The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The instructional materials present a Problem-Based Interactive Learning activity (PBIL) and a Visual Learning Bridge (VLB) within each lesson to develop conceptual understanding. However, the PBIL and VLB are teacher-directed and do not offer students the opportunity to practice conceptual understanding independently through the use of pictures, manipulatives, and models.

Overall, the instructional materials do not consistently provide students opportunities to independently demonstrate conceptual understanding throughout the grade level.

• In Topic 4 Lesson 4-3, the Overview of the PBIL states, “Students will use place-value blocks to add whole numbers.” In the teacher-directed PBIL activity, students use place-value blocks to add multi- digit numbers. The Develop the Concept: Visual section of the lesson describes three separate steps to add multi-digit numbers without place-value blocks. Step 3 states, “Add the hundreds, regroup, and then add the thousands.” The directions for the Independent Practice state, “In 9 through 24, find each sum.” Students do not demonstrate conceptual understanding of adding multi-digit numbers independently as sums are shown as sample answers in the 16 problems in the Independent Practice.
• In Topic 8 Lesson 82, the Overview of PBIL states, “Students will use arrays and an expanded algorithm to multiply two-digit numbers.” In the teacher-directed PBIL activity, students draw arrays to demonstrate partial products when multiplying two-digit numbers. The directions for the Independent Practice state, “In 5 through 8, find all the partial products. Then add to find the product.” Students do not demonstrate conceptual understanding of partial products independently as products are shown as sample answers in the four problems in the Independent Practice.
• In Topic 11 Lesson 11-4, the Focus question of PBIL states, “How can you name two fractions that name the same part of the whole?” Fraction strips are used during this teacher-directed activity to find equivalent fractions. In the Develop the Concept: Visual section of the lesson, fraction strips are used to check if fractions are equivalent. However, in the Guided and Independent practice, students multiply or divide to find equivalent fractions. The directions for the Independent Practice state, “For 9 through 16, multiply or divide to find equivalent fractions.”
• In Topic 13 Lesson 13-1, the Overview of PBIL states, “Students fold paper strips to investigate the representation of a fraction as a multiple of a unit fraction.” In the teacher-directed PBIL activity, students fold into fourths, color 3 out of 4 of the sections, and label. The process is repeated with a second strip, but each section is cut, thus showing students an equivalent fraction. The directions for the Independent Practice state, “For 7 through 22, write the fraction as a multiple of a unit fraction. Use fraction strips to help.” Students do not demonstrate conceptual understanding of fractions as multiples of unit fractions independently as expressions are shown as sample answers in the 16 problems in the Independent Practice.

The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Each topic includes at least one Problem Solving lesson that can be found at the end of the topic. These lessons offer students opportunities to integrate and apply concepts and skills learned from earlier lessons. Within each individual lesson, there is a section titled, Problem Solving, where students practice the application of the mathematical concept of the lesson. However, the applications of mathematics in Problem Solving are routine problems.

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. Examples of routine applications, where a solution path is readily available, are:

• In Topic 2 Lesson 2-6, students use the four operations to solve a multi-step word problem with whole numbers. Independent Practice problem 4 states, “Find the number of each kind of object in Anya’s collection. 6 minerals, 3 fewer gemstones than rocks. 15 objects in all.”
• In Topic 7 Lesson 7-5, students use the four operations to solve multi-step word problems involving whole numbers. Independent Practice problem 5 states, “Abby buys 15 sunflower plants and 12 petunia plants to plant in her garden. She plans to plant 3 flowers in each row. How many rows of flowers will Abby plant?”
• In Topic 8 Lesson 8-5, students use multiplication and addition to solve a multi-step word problem with whole numbers. Independent Practice problem 5 states, “Dave plans to retile his porch floor. He wants to buy 25 black tiles and 23 white tiles. Each tile costs $2. How much money, m, will it cost Dave to retile his porch floor? Write an equation for each problem and then solve.”
• In Topic 12 Lesson 12-3, students solve subtraction word problems involving fractions with like denominators. Problem Solving problem 31 states, “To avoid their predators, ghost crabs usually stay in burrows most of the day and feed mostly at night. Suppose a ghost crab eats 1/8 of its dinner before 10:00 pm. By midnight, it has eaten 5/8 of its food. How much of its food did it eat between 10:00 pm and midnight?”
• In Topic 13 Lesson 13-3, students use multiplication of fractions to solve real-world problems dealing with making fruit punch, batches of pudding, and bread. Problem Solving problem 22 states, “Sun is making 7 fruit tarts. Each tart needs 3/4 cup of strawberries and 1/4 cup of blueberries. What is the total amount of fruit that Sun needs for her tarts?”

The instructional materials reviewed for enVision Grade 4 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Mathematical Practice standards are identified in three places within the Teacher Edition: Problem Based Interactive Learning activity, Guided Practice exercises, and Problem-Solving exercises.

Throughout the teacher and student editions, there is a symbol that indicates that one or more MP is being used. Key phrases such as “Look for Patterns,” “Use Tools,” and “Reason” identify which practice is being highlighted. At the beginning of each lesson, all eight mathematical practices are listed. A check mark is placed beside each practice that is to be addressed within the lesson. For example:

• In Topic 9 Lesson 9-1, MP7 is identified with the icon and the key words “Use Structure.” Question 26 states, “At the North American Solar Challenge, teams use up to 1,000 solar cells to design and build solar cars for a race. If there are 810 solar cells in 9 rows, how many solar cells are in each row?” Teachers are told to “Guide students to look at the first 2 digits in the dividend and the 1-digit divisor. What basic fact can you use to help solve the problem? How will the answer to the problem be different than the basic fact?”
• In Topic 9 Lesson 9-5, MP4 is identified with the icon and the key word “Model.” Question 16 states “Tina is making flag pins like the one shown below. How many of each color bead are needed to make 8 pins?” Teachers are given the information that, “If students need additional help with this problem, they can draw a picture or use counters to model the number of beads of each color needed to make 8 pins.”

An example where the MPs are incorrectly labeled:

• In Topic 9 Lesson 9-4, MP5 is identified with the icon and the key words “Use Tools.” Question 37 states, “At the school concert, there were 560 people seated in 8 rows. If there were no empty seats, how many people were in each row?” Answer choices were: A. 553 people, B. 480 people, C. 70 people, D. 60 people

Overall, all eight math practices are included within the curriculum and are not treated as separate standards. However, the standards are not always used to enrich the content. They are aligned to some of the problems as an explanation to what math practice students might need to use to solve the problem.

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher Edition contains a Mathematical Practice Handbook which defines each math practice and includes question stems for each MP to help the teacher engage students. MP3 offers the following questions stems: “How can I use math to explain why my work is right?”, “How can I use math to explain why other people’s work is right or wrong?”, and “What questions can I ask to understand other people’s thinking?”

The materials label multiple questions as MP3 or parts of MP3; however, those labeled have little information assisting teachers to engage students in constructing viable arguments or to critique the reasoning of others. The information that the materials provide is not specific and are often hints or reminders to give students while they are solving a problem.

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• In Topic 2, Lesson 2-1, Problem Solving Question 18 states, “Writing to Explain. Balloons are sold in bags of 30. There are 5 giant balloons in each bag. How many giant balloons will you get if you buy 120 balloons? Explain.” No teacher guidance is given for this question.
• In Topic 4, Lesson 4-3, Guided Practice Question 7 states, “Construct Arguments. When adding 36,424 and 24,842 above, why is there no regrouping in the final step?” No teacher guidance is given for this question.

Examples where teachers are supported, although generally, to assist students in constructing viable arguments and analyzing the arguments of others include:

• In Topic 6, Lesson 6-5, Problem Solving Question 32 “Construct Arguments. Mr. Tran would like to buy a new sofa that costs $934. He can pay the total all at once, or he can make a $125 payment each month for 8 months. Which plan costs less? Explain.” The teacher notes for that question say “Guide students to understand that they need to compare exact amounts to solve this problem.”
• In Topic 7, Lesson 7-3, PBIL identifies MP3 as the mathematical practice being used in the activity. The teacher note states, “When students work with a partner they explain their thinking to others. This helps them both to construct arguments as well as critique the reasoning of others.”