2nd Grade - Gateway 2

Materials partially meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. There are good conceptual discussion pieces located throughout the work. Some good conceptual Home-Link and practice problems exist; however, these come before the focus lessons, which, without the lesson to understand the concept, could present possible issues for the students.

Below are lessons where the full depth of conceptual understanding is addressed.

• Unit 1: 1-2 Number Line Squeeze Game; 1-5 Open Response: Number-Grid Puzzles.
• Unit 2: 2-4 The Making-10 Strategy; 2-7 Open Response: Subtraction and the Turn-Around Rule; 2-8 Exploring Addition Tools, Odd and Even and Patterns and Shapes; 2-10 Name-Collection Boxes; 2-11 Playing Name that Number.
• Unit 3: 3-1 Open Response: Using Addition Strategies; 3-2 Subtraction from Addition: Think Addition; 3-6 -0 and -1 Fact Strategies and Subtraction Top-It; Unit 3 Open Response Assessment (which allows students to represent mathematical thinking with representations and verbalization).
• Unit 4: 4-4 Numeration and Place Value; 4-5 Using Place Value to Compare Numbers; 4-6 Open Response using Base-10 Blocks to Show a Number; 4-7 Playing Target; 4-11 Explorations Matching Facts with Strategies and Exploring Arrays.
• Unit 5: 5-5 Explorations Exploring Arrays: Time, and Shapes; 5-6 Mentally Adding and Subtracting 10 and 100; 5-11 Open Response Adding Multi-digit Numbers.
• Unit 6: 6-7 and 6-8 Partial sums addition; 6-9 Open Response Subtracting with Base-10 Blocks; 6-10 Explorations Exploring Arrays.
• Unit 7: 7-1 Playing Hit the Target; 7-2 Open Response Four or More Addends; 7-3 Playing Basketball Addition; 7-4 Measuring with Yards; 7-5 Measuring with Meters.
• Unit 9: 9-5 Reviewing Place Value; 9-6 and 9-7 Expand and Trade Subtraction.

In addition, the following routine also builds conceptual understanding.

• Routine 1: Students represent the number of the day on a class number line, count the days using straws or craft sticks bundled in tens and hundreds, record the number in expanded form, and represent and count the number using coins. There are also additional add-ons that build upon place value understanding including adding and subtracting 10s or 100s. The ongoing assessment of this routine (TE page 9) also includes questions addressing 2.NBT.A and 2.NBT.B including, "Can children use the number line to represent and read the number of the day? Can children identify the value of the ones, tens, and hundreds digits? Do children understand the structure of expanded form and use it to write the number of the day?"

Lessons which partially meets the requirements for conceptual understanding are listed below.

• 1-12 Exploring Base-10 Blocks, Area, and Dominoes: The base-10 building activity with the recommended sentence frame does not encourage open-ended conceptual conversation between students to explain the whys or hows of their mathematical representations. Unit 1 Assessment provides students some opportunity to verbalize their understanding but only in the challenge, which is optional. The Challenge provides opportunity for students to create their own visual representations with showing how much money is spent.
• 2-1 Grouping by 10's- Playing the Exchange game: The game involves students in hands-on activities but misses opportunities for students to have conversations about the exchanges being made. The Unit 2 Assessment provides one opportunity for students to verbalize their mathematical thinking and no opportunities for students to create concrete or visual mathematical representations.
• The Unit 3 Assessment allows the student to verbalize their mathematical thinking but does not provide opportunity for concrete or visual representation.
• 4-8 How Big is a Foot, 4-9 The Inch, 4-10 The Centimeter and 4-11 Exploration for measuring a path: These lessons do not get to the full depth of the standard which calls for students to measure and estimate lengths in standard units. Unit 4 assessment gives students opportunities to verbalize their mathematical thinking but no opportunities for students to demonstrate their understanding through concrete or visual representations.
• The Unit 5 Assessment provides students some opportunity to verbalize their understanding, but only the Open Response Assessment provides opportunity for students to create their own visual representations with showing how much money spent.
• 6-10: Exploring lengths does not meet the full depth of the standard to measure and estimate lengths. Unit 6 assessment gives students opportunities to verbalize their mathematical thinking but no opportunities for students to demonstrate their understanding through concrete or visual representations.
• The Unit 7 Assessment provides students some opportunity to verbalize their understanding, but only the Open Response Assessment provides opportunity for students to alter a visual representations of base-10 blocks.
• In Unit 9, only the Open Response Assessment provided opportunity for students to verbalize their understanding of place value concepts.

Lessons which miss opportunities to develop conceptual understanding are listed below.

• 1-11 Comparing Numbers: The lesson does not employ visual representations of numbers to illustrate the difference in amounts when comparing numbers. Students should be able to see the difference using base-10 blocks or other groups of objects.
• Lesson 1-3 does not work on building conceptual knowledge. Although labeled with 2.NBT.A, students are not developing understanding of place value. In this lesson students are learning to use various math tools.
• 5-2 through 5-4, using and calculating with coins: These activities are more about measuring value with money and do not clearly connect to learning about the concepts of place value using coins as visual representations of numbers.

A lack of instructional time devoted to some topics and the late introduction of some topics lead to a lack of development of conceptual understanding for students. For example, students may not be given enough time to build a conceptual understanding of a yard and a meter based on the lessons provided. Students also only spend lesson 7-9 exploration (10 minutes) measuring the same object in two different units. In the lessons for length measurement, there is no time spent determining how much longer one object is than another. One thousand is not introduced conceptually with base ten blocks until lesson 9-5. Addition strategies based on models are not introduced until focus lessons 6-7 and 6-8. Finally, conceptual subtraction strategies for multi-digit numbers are not shown until lessons 9-6 and 9-7

The materials partially meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

Each unit contains a two-day "Open Response" lesson which engages students in application of mathematics. For example, lesson 9-3 has students engaging in application of the mathematics where students are asked to find a way to fairly share four muffins with five people. Online in the resource section, several projects are available to help students with application of mathematics.

Word-problem contexts are generally familiar to Grade 2 students including children playing, pencil cups, markers, stickers, and books. Add to and take from result unknown problems are the most frequently presented. There are limited opportunities for students to solve take-from, start-unknown, and change-unknown problems. Compare problems are frequently represented within the curriculum, sometimes connected to a graph (Student journal, page 149). However, there is not enough instruction for application for two-step addition (only one lesson directly focuses on this) and no instruction for application for two-step subtraction. Examples of application in number stories include:

• 2-2 Addition Number Stories
• 2-7 Subtraction and the Turn Around Rule (writing number stories and number models)
• 3-2 Subtraction from Addition
• 3-9 (Practice) Solving Subtraction Stories
• 5-8 Change to More Number Stories
• 5-9 Parts-and-Total Number Stories
• 6-2 Comparison Number Stories
• 6-3 Interpreting Number Stories
• 6-4 Animal Number Stories
• 6-9 Subtracting with Base-10 Blocks
• 8-8 Equal-Groups and Array Number Stories
• 8-9 More Equal Groups and Arrays
• 9-10 Connecting Doubles Facts, Even Numbers, and Equal Groups

Examples of 2-step application number stories include:

• 6-5 Two-Step Number Stories including home-link page.
• These lessons provide some instruction and focus on multi-step addition: 6-7 Finding partial sums involving multi-step addition strategies for adding 2-digit numbers, 7-2 Four or More Addends, 9-9 Estimating Costs in which students must choose at least three items from a market to mentally estimate which items they can purchase for $100.

The Grade 2 Everyday Mathematics 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. However, the lack of lessons on application do not allow for a balance of the three aspects.

• 36 lessons focus on or partially address conceptual understanding.
• "Daily Warm Ups," including "Mental Math and Fluency," and "The Number of the Day Routine" directly focus on math fact fluency.
• Seventeen lessons focus on Application with only one lesson directly focused on two-step addition and no lessons directly focused on two-step subtraction.

In terms of balance, the lessons do not provide as much instruction for application as they do for conceptual understanding. Conceptual understanding gets much more emphasis than the other two aspects of rigor.

One cluster where you can see the expectations for balance handled appropriately is 2.MD.A, measure and estimate lengths in standard units.

• Lesson 4-8, students learn about the importance of using consistent measurement lengths and measure classroom objects by iterating units without overlaps or gaps. In this lesson, students engage in all of the aspects of rigor.
• Lesson 5-8, students look for objects in the room that are about an inch, centimeter, 10 inches, and 10 centimeters and measure the objects. In this practice lesson, students engage in two aspects of rigor, procedural skills and application.
• Lesson 6-4, students understand length versus height and use lengths (in inches and feet) to compare length and/or height of animals. Students also write and solve comparison problems using the same context. In this lesson, students engage in all of the aspects of rigor.
• Lesson 6-10, students measure four different objects and compare their lengths, engaging in one of the aspects of rigor, procedural skill.
• Lesson 7-4, students estimate lengths, explore standard and non-standard units, compare measuring tools such as a yard stick and tape measure, select and measure familiar objects and measure distances. In this lesson, students engage in all of the aspects of rigor.

The instructional materials reviewed for Grade 2 partially meet the expectations for identifying the MPs and using them to enrich the mathematics content. Each of the standards is identified in the Grade 2 materials. The practices are not over-identified or under-identified. For example, Unit 1, page 55 discusses how MP2 and MP5 unfold within the unit and lesson. Within the lesson are spots where the MPs are identified. However, within the lessons, limited teacher guidance is given on how to help students with the MPs. Because there is limited guidance on implementation, it is difficult to determine how meaningful connections are made. MP3, MP4 and MP8 are the least identified in the Grade 2 materials.

The Assessment Handbook includes "Mathematical Practices for Unit(s) Individual Profile of Progress" that can be used to assess practice standards. Unit Assessments do not identify content or practice standards being assessed. The Beginning of Year assessments identifies MP1 and MP7 for assessment, and the Middle of Year assessment identifies MP1, MP2, MP3, MP4, MP6 and MP7 for assessment. The End of Year assessment identifies MP1, MP2, MP3, MP4 and MP6 for assessment. MP5 is not identified with any formal assessment.

The materials partially meet the expectation for prompting students to construct viable arguments and analyze the evidence of others. MP3 is not explicitly called out in the student material. 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 on assessments and in the materials. Sometimes there are questions asking them to look at other's work and tell whether the student is correct or incorrect and explain. Little direction is provided to make sure students are showing their critical thinking, process or procedure, or explaining their results. Many questions that prompt students to critique the reasoning of others tell the student if the reasoning was originally correct and incorrect. It should be noted, though, that student materials never explicitly call out entire MPs at once; MP3 is broken into GMP3.1 and GMP3.2 in the materials.

The open response lessons could be opportunities for students to construct arguments for or against a mathematical question. However, besides just working in groups, there is little prompting from the teacher for students to discuss the answers of other groups or students.

The following are examples of MP3 in the assessments:

• Unit Assessment 1: questions 4b and 7e, "Explain how you knew."
• Unit 1 Open Response Assessment: question 3, "Explain what you did to solve problem 2."
• Unit 2 Challenge: "Explain how she can use to double to solve."
• Unit 3 Assessment: question 2, "How do you know? Explain your thinking." question 4c, "Explain how you solved one of the facts above." question 6, "Explain Martin's thinking."
• Unit 3 Challenge: "Write an argument to explain why you agree or disagree with Zoe."
• Unit 3 Open Response: "Show and explain how to use Grace's strategy."

The following are examples of opportunities to construct viable arguments:

• Student Math Journal Volume 1 and 2: Most Math Boxes pages ask "what do you notice" and "explain" and other how and why questions.
• The Student Math Journals include at least 50 items that ask students to explain their solution, their strategy, how they know, etc.

The following are examples of opportunities to analyze the arguments of others found in Student Math Journal Volume 1:

• page 74, item 7: Student has to critique and explain student with wrong answer.
• page 79, item 8: Do you agree with Marta? Explain your answer.
• page 81, item 6: Why were your measures and your partner's measures the same for each object?
• page 91, item 5: For Problem 1, suppose your teacher asked your class to write the number in expanded form. Your friend Cassie wrote 700+2+5. Do you agree? Why or why not?

The materials partially meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. In the materials, usually only one right answer is available, and there is limited teacher guidance on how to lead the discussion given besides a question to ask. Many missed opportunities to guide students in analyzing the arguments of others exist. Students spend time explaining their thinking but not always justifying their reasoning and creating an argument.

The following are examples of lessons aligned to MP3 that have missed opportunities to assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others:

• In Routine number 4, the teacher is asked to have children support their arguments about weather trends based on data from the weather bar graph. No further direction.
• In Lesson 2-6 on page 183 TE, teachers are given the question to ask, but there is no follow-up on how to direct the discussion.
• In the summarize section of lesson 1-2, children discuss how sharing strategies with a partner can help with learning mathematics. There is no further direction for the teacher.
• Lesson 1-9 directs the teacher to call students up and for children to make predictions, but it does not guide the teacher in asking students to explain their thinking.
• Lesson 2-5 explains how to use nearby doubles and then apply the strategy. Not enough information is provided to the teacher.

The instructional materials reviewed for Grade 2 partially meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics; however, often the correct vocabulary is not used.

• Each unit includes a list of important vocabulary in the unit organizer which can be found at the beginning of each unit.
• Vocabulary terms are bolded in the teacher guide as they are introduced and defined but are not bolded or stressed again in discussions where students might use the term in discussions or writing.
• Each regular lesson includes an online tool, "Differentiating Lesson Activities." This tool includes a component, "Meeting Language Demands," that includes vocabulary, general and specialized, as well as strategies for supporting beginning, intermediate, and advanced ELLs. An example of this from Lesson 1-4 includes "For beginning ELLs use ... Visual aids, gestures, modeling, and guided practice to teach the meanings of words and phrases."
• Everyday Math comes with a reference book that uses words, graphics, and symbols to support students in developing language.
• Correct vocabulary is often not used. For example, turn-around fact is used rather than the term commutative property, number sentence is used instead of equation, name-collection box instead of equivalent equations or equivalent expressions.