2nd Grade - Gateway 2

The instructional materials for enVisionMATH California Common Core Grade 2 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 8 Lesson 8-4, the Overview of the PBIL states, “In this activity, children use connecting cubes to model adding two two-digit numbers and then record the sum. They will decide if regrouping is needed.” The Independent Practice directions state, “Use connecting cubes and your workmat. Add. Did you need to regroup? Circle yes or no.” There are four problems within the Independent Practice that has students using cubes to demonstrate conceptual understanding independently.
• In Topic 10 Lesson 10-2, the Overview of the PBIL states, “Children will use place-value models to represent numbers up to 1,000. The Independent Practice directions state, “Write the numbers. Use models and your workmat if needed.” Students do not use place-value models independently to represent numbers as the pictures are given for the numbers and are shown in the exact same order of hundreds, tens, and then ones for all problems. Students can count the pictures and place the corresponding number in the chart.
• In Topic 11 Lesson 11-4, the Overview of the PBIL states, “In this activity, children will use place-value blocks to add three-digit numbers.” The Independent Practice directions state, “Add. Regroup if needed. Use models and your workmat to help you.” Students do not demonstrate adding three-digit numbers independently using place-value blocks as they fill in a template of a given problem.

The instructional materials for enVisionMATH California Common Core Grade 2 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 instructional materials provide opportunities for working with the applications of mathematics through routine problems within the Problem Solving lessons and the Problem Solving section within all lessons.

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 4 Lesson 1-4, students write an addition number sentence to represent a story problem. Independent Practice problem 4 states, “There are 4 children with balloons. If each has 3 balloons, how many balloons are there in all?” The problem has a blank repeated addition number sentence below the story problem. The students fill in the blanks with the numbers.
• In Topic 8 Lesson 8-9, students write an addition number sentence to represent a story problem. Independent Practice problem 4 states, “Tina picks 39 blueberries. She picks 14 strawberries. How many pieces of fruit does Tina pick?” The problem has a blank addition number sentence below the story problem. The students fill in the blanks with the numbers.
• In Topic 11 lesson 11-9, students write a subtraction problem in a place-value template to solve a story problem. Problem Solving problem 6 states, “Jeff has 517 baseball cards. He has 263 football cards. How many more baseball cards than football cards does he have?” The problem has a blank place value template with a subtraction symbol next to the story problem. The students fill in the blanks with the numbers.

The instructional materials reviewed for enVisionMATH California Common Core Grade 2 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. 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 connections between the MPs and the content standards are not always clear.

• There are multiple places for finding the MPs in the materials: Content Guide in the Program Resources Tab, the beginning of each Topic, sometimes in the Math Background section within each Topic, and at the beginning of each lesson.
• Within each lesson there is a check list of MPs, but not all of the checked MPs are explicitly labeled within the lesson itself.
• In the Content Guide and the check lists, the MPs are labeled and addressed. Within enVisionMATH California Common Core Grade 2 lessons, the MP is abbreviated.
  • MP1 - Make Sense of Problems (no perseverance)
  • MP2 - Reason Quantitatively or Reason Abstractly (treated separately)
  • MP3 - Communicate or Critique the Reasoning of Others (treated separately)
  • MP4 - Model with Mathematics
  • MP5 - Use Appropriate Tools
  • MP6 - Attend to Precision
  • MP7 - Use Structure
  • MP8 - Check for Reasonableness or Make Generalizations (treated separately)
• In Math Background (page 1D) for Topic 1, the teacher note states, “Mathematical Practices: Make Sense of Problems - Remind children to first read a story problem in its entirety at least once, and then to think about it one sentence at a time.” The portion of MP1 about perseverance is not addressed.
• In Lesson 11-2, MPs 1 and 5 are listed. MP1 is identified in the lesson (page 394, item 15), but MP5 is not identified. Even though MP5 is not explicitly identified, there is a note for teachers that states, “How do the blocks help you understand the number 243? How can you show 200 using place-value blocks?”
• In Lesson 16-1, MP7 is on the check list but not explicitly denoted within the lesson. In the Topic 14 Overview, MP7 is checked, but no lesson identifies the MP as being used in teaching. In Lesson 15-5, MP7 is checked, but nowhere in the lesson is the MP explicitly taught.

The instructional materials reviewed for enVisionMATH California Common Core Grade 2 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 throughout the material 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.

Materials provide little assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others.

• Lesson 4-1 exercise 2, page 102, is an Error Intervention. “If children have difficulty seeing that repeated addition sentences have equal groups, then have them model the sentences with counters on the page.” This is a missed opportunity for teachers to use MP3 to share strategies and critique other’s thinking about addition in relation to equal groups.
• In Lesson 6-1 page 157, “Critique the Reasoning of Others: Encourage students to examine if their partner’s reasoning makes sense.” In the Small Group Interaction, one partner is to think of a two-digit number and build that number using base-10 blocks. Students are to then add a multiple of ten to the number created. Children record their answer on the worksheet. There are no directions for teachers to encourage children to give the partner feedback or ask how they know if the answer given is correct.
• In Lesson 8-2 exercise 5, page 220A, teacher directions for Quick Check Master state, “Writing to Explain: Children should write a two-digit number and a one-digit number on the table in the appropriate places. They should write an explanation of how they found the sum.” No direction for teachers to have students share their explanations and critique others is given for MP3.
• In Lesson 10-3 page 305, the teacher note for MP3 states, “By listening and reacting to volunteers, students will understand the different forms and how the forms are similar.” The teacher is to Pose the Problem: “What are three different ways to write the number 519? Lead a class discussion, inviting children to share their answers.” This is a missed opportunity for teachers to have students share their answers as well as the reasoning used to arrive at the answer and what other students think of the given reason.
• In Topic 12 Math Background, the following explanation on how to use MP3 within the topic is provided, “Mathematical Practices: Communicate - Encourage children to share their observations about shapes with one another. By describing what they see, they will be identifying the various properties of the shapes.” Critiquing others’ reasoning is not mentioned.