3rd 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.

Standards 3.OA.1 and 3.OA.2 focus on interpreting products of whole numbers and interpreting whole-number quotients of whole numbers.

• Lesson 1.8 begins students on 3.OA.1 by using pictures and discussing grouping. In lesson 1.10, students subitize and practice doubling, then fact families. In lesson 1.12, there is more work with 2s, 5s and 10s. In lesson 2.6, students practice making sense of equal groups using pictures, counting, skip counting, arrays, and repeated addition. Lesson 3.9 begins with word problems to reinforce the mathematics of 3.OA.1. Lesson 3.11 has students build arrays with counters. Lesson 5.6 returns to doubling; this time using area. In lesson 7.2, there are arrays and estimation. Few questions directly address students' conceptual understanding. Rather, it appears the totality of the activities is designed to encourage students to develop understanding. Teachers are not provided many opportunities to check this understanding.
• Lesson 1.9 begins students on 3.OA.2 by posing leading questions and facilitating students procedures and explanations. Opportunity is not provided for students to really question their strategy nor to relate it in a meaningful way.

Cluster 3.NF focuses on developing understanding of fractions as numbers.

• Fractions first begin in lesson 1.12 in Exploration B. Here students are asked to cut out circles, use dice to determine the number of pancakes and the number of people, and then answer "How much does everyone get if everyone gets an equal share?" Depending on the help teachers provide, this could develop conceptual understanding. The practice section then has students work with number stories involving halves. The progression is fragmented and does not lend itself to students developing an understanding but rather a need to rely on a procedure. For example, the "Equal Shares at a Pancake Breakfast" activity provide an answer of "one-half of a pancake" and then states that "drawings vary." The teacher is not provided with sample answers to see examples of student conceptual understanding.
• Lesson 2-9 "Math Message" asks "4 friends equally share 6 granola bars. How many granola bars will each friend get?" It encourages students to use sketches to show their thinking. This problem lends itself to conceptual understanding if teacher's focus on students' thought processes during the follow-up.
• The best conceptual understanding problems generally occur in the "Open Response" problem in each unit. However, much of the conceptual understanding is limited due to heavy teacher involvement, direct instruction, leading questions, and emphasis on procedures.
• Lesson 2-12 focuses on 3.NF.1 but has students work on vocabulary and familiarity with fraction circles instead of developing understanding from any meaningful manipulation or questioning with the fraction circles. Teachers are prompted to ask "What fraction of a ______ piece is a ________piece? How do you know?" However, the example answer is only "a yellow piece is one-fourth of the red circle."
• "Exploring Fractions" in lesson 5-1 sets the stage for students conceptual understanding by having students note what makes one-fourth.
• Lesson 5.2 allows students to demonstrate conceptual understanding in the Math Message by explaining one-third. The remainder of the lesson is procedural in nature as it ends the Math Message with a note to "(t)ell children that today they will continue to represent fractions with fraction circles, words, and numbers."
• Lesson 7.4 allows students to develop conceptual understanding as students divide shapes into equal parts and connect those to fractions.

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

• On page 452 of the Teacher Edition, the Professional Development box explains unit fractions and shows an example of how representing non-unit fractions by counting unit fractions can help build student understanding of the relationship between the numerator and denominator.
• On page 660 of the Teacher Edition, the Professional Development box explains that children are already familiar with two area models of fractions, fraction circles and fraction strips, and this lesson will introduce the number line as a different model for fractions.

The instructional materials reviewed for Grade 3 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 Grade 3 level, the lessons do not build upon each other to help students reach fluency for all facts, particularly those associated with 3.OA.7.

The instructional materials lack activities to build fluency multiplying and dividing within 100, 3.OA.7. The online spiral tracker shows 125 exposures to 3.OA.7 in focus lessons. When analyzing the lessons, many of the instances noted in the tracker show multiple exposures for the same lesson, and a few lessons were noted as Grade 4 lessons. For example, lessons 2.8, 2.9 and 2.10 all have students using remainders in division which is a Grade 4 standard. Twenty-eight lessons have students multiplying and dividing. Only three (lessons 6.6, 8.2 and 8.6) have students dividing, and in one (lesson 8.2) of those, there are only two division problems. Additionally, in the other 25 lessons, only the multipliers 0, 1, 2, 5, 9 and 10 are explored specifically. Since there is not a consistent progression of learning, it is difficult to be assured all students will have the teaching available to them to reach mastery of fluencies and skills.

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 3.
• 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.
• Some games on the Activity Cards develop fluency, for example: Roll to 1000 on page 153, Beat the Calculator on page 721, and Multiplication Top-it on page 823 help build fluency. These appear throughout the year, sometimes in unrelated lessons.
• 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 teacher's to mark when mastery of facts is accomplished.

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 4-11 has students engaging in application of the math building a rabbit pen. Online in the resource section, some "Projects" are available to help students with application of math.

Standard 3.OA.3 has 161 exposures within the curriculum and is listed as the focus of 27 days of Focus lessons.

• The Focus portions of Lessons 1-8, 1-9, 2-5, 2-6, 2-7, 2-8 (2 days) 2-9, 2-10, 3-10, 3-11, 3-12, 5-5, 5-10 (2 days) 5-11, 6-6, 7-2, 7-3, 8-2, 8-3, 8-4 (2 days) 8-6, 9-2, 9-3, and 9-5 are aligned to 3.OA.3.
• In Lesson 1-8, the first Focus lesson addressing 3.OA.3, students are given one-step multiplication word problems. At the end of the lesson, students write their own number story to match a number sentence. The activities in this lesson requiring students to write their own word problems takes away from the time that students would spend applying this standard and multiplying and dividing to solve word problems.
• In Lesson 2-9, students are given one-step division word problems in the "Equal-Sharing Number Stories" activity. Of the three problems, two problems have a dividend that is a multiple of 10 and 1 has a dividend that is a multiple of 5. The problems are very similar to the sample problems done during the Focus portion of the lesson, so true application of the standard is not required.
• Lesson 5-11 is aligned to 3.OA.3. In this lesson, students learn to divide rectangles/arrays in different ways. Although some of the situations are presented with a context, these problems are not true application problems, and the focus of the lesson is not on multiplication and division within 100.
• Lesson 6-6 contains a "More Number Stories" Math Journal worksheet. The worksheet contains multiplication and division word problems. The worksheet begins with a bulleted set of directions for approaching each problem, and the problems themselves are scaffolded. For example, each problem includes a table to fill in before providing the solution. 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.
• Lesson 7-2 contains an "Estimating the Number of Plants" activity that is aligned to 3.OA.3. This activity is only one problem. The context of the problem is thin, and the problem is really more about estimation than multiplication and division within 100. In the problem, students are provided with a diagram, and the solution path is clear. Students can get the answer of 40 by skip-counting.
• The "Solving Number Stories with Measures" activity in Lesson 7-3 is aligned to 3.OA.3. However, problems 3 and 4 do not require multiplication or division.
• Lesson 9-3 includes the "Using Mental Math to Multiply" and "North American Bird Number Stories" activities. Both of these activities include one-step problems. The focus of this lesson is not on using multiplication and division within 100 to solve word problems. Many of the products are above 100. This lesson focuses on strategies for "breaking apart factors into numbers that can be multiplied mentally." As a result, the focus of these activities is on strategies for mental multiplication and not application of this standard.
• Lesson 9-5 only contains one worksheet addressing 3.OA.3. The "Jonah's Garden" activity is asking students to determine how many seeds can be planted if nine seeds are planted in each of 16 rows. The problem is very scaffolded. Students are first provided with a rectangle and asked to divide it into two sections: one section of 10 rows and one section of 6 rows. Although dividing this garden and using the scaffolding does allow students to work with two multiplication equations that are within 100 as required by the standard, if a student attempts to solve the word problem without using the provided scaffolding, the multiplication is not within 100 as required by the standard.

Standard 3.OA.8 has 129 exposures within the curriculum and is listed as the focus of 21 days of Focus lessons.

• The Focus portions of Lessons 2-2, 2-3, 2-4, 2-5, 3-2 (2 days), 3-3, 3-4, 3-5, 3-6, 4-12, 5-10 (2 days), 6-1, 6-7, 6-8, 6-9 (2 days), 6-10, 6-11 and 7-2 are aligned to 4.OA.3.
• Lessons 2-2 and 2-3 are aligned to 3.OA.8, but the lessons only include one-step word problems, not two-step word problems.
• Lessons 2-4 and 2-5 both include two-step word problems. However, problems are scaffolded for students, thus the problems limit the entry points for students. For example, on the Lesson 2-4 Math Journal worksheet "Multistep Number Stories, Part 1," students are required to write a number model before they write their answers.
• Lesson 3-3 is aligned to 3.OA.8. Students do not solve two-step word problems in this lesson. Although estimation is used during the process of learning partial-sums addition, there is no evidence of application of standard 3.OA.8 in the Focus portion of this lesson.
• Lesson 3-4 is aligned to 3.OA.8. Students do not solve two-step word problems in this lesson. Although estimation is used during the process of learning column addition, there is no evidence of application of standard 3.OA.8 in the Focus portion of this lesson.
• Lesson 3-5 is aligned to 3.OA.8. Students do not solve two-step word problems in this lesson. Although estimation is used during the process of learning counting-up subtraction, there is no evidence of application of standard 3.OA.8 in the Focus portion of this lesson.
• Lesson 3-6 is aligned to 3.OA.8. Students do not solve two-step word problems in this lesson. Although estimation is used during the process of learning "expand-and-trade subtraction," there is no evidence of application of standard 3.OA.8 in the Focus portion of this lesson.
• Lesson 6-1 is aligned to 3.OA.8. Students do not solve two-step word problems in this lesson. Although estimation is used during the process of learning "trade-first subtraction," there is no evidence of application of standard 3.OA.8 in the Focus portion of this lesson.
• Lessons 6-8 and 6-9 include parentheses in number sentences. This is not appropriate for grade 3; parentheses are not introduced in the Standards until grade 5. Although Lesson 6-8 is aligned to 3.OA.8, there are no word problems in the lesson. Lesson 6-9, "Connecting a Number Story and a Number Model," includes a two-step word problem, but the word problem is very scaffolded. The number sentence is already written out for students, and the provided number sentence included parentheses.

The Grade 3 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.

The teacher's guide states that conceptual understanding, procedural skills and fluency, and application are all dimensions of Everyday Math which is certainly true. This curriculum most emphasizes procedural skill and fluency frequently through the spiral curriculum and daily practice. Conceptual understanding is developed in some clusters but lacking in other clusters. Application is minimally present in the curriculum. The evidence for these conclusions are stated in each of the earlier indicators (2A, 2B and 2C).

The unbalanced aspects of rigor in lessons and assignments lead to a heavy emphasis on procedural skills and fluency. 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 3 partially meet the expectations for identifying the MPs and using them to enrich the Mathematics content.

The MPs are identified in the Grade 3 materials for each unit and the focus part of each lesson.

• For Unit 1, page 13 discusses how MP4 and MP5 unfold within the unit and lesson.
• For Unit 3, page 219 discusses how MP2 and MP7 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 7, page 633 explains the development of MP4 and MP5 in this unit.
• 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. An Implementation Guide is provided on pages 7-16; however little guidance is provided throughout the lessons. 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 Grade 3 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.

• MP5: Lesson 5.5 cites MP7, look for and make use of structure; however, in the lesson, students are simply doubling and not looking for and making use of structure. Lesson 7.1 cites MP5, use appropriate tools strategically. The intent of this MP is for students to choose their own tools and not be given the tool. In this lesson, students are given the tools to use, so it doesn't meet the intent.
• MP 6: Lesson 6.4 cites MP6, attending to precision, and during the lesson, one of the places where the MP is highlighted has students deciding if a calculator would be faster.
• MP7: Lesson 8.8 cites MP7, look for and make use of structure. Simply asking what is the same about all of these shapes doesn't meet the intention of a student looking for and making use of structure.

In some Lessons, the MPs are treated in an accurate and meaningful way. For example, in Lesson 5.3, students are modeling with mathematics using fractions.

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. Little direction is provided to make sure students are showing their critical thinking, process or procedure, or explaining their results. Sometimes there are questions asking them to look at other's work and tell whether the student is correct or incorrect and explain. 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 2 assessment on page 18, question 7 asks students to decide if Jeremiah's number model fits the number story.
• In the Math Journal on page 80, problem 5 asks students how the number model they created fits the story problem.
• In the Math Journal on page 117, problem 5 asks students if they agree with Nicholas' reasoning; Nicholas is a fictional student.
• In Lesson 7-2, on the "Exploring Equivalent Fractions" Math Masters worksheet, students are asked "Do you agree or Disagree? Explain." However, this worksheet has two fractions cards, and the conjecture that students are analyzing is simply that the two cards show equivalent fractions.
• In Lesson 7-3, the Math Message follow-up says to "(h)ave partners share their problem-solving strategies with each other, and then invite a few volunteers to explain how their partner solved the problem." Although some students might analyze the arguments of others, the prompt does not require it, and only volunteers will participate in the activity.

There are many missed opportunities for students to construct viable arguments and/or to analyze the reasoning of others. An example of this is in lesson 7.7 where students read a journal page about the volume of a 1-liter container. They are discussing the conservation of mass (in this case liquid). The teacher is prompted to have the students complete the problem independently and then have a class discussion and listen to students answers. Teachers are instructed to provide support for answers that state all containers hold 1-liter of liquid. The opportunity missed here is encouraging the rich conservation students could have to defend answers by constructing reasonable arguments and defending arguments of others. The materials do not include explicit directions to prompt this conversation.

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 3 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.

Usually only one right answer is available, 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 2.7 states "Could those numbers work as factors of 24? Explain." The missed opportunity here is for teachers to guide students in a rich discussion about what strategies they used and why. There is not that type of guidance for teachers.
• Lesson 3.4 states to have the students trade journals and make sense of their partners work. Again, there is no instruction or guidance for the teacher to support students as they complete the activity.
• Lesson 5.5 has students sketch their thinking, which is not engaging in creating or analyzing arguments. Teachers are not given specific guidance around what to do with the sketches in order to really help students construct viable arguments and analyze the arguments of others.
• Lesson 6.7 states "work together to resolve discrepancies by showing and making sense of their solutions." Again, there is no instruction or guidance for the teacher to help the students do the work.
• Lesson 8-4, "Setting Up Chairs," is a 2-day lesson designed to get students to make, discuss, and revise conjectures. More teacher guidance is needed in order for teachers to support students. For example, on page 753, the text states "Once children have begun working on their conjectures and arguments, try to minimize intervention." The second day of this lesson does provide some sample student work with some sample answers for teachers in the "Planning a Follow-Up Discussion" section on pages 754-755. However, one sample answer for each question is provided, and teachers are not given guidance on how to handle different answers.

The instructional materials reviewed for Grade 3 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 contains vocabulary, general and specialized, as well as strategies for supporting beginning, intermediate, and advanced ELLs. An example of this, from Lesson 3-5 includes "For beginning ELLs use visual aids to scaffold understanding of task directions."
• 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.  Other non-mathematical vocabulary includes “closer but easier number”, “expand and trade subtraction”, “helper fact”, “break apart strategy”, “near squares”, and “fact powers”.