Kindergarten - Gateway 2

The instructional materials reviewed for Kindergarten enVision Math 2.0 partially meet the expectations for giving attention to conceptual understanding. The materials rarely develop conceptual understanding of key mathematical concepts where called for in specific content standards or cluster headings. The lessons do not devote a lot of time to hands-on learning which would lend itself to building the conceptual understanding of a student in Kindergarten.

Rarely are students asked to work with manipulatives when the materials would lend themselves to it. Most of the regular grade-level work consists of pages in the Student book.

Also, these lessons are sometimes clustered in a way that may be problematic if students don't grasp the concepts within the specific topic. For example, students work with place value (K.NBT) in Topic 10. There are seven lessons devoted to K.NBT.1. The remaining lessons after Topic 10 focus on other domains such as counting and cardinality, geometry, and measurement and data. Students are not provided the opportunity to apply or further develop understanding of number and base ten within the lessons.

Standard K.OA.1 focuses on representing addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.

• Topics 6, 7 and 8 specifically address K.OA.1.
• Lesson 6-1 is the first lesson that addresses K.OA.1. The whole class portion of this lesson begins with a page from the student book showing pictures of flowers and pictures of hands. Although the directions on page 288 state that students should “do all of the following to show each part to find how many in all: clap and knock, hold up fingers, and give an explanation of a mental image,” the students are doing these things in order to complete pages from the student book. Although crayons, erasers, and blocks are all pictured on the pages from the student book, students are not building their understanding through hands-on work with any of these objects mentioned in the lesson.
• On page 293, Lesson 6-2 begins with a hands-on activity that allows students to represent addition using connecting cubes. Although the rest of the lesson includes directions to use connecting cubes or counters to model the problems and write addition sentences, students are shown pictures and given a sentence with blanks to fill beneath the groups of pictures.
• Lesson 6-3 begins with a page from the student book. The page from the student book has a picture of two tomato plants with tomatoes on them, and students fill in blanks in a sentence. From the beginning the lesson focuses on completing pages from the student book and lacks hand-on activities to build conceptual understanding of addition. Although the directions in the lesson state to use connecting cubes or counters to model the problems and write addition sentences, students are shown pictures and given a sentence with blanks to fill beneath the groups of pictures.
• Lesson 6-4 focuses on the idea that the plus sign means “and.” Students complete pages from the student book recopying numbers and replacing “and” with a “+” sign.
• Lesson 7-1 is the first lesson focused on subtraction. The lesson begins with five problems with pictures, and then students begin solving problems on pages from the student book. Page 367 of the Teacher Edition does state that for items 3-6, teachers should “(g)ive each student 10 connecting cubes so that they can act out each problem using objects.” The lesson and the directions do not require hands-on activities in order to complete the pages from the student book until the independent practice portion of the lesson. The lesson is developed using pages from the student book.
• On page 371, Lesson 7-2 begins with a hands-on activity that allows students to represent subtraction using connecting cubes. In problems 1-8 students are shown pictures and given blanks to fill beneath the pictures. Problems 9 and 10 suggest that student “draw counters to show a group,” not actually use physical counters.
• Lesson 7-3 begins with a page from the student book. The page from the student book has a picture of four ladybugs, and students fill in blanks in a sentence. From the beginning, the lesson focuses on completing pages from the student book and lacks hand-on activities to build conceptual understanding of addition and subtraction. Students are shown pictures and given a sentence with blanks to fill beneath the groups of pictures throughout the lesson.
• Lesson 7-4 focuses on the idea that the minus sign means “take away.” Students complete pages from the student book recopying numbers and replacing “take away” with a “-” sign.
• Lesson 7-5 focus on subtraction with equations. Although the directions state that students should use counters to solve the problems, the page from the student book includes drawings of objects. For all but one of the problems, the equations are written by recopying the numbers and filling in blanks beneath the words “take away” and “is” with the “-“ and “=” signs. Students are following this procedure as they complete nine problems.
• In Lesson 8-3, students draw their own pictures to represent addition and subtraction equations and then write the corresponding equations. This lesson allows students to show their understanding of addition and subtraction.

Cluster K.NBT.A focuses on working with numbers 11-19 to gain foundations for place value.

• Topic 10 specifically addresses K.NBT.A.
• In Lesson 10-1 students compose the numbers 11, 12 and 13. Students begin with a hands-on activity using 13 counters using a ten-frame. However, after that activity students are completing pages from the student book. Some questions provide students with pictures of 10-frames filled in with additional ones drawn underneath, and students fill-in the equations in the blanks beneath. Some questions provide students equations and ask them to draw the pictures to represent the equations. A couple of problems give students equations with a missing addend and ask students to fill in the missing number and draw the pictures to represent the equation. For two problems students are given a number and have to draw counters and write an equation to show how to make the number. The last problem of the independent practice asks students to decompose the number 13; this is the only time that a number is decomposed in the lesson.
• In Lesson 10-2 students compose the numbers 14, 15, and 16, and in Lesson 12-3 students compose the numbers 17, 18 and 19. Students begin these lessons with a hands-on activity using counters and a 10-frame. However, after that activity students are completing pages from the student book by either drawing counters or filling in blanks to write equations. The last independent practice problem in Lesson 10-2 asks students to decompose the number 16; this is the only time that a number is decomposed in the lesson. The last independent practice problem in Lesson 10-3 asks students to decompose the number 19; this is the only time that a number is decomposed in the lesson.
• In Lessons 10-4 thru 10-6, students are decomposing the numbers 11-19. These lessons begin with a hands-on activity using 10-frames and counters. However, after that activity, students are completing pages from the student book by either drawing counters or filling in blanks to write equations, although occasionally students are prompted to use counters before drawing the counters.

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.

• Lessons 6-3, page 303A, and 6-10, page 345A, have interventions developing conceptual understanding for K.OA.A.
• Lesson 8-5 on page 463A, Lesson 8-7 on page 475A and Lesson 8-9 on page 487A have interventions developing conceptual understanding for K.OA.A.
• Lesson 10-4 on page 589A and Lesson 10-6 on page 601A have interventions developing conceptual understanding for K.NBT.

The materials reviewed in Kindergarten for this indicator partially meet the expectations for being designed so that teachers and students spend sufficient time working on engaging the applications of the mathematics. In general, some lessons designed to emphasize application do not always provide opportunities for students to apply mathematical knowledge and skills in a real-world or non-routine context.

Most topics have at least one lesson designated to application. However, the emphasis of these lessons is on the standards addressed in the rest of the topic and not necessarily application. Some of these lessons do not provide opportunities for students to apply mathematical knowledge and skills in a real-world or non-routine contexts. For example, Lesson 1-11 is designated as an application lesson. In this lesson, students are shown five circles on the page, and the teacher says “Alex needs to count the group of shapes. How can you count these shapes? Use objects or words to help. Write the number to tell how many shapes. Tell why your number is correct.” Although the problem includes a person’s name, students are simply counting circles on the page. Another example is lesson 11-7. The objective of this lesson is to count on from any number counting by tens and by ones. The lesson emphasizes patterns using a hundreds chart and does not include real-world or non-routine contexts.

Standard K.OA.2 is focused on solving addition and subtraction word problems and adding and subtracting within 10.

• Only seven lessons specifically target K.OA.2.
• Lessons 6-7, 6-8 and 6-10 include addition stories. 6-7 includes ten add-to result-unknown problems and one add-to change-unknown problem within the problem based learning, guided practice, and independent practice sections. Lesson 6-8 includes nine put-together problems and one add-to result-unknown problem within the problem-based learning, guided practice, and independent practice sections. Lesson 6-10 includes ten word problems with a variety of add-to and put-together problems. In Lesson 6-10, most of the word problems are accompanied by pictures of the objects. Once students learn the procedure needed to solve the problem, the context of the word problem is irrelevant.
• Lessons 7-3, 7-7, and 7-9 include subtraction stories. 7-3 includes thirteen take-from result-unknown problems within the problem based learning, guided practice, and independent practice sections. The problems in Lesson 7-3 are accompanied by pictures of the objects. Once students learn the procedure needed to solve the problem, the context of the word problem is irrelevant. Lesson 7-7 includes ten take-from result-unknown problems within the problem based learning, guided practice, and independent practice sections. Lesson 7-9 includes four take-from and three add-to problems within the problem based learning, guided practice, and independent practice sections. Lesson 7-9 is the only lesson in which students have to listen to the story problem and consider which operation to use. Other lessons focus solely on addition or solely on subtraction problems.
• Lesson 8-8 includes both-addends-unknown story problems. Most of the word problems are accompanied by pictures of the objects. Once students learn the procedure needed to solve the problem, the context of the word problem is irrelevant. The last problem does not include pictures, but it is not a both addends unknown problem.

Real world situations are often found in the solve-and-share and visual learning components of lessons, but the contexts are not always relevant or familiar to Kindergarten age students. For example, the solve-and-share for Lesson 4-1 is about a chicken farm. Another example is the use of shells in the Lesson 3-2 visual learning component. Word problem contexts may also not be familiar to some kindergarten students including flamingos and food bars and baby alligators in the marsh in Lesson 7-9.

The 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 most lessons focus on one aspect of rigor at a time. Procedural skill is treated separately from fluency with a small number of activities dedicated to fluency, and the lack of lessons on fluency does not allow for a balance of the three aspects.

In Topic 11, of the 7 lessons, four target conceptual understanding, one targets procedural skills, one targets application, and one targets both conceptual understanding and procedural skills. Often when more than one aspect of rigor is the focus of a lesson, the aspects are conceptual understanding and procedural skills. For example, in Topic 1, of the 11 lessons, eight target conceptual understanding and procedural skills, two target conceptual understanding, and one targets application. 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 40 lessons.
• MP 2: approximately 70 lessons.
• MP 3: approximately 50 lessons.
• MP 4: approximately 60 lessons.
• MP 5: approximately 50 lessons.
• MP 6: approximately 60 lessons.
• MP 7: approximately 40 lessons.
• MP 8: approximately 30 lessons.

Since each MP is identified in so many lessons, each lesson has 3-5 practices identified in it. 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 the MP not enriching a student's opportunity to learn the content of the lesson. For example, the "Do You Understand? Show Me!" item on page 202 in lesson 4-1 is labeled "MP4 Model with Math: Show students a row of 8 counters and a row of 7 counters. Is a group of 8 counters greater or less in number than a group of 7 counters? How can you tell?" In this example, students are not modeling with math because they are simply answering questions about something the teacher is showing them.

In some instances, more guidance to teachers could enrich the content, and in other instances, the connection is limited or the MP may be misidentified.

• In Topic 4 on page 204, item 8 is labeled "MP8: Generalize The first step is for students to draw a group that is the same in number. Once the two groups match, then students can add more to their drawing to make a group that is greater in number. How many are in the group? How can you draw a group that has more?" This item does not have students express regularity in repeated reasoning as there is only one step in the problem.
• In Topic 6 on page 330, part of the "Guided Practice" is labeled "MP5: Use Appropriate Tools Strategically What could you use to solve the problem? How did you use the cubes? How does the picture of the cubes help you find the answers?" In this example, students aren't using any tools; they are responding to questions about a picture that shows cubes being used.

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 10 indicators to assess MP1, "Listen and look for the following behaviors to monitor students' ongoing development of proficiency with MP1" A proficiency rubric is also included.

The materials partially meet the expectation for prompting students to construct viable arguments and analyze the evidence 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.

There are some questions that do ask students to explain their thinking in the materials. MP3 is identified 52 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, in lesson 11-4, MP3 is cited; however, in the student materials students are not asked to construct an argument or analyze the arguments of others. Additional examples of this can be found in the following lessons: 1-3, 2-4, 3-2, 4-3, 6-1, 7-3, 9-2, 10-7, 12-8, 13-4 and 14-2.

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

• Topic 1, page 8. Show students a group of three objects, such as buttons. Say: Tell how you know how many objects are in this group.
• Topic 1, page 61. Marta is thinking of two numbers - one is the number that comes just before 4 when counting, and the other is the number that comes just after 4 when counting. Write the two numbers Marta is thinking of. Show how you know you are correct.
• Topic 1, page 67. Alex needs to count the group of shapes. How can you count these shapes? Use objects and words to help. Write the number to tell how many shapes. Tell why your number is correct.
• Topic 2, page 99. Have them draw a circle around the group that is greater in number than the other group, and then explain why they are correct.
• Topic 4, page 202. Let's count the chicks together. Point to each chick as you count aloud with students. Is a group of 7 chicks greater in number or less in number than a group of 10 chicks? How do you know?
• Topic 6, page 329. Give pairs of students three connecting cubes of one color and two of another color. Have them place the cubes on the workman in groups arranged by color. Say: Daniel's teacher is making name tags for her students. She makes 3 name tags for boys. She makes 2 name tags for girls. Now she has 5 name tags. How does Daniel's teacher know that she has 5 name tags? Explain and then show how you know.

Examples of opportunities to analyze the arguments of others:

• Page 133, item 2. David says that his group of toy cars is greater than his group of alphabet blocks. Do you agree with him? Have students draw a circle around yes or no, and then have them draw a picture to explain their answer.
• Page 271, item 1. Jared says that the category of green crayons is greater than the category that is NOT green. Does his answer make sense?

Most of the time when students are asked to critique the reasoning of others it is on paper, not with a partner.

The instructional materials reviewed for Kindergarten partially meet the expectations for explicitly attending to the specialized language of mathematics.

• Each lesson includes a list of important vocabulary in the topic organizer which can be found at the beginning of each topic. These vocabulary words are also noted in the "Lesson Overview" at the beginning of each lesson. While the identified vocabulary words appear within the blue script that teachers may use, the words are not highlighted or identified in any way.
• Each unit includes two-sided vocabulary cards in the student edition with a word on one side and definition and/or representation on the other. The teacher's edition includes vocabulary activities at the start of each topic.
• Each topic opener has a vocabulary review activity, and each topic ends with a vocabulary review activity.
• There is an online game for vocabulary, Save the Word.
• There are instances in the materials that the definition may be vague or unclear for kindergarten age students. For example, on page 312, "Point out to students that the number of circles on either side of the is and the + is the same. Say: This word and symbol are the balance points in the equation."
• Correct vocabulary is sometimes not used. For example, "addition sentence" and "subtraction sentence" are used instead of "equation" and "same number as" is used instead of "equal."