6th Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Does Not Meet Expectations | 55% |
|---|---|
Criterion 2.1: Rigor | 5 / 8 |
Criterion 2.2: Math Practices | 5 / 10 |
The instructional materials reviewed for Grade 6 do not meet the expectations for rigor and mathematical practices. The instructional materials partially meet the expectations for rigor and do not meet the expectations for mathematical practices.
Criterion 2.1: Rigor
Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
The instructional materials reviewed for Grade 6 partially meet expectations for rigor and balance. The materials include specific attention to both conceptual understanding and procedural skill and fluency; however, there are limited opportunities for students to work with engaging applications. As a result, the materials do not exhibit a balance of the three aspects of rigor.
Indicator 2a
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The instructional materials for Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
Cluster 6.RP.A focuses on understanding the concept of ratios and of a unit rate.
- Teacher Resources Part 1 Unit 1 Lessons 6-9, 11, and 12 have students develop their understanding of ratios and unit rate through ratio tables and using ratio tables to find equivalent rates and a unit rate.
- Teacher Resources Part 1 Unit 1 Lesson 10 introduces double number lines as a way to model equivalent rates and further develop students' understanding of ratios and unit rate.
- Teacher Resources Part 1 Unit 5 Lesson 13, 15, and 17 introduce percents of quantities as a rate per 100; discuss the relationship between fractions, decimals, and percents; and use base-10 blocks to model the relationship between decimals and percents. The content of these three lessons focuses on students' understanding of ratios and unit rate by illustrating the connections between ratios written in different forms.
Cluster 6.NS.C focuses on applying and extending previous understandings of numbers to the system of rational numbers.
- Teacher Resources Part 2 Unit 4 has students develop their understanding of negative numbers through the use of number lines, drawings of protons (positive) and electrons (negative) as counters, and different contexts that promote an understanding of negative numbers, such as credits/debits and temperatures.
Standard 6.EE.3 has students applying the properties of operations to generate equivalent expressions.
- Teacher Resources Part 2 Unit 6 Lesson 16 has students use area of rectangles to compare two expressions. The teacher draws rectangles and asks students to find the area of them. The teacher then adds more rectangles to the original, and students find the new area and write equivalent expressions in a table. Students then go on to use properties of operations- Commutative, Associative, and Distributive- to write equivalent expressions as well as use a GCF to multiply to factor the expression.
Indicator 2b
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
The materials for Jump Math Grade 6 meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.
- The teacher's edition gives strategies for mental math starting on page A-30. The strategies are not incorporated into the lesson plans for the teacher.
- There is a game in the teacher's edition pages A42-A43 that helps to build student fluency. This games focuses on addition and subtraction, but it is not mentioned in any of the lessons.
Standard 6.NS.2 expects fluency in dividing multi-digit numbers using the standard algorithm.
- Teacher Resources Part 2 Unit 1 Lessons 57-62 offer students opportunities to develop fluency in dividing multi-digit numbers using the standard algorithm through practice dividing by single-digit positive integers, decimals, and multi-digit positive integers.
- There are further opportunities for students to develop their fluency with dividing multi-digit numbers using the standard algorithm within the Assessment & Practice books on pages 29-40 for Part 2.
Standard 6.NS.3 expects students to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- Teacher Resources Part 1 Unit 4 Lesson 33 offers students an opportunity to fluently add and subtract multi-digit decimals using the standard algorithm. The lesson involves decimals to the hundredths place, and it also gives students the opportunity to reinforce their understanding of adding and subtracting multi-digit decimals through the use of number lines and base-10 blocks.
- Teacher Resources Part 1 Unit 4 Lesson 41 helps to develop fluency by having students multiply decimals to the thousandths place by whole numbers.
- Teacher Resources Part 2 Unit 1 Lessons 48, 57, and 58 offer students opportunities to fluently multiply and divide multi-digit decimals using the standard algorithms. In addition to the standard algorithms, students' fluency is also developed through the use of number lines and the comparison of multiplying decimals to the process of multiplying fractions.
- There are further opportunities for students to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation within the Assessment & Practice books on pages 105-108 and 125 for Part 1 and pages 13-14 and 29-32 for Part 2.
Indicator 2c
Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
The instructional materials do not meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade. Overall, there is little evidence of the opportunity to work with engaging applications of the mathematics. There are few non-routine problems throughout the year. Word problems are present in the materials, but the context has limited bearing on the mathematics. There are ten Problem Solving Lessons designed to help students "isolate and focus on [problem solving] strategies."
6.RP.3 asks students to use ratio and rate reasoning to solve real-world and mathematical problems.
- In Lessons 8 and 9 of Unit 1 in Teacher Resources Part 1, there are word problems that include real-life contexts, but the word problems follow teacher-led examples which the students can mimic when completing their word problems. By following the teacher-led examples, the word problems become routine, and the real-world context does not have any bearing on students chances of successfully completing the word problems.
- In Lesson 19 of Unit 5 in Teacher Resources Part 1, students are asked to complete several word problems involving percents. In the exercises where different contexts are presented, the wording of the exercises is very similar to the teacher-led examples, so students still have a procedure for how to solve the problem in the new context.
- In Unit 2 of Teacher Resources Part 2, Lessons 25-30 include many word problems. In these lessons, the problems are routine because students are presented with procedures or specific models to use to solve the problem through teacher-led examples. In the instances where the context could have bearing on the problem, the structure of the wording of the problem closely resembles the structure of the examples, so the context does not have bearing on solving the problem. For example, in Lesson 25, there is an exercise that states "There are 12 cats for every 10 dogs. If there are 15 dogs, how many cats are there?" This follows a teacher-led example that states "In a pet shop, there are 6 cats for every 8 dogs. There are 12 dogs in the shop. How many cats are there?"
- The performance task In Problem Solving 6-5 engages students in a real-world task for "Making Punch;" however, the placement of this task after problem solving lessons on the use of a specific strategy removes application of math from the task.
Standard 6.EE.7 asks students to solve real-world and mathematical problems by writing and solving equations of the form x+p = q and px=q for cases in which p, q and x are all nonnegative rational numbers.
- In Lessons 14, 15, and 18 of Unit 6 in Teacher Resources Part 1, students are asked to solve problems involving the area of geometric figures. Before solving these problems, students are reminded of the formulas they would use to find the area of different shapes, and there are teacher-led examples completed for them. In addition, students are given a specific way to organize information from the problems and shown how to use key words to substitute specific values into the formulas in order to solve the problems.
- In Lesson 10 of Unit 3 in Teacher Resources Part 2, there is a set of four exercises where students are supposed to complete the four word problems on their own, but the directions before them tell teachers to "work through the first problem together, then have students work individually." The wording of the remaining three exercises matches the wording of other teacher-led examples found in the lesson.
Standard 6.EE.9 has students use variables to represent two quantities in a real-world problem that change in relationship to one another and write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
- Lesson 19 of Unit 6 in Teacher Resources Part 2 is supposed to address 6.EE.9, but the exercises that are presented for students to complete do not involve any real-world contexts. The extension problem in the lesson includes money, but the use of money as a context does not have any bearing on students completing the extension problem.
- In Lesson 20 of Unit 6 in Teacher Resources Part 2, students are presented with one set of exercises to complete, but the units for the independent variable are the same as in the teacher-led example, and although the units for the dependent variable are different from the teacher-led example, the scale and range (from 0 to 10 counting by 1s) on the graphs provided are the same.
Indicator 2d
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
The instructional materials partially meet the expectation that the materials balance all three aspects of rigor with the three aspects almost always treated separately within the curriculum including within and during lessons and practice. Overall, many of the lessons address procedural skills and fluency with few opportunities for students to apply procedures for themselves. There is a not a balance of the three aspects of rigor within the grade.
- The three aspects of rigor are not pursued with equal intensity in this program.
- Conceptual knowledge and procedural skill and fluency are evident in the instructional materials. There are multiple lessons where conceptual development is the clear focus.
- The instructional materials lack opportunities for students to engage in application and problem solving in real world situations.
- There are very few lessons that treat all three aspects together due to the relative weakness in application. However, there are several lessons that include conceptual development leading to procedural practice and fluency.
- There are minimal opportunities for students to engage in cognitively demanding tasks and applications that would call for them to use the math they know to solve problems and integrate their understanding into real-world applications.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for Jump Math Grade 6 do not meet the expectations for practice-content connections. Although the materials meet expectations for identifying and using the MPs to enrich mathematics content, they do not attend to the full meaning of each practice standard. Overall, in order to meet the expectations for meaningfully connecting the Standards for Mathematical Content and the MPs, the instructional materials should carefully pay attention to the full meaning of each MP, especially MP3 in regards to students critiquing the reasoning of other students and teachers engaging students in constructing viable arguments and analyzing the arguments of others.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials reviewed for Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The Standards for Mathematical Practices (MPs) are identified in Teachers Resources Parts 1 and 2 in most lessons. The MPs are not listed in the beginning with the lesson goals but noted in parentheses and with an arrow within the lesson at the part where they occur. As stated on page A-22 in Teacher Resources Part 1, “We guide students to develop the Mathematical Practice Standards by explicitly teaching the skills required. While the development of these practices occurs in virtually every lesson, only some lessons have grade-level applications of the standards. These grade-level applications are identified in the margin.”
Overall, the materials clearly identify the MPs and incorporate them into the lessons. The MPs are incorporated into almost every lesson; they are not taught as separate lessons. All of the MPs are represented and attended to multiple times throughout the year, though not equally. In particular, MP5 receives the least attention.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 6 do not meet the expectations for carefully attending to the full meaning of each practice standard. The publisher rarely addresses the Mathematical Practice Standards in a meaningful way.
The materials identify examples of the Standards for Mathematical Practice (MPs), so the teacher does not always know when a MP is being carefully attended to. MPs are marked throughout the curriculum, but sometimes the problems are routine problems that do not cover the depth of the Math Practices. Many times the MPs are marked where teachers are doing the work.
Examples where the material does not meet the expectation for the full meaning of the identified MP:
- MP1: In Teacher Resources Part 2 Unit 1 Lesson 45, students are given exercises and asked to use multiplication to check the answer to each division problem. Also, Teacher Resource Part 2 Unit 1 Lesson 46 asks students to "use a calculator to check your answers to the questions on AP Book 6.2 p. 9." While students need to be mathematically accurate, they are not making sense of problems or persevering in solving them.
- MP1: In Lesson 11 of Unit 2 in Teacher Resources Part 1, this MP is noted next to three problems where students are supposed to place improper fractions on a number line that is already subdivided into equal intervals. These three problems follow multiple teacher-led examples that have the same structure and format. The students do not have to make sense of the problem because the number lines are already subdivided, and they do not have to persevere because examples like their problems have already been completed in the lessons.
- MP2: In Lesson 5 of Unit 6 in Teacher Resources Part 1, there are three problems where students plot points on a coordinate grid and determine either coordinates for missing vertices of a quadrilateral or what type of quadrilateral is defined by four coordinate pairs. In these problems, students do not have to reason quantitatively because there are no units involved in the problem, and they also know that the quadrilateral will be either a square or a rectangle.
- MP4: In Lesson 32 of Unit 8 in Teacher Resources Part 2, the second extension problem is labeled with this MP, but the students are provided with diagrams that helps them visually model the problem. Also, since all of the problems in the lesson are about finding the volume of rectangular prisms, students would already know which calculations need to be made. Lastly, the statement of the problem limits the way in which some prisms can be arranged, so there is no reason for students to determine if other solutions might be better than the first one they obtain.
- MP4: In Teacher Resources Part 2 Unit 2, Lesson 26 has a section titled “Word problems involving ratios.” In this section, the teacher leads students through solving a word problem using a ratio table as had been done in a previous problem. In this problem, as with many other problems where MP4 is noted, students do not have the opportunity to make assumptions or approximations in a complex situation, identify important quantities and represent their relationships, draw conclusions, or interpret the results of a problem and make improvements if needed.
- MP5: In Teacher Resources Part 2 Unit 8, Lesson 33 asks students to use a calculator to check their answers. In instances where MP5 is noted, students do not get to choose and use appropriate tools strategically.
- MP7: In Teacher Resources Part 1 Unit 1 Lesson 7, students are learning how to create a ratio table. Students do not have look for or make use of structure as the rules used to create the ratio table are clearly shown (x2, x3). Because the teacher has given them the answer, students are not required to look closely to discern a pattern or structure.
- MP8: In Teacher Resources Part 2 Unit 4 Lesson 67, students are asked how far apart given integers are. Students do not have an opportunity to look for repeated reasoning when understanding the distance between numbers on a number line.
Indicator 2g
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Indicator 2g.i
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
The instructional materials reviewed for Grade 6 partially meet expectations that the materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
Materials occasionally prompt students to construct viable arguments or analyze the arguments of others concerning key grade-level mathematics detailed in the content standards; however, there are very few opportunities for students to both construct arguments and analyze the arguments of others together. In the lessons provided in the Teacher Resources Parts 1 and 2, examples are identified as MP3 when the material is asking the students questions. Students rarely have the opportunity to either construct viable arguments or to critique the reasoning of others in a meaningful way because of the heavy scaffolding of the program. For example, Teacher Resources Part 2 Unit 4 Lesson 69 asks students in the exercises "Which is closer to sea level: the bird or the fish?" This question leads to understanding of absolute value, but it does not address MP3 by having students construct their own arguments and/or critique the reasoning of others. Also, in Teacher Resources Part 2 Unit 8, Lesson 41 has students find the mistake made when simplifying a numerical expression that represents the surface area of a cube, but the students are notified there is a mistake before looking at the expression. This does not allow students to critique the reasoning of others because they already know there is a mistake.
Indicator 2g.ii
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
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 concerning key grade-level mathematics detailed in the content standards.
Within lessons, the teacher materials are not always clear about how teachers will engage and support students in constructing viable arguments or critiquing the reasoning of others. Materials identified with the MP3 standard often direct teachers to "choose a student to answer" or "have a volunteer fill in the blank." Questions are provided but often do not encourage students to deeply engage in MP3. In addition, although answers are provided, there are no follow up questions to help re-direct students who didn’t understand. Examples of how the materials supply some questions for teachers to ask but have limited additional support include:
- In Teacher Resources Part 1 Unit 1 Lesson 10, teachers are instructed to draw a ratio table on the board, and then they are directed to "ASK: Is this a ratio table? (Yes) How do you know? How did I find the second row from the first? (multiplied both numbers by 2) Ask a volunteer to come and complete the third row in the ratio table." The teacher is not given any possible answers for the second question, and there is not any assistance for the teacher as to how students might discus the volunteer's solution once it is presented.
- In Teacher Resources Part 1 Unit 6 Lesson 12, teachers are instructed to draw a triangle on the board during the third extension problem and ask the students how to draw the height in the triangle. The materials tell teachers that if the solution does not arise then they should tell the students the correct answers. There are no follow-up questions or prompts provided that could help students in constructing their own argument as to how the height could be drawn.
- In Teacher Resources Part 2 Unit 1 Lesson 52, teachers are instructed to “ASK: How many 1/5s fit into 6?” and “How many 3/5s fit into 6? How do you know?” There are no additional questions or prompts for teachers to ask if students are not able to initially answer, and there is no assistance given for teachers to help students analyze each others' arguments.
- Teacher Resources Part 2 Unit 4 Lesson 69: “Answer these questions. Did you compare the absolute values or the actual values of the integers?” The materials provide teachers with answers to the questions, but there is no assistance given to teachers for helping students construct their own arguments to support their answers or analyze the arguments of other students with which there might be disagreements in the answers.
Overall, some questions are provided for teachers to assist their students in engaging students in constructing viable arguments and analyzing the arguments of others; however, additional follow-up questions and direct support for teachers is needed.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Jump Math Grade 6 partially meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit, but there are times the materials do not attend to the specialized language of mathematics.
Although no glossary is provided in the materials, each unit introduction includes a list of important vocabulary, and each lesson includes a list of vocabulary that will be used in that lesson. The teacher is provided with explanations of the meanings of some words.
- In Teacher Resources Part 1, page A-21 states: “Vocabulary words are listed at the beginning of each lesson plan. Make sure students are familiar with the vocabulary words.”
- Vocabulary words are listed at the beginning of each lesson plan in the Teacher’s Guide, but definitions, if any, are within the lesson.
While the materials attend to the specialized language of mathematics most of the time, there are instances where this is not the case.
- Often students are not required to provide explanations and justifications, especially in writing, which would allow them to attend to the specialized language of mathematics. For example, in Teacher Resources Part 2 Unit 7 Lesson 3, the teacher begins this lesson by introducing the term frequency with students. The students are not required to provide an explanation or justification for their answers that would allow them to use the word frequency in this lesson.
- Many of the discussion prompts provided are guided by the teacher so that the student is merely repeating the teacher's language. This limits student ability to actively use mathematical language. For example, Teacher Resource Part 2 Unit 5 Lesson 27 starts with the teacher introducing new vocabulary (line of symmetry, reflect, image) to help students understand where the lesson is heading.