2nd Grade - Gateway 2

The materials reviewed in Grade 2 for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.

The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open ended questions that require evidence to show their thinking and reasoning.

The following are examples of attention to conceptual understanding of 2.OA:

• Unit 1, Module 2, Session 2 investigates the number rack structure of units of 5 and 10 with sums to 20. Students build numbers through a chain of reasoning relating the representation of one sum (5+5=10 so 5+6=11) to another sum using the number rack.

The following are examples of attention to conceptual understanding of 2.NBT:

• In Unit 3, Module 4, Session 4 in Problems & Investigations, students complete a "Sticks and Bundles" activity where they discuss estimates and then count the number of sticks using double-digit adding.
• In Unit 5, Modules 1-3, students use hundreds charts, stick bundles, base 10 blocks, and Unifix cubes to represent place value and skip counting up to the hundreds place to solidify conceptual understanding of clusters 2.NBT.A and 2.NBT.B.
• In Unit 8, Module 1, students use number lines, place value squares, and expanded notation to represent place value and skip counting up to hundreds place to solidify conceptual understanding of clusters 2.NBT.A and 2.NBT.B.
• In the December Number Corners, in Activity 2, students are working with craft sticks in singles and bundles of 10 and 100, the number line, and a greater than/less than chart. Students are first asked to represent a number (214) on the number line with the craft sticks and explain their thinking/observations of the final amount built on the place value chart. Next, they explore jumping the number line , starting with jumps of 10 forward and backward then moving to jumps of 100 (214, 314, 414, 514...). Finally, students guess a secret number from the number line; with each guess they are told "greater than, or less than" and the numbers guessed are recorded on a greater than, less than chart. At the same time, students are moving clips on the number line to mark the numbers that have been guessed.

The following are examples of attention to conceptual understanding of 2.MD.A:

• In Unit 1, Module 4, Session 1, students are using the "Number Line to 20 Mat" to represent sums of various numbers within 20.
• In Unit 1, Module 4, Session 2, students are given the number line as one of three tools of choice (number line, bead string, number rack), to solve addition/subtraction word problems.
• In Unit 8, Modules 2-4, students create ramps and roll marbles from various heights and then measure distance that the marble rolled to represent real world measurement to solidify conceptual understanding of cluster 2.MD.A.

The Grade 2 materials meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.

Fluency is developed throughout the sessions of the Grade 2 instructional materials.

• In Unit 3, Module 2, Session 4, the number line is used to subtract within 30, fluently leading to mental strategies involving place value.
• In Unit 4, Module 2, Session 5, students are introduced to the Work Place "Climb the Beanstalk." Fluency is practiced as students roll two dice and add to get the sum. Then they spin the spinner and subtract the amount on the spinner from the sum and move to that space on the game board.
• As students play the Work Place game "Star Power" in Unit 3, Module 1, Session 3, they are building their fluency while seeing who can reach 100 first. Students get to choose the order in which they add, promoting the strategy of anchoring on 10's. The game also allows students to explore the associative and commutative properties of addition.
• In the October Computational Fluency Number Corner, students review and practice combinations to 10 using 10-frames and the activities, "Make Tens" and "Break Tens."
• In the March Computational Fluency Number Corner, students use their "Quick Facts" forms to practice their fluency each day. Students work on various strategies: Count On/Make Ten, Quick Fact Doubles/Doubles Plus or Minus One, Quick Facts Add Nine/Add Ten.
• The April Number Line Number Corner routine reinforces procedural skill leading to fluency by adding and subtracting units of 1, 5 and 10 on a number line to reach a specific number.

Materials meet the expectations for having engaging applications of mathematics as they are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of the grade.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. Examples of these applications include the following:

• Unit 1, Module 3, Session 3: Students decompose numbers in a variety of ways using the context of a dinner party. Equations and visual representations are used to record combinations for various scenarios related to the context of the dinner party and how to seat the guests.
• Unit 1, Module 4, Session 2: Students engage in application of using number lines and number racks to assist in solving story problems.
• Unit 3, Module 2, Session 3: Students compare and contrast three different solutions to a story problem involving length.
  
• Unit 7, Module 3, Session 2: Students apply place value and addition knowledge and skills to story problems involving money.
• The Number Corner March Number Line on days 5, 10, 15 and 20: When students play "Put It on the Line," they are solving addition and subtraction word problems on the open number line. Examples of problem types include add to and start unknown.
• The Number Corner January Calendar Grid: As students complete various graphs that represent class data, the teacher creates word problems for students to solve.

The materials reviewed in Grade 2 meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Grade 2 materials, all three aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the unit sessions and in Number Corner activities. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve understanding and conceptual understanding.

Although some sessions focus on individual aspects of rigor, some session do combine the three aspects of rigor. For example, in Unit 3, Module 1, Session 2, the session includes all aspects of rigor for domain 2.MD.B and 2.NBT.A. Students investigate measurement through a contextual problem leading to understanding and application of using a number line tool to represent their thinking. Another example is Unit 3, Module 3, Session 5. This lesson includes all aspects of rigor as students create and solve their own story problems. Strategies and solutions are shared as students use various tools and mathematical representations to show evidence of their thinking.

The instructional materials reviewed for Grade 2 meet the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level Standards for Mathematical Content, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Grade 2 Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The chart is broken down into four categories: Habits of Mind, Reasoning and Explaining, Modeling and Using Tools, and Seeing Structure and Generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Grade 2?" guide for teachers (AG, page 17).

Each Session clearly identifies the MPs used in the Skills & Concept section of the session. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation of the connection between the indicated MP and the Standards for Mathematical Content for the teacher. Examples of the MPs in the instructional materials include the following:

• In Unit 1, Module 2 each of the 6 Sessions address MP7. There is a "Math Practices In Action" reference in two of the six sessions.
• In Unit 2, Module 3 in the Skills & Concepts section, Session 5 lists MP1, Session 1 lists MP2, Session 6 lists MP3, Sessions 1, 2, 4, 5 and 7 list MP7, and Sessions 3 and 6 list MP8.
• Unit 3, Module 3, Session 2 references MP2 within the Problems & Investigations portion of the session in "Math Practices in Action." Students are engaged in the "Presents & Parcels" activity. The teacher helps transition students from quantitative to abstract reasoning when she labels the sketches as "10" instead of showing each of the 10 presents. The labels remind students of the quantities without actually seeing the presents.
• In Unit 7, Module 1, Sessions 2 and 3 reference the MPs within the Problems & Investigations portion of the session as, "Math Practices in Action."
• In the September Number Corner MP1 is referenced in the Calendar Grid; MP2 is referenced in the Computational Fluency and Number Line; MP4 is referenced in Calendar Grid, Daily Rectangle, and Number Line; MP6 is addressed in Calendar Collector; MP7 is addressed in Calendar Grid, Calendar Collector, and Daily Rectangle; and MP8 is addressed in Computational Fluency.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These sessions that focus entirely on MPs include the following:

• Unit 1, Module 1, Session 1
• Unit 7, Module 2, Session 1
• Unit 8, Module 4, Session 2

The materials meet the expectations for attending to the full meaning of each practice standard. Each session clearly identifies the MPs used in the Skills & Concept section of the session. Typically there are two standards for MP listed for each session, so there is not an overabundance of identification.

Each Session clearly identifies the MPs used in the Skills & Concept section of the session Typically there are two MPs listed for each session, so there is not an overabundance of identification. Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, the MPs are not discussed or listed in unit overviews or introductions (Major Skills/Concepts Addressed); however, they are listed in section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.

In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Daily Rectangle, Computational Fluency, Number Line). The MPs are also listed in the assessment section of the Introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of the MPs within Number Corner.

The following are examples of times that the instructional materials attend to the full meaning of an MP:

• Unit 1 Module 2 Session 2 attends to MP7. Students are shown a number greater than 10 on a number rack for less than 3 seconds. This elicits the students to make sense of the structure of groups of 5 and 10 since they are unable to count unit by unit.
• Unit 4, Module 3, Session 1 attends to MP6. Students are comparing two different measuring tools: the student-made inchworm and footworm. As they measure things around the room, they first make estimates and then are asked to measure using each tool. After measuring they are asked to discuss the difference between using the footworm and inchworm rulers. They then work in their Student Books, choosing which tool, the inchworm or footworm to measure lengths of various objects, explaining their choices.
• Unit 8, Module 1, Session 3 attends to MP1. In the Problems & Investigations section of the session, students are solving story problems. The teacher models how to restate what the problem is asking, identify the information required to solve it, and make a reasonable estimate before starting to solve the problem. The teacher helps students get in the habit of orienting themselves before beginning their computations; making sense of problems and persevere in solving them.
• In the May Daily Rectangle Number Corner, MP7 and MP2 are attended to as students investigate the relationship between rectangles and area through rectangular arrays. Repeated addition equations are written to express the total as a sum of equal addends.
• At times MP4 was attended to fully in the sessions as students are posed contextual problems that require the use of mathematical understanding and application of strategies. Examples include Unit 3, Module 2, Session 1; Unit 4, Module 4, Session 1; Unit 4, Module 4, Session 3; and Unit 7, Module 3, Sessions 2 and 3.

However, at times the materials only partially attend to the meaning of MP4. Examples include Unit 6, Module 1, Session 2 and Unit 6, Module 2, Session 1.

The materials reviewed for Grade 2 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.

• In Unit 3, Module 2, Session 2, students reflect upon a variety of strategies and efficiency of jumping by larger numbers on a number line. The context of the problem involves a comparison with the difference unknown. The problem can be solved with both addition and subtraction. This leads to a discussion with students analyzing and critiquing each other’s approach to the solution.
• In Unit 3, Module 3, Session 5 strategies and solutions are shared as students use various tools and mathematical representations to show evidence of their thinking in solving student written word problems. Strategies are compared and contrasted as students investigate effective and efficient ways to solve the problem. Students listen to one another present their strategy and are encouraged to ask questions and add new ideas to their own paper based on the discussion.
• In Unit 4, Module 3, Session 1, students are given inchworms and a footworm to measure various lengths then they are asked their estimations, measurements, strategies, and why they would use inchworms or footworms to measure various sized objects.
• In Unit 5, Module 4, Session 1, students use Unifix cubes to search for patterns in various arrangements. Students are encouraged to share their own ideas about how the pattern will extend beyond what is shown while also considering their classmates' ideas. This provides the opportunity for students to construct viable arguments and critique the reasoning of others.
• In Unit 6, Module 2, Session 3, students work in pairs in their Student Books and with Geoboards to partition rectangles into rows and columns of the same size squares. They show/share their reasoning, expressing their arguments verbally as well as using models, sketches and symbolic notation (Math Practices in Action).
• In Unit 6, Module 2, Session 5 students engage in discussion with the teacher and critique strategies used to find the area of rectangles. Then students are directed to use efficient strategies in pairs and record the areas of the rectangles.
• In the Number Corner February Daily Rectangle during Activity 2, "Daily Deposit," students are adding the base ten blocks in the chart. They are asked to share their solutions and explain their thinking. Again, there is sample dialogue; however, none include samples encouraging students to analyze the thinking of others.

The instructional materials reviewed for Grade 2 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 that is present throughout the materials.

The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon, and trapezoid and using operations and algebraic thinking terminology such as equation and difference.

• Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
• The online Teacher Materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
• At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
• In Unit 6, Module 2, Section 2, students are exploring area with pattern blocks and use terms such as area, half, triangle, quadrilateral, hexagon, and rhombus.
• In the October Number Corner Daily Rectangle, students are introduced to the terms rectangular array and column while describing units of one as they build conceptual understanding of early multiplication.