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MathGrade 6 | 2015
Publisher: Houghton Mifflin HarcourtSeries includes:
The quality review is the result of extensive evidence gathering and analysis by Texas educators of how well instructional materials satisfy the criteria for quality in the subject-specific rubric. Follow the links below to view the scores and read the evidence used to determine quality.
Section 1. Texas Essential Knowledge and Skills (TEKS) and English Language Proficiency Standards (ELPS) Alignment
TEKS Student %
TEKS Teacher %
ELPS Student %
ELPS Teacher %
Section 2. Concept Development and Rigor
Section 3. Integration of Process Skills
Section 4. Progress Monitoring
Section 5. Supports for All Learners
Section 6. Implementation
Section 7. Additional Information
|Grade||TEKS Student %||TEKS Teacher %||ELPS Student %||ELPS Teacher %|
The materials concentrate on the development of the primary focal areas for sixth grade. A majority of concept development of these primary focal areas follows grade-level specific TEKS. The materials strategically and systematically develop student content knowledge appropriately as outlined in the TEKS, and practice opportunities are provided for students to master content.
Evidence includes but is not limited to:
The materials devote the majority of lessons to the focal areas, as outlined in the TEKS, clearly and consistently showcasing curriculum alignment in the grade level. Six of the seven units focus on sixth-grade level focal areas: operations with integers and positive rational numbers, ratios, and rates, including equivalent ratios to represent proportional relationships, expressions, equations to represent relationships, and data representation.
Each unit begins with a Unit Pacing Guide, outlining the order of TEKS taught during the unit. The introduction to each module in the Teacher Edition provides “Mathematical Background” for TEKS, providing further guidance and a brief explanation of the standards and accompanying visuals. Teacher editions provide a primary example of the focal area as it applies to a career; for example, in Unit 1, the introduction explains how a climatologist uses mathematics related to integers and rational numbers. In both the teacher and student editions, “Front Matter” provides an outline of grade-level TEKS taught in each unit, module, and lesson. Introductory materials also include an Essential Question for each module, an “Are You Ready?” activity to build students’ knowledge from the previous grade level, a “Reading Start-Up” activity with review and preview vocabulary terms, and “Unpacking the TEKS” with examples, visuals, and key vocabulary.
The instructional materials note a systematic philosophy around the introduction of key concepts as each lesson consistently follows the 5E model (Engages with Real-world video, Explores the math concept, Explains with examples in guided practice, Elaborates with math talk, and Evaluates through independent practice.) For example, in Lesson 2.3, “Comparing and Ordering Rational Numbers,” teachers pose an essential question for engage followed by a “Motivate the Lesson” question to transition into the explore. Next, the explore activity in the student edition uses a number line to connect fractions to decimals, including an “Animated Math” activity where students virtually manipulate and place fractions and decimals as points on a number line. For Lesson 12.1, “Writing Equations to Represent Situations,” students explain multiple examples and related questions with applicable solutions to build understanding. Then in Lesson 17. 3 “Dot Plots and Data Distribution,” students elaborate by providing answers to questions like “What measures of spread can you find from a dot plot?” At the end of Lesson 18.4, “Wages, Salaries and Careers,” students evaluate by working through guided and independent practice (available in paper and online formats) as well as a short “Lesson Quiz.”
The materials provide various practice opportunities in multiple settings or modalities as well as a systematic philosophy around key concepts. At the beginning of each unit, professional development videos created by the authors demonstrate the teaching and learning of math concepts. For example, in Module 10, “Generating Equivalent Numerical Expressions,” the author Juli Dixon models successful teaching practices as she explores prime factorization in an actual sixth-grade classroom. Each module also contains a “Go Digital” section, providing a variety of settings for focal area skills practice, such as the “Personal Math Trainer,” where feedback is given on online practice sets; the “Animated Math,” where students explore key concepts online; and the “Interactive Whiteboard,” which offers quick activities to draw on a whiteboard. In the majority of lessons in Grade 6, students model the concept before moving into the algorithm and other complex concepts. For example, in Unit 2, students model multiplying and dividing fractions and multiplying and dividing decimals, then move into the algorithm, and in Unit 4, students model expressions, write expressions, evaluate expressions, then finally evaluate real-world expressions.
The materials, therefore, build upon previously taught concepts to increase rigor and ensure students grasp the full intent of the concept, offering numerous opportunities for mastery. For instance, in Unit 2 Module 3 “Are You Ready?,” students practice converting between improper and mixed fractions as well as basic multiplication and division facts before moving on to Lesson 3.1: Multiplying Fractions. They multiply fractions in the context of real-world situations and build upon previously taught concepts before moving on to the grade-level concept. The skills learned in Module 3 are used again in Module 16 when solving area and volume questions using positive rational numbers. In Module 5 “Adding Integers,” students practice this primary focal area skill through the use of counters, number lines, tables, expressions, verbal descriptions, and real-world problems in the “Independent Practice” at the end of each lesson. Lessons provide additional opportunities with “Engage with the Whiteboard” and “Talk About It” and offer multiple versions of practice and problem-solving PDFs, editable documents, a quiz, and a reteach document. Focused on Higher-Order Thinking Skills, Guided Practice is provided where students answer questions, fill in tables, chart graphs, and respond to an Essential Question Check-In prompt for each lesson; Independent Practice provides open-ended response questions. In Lesson 13.1 “Writing Equations,” students develop real-world situations connected to the displayed content by answering the following: “The number line shows an inequality. Describe a real-world situation that the inequality could represent.” Units conclude with a Study Guide Review where students complete exercises over primary focal point areas from each module within the unit. Students also complete a Unit Performance Task that ties back to the career application introduced at the beginning of the unit. Lastly, students complete the “Module Quiz” to demonstrate mastery and a “Texas Test Prep,” which is a mixed review that spirals concepts.
Most of the provided materials sequence concepts from concrete to representational to abstract (CRA) as is appropriate for the sixth grade; however, the explanation for the concrete is minimal. Materials include a variety of concrete models and manipulatives, pictorial representations, and abstract representations. The primary focal areas for Grade 6 are limited in concrete examples of integers, expressions, and equations. Materials do offer teachers some support in understanding and appropriately developing students’ progression along the CRA continuum, but with limited explicit instructional support and guidance to respond to misconceptions.
Evidence includes but is not limited to:
The materials include a variety of concrete models and manipulatives, pictorial representations, and abstract representations to introduce and practice mathematical concepts. Within each module, there are lessons that include one or more “Explore” activities where students “select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.” For example, in Module 4, some lessons contain an “Explore Activity,” but others do not. In Lesson 4.1 “Multiplying Decimals,” the Explore Activity has students “use decimal grids or area models to find each product,” yet Lesson 4.3 “Applying Multiplication and Division of Rational Numbers ” does not contain an Explore Activity as the content is more abstract and builds on previous student understanding. When looking at manipulatives, Unit 1 resources in both the teacher and student editions provide links to integer counters, fraction bars, fraction decimal grids, bar models, geometry sketcher, algebra tiles, graphing calculator, and scientific calculator. These virtual tools provide a varying level of guidance on how to use the tool, not explicitly teaching students how to work with the concrete models. For example, the fraction/decimal grid “Help” menu provides a detailed explanation about how to use this tool for adding, subtracting, and multiplying; whereas, the “Bar Models” tool provides a “key” to the different buttons without guidance related to application or use. These materials are not interactively linked with a specific lesson. They are strictly digital manipulatives.
A primary focal area of grade 6 states that students should “represent integer operations with concrete models and connect the actions with the models to standardized algorithms.” Module 5 focuses on these operations with integers and begins with two examples demonstrating adding or subtracting integers using two-color counters. The lessons also ask students to verify sums using a number line. As the students explore the manipulatives, they create pictorial models to support their thinking and make conjectures to move toward solving integer operation problems abstractly. Explicit examples in the student edition guide teachers in supporting students as the materials provide lessons that build through the phases of the CRA continuum, including guidance related to “Avoid Common Errors” and suggestions to “Focus on Models.” For example, in Lesson 5.1 “Adding Integers with the Same Sign,” students use two-color counters and number lines to model adding integers with the same sign. In the teacher notes, it suggests that teachers “guide students to understand that each yellow counter represents a positive integer and has a value of +1. Each red counter represents a negative integer and has a value of -1.” In addition, teachers are notified that “some students may forget to attach the sign of the integers to the sum after adding. Remind students that the sum of two negative numbers must include the negative sign.” Materials illustrate a variety of real-world examples using integers, such as temperature thermometers, bank account transaction visuals, pictures of elevation among the mountains, and examples of divers diving into a body of water. In the Independent Practice set, though, some questions ask students to use counters to solve, without increasingly rigorous practice with the manipulatives.
Another primary focal area of grade 6 states that students should “determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.” In Module 11, a scale becomes a concrete model for equivalent expressions showing the balance needed when modeling equivalent expressions. A bar model, as well as algebra tiles, is then used to model algebraic expressions. Students create bar models equivalent to expressions to demonstrate their understanding of an algebraic representation (and vice versa).
Some other opportunities to use manipulatives and instruction on how students use/create representations of math concepts are as follows: Folded number lines, creased in half so the crease goes through zero, show the opposite of an integer. Hundreds grids introduce modeling percentages. Bar models and number lines make connections and extend understanding of percentages. In applying percent skills, circle graphs are also used throughout word problems. Pictorial models are used to represent areas of parallelograms, trapezoids, triangles, and composite figures; pictorial models also represent the volume of rectangular prisms, and various real-world illustrations are included to make connections with area and volume. Two-color counters and equations introduce measures of center. Number lines illustrate box plots and dot plots. Furthermore, visuals represent stem-and-leaf plots, histograms, and bar graphs. A basketball scoreboard introduces writing equations to represent situations. Then algebra tiles and number lines model solutions to one-step equations and inequalities, and number lines graph solutions that represent one-step inequalities.
When looking at the concrete to representation to abstract (CRA) continuum, some guidance and support are present, though not always consistent. “Professional Development'' videos included in each module guide teachers with “best practice” visuals specifically related to the content being developed in the module. For example, the video in Module 9 shows a teacher in a classroom of students building the equivalence of fractions, decimals, and percents. Through watching this video, teachers witness the development of the lesson, how grid models are incorporated to make connections, questioning strategies to guide students’ thinking, and “teacher moves” (i.e., encouraging students to accurately read decimals “thirteen hundredths”) to successfully progress from pictorial models to an equivalent abstract representation of decimals, fractions, and percents in this lesson. Also, the “Personal Math Trainer” provides teachers some feedback related to where students are in the phases of the CRA continuum by incorporating varied problem types that address the phases of the CRA continuum. However, minimal evidence was located to provide insight into which activities correspond to progressive points along the continuum. Teachers evaluate individual lessons or refer to the TEKS Correlation for Grade 6 information in “Front Matter'' to see which lessons align to which TEKS, but, in this document, no specificity is included related to where on the progression the activities align.
The materials provide some coherence and connections within content at the grade-level but do not support coherence and connections across grade levels. Students build their vertical content knowledge by accessing prior knowledge and understanding of concept progression. Throughout the lessons, the materials include some tasks and problems that intentionally connect two or more concepts, so students are able to explore relationships and patterns within and across concepts. However, the materials lack a significant number of tasks that require students to recognize mathematics in contexts outside of the classroom. Teachers’ support in understanding vertical and horizontal alignment guiding the development of concepts is not substantial; the materials are limited in the vertical progression of concepts and lack specificity related to how content builds on prior knowledge from previous grades. The materials do not provide enough evidence to enable teachers to connect what students have learned and where the content aligns above grade 6 to the breadth, depth, and complexity of high-quality materials.
Evidence includes but is not limited to:
Materials consistently contain one task to direct teachers to build on students’ prior knowledge before presenting a new concept aligned to a grade-level focal area. At the beginning of each module, “Are You Ready?” provides support for review skills needed for the chapter. The Teacher Edition provides assessments to determine if students need intensive or strategic intervention for the module’s prerequisite skills. If students are not ready, teachers reteach specific areas with Skills Intervention worksheets, which include two lessons and student practice focusing on intervention. For example, in Module 7, Are You Ready? assesses students’ understanding of simplifying fractions and writing equivalent fractions as foundational content for “Representing Rates and Ratios.” The second lesson for each of the concepts in the Skill Worksheets includes an “alternate teaching strategy,” which focuses on modeling simplifying fractions and equivalent fractions using fraction circles. After mastering the review, students are ready to continue with this sixth-grade focal area, “understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships.” Other than Are You Ready?, teachers are not consistently provided review material of previously learned concepts, nor do they receive further guidance on intensive or strategic interventions.
The Teacher Edition contains a “Grade 5 Review Test” in Front Matter that gives an overview of when TEKS are taught and when they are reinforced. It also includes skills and standards students should have mastered in fifth grade. The materials do not include specifics related to how the beginning units and modules build on students’ prior learning. The materials also do not contain an overview of how the sixth-grade objectives connect to previously learned concepts and concepts to be learned. Teachers are not provided with information regarding how students should progress in their knowledge and skills throughout future grade levels. What each unit does include is a brief breakdown of the progression from one unit to another horizontally—within the grade level. This progression at the beginning of the unit quickly shows teachers what students learned in the previous unit, what students learn in the current unit, and what students will learn in the upcoming unit. Additionally, each module includes a brief breakdown of the progression from one module to another horizontally—within the grade level—which quickly shows teachers what students learned in the previous module, what students learn in the current module, and what students will learn in the upcoming module. Also, at the beginning of each module, Unpacking the TEKS restates the TEKS, clarifying what students are expected to learn, providing an example related to the TEKS, and highlighting key vocabulary for the TEKS. This information is also available for all TEKS, not just those covered in a specific unit or module, via a QR code in the Teacher Edition.
Throughout the materials, each module does provide teachers insight into how the concepts progress in rigor in the “Grades 6–8 TEKS” section, which is divided into a three-column table with what the students will be doing Before/In This Module/After. The “Before” portion states the prior knowledge, so in Module 7, “Students understand ratios,” which is described as “write a ratio as a fraction” and “write equivalent ratios.” The middle column called “In This Module” states specifically how the module builds an understanding of proportional relationships. The final column, “After,” states connections that will be made in future lessons, such as “rates and unit rates” and “constant rates of change given a table, verbal description, table or graph.” However, it’s important to note that the “Grades 6–8 TEKS” section focuses on grade-level content and does not show a true vertical alignment across grade levels.
In each lesson in the Professional Development section, there is a Math Background description, which outlines all the TEKS for that unit and the math background of TEKS. Some of these just give the mathematical reasoning for why the TEKS is solved the way it is. For example, Module 2 explains the process for writing equivalent fractions and the two properties but does not give learning from previous grades. Module 14 Math Background reminds teachers that “although the term function is not mentioned in this lesson, the tables in the lesson represent functions. A function is a rule that relates two quantities so that each input value corresponds to one output value exactly.” This information provides a very brief vertical look at how the concepts covered in this lesson are foundational for developing the concept of functions introduced in subsequent grades.
Within the grade level, some materials provide tasks that help students connect concepts that are appropriate to their grade level. For example, in Module 2, students are “Comparing and Ordering Rational Numbers.” They apply their knowledge of ordering integers from Module 1 to questions posed in the Independent Practice: one question requires ordering the change in gas prices from least to greatest. (Since the prices at the different gas stations both increased and decreased, the changes are positive and negative rational numbers, which builds on their understanding of ordering integers.) Although the task uses familiar models and strategies from previous units, the depth of explanation is weak for both the teacher and the student.
The materials provide some opportunities for students to make connections within and across math concepts. These interconnections are supported for students, but teachers are provided little guidance on how to fulfill the connections. When teachers ask, “Can the points on the graphs only have whole number coordinates?” the provided teacher response says, “No, the coordinates can be any pair of rational numbers that satisfy the equation. For example, (0.5, 1.5) is on the graph of y = x + 1.” More guidance could be provided for new teachers to make better connections with previous skills taught, including an explanation of integers, although they would mainly be positive when graphing equations in sixth grade. Additionally, many lessons include exploration to examine relationships and patterns. For example, Module 2 asks students to use a number line to place decimal and fraction values in order. Then the students reflect and communicate their understanding of equivalent decimal and fraction values. In Module 3, students are encouraged to draw rectangular arrays to model the multiplication of fractions. The materials support interconnections with the question, “Will the product of 1/2 and 2/3 be greater or less than 2/3? Explain.” This question pushes students to make connections between what they understand about multiplication and rational numbers. In Module 14, the materials include the note, “Point out to students that the ordered pairs from each table are used to make the graphs. However, after the lines are drawn on the graphs, they represent a more complete picture of the relationship than the tables do because all positive real numbers, not just integers, are included in the graph.” This connection to rational numbers reminds students that even on coordinate planes, there are values between the scale values.
In one Explore activity of Module 7, students connect their understanding of ratios to create ratio tables to compare two different lemonade recipes. Students’ understanding of ratios at a conceptual level is clarified with the “Explain the Error” scenario presented, where students are challenged to understand the relationship between the parts of a ratio (i.e., the amount of lemonade concentrate to the amount of water). The scenario says, “Marisol makes the following claim: ‘Bailey’s lemonade is stronger because it has more lemonade concentrate. Bailey’s lemonade has 3 cups of lemonade concentrate, and Anna’s lemonade has only 2 cups of lemonade concentrate.’ Explain why Marisol is incorrect.” Students must understand the relationship between the lemonade concentrate and the water to understand Marisol’s error in thinking. Later, the teacher guide reminds teachers to ensure students make the connection between equivalent fractions and equivalent ratios by asking, “How is finding equivalent ratios like finding equivalent fractions?”
In Module 8, students complete a table during the Explore phase of the lesson, and the Professional Development notes describe how students may already be familiar with input/output machines that relate to two-quantity tables. In Module 12, the Teacher Edition includes some reminders, such as, “Remind students that addition and subtraction are inverse operations” and “Remind students that to add or subtract fractions with different denominators, it is necessary to rewrite the fractions with a common denominator.” Again, although the tasks use familiar models and strategies from previous units, the depth of explanation is weak for both the teacher and the student.
Another connection opportunity comes at the beginning of each module in Active Reading, where a Reading Start-Up page suggests graphic organizers to help students connect mathematical vocabulary to the content throughout the module. Some modules include a foldable, such as Module 2, where students create a trifold and record what they know, what they need to know, and what they learned.
The materials provide real-world problem solving and mathematical processes, helping students connect math concepts to life outside of the classroom. Students explore different realistic situations, preparing the lesson, and setting up conceptual understanding. For example, in Module 14, students are asked to observe a table that shows how much Amanda earns for walking different numbers of dogs. Students are then asked to complete the missing portions of the table and determine the pattern. Using the pattern, students are asked to write an equation to represent the number of dogs Amanda walks to the amount she earns. Another example asks students to determine the amount of the donation if a painting is sold for $1,200. The Teacher Edition prompts teachers to walk students through underlining and circling the important information in the problem. Teachers are then prompted to engage students on the whiteboard to extend the table that contains the painting prices and amounts for donations. Questioning Strategies guide teachers to help students identify the independent variable in the situation, a skill learned in a previous lesson within the same module. Teachers then prompt students to discuss how else the problem could be solved.
Each module also contains a Challenge problem. The Challenge problem for Module 7 requires students to apply the concepts of unit rates to a real-world scenario involving a track team. The students determine the rate of four runners given their times and distances in various units in order to determine the fastest runner and the slowest runner; next, they apply this understanding to related questions. An example of one question is as follows: “Suppose each runner ran at the rate given in the table above for 3.1 miles. How much time will elapse between the first-place finisher and the last place finisher?” (Now, It’s important to note that the task does not require students to recognize the mathematics needed on their own, as it guides the students’ thinking through direct questions, implying the content to apply to the situation, such as “Find the rate for each runner in miles per hour.”)
Each of the seven units begins with a section entitled Careers in Math. The units end with a Performance Task that incorporates concepts from these careers to real-world scenarios. For example, in Unit 1, the career is a climatologist; the ending performance task asks students to explore like a climatologist by measuring tree rings and answering related questions. In Unit 6, the career is a geneticist, and in the performance task, students collect data about eye colors and answer questions using that data.
For seasoned teachers, these materials might be sufficient; however, a new teacher may struggle with how to present the materials. All of the teacher supports provide questions to ask, reminders of common errors to avoid, and suggestions to have students go to the whiteboard. The “extend the math” sections are primarily just more practice of the same concept, not an extension of the concept to the next grade level.
Throughout the lessons, the materials include tasks partially designed to engage students in the appropriate level of rigor (conceptual understanding, procedural fluency, or application) as identified in the TEKS and as appropriate for the development of the content and skill; however, the materials do not always reach sixth-grade depth and complexity nor is the concrete-representation-abstract (CRA) continuum consistently developed throughout. For the teacher, materials clearly outline mathematical concepts behind each task; however, they exclude evidence explaining how each task builds student efficacy toward the goal of demonstrating mastery. Teacher guidance on anticipating student responses and strategies often limits student responses, and teacher guidance is weak when facilitating strong student discourse grounded in the quality tasks and concepts. Additionally, no rubrics or keys are provided to assist teachers in evaluating and providing feedback to students while engaging in discourse. The integration of contextualized problems throughout provides students the opportunity to apply math knowledge and skills to new and varied situations, but with no regard to student interest.
Evidence includes but is not limited to:
The materials engage students in the rigorous tasks aligned to TEKS and are developmentally appropriate for the grade-level content and skills. Throughout each lesson, the 5E model (engage, explore, explain, elaborate, and extend) develops the concept with increased rigor through the engage, explore, and explain. Lessons begin with an understanding of naming and identifying mathematical terms and move to application and problem solving as students use CRA tools and models, increasing in depth and complexity. However, materials used are limited to two-color counters, number lines, bar models, algebra tiles, and fraction/decimal grids; their use is inconsistently guided in the materials. Some tools, like two-color counters, are supported with clear instructions, while others, like the bar models, are presented but not specifically taught. This lack of depth of understanding of the tools jeopardizes building the foundation necessary for students to fully master content. “Focus on Models” allows students to see connections between the concrete models and the pictorial models while providing a foundation for understanding before moving to solve the problem; however, when students reach the elaborate portion of the lesson, the variance in the problems is not evident. Students continue to solve real-world contextualized problems that relate to the content, but they are rarely challenged to make generalized conjectures or apply their thinking beyond this level. To extend, problems tend to be a repetition of the HOT (higher-order thinking) Questions that are in “Independent Practice.” The materials contain Performance Tasks for every unit in the Assessment Resources that include teacher guides, a rubric, and teacher guidance for the rubric.
For example, Module 1 takes the students through activities and questions around win/loss records for a sports team. To explain, a number line is introduced as a way to order integers. The students then move into writing inequalities to compare integers. In elaborate and evaluate, students solve HOT problems to compare and order integers. Module 2 uses number lines to compare and order rational numbers by modeling the fraction and decimal equivalents on the same number line. This model builds on students' understanding of integers and also pictorially connects their understanding of fractions and decimals. However, the student understanding only focuses on the strategy of writing fractions as decimals or decimals as fractions without providing alternate strategies. Module 7 asks students to explain by modeling a ratio. Next, students complete a table to compare rates, and then students extend their knowledge as they use their knowledge of ratios and rates to solve problems.
As students work through units, modules, and lessons, materials include explanations of the mathematical concepts and goals behind each task and serve to build teacher content knowledge. Each unit contains “Math Background,” which clarifies which TEKS are taught in the unit, what lesson they are taught in, and the background to those TEKS. Some examples are provided of what students may wonder about the TEKS, the mathematical reasoning for the TEKS, or what students should have learned previously. Vocabulary is also clarified for each concept, and where appropriate, visuals are provided.
The materials provide students the opportunity to apply math to different situations and real-world contexts. In fact, each unit begins with “Careers in Math,” offering specific jobs that use the concept in life; then, at the end of the unit, a “Performance Task” about the career provides students an opportunity to solve the career-challenges. For example, in Unit 1, the career is a climatologist; the end task relates to measuring tree rings, asking students to answer questions about the measurements. Unit 3 asks students to solve scaling and estimation problems similar to those of a residential builder, and in Unit 6, students think like a geneticist while collecting data about eye colors and answering questions about that data.
Besides the Careers in Math, each lesson contains multiple opportunities to answer questions in real-world contexts. For example, in Module 2, as students compare and order rational numbers, they move from comparing lists of unrelated rational numbers to comparing the weights of aluminum can donations. In Module 3, students calculate how many students have a pet when they know that there are 18 students and a proportion of 5 to 9 have a pet. In this same lesson in the independent practice, 6 of the 10 questions relate to real-world contexts. In Module 8, students make a table to show the distances that a train travels over various amounts of time. In this same lesson in Independent Practice, 5 of the 11 questions relate to real-world contexts. Then in Module 14, students analyze relationships in two variables when answering questions about the amount of clay that an art teacher procures or the amount of time it takes a train to travel a given distance. These materials do provide students with opportunities to apply math to different real-world situations, but support is not provided for teachers to modify tasks to specific student interests and backgrounds. The materials provide editable documents, such as leveled quizzes/tests and skill intervention worksheets. There is no guidance for how to revise the content; therefore, the revision of content is left to the insight and knowledge of the teacher without guidance from the materials.
The materials provide some guidance to the teacher in supporting student discussion and responding to student strategies as they use problem-solving to support the development of skills. The teacher’s edition offers “Essential Questions,” “Questioning Strategies,” “Focus on Reasoning,” and “Talk About It” to stimulate discourse; however, according to NCTM, discourse in the mathematics classroom incorporates “ways of representing, thinking, talking, agreeing, and disagreeing.” Using these materials, the questions may provoke thinking, but they do not build talk that fosters agreement and/or disagreement.
Each lesson includes an Essential Question related to the covered TEKS that students answer. This also addresses the knowledge and skill (6)(1)(f), which states, “the student is expected to analyze mathematical relationships to connect and communicate mathematical ideas.” The essential question posed in one lesson from Module 2 asks, “How do you compare and order rational numbers?” The corresponding teacher answer states, “You can write them as decimals and then compare them.” This response limits teacher support for alternate responses from students. What if a student mentioned writing them as a fraction and then comparing or drawing them as grid models? Also in Module 2, teachers ask, “How can you compare two fractions? Compare their equivalent decimals or rewrite them with common denominators and compare the numerators.” “How can you compare a fraction with a decimal? Rewrite them so both are decimals or both are fractions.” The final question asks, “In B Step 2 if you use a different common denominator other than 60, will the order be the same? Justify your answer.” Even though this appears to be a more open-ended question, there is one expected, correct response the teacher is looking for; most student discussion is used to summarize understanding, not to guide students’ own thinking towards choosing strategies that best fit their learning needs.
In a lesson from Module 15, teachers ask, “If three side lengths do not form a triangle, how many sides do you need to change? Explain using an example.” Teachers then ask, “What is another way to check whether segments can form a triangle?” Immediately following the answer, the materials move to Focus on Patterns, where teachers guide students to recognize that they only need to test that the sum of the two smallest sides is greater than the longest side. There is no suggestion for other topics, questions, or statements that students may generate naturally in a more spontaneous discussion. No additional solutions are offered. The anticipated strategies presented in the material keys typically align with what was presented in the lesson, with no evidence of additional “anticipated” strategies being sequenced or incorporated into the materials. No teacher guidance explains which strategies are appropriate for tasks based on grade-level expectations; the strategy presented is the one expected. Teachers are simply prompted to ask specific questions with specific expected student responses and are not provided guidance on asking probing questions to assess student thinking. Open-ended response-type questions are not provided for teachers, so an inexperienced teacher might struggle with evaluating and providing feedback.
Materials also foster discourse through “Math Talk” and Talk About It discussions. In Lesson 7.1, the Math Talk question asks students how to find the amount of each juice needed if a recipe is cut in half, and a possible student-response strategy is provided. These sample answers provide teachers with guidance on how to direct the discussion toward the correct explanation if students struggle. Talk About It includes a question to summarize and check for understanding. For example, Lesson 7.3 asks the question, “How can finding an equivalent ratio help solve Exercise 2?” Again, the materials provide a key for all questions, but very little flexibility is allowed in discussions. No rubrics exist for evaluating and providing feedback for student discourse so that teachers can assess student understanding.
“Avoid Common Errors” sections in both the lesson notes related to teaching the content and later in Guided Practice offer the teacher insight in identifying common errors made by students. These notes guide the teacher to point out or to remind students of mathematical processes to use during practice but do not provide questions or prompts to get students to understand their own mistakes. For example, in Module 7, teachers “remind students to multiply or divide both terms of a ratio by the same number to find equivalent ratios. Point out that multiplying and dividing only one term does not produce an equivalent ratio.” This strategy is reiterated in reference to Guided Practice exercises 1 and 2, which have students completing tables to find equivalent ratios. In Module 14, as students graph on the coordinate plane, teacher notes provide insight that “students may give incorrect coordinates for a point because they transposed the x- and y-coordinates. Remind students that the x-coordinate is the first number in an ordered pair.” Later in this same module, the teacher notes suggest, “if students have difficulty determining whether a relationship is additive or multiplicative remind them that in a multiplicative relationship the graph will pass through the origin, but in an additive relationship the graph will not pass through the origin.” While there are multiple opportunities to anticipate student responses, there are no strategies to combat any misconceptions other than within the Avoid Common Errors section. Also, some inexperienced teachers might struggle with guiding students through problem-solving if they are always pointing out the common misconception straight away and not allowing students to determine errors on their own.
The materials do not include a year-long plan for building fluency as appropriate for the concept development and grade, nor do they integrate fluency at appropriate times and with purpose as students progress in conceptual understanding. No significant scaffolds and supports for teachers to differentiate fluency development for all learners are evident in the materials, and guidance does not provide suggestions for the next steps to support all student learning.
Evidence includes but is not limited to:
While there are isolated fluency tasks, the materials do not provide a year-long plan for building fluency. The materials do not provide guidance for tracking the fluency progress of students across the year. No clear directions exist to support teachers for how/when to conduct fluency activities and practice with students. There are few lesson notes for teachers to describe the fluency practice and how it supports students’ access to the concept in each lesson, but they lack clarity.
Materials do not specifically address fluency within the lessons. This is illustrated by a search through materials for variations of the word “fluent/fluency/fluently.” In Unit 1, fluency is mentioned only two times. Once in the “Professional Development” section, the other in the “Animated Math” section for one lesson. In Unit 2, results show this root is mentioned 52 times. Of these 52 times, 35 are strictly from the wording of the TEKS, 14 are from the “before/in this module/after” graphic, and three are from the Animated Math section. An Animated Math example in Module 5 indicates that students will build fluency by playing a game, yet there is no evidence that this animated math game will build fluency. When students place a number on the number line, it cannot be corrected. If the “pin” is dropped in the wrong place, it cannot be moved. There is no guidance for fluency development and no mention as to when or how this activity should be conducted.
There is no evidence of instructional routines for building fluency and no evidence of the materials supporting the quick recall of facts. The following examples show how materials address the development of conceptual understanding, but these examples are not integrated into materials as part of a year-long plan. “Math Background,” at the beginning of each unit in the teacher’s edition, provides connections to the development of conceptual understanding. For example, Unit 2 focuses on number operations, specifically “Multiplying Fractions” in Lesson 3.1. Teacher notes say that “although students may have little difficulty applying this algorithm, they may not understand the underlying rationale.” The Professional Development section of this lesson, “Integrate Mathematical Processes,” explains the progression of the use of TEKS 6.1C. “In the Explore Activity, students use a model to find the product of two fractions. Then, in Example 1, students use an algorithm to multiply two fractions. Finally, in Example 2, students estimate and then use the algorithm to multiply a fraction and a whole number by rewriting the whole number as a fraction.” Also in Unit 2, Lesson 4.1’s Math Background explains the conceptual understanding behind multiplying decimals. This explanation includes information on the algorithm used to multiply decimals, as well as information on place value when working with decimals. Again these examples, all from the same unit, are not integrated into materials as part of a year-long plan for developing fluency; they do not specifically offer routines to build that fluency.
Numerous assessments are provided to determine each student’s development of procedural fluency, but there is nothing to suggest the next steps for supporting student learning. The materials do not suggest how results are to be used to support student learning beyond the assessments. Looking specifically at the Beginning-of-Year Diagnostic Test, which has 96 problems covering sixth-grade content, the purpose is “to assess knowledge of the key objectives that will be taught in the current school year” and can be used as “a baseline for a student’s mastery of math concepts and skills, and to evaluate growth during the school year.” Fluency support is limited and inconsistent. This diagnostic tool focuses on mastery versus assessing students developing procedural fluency. For example, question 5 from the diagnostic test says that “Jason plotted points on a number line” at four different values, with two fractions and two decimals. The question asks, “Which of these values is farthest from zero?” The reasoning for incorrect answers is, “Students who answered A, B, or C may have graphed the numbers incorrectly.” The answer key doesn’t diagnose where the problem is, nor does it offer suggestions on ways to build fluency for those students who lack understanding.
The materials indirectly integrate fluency activities with the development of conceptual understanding. For example, the “Explore” activity and “Guided Practice” introduce conceptual understanding; however, the activities are pre-scripted and fall within the lowest cognitive level when mapping fluency to Bloom’s Taxonomy. In Module 3, “Communicating Math” guides teachers to present two equations for students to explore: 8 ÷ 2 = 4 and 8 × 1/2= 4. Teachers “invite students to explain why the two expressions have the same result. Guide students to understand that dividing 8 by 2 is finding the size of each group when 8 is divided into 2 groups. Point out that this is really the same as finding half of 8, or 8 × 1/2. So, dividing by 2 and multiplying by its reciprocal, 1/2, gives the same result.” However, while this could be interpreted as building students' fluency through students knowing and understanding a variety of strategies, the materials do not build on this idea or extend it beyond this isolated incident. Then in Module 5, “Focus on Communication using the Mathematical Processes” tells the teacher to “Have students discuss whether number lines or colored counters would be a good way to model each sum. Ask students to explain their reasoning.” Although this question appears to offer flexibility, students are not given the opportunity to strategically and flexibly choose their own appropriate strategies and must only select from those provided. Again, students are guided through steps and are not allowed to strategically and flexibly choose the appropriate strategies for grade-level tasks.
Students are given ample practice opportunities, but none of them specifically lean toward fluency. The student edition online materials include “Math on the Spot” tutorial videos in each lesson for students to watch and build their conceptual understanding. Also included in some lessons is “Animated Math,” which engages students in interactive “Explore” activities to practice key math concepts and skills. The “Personal Math Trainer” provides a variety of learning aids, including videos, guided examples, and step-by-step solutions. However, there is no specific guidance for teachers on how fluency practice is addressed within the materials. The only “guidance” comes from the pages that show a description of each resource that can be found in the print materials and online materials.
The materials provide strategic discourse opportunities around the conceptual understanding, but there is no evidence of this discourse as a support for fluency. In Module 5, there is some discourse around the conceptual understanding of adding and subtracting integers, such as, “How would you explain in your own words how to subtract integers?” However, although the materials do not provide a specific fluency practice, questions such as this build fluency as students build understanding related to when it is appropriate to use a specific strategy.
Materials do not include scaffolds and supports for teachers to differentiate fluency development for all learners; there are additional skills worksheets for struggling learners. These worksheets and online options address accuracy and efficiency (more practice of the same concept), not fluency. For students who have mastered the content and need an extra challenge, “Extend the Math” of the teacher’s edition includes opportunities to extend fluency; however, these opportunities to extend fluency are infrequent throughout the materials. Overall, materials do not provide an integrated, cohesive plan for developing fluency.
Some materials support students in the development and use of mathematical language. The materials include embedded opportunities, mostly at the beginning of lessons, to develop and strengthen mathematical vocabulary, but only some materials include guidance for teachers on how to scaffold and support students’ development and use of academic mathematical vocabulary in context.
Evidence includes but is not limited to:
Materials include some opportunities to develop and strengthen mathematical vocabulary. Each unit begins with a Vocabulary Preview, which provides puzzles like word searches and word scrambles as a way to introduce new vocabulary. For example, Unit 3 provides a crossword puzzle as a way to preview “important concepts in this unit;” the puzzle’s clues describe terms associated with rates and ratios. Another section at the beginning of each unit is Math Background, which provides teachers guidance on the academic vocabulary being introduced in each lesson in the unit, including clarification of the term’s definition. For example, Unit 3 focuses on the development of ratios; it includes a definition, examples, and reminders to guide the teacher’s introduction of the term, like “It is helpful for students to have different ways of describing and picturing ratios.”
The teacher and student editions provide an outline of the mathematical vocabulary within modules. Each module begins with an Unpacking the TEKS page where key vocabulary, definitions, and the Spanish words are all given. Additionally, each module offers a Professional Development Video with guidance on “teacher moves,” including ways to encourage language development. In one example from Module 4, teachers are guided to use questioning techniques to encourage students' use of academic vocabulary with questions such as “What is this called using decimal language?” The Reading Start-Up, also at the beginning of each module, describes the development of mathematical vocabulary, including a review of previously introduced vocabulary and a preview of key terms for the module. This section includes three parts. Part one: Visualize Vocabulary uses graphic organizers, charts, and diagrams as a way for students to review vocabulary that will be used in the lessons. Here, the Teacher Edition tells teachers to discuss these terms as a class. Part two: Understand Vocabulary, in the Teacher Edition, provides a detailed “explanation to help students learn the preview words” by asking them to complete sentences using these bolded words outlined on the page. Part three: Active Reading provides students with “reading and note-taking strategies to help them organize and understand new concepts and vocabulary.” The materials provide instructions on how to use the foldable to take notes. Here is an example of what all three parts of Reading Start-Up look like for one module: In Module 7 “Representing Ratios and Rates,” Visualize Vocabulary asks students to choose from review words in a word bank to complete the chart associated with multiplication and division. In Understand Vocabulary, a word bank provides preview words, and students complete sentences incorporating their definitions. In Active Reading, a two-panel flip chart for “ratios” and “rates” provides a method to organize students’ learning. The chart encourages a place for students to “write important ideas under the appropriate flap.” The flip chart provides students a place to “include information about unit rates and any sample equations” as a resource for remembering concepts.
Lessons identify the vocabulary to be introduced and used within. Each lesson highlights the term being introduced and uses bold print to put emphasis on these academic words. For example, Module 3 introduces the term “reciprocal” when teaching about dividing fractions. (This term was also included as a preview word in Reading Start-Up at the beginning of the unit. In addition, the teacher’s edition includes a Focus on Math Connections section, which guides teachers to “remind students that two numbers are reciprocals if their product is 1. Note also that every number except 0 has a reciprocal.”) For each lesson, the materials include a Reading Strategies worksheet as an additional resource for struggling students. This material includes additional visuals for students to connect with the new words. Activities like Using the Context help develop the key vocabulary of the lesson and build an understanding of the vocabulary by providing a simplified definition and additional models. For example, in Module 7, ratios are defined as “a multiplicative comparison of two quantities expressed with the same units;” however, in the Reading Strategies worksheet, to support this concept, ratios are introduced as “a comparison between two similar quantities” followed by an explanation that breaks the process down into small steps. Some questions are also provided as an opportunity for students to display, describe, and communicate mathematical ideas using precise mathematical language.
Many lessons include vocabulary connections within the Explore and Explain activities. For example, in Unit 1, Lesson 1.1, the vocabulary connection is to make a list of opposites. The teacher’s edition includes Connect Vocabulary for some lessons, which prompts teachers to deepen students’ understanding of vocabulary. For example, Lesson 17.1 encourages teachers, saying, “to help students distinguish the terms median and mean, give them another definition of median: a strip of land in the center of a road that separates the lanes of traffic going in opposite directions. Point out that just as a median divides a road down its center, the median of a data set divides the data values in two at the center.”
Some other lesson activities and practice include specific questions that push students to use and apply an understanding of academic vocabulary in mathematical contexts. These activities are Analyze Relationships, Justify Reasoning, Make A Conjecture, Critique Reasoning, Communicate Mathematical Ideas, and Essential Question Check-In. One example comes from Module 1, where students are asked to evaluate and explain why a given scenario is a misinterpretation of absolute value.
Some lessons also include vocabulary as part of the Guided Practice, such as the fill-in-the-blank vocabulary questions in Lesson 1.3, which asks students to express their understanding of absolute value by completing the sentence, “If a number is __________, then the number is less than its absolute value.” Then, Lesson 14.1 guides students to describe their understanding of ordered pairs as they “describe how an ordered pair represents a point on a coordinate plane” by including “the terms x-coordinate, y-coordinate, and origin” in their answer.
Overall, several components are there for the development and use of academic mathematical vocabulary in context, but there are missing supports for teachers on how to scaffold and support students. There are not enough opportunities for students to use academic vocabulary to listen, speak, read, and write. Some questions ask students to write their reflections, but there is no guidance on including precise mathematical language when doing so. Most of these vocabulary connections are limited to “remind the students” statements or statements such as “discuss the meaning of counter (a game piece) and board (a game tool, not a piece of wood).” The “development” of the language and vocabulary is not readily demonstrated for teachers as materials do not build on students’ growth from informal language to the formal. No explicit instructions are included for repeated opportunities to use and develop terms. A strategic approach to developing math vocabulary is lacking; other than the Reading Start-Up page, addressing vocabulary development strategically is missing.
The materials provide some 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 problems focus on isolated content knowledge and skills throughout the materials, although they lack opportunities for students to integrate multiple knowledge and skills together to successfully problem solve and use mathematics efficiently. The materials do not include opportunities for students to analyze data through real-world contexts.
Evidence includes but is not limited to:
The materials provide multiple opportunities for students to solve real-world problems from a variety of contexts. Each module begins with an Essential Question that pertains to the real world; many lessons include application to the real world via the Explore section. In Unit 1, Module 1 focuses on integers, and the Explore activities include contexts such as elevation, temperature, wins/losses, and credit card balances. In Unit 2, the Guided and Independent practice sets include real-world contexts about the cost of party supplies and recipes. Other problems arising in everyday life presented in the sixth-grade units are planning a family trip, determining how much water you use in the shower, comparing sports statistics, and calculating profit.
In addition, the opening of each unit contains Careers in Math, which introduces how the mathematical concepts in the unit are applicable to a specific career that ties to the student edition Performance Task. These open-ended tasks help students better understand the application of mathematics in the specific career, integrating knowledge and skills obtained previously and in the modules throughout the unit. For example, in Unit 2, the performance task asks students to find the cost of renovating a house using fractional measurements. This task also asks students to determine the efficiency of concrete performance against hourly temperatures of positive and negative values. In Unit 3, students are introduced to the primary focal area of ratios and rates using the career of a residential builder. They create and compare ratios determining paint budgets and gallons needed. Within this unit, other rate/ratio questions ask students to compare prices of juice at the grocery store and determine the number of cabins needed at a summer camp. Then, in the next lesson, students use ratios and rates to interpret and compare profits from a lemonade stand and to scale up a recipe. They must also determine equivalent ratios of projects at a science fair and describe driving rates and total distance. Later, Unit 5 focuses on “Relationships in Geometry” and how theater set construction workers use geometry when “scaling and building sets based on the dimensions of the models.” In the Unit 5 Performance Task, there is a pictorial model of a scenery design incorporating triangles and rectangles. Students focus on determining if a specific sized piece of wood would be enough to create the scenery, the actual number of square feet in the design, and the amount of paint needed to paint one side with two coats, given the amount of coverage provided by a can of paint.
These opportunities allow students to connect math concepts to the real world, and problems are routine in nature, providing students a clear path to replicate step-by-step methods previously learned within the unit—in isolated cases. Non-routine problems that require students to apply mathematics learned to this context are not included.
Because of isolated practice, the program incorporates minimal opportunities requiring students to integrate knowledge and skills to make sense of a context and lacks the opportunity for students to develop an efficient and successful solution strategy. In fact, the Front Matter of both the teacher and student editions lays out the Mathematical Process Standards. It gives a suggested problem-solving method of the following steps: 1. Analyze Information, 2. Formulate a Plan, 3. Solve, and 4. Justify and Evaluate. However, its application and use are not visible or supported consistently throughout the materials. For example, in Unit 2, Lesson 4.3, “Example 1,” the task compares finding half of 3.75 using decimal multiplication versus fraction multiplication. The materials guide teachers to point out the efficiency of one method since “the second solution seems easier because the steps are shorter. There is no need to simplify the fractions after you multiply.” While this supports students finding efficient strategies, it is an isolated example, and efficiency is not a significant focus of the materials, nor is the problem-solving model explicitly a focus of instruction or learning.
The Teacher Edition offers a “performance task” in Assessment Resources (not to be confused with the Performance Task connected to Careers in Math), which includes a rubric to evaluate students in making sense of the problem, creating a successful and efficient solution strategy, integrating knowledge and skills, and clearly communicating their reasoning. The first statement of the Student Scoring Rubric says, “Make a plan. If the plan does not work, change it until it does work.” For the Unit 4 Assessment Resources performance task, students integrate knowledge and skills within the focal point of expressions and equations to represent relationships. Students use previous understanding of perimeter and area to write expressions and equations that represent rectangular flower beds as lengths and widths are adjusted.
For the Unit 6 Assessment Resources performance task, students are provided a table of data and must determine mean, median, and interquartile range. Graphical representations—including a stem-and-leaf, frequency histogram, and box-and-whiskers plot—are created to display the data. One question within the task requires students to “explain why a bar graph would not be a good choice to represent the data.” These tasks isolate the knowledge and skills presented throughout the unit and focus on routine problems and guide the students to replicate previously learned procedural methods, limiting integration and application—the next steps in the learning progression.
At the beginning of each module, Real-World Videos are provided to “engage students with interesting and relevant applications of the mathematical content of each module.” Although materials offer students a chance to read and use real-world data, no opportunities for students to analyze data are provided. They do not have opportunities to connect with and communicate findings; in fact, what is called analyzing is simply calculating and following steps. Students do not use graphs and tables in a way that helps them better understand or draw conclusions about their world. Students mainly “examine” data to complete a specific task. The following examples provide evidence: For example, in Module 14, students are provided tables and graphs from which they identify independent and dependent variables. Students use real-world data pertaining to the constant speed of a freight train, the relationship between the amount of clay bought and clay used in art class, and the amount of savings in a bank account. Students then create their own real-world situations to represent data from tables and graphs. In Module 17, the Real-World Video shows how field biologists gather data and use statistics to make determinations about a species, and although a graph and specific data are shared, the opportunity for this data to be analyzed by students is not provided; the data’s use is limited to procedural tasks. Also in Module 17, Guided Practice provides RBI (runs batted in) data from the 2010 Seattle Mariners. While this data is real-world, students use it to create a box plot and determine the median, upper quartile, lower quartile, IQR, and range without an analysis of generalizations or conclusions which could be drawn based on the data presented. Additionally, students are not asked to collect real-time data in lessons.
The materials are not supported by research on how students develop mathematical thinking. The materials do not include cited research throughout the curriculum that supports the design of teacher and student resources. The materials do not provide research-based guidance for instruction to enrich educator understanding of mathematical concepts, and there is no research to support the validity of any recommended approaches. The program does not cite research to understand skill development in mathematics and does not reference application to Texas-specific context or demographics. There is no bibliography present in the materials.
Evidence includes but is not limited to:
The materials do not include a description of the design of the program. While the materials include a limited concrete-representational-abstract (CRA) approach to developing students’ conceptual understanding and procedural fluency throughout the units, there is no reference of cited research from the field of mathematics education or special education justifying this approach for lesson design. For example, in Unit 2 Lesson 5.2, adding integers with unlike signs is introduced by moving students from number lines and two-color counters to the abstract; however, no research-based information is provided related to this approach. The program does not provide cited research about the effectiveness of a specific model for teaching, such as graphic organizers. The program does not cite research on effective blended instruction.
The materials provide descriptions of the mathematical concepts within the materials to support educators in deepening their own understanding of the mathematics being developed but do not provide research to support these descriptions. For example, at the start of each unit in the Teacher Edition, the materials provide educators with detailed explanations and visuals for the trajectory of learning mathematics within each unit, including representations and contexts used during instruction, but do not cite supporting research. Additionally, at the beginning of each unit, Math Background provides educators with detailed explanations related to the mathematical concepts that will be introduced in the unit, including connections to prior understanding, vocabulary, examples/non-examples, pictorial models, and common misconceptions which guide teachers’ instruction; however, the information provided does not include any references to research that supports these approaches nor are contexts specifically included in the information. In Unit 1, while teaching comparing and ordering rational numbers, the only guidance to support teacher and student understanding says, “According to the Law of Trichotomy, given any two rational numbers a and b, exactly one of the following relationships must be true: a < b, a > b, a = b.” There is no evidence of mathematical research to support this information and no resources to consult for further explanation. While there is guidance in each unit’s Pacing Guide (how long to spend on each lesson), Program Resource page (how to plan for the lesson, introduce the lesson, teach the lesson, and assess the lesson), and Front Matter (how the Mathematical Process Standards are met), research is not cited to explain reasoning/justification for these suggestions.
The materials do not explain the validity of their approach to the development of mathematical understanding and the process standards and do not reference research-based instructional techniques. Lessons follow the 5E model: Engage, Explore, Explain, Elaborate, and Evaluate. Again, no research is cited to support the “validity” of this approach. The materials include specific tasks related to their integration of the mathematical process standards. For example, the “HOTS” (Higher Order Thinking Skills) and Reflect sections are included in the lessons throughout the materials and support TEKS 6.1.F related to analyzing relationships; however, the validity of these types of tasks is unconfirmed due to no cited evidence.
The TEKS Correlation within the materials provides citations specific to the “instructional resources that support all of the Texas Essential Knowledge and Skills for Mathematics Grade 6.” In addition, this section includes references within the materials specific to the English language proficiency standards (ELPS), which “outline English language proficiency level descriptors and student expectations for English language learners.” While this information supports a Texas-specific context and relates to the demographics of Texas students, the materials lack specific evidence of research used to design the program around these parameters.
The materials do not contain a bibliography that cites research presented throughout the instructional materials. The only items found in the Back Matter are a glossary, index, table of measures, formula chart, and a list of commonly used mathematical symbols. There is no bibliography or appendix to list cited research.
Although materials provide a problem-solving model that is transferable across problem types and grounded in the TEKS, the development of student ability to use and apply the model is lacking. Throughout the lessons, materials do not consistently prompt students to apply a transferrable problem-solving model. The materials do provide some guidance, prompting students to reflect on their sample problems, and some guidance provides teachers support for student reflection of problem-solving, but not of their own problem-solving applications.
Evidence includes but is not limited to:
The Mathematical Process Standards pages introduce a problem-solving model in Front Matter of the teacher and student editions, which is given within the context of the process standard TEKS 6.1.B and divided among the four parts. The process is listed as “Analyze Information,” “Formulate a Plan,” “Solve,” and “Justify and Evaluate.” Each category of the problem-solving model also includes guiding questions to better explain what the model means. For example, Analyze Information asks the following guiding questions: “What are you asked to find? What are the facts? Is there any information given that you will not use?” Next, the Formulate a Plan step asks, “What strategy or strategies can you use? Have you solved any similar problems before?” The third step in the problem-solving model, Solve, includes statements to remind students to follow their plan and show the steps in their solution. Finally, the Justify and Evaluate step asks, “Did you answer the question? Is your answer reasonable? Are there other strategies that you could use?” However, the problem-solving model in its entirety is only sometimes present, not used throughout every unit for all problem types. The prompts to analyze and justify are throughout, but the materials do not support the student to develop and practice the model using all four sections. These prompts are more aligned with 6.1F (analyzing mathematical relationships) and 6.1G (display, explain, and justify mathematical ideas) rather than 6.1B (using the problem-solving model). Some questions in the Higher-Order Thinking (HOT) section are labeled “Problem Solving,” but the support to practice the four parts is missing. The first time the problem-solving model is clearly introduced in the materials is Example 2 of Unit 2, Lesson 4.3. While Example 2 models the problem-solving steps for students “through a four-step problem-solving plan to solve a multistep word problem,” it is one of only four example problems (Lesson 4.3, Example 2; Lesson 5.4, Examples 2 and 3; Lesson 6.3, Example 1) provided throughout the 14 lessons in Unit 2 which explicitly integrate and provide opportunities to develop the problem-solving model.
The materials provide few opportunities for students to practice and apply the problem-solving model. On page TX2, the Correlation for Grade 6 includes a list of pages within the student edition where specific TEKS are addressed. TEKS 6.1.B, the use of a problem-solving model, can be found in ten lessons within the following modules: Modules 4, 5, 6, 12, 13, 14, 15, 16, and 18. This does not support the consistent development and practice of a problem-solving model for students. Additionally, students are not asked to complete this model on their own during independent practice. For instance, Module 4’s example uses the problem-solving model to solve a problem with rational numbers. Students analyze information by identifying the important information, which has been identified and listed for them. Then, students formulate a plan, which is also there for them to read. Next, students are prompted to solve; the materials provide step-by-step instructions on how to solve the problem. Finally, students are asked to justify and evaluate. A justification is provided but does not include an opportunity for students to evaluate the reasonableness of the solution. Immediately following the problem-solving model is an additional question labeled “Your Turn.” Students are expected to mimic the problem-solving model here, but no further guidance is provided for them to truly practice each part of the problem-solving model on their own.
Materials contain few guiding prompts for students (or for teachers) to apply a transferrable problem-solving model. No anchor charts are included of the problem-solving model, which would allow students a quick reference throughout the year. The Front Matter of the materials in the student and Teacher Editions include a graphic of the four-step problem-solving model as a reference, which includes question prompts for each step; however, no prompts encourage the use of this resource. For example, “Practice and Problem Solving” worksheets do not have prompts or guides to help students remember to use the problem-solving model for all problem types. These prompts are only used in specific lessons where the problem-solving model is presented. Students are frequently asked to answer questions involving problem-solving but make no mention of using the model to analyze information, formulate a plan, solve, then justify and evaluate. There were three total instances found in the materials where the students were prompted to utilize a problem-solving model. In Unit 2 of the materials, two of the 186 Independent Practice questions explicitly encourage students to “use a problem-solving model.” Although these two problems focus on the entire problem-solving model, it is important to note that there are numerous problems within each lesson that focus on a specific part of the model, although the connection to the model is not explicitly made. The answer key does not walk the teacher through the problem-solving model steps; it just gives the answer. At other times, the opportunities provided are in segregated pieces. For example, in Module 1, after the “Explore” activity, the student is prompted to “Analyze Relationships.” Later, in the same lesson, students are prompted to “Justify Reasoning.” Students focus on justifying their reasoning in one Module 6 question, which is a part of the problem-solving model; however, a connection between justification and the problem-solving model is not made. Later, in one Module 8 question, the teacher guide includes prompts, encouraging students to analyze information saying, “sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.” Although this is the initial step in the problem-solving plan, the materials do not explicitly make this connection.
The materials provide minimal prompts for students to reflect on their approach to problem-solving. When the materials do prompt reflection, students are not reflecting on their own personal processes for solving a problem; the reflections address the context and not the approach to problem-solving. Although the guiding questions associated with the problem-solving model include “Are there other strategies that you could use?” (in the Justify and Evaluate portion), no evidence was found of prompting students to reflect on their own approach to problem-solving. However, the materials do provide opportunities for students to reflect on provided problem-solving approaches. For example, in Example 1 of Module 4, “students examine two possible solution methods and decide which method they prefer.” Also, Module 5 “Math Talk” prompts students to reflect to determine if they can “use the same procedure you use to find the sum of two negative integers to find the sum of two positive numbers?” Additionally, throughout the materials, “Reflect” problems are incorporated, requiring the application of conceptual understanding to a new problem-solving situation, but not on the approach used. For Module 2, one of the reflections asks students to “name two integers that are not also whole numbers.” In the next lesson, one question asks, “How would you graph -2.25? Would it be to the left or right of point D?” Later in Module 14, Reflect asks students to “identify the dependent and independent quantities” in the scenario provided. These reflections do not ask students to think about their problem-solving approach.
Almost all guidance provided for teachers, as stated above, encourages a reflection on contextual understanding and not on the student’s ability to use and apply a problem-solving model.
The materials provide some opportunities for students to use appropriate tools for the task and concept development. Throughout the materials, there are opportunities for students to use real objects, manipulatives, representations, and algorithms as appropriate for the stage of concept development, grade, and task; however, the selection of these tools is not left to the students. In addition, the materials provide opportunities for students to use technology (e.g., calculator, graphing program, virtual tools) as appropriate for the concept development and grade; again, their selection is not clearly provided within the tasks. Additionally, the materials provide teachers with little guidance on tools that are appropriate and efficient for the task.
Evidence includes but is not limited to:
Students use representations from the grade-level TEKS to solve tasks and enhance their understanding of concepts by exploring mathematical ideas and making/testing conjectures. For example, in Module 3, students use paper and two colored pencils to explore and understand how to represent multiplying fractions. Later, in this module, they use pattern blocks for dividing fractions. In Module 5, students complete sentences that describe the action when using a number line to model the addition of integers. In the “Explore” activity, students learn to use the number line tool and follow step-by-step instructions for modeling sums of integers using two-color counters marked with “+” and “-” signs.
The materials do not explicitly provide students opportunities to select grade-appropriate tools for solving tasks. In “Multiple Representations” in Lesson 1.1, the task asks students to “explain how to graph the elevation of Morning Glory Stream on a number line.” It is telling students which tool to use, not allowing students to select the appropriate tool. In Lesson 1.2, the “Professional Development” section states that “This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C. In the “Explore Activity” and in both “Examples,” students use a number line to order and compare integers….” In the same lesson, one question asks the student, “How can you use a number line to compare and order numbers?” Multiple Representations again tells the students exactly which tool to use when it says, “explain how to use a number line to find the opposites of the integers three units away from -7.” In Lesson 3.1, the Professional Development box says that this lesson addresses TEKS 6.1C, “‘select tools…including mental math, estimation, and number sense…to solve problems.’ In the Explore Activity, students use a model to find the product of two fractions. Then, in Example 1, students use an algorithm to multiply two fractions. Finally, in Example 2, students estimate and then use the algorithm to multiply a fraction and a whole number by rewriting the whole number as a fraction. These examples given do not allow the student to select the tool because they are told what to do. Then in Lesson 9.1, the materials note that one particular question addresses 6.1C; however, the problem tells the student which tools to use to explore percentages: grids, percent bar models, and proportions. Students, though, are not prompted on how to select between these tools. Instead, questions tell students to use a specific tool versus making a choice: “Shade the grid to represent the ratio” and “Use the percent bar model to find the missing percent.” This same scenario frequently occurs where the materials tell students which tool to select.
There are a few opportunities that allow students to choose the most appropriate tool for a task. In Lesson 6.1, when students are multiplying integers, three Guided Practice questions suggest “Students can choose to use counters or number lines or both to model the products.” In Lesson 15.1, students are asked in “Reflect” to “try to make triangles using real-world objects such as three straws of different lengths.” Later in the lesson, students are asked in “Independent Practice” to “choose a real-world object that you can cut into three different lengths to form a triangle.” In Lesson 18.4, students must calculate the effects on lifetime income in Example 1 of the lesson. Students answer this Reflect question: “What are two tools you could use to find the total income over 30 years?”
The materials provide students some opportunities to learn to use grade-appropriate tools for solving tasks and understanding concepts. In Unit 1 in the teacher and student editions, there are links to integer counters, fraction bars, fraction decimal grids, bar models, geometry sketcher, algebra tiles, graphing calculator, and scientific calculator. However, not all tools are explicitly taught, and choice in regards to which tool to use is not provided for students within the context of the tasks/questions. In Lesson 1.2, the statement says the lesson addresses 1C (select tools) but are there no opportunities to learn to use tools. The student can click on a question mark, which brings up instructions on how to use each tool to model a task. The only clear guidance provided is to explain how to use virtual two-color counters. Other tools, such as the bar models used for fraction division in Module 3, are presented but not specifically taught. Without a depth of understanding, students cannot effectively use the tools to explore mathematical ideas and make conjectures.
Each module does include one “Animated Math” activity, which provides online, interactive tools and simulations for students. The materials provide tutorials and prompts to support student learning of the tools: dynamic number lines, bar models, calculators, tables, dynamic graphs, dynamic percent bars, algebra tiles and dynamic parallelograms/trapezoids, dynamic box plot creator, and dynamic check registry. Activities in Animated Math use the tools to explore concepts such as rational/irrational numbers, dividing fractions, estimating products, unit rates, ratios, percents, equivalent expressions, area formulas, box plots, and check registries. For example, in Module 2, the teacher guide shows where to use Animated Math during the lesson and provides an explanation of the tool: “Students build fluency with ordering rational numbers in an engaging scoring game with a dynamic number line.” While these models are representations from the grade-level TEKS, it’s important to note that these tools are specific to a prescribed task and include minimal instructions on their use, which limits their application across concepts, as well as an understanding of their use.
The materials provide some teacher guidance about each tool, but the teacher guide does not explain the purpose of each tool, when it is introduced within each grade, and how it connects to the TEKS. For example, Unit 1’s “Professional Development” offers guidance on using number lines for use with integers, and in Unit 2, the guidance is on using counters for operations with integers. Other units guide the teacher through using pencil and paper as well as mental math, but the materials do not explain which tool is more appropriate or more efficient for a task. In Lesson 5.1, teachers ask students, “What are some advantages and disadvantages of a number line model over a counter model?” The sample answer states, “It would be easier to add or subtract larger numbers on a number line. With counters, you don’t need to place a number correctly or decide which way to move.” This explanation states which tool is more efficient but does not explain which tool is appropriate and does not provide a detailed explanation of the affordance and constraints of each tool.
“Math Background,” in the teacher’s materials, provides a description of tools introduced in each lesson; however, the teacher notes lack the detail to help guide supporting students choosing a tool. For example, a description of a grid model in Unit 3 is provided, telling teachers why this is an appropriate tool for percentages. The material states, “Percents compare numbers to 100 so that 12% means “12 out of 100” or 12/100. This can be represented by shading 12 of the 100 squares on a 10-by-10 grid. Students can use this representation to help them develop a mental image of 12%. For example, suppose 12% of a state park is a redwood forest. This means that if the area of the entire park were represented by 100 squares, 12 of the squares would be covered with redwood trees.” In Unit 2, guidance is provided for the integration of a calculator as a tool within the materials. Teacher materials from Lesson 5.3 encourage providing the opportunity for students to “explore addition and subtraction of integers on a calculator. Help them to distinguish between the calculator’s subtraction key - and the opposite or negative sign (-). Point out that most calculators color code the operation keys so that the addition and subtraction keys look the same but very different from the opposite key (-). Encourage students to try subtracting with the opposite key to reinforce how the keys are distinct. Then have students rewrite each difference and check both expressions on a calculator.” Also, in Unit 3 Lesson 9.3, the “Focus on Technology” section of the teacher materials prompts teachers to “discuss with students how they might use a calculator to find a percent of a number. Guide students to see that they can write the percent as a decimal and then multiply the given whole number by the decimal. For example, 20% of 360 = 360 × 0.2 = 72. Most calculators also have a % key. You can enter 360, press ×, enter 20, and press %.”
The materials do not meet the requirements of this indicator. Although the materials provide multiple appropriate strategies for students to solve mathematical problems, opportunities for students to self-select appropriate strategies or techniques for given tasks do not exist. Instead, students are often prompted on how to solve problems. Additionally, they do not have opportunities to use two or more approaches at once, nor are students asked to describe similarities and differences between approaches without teacher guidance. The materials provide minimal support for teachers in understanding the appropriate strategies or guiding students to apply more efficient ones.
Evidence includes but is not limited to:
Mental math, estimation, number sense, generalization, or abstraction techniques are used sparsely throughout the materials, and when they are used, the materials do not prompt students to select an appropriate technique for solving tasks. Students are provided the specific technique to use, without an opportunity to choose their own. For example, the use of estimation as a technique is encouraged through prompts in the example and practice problems in Unit 2. Lessons instruct students with the following: “Estimate. Then solve.” or “Estimate to check whether your answer is reasonable.” Later in the unit, students estimate products or are prompted to determine how estimation can be used to check for reasonableness. None of these allow students to select a technique. (It is important to note that estimation is limited mainly to this unit and is rarely present throughout other units. In the remaining six units, the materials include very few prompts for students focused on the use of estimation.)
Evidence was found to support teachers prompting students to use a specific technique, but, again, there was no evidence found related to support students to select an appropriate technique. In Unit 2, where estimation is the main provided technique, materials prompt the teacher to provide the strategy. A few examples from the unit include “Remind students to estimate the product first,” “Encourage students to always estimate the quotients when dividing mixed numbers before they actually solve the problem,” and “Encourage students to estimate the answer before actually solving it.” Later in Unit 2, the teacher guide prompts teachers to “Encourage students to use mental math for this type of exercise.” Another states, “When estimating, if you round both of the factors up, what can you say about your estimated answer?” “Focus on Reasoning” in this unit reminds teachers to “encourage students to estimate the answer before actually solving it.”
Although these prompts offer support throughout the Teacher Edition, materials do not outline the importance of the strategy in a student’s mathematical learning trajectory with explicit connections to the TEKS. Teachers do not learn the reasoning of early less-efficient strategies in supporting students’ early conceptual work. There is never a mention of why one is more appropriate than another. The “Math Background” at the beginning of each unit provides teachers with some strategies to use within each concept but does not discuss the importance of beginning with specific visual models before moving on to the abstract and conceptual strategies to develop students’ understanding. Throughout the materials, efficiency is addressed one time in Unit 3, where Math Background explains that for TEKS 6.4.H, “conversion factors are an efficient tool for converting units. In this lesson, they are used to perform conversions within the customary system (for example, inches to feet or pounds to ounces), but they may also be used for conversions within the metric system or for conversions between systems.” Another Unit 3 Math Background provides guidance on how estimation can be used to estimate tips. However, the guidance includes “tricks” such as “move the decimal point of the amount of the bill one place left to find 10%” without building understanding.
The materials do provide questioning prompts for the instructor but only assist in generalizing solving techniques. An example from the teacher guide for Lesson 5.1 prompts teachers to ask, “What do you think is the most efficient way to add two integers that have the same sign?” Then, in Lesson 5.2, teachers have students compare the answers in sets of exercises for adding integers. Teachers prompt students to make generalizations about the rules for adding integers with different signs. Then they have students compare generalizations until they reach a consensus. In Lesson 12.3, “Number Sense” in the teacher guide encourages “Have students use a fraction bar to indicate division of both sides of the equation” and in “Questioning Strategies,” the materials prompt teachers to ask, “Using number sense, what can you determine about the value of x given that the product of two numbers is 72?” In Lesson 13.2, Number Sense guides teachers to encourage students to use number relationships to write equivalent inequalities. These examples fail to develop teachers’ understanding of strategies appropriate for solving a task.
The materials include opportunities for students to learn multiple appropriate strategies for solving problems. For example, in Lesson 5.1, students use counters to add positive and negative integers, a number line to add negative integers, and absolute value to add two same-sign integers. In Lesson 9.3, students use bar models, proportional reasoning, 100’s grids to solve percent problems. In Lesson 12.2, students use algebra tiles and number lines to model the solutions to one-step equations. They proceed to solve algebraic equations symbolically by using inverse operations to isolate the variable. In Lesson 16.1, connections to area of rectangles and area formulas are used to determine the area of parallelograms and trapezoids; then, in Lesson 16.2, students use grid paper to derive area formulas for triangles. Next, students use words, symbols, diagrams, and the area formula to represent and calculate the areas of triangles.
However, there are no tasks requiring students to solve a problem using at least two different approaches learned within the unit, and students are not directly asked to describe the similarities and differences between the two approaches. The only evidence found involves teachers asking students to compare approaches. For example, in Lesson 3.1, teachers ask students, “How are multiplying two fractions and multiplying a fraction and a whole number similar and different?” In Lesson 11.3, teachers ask students, “How are algebra tiles similar to counters? How are they different?”
Overall, materials are teacher-driven, and students do not have the opportunity to select appropriate strategies, thinking through grade-level math work and concept development.
The materials do not meet the criteria for this indicator because the materials do not support students seeing themselves as mathematical thinkers who can learn from solving problems, make sense of mathematics, and productively struggle; the materials do not support students in understanding that there can be multiple ways to solve problems and complete tasks, and the materials do not support and guide teachers in facilitating the sharing of students’ approaches to problem-solving.
Evidence includes but is not limited to:
The materials provide a guide in Front Matter—for students to see their role—called “Succeeding with...Math.” Students see the components of the materials where they “actively participate” in lessons, engage in “Explore” activities, try “Your Turn” exercises to check for understanding, scan QR codes to watch “Math On the Spot” tutorials, and check for mastery by completing “Texas Test Prep” questions. Additionally, the guide shows students how to enhance their learning with an online assessment and intervention tool, “Personal Math Trainer,” and “Animated Math” activities for interactive exploration of key math concepts and skills. However, the materials do not support the development of a growth mindset. Classroom norms are not provided to support a shift in the authority of mathematics from the teacher-and-textbook to the classroom community. Students rarely share their strategies and approaches with classmates, nor do they communicate or respond to their peers’ thinking or problem-solving. While the materials do provide opportunities for some collaboration, the development of a mathematical community is not inherently introduced or supported as there is no guidance for teachers on how to engage all students, especially those reluctant to participate. There is no guidance on small group discussions, and the whole group discussions do not involve making sense of mathematics. The following is a rare example guiding teachers to provide discourse around solution strategies: The “Differentiate Instruction” section of the teacher guide includes “Cooperative Learning” in some of the lessons, which provides an additional grouping activity; however, the opportunity to focus on alternate strategies or compare strategies was rarely found. In Lesson 7.2, Cooperative Learning guides teachers to “provide prices for different packages of competitive products from several stores, for example, 24 oz for $4.80 or 36 oz for $5.40. Have students work in pairs to find the unit rate for each item and to determine which items are the best buys. Then have pairs compare their work and explain their steps.”
The materials do not include tasks designed to support the development of confident problem solvers that struggle productively, making sense of the problem, and solving it. The materials do not provide low floor, high ceiling tasks for students to make mistakes and ask questions. The materials only provide opportunities to correct mistakes in word problems from other “textbook students,” but the tasks themselves are suggestions of what students might do and not actual errors current students have made. For example, “Avoid Common Errors” highlights common, anticipated errors/misconceptions and provides guidance for alerting and/or guiding students to avoid these misconceptions. Lesson 1.2 warns, “when students work with negative numbers, they often think that the number with the greater absolute value is the greater number. You may want to remind them that for negative numbers, the number with the greater absolute value is actually the lesser number because it is farther away from zero in the negative direction.” Another example from Lesson 17.1 reminds teachers that “students may forget to list 8 two times when ordering the data values. Suggest that they count the total number of numbers in the original data set and then count the number of numbers in their reordered list. Each list should include the same number of data points.” Because teachers are not prompted to address “real-time” errors, the characteristics of being a learner of mathematics are not specifically addressed and do not foster an intentional growth mindset as students may not connect these errors to their own.
The materials highlight mathematical careers focused on the work of mathematicians, but these do not necessarily enable students to see themselves as mathematical thinkers. At the beginning of each unit, materials include “Careers in Math,” which provides an opportunity “to apply mathematics to problems arising in everyday life, society, and the workplace.” “Real-World Videos” are provided for students to see and hear these careers in action. However, these are not focused on developing efficient ways of solving problems. A “Performance Task” is also included at the end of each unit for students to work through problems involving the highlighted career, concepts, and skills. For example, in Unit 1, the career is a climatologist. The question states: “Each year a tree is alive, it adds a layer of growth, called a tree ring, between its core and its bark. A climatologist measures the width of tree rings of a particular tree for different years: The average temperature during the growing season is directly related to the width of the ring, with a greater width corresponding to a higher average temperature.” The questions are related to a table of tree ring widths compared to the year that a climatologist measured them. The students list years in a) increasing width, b) which year was hottest, c) which year was coldest. Another example from Unit 2 Careers in Math highlights how chefs “use mathematics when scaling recipes and converting units of measure, as well as in budgeting and financial planning” and includes an image of chefs from different ethnicities; however, beyond the image, there is no specific connection to these individuals in the materials. Additionally, there is no indication the purpose of this section is to encourage students to see themselves as doers and thinkers of mathematics. The teacher guide includes a link to the American Mathematical Society website as an additional resource “for more information about careers in mathematics as well as various mathematics appreciation topics;” the link is to the home page for the site without guidance to the specific materials referenced.
The materials provide very few tasks designed to allow for multiple pathways to a solution by applying the tools and procedures they are learning. Students see multiple pathways for solving a problem through the examples in the materials. However, there are no discussion questions or prompts to compare the pathways between students and to orchestrate a productive mathematical discussion. For example, in Unit 4, Lesson 11.1, students are shown how to simplify an expression using a balance scale. In the teacher notes, an additional suggestion is shown to use a strip diagram to simplify an expression. The Unit Performance Tasks (separate assessment Performance Tasks, not related to Careers in Math) include open-ended questions for students to apply their prior knowledge, and according to the student rubric, the students should use different methods and models to help find the solution. However, the teacher rubric does not mention multiple methods, nor do the performance tasks actually require or ask for multiple methods. Given that multiple strategies are not taught or encouraged throughout the materials, these tasks tend to lean toward mimicking procedures versus deep exploration of alternate strategies.
Instructional routines are not explicitly included in the materials. However, it is important to note that most of the lessons follow the “I do—We do—You do” routine which the National Council of Teachers of Mathematics (NCTM) notes, “focuses on doing processes and procedures with little understanding of how and why they work or the appropriate use of different processes and procedures and how they can be applied in varied mathematical situations. In addition, lessons that follow this routine tend to “focus on mimicry and memorization.” In addition, the problem-solving process outlined in the Front Matter is only used a few times throughout the materials. The materials do not provide instructional routines designed to provide greater access to a problem. The Explore Activity typically models a rudimentary breakdown of a specific strategy used for the concept or skill. This suggests a focus for students to align with few solution strategies with no suggestions for setting up a task in ways that encourage divergent solution strategies from students. Mathematics as an act of creativity and experimentation does not exist. For example, in Lesson 12.2, algebra tiles are used to solve one-step addition and subtraction equations where guidance is provided on the use of the tiles earlier in the unit. No alternative strategies are provided in the materials. After the one Explore Activity, inverse operations are used, and no connection is made to the algebra tiles or the multiple solution paths that could be used when solving one-step equations.
Some opportunities exist for students to effectively communicate mathematical ideas, reasoning, and their implications using multiple representations. However, the use of these representations to communicate their thinking with others is not prompted; in fact, there is no indication that students’ communication extends beyond self, other than whole class discussion. In addition, the opportunity to develop vocabulary exists, but there is a lack of teacher guidance to develop written mathematical communication since there are no consistent prompts for students to communicate ideas and reasoning. Because this indicator focuses on communication, including writing, it only partially meets grade-level expectations.
Evidence includes but is not limited to:
The materials provide numerous opportunities for students to communicate mathematical ideas and representations using visual, physical, contextual, verbal, and symbolic representations. For example, in Module 2, students use number line models to represent positive and negative numbers. Students also use number lines to explain real-world situations using positive and negative numbers. Students begin to organize values in a table, on horizontal number lines, and on vertical number lines to identify opposites and absolute value of rational numbers. Students present their understanding of graphing integers and rational numbers on the number line in writing when answering the “Reflect” questions. Then in Module 3, students use bar models to represent fraction division. Students write about reciprocals, and teachers provide additional questioning strategies for students to summarize their connections between multiplying and dividing fractions. Additionally, students use pattern blocks to model division when teachers follow the differentiated instruction plans, and students are invited to explain orally why two expressions are equivalent.
The “Multiple Representations” tasks that appear throughout the material provide opportunities for students to represent their thinking in multiple ways and can be solved using a variety of representations. Students can use models, pictures, or verbal descriptions to solve tasks. For example, in Module 1, students use number lines and integers to represent opposites; however, the material lacks prompting for these representations to be shared with others. In Module 3, students draw a model to illustrate the multiplication of fractions. While this lesson contains tasks that ask the students to communicate their thinking, there is no evidence of how students show their thinking to others. In Module 5, students use two-color counters to support their understanding of integers, yet they are not using these counters to communicate with others. In Module 11, students use bar models to represent the temperatures of two cities as algebraic expressions. The materials ask students to justify the equivalency of the two different models; however, the visibility of their thinking to others is undefined.
The materials support teachers in developing students’ use of mathematical vocabulary at the beginning of each unit in “Reading Start-Up,” where important mathematical vocabulary is listed. Graphic organizers, such as main idea webs, introduce vocabulary. Students visualize and understand vocabulary and create a foldable to add important ideas and vocabulary as they work through the module. In Module 11, students create a “Key Term Fold” as a tool for learning vocabulary. Guides include prompts for the teacher related to the use of the graphic organizer. For example, in Unit 2, Module 4, the teacher guide prompts teachers to “discuss other attributes of division that can be added to the chart.” Each unit also contains a vocabulary preview that is a word puzzle of some kind like a word search, an “unscramble the word,” or a crossword puzzle. Additionally, the teacher guide includes “Connect Vocabulary,” which guides the teacher in making connecting relationships for vocabulary development. An example from Module 1 prompts, “Point out to students that when working with money, a loss or a debt can be represented by a negative number.”
Students are provided opportunities to share thinking with “Communicate Mathematical Ideas” tasks, which are included in the introductory activities, in examples, and in “Independent Practice.” In Module 1, students share their understanding of integers by answering, “Which number is farther from 0 on a number line: -9 or 6? Explain your reasoning.” In Module 10, students communicate an understanding of prime factorization when they answer, “If one person uses a ladder diagram and another uses a factor tree to write a prime factorization, will they get the same result? Explain.” Module 17 asks students to describe how the data are distributed, while another question asks students to explain their thinking as to whether the mean or the median is a better description of exam scores. In the same lesson, a question asks students to make a recommendation. In Module 18, students express their understanding of ways to pay for college: “Which methods would you choose to pay for college? Explain why you would choose those methods.” Additional communication opportunities come from “Math Talk,” where students answer questions like “Is there a greatest integer? Is there a greatest negative integer? Explain.” and “The ratio of apple juice to grape juice in a recipe is 8 cups to 10 cups. How can you find the amount of each juice needed if the recipe is cut in half?”
The materials also provide opportunities for students to take notes and share their mathematical ideas in the online, write-in Student Edition. This version of the student edition allows students to digitally take notes and answer questions posed in lessons. The materials allow students to save their notes and responses to their online “My Notebook” tool by clicking the checkbox as they use the digital write-in feature.
The materials provide some suggestions for teachers on ways to support students in orally expressing their mathematical ideas; however, there is no guidance for teachers on how to use writing to develop reflection. For example, the teacher guide includes a “Focus on Math Connections” for teachers to point out important steps and information throughout a lesson. The Teacher Edition includes “Avoid Common Errors,” where teacher prompts remind students of pertinent protocols when working with numbers and operations, especially when using symbols. The “Essential Question” and “Questioning Strategies” included in the teacher guide provide opportunities for teachers to prompt students’ sharing their mathematical ideas orally through the inclusion of open-ended questions that often ask students to explain or justify their thinking. Also, questioning strategies exist that support students’ reasoning with representations, but they generally do not include multiple representations. For instance, Unit 1 Talk About It asks, “How do you find the opposite of an integer?” Talk About It in Lesson 1.2 asks, “How is a number line used to compare and order integers?”
At times, Focus on Communication (also in the teacher guide) provides discussion prompts such as this one provided in Lesson 5.2: “Have students discuss whether number lines or color counters would be a good way to model each sum. Ask students to explain their reasoning.” In general, though, this section guides teachers with reminders for students more often than prompts for discussion. In Unit 2, five of the seven prompts relate to sharing mathematical ideas orally, whereas the other two prompts do not elicit sharing of ideas orally or in writing and were simply reminders to students, such as “remind students that the multiplication sign means ‘groups of.’” In the other units, the Focus on Communication sections prompt discussions less. In Unit 1, one of three prompts suggests discussion, one of four in Unit 3, one of six occurrences in Unit 4, zero of four in Unit 5, etc.
Another guide, “Curriculum Integration,” includes few prompts to incorporate writing as a way to express students’ mathematical ideas. For example, in Lesson 18.2, the materials suggest “students research credit scores. Then have students write a paragraph that summarizes the factors that may lead to a lower or higher credit score and factors that can improve a credit score.” Most prompts that are designated as writing assignments could just as easily be completed orally because no specific details guide the teacher in written instructions, expectations, or scoring.
The materials do not meet the criteria for this indicator. Although there are opportunities for students to engage in mathematical discourse, discussions are not structured to develop and strengthen content knowledge and skills. Some opportunities allow students to engage in a variety of settings (e.g., whole group, small group, peer-to-peer), but not necessarily to develop understanding. Additionally, these opportunities are sporadic and lack guidance for teachers on how to facilitate and integrate discussion. Materials do not integrate discussion throughout lessons to support students’ development of content knowledge and skills as appropriate for the concept and grade level.
Evidence includes but is not limited to:
Discussion prompts are ample. A few come from “Focus on Communication,” which prompts teachers, in Lesson 1.3, to “Discuss with students why an absolute value of $25 may be used to describe a −$25 change to the balance of a gift card.” Lesson 2.3 prompts, “For Exercises 12–20, have students discuss the methods they used to order the numbers.” In Lesson 5.2, teachers say, “Have students discuss whether number lines or colored counters would be a good way to model each sum. Ask students to explain their reasoning.” However, the design lacks intentional opportunities for all students to discuss mathematics during every lesson with partners, small groups, and/or the whole class.
Though grouping strategies are suggested at times, they are always mentioned as an option and not fully developed as a strategy. For example, in Module 5, an “Extend the Math” Pre-AP section suggests students play a game in pairs; the final statement of the directions has students compare and discuss their final answers. “Connect Vocabulary” says, “Have students discuss whether number lines or colored counters would be a good way to model each sum. Ask students to explain their reasoning” (Module 15). At times, “Visualize Vocabulary” recommends “As a class, brainstorm other terms that can be added to the chart and discuss their definitions.” Teachers would need training beyond the materials to implement grouping strategies. In “Questioning Strategies” and “Talk About It,” notes suggest the teacher engage students in discussion, but no mention is made of grouping for these activities. Each module begins with a “Vocabulary Preview,” but the teacher guide includes generalized grouping prompts like “Students may work individually, in pairs, or in groups.” There is no specific guidance. While each lesson begins with an “Essential Question” that could serve as a springboard for discourse, the materials do not include explicit guidance related to the integration of different groupings for discussion. (In addition, the Essential Question sometimes focuses on procedures and not developing and strengthening understanding—like Lesson 3.3, “How do you divide fractions?”) “Communicating Math” recommends, in Module 17, having students create a presentation to compare and contrast the mean and the median of a data set. The materials state that “students can work alone or in groups.” There is no specific guidance. When Integrating the ELPS, the teacher guide suggests teachers may want to pair up English Learners with a partner for portions of the lesson to help them develop their language skills; no further suggestions are made on what type of students should be chosen for the pairs. When differentiating instruction, the materials suggest in some lessons that teachers orchestrate cooperative learning and opportunities for the use of manipulatives by having students work in pairs, but, again, there is no set structure as to when grouping will take place or how.
The materials do not intentionally support discussion throughout all phases (beginning, middle, and end) of content and skill development. In Unit 1, for example, (the first time “discuss” is used in the materials), the prompt is in the Explain phase of the 5E model (Engage, Explore, Explain, Elaborate, Evaluate). There is another prompt to discuss loans during the “Home Connection” in the “Differentiated Instruction” piece of this same lesson, but it is not part of the 5E model/lesson. “Discuss” is seen later in Unit 1 in the Elaborate phase of the lesson. In other words, the discussion pieces are there, but they are sporadic throughout the lessons. Additionally, when introduced to a concept, students are asked an Essential Question but are not guided on how to engage in a discussion. The teacher is prompted to ask a question to motivate the lesson but is not directed in how students should discuss the topic. There is no mention of whether students should answer these questions on their own or discuss in small groups. An effort is made to integrate discussion throughout the lessons but does not necessarily address the development of content knowledge and skills. Then “Talk About It” prompts teachers to ask a question to summarize the lesson; the questions make an effort to conclude the learning but do not focus on the development of efficient and accurate skills for solving problems. For example, in Lesson 5.1, students begin developing their understanding of integer addition through the use of two-color counters. “Reflect” tasks prompt students to independently think about adding integers with like signs; however, the materials do not include guidelines for the discussion or sharing. As students move to adding integers with like signs on a number line, the teacher guide includes more Reflect questions and additional questions related to “advantages and disadvantages of a number-line model over a counter model” and adding “two positive integers on the vertical number line shown,” which could be used for discussion, but guidance for their use is not included in the materials. Talk About It prompts teachers to ask, “If the temperature is -3 °F, why is a drop of 4 degrees like adding -4 °F to -3 °F?” As students build toward solving integer expressions with the same sign, the teacher guide includes additional questions that prompt students to determine the most efficient way to add two integers that have the same sign. While these questions are available at the beginning, middle, and end of the lesson, there is no evidence that students discuss these questions with others.
The materials do not offer guidance for teachers on how to structure discussion that is appropriate for the grade level. Materials do not provide a guide for creating norms and expectations for classroom discussions about math. Materials do not offer guidance for a grouping structure for class discussions or building community. Materials do not offer anchor charts or suggest posters to teach the class how to “actively listen” or “respond to others’ ideas.” Materials do not include sentence stems or prompts for students to use for different types of responses in math discussion; there are no stems such as, “I agree with...because….” or “I noticed/wondered….” Materials do not provide teachers with a rubric to utilize to provide feedback for students when discussing mathematical concepts and ideas. The materials lack best practice guidance related to key elements of classroom discussions, as suggested by John A. Van de Walle et al., in Elementary and Middle School Math: Teaching Developmentally. There are no productive talk moves (i.e., revoicing, reasoning, waiting, rephrasing, elaborating), teacher actions (i.e., anticipating, selecting, connecting, monitoring, sequencing), or encouraging student dialogue and questions.
The materials provide some opportunities for students to construct arguments and justify mathematical ideas using multiple representations and precise mathematical language. However, the materials lack support for teachers to facilitate students’ construction of their arguments using grade-level appropriate mathematical ideas.
Evidence includes but is not limited to:
Throughout the materials, tasks are identified as aligning to TEKS 6.1.G, “incorporate using precise mathematical language in written or oral communication.” There are numerous opportunities throughout the materials for students to justify mathematical ideas, but almost all are represented with words, not multiple representations. For example, in Lesson 1.3, a question in “Independent Practice” says, “Angelique says that finding the absolute value of a number is the same as finding the opposite of the number. For example, |-5 | = 5. Explain her error.” In Lesson 3.1, one question prompts students to “compare simplifying before multiplying fractions with simplifying after multiplying the fractions.” Another question from this same lesson asks, “When multiplying a whole number by a fraction, the whole number is written as a fraction by placing the value of the whole number in the numerator and 1 in the denominator. Does this change the final answer? Explain why or why not.” During Lesson 5.4, students solve problems with integers, and following one example in the lesson, students are asked to write a description of another way to find the solution. In Lesson 13.4, students complete a table and determine patterns when investigating inequalities. Next, students are asked to write what they noticed and also write how to make inequality statements true. Prompts like this provide students the opportunity to construct arguments based on their own mathematical understanding and experience. In addition, “Communicate Mathematical Ideas” and “Critique Reasoning” provide students opportunities to explain and justify their thinking with words. In Lesson 15.2, a question prompts students to “explain how you can use the figure to find the sum of the measures of the angles of quadrilateral ABCD” and then “How can you show that your guess is correct?” The students also have the opportunity to construct and present arguments using multiple representations during the “Explore and Explain,” “Justify Reasoning,” “Analyze Relationships,” “Persevere in Problem Solving,” and “Critical Thinking” tasks.
The materials include rare opportunities for students to justify using methods other than words, such as in Lesson 7.1, where a question says, “Draw a model to represent the ratio 1 to 3. Describe how to use the model to find an equivalent ratio.” Then in Lesson 16.1, the Explore Activity and Examples ask students to use symbols, geometric labels, definitions, and consistent and precise language to determine and describe the areas of parallelograms, trapezoids, and rhombuses.
The materials support students sharing their ideas with peers orally by providing students opportunities to respond to questions, but these are geared toward answering questions as a whole group. Also, the materials lack the guidance of clearly defined structures and routines to help students think metacognitively about their argument and the arguments of others; in fact, they are not prompted to think and write before sharing ideas. The materials do not provide opportunities for students to engage in discussions where they present their fully developed arguments. While some of the Explore phases do guide students through justifying their thinking, there is no guidance for students to process their thinking through discussion with others, and the structure for the sharing process is not explicit in teacher materials. Although the problem-solving model incorporated in the materials includes a Justify and Evaluate step, given that the model is not explicitly reinforced, the relevance of these occurrences is lessened.
“Questioning Strategies” guide a student's understanding of the topic being presented and the strategies and processes used to find the solution. Before moving on to the next example or activity, students are provided an opportunity to “Reflect” and communicate their mathematical ideas. However, materials do not list discussion questions and sentence stems to elicit different types of responses from students as they present their arguments. Students are not asked additional questions during the discussion to deepen their understanding, to critique an idea, or to develop their explanations. Beyond the Questioning Strategies, no additional supports assist teachers in facilitating students to construct arguments. Students are often asked to justify their reasoning within the independent practice, but there are no guidelines for the teacher on how to help students develop their argument when justifying their reasoning. Lessons do not suggest time be set aside to have students read from their written arguments to partners before whole-class discussions. Discussion questions and sentence stems eliciting different types of responses from students do not exist. There are no prompts such as, “When will that strategy work?” or “Why do you think that is true?” or “Do you agree or disagree?”
Materials include limited developmentally appropriate diagnostic tools (e.g., observational, anecdotal, formal) to measure all content and process skills for the grade level, as outlined in the TEKS and Mathematical Process Standards. Very little guidance is available for teachers and students to monitor progress, and there are no tools for students to track their own progress and growth. No guidance is provided to ensure consistent and accurate administration of diagnostic tools. In all, while the materials provide some tools, there is a lack of variety, student-tracking, and guidance for teachers to ensure consistency.
Evidence includes but is not limited to:
The materials include some formal assessment measures designed to support the teacher in determining a student’s understanding and fluency with critical content and skills. In fact, according to the “Assessment Resources” in Unit 1, the assessment options for diagnostic purposes are the “Placement Test,” “Beginning-of-Year Diagnostic Test,” and “Are You Ready? Intervention.” However, there is no indication that the tools are designed to allow students to demonstrate understanding in a variety of ways and settings. Additionally, the “Mathematical Process Standards” are not measured in the formal assessments, nor are there assessment tools that provide information on the development of mathematical reasoning or the use of mathematical discourse.
In Unit 1, the Placement Test is a formal assessment measure used “to assess prerequisite skills mastery before beginning the school year.” Students answer 37 questions, and given that the Placement Test covers fifth-gradeTEKS, this tool supports teachers in determining which students may need intervention. The test includes an “Individual Student Profile” that allows the teacher (or student) to record data collection by providing a column to record proficiency for each question from the tests available. It is important to note, though, there are only three TEKS that are presented twice; a student’s “proficiency” is determined by one single question, except for 5.4H, 5.7, and 5.8C.
Also in Unit 1, the Beginning-of-the-Year Diagnostic Test is a formal assessment measure included “to assess knowledge of key objectives that will be taught in the current school year.” In addition, the Beginning-of-the-Year Diagnostic Test is used as “a baseline for a student’s mastery of math concepts and skills, and to evaluate growth during the school year.” Many of the sixth-grade math TEKS are represented on the diagnostic test multiple times; most have at least two items. (One even has 12 items correlated to it.) An Individual Student Profile accompanies the test and allows the teacher (or student) to indicate proficiency in each of the tested math TEKS. It also matches the test item to the module in which it is taught. (The last column of the profile provides a place for teachers to designate if a student is proficient or not. While the tool could be used by students, the minimal guidance directs teachers, which keeps students from self-monitoring. The use of this tool by students would only occur if a teacher made a purposeful move to share it with them and provided more detailed insight into its relevance.) Also included is an answer key with “Test Prep Doctor” comments related to why a student may have chosen an incorrect answer, such as on Question 2, where it provides, “Students who answered B might have ignored the negative sign for -5℃. Students who answered C might have misunderstood the inequality symbol. Students who answered D might have understood that 5 is less than 2 or 1, but neglected to compare 2 and 1.” This is helpful guidance for the teacher to understand the misconception. Most of the time, though, the reasons are vague and simply computational-related or content-specific. “Students who answered B, C, or D may not know what the opposite of a number is” or “Students who answered A, or B, or C may have graphed the numbers incorrectly” or “Students who answered A, B, or C need to review how to divide a number by a fraction.” Overall, the test is very limited in allowing students to demonstrate their competence given that almost all 96 questions require students to demonstrate their understanding in an abstract way: only one question asks students to demonstrate their solutions pictorially. The remaining 95 questions require students to provide an abstract representation of the content and skills being assessed. (Note: in addition to the paper version, another Beginning-of-the-Year Assessment is found online through the student edition “Personal Math Trainer,” although guidance related to student performance or feedback related to their misconception was not found in the materials. The online option does provide opportunities to incorporate concrete, pictorial, and abstract representations as well as written descriptions, which makes an important difference. For example, question 5 (online) provides a scenario that involves the multiplication of fractions. Students are prompted to choose a pictorial model that represents the situation and share the solution. In the next step, the materials prompt students to write an equation to represent the situation. Finally, the materials prompt students to use dropdown menus to complete an explanation if the “fraction representing the sale price” will “increase or decrease when multiplied by ½.”)
Each module begins with an assessment, and “For students who require intervention, use the online Are You Ready? Intervention.” This assessment aligns concepts with the online Personal Math Trainer, which is intended to provide intervention through examples, step-by-step procedures and links to the textbook to build students’ understanding. It also allows the student to print helpful materials. An example from Module 1 Personal Math Trainer covers comparing numbers and models beginning support. If the student answers incorrectly, the Personal Math Trainer gives immediate feedback like, “Try again. Check the digits in the tens place again and compare.” When the “Are You Ready?” is completed, according to the teacher guide, it will “automatically prescribe a targeted, personalized intervention path” for students; however, evidence was unavailable relating to how an intervention path is determined.
Throughout the materials, each module provides a “Ready to Go On?” quiz in the student edition, as well as “Leveled Module Quizzes” in the Teacher Edition, providing correlations to sixth-grade TEKS to assist in providing intervention support. In addition, these tools measure the content and process skills for sixth grade, as outlined in the TEKS and Mathematical Process Standards. However, it is important to note that the materials lack specificity related to the alignment to the sixth Grade TEKS and Mathematical Process Standards.
The “Assessment Resources” provided for the teacher show an overview of the options available and whether the source can be found in the Assessment Resources section, the student or Teacher Edition, or online. The Assessment Resources break down each of the assessments by what they are used to assess, but there are no recommendations as to how to use the assessment past the “diagnostic” or “formative” label given. This resource does suggest when to administer the benchmarks, the purpose of the placement and diagnostic tests, and the levels for the different quizzes and unit tests. However, it lacks guidance related to assessment administration (i.e., time expectations, scripted instructions, etc.); no guidance is provided regarding administering the 96-question test to ensure validity or in a developmentally appropriate way. Additionally, the materials do not provide guidance for collecting observational or anecdotal data, and the placement test does not guide the teacher in recommendations from the student results on the test. For example, the Individual Student Profile included with the Placement Test and the Beginning-of-Year Diagnostic Test are simply yes/no checklists. The materials do not include guides for each domain area to support teachers in understanding examples of student language and behaviors which demonstrate progress toward identified outcomes.
The materials do not include opportunities for the students to track their own progress and growth with teachers or guardians. No student portfolios illustrate progress toward goals. Students do not collect samples and share reflections on their selected work. No student reflection tools are offered for assessments; students are not prompted to revisit errors to confirm their understandings and misunderstandings. The materials do not include on-going informal diagnostic tools or guidance related to the use of checklists, observations, anecdotal notes, interviews, etc. The materials do not include family input questionnaires, nor do they provide information to families to support their understanding of students’ learning needs or provide learning opportunities at home. The materials do include “Reflect” tasks throughout lesson examples, “Guided Practice” and “Independent Practice;” however, these tasks do not provide students opportunities to confirm their understanding and determine what they need to learn next, nor do they help students identify their strengths and areas of growth. The Reflect tasks in the materials are generally more focused on the specific lesson content, such as in Lesson 7.1, Example 1 prompts, “Write a rule that you can use to find the number of star beads when you know the number of moon beads.”
Materials do not include guidance for teachers and administrators to analyze and respond to data from diagnostic tools, nor do they yield meaningful information for teachers to use when planning instruction and differentiation beyond the “Placement Test” and the “Beginning-of-Year Test.” While assessments exist, there is a lack of support for teachers with a variety of suggestions and activities to respond to data from diagnostic tools. The materials do not provide guidance to respond to individual students’ needs in all areas of mathematics, based on measures of student progress appropriate to the developmental level. There is no evidence of guidance for administrators to support teachers in analyzing and responding to data.
Evidence includes but is not limited to:
The materials do not include guidance that supports the teacher in utilizing results from a variety of assessments to support purposeful planning of the appropriate grade-level experience. Also, the answer keys do not contain suggestions of activities as a way to respond to students’ demonstration of mastery on assessments. The data from these diagnostic tools included in the materials (Placement Test, and Beginning-of-the-Year Diagnostic Test) could guide teachers in differentiating instruction, but the materials do not include guidance showing the correlation between students’ performance to grade-level TEKS and the level of support they may need. None of these assessments provide guidance to understand the results, other than “proficient” or “not proficient.” There is also no indication of what “proficient” looks like, other than getting a question correct. The “Assessment Resources” Beginning-of-Year Diagnostic Test does provide a “Test Prep Doctor” that explains reasons for incorrect answers. Sometimes these reasons are diagnostic in nature, sometimes they are content/computational specific, but the relevancy for the teacher is minimal given that similar guidance is not evident on other measures of their progress throughout the materials. In the Assessment Resources, the table describes the purpose of the placement test, then goes on to suggest, “for students who require intervention…” but there is no guidance determining which students need intervention. In fact, the Placement Test “Individual Student Profile” and the Beginning-of-Year Individual Student Profile are the only two resources that allow individual student data to be checked, based on proficiency in each standard assessed, but further resources for observing student mastery are not provided.
While the materials may not frequently guide teachers to use certain activities based on student performance, the materials do incorporate materials that would be appropriate for reteaching or additional content support throughout the materials. Throughout each lesson, the teacher guide offers suggestions for differentiation in the following sections: “Differentiate Instruction,” “Success for English Learners ELL,” “Reteach,” “Leveled Practice and Problem Solving,” and “Challenge” activities. No scoring guide is provided for differentiation, and these are not diagnostic in nature. (For example, in the leveled module quizzes and leveled unit tests, there is no guidance as to how to assign the different levels other than “slightly below level,” “on level,” “advanced,” and “considerably below level.” No assessment instructions explain how students reach different levels.) Teachers are prompted within the Teacher Edition to use differentiated materials, but no specific guidance offers best practices for these materials (small group, one-on-one, etc.) The Differentiate Instruction section addresses critical thinking, modeling, multiple representations, graphic organizers, number sense, cognitive strategies, technology, and more, but nothing as a result of data. These worksheets are typically more problems similar to those in the unit and do not include recommendations for how to respond to student needs; these tools are simply more of the same practice. Prompts are not purposefully related to student needs in terms of additional time or alternate strategies. “Questioning Strategies” throughout these lessons are specific to the lesson and do not support students or teachers in using a variety of strategies toward pursuing solutions. Some lessons include “Curriculum Integration,” a section of the Differentiate Instruction, which provides connections to other content areas, but they are very specific and guided. For example, in Lesson 14.1, Curriculum Integration prompts teachers to “have students draw coordinate grid lines on maps of Texas. Instruct students to draw the x- and y-axes through the state capital and the other lines at ½-inch increments above, below, to the left, and to the right of the axes. Have the students label the grid lines, beginning with the axes, with the appropriate numbers, and give coordinates for various cities and towns on the map.” While this is an example of a connection to geography, it does not provide an opportunity for students to explore independently. Rather, it guides them to complete a specific task. Also, there are only 11 out of 63 lessons in the sixth grade materials with a Curriculum Integration section.
All lessons include Reteach activities and practice, which are provided to support deficits in a particular skill or concept while also providing a different lens with which to explore the concept. For example, in Lesson 9.1 on “Understanding Percents,” the concept is taught through the use of bar models and grids, whereas Reteach materials focus on writing percentages as fractions out of 100 and then using division to write a fraction as a decimal. In addition, the materials offer “Performance Tasks,” an alternative method for assessing students’ mastery of concepts, that provide enrichment at the end of each unit. These tasks provide students with the opportunity to apply learning from each unit in real-world problem situations, but they do not connect to other content areas. While the rubric measures students’ critical thinking skills, reasoning, and constructing arguments, there is no guidance to support the teacher in understanding this as a diagnostic tool.
The “Math Background” section at the beginning of each unit includes specific trajectories to support the teacher in understanding the progression of content and skill development. For example, in Unit 2, this section provides a pictorial area model to represent the multiplication of fractions and relates this model to the algorithm. The section also makes teachers aware of common misconceptions; however, little to no support is offered to help teachers interpret assessment results for individualized instruction. Each module also contains an “Are You Ready?” skills check, an alternative teaching strategy for students who need additional targeted intervention lesson activities and practice. “Skills Worksheets” support deficits in a particular skill or concept. Each Skills Worksheet includes an outline of how to teach this skill, including common errors students may exhibit. To help, each Skills Worksheet includes a group practice with each step listed in how to solve the problem and ends with a “Practice on Your Own.” However, these do not directly support teachers’ planned instruction based on data from diagnostic tools. Again, the resources are there but not as a means to provide support based on student assessment data.
There is no evidence of guidance for administrators to support teachers in analyzing and responding to data. In fact, the materials do not provide any evidence of guidance for administrators to support teachers in designing instruction to respond to data. While it seems likely that the assessments completed in the student online edition through the “Personal Math Trainer” would have data that would be accessible for analysis for individual students, classes, and the school, evidence of this data was not found.
Materials provide some frequent, integrated assessment opportunities. Throughout the materials, routine and systematic progress monitoring accurately measures student progress; however, those opportunities lack explanations of measurement and tracking student progress.
Evidence includes but is not limited to:
Materials include routine and systematic progress monitoring opportunities that accurately measure and track student progress and are content specific as well as age-appropriate. According to “Assessment Resources” in Unit 1 Teacher Editions, summative assessments are used for assessing mastery, including leveled “Unit Tests,” “Unit Performance Tasks,” “Quarterly Benchmark Tests,” “Mid-Year Test,” “End-of-Year Test,” “Unit STAAR Test Prep,” “STAAR-Aligned Practice Tests,” and “Online Quizzes and Tests.” More specifically, Assessment Resources highlight Unit Tests as a way to “assess mastery of concepts and skills taught in the Unit.” Materials prompt teachers to use “Level A for students who are slightly below level; Level B for students who are on level; Level C for advanced students; and Level D for students who are considerably below level and require modified materials.” The Unit Tests Include 15 to 24 questions in a multiple-choice and open-ended format and are provided at the end of each Unit through the online teacher guide resources for that unit and an online version that is also available via the student “Personal Math Trainer.” Also, “Benchmark Tests” are available for teachers to “use for prep for the STAAR Test.” Within the materials, there are four Benchmark Tests: two quarterly tests (one covering Modules 1–6, the other covering Modules 10–16), the “Mid-Year Test,” and the “End-of-Year Test.” There is little guidance as to when these are given other than the name of the test (Mid-Year, End-of-Year), and according to the link for the quarterly tests, they are intended to be given after Modules 1–6, and again after Modules 10–16.
According to the Assessment Resources, the formative/progress monitoring tools are the leveled “Module Quizzes,” “Your Turn,” “Math Talk,” “Reflect,” “Questioning Strategies,” “Essential Questions,” “Lesson Quizzes,” “Ready to Go On? Quizzes,” “Module Mixed Review: STAAR Test Prep,” and the online pieces of Ready to Go On? Intervention and Enrichment, “Online Homework,” Module Mixed Review: STAAR Test Prep, Online Quizzes, and Tests. These provide a variety of question formats, such as multiple-choice, short and expanded answer, and gridded response. The questions mostly assess student knowledge to recall and apply their knowledge and skills, sometimes asking students to explain their thinking. Students are rarely asked to reflect on their thinking and justify their reasoning on formal progress monitoring assessments. For instance, Module Quizzes are explained as routine, formative assessments used to “assess mastery of the concepts and skills taught in the Modules.” The materials prompt teachers to “use Level D for students who are considerably below level and require modified materials” and to use Level B “for all other students.” The Module Quizzes include from 12 to 28 questions in a multiple-choice and open-ended format at the end of each Module. The materials include “Are You Ready?” sections at the beginning of each module and Ready to Go On? sections at the end of each module, which help to identify students who would benefit from receiving the intervention opportunities. The “Program Resources” page also explains that the Personal Math Trainer is used to “monitor student progress through reports and alerts,” and teachers can “create and customize assignments aligned to specific lessons or TEKS.” (The materials provide no further guidance on how the Personal Math Trainer can be used to accurately measure and track student progress.) An online version is also available via the student Personal Math Trainer, which “provides online practice, homework, assessments, and intervention.” Teachers “monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or TEKS.” Assessments taken through the online Personal Math Trainer provide “instant scoring, feedback, and customized intervention or enrichment.” (That evidence was not provided given that access to the “teacher-side” of Personal Math Trainer is not available for review; there was no guidance as to how to access any of these reports and alerts or how to get the personalized intervention path.) The materials do include suggestions for more frequent monitoring of students who are not demonstrating progress and include instructional interventions to support students who are struggling learners. Online and print resources are available to differentiate instruction, as well as differentiate assessments. Resources to support struggling learners include “Reteach” worksheets, “Reading Strategies” handouts, and “Success for English Learners” worksheets.
Additionally, throughout the teacher guide, Your Turn, Math Talk, Reflect, and Questioning Strategies sections are included to guide informal monitoring of student progress and understanding related to the content. Each of these sections of the teacher guide provides specific questions to check for understanding and application of knowledge. For example, in Lesson 7.3, the materials prompt teachers in the Your Turn section to ask, “How can finding an equivalent ratio help solve Exercise 2?” Questions like these provide opportunities for teachers to routinely check for understanding and address misconceptions. In each lesson, Math Talk opportunities are also used to “continually monitor and assess student progress with integrated formative assessment.” Sample answers to Math Talk questions are provided. Although Math Talk is labeled as a formative assessment, there is no guidance on how to track student progress or collect data using these questions. Then, in the student and Teacher Editions, each module concludes with a Module Mixed Review: STAAR Test Prep with STAAR-formatted questions covering current content as well as “mixed review concepts from previous modules or a previous course.” For example, Module 13 Mixed Review requires students to use their understanding of integer operations, a concept introduced in Unit 2.
Formative and summative opportunities are plentiful. However, while the materials have a variety of assessment opportunities, both formative and summative as well as observational, the materials provide no guidance as to a way to track student progress. There is no progress monitoring to provide teachers with feedback for identifying each student’s content and skill level and how they change over time. Tracking and measuring the data are unknown since there is no access to the actual reports. No checklists exist for documenting and tracking observations and individual student growth. The materials state that the teacher can view the reports to track progress, and one of the reports mentioned is a Knewton Analytics Report. However, the reports were not available to preview.
Materials include guidance, scaffolds, supports, and extensions that maximize the learning potential for some students. Struggling learners receive targeted instruction and activities in order to successfully master sixth-grade content. However, materials are limited for students who have mastered the content, offering little extension and push to maximize their potentials. Additional enrichment activities provide some exploration and application opportunities, but not much variety.
Evidence includes but is not limited to:
The materials provide several supports for differentiated instruction for students who struggle to master the content; these include guidance for motivating learners using connections to daily life, modeling using mathematical process standards, connections to vocabulary beyond their mathematics definitions, questioning strategies, checks for understanding, engagement strategies, focus on patterns, avoidance of common errors, and additional examples. In the teacher’s edition, each module begins with a “Are You Ready?” assessment (also available online) for instant feedback and to “determine if students need intensive or strategic intervention for the module’s prerequisite skills.” Here, teachers are provided guiding questions, common errors, and alternative teaching methods using concrete examples. Within each module, “Skills Intervention” worksheets are available to target specific skills, including prerequisite skills. These mini-lessons also include common errors, practice problems, alternate approaches, and “Practice on your Own.” Included in each lesson of the modules, “Practice and Problem Solving” provides a variety of ways for students to develop skills using different types of questioning. “Differentiate Instruction” offers specific ideas for visual clues, critical thinking, kinesthetic experience, multiple representations, manipulatives, communicating math, number sense, “Cooperative Learning” suggestions, and/or additional resources. In addition, printable “Reteach” materials start with a mini lesson offering examples and end with several questions for students to apply what they learned. Within each module, “Professional Development” videos include models and provide guidance to teachers for scaffolding instruction. Each unit includes leveled unit tests, including tests specifically designed for students who are “slightly below grade level” and students who are “considerably below grade level and require modified materials.” Other supplemental resources are “Reading Strategies,” which uses real-world context to support the focal point of the lesson, and Reteach, which includes extra examples for struggling students to grasp and make connections to the academic vocabulary being used in the focal point. “Success for English Learners” is another resource, which includes visuals for making connections to the real world and supporting language as it pertains to the focal point.
For students, the “Personal Math Trainer” is an available intervention tool that prescribes “a targeted, personalized intervention path” by scaffolding content for students. It provides more questions over the content that allows the students to check their answers in real-time, but it also gives the option for a step-by-step example. A few “Animated Math” problems in select lessons provide a more tactile way for struggling learners to grasp the focal point, and lesson tutorials, called “Math on the Spot,” offer videos that allow students to pause and take notes over the concept with which they are struggling. The student also can hear the information from an expert other than their teacher. Additionally, visual aids are found as graphic organizers to help explain vocabulary; concrete objects help students “Explore” the concept, and “Engage with the Whiteboard” allows students to illustrate learning.
Materials provide minimal recommended targeted instruction and activities for students who have mastered the content. Materials do not guide teachers in specific ways to extend grade-level content and skills. Materials in each lesson do ask students to engage in “HOT” questions (Higher-Order Thinking) to communicate mathematical ideas, analyze relationships, critique reasoning through error analysis, and use multiple representations to justify their understanding. Materials also recommend that teachers include an “Extend The Math” and “Challenge” activity asking students to answer a Pre-AP level question as an extension of the lesson, yet this resource can be found only in the online materials and Teacher Edition, meaning students do not have independent access. In addition, these extensions do not engage students in more challenging tasks, nor do they ask students to elaborate on their responses by making connections to larger mathematical ideas. Though “Leveled Practice and Problem Solving” includes lessons for “on-level” or “advanced” students, the questions seem to be more of the same level, lacking higher-level exploration. Following Independent Practice, each lesson contains a short, Extend the Math Activity, which is sometimes a game and sometimes a problem to solve, but these activities do not extend grade-level content and skills. There is little evidence of ways to enrich content to support students who have met mastery; there is no evidence of project-based exploration. Materials do not provide students with opportunities to use learned concepts in new ways, creating a depth of knowledge.
As for enrichment activities for all levels of learners, some materials include activities that allow all students to explore and apply new learning in a variety of ways, including technology supports. Each unit does conclude with a “Performance Task” (“accessible to all students and suitable to be completed in a classroom”) that engages students in a career application situation using the cumulative skills needed throughout the unit. Additionally, each module includes a “Real-World Video” that allows students to make connections between the focal points and their application to daily life. The videos, though, do not add depth to the learning or provide additional activities.
Lessons include HOT problems within independent practice that focus on higher-order thinking skills and the TEKS process standards, though this independent practice is more of the same type of problem, not an extension or project-based exploration. Some lessons include a Single Representation technology tool called Animated Math that engages students in utilizing interactive examples and visuals of the content skills in the lesson, but not all units have them. Each lesson does include an Extend the Math activity in the teacher’s edition and accessible online, offering students additional practice and sometimes partner work, yet the additional practice does not truly extend the learning. Several lessons begin with an “Explore Activity” that asks students to use concrete objects or to engage in Math Talk, but again, these activities do not really provide enrichment. Each lesson also includes a Challenge page, geared toward Pre-AP students; however, these problems are more practice on the lesson content, not the application of new learning.
Materials provide instructional methods that offer some variety of learning and appeal to learner interests and needs. Instructional approaches offer consistency and continuity in approaching mastery of content; however, the variety of strategies and activities remains minimal. Although it mentions flexible grouping, the materials do not provide teachers with guidance as to when students should participate as a whole class, small groups, or as individuals. Not enough new teacher supports exist to achieve effective implementation.
Evidence includes but is not limited to:
The materials provide routines for whole group instruction, small group instruction, pairs, and independent work, but contain little guidance on when to use specific grouping structures based on the needs of students or how to address misconceptions from previous class instructions. During the “Vocabulary Preview” and “Reading Start-Up,” for example, materials suggest that students may work individually, in pairs, or in groups, but do not guide teachers. For intervention and enrichment activities, no grouping recommendations guide teachers on how to group students to complete the work. Materials do include lessons to support skill acquisition for students who require intervention but make no suggestions on how many students should be grouped to maximize these resources. The materials do not guide teachers on how to support small group work with peers, including norms and classroom routines. There is no evidence of students learning a routine for independent work throughout the lessons even though individual exploration is suggested for “Animated Math,” the “Personal Math Trainer,” and blackline masters of practice sets. At the beginning of each module, “Are You Ready?” activities offer teachers targeted intervention lessons and practice worksheets for use with a targeted small group but don’t make suggestions. “Differentiate Instruction” also includes activities to engage small groups or pairs, but again, teachers lack group formation guidance.
The materials do include guidance to support teacher understanding of developmentally appropriate instructional strategies. “Math Background” includes information about the meaning of the TEKS and how they are addressed. Models and visuals offer concrete ideas to avoid student errors and misconceptions. Each module also includes “Professional Development Videos” that model successful teaching practices and strategies in actual classroom settings. Differentiate Instruction offers suggestions for manipulatives, historical examples, cooperative learning, and critical thinking.
All lessons provide a consistent format using the 5E model. Teachers first engage students with an essential question to motivate the lesson and spark interest for the opening exploration. “Careers in Math” connects the new concept to a real-world career as “Unpacking the TEKS” provides examples of what students will be learning and how it relates to the TEKS. “Assess Readiness” enables teachers to determine if students need intensive or strategic intervention for prerequisite skills. Then, as students explore by engaging with a whiteboard or using models, the teacher’s “Avoid Common Errors” suggests mistakes students may make. Module 4 warns the teacher that “when dividing decimals, students sometimes align digits incorrectly and produce an answer that has the decimal point in the wrong place. Have students use grid paper (or lined paper turned sideways) to help align the digits correctly.” In addition, “Focus on Models” guides teachers through building conceptual knowledge with specific directions like “give each student 3 paper squares and a pair of scissors to work through the Explore Activity with physical models.” Students have an opportunity to explain concepts through “Talk About It” and “Focusing on Communication,” where they connect vocabulary to similar terms, clarify its meaning, and provide connections. (Teacher supports for vocabulary also include “Visualize and Understand Vocabulary” and a “Glossary.”) Teachers explain content using provided questioning strategies that connect materials to daily life and guide students to talk about their understanding. Some provided supports include “Focus on Math Connections,” “Connect to Daily Life,” “Focus on Patterns,” and “Connect Multiple Representations.” “Active Reading” provides ideas for foldables to aid in note-taking, understanding, and organization of new concepts. To elaborate, students summarize the lesson and work through guided practice. “Reflect” questions allow students “to analyze mathematical relationships [and] to connect and communicate mathematical ideas.” When students complete these questions online, their responses can also be saved to “myNotebook,” which teachers can view. The online “Personal Math Trainer” provides content practice using examples, video tutorials, access to the online textbook, and other “similar” problems, with step-by-step directions for reinforcement. “Depth of Knowledge” employs higher-order thinking questions, encouraging students to evaluate their level of understanding. Each lesson provides support with “Questioning Strategies,” think-aloud processes, and patterns within the content, as well as “Math Talk” opportunities to formally assess student progress. Another part of evaluate, “Extend the Math,” helps students take their current knowledge and extend it to a higher understanding. (However, many of the extensions provide more of the same type of practice and do not truly extend the meaning of the concept.) Although consistency exists in the lesson format, lessons do not have a large variety of different instructional approaches.
Some instructional strategies include engagement with real-world context, exploration with concrete, hands-on materials to model examples, teacher-modeling of a new concept, opportunities to discuss common errors, and language with pictorial supports. Unit 1 “Resources” include virtual tools for student use. Lessons sometimes include concrete practice with interactive materials, such as colored integer counters, fraction bars, fraction/decimal grids, bar models, geometry sketcher, algebra tiles, a graphing calculator, and a scientific calculator. The materials include visual representations or symbolic abstractions. However, even though students have online access to all virtual manipulatives, not all lessons use manipulatives, and variation of materials is minimal. The following examples detail common manipulatives: In Modules 5 and 6, students use two-color counters for exploration of integer operations. They manipulate number lines to solve subtraction problems involving integers and create integer multiplication problems modeled on a number line. In Module 9, students shade grids and draw bar models to create visual representations of percentages, and in Module 11, students use algebra tiles to determine if two expressions are equal. Module 15 asks students to use equations to represent the relationship between the angles of a triangle.
The following examples provide insight into the implementation instruction teachers receive in various activities: Every lesson includes an “Engage with the Whiteboard” where students explain and illustrate answers to questions (it is unclear they are online or actual whiteboards in the classroom). “Talk About It” allows students to summarize the lesson and their understanding. In Module 15, students evaluate triangles by experimenting with different side lengths to create one; they make conjectures based on their experience, which leads to the Triangle Inequality Theorem. “Guided Practice” gives the teacher sequenced steps to walk students through the questions. For example, in Module 17, students use manipulatives to illustrate how a box plot is made, then extend their learning with an introduction to Mean Absolute Deviation, an eighth-grade concept. For students who struggle to master the content, “Math On the Spot” videos reteach the lesson. Module 15 provides a reteach video for steps in finding unknown angles in a triangle. Differentiate Instruction includes “Cooperative Learning” opportunities, sometimes having students work in pairs to complete a task. For example, Module 7 groups students in a cooperative learning activity where they toss two number cubes to create a ratio; group members then name one equivalent ratio. Then in Module 17, students model finding the mean of a set of data using counters; an additional option includes students working independently or in groups to create a presentation to compare the mean and median of a set of data.
The materials include some supports for English learners to meet grade-level learning expectations. The materials provide a few scaffolds for English learners but do not specifically provide scaffolds for each level of English language proficiency beyond the beginning teacher materials. No opportunities are provided that 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). The English Language Proficiency Standards (ELPS) can be found; however, language support is limited for languages other than Spanish.
Evidence includes but is not limited to:
The Front Matter states that the material “supports English language learners at all proficiency levels. The Student Edition provides integrated resources to assist all levels of learners. In addition, students at various levels may benefit from additional program support: Students at a Beginning level are supported by “Spanish Student Edition,” “Spanish Assessment Resources,” “Success for Every Learner,” and “Leveled Practice A” worksheets in “Differentiated Instruction,” “Math on the Spot” videos with Spanish closed-captioning, and the “Multilingual Glossary.” Students at the Intermediate level may use any of the resources above and may also use Reading Strategies in Differentiated Instruction. Advanced and Advanced High Students at these levels will be successful as the Student Edition promotes vocabulary development through visual and context clues. The Multilingual Glossary may also be helpful.” (However, many of these supports, such as the Spanish Student Edition, the Spanish Assessment Resources, and the Math on the Spot videos with Spanish closed-captioning are specific to native Spanish speakers and do not support all ELs.) No additional reference or guidance specifically addresses Beginner, Intermediate, Advanced, or Advanced High ELs; the other resources lump all ELs together when addressing options.
Also in the Front Matter, a table lists all of the ELPS and where that standard is met in the student edition, yet most of these citations do not match what the ELPS says. For example, C.2.C says, “learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions;” one citation given says, “Explain what the data tells you about the win/loss records of the teams in the league.” Then in example C.3.E, directions say, “Share information in cooperative learning interactions;” one citation says, “Is the value of 23the same as the value of 32? Explain.” Also, the “Differentiated Instruction” section, mentioned as one support for Intermediate learners, is never mentioned in the table as meeting the ELPS, and some of these areas listed are for the purpose of engaging with the content and not for linguistic accommodations.
Each lesson in the online materials contains a “Success for English Learners” task, which introduces the lesson content in a format that includes pictorial models, graphics, pictures, short text, and simplified vocabulary. Each task provides two sample problems and some practice questions related to the content covered in the lesson. While these materials are available, the only reference for their use is included in the teacher guide under Differentiated Instruction, which encourages ALL students, not just ELs, with modeling, kinesthetic activities, cooperative learning, and sometimes language development. The materials do not include specific guidance related to language development. For example, Lesson 3.1 materials provide two examples that use a circle model to build an understanding of multiplying fractions. Each step of the process is modeled, written directions and guidance are succinct, and two follow-up questions specifically focus on multiplying fractions compared to adding fractions. No specific guidance to help ELs exists.
The materials include some suggestions for appropriate scaffolds to support students learning English. At the beginning of each module in the student materials, the “Reading Start-Up” section focuses on reviewing and previewing important vocabulary. Three sections are included: “Visualize Vocabulary,” “Understand Vocabulary,” and “Active Reading.” To begin, Visualize Vocabulary provides diagrams and graphic organizers to help students review vocabulary in the module. (The Reading Start-Up page calls attention to ELPS and states that utilizing the vocabulary graphic addresses the standard listed.) Next, Understand Vocabulary provides tasks intended to help students learn the preview words. Finally, Active Reading supports ELs through the integration of “reading and note-taking strategies to help them organize and understand new concepts and vocabulary.” For example, in Module 15, students create a pyramid foldable as a way to organize their knowledge of angles, triangles, and equations. Unfortunately, strategies do not explicitly teach the process; in fact, the teacher guide does not suggest specific models or information to include on the pyramid to support ELs. At a beginner level, ELs would need to see modeled step-by-step instruction, explaining important information to include. No explicit connections are made between English and the students’ home language.
Another appropriate scaffold occurs at the unit start: “Vocabulary Preview” uses “puzzles to give students a preview of important concepts in [the] unit.” The teacher guide prompts teachers to allow students to “work individually, in pairs, or in groups.” Within the task, students are provided a definition and a reference to the vocabulary introduced, which is bolded within the student edition. Unit 1 Vocabulary Preview begins with a word search puzzle. In Unit 6, Vocabulary Preview asks students to unscramble vocabulary words to solve a riddle.
The materials include opportunities to support a student’s development of English in speaking or writing through the use of mathematical language tasks, such as those provided in the “Math Talk” section. Through the question posed, “the student is expected to display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” For Lesson 18.1, the Math Talk asks students, “Margaret wants to open a new checking account. What are some things she should consider when choosing a bank?” Questions like this one provide students the opportunity to respond orally or in writing using content-specific understandings. Some lessons encourage teachers to engage the class in discussion; here, English learners benefit from hearing and participating.
Integrating the ELPS provides very generalized guidance for teachers related to supporting students and their language development. Many lessons say, “You may want to pair English learners with a partner for Explore Activity 1 to help them develop their language skills.” There is no guidance on how to partner students if the partner should also be an EL, etc. This guidance is lesson/content-specific and does not include general teaching strategies that would be effective across all the units. For example, in Lesson 11.2, the materials guide teachers to “be sure English learners understand the references to Celsius and Fahrenheit in Example 3. You may want to point out that both scales measure temperature, but the Celsius Scale is part of the Metric System.” The Lesson 17.5 reminder says that “English learners will benefit from hearing and participating in classroom discussions” but doesn’t provide appropriate questioning strategies or sample questions for a variety of learners.
“Focus on Modeling” provides effective strategies for teachers to model content; however, while this strategy is an appropriate support for ELs, the materials do not identify modeling as a way to support ELs. In Lesson 10.2, the materials explain that “students who find mental math difficult may find ladder diagrams to be challenging. Model other methods for dividing by 2, such as using long division or a calculator, to show that the ladder diagram is a useful organizational tool. Emphasize that they can use a combination of division methods when using ladder diagrams.”
“Connect Vocabulary” in the teacher guide provides effective strategies for building content vocabulary and is also denoted within the materials as supporting the ELPS. For example, in Lesson 1.1, the materials guide teachers “to help students understand the concept of opposite in math and in other contexts, make a list with students of pairs of opposites, such as hot and cold, black and white, up and down, left and right. Clarify that left and right are used in the math concept of opposite with negative numbers to the left of 0 and positive numbers to the right. Zero is its own opposite.”
The materials include a glossary that inconsistently supports the use of ELs’ first-language as a foundation for English vocabulary development. Within Unit 1 materials, the “Multilingual Glossary” provides an audio recording in English and Spanish of the pronunciation of each vocabulary term, a written definition in English and Spanish, and an example (as applicable) in English. This ensures the proper pronunciation of terms as a foundation for building accurate verbal pronunciation. In addition, a translation of definitions is available in 12 additional languages (Arabic, Armenian, Chinese, Haitian Creole, Hmong, Khmer, Korean, Punjabi, Russian, Tagalog, Urdu, and Vietnamese); however, it is important to note that the definition translations were not available for all terms in each of the additional languages. For example, Haitian Creole is included as a translation option; only 18 of the 33 terms under “A” are in that glossary. A translation for the definitions in Haitian Creole is not provided.
Additionally, the materials include little instruction to support students at varying levels and nothing after the Front Matter that specifically addresses each student proficiency level. In Unit 1, the teacher notes state to encourage the EL to use the photo to help them understand the scenario, and another example states the teacher should encourage the EL to ask for clarification on terms they don’t understand. At the end of Unit 2, the “Study Guide Review” suggests encouraging the EL to refer to their notes and the illustrated bilingual glossary to review concepts. While some materials attempt to support ELs, there is no instruction that is repetitive, playful, and interactive. Materials do not always help ELs meet grade-level learning expectations.
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, but there is little guidance for the vertical alignment. The materials provide some opportunities to review and practice mathematical knowledge and skills throughout the curriculum, but most spiral review comes from module and unit assessments, not daily practice.
Evidence includes but is not limited to:
Materials include a cohesive, year-long plan to build students’ foundational literacy skills. A “Unit Pacing Guide,” provided at the beginning of each unit, includes a breakdown to show the timelines and sequence of instruction. Five of the seven units contain a pacing guide for 45-minute classes and 90-minute classes. (The other two units (Units 2 and 4) contain only the 45-minute class period pacing guide.) The Unit Pacing Guide specifies teaching all lessons over a two-day span when following the 45-minute class outline. For example, the Unit 3 Pacing Guide recommends teaching Unit 3 over 23 days with 45-minute classes or 13 days with 90-minute classes. Equal amounts of time are given to each lesson. The Pacing Guide provides guidance for 144 days out of the 180 instructional days without specific guidance given for longer times to ensure mastery of the focal areas. There is also no guidance for inclusion of the “Beginning-of-the-Year Diagnostic Test,” the seven “Unit Tests” and their accompanying “Study Guide Review,” the seven “Unit Performance Tasks,” the “Quarterly Benchmark Tests,” the “Mid-Year Test” or the “End-of-Year Test.” Additionally, the pacing guide does not list TEKS to be covered; it simply includes the module and lesson number.
Another pacing component is “mySmartPlanner,” which creates and schedules the entire year based on user input. This tool allows the user to input a personal school calendar, including days off and holidays. The “Auto-Schedule” button then schedules the entire year according to how many minutes the user says are in each class period. When ‘details’ is clicked on the date in mySmartPlanner, it brings up the standards taught for that day as well as all of the lesson resources that could be utilized.
The TEKS are listed on the “Unit Content” page before the Unit Pacing Guides but do not recur in subsequent Unit Contents to allow the teacher to see how concepts spiral throughout the year. Each module outlines the included lessons and lists the TEKS that connect to each lesson. The majority of units include instruction in the primary focal areas. In grade 6, the primary focus of six of the seven units is in the focal areas of operations with integers and positive rational numbers, ratios, and rates, including using equivalent ratios to represent proportional relationships, expressions and equations to represent relationships, and data representation. More specifically, students’ knowledge of integers and rational numbers builds throughout the materials. Students use rational numbers to solve real-world problems involving multiplication and division of rational numbers, and then they use that deep understanding of operations with rational numbers to apply to two-dimensional figures and solving equations.
The materials do not include a vertical alignment chart that shows how activities directly align to concepts and skills outlined for students in preceding and subsequent grades. For example, grade 6 materials do not develop teacher understanding of how previous and subsequent grade levels vertically align for all focal areas in Math Texas Essential Knowledge and Skills. The activities, though, are clearly connected within each unit, and the introduction of new concepts builds upon prior knowledge from the “Are You Ready?” activity at the beginning of each module. Are You Ready? assesses readiness in the modules’ prerequisite skills but does not label specific TEKS, so it is unclear which grade level or standard these skills address. While the activity checks for understanding for prior learning, it provides no teacher guidance as to how these skills are vertically aligned.
The Teacher Edition includes a “Unit Vocabulary Preview,” which has an alignment guide listed at the bottom. The guide introduces the concepts that students understand from previous instruction, what students will learn about in the current unit, and what students will connect their learning to in subsequent units. However, it does not list specific TEKS; it outlines only brief descriptions of content and skills in bullet points. The “Reading Start-Up” resource follows this format. At the bottom of each Reading Start-Up page, the Grades 6–8 TEKS alignment guide mentions concepts from previous instruction, current module concepts, and connections for subsequent modules. Again, it does not list specific TEKS; it only outlines brief descriptions.
The materials include some guidance that supports the teacher in understanding the vertical alignment for all focal areas in Math Texas Essential Knowledge and Skills. The beginning of each unit includes a chart labeled “Before,” “In This Unit,” and “After,” which shows the vertical alignment of concepts across grade levels. “Before” notes prerequisite knowledge and skills from within the current grade and preceding grade levels that students need to understand prior to beginning the unit. “In This Unit” notes the knowledge and skills students will learn about in the unit. “After” shows upcoming knowledge and skills from within the grade level and subsequent grade levels, which will build on the understanding of the unit knowledge and skills. For example, in Unit 4, the Before section notes that “Students understand operations with whole numbers, decimals, and fractions; order of operations; properties of operations including inverse, identity, commutative, associative, and distributive properties; and graphs in the first quadrant” (which are Grade 5 and Grade 6 TEKS). “In This Unit” section states, “students will learn about exponents; prime factorizations; numerical and algebraic expressions; equations and inequalities and the coordinate plane” (which are knowledge and skills from Grade 6 TEKS). Following this unit, the “After” section notes that “students will connect how to evaluate algebraic expressions with more than one variable; write two-step equations and inequalities to represent real-world problems and write a real-world problem to represent an equation or inequality; solve two-step equations and inequalities; and graph linear equations in the form y = mx + b on the coordinate plane” (which are Grade 7 and 8 TEKS). However, it is important to note that the materials do not include notation of specific TEKS within this section, but only a list of knowledge and skills, so it is up to the teacher to know if these are concepts from preceding grades, the current grade, or subsequent grades.
Another content plan is cohesively designed to build upon students’ current level of understanding with clear connections within and between lessons. Similar to the beginning of each unit, each module begins with a chart, also with three sections labeled Before, In This Module, and After. Likewise, this chart shows connections within and between lessons. For example, Module 10, the Before section notes that “students understand operations with whole numbers, decimals, and fractions; prime numbers and order of operations” (which are Grade 5 and Grade 6 TEKS). In the In This Module section, “students will learn to generate equivalent numerical expressions using exponents; generate equivalent numerical expressions using prime factorization and simplify numerical expressions using the order of operations.” Following this module, the After section notes that “students will connect “order of operations and numerical expressions and numerical and algebraic expressions,” which are knowledge and skills for subsequent lessons. Again, the materials do not include notation of specific TEKS, just a list of knowledge and skills, so it is up to the teacher to know these concepts and grade levels.
Materials provide some review and practice of foundational skills throughout the curriculum and at the appropriate level of rigor, but very little daily spiral review. The “Guided Practice” and “Independent Practice” provided within each lesson of the student edition contain practice problems that directly align with the concept taught in that lesson. For example, Lesson 3.1 teaches students to multiply fractions. All of the problems included in Guided Practice and Independent Practice can be solved by multiplying fractions without the inclusion of cumulative practice problems. Then, in the online Teacher Edition, each lesson includes a “Practice and Problem Solving” task that aligns directly with the concept taught in the lesson. Just like the Guided Practice/Independent Practice in the student edition, the Practice and Problem-Solving task focuses only on the concept taught in the lesson and does not include cumulative practice problems. The materials contain numerous other print and online resources for additional practice. Some of these include, but are not limited to, “Reteach” and “Challenge.” However, these extra practice problems cover concepts from the lesson; they do not include opportunities for spiraled review. Also, the textbook suggests that teachers can use the PowerPoint presentation to present or review concepts; however, these presentations only cover the lessons in which they are found and do not build on previously taught content.
However, at the end of each module in the student edition, there is a one-page “Texas Test Prep Mixed Review” that includes questions from the current module, as well as questions that address mixed review concepts from previous modules or a previous course. In the teacher guide, the materials provide Grade 6 TEKS alignment, as well as Mathematical Process TEKS alignment, for each question in the Texas Test Prep Mixed Review. Questions that are from previous modules or a previous course are denoted with an asterisk. Similarly, at the end of each unit in the student edition, there is a Texas Test Prep Mixed Review, which includes questions from the current unit as well as questions that address mixed review concepts from previous modules or a previous course. Again, the materials provide TEKS as well as Mathematical Process alignment for each question, and questions that are from previous modules/courses are denoted with an asterisk. (Four periodic Benchmark Assessments assess students’ knowledge of material taught previously.) The materials have plenty of resources that could be used as a review but do not suggest or guide the teacher in what this might look like or how to shape the extra worksheets into a spiraled review.
The materials include some implementation support for teachers but have no guidance for administrators in supporting teachers or recognizing best practices in the math classroom. The materials are accompanied by a TEKS-aligned scope and sequence outlining the essential knowledge and skills that are taught in the program, the order in which they are presented, and how knowledge and skills build and connect across grade levels. A school years’ worth of math instruction is evident, including realistic pacing guidance and routines.
Evidence includes but is not limited to:
Materials are accompanied by a TEKS-aligned scope and sequence, the “Pacing Guide,” outlining the concepts taught in the program and the order in which they are presented. However, while the Pacing Guide does not include specificity related to the Grade 6 TEKS, each unit, module, and lesson within the materials includes the TEKS addressed in them. The Pacing Guide does show the timelines and sequence of instruction for using the materials in a 45-minute class, as well as in a 90-minute classroom, which is sufficient for a full year of instruction. (Note: two units do not have the 90-minute option.) The Pacing Guide for sixth grade contains 63 lessons (53 of these specifically teach focal areas), and the materials suggest two days per lesson for the 45-minute classes. With a total of 126 days to cover the lessons, this allows for additional days spent on assessment, reteach, and implementing the “Are You Ready?” skills practice or the “Reading Start-Up” vocabulary activities. The pacing guide allows for flexibility and the ability to implement each lesson provided in the materials. The materials contain a “mySmartPlanner” that allows the user to create and schedule the entire year. The Smart Planner allows input for personal school calendars, including days off and holidays. The “Auto-Schedule” button then schedules the entire year based on the calendar and the specified length of each class period. When the user expands the lesson details in the mySmartPlanner, it will list the Standards being taught as well as the Resources used.
In order to see a more in-depth TEKS guide, teachers must use other materials. For example, the Front Matter includes a TEKS for Mathematics Correlation for Grade 6 table, which lists each TEKS for the grade level in numerical order by standard (with the standard descriptor) and includes page references where the content is taught or reinforced. For example, TEKS 6.11, which expects students to “graph points in all four quadrants using ordered pairs of rational numbers,” is taught on pages 379–382 in the student edition and is reinforced on pages 383–384, 405–406, 412–413, and 416 (although the chart incorrectly lists 414–415 and 418 instead of 412–413 and 416 where graphing on a coordinate plane is actually reinforced). Also, the Grade 5 Review Test in the Front Matter provides teachers a way to assess if students “have mastered the concepts” from fifth grade, which is essential to making new connections. This assessment tool includes a TEKS alignment chart denoting the TEKS assessed on each question of the test.
The materials include other guidance that supports the teacher in connecting the learning of essential knowledge and skills across multiple grades within the program. “Math Background” begins each unit and clarifies each standard along with an example of how the standard is tested or should be presented. It also guides teachers in making direct connections between prior knowledge and the new concept in a lesson. It highlights concrete models/representations from prior lessons, showing the relationship between what students know and what they are learning. For example, Math Background in Unit 2 Lesson 3.3 uses a bar model to display the division of fractions; students have previously used bar models to represent fractions, so this connection is familiar and activates their prior knowledge and application to this concept.
The beginning of each unit provides a chart with three sections: “Before,” “In This Unit,” and “After.” It shows the vertical alignment of concepts across grade levels. The Before section notes prerequisite knowledge and skills. Next, In This Unit notes the content students will learn about in the unit. Finally, the After section shows upcoming knowledge and skills from within the grade level and subsequent grade levels, yet it has no specificity to a grade level. It is unclear if the skills learned before and after the unit are within the same grade level or the grade levels prior or subsequent. It is unclear how the skills build and connect across grade levels.
While the materials include a “Program Resources” guide to support teachers in understanding the included resources, and while the components routinely include Plan, Engage and Explore, Teach, and Assessment and Intervention, some of these supports lack guidance. For example, the materials note that teachers can “present engaging content on a multitude of devices, including tablets and interactive whiteboards.” However, the materials offer no teacher implementation guidance for interactive whiteboards. Also, “Math Talk” and “Differentiated Instruction Print Resources” offer supports like “Leveled Practice and Problem Solving,” but again, they include no teacher guidance regarding their specific implementation. The “Personal Math Trainer” can “create and customize assignments aligned to specific lessons or TEKS,” but the reviewer could not confirm these features.
The materials use inconsistent naming conventions, which causes confusion when locating different materials. For instance, each unit contains two different tasks titled “Performance Task.” One Performance Task is in the student edition at the end of the chapter. There is an additional and completely different Performance Task in the teacher online materials. No guidance indicates the difference between the two. Also, each unit and module contain a Before/In This Module (Unit)/After table. The unit table is a type of vertical alignment; the module table is an alignment within the same grade level. Nowhere do the materials state this. Furthermore, the Teacher Edition and student edition both contain a “Benchmark Test” in the Front Matter. The “Assessment Resources” also contain Benchmark Tests, but they are not the same as the one in the teacher/student editions.
The teacher guide does not mention the online Assessment Resources document, yet it serves an important purpose: denoting all assessments included in the materials. The Assessment Resources document also explains the “level” options. For Unit Tests, Level A is for students who are slightly below level; Level B, for students who are on level; Level C, for advanced students; and Level D, for students who are considerably below level and require modified materials. Then for Module Quizzes, Level D is for students who are considerably below level and require modified materials; all other students use Level B. Even though differentiated tests are necessary, no guidance explains how to implement these levels or which students qualify for each.
The materials provide some implementation guidance to meet variability in programmatic design and scheduling considerations. The materials provide some guidance for strategic implementation without disrupting the sequence of content that must be taught in a specific order following a developmental progression. The materials are designed in a way that partially 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 provide a suggested sequence of units that considers the development of conceptual understanding, although it does not explain the reasoning behind the sequence related to the developmental progression of mathematics. By following the structure inherent in the materials (by unit, by module, by lesson), teachers ensure that students learn prerequisite concepts first, and the development of conceptual understanding usually follows the CRA continuum. For example, in following the materials in the order presented, students in grade 6 will learn prerequisite skills related to ratios and rates in Unit 3 Module 7 and then apply this understanding to solving proportions in Unit 3 Module 8. At other times, the prerequisites mentioned are not related to previous units. For example, in Unit 2, the introductory pages include a sidebar of what students will learn in the unit and what students learned before. However, the skills listed in the “Before” section are not skills that students learned in Unit 1. Moving into Unit 3, the “Before” section of the sidebar lists skills that were not taught in Units 1 and 2. As a result, the materials do not provide guidance about the flexibility of the placement of specific lessons, modules, or units.
Throughout the materials, 53 of the 63 lessons specifically focus on the primary focal areas for Grade 6, which include operations with integers and positive rational numbers, ratios, and rates, including using equivalent ratios to represent proportional relationships, expressions, and equations to represent relationships, and data representation. While the materials provide no guidance to support teaching the focal areas without disrupting the recommended sequence, given the design of the materials and following the order of the materials, the lessons covering the focal areas could be used effectively given that lessons do not include any spiraling in daily practices. However, it is important to note that “Texas Test Prep” and “Unit Tests” do involve spiral questions, so their use is limited if the materials are not fully implemented in the order presented.
The materials provide some support for LEAs to consider how to incorporate the materials into a variety of school designs. The beginning of each unit includes a “Pacing Guide” in the Teacher Edition to support LEAs with a timeline and sequence of instruction for using the materials in a 45-minute class, as well as in a 90-minute (block) classroom. The Pacing Guide is very basic, providing only a lesson reference (i.e., Lesson 1.1) for each recommended day of instruction for the unit. It is important to note that, in the grade 6 materials, Units 2 and 4 do not include a Pacing Guide for 90-minute classes, only 45-minute classes. Also, the materials do not include suggestions regarding the implementation of the materials beyond their use within a 45-minute class or a 90-minute class, nor do they include suggestions for implementation related to co-teaching, multi-grade classrooms, and/or online schools. While the materials do include Intervention/Reteach tasks, such as the “Skills Worksheets” and the “Reteach” found in the online teacher and student editions, they provide no guidance for how to incorporate full class as opposed to small group intervention times, co-teaching, multi-grade classrooms, or online school.
Various school settings (magnet schools, charter schools, and public schools) should be able to utilize these materials in their entirety. In fact, the materials contain a “mySmartPlanner” that allows the user to create and schedule the entire year. The Smart Planner allows input of personal school calendar information, including days off and holidays. The “Auto-Schedule” button then schedules the entire year based on the calendar and the length given for class periods.
The materials provide little guidance on fostering connections between home and school. Throughout, the materials lack the development of strong relationships between teachers and families; there are no recommendations to build relationships. The materials only specify online activities for use at home to support students’ learning and development but do not provide explanations for how to use the resources from home to improve connections.
Evidence includes but is not limited to:
The materials include online access to resources parents can use at home, but the materials only specify online activities for use at home to support students’ learning and development and do not provide suggestions for meaningful, authentic opportunities. (Additionally, there is no evidence that parents can access the resources.) The materials include a “Student Online Edition,” which allows them to “explore concepts, take notes, answer questions, and complete homework.” “Animated Math” activities provide students the opportunity to “interactively explore and practice key math concepts and skills.” The online “Personal Math Trainer” “provides a variety of learning aids that develop and improve your understanding of math concepts, including videos, guided examples, and step-by-step solutions.” Both the online and paper versions include “Math On the Spot” QR codes that students can scan with a “smartphone to watch Math On the Spot tutorial videos for every example in the book” with “step-by-step instructions of the math concepts covered in each example.” Printable versions are available of all the practice worksheets, including “Challenge,” “Are You Ready?,” “Reteach,” and “Practice and Problem Solving.” It also includes a library of online manipulatives, TI Activities (although the link is to a “no page found” page on the Internet), and STEM Projects by Spacemath@NASA. However, the materials do not provide any guidance for teachers to communicate the accessibility of any of these materials to families.
For some English Learners, the “Spanish Student Edition” mirrors the layout and materials provided in the original student edition with all the text translated into Spanish. The online student edition includes a page with each lesson entitled “Success for English Learners,” which can be downloaded and printed. Math On the Spot video tutorials include a Spanish translation, both auditory and Spanish closed-captioning. The End Matter of the student edition includes a Glossary which provides a written definition for each term in English and Spanish and has an example (as applicable) in English. The Multilingual Glossary provides an audio recording in English and Spanish of the pronunciation of each vocabulary term, a written definition in English and Spanish, and an example (as applicable) in English. In addition, a translation of definitions is available in 12 additional languages. While the materials include resources that could be used as school-to-home supports, it is important to note that the materials do not include any guidance related to communicating the availability of these resources to families.
The materials do not guide teachers on communicating with families about mathematics in Grade 6, the structure of the mathematics classroom, or how families can support the class (i.e., practicing math facts, playing games, posing complex problems) which according to the National Council of Teachers of Mathematics (NCTM) are foundational components of a strong relationship between teachers and families. There are no family connection ideas related to each unit that explain answers to common math questions such as pedagogy, content, and learning outcomes. The materials do not make suggestions for planning family math nights or other similar activities. They provide no resources to bring attention to how families could engage with the main focal areas from each grade level at home. The materials do not include opportunities for students to engage in assessment reflections to plan for remediation and enrichment on particular skills both at home and at school, nor do they include suggestions for helping parents to participate in decision making and goal setting for their child.
“Differentiated Instruction” includes a “Home Connection,” which suggests real-world examples of using math at home. (Only nine of the 63 lessons offer this connection.) Although the section is titled Home Connection, the tasks included in this section are an extension of the lesson and rarely provide teachers a meaningful way to engage parents in the mathematics content at home to build relationships. For example, Lesson 1.3 states, “Students may be unfamiliar with how loans work. Discuss that many people borrow money they need to buy expensive items like cars, furniture, computers, and homes. Discuss that people pay back the money they borrow over a period of time, and they pay fees for that privilege. Invite students to cite some examples with which they are familiar. Then have them define absolute value in their own words and then explain how it is used to express the amount of money borrowed.” This could easily be done at school and does not foster home connections. Then, four of these tasks encourage students to locate items at home: Lesson 8.2 says, “Have students create a table like the one below. Ask them to find 4 items at home that have a given measure. Have them record this information in the table and then calculate and record an equivalent measure in a different unit.” Again, this could be an independent activity where family members are not involved. Twice students are encouraged to record real-world situations using experiences from their home or family. In Lesson 8.1, the Home Connection says: “Ask students to describe both an additive relationship and a multiplicative relationship that involves their family. Sample answer: ‘I am 2 years older than my sister’ describes an additive relationship. ‘I can do twice the number of sit-ups that my brother can’ describes a multiplicative relationship.” (This is not something that requires the student to be at home.) In Lesson 11.2, the materials guide teachers to “have students record real-world math situations they experience at home, using both words and mathematical symbols.” (The sample answer given says, “Mom works out for the same length of time each day. How long does she work out in a week? 7t, where t represents the length of time she works out in a day.” This example does not require the student to be at home or have interaction with a parent/family member.) Two other times, students are encouraged to survey their family, like in Lesson 18.1, where the activity suggests “have students survey their family, relatives or friends about whether they have checking accounts, ATM cards, debit cards, or credit cards. If they do, have students ask them to explain their preference for each. Students can then share real-life reasons people use these banking services.” (This is the only activity that truly would need a family member to assist. However, the only “assistance” offered is answering questions on a survey. The family member is not helping the student to understand the material better, reinforcing anything, or practicing the new skill.) Lesson 17.1 instructs, “Have students search through a newspaper or magazine at home with a family member to find a set of data, such as prices of houses, high temperatures, or points scored by the players on a sports team. Students can then find the mean and median for the data set.” While this does say to do this with a family member, it doesn’t have to be, nor does it even have to be done at home. The teacher could bring in newspapers and magazines and complete this activity from the classroom just as easily with the same outcomes. These examples are all specific to the content in the unit; they are simply suggestions for the student to extend learning beyond the classroom and do not develop the relationship between teacher and family.
The visual design of student and teacher materials (whether in print or digital) is neither distracting nor chaotic. Throughout the materials, an appropriate use of white space and design that supports and does not distract from student learning is included. Pictures and graphics are supportive of student learning and engagement without being visually distracting.
Evidence includes but is not limited to:
The teacher materials are designed in a way that most teachers can locate important information for lesson planning and implementation. For example, the “Teacher Resource Page” lists all Units, Modules, and Lessons, with links to all of the ancillary materials that can be used to support differentiated learning. Materials for the Unit, such as “Unit Tests” and “Performance Tasks,” are always provided under the Unit heading. Module components such as “Real-World Videos,” “Module Quizzes,” “Challenge” activities, and “Professional Development Videos” are nested under each Module heading. Lesson components such as “Math on the Spot” examples, “Extend the Math,” “Differentiated Instruction,” and “PowerPoint Presentations” are nested under the Lesson heading. (While this consistency is helpful to teachers in navigating the materials, it should be noted that the “Online Teacher Edition” lacks a user guide to point out supports such as the “Assessment Resource” document, the virtual manipulatives, and the Multilingual Glossary found in Unit 1.) The “Contents and Unit Pacing Guide” pages, though, are clear and uncluttered. The “Program Resources” page includes a breakdown of tools and resources available throughout the materials that are used to support the implementation of paper-based and online content.
The materials consistently include items to support teachers in planning and implementing lessons. Within each module, Professional Development videos include models and guide teachers to build understanding related to the Module content. The materials consistently include these videos, and they are easily accessible via the QR code in the Teacher Edition or by clicking the Professional Development hyperlink in the Module menu of the Online Teacher Edition. The Teacher Edition each lesson includes a Professional Development section that includes guidance related to how the lesson integrates the “Mathematical Processes” as well as a “Math Background section,” which reinforces vocabulary and provides alternate ways to think about the content and scaffolding guidance related to the content (i.e., what concepts are being built on and how this content will be used later).
The teacher material is designed to emphasize the 5E model and is found in the same order for each lesson so that teachers can reach each component easily. An “Essential Question” guides teachers to the focus of the lesson as part of the Engage section. The Explore section offers specific notes related to implementing the provided Explore Activity. Explain includes multiple examples, sample questions with responses, and common misconceptions for teachers to consider in preparing and implementing the lesson. Elaborate guides the teacher to summarize the lesson through the provided question(s) and “Guided Practice.” In the sidebar of the Teacher Edition, a Differentiated Instruction section provides guidance on ways to adapt the instructions to a Kinesthetic Experience, Number Sense, and numerous other ways to differentiate the instruction. Evaluate includes “Independent Practice” as well as tables for the teacher to understand the focus of questions within the Guided Practice and Independent Practice. The materials include a “Lesson Quiz” and additional resources to support students’ understanding. The Extend the Math section at the end of the lesson provides teachers guidance on an activity that could be incorporated to build on the provided lesson.
The materials follow the guidelines of User Interface Design. The materials include visibility of system status as well as user control and freedom. In the Teacher Edition, a page is devoted to the unit title and a table of contents. The lines of text are spaced appropriately, and the eye is drawn to the unit title. The text on the Math Background pages contains the right white space, and tables and models are spaced appropriately. Throughout the unit, the teacher can locate teacher supports in the same area at the bottom of the pages as well as the right and left margins. Each page of the introductory material is designed with the same font and color scheme in each unit. The color scheme and use of fonts are consistent, appealing, and appropriate. However, the pages of Program Resources are highly distracting and do not use white space appropriately. The font is too much bold text and not enough white space surrounding the text. The visuals are distracting.
Items with photographs and colorful pictures do not distract from the text on the page or interfere with learning. Display charts, tables, and graphs are also clear and easy to read and understand. For example, the tables included in the materials use a consistent, easy-to-read font and include translucent shading to highlight the titles for the independent and dependent variables displayed in the table as well as the title of the table (when included). Although the materials use a variety of colors for highlighting tables (i.e., green, blue, and purple), this difference does not impact the effectiveness or ease of use of the tables. Additionally, number lines and coordinate graphs are incorporated, which include clear labels and consistent spacing to support student learning. Pictures and graphics are supportive of student learning and engagement without being visually distracting, and where appropriate, real-world photos are placed to support the context of the problem. For example, the Glossary included in the Student Edition and the Multilingual Glossary included in the Online Student Edition include clear and authentic pictures and drawings to support understanding and reinforce highlighted vocabulary within the materials. In both the Student and Teacher Edition, the unit introductory page contains a full-page color real-world photograph with the use of white space to place introductory text.
The lessons follow the same general design with the same pieces being highlighted the same way. The icons used to represent included components in the materials are consistent throughout the materials. For example, the Animated Math “online interactive simulations, tools, and games” are consistently notated by the icon labeled x2. The Math on the Spot video tutorials are noted throughout the materials with a small television icon. The Personal Math Trainer, which “provides online practice, homework, assessment, and intervention,” is denoted by an icon of the four operation symbols (i.e., addition sign, subtraction sign, multiplication sign, and division sign) wearing a cap and whistle. All examples use a blue-colored font to show each step of the solution; pink writing (in the Teacher Edition) indicates correct answers/thinking for each task presented, as well as notes about sample questions and student responses to support instruction and discussion.
Materials contain features that allow students to interact digitally with tasks, receive immediate feedback while completing activities, and listen to high-quality examples before practice. For example, students have access to materials such as “Math on the Spot” QR codes, which can be scanned with a “smartphone to watch Math on the Spot tutorial videos for every example in the book.” The Math on the Spot videos include “step-by-step instructions of the math concepts covered in each example,” which supports student understanding. Students can also use online manipulatives such as algebra tiles and integer counters, and the student can practice concepts learned using these tools, following the progression of the math content. Number lines and coordinate planes represent various real-world situations and are used throughout the text. “Independent Practice” has a live link to “Selected Answers.” Students also have access to a “Personal Math Trainer,” which allows students to practice skills and complete homework online. The Personal Math Trainer “provides a variety of learning aids that develop and improve...understanding of math concepts, including videos, guided examples, and step-by-step solutions.” “Animated Math” provides students the opportunity to “interactively explore and practice key math concepts and skills.” These activities provide immediate feedback as students work on problems similar to those in the student edition of the textbook. If answers are incorrect, students are given step-by-step instructions to guide them to correct answers with additional practice. Students are also guided through the use of online manipulatives such as number lines, graphing tools, etc. In addition, the teacher materials include digital access to the Teacher Edition and resources, as well as a digital calendar and planner, reports, and the ability to create assignments digitally.
The technology materials are somewhat aligned to the scope and sequence of the program. The materials do not provide recommendations for teachers on which days to utilize technology with students, but there is a time during lessons that the technology would enhance student learning. The “Unit Pacing Guides” do not include guidance or overviews on technology use. The digital planning guides have live links to the other online resources to facilitate planning and ease of use. However, the materials include a “Student Online Edition” that mirrors the Student Edition and follows the scope and sequence as outlined by the program in the Unit Pacing Guide at the beginning of each Unit. These materials follow the same progression for content as the lesson and are embedded within specific lessons to ensure continuity with the scope and sequence. For example, the Math On the Spot videos supports student understanding of the TEKS covered in the lesson. The Animated Math activities (accessible through the Student Online Edition) provide interactive exploration and practice of key math concepts and skills that are taught in each lesson. The Personal Math Trainer allows students to practice skills and complete homework online. The Personal Math Trainer aligns with the scope and sequence outlined in the materials. Within the resources, the teacher has links to assessments, practice pages, reteach pages, and other worksheets that are editable or available to print as a PDF. Each unit contains these same links and follows the materials’ sequence of concepts.
The materials contain digital features that enhance and do not replace or detract from classroom learning. Students can navigate through the units, modules, and lessons with ease, and buttons to visit videos and online skills practice are consistent and easy to use. Students’ individual, interactive, write-in edition includes a navigation menu to allow students to easily move throughout the online materials. The Content menu button provides access to the Student Edition text, tutorial videos, online skills practice, and highlighting/note-taking opportunities within the online text. The Resources menu button provides students access to lesson-level resources through the Personal Math Trainer, as well as access to the digital manipulatives, which include integer counters, fraction bars, fraction-decimal grids, bar models, Geometry Sketcher, Algebra tiles, graphing calculator, scientific calculator, and Multilingual Glossary. The Bookmarks button allows students to view and select pages that they have previously bookmarked. The Notes section allows students to see any notes they have taken online or any text that they have highlighted. The Page View button allows students to change their page view to a portion of a page zoomed in, one full page, or two full pages at a time. The Search button allows students to search online materials. Finally, the More navigation menu item allows students to access “My Notebook,” Print, review the Quick Start Guide for understanding and using the online materials, and find out about the edition of the materials they are using.
The materials do not provide support for teachers to successfully integrate technology. Aside from the “Quick Start Guide” in the Online Student Edition, the materials do not provide guidance for teachers to successfully integrate technology. Sidebars within the Student and Teacher Editions include references to the technology included; however, no guidance is available to assist teachers with the integration of the technology. The teacher materials do not guide on how to support students other than suggesting the student uses the tools. There is no communication for families in how the student should utilize the digital components.
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