word. Topic 15 vocabulary cards include words such as bar, graph, and data. Students look at the words, fill in the blanks on the backs of the cards, and then write their own sentences using the words. The first lesson has the students using data and bar graphs to answer questions. Teachers ask the recommended questions, such as “What do you already know that can help you?” and “After you fill in the table, will it show the same information that is in the graph?” “Share and Discuss Solutions” provide an opportunity for students to use the same vocabulary that they learned on the vocabulary cards and that the teacher modeled for them.

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, including problems arising in everyday life, society, and the workplace, 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: 

There are many developmentally-appropriate opportunities for students to solve real-world problems in a wide variety of situations. The materials provide multiple opportunities for students to make sense of open-ended, real-world contexts involving mathematics. As students practice the problem-solving process in activities such as “Solve and Share,” “Math and Science Projects,” independent practice, and reteach lessons, they apply their math knowledge and skills using real-world information or in real-world situations. In Topic 7, Lesson 9, students solve a math problem in which they begin with a number of library books they have checked out, return some and check out some more, and then solve to find how many library books they now have checked out. Topic 8 begins by reviewing two-digit addition and subtraction as well as ones, tens, and hundreds places to activate prior knowledge. The topic then has an “Interactive Math Story” where students read about a group of kids doing three-digit addition without regrouping. The “Math Science Activity” for the topic talks about forest animals and what they eat; students make up their own three-digit addition or subtraction problems about the subject. Materials provide an example: “A tree has 485 leaves, and a deer eats 132 of them. How many leaves are left on the tree?” This nature theme continues in Topic 8, Lesson 6, where students learn about ants and create three-digit addition and subtraction problems about them. Topic 8, Lesson 10, also has a Math Science Activity, where squirrels are taking acorns from a tree. Students solve the given problems and create additional three-digit problems on this theme. In Lesson 5, Topic 11, students solve a problem about the number of songs a child has on a music player after the child purchases new songs. 

The materials provide opportunities for students to analyze data from another content area (e.g., science). Each topic begins with an Interactive Math Story and Math and Science Activities that are of interest to the students. Topic 3, Lesson 1 has students create hundreds, tens, and ones charts to practice place value. This lesson is a way to display data on a chart. Topic 2 has students using hundreds charts to help with skip counting. Topic 3 reviews hundreds charts before beginning the new topic. Topic 6 includes multiple opportunities for students to analyze data. The Topic 6 Interactive Math Story discusses characters on a trip. Students solve addition problems based on pictures of the different activities the characters participate in: playing by the river, climbing a mountain, swinging on a tire swing, and sleeping in sleeping bags. The Topic 6 Math and Science Project asks students to read and record the forecast for the day’s weather. Students collect the information for the day’s temperature, wind speed, and amounts of rain or snow. In the Topic 10 Math and Science Project, students classify matter by physical properties, including shape, relative mass, relative temperature, texture, flexibility, and whether the material is a solid or liquid. At the end of some lessons, a problem-solving reading activity provides a page of math data on the mat. In Topic 14, Lesson 1, students describe real-world activities they might do at 6:15 a.m. and 6:15 p.m. Students discuss other hours of the day, such as 10:15 a.m. versus 10:15 p.m.

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

The materials include some cited research throughout the curriculum that supports the design of teacher and student resources and provides some research-based guidance for instruction. The materials do not, however, provide an overall research-based justification for the design of the resources, nor do they provide a bibliography to verify that the materials are academically vetted or aligned to a Texas context. 

Evidence includes but is not limited to: 

The materials do not include a description of the program’s design that cites research about the design of instructional materials and how students learn mathematics. 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. However, it does not cite research about the design of the instructional materials.

The “Lesson Overview,” “Math Background,” and “Essential Understanding” sections within the Teacher Edition (TE) provide guidance for instruction to help enrich an educator’s understanding of the mathematical concepts and the validity of the publisher’s recommended approach. The “Focus on Content” section in the TE provides information regarding different math concepts and skills taught within the topic. Two of the lessons provide research-based guidance for instruction that enriches educator understanding of the mathematical concepts and validity of the publisher’s recommended approach. Topic 7, Lesson 1 quotes the National Research Council (2001) in a discussion about understanding subtraction and the purpose of mathematical procedures such as regrouping. The same source is cited in Topic 7, Lesson 5, addressing how subtraction algorithms require more time and support to learn than addition algorithms. In Topic 15, Lesson 5, the materials state, “Research says that graph comprehension is based on knowledge of the components based on the graph and facility in relating information back to its context” (Friel, Curcio, & Bright 2001). 

Cited research is not particularly current, nor is bibliographical information available to determine relevancy for this curriculum. The only source quoted is the National Research Council (2001). It is not possible to determine if the research is academically vetted or demographically aligned with the Texas-specific context.

The materials do not provide a bibliography.

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 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 include a transferable problem-solving model grounded in the TEKS and encourages students to analyze, plan, solve, justify, and evaluate throughout the curriculum. The materials also prompt students to apply a transferrable problem-solving model and provide guidance for teachers to support student reflection on their problem-solving approach. 

Evidence includes but is not limited to: 

The materials support the development and practice of a consistent problem-solving model across the topics. The problem-solving model, introduced in Topic 1, Lesson 5, is grounded in the mathematical process standards of analyzing the information, making a plan, forming a strategy, solving the problem, justifying the solution, and evaluating the problem-solving process. Topic 2, Lesson 6, begins by guiding students to analyze the word problem and plan for how to solve the problem by using counters and a part-part-whole model. Then, students solve, justify, and evaluate the problem, continuing to use counters and the part-part-whole model. Consistent opportunities to practice with the model can be seen in Topic 5, Lesson 8. Throughout the lesson, students analyze, solve, justify, and evaluate multi-step problems related to adding and subtracting two-digit numbers. For students to be successful, they must first analyze the problem and identify the hidden question they need to solve. Then, they must solve each step of the problem by writing and solving addition and subtraction number sentences. Last, students evaluate their solutions for each step in order to know they are correct. In Topic 10, Lesson 5, students analyze different ways to make 30 cents with guided instruction. The lesson instructs students to make a plan to solve the problem by making a chart with tally marks to keep track of the different coins they use to make 30 cents; to solve the problem by making the actual chart; and then to evaluate their problem by explaining their chart. In Topic 12, Lesson 7, students learn how to divide an irregular shape into equal parts to represent a solution to a word problem. They begin by analyzing; the teacher asks, “Why must there be 4 equal parts?” They then formulate a plan; the teacher asks, “What makes this shape different from most of the shapes we've worked with before when making equal parts?” Finally, the students solve, justify, and evaluate; the teacher asks, “How do you know the parts in each shape are equal? How are the parts in Way 1 the same as the parts in Way 2? How are they different?”

The materials use prompts in teacher and student materials to encourage the use of the problem-solving model. Topic 5, Lesson 8, explains that students need to solve the hidden question in the problem first. For guided practice and independent practice, students write number sentences to solve both parts of the given math problems. In Topic 7, Lesson 9, the problem-solving steps of “Analyze, Plan, Solve, Justify, and Evaluate” are written in red letters at the top of each step of an example problem. As students use the problem-solving model, teachers ask questions such as, “How do you solve a two-step problem? How can you make sure the answer makes sense?” Topic 13, Lesson 9 directs students first to formulate a plan. In the “Solve and Share” problem, students use reasoning to determine what new shapes can be made by cutting apart an existing shape. Teachers then ask, “What are you asked to find out? What tools do you have to help you solve the problem?” Teachers give hints as needed, asking, “Can modeling help you? Can you make more than one new shape?” Finally, students share and discuss solutions, summarize, and generalize.

The materials provide areas for students to reflect on their approach to their problem solving. Topic 7 includes a problem that instructs students to discover what is wrong with the two-digit subtraction problem, rework the problem to find the correct answer, and then explain how they fixed the problem. In Topic 11, Lesson 6, the teacher asks, “What strategies could you use to help solve this problem? What can you write or draw on your work mat?” In Topic 14, Lesson 13, the lesson includes questions for students to analyze and formulate a plan; for example, “How is measuring one of these paths different than other objects you have measured? How can you find the length of one of these paths?” Suggested questions to help students solve and justify include “Why do you need to find the two parts of the blue and red paths? How do you line up the ruler? What are the lengths of the two parts of the blue path? The red path? What is the length of the blue path? The red path?” Finally, students evaluate with the assistance of questions such as “Which path is longer? How do you know? How can you find how much longer the red path is?” In Topic 14, Lesson 4, a lesson demonstrates the problem-solving model; a “Do You Understand?” section asks, “If the areas of two shapes are the same, does the distance around have to be the same?” 

The materials provide teacher prompts and questions to use in each lesson when guiding students to reflect on problem-solving. The Solve and Share portion of Topic 4, Lesson 1 has a “Build Understanding” section, which provides questions for the teacher to ask the students, such as “What are you asked to find? What do you already know about the number line?” In Topic 6, Lesson 8, the materials provide questions to build understanding, including “What are you asked to find out? What do you already know about Kim?” Teachers give hints as needed, such as “How can you use the strip diagram to show this problem? What will be the addends in your number sentence?” Students then have an opportunity to share and discuss solutions, and finally to summarize and generalize. Another example is in Topic 14, Lesson 8. Teacher prompts include “What are you asked to do in this problem? How can we use cubes to measure an object?” Teachers encourage students to analyze the given problem.

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

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. Evidence provides opportunities for students to select and use technology (e.g., calculator, graphing program, virtual tools) as appropriate for the concept development and grade. The material provides some teacher guidance on tools that are appropriate and efficient for the task. There is not, however, any specific teacher guidance on the reasons for using tools or which tools would be the best for any specific problems. 

Evidence includes but is not limited to: 

The instructional materials provide many opportunities for students to learn to use grade-appropriate tools for solving tasks and understanding concepts. Throughout the lessons, materials provide pictorial models, and students also have opportunities to use counting cubes, counters, hundreds charts, etc. Topic 1, Lesson 1 introduces connecting cubes. In Topic 1, Lesson 4, students use ten-frames and counters to help show how thinking about 10 helps you solve 9+3. Topic 2, Lesson 6 brings in strip diagrams. In Topic 3, Lesson 1, students use place value mats and place value blocks to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value. Topic 3, Lesson 3, introduces number cubes. In Topic 4, Lesson 5, students use the hundreds chart for the first time in the second-grade curriculum. In Topic 7, Lesson 1, students use cubes to show how they can use tens and ones to solve 23 - 6. Topic 12, Lesson 1 introduces pattern blocks for fractions. Topic 17, Lesson 1 asks students to explain all the tools they could use to show a number. 

The materials provide opportunities for students to use and select grade-appropriate technology. There is a wide selection of technological tools, games, and online assessments for the students to use on the computer. The assessments all have a “listen” option for those students who need help reading. Some 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 online games help the students practice the skills they are currently learning. The materials provide tasks that allow students to select from a variety of tools. For example, in Topic 1, Lesson 4, students select tools and techniques to model addition. In Topic 6, students complete a digital math tool activity to reinforce the lesson content or prerequisite content using a suite of digital math tools. 

Students have opportunities to use a variety of tools throughout the lessons, particularly for the “Solve and Share” problems where digital tools are always available. Through this and the discussion and sharing of solutions, students gain experience in deciding which tools work best for different situations and see 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 that can encourage further discussion as needed.

The materials use many tools and recommend specific tools for use in each lesson. There is no guidance provided to the teacher about why the designated tool is being used or which tool might be more appropriate than any other tool. The materials do not specifically provide explanations about which tools are appropriate and efficient for a task.

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

Materials support teachers in understanding how appropriate strategies can be applied and how to guide students to more efficient strategies. They provide student prompts for appropriate techniques to use to solve problems. Evidence also shows that the material provides opportunities for students to solve problems using multiple appropriate strategies.

Evidence includes but is not limited to: 

Materials prompt students to select a technique (mental math, estimation, number sense, generalization, or abstraction) appropriate for the grade level and the given task. In Topic 1, Lesson 4, students make ten to add; in Topic 1, Lesson 5, students write to explain their adding. In Topic 2, Lesson 1, students write and solve a few problems using their preferred techniques. Topic 2, Lesson 5, the “Visual Learning Bridge” section, reminds students to “think of strategies to help (them) practice the facts.” For Topic 8, Lesson 1, the materials provide guidance for the teacher, such as, “Students discuss different strategies to solve a word problem involving three-digit numbers. Students continue to explore different strategies as they add any way they choose to solve addition problems, including word problems.”

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. For example, in Topic 4, materials guide teachers to help students add by using doubles, near doubles, and using the property that states that the order of the addends, if changed, does not change the sum (commutative property of addition). The “Select and Use Tools” section of the Teacher Edition states: “For Topic 5, Lesson 3, students select from models, mental math, and ten-frames as they develop different strategies for adding tens and ones.” 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. The Math Background section of Topic 5, Lesson 1, states, “Building on their understanding of place value and the practice of making tens to add, students use mental math strategies to add two-digit numbers. They count on by tens or add the tens to solve addition sentences containing a two-digit number in multiples of ten and a one-digit number.” In Topic 6, Focus on Process details two of the process standards: students can use a strip diagram to help them formulate a plan; the topic also “builds on students’ understanding of addition and subtraction as they use various representations such as connecting cubes and addition frames in order to add one- and two-digit numbers.” Teachers are also informed of how students use a number line to solve addition problems. The Focus on Content section explains how using place value helps students add and subtract two-digit numbers using the traditional algorithm. It also explains how using models to show regrouping can assist students in understanding “what happens when you add to a two-digit number and need to regroup.” 

The materials support students in learning multiple appropriate strategies to solve problems. Topic 1 reviews the following different addition strategies: using doubles and near doubles, adding in any order, adding three numbers, and making ten to add. In Topic 2, students learn to subtract by making ten to subtract, connecting addition and subtraction, thinking addition to ten to subtract, and addition and subtraction facts. In Topic 8, students use tools to help solve problems involving adding and subtracting three-digit numbers. In Topic 8, Lessons 1 and 5, students can use place value blocks to explore different ways to add and subtract. In Topic 8, Lessons 3 and 7, students use place value blocks and the standard algorithm together. The use of place value blocks in this way helps students visualize the regrouping process and strengthen their understanding of both algorithms and place value. They can use objective and pictorial models or join and separate connecting cubes to model a problem.

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 learn from solving problems. They can make sense of mathematics and productively struggle. The materials support students in understanding that there can be multiple ways to solve problems and complete tasks. The curriculum helps teachers to ensure students are given many opportunities to share their thinking in problem solving. 

Evidence includes but is not limited to: 

The formatting of the lessons ensures that all students are participating in solving problems and making sense of mathematics. The “Solve and Share” sections foster a mathematical community that, with the teacher’s guidance, can ensure all students participate and engage as mathematical thinkers. Students solve the problem in any way they choose. Teachers then choose some of the students to explain their solution to the class. For example, in Topic 2, Lesson 2, materials ask, “How can you show 12? Write addition and subtraction sentences for 12.” This question 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 ultimately solve the problem. The Solve and Share for Topic 7, Lesson 4, asks students to complete a double-digit subtraction problem and to determine if they need to regroup or not. A guiding question is provided to the teacher to help foster the students’ thinking: “What tools do you have to help you solve the problem?” In Topic 10, Lesson 4, students share their solutions and discuss them with the teacher. Questions allow all students to participate; for example, “Look at the front and the back of the dollar bill. What is next to it? What is the value of the bill? What is the value of the coin? What is the value of the set of coins shown? Could you trade these for a dollar bill? Look at the dollar sign. How would you write this symbol? Point to the decimal point. What does this separate? How would you write 88 cents in this way?” The materials use drawings of children from all races and genders to give guidance in the Student Edition. This art 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. For example, in Topic 1, Lesson 1, students add using doubles and near doubles. Later in Topic 1, students add by changing the order of addends and by making 10. In Topic 2, Lesson 3, students use addition facts to solve a subtraction problem mentally. In Topic 9, Lesson 6, students work with partners to solve the given problem. They are told that there are 12 students in the classroom and that they sit in equal groups at four different tables. They must state how many students are at each table and how they know. Students learn multiple ways to solve problems, such as repeated addition, repeated subtraction, using strip diagrams, and writing number sentences. Topic 10 gives teachers the following questions to guide students in solving problems: “How does this problem connect to previous ones? What is my plan? How can I use tools? How can I use number sense? How can I communicate and represent my thinking? How can I organize and record my information? How can I explain my work? How can I justify my answer?” 

The materials provide many problems that encourage divergent solution strategies. Each lesson has a Solve and Share problem. Materials introduce a lesson by giving students problems that embed some important math ideas. Students solve the problem in any way they choose. Students share their strategies almost daily as a part of the initial Solve and Share problem, during guided work, and during intervention or center work. Another example of this is in Topic 6, Lesson 8, where pairs of students receive 40 connecting cubes and a two-part mat. They must solve the given problem using some type of model and a number sentence. The problem states, “Kim puts 15 toys in the toy box. Then she puts 17 more toys in the toy box. How many toys are in the toy box in all?” In Topic 7, Lesson 9, teachers guide students to formulate a plan and connect ideas: “What is the first step? What is the second step? How do you solve a two-step problem? How do you solve the first step? How many birds did Mia see before some flew away? How do you solve the second step? How many birds are left? How can you make sure the answer makes sense?” The materials also provide suggestions for sequencing the discussion of student strategies for solving the problem. For example, the teacher guide throughout the curriculum provides mathematical misconceptions and ways to address them. In Topic 12, Lesson 4, the materials point out: “Some students may write four fours to show a whole. Remind students to use fourths for the whole.”

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

Materials include opportunities for students to communicate mathematical ideas and solutions using multiple representations. The materials also provide prompts to teachers to help guide students in their communication of mathematical ideas and reasoning in multiple representations, including writing and mathematical vocabulary, as appropriate for the task.

Evidence includes but is not limited to: 

The materials include opportunities for students to explain mathematical ideas in the “Solve and Share” section at the beginning of each lesson. For example, in Topic 3, Lesson 2, students work in pairs and use flats, tens rods, and ones cubes to find different ways to write 231, using their knowledge of the value each digit represents. In Topic 5, Lesson 1, students explain a strategy to find the sum of an addition problem involving 44 and 20. Students explain how their mental math strategy can help find the sum when adding tens to a two-digit number. In Topic 5, Lesson 3, students show how they would solve an addition word problem. The teacher gives students base-ten blocks to solve the problem and find their answer. In Topic 9, Lesson 5, students use counters, pictures, and stories to communicate a division problem. In Topic 11, Lesson 4, students solve a missing-part problem any way they choose. They then explain their work to the class. 

Materials provide ways for students to communicate mathematical ideas and reasoning using multiple representations, including writing and the use of 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. The materials also provide the teacher with prompts to encourage students to communicate mathematical ideas, both in written and oral form. In Topic 1, Lesson 5, students write to explain the make a ten strategy to add 7 + 9. Teacher prompts are available to support students’ writing with multiple representations, such as using words and drawings. Students then orally share their explanations. In Topic 4, in the “Interactive Math Story,” the teacher reads a story aloud and asks students to read some of the seat numbers, asking, “What seat numbers can you read in the row where Polly and her family are sitting?” Students then point to the seat numbers in Polly’s row as they count aloud. Students are given Interactive Math Story books, circle the number in the question, and color the corresponding seat number. In Topic 5, Lesson 2, students find the missing numbers that make the next ten. Students explain their thinking in writing. Students also make up and solve subtraction problems about the supply of fresh water on a ship. In Topic 16, Lesson 6, the teacher ensures that students know the difference between consumers and producers. After the students watch the “Visual Learning” that introduces consumers and producers, the class has a guided lesson. The teacher asks, “What is the difference between a producer and a consumer? Who is a producer in the problem? Explain. Who is a consumer in the problem? Explain.”

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 with partners, small groups, and the whole class. There is evidence that the materials integrate discussion throughout to support students’ development of content knowledge and skills as appropriate for the concept and grade level. 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 (e.g., whole group, small group, peer-to-peer). The “Solve and Share” problems provide opportunities for students to share their problem solving. Students solve the problems in any way they choose. This activity can be done in small groups or whole groups, which provides opportunities for students to share and discuss with others. 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. The problem-solving reading activities can be completed in large or small groups; they give the students several opportunities to discuss their thinking as they complete the mat and the activities that go along with the mat. A good example of this is Topic 15, Lesson 6. The Solve and Share problem at the beginning of the lesson has students working in pairs to create a graph using counters, cubes, and place value ones cubes. Then they write a problem based on the graph they created. Teachers select some of the pairs with different problems to share their solutions with the class. The whole class then does guided practice together, using bar graphs and provided problems. After the teacher assesses their independent practice, students either work with the teacher in a small group for intervention practice with picture graphs on favorite sports or with small groups working on reading mats. 

Materials integrate discussion throughout to support students’ development of content knowledge and skills as appropriate for the concept and grade level. They include opportunities for discussion in all phases of concept and skill development (i.e., beginning, middle, and end). The Solve and Share sections in the Student Editions introduce a lesson by giving students problems in which some important math ideas are embedded. Students solve the problem in any way they choose. 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. For example, Topic 1, Lesson 4 starts with pairs of students using ten-frames to add 9 + 3. Some of the pairs then discuss their solutions with the class. During guided practice in the middle of this lesson on making 10 to add, 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 in centers. In another example, in Topic 8, Lesson 3, the students begin with the Solve and Share problem. Students use place value blocks to model a word problem involving three-digit addition. Teachers ask, “What are you asked to find out? What tools do you have to help you solve the problem?” In the middle of the lesson, students answer questions and discuss: “Which two numbers are you adding? Do you need to regroup the ones? How do you know? Do you need to regroup the tens as one hundred? Why?” Then, they complete guided questions. At the end of the lesson, students work on an intervention activity to discuss questions posed to them. 

The materials provide guidance to the teachers on how to structure discussion that is appropriate for the grade level. The Teacher Edition (TE) provides guiding questions with the Solve and Shares as well as within the problem-solving reading activity guides; these include questions to ask the students to discuss with their peers. For each Solve and Share problem, there are “Build Understanding” and “Give Hints as Needed” sections in the TE. Questions can be found within the center games, which teachers can use for guided small groups. For example, in Topic 7, Lesson 7, before students discuss and share solutions, the materials ask teachers to give hints as needed to make sure each student can share and discuss their solution with the class since they can be called upon to share. Another example of this is found in Topic 10, Lesson 3. In this lesson, students are counting coins. The teacher prompts the students, “What is the number of coins in the first bank? The second? What do you have to do before you can see which set is greater?”

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

The instructional materials provide opportunities for students to 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 mathematical ideas using multiple representations. The “Solve and Shares” provide a strategy idea to be used to help guide a student in solving the problem; however, students are allowed to solve the problems how they choose. For example, in Topic 3, Lesson 6, students use clues to identify a four-digit number. They justify their answer by ensuring their number is the only choice that matches each of the clues. In Topic 6, Lesson 8, students use connecting cubes, strip diagrams, and number sentences to solve and justify an addition problem to their partners. They can then share and discuss their solution with the class. For Topic 10, Lesson 4, students show a way they can make 100 cents if they are given one half dollar, five quarters, five dimes, and five nickels. Students discuss and share the coins they used to make 100 cents and why they chose those specific coins. 

The materials assist teachers in facilitating students to construct arguments using grade-level-appropriate mathematical ideas. In Topic 9, Lesson 6, teacher prompts facilitate students’ arguments. The teacher asks, “What strategy will you use?” (Draw a picture) “Why are there 3 boxes in the strip diagram?” For Topic 10, Lesson 4, there are steps and guiding questions to use with the Solve and Share. The steps are to pose the problem, build understanding, give hints as needed, share and discuss solutions, summarize and generalize, and then extend the lesson with a new problem or question. Questions provide hints for the students and include, “Can you make 100 cents using only one of the coins you have been given? Is there only one way to make 100 cents using coins?” In Topic 13, Lesson 9, students use lines to cut shapes into smaller shapes. Teacher prompts help students to justify their answers: “Is there another line you can draw to cut the square into two triangles? What is the other line?” The materials also provide prompts for the teacher to assist students when constructing arguments. For example, Topic 12, Lesson 1, “Construct Arguments,” states that when students have finished this problem, the teacher should use the answer choices as non-examples to further develop the concept: “Explain why the other answer choices do not show halves.” In Topic 14, Lesson 12, the materials provide questions to help elicit different types of responses from students as they present their arguments: “Is the desk actually more than one meter or less than one meter long? Why do we say that it is about one meter? Are there any other units you have learned how to use to measure something like the length of a desk? What rule can we make about measuring with two different sized units?”

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 do not provide teacher tips to ensure consistent and accurate administration of diagnostic testing, nor do they provide student tracking systems to show personal growth or areas on which to focus. They include diagnostic tools to measure all content and process skills for the grade level, as outlined in the TEKS and Mathematical Process Standards. Materials do not include tools for students to track their own progress and growth.

Evidence includes but is not limited to: 

The materials include frequent assessment opportunities to closely monitor student progress to know when extra support is needed. These tools are formal and informal and grade-level appropriate. There are two grade-level diagnostic test forms: Form A and Form B. Materials also include digital assessments that are taken online and auto-scored. Assessments can be customized online; teachers can upload a district assessment. There is an online placement test at the beginning of the grade 2 instructional materials, including 20 questions. 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. 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 report for the teacher to assess student mastery of the skill. The Teacher Edition explains how to grade the quick checks and sort students into groups that need intervention, are on level, or are advanced. For example, in Topic 7, Lesson 3, the teacher uses independent practice as a quick check. Students who score 0–3 points go to interventions, 4 points are on level, and 5 points are advanced. Follow-on activities are provided to challenge each group on the topic of the lesson. Within the “Teacher Resources” section, there are 12 basic fact timed tests. Each of these includes a mixture of addition and subtraction facts. The materials include a “Math Diagnosis and Intervention System” (MDIS). 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. Further teacher support for the formal and informal assessments in the program is provided at the publisher training link, 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 to support consistent and accurate administration of the diagnostic tools. Before the assessment, the materials guide teachers to ask questions, direct students to answer the essential question (verbally or in writing), and direct students to give examples that support their answers. Teachers make these elements explicit when discussing students’ answers; the materials inform them of what to look for when observing students. In Topic 2, for example, the materials explain: “Use ten to help you subtract. Example: Find 14 - 5. Start with 5 and add 5 to get to 10. Next, add 4 more to make 14. Add 4 + 5 = 9. This means that 14 - 5 = 9.” The independent practice section is used as a diagnostic check to determine mastery of the concept taught in that lesson. In Topic 14, Lesson 13, the teacher manual explains how to grade the quick check and how the questions the students answer correctly determine which category (intervention 0–3, on level 4, or advanced 5) the student is placed in for the remainder of the lesson. 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.

The materials do not include tools for students to track their own progress and growth. The materials include informal and formal assessments aligned to the grade-level TEKS and Mathematical Process Standards. The materials include diagnostic tools to measure all content and process skills for the grade level, as outlined in the grade-level TEKS. For example, at the start of the year, there is a placement test and diagnostic test measuring all content skills for that grade level. There are also TEKS correlations provided from prior grade levels to assist teachers in developing plans for additional support. 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 MDIS to track student mastery of skills.

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 measures of student progress appropriate to the developmental level. The diagnostic tools yield meaningful information for teachers to use when planning instruction and differentiation. The materials provide a variety of resources and teacher guidance on how to leverage different activities to respond to student data. The materials provide administrator-level data and guidance.

Evidence includes but is not limited to: 

The instructional materials give guidance and direction to teachers to respond to individual students’ needs in all areas of mathematics based on measures of student progress that are developmentally appropriate. The “Texas Assessment Resources” guide explains that comprehensive test reports are available to use with each of the end-of-course tests. In the report, each item tested is referenced to 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 that a student has not mastered, they 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 25 on the practice test, which asks a student to read a chart about money and decide which statement is not true, the “Practice Test Report” shows that this question tests TEKS 2.5B, and the teacher can use MDIS lessons A68 to have the student review and practice this skill. Lesson A68 includes exercises to practice ways to show the same amount (in dollars and cents). At the end of each lesson, there is an “Assess and Differentiate” section. The quick check assessment for each lesson (taken online or paper and pencil) separates students into intervention, on-level, and advanced groups for the remainder of the lesson. For example, in Topic 15, Lesson 1, students who get three or fewer points on the quick check receive additional intervention. They work in a small group to create a bar graph about how they all get to school. They then work independently to read data from a bar graph on a page. 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. In the end-of-course tests, a comprehensive test report provides information for each student once an assessment has been taken. Each item in this assessment is referenced to a TEKS standard and 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 students who have not yet reached proficiency with additional reviews and practice. 

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 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, the Practice Test Report in the Texas Assessment Resource guide instructs the teacher to use the MDIS lessons A61 and A64–A67 for a student who did not answer test questions 5, 7, 10, and 11 correctly, which cover TEKS 2.5A. Online reports, based on the online activities students complete, allow teachers to see individual and class views of progress. TEKS reports 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 that students need to review and practice. The lessons and activities within the instructional materials list the corresponding TEKS, which help the teacher 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 centers’ activities. If a student has mastered a skill, a teacher can look in the “Scope and Sequence” within the “Content Guide” to find the related higher-level skill the students work on next so that the student can begin working toward those skills. A “Leveled Assignment Guide,” in the Teacher Edition, provides intervention, on-level, and advanced assignments. The materials have a data section that produces graphs to show the results of diagnostic tools. The mastery section of data provides links to additional content that assists struggling students. The Texas Assessment Resources 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 materials also include a quick check at the end of each lesson. In the independent practice handout, the checkmark indicates exercises for prescribing differentiation on the next page in the student’s workbook. In Topic 9, Lesson 1, Exercises 6 and 10 are worth one point, and Exercise 13 is worth up to three points. The teachers use the quick check data on the previous page to prescribe differential 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.

Materials provide a variety of resources and teacher guidance on how to leverage different activities to respond to student data. The materials provide a variety of suggestions and activities for teachers to use to address the results of student assessments. In the “Analysis for Diagnosis and Intervention” section, each item or problem is correlated with a grade-level TEKS; most have an intervention system with additional support for students who need it. The materials provide intervention that can be auto-assigned at the online resources. Customizable digital intervention is available through the use of online resources, eText pages, printable PDFs, animations, online tools, and math games. 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 measuring area, which 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 7, Lesson 7, the quick check suggests that the teacher use connecting cubes and strip diagrams to review the concept with those who are still struggling. Once they grasp the concept with the manipulatives, an individual worksheet is available to solve the problems with drawings rather than concrete items. 

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 materials include routine and systematic progress monitoring opportunities that accurately measure and track student progress. These assessments provide ways to track student progress through a variety of assessment types (observational, anecdotal, formal, and informal) and question types (short answer, multiple-choice, and open-ended/griddable). 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. The “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 variety of grade-appropriate question formats. In the second grade tests, there are word problems, multiple-choice problems, gridabble (open-ended) problems, and short-answer problems. 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 have not 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 give the ability for assessments to be customized. Teachers can customize assessments or upload district-created or teacher-created assessments for data tracking. Online data tracking of the instructional materials allows teachers to monitor student progress closely and 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 report 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 “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 materials provide guidance for scaffolding instruction and differentiating activities based on targeted areas. Each lesson provides a “Reteach to Build Understanding” worksheet that provides additional problems to address the content in the lesson. Each topic also gives “End of Topic” reteach ideas related to the topic, number sense, and mixed problem-solving. The material provides resources to support additional practices in the teacher book using differentiated pages. These pages provide resources for intervention, on-level, and advanced learners. 

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. The reading level of the challenge questions is on grade level and accessible to the students with little assistance. The Teacher Edition also provides suggestions for early finishers in each lesson. For example, in Topic 6, Lesson 7, the teacher is instructed to have early finishers pick seven other numbers, three from one list and four from another. This practice helps students continue practicing the lesson goal of making 10. 

Students can expand their thinking in most of the student lessons, including via the Today’s Challenge, “Extend Your Thinking,” and “Extensions for Early Finishers” sections found throughout each topic and lesson. A wide variety of activities, online games, and center games allow for concept practice for all students. In Topic 3, students point to a number on the activity page. They say the number, air write the number, and ask their partner to tap on the place value blocks to show the number. The partner says the number in expanded form. Then, the number is covered with a square or counter, and it is the other player’s turn. The game continues until all numbers are covered. A center game in Topic 14, Lesson 2, has students use number tiles to create times on a clock and then determine an activity the student might do at that time. There are also “Reading” or “Math Science” activities to show real-world applications of the content. One example is in Topic 13, Lesson 7, where students have an extension math-and-science activity on creating shapes. 

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.”

Online resources can auto-assign differentiation based on the results of online Topic Tests and online Benchmark Tests. These include both for students who struggle to master content and for 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 guidance to support the teacher in meeting all students’ diverse learning needs, specifically addressing teaching approaches, instructional strategies, and flexible settings utilized to support the mastery of content. Materials support developmentally appropriate instructional strategies, 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 different instructional approaches and teaching strategies to meet students’ needs. 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. In each topic, there are visual learning components, games, online activities, independent work, guided work, and lessons that can be used for small group or whole group. For example, in Topic 3, Lesson 1, the problem-based learning instruction involves repetitive interaction of the learner with the content. Students work in pairs and consider place value to find a way to show 125 with place value blocks. Students build their understanding by comprehending that there are different sizes with different numbers of cubes. Teachers provide help as needed through hints such as “How many cubes make up a rod?” and “How many rods make up the largest block?” Students share and discuss solutions. In another example, the “Solve and Share” for Topic 12, Lesson 4, allows for a small group or whole group lesson. It directs students to fold a paper strip into eighths and then asks how many eighths should be colored to make the whole paper blue. Students fill in the blank on the page, stating that eight-eighths are blue; they read the sentence “One whole is blue,” connecting the idea that eight-eighths is a whole. 

The materials support the use of a variety of instructional strategies to support delivery and guide teachers toward appropriate teaching strategies. The Teacher Edition (TE) 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 that are above level. For example, in Topic 6, Lesson 7, the lesson starts with partners attempting to add a series of two-digit numbers. The teacher asks questions to guide their thinking. Advanced students can choose seven other numbers to try to add. The class completes a guided learning section with the teacher. After this, the students complete some independent practice on adding two-digit numbers. The teacher then checks this worksheet. Students who are on-level or those who mastered the content can then do a problem-solving reading activity or online math activities or games. Those who struggle can do an intervention activity with the teacher, going back to single-digit addition. 

Materials support flexible grouping. 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, the Topic 11, Lesson 1 Solve and Share shows a workmat with the numbers 1–20 listed below. Students use cubes to make the numbers and see which groups of cubes they can break into two equal groups. If the number can be broken into two equal groups, the students color that number on the chart. The framework of each lesson in the TE starts with a guided activity followed by independent work that is assessed for content mastery. Those 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. For example, in Topic 14, Lesson 3, students working in pairs first estimate and then use manipulative squares to cover a rectangle on their worksheets. The teacher then walks them all through the guided practice, estimating squares that fit in different rectangles. Students then complete an independent sheet to assess their comprehension. Those who need intervention can do another activity with the teacher using graph paper and a reteach worksheet. Those who grasped the concept can do a reading mat activity or online math practice and math games. The teacher assigns homework problems based on the student’s level of understanding

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. They also include an “ELPS Toolkit” that offers research-based scaffolds that are 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 ELs at the 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 through the “Visual Learning Animation Plus,” “Visual Learning Bridge,” animated glossary, and visual learning in exercises. At the beginning of each lesson, guidance tells the teacher which ELPS is 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?” For example, in Topic 4, Lesson 1, the ELPS focus on learning strategies (1a) and speaking (3b). For Beginning ELs, the teacher circles 99 and points to 98 and 100; students say before or after as the numbers are pointed to using the following sentence frame: “98 comes...100.” Intermediate ELs define what between means as the teacher, after circling 99, asks, “What number comes before 99? What number comes after 99? What number comes between 96 and 98?” Advanced ELs take turns asking a partner questions about the numbers on the number line. Advanced High ELs explain how they would extend the number line to the left and use the word before. The student then repeats the activity, extending the line to the right, and using the word after. In Topic 7, Lesson 3, materials include prompts for a variety of proficiency levels. For Beginning ELs, teachers ask, “Do we need to regroup? (Yes) What number goes in the box above the 3? (2) What number goes in the box above the 1? (11).” For Intermediate ELs, teachers point to the three and ask, “When we regroup, the number in the tens box is how many less than the tens digit? (1).” Teachers point to the one and ask, “The number in the ones box is how many more than the ones digit? (10).” For Advanced ELs, teachers ask, “How does the number in the ones box compare to the ones digit? (It is 10 more).” Advanced High ELs solve the problem. A volunteer explains why students write an 11 in the box above the one. Another example of this is in Topic 9, Lesson 4, where the ELPS focuses on teaching ELs the concept of division by taking away equal groups. Students receive the problem 14 divided by 2. Beginning ELs receive 14 counters and put them into groups of two. The teacher takes away one group at a time and asks them how many groups they took away. Intermediate ELs do the same activity and then explain why they were taking away groups of two. Advanced ELs explain how to use repeated subtraction to solve the problem. Advanced High ELs draw a 0-to-14 number line and then hop back seven times. Students discuss how this is the same as repeated subtraction. 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, including modeling thinking aloud, 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. Topic 6, Lesson 1, suggests that Advanced High ELs solve the problems one at a time. The teacher divides students into groups of four and instructs two students in each group to choose and solve problems that do not require regrouping and explain why regrouping is not needed. Then the teacher asks the other two students in each group to choose problems that involve regrouping and explain why it is required. In Topic 8, Lesson 8, teachers model writing a problem on the board: “New gym holds 600 people. Only 120 tickets left for the state basketball playoff game.” The students use demonstration sentence frames: “I can subtract...from...to find how many tickets have been sold.” Another example is in Topic 12, Lesson 2, where students review the concept of a fraction. The students are broken into their capability groups. Two boys and two girls are brought to the front of the room. Beginning ELs are asked how many are in the group (4); after one person sits down, they are asked which fraction is sitting (¼). Intermediate ELs are asked how many girls there are (2) and then what fraction of the group is girls (½). Advanced ELs explain to a partner how they can use the group of students to depict ½ and ¼. Advanced High ELs add two more girls and two more boys. They then explain to a partner how they can use the group to display ¼ and ⅛, and what the difference is between the two. 

The materials encourage the strategic use of the students’ 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 teachers can use and that are effective with ELs. 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 and vertically align instruction that builds year-to-year, within and between the lessons and the grade levels. 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 116 days, with an additional 10 days for the “Step Up to Grade 3” 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 grade levels K–5 and the grade levels in which they are introduced, practiced, and applied. For instance, under “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,” which shows how topics align, both directly and indirectly, to topics outlined for students in preceding and subsequent topics. For example, in grade 1, Topic 11, under “Looking Back TEKS 1.4A,” the materials state that students describe the relationship between pennies and nickels. In grade 2, Topic 10, the materials state that with TEKS 2.5A, students compare the total value of two collections of coins. In grade 3, Topic 1, students count money.

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. “Online Games” can cover more than one lesson concept and are found within many topic lessons. For Topic 5, Lesson 4, the “Math Tools Activity” can be used with both that lesson and the preceding one. The online game “Add It—2-Digit Numbers” is included in many lessons in Topic 8, including Lessons 3, 4, and 7. “Center Games” also provide opportunities for students to use and practice newly taught skills as well as previously taught skills. For the Center Games for Topic 7, Lesson 6, students use number lines to count the spaces after they roll a die. The first page shows number lines with all the numbers listed 1–89. The second page shows number lines for the numbers 9–92; however, not all the numbers are included. Instructions for the games tell students to count forward and backward. The Skills Trace section details how the materials build upon previously taught content and prepare for future content. For Topic 8, it shows how TEKS 2.4B (use models or a number line to add up to two-digit numbers) was addressed in Topics 6 and 7 and how this leads into TEKS 2.4C (use strategies based on place value to add three-digit numbers) in Topic 8. For the “Looking Ahead” for grade 2, Topic 8, it shows that these TEKS practiced support the TEKS addressed in grade 3, Topic 3, such as TEKS 3.4A (use models to add whole numbers within 1,000). “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 TEKS-aligned scope and sequencing, which 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,” and “Skills Trace” for each grade 2 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” gives guidance for 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 resource 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 2.4A. The Big Idea that matches this TEKS is also detailed. Essential Understandings are provided for Lessons 1 through 4 in Topic 1, in which this TEKS is taught. Focus on Process provides detailed information for the two process standards featured in Topic 1, “Formulate a Plan” and “Communication.” Focus on Content provides information on using doubles, using near doubles, adding in any order, and adding three numbers.

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 116 days. Additional time may be spent on review, remediation, differentiation, and assessment as needed. “Step Up to Grade 3” includes an additional 10 days. In the “enVisionMATH 2.0 and Texas” guide, a year-long “Pacing Guide” is located in the materials. This guide proposes five days for Topic 1, eight days for Topic 5, six days for Topic 11, and 13 days for Topic 14. The guide proposes 116 days in all for Topics 1–16 and 10 days for Topic 17. The lessons are all formatted similarly 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 the teacher’s own 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 grade 2 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 14 addresses TEKS 2.9G regarding telling time. The Looking Back column shows TEKS 1.7E, regarding time, was covered in first grade; the “Looking Ahead” column shows how TEKS 3.7C relates to elapsed time, addressed in grade 3. The topics are organized in sequential order. In grade 2, Topic 5 addresses exploring addition and subtraction, Topic 6 addresses adding two-digit numbers, Topic 7 addresses subtracting two-digit numbers, and Topic 8 addresses three-digit addition and subtraction.

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 to allow the ability to incorporate the curriculum into the district, campus, and teacher programmatic design and scheduling considerations. The materials allow the rearrangement of topics 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 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 level homework for each lesson for which there is a “Home Connection” section. The materials provide suggestions and examples of exemplary family engagement practices. For example, in Topic 3, Lesson 2, Home Connections reads, “Your child identified and wrote the expanded form, the number word, and the standard form for three-digit numbers.” The “Home Activity” suggests the parents say a three-digit number, such as 851, and write it down in number-word form. Parents then ask their child to write the number in both standard form and expanded form. The materials also 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 Home-School Connection for Topic 7 shares how students are learning subtraction skills, such as regrouping, and how to use tools like models and number lines to help subtract. The “At-Home Connection” also shows how to use addition to check subtraction and contains an activity for the students to practice regrouping one ten as ten ones. The Home-School Connection for Topic 14 explains how students are estimating and measuring items to the nearest inch, foot, yard, centimeter, and meter, as well as finding the area of shapes and telling time to the minute. The sheet suggests that families can help their students look for clocks around the house, both digital and analog, and tell the time on each clock. A student progress report is 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 form ensures that parents are aware of how their students are progressing in math. 

The materials include online access to resources that parents can use at home to work with their children on specific skills. 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 5, Lesson 1, Home Connection informs the parents that their child used models and a mental math strategy to add multiples of ten to a two-digit number. The Home Activity suggests parents ask their child to count by tens to find the sum for 54 + 20. The materials include Home-School Connections pages that give families an overview of the content in each topic. The Home-School Connection at the beginning of Topic 12 explains to families that their children are learning about unit and non-unit (number of parts is greater than one) fractions. It suggests having students look for items in their homes broken into equal parts, such as window panes; students can draw, count, and write down which fraction each part represents. 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 15, where students discuss with their family and friends about different situations where they would need to organize data. Students survey their classmates about which of two animals is their favorite and organize the data on a picture graph. 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. 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 make for 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 assignment 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 the 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, including 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 Topic 10, Lesson 4, Visual Learning Activity 1, students watch a review video of how to show and write one dollar. This video shows and explains a dollar bill and a dollar coin. It also shows a mixture of coins with the total value of one dollar that students count; the video explains how 100 cents is equal to one dollar. For Activity 2, students compare a dollar bill and a dollar coin and use the text box and writing tools to explain if one is worth more than the other. 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. MyPearsonTraining.com features many online tutorials and quick-start guides to help teachers jumpstart their “enVisionMath Texas 2.0” training.