3rd Grade - Gateway 2

The instructional materials reviewed for Grade 3 enVision Math 2.0 partially meet the expectations for giving attention to conceptual understanding. The materials sometimes develop conceptual understanding of key mathematical concepts where called for in specific content standards or cluster headings.

Most of the lessons in the materials have students filling out student pages in a very procedural manner. Rarely do the materials feature high quality conceptual problems or conceptual discussion questions. Some of the lessons start with a problem which could develop conceptual understanding; however, the lessons quickly transition to simply filling out the pages in the student book. For example, lesson 1-1 begins with the question "Ms. Witt bought 3 boxes of paint with 5 jars of paint in each box. What is the total number of jars Ms. Witt bought?" Students are given time to solve this; however, instead of spending time exploring this concept, the lesson quickly moves to filling out pages in the student book in a prescribed manner. In lesson 3-3, the standard is to understand properties of multiplication and the relationship between multiplication and division. The opening question is a good question if students were given time to explore; however, students are quickly immersed into the pages in the student book and a procedure. In lesson 4-2, the standard is to understand properties of multiplication and the relationship between multiplication and division; however, instead of students exploring the conceptual understanding, students are simply doing a procedure using fact families.

Standards 3.OA.1 and 3.OA.2 focus on representing and solving problems involving multiplication and division and understanding properties of multiplication and the relationship between multiplication and division.

• Eleven lessons are focused specifically on 3.OA.1 and 3.OA.2. Lessons 1.1, 1.2, 1.3, 1.5, 1.6, 1.7, 2.1, 2.2, 2.3, 2.4 and 2.5 specifically address standards which are explicitly outlined as conceptual standards.
• In Lesson 1.1 students use counters to create arrays to show the relationship between repeated addition and multiplication. However, students get very little time to really explore the connection between repeated addition and multiplication before the lesson transitions into more procedural type problems in the student book.
• In Lesson 1.2, students use number lines as a way to represent multiplication.
• In lesson 1.4, students draw arrays using the same number of counters and write the corresponding equations to demonstrate understanding of the commutative property of multiplication.
• In lesson 1.5 students use counters and drawings to show equal groups and represent division as sharing.
• In lesson 1.6 students use counters and pictures to show division as repeated subtraction.
• Topic 2 begins each lesson with a problem or two to develop conceptual understanding, but the topic is really focused on patterns, not conceptual understanding. The title of the topic is “Multiplication Facts: Use Patterns.” For example, in lesson 2-2 students use patterns to multiply by 9.

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

• Cluster 3.NF.A is the focus of Lessons 12-1 thru 12-5, Lesson 12-8, and Topic 13.
• Lesson 12-1 is the first lesson addressing 3.NF.1. Page 609A of the teacher edition explains that “The lesson explores the basic idea of a fraction and a unit fraction. Knowing what makes up a fraction -- the numerator and the denominator -- helps students understand the concept of number of equal parts and total number of equal parts.” Lesson 12-5 focuses on number lines and fractions greater than 1. On page 633A in the teacher’s edition, the rigor portion states “Some students may find it challenging to consider a number greater than 1 as a fraction. However, if students keep in mind that a fraction is made up of a numerator and a denominator, either of which can be the greater number, they should have few problems remembering that fractions can represent numbers greater than 1 as well as numbers less than 1.” Careful attention should be paid to ensure that students are not given only these descriptions of fractions. Standard 3.NF.1 clearly states that students should understand a fraction 1/b as the quantity formed by 1 part when a whole is portioned into b equal parts and understand a fraction a/b as the quantity formed by a parts of size 1/b.
• Lesson 12.1 jumps quickly into representing fractions in numerical form and using numerator/denominator vocabulary; students may need more time to develop concept of equal parts, partitioning, and using fractional unit vocabulary.
• In Lesson 12-5, students are representing fractions greater than 1 on a number line. The pages in the student book all provide number lines with some fractions already filled in making it difficult to determine if students have truly understood the concept.

There are some interventions that encourage the development of conceptual understanding; however, these interventions are not meant for all students, only those not meeting the standard.

• For example, in lesson 1-1 students in the intervention activity are actually using counters to make equal groups instead of just being shown a picture in the lesson.
• In the lesson 1-3 intervention students are using graph paper to make arrays.
• In the lesson 1-4 intervention students explore the Commutative Property using color-counters.
• On page 203A in the teacher edition students explore division using color-counters.
• On page 323A, the intervention for teaching area gets more to the concept than the one problem on page 320 in the student edition that shows that it is the square units in the square or rectangle.

The materials partially meet the expectations for procedural skill and fluency by giving limited attention throughout the year to individual standards, which set an expectation of procedural skill and fluency. The majority of the lessons are done in a very procedural manner.

Fluency practice activities do not always focus on Grade 3 fluencies. For example, the fluency practice activity on page 49 in Topic 1 addresses addition and subtraction within 20; this is a Grade 2 fluency. The fluency practice activity on page 97 in Topic 2 addresses addition within 100; this is a Grade 2 fluency. The fluency practice activity on page 157 in Topic 3 addresses subtraction within 100; this is a Grade 2 fluency.

3.OA.7 is fluently multiplying and dividing within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of Grade 3, students should know from memory all products of two one-digit numbers.

• Only 1 lesson deals with division, 5-7, and one lesson, 5-4, has both division and multiplication.
• Six lessons are multiplication and align to 3.OA.7.
• Fluency Practice Activities aligned to 3.OA.7 are found at the end of Topics 5-8, 11, 13 and 15. These activities are all either "Point & Tally," “Follow the Path,” or "Find a Match" activities. These seven pages are found at the end of each topic, not within a lesson, so teachers would have to intentionally incorporate these activities into the lessons. Also, the activities often focus on multiplication and division separately. In Topic 5 on page 285, the activity focuses only on multiplication within 100. In Topic 6 on page 343, the activity focuses only on division. In Topic 7 on page 389, the activity focuses only on multiplication. In Topic 8 on page 459, the activity focuses only on division.
• Six Fluency Practice/Assessment pages in the student book aligned to 3.OA.7 are included in the instructional materials. These pages in the student book can be seen on page 71 of the teacher's edition. These pages in the student book each have 25 problems.

3.NBT.2 is fluently adding and subtracting within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

• 3.NBT.2 is addressed in Topics 8 and 9.
• In Topic 8 students learn concepts and procedures to add and subtract.
• Topic 9 focuses on fluency with addition and subtraction within 1,000. Although two lessons focus on using partial sums to add and subtract, in most of the lessons in this topic students learn and use the standard algorithm for addition and subtraction.
• Fluency practice activities aligned to 3.NBT.2 are found at the end of Topics 9, 10, 12, 14 and 16. These activities are all either "Point & Tally," “Follow the Path,” or "Find a Match" activities. These five pages are found at the end of each topic, not within a lesson, so teachers would have to intentionally incorporate these activities into the lessons. Also, the activities sometimes focus on addition and subtraction separately. In Topic 9 on page 523, the activity focuses only on addition within 1,000. In Topic 14 on page 793, the activity focuses only on subtraction within 1,000.

Page 235M in the teacher's guide outlines the framework for assessing, providing practice and intervention, and providing summative assessment information at the end of the year to check mastery of fluencies. There are also resources for students to self-monitor and track their own growth for multiplication mastery.

The Game Center at PearsonRealize.com provides online mathematics games to help build fluency. There is one game for multiplication and division fluency and one for addition and subtraction fluency.

The Grade 3 enVision Math 2.0 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, but a balance of the three aspects of rigor within the grade is lacking.The majority of the lessons are very procedural. However, the fluency of facts is not thoroughly addressed, and division fluency is only given two lessons in the materials, one of which is shared with multiplication. Application is found throughout the materials. Most lessons only focus on one aspect of rigor at a time, and there are many missed opportunities to connect the different aspects of rigor.

The materials partially meet the expectations for identifying the MPs and using them to enrich the mathematics content within the grade. Overall, the MPs are identified and used in connection to the content standards, but the materials do not always use the MPs to enrich the mathematics content. In the materials, the MPs are over-identified, and the connections between the MPs and the content standards are not clear.

According to the teacher overview, the MPs are identified as follows:

• MP 1: approximately 75 lessons.
• MP 2: approximately 75 lessons.
• MP 3: approximately 90 lessons.
• MP 4: approximately 70 lessons.
• MP 5: approximately 35 lessons.
• MP 6: approximately 50 lessons.
• MP 7: approximately 50 lessons.
• MP 8: approximately 40 lessons.

The total number of lessons identified for the 8 MPs is approximately 485, with 110 lessons total in the materials, so this would lead to approximately 4 to 5 MPs per lesson. With this many practices identified in each lesson, there are many times when the entire meaning of the MP is not evident in the lesson, which leads to students not being able to develop a complete understanding of the MP and its connection to the grade-level content.

• In Topic 12, Lesson 8 cites that MP1, "Make sense and persevere," is the focus MP and 3.NF.1 is the content objective. The lesson focuses on students working to figure out which information from a word problem is necessary to solve. It doesn't enhance the understanding of 3.NF.1.
• In Topic 8, Lesson 1 cites MP8, "Generalize Listen and look for students who correctly ... regardless of the order of the numbers." The meaning of MP8 is not evident in the "Solve & Share" item because students ultimately are using the associative and commutative properties of addition to answer a question, and by using the properties, the students are engaging in MP7, "Look for and make use of structure."
• The directions to help teachers make connections are often vague. For example, lesson 12-6 cites MP1 but does not give teachers any guidance on how to help students make sense of the problem or persevere.

The Math Practices and Problem Solving Handbook in the front of the teacher's edition is a resource for understanding the MPs and knowing what to look for in student behaviors. For example, page F23A lists six indicators to assess MP3, "Listen and look for the following behaviors to monitor students' ongoing development of proficiency with MP3." A proficiency rubric is also included.

The materials partially meet the expectation for prompting students to construct viable arguments and analyze the arguments of others. 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. In most lessons, when students explain their thinking, the explanations are directed by the teacher and are not independent student thinking.

MP3 is identified 90 times in the student edition. In many of the places where MP3 is identified, the students are not attending to the full meaning of the MP. For example, "What You Write" in lesson 13-4 on page 692 cites MP3, however students read an already-written argument and are not asked to either create an argument or analyze the argument already given. Additional examples of not attending to the full meaning of the MP can be found in the following lessons: 2-1, 3-3, 4-2, 5-5, 8-4, 9-7, 10-1, 12-5, 13-6, 14-2, 15-3 and 16-4.

Examples of opportunities to construct viable arguments but not analyze the arguments of others:

• In lesson 9-8, students are asked to create an argument explaining whether addends in a different order would have a sum greater than, less than, or equal to the original sum.
• In lesson 12-4, students construct an argument for approximately locating 2/3 if they know where the point for 1/3 is located.
• In lesson 13-8, students must explain if the two fractions they wrote are equivalent.
• In Topic 9, performance assessment question 5B asks students to explain how they determined the number of green tokens used.
• In Topic 12, performance assessment question 4 asks students to explain what fraction represents a whole cake.
• In Topic 13, performance assessment question 6 asks students to construct an argument to justify a given conjecture.

Examples of opportunities to analyze the arguments of others.

• Topic 8, item 17 on page 420. Zoe says 247 rounded to the nearest hundred is 300 because 247 rounds to 250 and 250 rounds to 300. Is Zoe correct? Explain.
• Topic 10, "Solve & Share" on page 545. Three students found 5 X 30 in different ways. Which student is correct? Explain.
• Topic 13, "Convince Me!" on page 692. Julia says 1/8 is greater than 1/4 because 8 is greater than 4. Is she correct? Explain.

The instructional materials reviewed for Grade 3 partially meet the expectations 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. Usually questions have one correct answer, and there is not much guidance for teachers on how to lead discussions beyond the provided questions. There are many missed opportunities to guide students in analyzing the arguments of others.

Teacher materials sometimes prompt students to have discussions.

• In Lesson 13-2 on page 680 of the TE, the teacher is instructed to ask questions about comparing fractions.
• In Lesson 14-1 on page 740 of the TE, the teacher is instructed to ask students about the hour hand.
• In Lesson 15-2 on page 818 of the TE, students must draw a quadrilateral and explain their reasoning about why it doesn’t fit in a group. Teachers are encouraged to have students come to the board, share their drawings, and explain their reasoning.