Kindergarten - Gateway 2

The instructional materials for enVisionMATH California Common Core Kindergarten 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 5 Lesson 1, the Overview of the PBIL states, “Children will read and write the numbers 11 and 12 and identify or make matching sets of objects.” The Guided Practice section of the lesson states, “Remind children that there is a special symbol for the number eleven and for the number twelve.” Students then count a given set of pictures and trace over the given numbers of eleven and twelve. The directions for the Independent Practice state, “Children count 11 and 12 pictures and write the number that tells how many there are in each group.” Students do not demonstrate conceptual understanding of writing the numbers 11 and 12 independently. Within the Independent Practice, students count a given set of pictures and trace over the given numbers of 11 and 12.
• In Topic 7 Lesson 6, the Overview of the PBIL states, “Children will join groups of connecting cubes to show a number story. They will write numbers to record each group and how many there are altogether.” In the teacher-directed PBIL activity, students use cubes to represent the number of boats from a given problem. The Guided Practice section of the lesson states, “Guide children to use cubes to show each group of boats and place them on the appropriate pictures. Then have them connect the cubes to find out how many boats there are altogether.” The directions for the Independent Practice state, “Children use cubes to show each group of boats. They connect the cubes to find out how many there are altogether. Children record each number sentence.” Students do not independently demonstrate conceptual understanding of adding groups as the boat pictures are given within the Independent Practice and is the same exercise as in the Guided Practice.
• In Topic 8 Lesson 6, the Overview of the PBIL states, “Children will act out a subtraction story with counters. Then they will complete a subtraction sentence to show the story.” In the teacher-directed PBIL activity, students use counters to represent the number of puppets in all from a given problem. The Guided Practice section of the lesson states, “Have children cross out the puppets in each exercise and read each subtraction sentence with you.” On the student page, some puppets have a dotted X printed on top of them so the students can see how many are to be subtracted. The directions for the Independent Practice state, “Children write how many puppets there are in all. They cross out three puppets in exercise 3 and one puppet in exercise 4 and write the subtraction sentences. Ask children to read the completed subtraction sentences aloud.” Students do not independently demonstrate conceptual understanding of subtraction as the puppets already have a dotted X printed on top of them that the students can see. This exercise within the Independent Practice is the same exercise as in the Guided Practice.

The instructional materials for enVisionMATH California Common Core Kindergarten 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 10, students count on two more from a given number. Independent Practice problem 5 states, “Jacob sees birds in a tree. Then he sees 2 more. How many birds are there now?” All four of the practice problems have this same format.
• In Topic 11 Lesson 5, students decompose a teen number into a ten and some ones. Independent Practice problem 3 states, “Write the numbers that complete the number sentence.” There are two ten frames shown with counters already in them.
• In Topic 9 Lesson 6, students write two different number sentences that describe a given picture. Independent Practice problem 3 presents two yellow counters and seven red counters to students.

The instructional materials reviewed for enVisionMATH California Common Core Kindergarten 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 1 lessons, the MP is abbreviated.
  • MP1 - Make Sense of Problems (not 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)
• Lesson 16-3, page 307A, identifies MPs 1, 4, 5, 6 and 7. The Problem-Based Interactive Learning on page 307 is labeled with “MP7 Use Structure: Remind children to use attributes such as curved or flat surfaces to determine which three-dimensional (solid) figures can roll, stack and slide.” In this example, students are not engaged in MP7 because the teacher rolls the sphere and the cone before students have had a chance to search for themselves.
• Each Topic has a list of MPs checked off if done within the Topic. Each of the lessons within the Topic has a list of MPs checked off if done within the lesson. If an MP is checked, there is not necessarily explicit instruction for teachers as how to use the MP effectively when teaching the lesson. In lesson 3-1, MP3 is one of six MPs checked, but MP4 has explicit directions for teachers within the lesson.