2nd Grade - Gateway 2

The instructional materials reviewed for Grade 2 enVision Math 2.0 do not 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.

Most all of the lessons in the materials have students filling out student pages in a very procedural manner. For example, in Lesson 3-4 has students using place value to solve addition problems; however, instead of truly using place value properties, students are given a procedure to use. In Lesson 5-2 students are asked to use an open number line to solve a subtraction problem; again students are quickly led to a procedure instead of exploring the conceptual understanding needed.

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 pages in the student book.

Clusters 2.NBT.A and 2.NBT.B focus on understanding place value and using place value understanding and properties of operations to add and subtract.

• Lessons 8-1, 8-2, 8-3, 8-6, 8-7 and 8-8 focus on MD.C and work with time and money, but an additional alignment to 2.NBT.2 is included. Skip-counting by 5s, 10s, and 100s based on place value is not addressed until Topic 9, so these six lessons are not truly developing conceptual understanding of skip-counting. These lessons focus on a procedure for figuring out the value of a group of coins or bills or telling time. For example, Lesson 8-1 is the first lesson focused on coins. The lesson begins with a word problem about cents; although the teacher directions on page 443 of the teacher’s edition state that “(y)ou may want to give students coins…to use during this activity” and “(e)ncourage students to use coins or drawings of coins,” students are not required to use manipulatives in this first problem. The example of correct student work for this problem features addition equations, not skip-counting. On page 444, students are introduced to coins and counting-on to find total values. The coins are drawn, and students are told the value of each coin. As students begin solving problems, they are using pictures of coins and the defined value of the coins to solve problems; no manipulatives are required in the lesson, including actual coins. Lesson 8-3 is the first lesson focused on dollar bills and students are provided limited opportunities to skip-count, again focused on procedure.
• Lesson 9-1 is designed to build student understanding of hundreds, but often the lesson focuses more on procedures than conceptual understanding. The first problem on page 511 of the teacher’s edition tells teachers that “(y)ou may want to give students place-value blocks to use during this activity.” Many students in Grade 2 still need the support of concrete models in building conceptual understanding. On page 512 of the teacher’s edition, the Essential Question is “How can you find the value of a group of hundreds?” The sample answer is “I can count by hundreds to find the total.” This focus on counting impedes conceptual understanding of hundreds and place value.
• Lesson 9-2 begins with students using place-value blocks to model 125, 259, and 395. Then the lesson transitions into pictures of place value blocks. The directions for the guided practice and independent practice problems say to “(u)se models and your workmat if needed.”
• In Lesson 9-3, students are identifying the value of the digits in 3-digit numbers. Although a place-value chart is provided for the first problem, the instructions on page 523 of the teacher’s edition state to “(p)rovide place-value blocks…for students to use, as needed.” Place-value blocks are drawn for two problems on page 524, but the emphasis is on using the place-value chart. The Essential Question asks “How does the position of a digit help you name its value?” The sample answer is “Each place has a value. In a 3-digit number, the first digit tells the number of hundreds, the second digit tells the number of tens, and the third digit tells the number of ones.” The lesson focuses more on a procedure of using the chart more than conceptual understanding of amounts of hundreds, tens, and ones.
• In Lesson 9-4, students are reading and writing 3-digit numbers. The directions on page 529 of the teacher’s edition say to “(m)ake available place-value blocks…for students to use throughout the lesson, as needed.” The pages in the student book focus more on correct procedures for writing numbers than conceptual understanding of place value. For example, the higher order thinking problem on the bottom of page 531 has students write a number in standard form and expanded form based on the following: “It has 5 hundreds. The tens digit is 1 less than the hundreds digit. The one digit is 2 more than the hundreds digit.” This problem does not emphasize the amount of hundreds, tens, or ones or conceptual understanding of place value.
• Lesson 9-5 cites 2.NBT.3 and 2.NBT.1a; however, in this lesson students are simply writing 3-digit numbers in different ways. For example, on page 536 the sample problem at the top of the page shows 123 written as 100+20+3, 120+3, and 100+10+13. This lesson is not focused on developing conceptual understanding of place value.
• In Lesson 9-8, students are comparing two 3-digit numbers. Although the lesson starts with students comparing numbers using place-value blocks, the lesson quickly transitions to writing the numbers in a place value chart vertically in order to compare digits. The lesson emphasizes the procedure of using the place value chart more than comparing the two 3-digit numbers based on the meanings of the hundreds, tens, and ones digits.
• In Lesson 9-9, students are completing a comparison of two three-digit numbers using a number line. The focus of the lesson is on understanding that numbers that are less than another number are found to the left on the number line and numbers that are more than another number are found to the right on the number line, not comparing the two 3-digit numbers based on the meanings of the hundreds, tens, and ones digits. Students complete the page from the student book by providing a sample number that is greater than or less than a given number with no explanation or model to show their thinking.
• Although 2.NBT.6 is cited in three lessons, Lessons 3-7, 4-5, and 4-6, this standard is only the focus of one of the lessons, Lesson 4-5. Lesson 3-7 focuses on strategies and procedures to add two 2-digit numbers, and Lesson 4-6 focuses on procedures and strategies to add numbers. In Lesson 4-5, although students are adding up to four two-digit numbers, the lesson focuses mostly on procedures. On page 218 in the teacher’s edition, the Essential Questions is “How can you add more than two 2-digit numbers?” The sample answer is “I can use the same method as adding two 2-digit numbers. I can write the numbers in two columns to align the tens and the ones. Add the ones, then add the tens. Regroup 10 ones as one ten, if needed.” The directions at the top of page 217 state that the opening activity “prepares them for the next part of the lesson, where they add up to four 2-digit numbers using the standard algorithm.” This is the only lesson focused on 2.NBT.6, and all of the pages from the student book are expected to be completed using the standard algorithm.
• Standard 2.NBT.7 is the focus of Topics 10 and 11. In Topic 10 students are adding within 1,000, and in Topic 11 students are subtracting within 1,000. The lessons begin with using open number lines, then mental math, then partial sums, and then models. Models are used “to reinforce conceptual understanding of the standard addition algorithm” and the standard subtraction algorithm (pages 609A and 661A), not to build conceptual understanding of using place value to add and subtract within 1,000.
• Although standard 2.NBT.8 is cited in four lessons--Lesson 9-6, Lesson 9-10, Lesson 10-1, and Lesson 11-1--this standard is only the focus of two lessons, Lesson 10-1 and 11-1. Lessons 9-6 and 9-10 focus more on patterns than using place value understanding to mentally add or subtract 10 or 100 to a given number. Lesson 10-1 focuses on adding 10 and 100, and Lesson 10-2 focuses on subtracting 10 and 100. The first problem states that the teacher should “if possible, distribute place-value blocks” (pages 585 and 637), so students may not use concrete models to develop their conceptual understanding of adding and subtracting 10 or 100. Place value drawings are provided for a few problems in both lessons, but the emphasis of the lesson is on developing procedures for finding the answers. For example, the Essential Question on page 638 of the teacher’s edition is as follows: “How can you use mental math to subtract 10 (or 100) from a 3-digit number?” The sample answer is as follows: “To subtract 10, I take away 1 from the tens digit of the 3-digit number. To subtract 100, I take away 1 from the hundreds digit of the 3-digit number. To subtract 10 from a number that has 0 in the tens place, I regroup 1 hundred as 10 tens before I subtract.”
• Although 2.NBT.9 is cited in many lessons, the only lesson that lists this standard as the main standard is Lesson 10-6, “Explain Addition Strategies;” however, although it is not listed as the main standard, Lesson 11-6 is titled “Explain Subtraction Strategies.” In these two lessons, students are provided opportunities to choose and explain any strategy to add and subtract. Often the other lessons aligned to 2.NBT.9 include a small number of problems that require students to explain an answer based on the procedure taught in the lesson.

Cluster 2.MD.A focuses on measuring and estimating lengths in standard units.

• Lesson 12-1 is the first lesson addressing 2.MD.A, and the standard cited is 2.MD.3. In this lesson students are using the length of objects to estimate the length of other objects before they have measured anything. On page 14 of the K-5 Progression on Measurement and Data (measurement part), the last paragraph states that “(a)fter experience with measuring, second graders learn to estimate lengths.” The progression document continues to say that “(s)killed estimators move fluently back and forth between written or verbal length measurements and representations of their corresponding magnitudes on a mental ruler.” This conceptual understanding is not possible because students have no experience with a ruler at this point.
• In Lesson 12-5, students begin using rulers to measure objects in centimeters. An initial activity uses centimeter cubes to measure an object. However, instruction on how to use rulers and the concepts behind measuring length are not included in the lesson. Students continue estimating and measuring objects in Lessons 12-6 and 12-7 without instruction on how to use and understand the measuring tools.

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 1-3 students in the intervention activity are actually connecting cubes to combine them instead of just counting cubes such as on the page from the student book from in the lesson.
• In the Lesson 1-6 intervention kids are using post-its to show the relationship between adding and subtraction.
• In the interventions for Lessons 2-3 and 2-4 interventions students are using counters to make arrays to find totals.

The materials do not give enough opportunities for students to develop fluency and procedural skill throughout the text and especially where it is specifically called for in the standards.

Standard 2.OA.2 is fluently adding and subtracting within 20 using mental strategies. By the end of Grade 2, students should know from memory all sums of two one-digit numbers.

• The teacher's edition program overview indicates 14 lessons address this standard, and all of these lessons are in Topics 1 and 2 which would fall within the first month of school. There are fluency practice activities at the end of each of topics 1-5, 7-8, and 11. If these are used, students would have an additional eight days of fluency instruction which is likely not enough to ensure all students achieve fluency adding and subtracting to 20.
• In Lesson 1-1 students are counting on and changing the order of addends; the sums are all within 14, not 20.
• In Lesson 1-3, only six out of the 18 problems have sums larger than 15, and three of the six problems are in the Math Practices and Problem Solving section at the end of the lesson.
• In Lesson 1-5, students are counting to subtract. All of the subtraction is within 16.
• In Lesson 1-6, students only subtract from numbers larger than 15 in 6 of the 20 problems.
• In Lesson 1-7, students only subtract from numbers larger than 15 in 5 of the 19 problems.
• Fluency practice activities aligned to 2.OA.2 are found at the end of Topics 1-5, 7-8, and 11. These activities are all either "Point & Tally," “Follow the Path,” or "Find a Match" activities. These eight 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, in some of these activities more problems are devoted to addition and subtraction within 15. For example, on page 111 only has three problems have numbers over 15, and only two sums are over 15. On page 177 every difference is less than 10, and only three problems include numbers greater than 15. On page 309, only four problems have numbers over 15, and all sums and differences are within 15. On page 679 only 7 problems out of 50 have sums over 15, and all differences are within 10.
• Six Fluency Practice/Assessment pages from the student book aligned to 2.OA.2 are included in the instructional materials. These pages from the student book can be seen on page TP-71 of the teacher's edition. These pages from the student book each have 19 problems. There are a total of 114 problems on these pages from the student book. Only 13 problems have sums or differences from 15 to 20; most of the problems focus on addition and subtraction within 20.

Standard 2.NBT.5 is adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• Page 83 of the teacher's edition program overview indicates 29 lessons address this standard, and all of these lessons are in Topics 3 through 6 which would fall within the half of the school year. Without continued, ongoing opportunities to practice and continue strategy instruction, it is not evident that all students will achieve fluency to 100.
• Topic 4, which specifically references fluency in the topic title, uses partial sums and algorithms to teach procedural skill but has no specific activities designed to increase student fluency other than these procedures. Topic 6, which also specifically references fluency in the title, uses algorithms for subtraction to teach procedural skill but does not specifically address fluency either.
• Fluency practice activities aligned to 2.NBT.5 are found at the end of Topics 9, 10, and 12-15. These activities are all either "Point & Tally," “Follow the Path,” or "Find a Match" activities. These six 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. Some of these activities provide a limited number of problems. For example, pages 747 and 839 each only have 8 two-digit addition and subtraction problems.

There are online mathematics games provided to help build fluency. Although these games are listed in the Math Background pages, they are never actually mentioned in the lesson to suggest to teachers when they may be beneficial. The "Flying Cow Incident" game is good for building fluency and the concept of using a number line to add and subtract. The "Launch the Sheep" game is focusing on multiplying and dividing and function tables which are not second grade skills. The "Robo Launch" game is confusing and above grade level as it using function tables. "Gobbling Globs "uses numbers larger than 1,000. It is a good game for students in Grade 3, but not Grade 2. The "Fluency Games" are good games for building fluency at a Grade 2 level. The "Add It" game focuses on using a procedure and not on the conceptual understanding that second graders need. The "Space Jump" game would be better used at a higher level; it would be difficult for most second graders to maneuver and understand.

There is minimal time spent on counting within 1,000 in the instructional materials. Although standard 2.NBT.2 is cited for Lessons 8-1, 8-2, 8-3, 8-6, 8-7, 8-8, and 9-7 these lessons focus on the second part of the standard, skip-count by 5s, 10s, and 100s.,

The instructional materials do not meet the expectations for balance. The majority of the lessons are very procedural. However, the fluency of facts is underdeveloped, and additional problems focusing on addition and subtraction from 15-20 are needed. There is a lack of conceptual understanding in the materials; most of the materials labeled as conceptual understanding miss opportunities to develop understanding and instead teach a procedure. Many lessons only focus on one aspect of rigor at a time. 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 3, of the 10 lessons, five target conceptual understanding and procedural skills, three target conceptual understanding, and two target application. In Topic 7, of the seven lessons, two target procedural skill and conceptual understanding, four target conceptual understanding, and one targets application. There are many missed opportunities to connect the different aspects of rigor.

The instructional materials reviewed for Grade 2 do not meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the materials do not treat each MP in a complete, accurate, and meaningful way.

The lessons give teachers very little guidance on how to implement the MPs, and many of the MPs are misidentified in the materials. Also, the materials often do not attend to the full meaning of some of the MPs.

• MP1: Lesson 9-2 cites MP1; however, there isn't a rich problem attached. Students don't have an opportunity to persevere or make sense of a problem; students are putting numbers into a chart. Lesson 12-7 cites MP 1; but, there is not a rich problem attached. Students are asked to decide how to measure objects. Although Lesson 14-5 cites MP1, students are filling out sentence frames about a graph, which is not having students persevere or make sense of a problem.
• MP4: Lesson 9-3 cites MP4 for items 2 through 6; however, students are not asked to provide a mathematical model as they are identifying the value of an underlined digit. Lesson 10-2 cites MP4; however, students are told to use an open number line as the model. Lesson 13-1 cites MP 4; however, students are told how to model the problem.
• MP5: Lesson 9-1 cites MP 5; however, students are told to use place value blocks instead of being allowed to choose their own tools. In Lesson 10-4, students are told to use place-value blocks, and MP5 is cited. Although Lesson 12-3 cites MP5, students are told to use rulers.
• MP7: Lesson 9-4 cites MP 7; however, students are not using structure to solve the problem. Lesson 12-5 cites MP7; however, students are not using structure to solve the simple addition problem. Lesson 15-2 cites MP7; however, telling students what to look for is not having students look for structure.
• MP8: Lesson 2-4 cites MP8; however, it tells the students to draw the array first. Lesson 10-1 cites MP8; however, with only one problem present, it is impossible for students to generalize. Lesson 11-7 cites MP8; however, with only one problem present, it is impossible for students to generalize.

The materials do not meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Usually questions have one correct answer, and there is not a lot of teacher guidance on how to lead discussions beyond the provided questions. There are many missed opportunities to guide students in analyzing the arguments of others.

• In Lesson 1-3, the leveled assignment on page 22 includes an item that is tagged with MP3. "Blanca wants to add 5+8. Describe how she can make a 10 to solve." There is no supporting commentary or questioning to assist teachers in helping students form or develop an explanation. This is the first place MP3 is tagged in the second grade curriculum, and support is not provided.
• In Lesson 1-7 on page 42, the teacher commentary states: "MP3 Construct Arguments. Students explain and justify whether they prefer to use addition or subtraction to make a 10." This is followed by a discussion of the mathematics that's being built upon, but no information for teachers on how to support students to construct an argument is provided. This is the second time MP3 is tagged at this grade level, again with no support or guidelines. Later in the same lesson, on page 43, teachers are provided with some support. Item 15 states, "Remind students to provide reasons why Carol was correct or incorrect. Suggest that they make a 10 to find 15-6 on their own to check Carol's work and find possible reasons she may be correct."
• In Lesson 1-8 on page 50, the teacher commentary states "MP3 Construct Arguments. Make sure students understand that the gray box represents the missing addend. Encourage students to use words and sentences to explain their thinking." While teachers are directed to tell students what to do, the materials don't provide guidance on how to do this. Teachers could be provided with examples of good explanations or components that they can support students to develop.
• In Lesson 5-2, students are asked "Why should you place the greater number in the problem on the far right side of the line?" Although a sample answer is provided, teachers are not provided assistance in helping students construct their answer.
• In Lesson 6-4 of the Teacher's Edition, the teacher is prompted to have students explain why they need to regroup. Although a suggestion for follow-up is provided, the teacher is not given assistance to help students construct their explanations.
• In Lesson 8-7, students are asked "What is the greatest number of minutes that a digital clock can show to the right of the colon? How do you know?" Although a sample answer is provided, the teacher is not provided any assistance in helping students construct viable arguments or analyze the arguments of others.
• In Lesson 10-1, students share their explanations for how they used mental math to add 100 and 10 to a 3 digit number and explain why their strategy works. Teachers are not provided with sample explanations or guidance.