6th Grade - Gateway 2

The instructional materials for Grade 6 partially meet the expectations to develop conceptual understanding of key mathematical concepts, especially when called for in specific content standards or cluster headings. Overall, the instructional materials present inquiry labs and visual examples as a way to develop conceptual understanding. However, when the materials present conceptual understanding, it is generally as part of class instruction and is rarely incorporated into student practice, so students miss the opportunity to fully develop their own understanding of mathematical concepts.

• Conceptual understanding is called for in 6.RP.A. This standard is covered primarily in chapters 1 and 2.
  
  • The first Inquiry lab shows students how to visualize rates using counters and multiplication tables.
  • Lesson 2 introduces ratios with many pictures and visual examples.
  • The second inquiry lab shows students how to use a bar diagram to find a unit rate. However, in lessons 3 and lesson 6 where students are expected to find equivalent rates and unit rates, there is no connection to the inquiry lab's examples. Students are expected to use division to find unit rates.
  • Lessons 4 and 5 use tables and graphs to further offer visual examples of ratios. However, some degree of conceptual understanding is lost because the materials provide students with all of the tables and graphs needed to answer the questions. As a result, students do not have to draw their own graphs or tables as a process to understanding a problem, rather they just fill in the blanks on partially filled in tables and graphs.
  • The third inquiry lab shows students how to use a bar diagram to solve ratio and rate problems.
  • In chapter 2, lessons 6 - 8 and an inquiry lab show students how to understand percents and solve grade-level percent problems. The materials have examples that include bar diagrams and double number lines to help students gain a conceptual understanding. However, students are rarely required to use those mathematics tools when completing the student exercises.
• Conceptual understanding is called for in 6.EE.A.3. This standard is covered in chapter 6. lessons 5-8 and includes two inquiry labs.
  
  • The inquiry lab and lesson 6 cover the distributive property. The inquiry lab uses area models and algebra tiles to develop the concept of distributive property. However, the student practice section in lesson 6 does not require students to use such tools.

The instructional materials for Grade 6 partially meet the expectations to give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, knowledge of how and when to use procedures is developed in specific content standards. However, students are not given opportunities to practice the individual standards that require procedural skill and fluency throughout the year.

• Procedural skill and fluency is expected in 6.NS.B.2. and 6.NS.B.3. These standards are primarily covered in chapter 3.
• Addition and subtraction of decimals is covered in lesson 1. The problems are both computation and story problems, giving students some practice with when to apply appropriate procedures.
• Multiplication of decimals is covered in lessons 2-4. Students are shown a variety of problem types including estimation to gain fluency on multiplying decimals.
• In lessons 5 and 6, students practice the division algorithm including estimating quotients. The division algorithm is continued in lessons 7 and 8 where student use the division algorithm to compute with decimals.The lesson on division gives students enough practice so that they can become comfortable with the division algorithm.
• Chapter 3 includes enough practice problems that students will develop procedural skill and fluency, but chapter 3 is the only chapter that does this. As a result, students will not get the continued practice throughout the year required to build fluency with decimal operations and multi-digit division.
• Decimals are occasionally incorporated into the chapters on expression and equations, geometry, and statistics, but there are only a few practice problems with decimals. After students have put so much work into becoming fluent with decimals, the expectation would be that they would be incorporated into subsequent mathematics practice. For example, chapter 11, lesson 1, shows student how to calculate the mean. In this lesson students have to utilize several operations to solve a problem; however, all but one of the practice problems involve only whole numbers.
• When decimals are incorporated into the chapters following chapter 3, they are not done to the full expectation of 6.NS.B.3. For example, in the chapter where student use division to solve equations, all the resulting quotients are whole numbers. Though students have had prior practice with decimals in the quotients, they are not expected to continue that practice and get comfortable when those situations arise.

The instructional materials for Grade 6 partially meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have multiple opportunities for application, but many of the application problems are one-step, routine word problems in which students are directed on the procedure to follow in order to solve the problem. There are few opportunities for students to reflect on their learning, and there are few open-ended questions that encourage higher cognitive demand.

• The materials incorporate the following application type of lessons throughout the chapters.
  • The "Power Up" performance tasks at the end of each chapter offer students multi-step abstract questions where they solve problems by using a variety of solution paths.
  • At the end of each unit, there is a unit project. This project gives students the opportunity to research a topic and relate that information to the mathematics of the unit.
  • The materials have problem solving investigations through-out each chapter. They give students step-by-step ways to use a problem-solving strategy
• Application is called for in 6.RP.A.3. This topics is covered primarily in chapters 1 and 2.
  • Chapter 1, lessons 2, 3 and 7 cover ratios and rates; the student practice includes story problems where students have to interpret tables and pictures to answer ratio and rate questions, giving students experience with non-routine problems.
  • Chapter 1, lessons 3 and 4 cover tables and graphs. Opportunities for modeling are provided to student in these lessons. However, the examples and questions involve fill in the blank tables and graphs, and students are heavily prompted on how to solve problems using tables and graphs.
  • The problem-solving investigation in chapter 1 is one of the few places where students engage in application problems and the path to find a solution is not prompted.
  • Chapter 2, lesson 6-7 incorporate some application problems in the student practice, but often the problems are one-step routine problems.
  • Chapter 2, lesson 8 gives students both the opportunity to model percent problems and opportunities to solve more complicated application percent problems.
• Application is called for in 6.NS.A.1. This topic is primarily covered in chapter 4, lessons 6 - 8.
  
  • These lessons cover division of fractions. Though there are some application problems incorporated in these lessons, the general focus is fluency. The included story problems are mostly one-step, and it is clear that division of fractions is required to find the answer to the story problems.
• Application is called for in 6.EE.B.7. This topic is primarily covered in chapter 7, lessons 2-5.
  • The four lessons each cover solving and writing equations that involve a different operation. Each lesson includes an example and some application problems. The included application problems are generally one-step and tell student exactly what to do the solve the problem. For example, question 8 in lesson 3 states, "Pete is 15 years old. This is 6 years younger then his sister Victoria. Write and solve a subtraction equation to find Victoria's age." (Example 2.) Students are told what they have to do to find the answer and they are told which problem in the examples to copy. They do not solve the problem on their own.
  • This section also includes a Problem Solving Investigation; it is one of the few places where students engage in application problems and the path to find a solution is not prompted.
• Application problems are called for in 6.EE.C.9. This topic is primarily covered in chapter 8, lessons 1 - 4.
  • These lessons cover functions tables, equations, and graphs. Even though each lesson includes some application problems, students are given blank tables and graphs to fill in. It is obvious how to get to the answer of each problem. Students do not need to plan or devise a strategy to solve a problem.
  • This section also includes a problem-solving investigation; it is one of the few places where students engage in application problems and the path to find a solution is not prompted.

The materials reviewed for Grade 6 partially meet the expectation for appropriately prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards. Overall, every lesson's problem set has one or more questions in which students have to explain their reasoning. However, students are only occasionally prompted within problem sets and application problems to explain, describe, critique, and justify.

• In the practice problems nearly every lesson includes questions that are specifically labeled with the heading "Justify Conclusions." These questions ask students to explain how they got their answers.
• In a few lessons, the questions are labeled in bold with the heading "Construct a Viable Argument." These questions often ask students to explain if something is true or not.
• In some lessons the questions are labeled in bold with the heading "Find the Error." In these error analysis problems, students are presented with someone's solution and asked to simply identify the error. This does not attend to the full meaning of the standard where students would need to refute claims made by others by offering counter examples and counterarguments. There were very few instances where students were asked to find a counter-example.

The materials reviewed for Grade 6 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, the materials direct teachers with many scaffolding questioning strategies asking higher level questions and offering some suggested activities that lead students to construct viable arguments and analyze the arguments of others. However, the materials lack suggestions or ideas that guide a teacher with setting up scenarios where students experiment with mathematics and, based on those experiments, construct and present ideas.

• In the side bar of the teacher edition, the teacher is provided with many scaffolding questions. The Beyond Level questions ask higher Depth of Knowledge level questions and provide some supportive structures to analyze student arguments.
• In the side bar of the teacher edition, there are suggested activities for teachers to use with students. Very often these suggested activities have students compare, critique, and analyze answers. For example, in chapter 1, lesson 3 "Find the Fib", students work on a team where one student creates three problems, two are solved correctly and one is incorrect. The other students find the one that is wrong and correct it.
• When it comes to student's independent practice, the higher order thinking problems in the students practice section of the materials incorporate some of the MPs that help students to construct viable arguments and analyze the arguments of others. Students are given occasional opportunities to be persistent in their problem solving, to express their reasoning, and apply mathematics to real-world situations. However, very little guidance is given to teachers on how to promote and support students in the development of these skills. This is coupled with the fact that many students are rarely given authentic opportunities to develop the true intent of any of the MPs mentioned above.

The materials reviewed for Grade 6 partially meet the expectation for attending to the specialized language of mathematics. Overall, the materials identify and define correct vocabulary, but there are only sporadic places where vocabulary is integrated into the lessons.

• At the start of every chapter, there is a list of related vocabulary words that will be used in the chapter. Students are given a box that outlines key concepts and key words are highlighted in yellow and immediately defined.
• In each lesson that introduces new mathematical vocabulary, there is a vocabulary start-up that frequently uses a graphic organizer to help students understand the new vocabulary. The materials offer related vocabulary at the start of the lessons, however, minimal reference is made back to them as the lesson progresses. In this way, students are not explicitly supported in coming back and revising/adding to their understanding of these terms. Assumption is made that mastery of vocabulary is immediate.
• At the end of the chapters, there is a vocabulary check included in the chapter review.
• Students are given sporadic opportunities to express mathematics vocabulary with the daily lessons. The materials lack consistent structures to make mathematics terms meaningful and incorporate high levels of mathematical language. There are few places where students are given the opportunity to write or explain in a way that the use of mathematical vocabulary is assessed. The vocabulary usually consists of key words highlighted for the introduction of the lesson with a given definition.