5th Grade - Gateway 2

Materials partially meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Frequently, opportunities are missed. Opportunities for students to work with standards that specifically call for conceptual understanding occur by use of pictures, manipulatives, and strategies, but frequently, these fall short by not providing higher-order, thinking questions to truly determine students' understandings.

Cluster 5.MD.C calls for conceptual understanding of volume and how volume relates to multiplication and addition.

• There are 12 focus lessons on volume. Many of the lessons are directed and explicit, so students do not have many opportunities to struggle with the understanding of the mathematics. There is only one Open Response lesson on volume in the year. There are some missed opportunities to connect conceptual understanding of measurement of volume to multiplication and addition.

• Cluster 5.NF.B focuses on applying and extending previous understanding of multiplication and division to multiply and divide fractions.

• Lessons 3.10-3.12 provide several opportunities to develop understanding, including use of manipulatives. "Fraction Capture" provides opportunity for students to create different combination of fractions to sum to a given fraction. Overall, however, the student work in these lessons prompts and promotes students to work with mathematics in a procedural manner.
• The Professional Development box on Teacher Edition page 446 discusses development of understanding in Lessons 5.1-5.4. These lessons provide students ample time and opportunity to work with a variety of solving problems with fractions using several strategies. Although many strategies are addressed in a procedural way, the amount of time spent on these strategies may provide an excellent foundation for developing understanding.

Clusters 5.NBT.A and 5.NBT.B focus on understanding the place value system and performing operations with multi-digit whole numbers and decimals to hundredths.

• In Lessons 4.1-4.5 students are frequently told how to think, sort, and label during problems, thus detracting from developing an understanding. The Teacher Questions for the "Fraction of" game in Lesson 4.2 allow students the opportunity to make mathematical sense of diagrams/manipulatives which could lead to understanding, and students are given time to express their thinking.
• In Lessons 6.1 and 6.3, the Math Journal provides problems to probe student understanding; however, problems simply address student "why?" without providing a task that challenges their thinking. Repetition of mathematical problems detracts from developing conceptual understanding.

Some attention to Conceptual Understanding is found in the Professional Development boxes throughout the Teacher Edition.

• On page 23 of the Teacher Edition, the Professional Development box explains that, in Grade 5, students should find the area of rectangles with fractional side lengths using tiling and applying the formula for area. The box emphasizes that students are not expected to use the area formula until later in the year.
• On page 414 of the Teacher Edition, the Professional Development box explains that the purpose of the lesson is to expose the class to several different decimal subtraction algorithms. The box emphasizes that the "most reliable or efficient algorithm may vary from student to student" and that students do not need to master every algorithm.

The materials partially 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.

Most problems are presented in the same way throughout the entire curriculum. There is little variety of problems or types of problems. Problems are presented as short, one-correct-answer problems. Some of the problems are tied together through concepts and ideas, but many times, lessons are completely disjointed from one another. Each unit contains a two-day "Open Response" lesson which engages students in application of mathematics. For example, lesson 6-5 has students engaging in application of the mathematics where students are asked to figure out how much breakfast casserole is shared with a class of Grade 5. Online, in the resource section, some "Projects" are available; however, several of the "Projects" have students doing activities which do not align to standards, such as Mayan math, Ancient Civilizations math, and Magic Computation Tricks.

Standard 5.NF.6 has 59 exposures within the curriculum and is listed as the focus of 13 days of Focus lessons.

• The Focus portions of Lessons 3-13, 3-14, 5-5, 5-6, 5-7. 5-9, 5-10, 5-12, 7-1, 7-2, 7-3, 8-1 and 8-3 are aligned to 5.NF.6.
• Standard 5.NF.6 is focused on solving real-world problems involving multiplication of fractions and mixed numbers. However, there is not enough instruction or practice of application of solving real-world problems involving multiplication of fractions and mixed numbers.
• On the Lesson 3-13 Math Journal worksheet "Fraction-Of Problems," students are solving routine one-step word problems. This worksheet is aligned to 5.NF.6, but students are not multiplying fractions and mixed numbers.
• On the Lesson 7-1 Math Journal worksheet "Multiplying Mixed Numbers," students are given one-step routine word problems to solve and one multiplication problem to use to write a word problem. These word problems do not require true application of the standard given that the title tells students what to do with the only two numbers in each of the word problems.
• Lesson 7-3 provides scaffolded application problems involving area and multiplication of mixed numbers by integers, not mixed numbers and fractions.
• Lesson 8-1 has a practice sheet with an alignment to 5.NF.B.6, but none of the word problems on the page require multiplying a fraction and a mixed number.
• On the Lesson 8-3 Math Journal worksheet "Buying a Fish Tank," students are given a multi-step word problem. However, the problem is scaffolded, and students are not provided an opportunity to multiply a fraction and a mixed number.

Standard 5.NF.7.C has 30 exposures within the curriculum and is listed as the focus of three days of Focus lessons.

• The Focus portions of Lessons 5-13, 5-14, and 7-4 are aligned to 5.NF.7.C.
• On the Lesson 5-13 Math Journal worksheet "Solving Fraction Division Problems," students are given one-step word problems requiring division of a fraction by a non-zero whole number. In this lesson, the word problems are very similar, and the directions and problems are so scaffolded that true application of the standard is not achieved.
• In Lesson 5-14, students continue to work the same types of one-step word problems that they encountered in Lesson 5-13. Additionally, students are asked to write one-step word problems to match division problems. Students are not provided with a variety of single- and multi-step contextual problems, including non-routine problems that truly require application of the standard.
• In Lesson 7-4, students again solve one-step word problems and write one-step word problems to match division problems. Students are not provided with a variety of single- and multi-step contextual problems, including non-routine problems.
• Student work with this standard focuses on routine problems. Even when students are writing their own word problems, the provided sample answers are typically one-step routine problems. For example, the "Multiplying and Dividing Fractions" Math Journal in Lesson 7-10 gives a one-step sample word problem involving drinks. Both of the sample answers for the "Fraction Division Problems" Math Journal in Lesson 7-4 are about meatloaf.

The Grade 5 Everyday Mathematics instructional materials partially meet the expectations for balance. Overall, the three aspects of rigor are neither always treated together nor always treated separately within the materials. The instructional materials meet expectations for procedural skill and fluency; however, the lack of lessons on conceptual understanding and application do not allow for a balance of the three aspects.

Despite efforts to include conceptual understanding and application, problems are all too often presented in a formulaic way. Questions give away the answers or prompt specific thought patterns. The order of questions often lead students to a specific procedure. Contexts are frequently routine, and problems are posed in a way in which students can solve them by relying on the procedural skill. All aspects of rigor are almost always treated separately within the curriculum including within and during lessons and practice.

The instructional materials reviewed for Grade 5 partially meet the expectations for identifying the MPs and using them to enrich the mathematics content.

The MPs are identified in the grade 5 materials for each unit and the focus part of each lesson.

• For Unit 3, page 217 discusses how MP5 and MP8 unfold within the unit and lessons.
• For Unit 5, page 433 identifies which MPs are in the focus parts of the lessons within the unit.
• For Unit 6, page 553 discusses how MP6 and MP7 unfold within the unit and lesson.
• For Unit 7, page 657 explains the development of MP2 and MP8 in this unit.
• Within the lessons are spots where the MPs are identified.

However, within the lessons, limited teacher guidance on how to help students with the MPs is given. Because there is limited guidance on implementation, it is difficult to determine how meaningful connections are made. Additionally, it is difficult to determine if the MPs have meaningful connections, since the materials break them into small parts and never address the MPs as a whole. The broken apart MPs can be seen on pages EM8-EM11.

The materials partially meet the expectation for prompting students to construct viable arguments and analyze the evidence of others. MP3 is not explicitly called out in the student material. Although the materials at times prompt students to construct viable arguments, the materials miss opportunities for students to analyze the arguments of others, and the materials rarely have students do both together.

There are some questions that do ask students to explain their thinking on assessments and in the materials. Sometimes there are questions asking them to look at other's work and tell whether the student is correct or incorrect and explain. Little direction is provided to make sure students are showing their critical thinking, process or procedure, or explaining their results. Many questions that prompt students to critique the reasoning of others tell the student if the reasoning was originally correct and incorrect. It should be noted though that student materials never explicitly call out entire MPs at once; MP 3 is broken into GMP 3.1 and GMP 3.2 in the materials.

The open-response lessons could be opportunities for students to construct arguments for or against a mathematical question. However, besides just working in groups, there is little prompting from the teacher for students to discuss the answers of other groups or students.

The following are some examples of where the materials indicate that students are being asked to engage in MP3:

• In Lesson 1-3, the teacher's guide provides the following student question: "Why is it important to make sense of others' mathematical thinking?" This question does not require students to analyze the evidence of others as indicated in the materials.
• In Lesson 2-9, on Math Journal page 59, problem 5 asks students which method they used to multiply and why?
• In Lesson 3-9, problem 5 on the Math Journal "Addition and Subtraction Number Stories" activity asks students to explain how they solved the problem.
• In Lesson 3-9, problem 5 on the Math Journal worksheet asks students to explain Morton's reasoning.
• In Lesson 3-14, teachers are told to "Look for partnerships using a successful strategy and have them share their strategies and representation." Although the selected students will explain their thinking, the other students will not. Also, students will not be analyzing the evidence of others.
• In Lesson 3-14, on page 307 of the teacher's guide, teachers are told "(a)fter each strategy is shared, encourage other students to explain it in their own words." Although students may critique the reasoning as they are providing explanation, they are not prompted to do so by the materials.
• In Lesson 5-14, on page 531 of the teacher's guide, teachers are told to "(h)ave students restate others' ideas in their own words to make sure that they understand why the quotients is larger than the dividend." Although students may critique the reasoning as they are providing restated idea, they are not prompted to by the materials.
• In the Math Message Follow-Up for Lesson 8-11, students are simply sharing their answers and explaining how they used the graph to make their predictions.

The materials partially meet the expectation for 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. The Grade 5 materials sometimes give teachers questions to ask students to have them form arguments or analyze the arguments of others, but typically, the materials do not give both at the same time.

In the teacher's guide and lessons, the teachers have very specific, almost scripted, directions for students. Most, if not all, of the Math Master worksheets are presented in a step-by-step directive that does not allow for students to evaluate, justify, or explain their thinking. Usually, only one right answer is available to the posed problem, and there is not a lot of teacher guidance on how to lead the discussion given besides a question to ask. There are many missed opportunities to guide students in analyzing the arguments of others. Students spend time explaining their thinking but not always justifying their reasoning and creating an argument.

The following are examples of lessons aligned to MP3 that have missed opportunities to assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others:

• In the Math Message Follow-Up for Lesson 1-6, students are asked to share ideas that they discussed with partners. Teachers are told to encourage them to "explain their own thinking clearly and to ask questions to be sure that they understand each other's thinking;" however, the questions are not provided for the teacher.
• In the Math Message Follow-Up for Lesson 3-6, students are sharing how they solved the math message. Teachers are told to "(b)e sure the discussion covers the following two strategies;" however, the materials do not offer assistance to teachers to ensure that the two strategies are incorporated into the discussion.
• Lesson 3-11 has students discussing their models and solutions to fair share problems; however, there is no guidance to the teacher on how to prompt rich mathematical discourse.
• In the Math Message Follow-Up for Lesson 6-8, teachers are told to ask students to share their conjectures and arguments. The teacher guidance states that teachers should "(e)xpect most to argue that 2.4*1.8 is greater than 2.4 because 1.8 is greater than 1." However, the teacher guidance does not offer any suggestions on how to guide the conversation if most students do not provide that conjecture.
• In the Math Message Follow-Up for Lesson 8-3, teachers are told to have students share their conjectures, but teachers are not given guidance to help students form the conjectures.

The instructional materials reviewed for Grade 5 partially meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of Mathematics; however, often the correct vocabulary is not used.

• Each unit includes a list of important vocabulary in the unit organizer which can be found at the beginning of each unit.
• Vocabulary terms are bolded in the teacher guide as they are introduced and defined but are not bolded or stressed again in discussions where students might use the term in discussions or writing.
• Each regular lesson includes an online tool, "Differentiating Lesson Activities." This tool includes a component, "Meeting Language Demands," that includes vocabulary, general and specialized, as well as strategies for supporting beginning, intermediate, and advanced ELLs. An example of this, from Lesson 2-5, includes "For beginnings ELLs, use visual aids and role play to scaffold comprehension of explanations."
• Everyday Math comes with a Reference book that uses words, graphics, and symbols to support students in developing language.
• Correct vocabulary is often not used. For example, "Turn-around fact" is used rather than the term commutative property, number sentence is used instead of equation, "name-collection box" instead of equivalent equations or equivalent expressions, "number model" instead of expression, and trade-first subtraction.
• Some units have a heavy load of required mathematical vocabulary. In Unit 7, there are 39 vocabulary words needed for students in Grade 5 to understand the unit. Some of these words include corresponding terms, fathom, hierarchy, great span, joint, relationship, subcategory and others. In contrast, unit 6 only has 14 vocabulary words for the unit which is a much more manageable number for students in Grade 5.