4th 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 they frequently fall short by not providing higher-order thinking questions to truly determine students' understandings.

The spiral tracker cites many instances of exposure to 4.NF.A, 4.NF.B and 4.NF.C, all of which require the development of conceptual understanding. Analysis of the lessons indicate only 13 lessons for 4.NF.A, 19 lessons for 4.NF.B, and 9 lessons for 4.NF.C. When looking at the lessons, the majority of the lessons do not develop conceptual understanding but instead give students a procedure to follow. Frequently, they work to develop conceptual understanding is through the math boxes, math journal, and games in the lessons. The teacher-provided directions and questions often remove the opportunity for students to develop conceptual understanding and create a procedural approach.

• In the lessons on fractions, there is one open response lesson in unit 5 about fractions. Students spend two days discussing a word problem about inheriting land. However, there is only one answer to this question and one entry point to the problem.
• In lesson 3-4, students are required to develop a rule for finding equivalent fractions. Instead of working with number lines and models, they are introduced to standard multiplication to find the equivalent fraction. With the way this lesson is set-up, students are simply employing a rule to find the answer.
• In lesson 3-7, students are required to put fractions in order. The first page is done with the use of visual fraction models. The second page is done with number lines. However, only one problem gives any fractions on the number line to help students reason about their size.

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

• On page 242 of the Teacher Edition, the Professional Development box reminds teachers to develop understanding instead of pushing multiplication by 1 (and explains why).
• On page 326 of the Teacher Edition, the Professional Development box has strong commentary on the development of the unit. However, the box indicates an intentional emphasis on math facts in an attempt to build towards understanding.

There are many missed opportunities for the daily math message to provide a problem which would lead to student questioning and conceptual understanding of key topics. For example, lesson 5-5 is about adding tenths and hundredths; while the math message is about tenths and hundredths, students only write the "number model." This could have been a rich problem filled with mathematical discourse before the lesson to build conceptual understanding, but instead it is treated very procedurally.

The instructional materials reviewed for Grade 4 partially meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. While lessons do exist to work on fluencies required at the fourth grade level, the lessons do not build upon each other to help students reach fluency, particularly with 4.NBT.4.

The instructional materials lack activities to build fluency adding and subtracting multi-dgit whole numbers using the standard algorithm. There are two lessons on the standard algorithm, which is the first time students are exposed to the standard algorithm. The online spiral tracker shows 81 exposures to 4.NBT.B.4 in focus lessons. When analyzing the lessons, many of the instances noted in the tracker show multiple exposures for the same lesson. Some of the lessons are not having students add and subtract multi-digit numbers. In lesson 1-5, students are rounding and then adding using friendly numbers. Lesson 1-10 has students converting yards, feet, and inches which involves multiplication but does not have students adding or subtracting. In chapter one, where the tracker showed 21 exposures, there are only eight actual lessons, and of those, only two align to the stated standard, lessons 1-7 and 1-9.

There are some places where fluency is given attention in the materials.

• Most lessons in the materials have a "Mental Math and Fluency" piece which allows for students to practice fluencies required in Grade 4.
• Several online games help students with the expectation of fluency, including: Baseball Multiplication, Multiplication Top-It, Beat the Computer, and Multiplication Bingo. It is important to note none of the online games have students practicing division.
• Online is a reference sheet called "Do Anytime Activities" with suggestions to help students practice fluencies at home.
• There is a fact check in the assessment book for teachers to mark when mastery of facts is accomplished.
• Math Boxes are used during each lesson which consist of an average of six problems for students to complete. These problems do not connect to each other and are pulled from several different clusters and/or domains for students to complete for practice and maintenance of previous skills.

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 anther. 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 create fruit baskets using multiple fruits. Online in the resource section, some "Projects" are available to help students with application of math.

Standard 4.OA.3 has 106 exposures within the curriculum and is listed as the focus of 14 days of Focus lessons.

• The Focus portions of Lessons 1-5, 1-6, 1-7, 1-9, 4-2, 4-12, 5-13, 6-5 (2 days), 6-8, 7-7, 7-12, 8-1, and 8-9 are aligned to 4.OA.3.
• Lesson 1-5 is aligned to 4.OA.3. The Focus portion of the lesson addresses estimation, but the problems are scaffolded and center more on the different strategies that were introduced in the lesson than on computation. The Math Masters "Using Estimation Strategies" worksheet includes a family note that states "Today students explored different ways of estimating...While all methods of estimation are equally valid, some may be more helpful than others for answering specific kinds of questions." The note never mentions computation, and the directions state "Read the number stories. Choose an appropriate estimation strategy."
• Lesson 1-6 also focuses on estimation. The "World's Tallest Buildings" Math Journal provides five multi-step word problems. These problems require students to provide estimates and answers. However, on page 46 of the Teacher's Lesson Guide, teachers are told to do the following: "Referring students to the Guide to solving Number Stories on page 26 of the Student Reference Book, guide a discussion of the problem-solving process." This procedure for students to follow when solving number stories along with the scaffolding accompanying the problems detracts from the true application of the standard.
• The focus portion of Lesson 4-2 is also aligned to 4.OA.3. Again, this lesson is addressing estimation without a focus on computation. The Math Journal activity "Finding Estimates and Evaluating Answers" requires students to write estimates and then "(u)se a calculator to solve the problem."
• The Focus portion of Lesson 4-12 also addresses estimation. Although the Math Journal "Solving Multistep Multiplication Number Stories" does provide four multi-step word problems, the Teacher's Lesson Guide again scaffolds the problem solving and detracts from the true application of the standard (page 398).
• The focus portion of Lesson 5-13 includes a "Planning a School Fair" Math Journal activity. The activity includes five problems, but only one of the five problems is not scaffolded for students.
• The Focus portion of Lesson 6-8 addresses solving division number stories with remainders. Students complete the "Interpreting Remainders" Math Journal activity. The activity is very scaffolded. The top of the worksheet has bulleted steps for how to solve each problem, and three of the four word problems are scaffolded for students.
• Lesson 7-12 includes a "Shopping at the Stock-Up Sale" Focus activity. This activity includes four word problems. Although the problems are multi-step, the word problems are very brief, and the context is very thin.
• Lesson 8-1 includes "Cracking a Number Story Code" Focus activity. The activity requires students to solve eight multi-step word problems to crack a code.

Standard 4.NF.3.D has 76 exposures within the curriculum and is listed as the focus of 15 days of Focus lessons.

• The Focus portions of Lessons 5-3, 5-4, 5-7, 5-8, 6-12, 7-6, 7-11, 7-12, 8-5, 8-6, 8-7, 8-8, 8-9, 8-10, and 8-11 are aligned to 4.NF.3.D.
• The Focus portion of Lesson 5-3 focuses on fraction addition number stories. True application of the standard is not achieved because the problems are clearly identified as addition problems, so students know that they simply need to add the two fractions in the word problems. Also, on the "Fraction Addition" Math Journal worksheet, the problems are scaffolded, and students are required to use different strategies to solve each of the two word problems. Also, the worksheet includes three fraction addition problems which also clue the student to add the numbers.
• The Focus portion of Lesson 5-4 focuses on mixed number addition problems. True application of the standard is not achieved because the problems are clearly identified as addition problems, so students know that they simply need to add the two fractions in the word problems. Also, on the "Adding Mixed Numbers" Math Journal worksheet, the problems are scaffolded, and students are required to use different strategies to solve each of the two word problems. Also, the worksheet includes four mixed number addition problems which also clue the student to add the numbers.
• The Focus portion of Lesson 5-7 focuses on fraction subtraction number stories. True application of the standard is not achieved because the problems are clearly identified as subtraction problems, so students know that they simply need to subtract the two fractions in the word problems. Also, on the "Fraction Subtraction Number Stories" Math Journal worksheet, the problems are scaffolded, and students are required to use different strategies to solve each of the two word problems. Also, the worksheet includes four fraction subtraction problems which also clue the student to subtract the numbers.
• The Focus portion of Lesson 5-4 focuses on mixed number subtraction problems. True application of the standard is not achieved because the problems are clearly identified as subtraction problems, so students know that they simply need to subtract the two fractions in the word problems. Also, on the "Subtracting Mixed Numbers" Math Journal worksheet, the problems are scaffolded, and students are required to use different strategies to solve each of the two word problems. Also, the worksheet includes four mixed number subtraction problems which also clue the student to add the numbers.
• The Focus portion of Lesson 6-12 focuses on addition and subtraction number stories with fractions and mixed numbers. The "Fraction Number Stories" Math Journal activity provides six word problems. Although they are routine problems, some of them are multi-step. Some of the problems contain additional questions for students to answer after answering the original question, but this follow-up question is separate from the original question.
• Lesson 7-6 includes a "Three-Fruit Salad" activity which requires students to create a recipe. This multi-step application problem has multiple solutions and entry points for students.
• Lessons 7-12 and 8-7 are focused on Decimal Number Stories. The included word problems in Lesson 7-12 are all focused on money, and the connection to fractions is only made when students convert decimals to fractions, and in Lesson 8-7 the decimals are all "simple" (tenths). The connection to standard 4.NF.3.D is only made if students follow the procedure to solve the problems that has been introduced in the focus portion of the lesson.
• Lesson 8-10 includes a "Making Sparkling Punch" activity. The application of addition and subtraction of fractions is limited because only two of the five ingredients in the recipe are fractions.
• Lesson 8-11 includes a "Puppy Feeding Guidelines" activity which allows students to apply both standards 4.NF.3.D and 4.NF.4.C.

Standard 4.NF.4.C has 64 exposures within the curriculum and is listed as the focus of 15 days of Focus lessons.

• The Focus portions of Lessons 6-13, 7-2, 7-3, 7-4, 7-5, 7-6 (2 days), 7-10, 7-11, 7-12, 8-7, 8-8, 8-9, 8-10, and 8-11 are aligned to 4.NF.4.C.
• The Focus portion of Lesson 6-13 focuses on multiplying a fraction by a whole number. Four of the five problems on the "Making Lip Balm" Math Journal worksheet are scaffolded, requiring students to fill in the blanks for addition and multiplication equations.
• The Focus portion of Lesson 7-2 focuses on multiplication number stories. True application of the standard is not achieved because the problems are clearly identified as multiplication problems, so students know that they simply need to multiply two numbers. On the "Baking Muffins" Math Journal worksheet, the fractions are all multiplied by either three or four. Most of the problems are of the same type making the activity routine.
• Lesson 7-3 is aligned to 4.NF.4.C. On the "Multiples of Unit Fractions" Math Journal, there are only two word problems. The seven other problems on the page cue students to solution methods to the word problems and do not allow for true application of the standard.
• In Lesson 7-4, the problems on "The Walking Club" Math Journal are too scaffolded to allow application of the standard. Each problem prompts students to write a multiplication equation.
• The Focus portion of Lesson 7-5 focuses on multiplying mixed numbers by whole numbers. True application of the standard is not achieved because the problems are clearly identified as multiplication problems, so students know that they simply need to multiply the two numbers in the word problems. Also, on the "Solving Number Stories" Math Journal worksheet, the problems are scaffolded, and students are required to use different strategies to solve each of the two word problems. Also, the worksheet includes four multiplication problems with an integer and a fraction which also clue the student to multiply the numbers.
• Lesson 7-6 includes a "Three-Fruit Salad" activity which requires students to create a recipe. This multi-step application problem has multiple solutions and entry points for students.
• Lesson 7-10 includes a "Burning 100 Calories" activity. Although the problems have thin contexts, students are able to choose their own solution methods.
• Lessons 7-12 and 8-7 are focused on Decimal Number Stories. The included word problems in Lesson 7-12 are all focused on money, and the connection to fractions is only made when students convert decimals to fractions and, in Lesson 8-7, the decimals are all "simple" (tenths). The connection to standard 4.NF.4.C is only made if students follow the procedure to solve the problems that has been introduced in the Focus portion of the lesson.
• In Lessons 8-7 through 8-11, contexts are often expected but require application of the standard. For example, Lesson 8-11 includes a "Puppy Feeding Guidelines" activity which allows students to apply both standards 4.NF.3.D and 4.NF.4.C. Often multiple problems focus on the same context, for example sewing, allowing problems to become more procedural and require less true application.

The Grade 4 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. 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 leads students to a specific procedure. Contexts are frequently thin, and problems are posed in a way in which students can solve them by relying on 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 4 partially meet the expectations for identifying the MPs and using them to enrich the mathematics content.

The MPs are identified in the Grade 4 materials for each unit and the focus part of each lesson.

• For Unit 3, page 219 discusses how MP3 and MP4 unfold within the unit and lessons.
• For Unit 5, page 429 identifies which MPs are in the focus parts of the lessons within the unit.
• For Unit 6, page 541 discusses how MP5 and MP7 unfold within the unit and lessons.
• For Unit 8, page 751 discusses how MP1 and MP4 unfold within the unit and lesson.
• Within the lessons, there 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. In a lesson, this can be seen in 2-8, page 168 TE.

The Grade 4 Everyday Mathematics instructional materials partially meet the expectation for treating each MP in a complete, accurate, and meaningful way. The lessons give teachers limited guidance on how to implement the standards.

Below are examples of where the full intent of the MP is not met.

• MP1: Lesson 8-1 cites MP1 and says "students will be solving stories that are more challenging but use the skills they already now;" however, in looking at the problems they are limited to two-step problems, a Grade 3 standard, which would not require students to persevere necessarily. Lesson 2-7 cites MP1; however, simply asking students "what happens when we go from a larger unit of time to a smaller unit" is not having students persevere in a problem. Lesson 3-2 cites MP1; telling students they should only use one color at a time and record a fraction to describe each of the different ways they find does not have student's persevering with problems.
• MP2: Lesson 1-2 cites MP2 but has students writing numbers in expanded form; this does not have students reasoning abstractly and quantitatively.
• MP4: Lesson 2-9 and 3-1 cite MP4; however, telling students how to model the problem does not meet the intent of practice. Lesson 3-6 cites MP4 and tells the students to use a visual fraction model, again not meeting the intent.
• MP5: Lesson 1-1 cites MP5; however, the students are told to use calculators. Lesson 2-7 cites MP5 and again tells students which tools to use. Lesson 7-4 cites MP5, use tools appropriately; however, in the lessons, students are given the tools with which they are to work and not allowed to choose the tools.
• MP6: Lesson 1-5 cites MP6; having students discuss real-life situations in which an estimate might be useful is not having students attend to precision. Lesson 2-3 cites MP6; however, reminding students that 2 and 7 are factors of 14 and asking for the other factor pair is not having students attend to precision. Lesson 3-9 cites MP6; students answering "what strategy did you use when comparing fractions and try to make a match" is not necessarily having students attend to precision.
• MP8: Lesson 7-5 cites MP8, but in the lesson, there is no indication students are looking for structure when playing "Divide and Conquer."

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. MP3 is not even called out until Unit 3 as a focus practice standard and then doesn't show up until lesson 6. Here, it is labeled next to the directions "Invite students to justify their conclusions." It should be noted, though, that student materials never explicitly call out entire MPs at once; MP3 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 the Unit 8 assessment on page 835, question 5, students are asked to explain how they solved problem 4. However, students are not asked to work with other students and really explain and defend their thinking.
• Math Journal page 72, problem 5, asks students if they agree with Sharita's reasoning; Sharita is a fictional student.
• Math Masters, page 111, problem asks students if they agree with Margot's reasoning; Margot is a fictional student.
• The first problem on the "Sharing Veggie Pizza" Math Masters in Lesson 3-5 homework has two answers, and students must choose the right answer. Students do not provide an explanation for their choice other than to add on to the provided drawing.
• In the Lesson 3-13 Math Message Follow-Up, teachers are told to "(h)ave students explain how they knew which decimal was larger." However, students are not given an opportunity to work with other students and really explain and defend their thinking.
• In Lesson 3-13, on page 305, the summarize problem asks students the following: "How can you help Vanna see her mistake?" A follow-up question states "What incorrect reasoning do you think Vanna used to get her answer?" These questions do not allow students to truly analyze the thinking of others because they are told that the thinking is incorrect.

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 4 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:

• Lesson 3-6 states "Justify their conclusions." The missed opportunity here is for teachers to guide students in a rich discussion about what strategies they used and why.
• Lesson 3-7 states "have the students justify their conclusion," but teachers are not given guidance to help students explore their justifications or the justifications of others.
• Lesson 3-10 cites MP3, and again it asks students to justify. Teachers are not given guidance to help students explore their justifications or the justifications of others.
• Lesson 4-11 has students explain how they solved the problem. Again, there is not instruction or guidance for the teacher to help the students explore the explanations of others.
• During the "Solving an Area Problem with Fractions" activity in Lesson 8-9, the teacher guide states on page 807 to "(h)ave partnerships solve Problem 1 and discuss responses as a class," but teachers are not given guidance to facilitate this conversation.

The instructional materials reviewed for Grade 4 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 5-5 includes "For beginning ELLs use visual aids and restatements to make task directions comprehensible and to explain word meanings."
• Everyday Math comes with a Reference book that uses words, graphics, and symbols to support students in developing language.
• Some units have a heavy load of required mathematical vocabulary. For example, in Unit 2, there are 39 vocabulary words needed for the students in Grade 4 to understand the unit. In contrast, Unit 7 only has 5 vocabulary words for the unit which is a much more manageable number for students in Grade 4.
• 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, trade-first subtraction.  Other non-mathematical vocabulary includes “close-to estimation”, “mirror image”, “rectangular numbers”, and “equivalent names”.