2nd Grade - Gateway 3

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include: 

• The Guide to Implementing AF Math provides a Program Overview for the teacher with information on the program components and scope and sequence. This includes descriptions of the types of lessons, Math Stories, Math Practice, and Cumulative Review.

• The Math Stories Guide (K-4) provides a framework for problem solving.

• Each Unit Overview includes a section called “Key Strategies” that describes strategies that will be utilized during the unit.

• The Teacher’s Guide supports whole group/partner discussion, ask/listen fors, common misconceptions and errors. 

• In the narrative information for each lesson, there is information such as “What do students have to get better at today? Where will time be focused/funneled?”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 7, Addition and Subtraction with 1000, Lesson 6 include:

• “What is new and/or hard about the lesson? Students need to mentally decompose numbers into hundreds, tens, and ones in order to make jumps on the number line to add two 3-digit numbers together. Some scholars may also make jumps of other quantities (ex. Jump of 3 from 237 to get to 240 instead of making 3 jumps of 1). They also make use of their understanding of part-part-whole relationships. Scholars need to understand that when adding on a number line, they can start at one of the parts and make jumps forward of the other part to get the whole or the total. If students are still struggling to add on a number line, they will likely struggle to subtract 3-digit numbers on a number line, especially if the misunderstanding is rooted in part-part-whole relationships.” 

• “Exemplar Student Response: I solved 436 + 249 on a number line. I started at 436. Then I made 2 jumps up of 100 b/c there are 2 hundreds in 249. Next I made 4 jumps of 10 b/c there are 4 tens in 249. Last I made a jump of 4 + 6 jump of 5 b/c 4 + 5 = 9. There are 9 ones in 249. I counted up on the number line and got 675. This proves 436 + 249 = 675.”

• “Potential Misconception: Not knowing the next number in the sequence when skip counting or counting by ones. Starting with smaller number (this works but is less efficient). Starting at one part and jumping up the same part (ex. Solving 356 + 291 by starting at 356 and making 3 jumps of 100, 5 jumps of 10, and 6 jumps of 1).”

• “Mid-Workshop Interruption: If > \frac{2}{3} of students are successfully solving, share out someone whose work is neat and organized and is clearly showing regrouping. Discuss the jumps they made and why they worked. If < \frac{2}{3} of students are successful, call students back together to clear up the misconception through a misconception protocol. Continue to circulate and check for students to apply the learning. Make note of student success in applying in your Rapid Feedback tracker to inform the path for the Discussion.”

• “Share/Discussion: Direct students to the Discussion work space in their packets as needed. Use workshop data to determine the appropriate discussion path: Facilitate a discussion around a major misconception, Share example where student skip counted incorrectly (likely over decades or centuries) or made incorrect jumps). Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR, 2-3 students share their work/strategies, What is the same about these strategies? What is different? OR, ask students to apply their learning in a new way with an additional exercise.”

Each lesson includes both “What” and “How” Key Point sections that describe what students should know and be able to do and how they will do it. Examples from Unit 7, Addition and Subtraction with 1000, Lesson 6 include:

• “What Key Points: We can solve 3-digit addition problems on a number line by starting at one of the parts and jumping forward the other part to find the whole/total. When adding on a number line, we can use place value to help us tell how many jumps of hundreds, ten, and one to make. Understand that if we start at a part and jump up the other part we will get the whole/total because part + part = whole.” 

• “How Key Points: When we add 3-digit numbers on a number line…. First we start at a part, Then we jump up the other part by making jumps of 100s, 10s, and 1s.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series. Examples include:

• Guide to Implementing AF Grade 2, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

• The Program Overview informs teachers “about how to ensure scholars have sufficient practice with all of the Common Core State Standards. Standards or parts thereof that are bolded are addressed within a lesson but with limited exposure. It is recommended that teachers supplement the lessons addressing these standards by using the AF Practice Workbooks to ensure mastery for all students. Recommendations for when to revisit these standards during Math Practice and Friday Cumulative Review are noted in the Practice section of each unit.”

• The Unit Overview includes a section called Identify Desired Results: Identify the Standards which lists the standards addressed within the unit and previously addressed standards that relate to the content of the unit.

• In the Unit Overview, the Identify The Narrative provides rationale about the unit connections to previous standards for each of the lessons. Future grade-level content is also identified.

• The Unit Overview provides a table listing Mathematical Practices connected to the lessons and identifies whether the MP is a major focus of the unit.

• At the beginning of each lesson, each standard is identified. 

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work. 

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series.

In the Unit Overview, the Identify the Narrative section provides the teacher with information to unpack the learning progressions and make connections between key concepts. Lesson Support includes information about connections to previous lessons and identifies the important concepts within those lessons. Examples include:

• Unit 7, Lesson 8 Narrative, “How does the learning connect to previous lessons? What do students have to get better at today? In the previous lessons, students added 3-digit numbers with regrouping in both places using flats, sticks, and dots, a number line, and expanded notation. Today, for the first time, students will subtract 3-digit numbers using flats, sticks, and dots, and expanded notation.”

• In the Unit Overview, the standards that the unit will address are listed along with the previous grade level standards/previously taught and related standards. Also included is a section named “Enduring Understandings: What do you want students to know in 10 years about this topic? What does it look like in the unit for students to understand this?” For example, in Unit 7, standards addressed are 2.NBT.7, 2.NBT.9. Previous Grade Level Standards/Previously Taught & Related Standards include 2.OA.2, 2.NBT.5, and 2.NBT.6. An example grade level enduring understanding is, “We can partition a number in different ways to solve multi-digit addition and subtraction equations in increasingly efficient ways.” An example for what it looks like in this unit is, “Students will partition numbers with number bonds to increase fluency and aid in solving problems with missing addends and subtrahends. They will also partition three-digit numbers concretely with place value blocks, pictorially with flats, sticks and dots, and abstractly with expanded form. This will help when adding larger numbers by adding ones and ones first, then tens and tens, and finally hundreds and hundreds. It will also aid in regrouping. Students will understand that 10 of any unit can be grouped into one larger unit (ex. 10 ones= 1 ten). Students learn that starting in the ones place allows them to regroup effectively and efficiently.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program. Examples include:

• The Implementation Guide states, "Our program aims to see the mathematical practices come to life through the shifts (focus, coherence, rigor) called for by the standards. For students to engage at equal intensities weekly with all 3 tenets, we structured our program into three main daily components Monday-Thursday: Math Lesson, Math Stories and Math Practice. Additionally, students engage in Math Cumulative Review each Friday in order for scholars to achieve the fluencies and procedural skills required."

• The Implementation Guide includes descriptions of “Math Lesson Types.” Descriptions are included for Game Introduction Lesson, Task Based Lesson, Math Stories, and Math Practice. Each description includes a purpose and a table that includes the lesson components, purpose, and timing. 

Research-based strategies are cited and described within the Program Overview, Guide to Implementing AF Math: Grade K-4, Instructional Approach and Research Background. Examples of research-based strategies include:

• Concrete-Representational-Abstract Instructional Approach, Access Center: Improving Outcomes for All Students K-8, OESP, “Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.”

• Principles to Actions: Ensuring Mathematical Success for All, 2014, “According to the National Council of Teachers of Mathematics, Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.”

• Problem-solving as a basis for reform in curriculum and instruction: the case of mathematics by Heibert et. al., “Students learn mathematics as a result of solving problems,” and that “mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Each lesson includes a list of materials specific to the lesson. Examples include:

• Unit 2, Lesson 1, Lesson Overview: “Materials: dot cubes, recording sheets, What’s Missing VA, and Rapid Feedback tracker.”

• Unit 9, Lesson 7, Lesson Overview: “Materials: handouts, scissors, glue, posters.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

Unit Assessments consistently and accurately identify grade-level content standards along with the mathematical practices within each unit. Examples from unit assessments include:

• Unit 2 Overview, Unit 2 Assessment: Addition & Subtraction to 100, denotes the aligned grade-level standards and mathematical practices. Question 2, “Solve and show your work. A. 16 + ____ = 74 b. 57 - ____ = 28” (2.NBT.5, MP5, MP6, MP7)

• Unit 5 Overview, Unit 5 Assessment: Length, Money, Graphing, and Time, denotes the aligned grade-level standards and mathematical practices. Question 6, “Dan has 35 cents. Emily says that he could have 3 dimes and a nickel. Jeremy says he could have a quarter and a dime. Who is right and how do you know?” (2.MD.8, MP2, MP3, MP4)

• Unit 9 Overview, Unit 9 Assessment: Geometry-Fractions, denotes the aligned grade-level standards and mathematical practices. Question 3, “Use lines to partition the rectangles into fourths in different ways:” Three unpartitioned, equal-sized rectangles are provided. (2.G.3, MP1, MP6, MP7)

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance. Examples include:

• Assessments include an informal Exit Ticket in each lesson and a formal Unit Assessment for every unit. 

• There is guidance, or “look-fors,” to teachers about what the student should be able to do on the assessments.

• All Unit Assessments include an answer key with exemplar student responses.

• The is a rubric for exit tickets that indicates, “You mastered the learning objective today; You are almost there; You need more practice and feedback.” 

Program Overview, Guide to Implementing AF Math: Grade 2, Differentiation, Unit-Level Errors, Misconceptions, and Response, “Every unit plan includes an ‘Evaluating and Responding to Student Learning Outcomes’ section after the post-unit assessment. The purpose of this section is to provide teachers with the most common errors as observed on the questions related to each standard, the anticipated misconceptions associated with those errors, and a variety of possible responses that could be taken to address those misconceptions as outlined with possible critical thinking, strategic practice problems, or additional resources.” Examples include: 

• Unit 3 Overview, Unit 3 Assessment: Story Problems, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 2.OA.1, “If the misconception is that the student does not understand the model they choose to represent: Encourage student to move back along CPA continuum. If the student is struggling to interpret their equation representation, are they able to decontextualize the problem with a tape diagram or number bond, and if yes, can they use those to solve? If no, students can represent with 1:1 pictures. To build understanding of more abstract models, see lessons 1 and 5 of this unit.”

• Unit 6 Overview, Unit 6 Assessment: Place Value- Three Digit Numbers, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 2.NBT.3, “If the misconception is that the student regroups incorrectly or is unable to flexibly decompose hundreds, tens, or ones to show a number in a different way: See lessons 8 and 9 of second grade unit 2 Refer to lessons 7 and 8 of this unit Move back on the CPA continuum: using place value blocks, represent a number and ask students to identify the number of hundreds, tens, and ones before composing the three-digit number. Then, decompose a hundred into 10 tens. Ask students to identify the number of hundreds, tens, and ones before composing the three-digit number. Ask students, ‘What did you see me do to show the number in a different way? How is it possible that 325 can have 3 hundreds, 2 tens, 5 ones or 2 hundreds, 12 tens, 5 ones?’ Repeat as needed and provide ample opportunities for practice.”

• Unit 8 Overview, Unit 8 Assessment: Geometry- Arrays, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 2.OA.3, “If students are struggling to prove if a number is odd or even: Refer to lessons 8-10 of this unit Move back on the CPA Continuum: Have students use cubes or other manipulatives to represent a number and have them organize them into groups of 2 or two equal groups to illustrate the concepts of pairs and teams. Connect to pictorial drawings of ‘pairs’ and ‘teams’ Create a visual anchor to reinforce vocabulary (odd and even) and examples of each Give students ample practice and feedback representing a number using cubes and drawings and seeing if they can be split into pairs and teams. Ask students to explain if the number is even or odd and how they know using their representation.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems. 

Assessments include opportunities for students to demonstrate the full intent of grade-level standards across the series. Examples include:

• In the Unit 3 assessment, the full-intent of standard 2.OA.1 (use addition and subtraction within 100 to solve one- and two-step word problems) is met. Item 6, “Mayshon’s string is 21 cm shorter than Bernadette’s string. Mayshon’s string is 18 cm long. Bernadette cut 12 cm off her string. Now how long is Bernadette’s string?” There are nine available items, varied addition and subtraction situations including two-step problems, and space is provided for students to use drawings and equations to solve.

• In the Unit 5 assessment, the full-intent of standard 2.NBT.3 (read and write numbers to 1000 using base-ten numerals, number names, and expanded form) is met. Item 2, “Write the value of 17 tens three different ways. Use the largest numerals possible (standard form, expanded form, unit form). Item 6, “Draw flats, sticks, and dots on the place value chart to show 348.”

• In Unit 9, the full-intent of standard 2.G.3 (partition circles and rectangles into two, three, and four equal shares) is met. Item 2, “1 whole = ___ halves; __ fourths = 1 whole; ___thirds = 1 whole.” Item 3, “Use lines to partition the rectangles into fourths in different ways. (three rectangles are provided).” Item 5, “Circle all the rectangles that are partitioned into fourths and cross out any rectangle that is not partitioned into fourths. (4 rectangles provided; 3 that are correctly partitioned into fourths and one that is not correctly partitioned into fourths). Explain why the rectangles that you crossed out do not show fourths.” 

Assessments include opportunities for students to demonstrate the full intent of grade-level practices across the series. Examples include:

• Unit 3 Assessment, Item 1, supports the full development of MP1: Make sense of problems and persevere in solving them and MP2: Reason abstractly and quantitatively. “Miss Taylor had a ribbon. She cut off 23 inches of the ribbon. Now the ribbon is 57 inches long. How long was Miss Taylor’s ribbon to start?”

• Unit 5 Assessment, Item 6, students engage with MP3: Construct viable arguments and critique the reasoning of others. “Dan has 35 cents. Emily says he could have 3 dimes and a nickel. Jeremy says he could have a quarter and a dime. Who is right and how do you know?” 

• Unit 7 Assessment, Item 5, supports the full development of MP3: Construct viable arguments and critique the reasoning of others, MP6: Attend to precision, and MP7: Look for and make use of structure. “Ava solved the problem below using expanded notation. What mistake did she make? How can she fix it? 603 - 246 = ____(work shown of the vertical form of expanded notation with the mistake made of regrouping the tens place of 603 to 100).”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each unit overview. According to the Program Overview, Guide to Implementing AF Math: Grade 2, Differentiation and Working with Special Populations, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning mathematics and feel unsuccessful. Therefore, it is critical that strong curricular materials are designed to provide support for all student learners, especially those with diagnosed disabilities (Hott et al., 2014). The Achievement First Mathematics Program was designed with this in mind and is based on several bodies of research about best practices for the instruction of students with math disabilities, including the work of the What Works Clearinghouse (an investment of the Institute of Education Sciences within the U.S. Department of Education) and the Council for Learning Disabilities (an international organization composed of professionals who represent diverse disciplines). Unit Overviews and lesson level materials include guidance around working with students with disabilities, including daily suggested interventions in the Workshop Section of the daily lesson plan. Teachers should reference these materials in conjunction with the information that follows in this Implementation Guide when planning instruction in order to best support all students.” Within Daily Lesson Plans there are two versions of Independent Practice Problem Sets, “one set is more scaffolded and can be used for all students and in combination with intervention as needed; the other is less scaffolded.”

Examples of supports for special populations include: 

• Unit 2, Lesson 4, Workshop, Suggested intervention(s), “Provide students with linking cubes or other manipulatives. Teachers should model and think aloud how to use doubles facts to solve near doubles. Provide students with frequent support and feedback. Students should use card set #1.”

• Unit 4 Overview, Data, Differentiating for Learning Needs, “As students engage with this data unit, this will be their last opportunity to learn about how to represent categorical data in graphing, setting the stage for using different kinds of data and deeper data analysis in the future. It is likely that they will bring a variety of experiences from first grade. In first grade, the expectation is that students are able to organize, represent, and interpret data with up to three categories and ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than another (1.MD.4). Some students will enter second grade with a complete understanding of these concepts and will be able to apply them to a larger data set with more categories without difficulty, while others may not be solid with the skills, strategies, or terminology. While students represented data in first grade, it wasn't required that they represent using particular types of graphs. Students who used this curriculum in first grade will be familiar with tally charts, simple tables, picture graphs, and bar graphs, but they will never have made them fully from scratch. (They will always have had a template.) Some students will be able to make the leap to defining categories, scale, and title without difficulty, but others may need more support. Due to this, teachers will need to know their students’ data and use that to differentiate both up and down while ensuring that students are all engaging in solving the same grade-level problems, no more and no less.” Suggested Interventions, “Encourage students to represent data sets first with simple tables and then use the tables to create graphs or charts by explicitly modeling this process. When answering questions that require the comparison of two categories of data, students may re-represent with cubes or pictorial 1:1 tape diagrams to compare directly by matching one-to-one. Explicitly model this and lead the small group in a discussion of how/why this works.”

• Unit 7 Overview, Addition and Subtraction with 1000, Differentiating for Learning Needs, “As students work with place value concepts to add and subtract within 1000 in this unit, it is likely that they will bring a variety of experiences from first grade and unit 2 of second grade (adding and subtracting within 100). In first grade, the expectation is that students are able to add and subtract within 20 (1.OA.6), use their understanding of place value to add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10 (1.NBT.4), and subtract multiples of 10 in the range of 10-90 from multiples of ten in the range of 10-90 (1.NBT.6). Second graders should be familiar with using models such as place value blocks, pictorial models of flats/ sticks/ dots, and expanded notation from unit 2. They also have experience using a number line as a tool to help them reason about addition and subtraction within 100. Using those tools and models, students will have experience and exposure to a variety of solution strategies including counting all, counting on/back, making tens, and applying known facts with place value understanding. Students likely have varying levels of comfort with each of these models and strategies -- a handful of students will have worked primarily in the concrete/ pictorial and still be developing fluency, while others may be comfortable solving within 100 mentally. In this unit, students will apply these models and strategies to a larger range of numbers up to 1000; the level of intervention/ extension they require will depend greatly on their comfort with the strategies and models previously described. Some students may begin this unit with a complete understanding of these concepts, while others may not be solid with the skills, strategies, or terminology. Regardless of the knowledge and experiences that children have at the start of this unit, teachers must meet their students where they are and ensure that all students are learning and deepening their understanding of the math concepts introduced in this unit. Teachers will need to know their students’ data and use that to differentiate both up and down while ensuring that students are all engaging in solving the same grade-level problems, no more and no less.” Suggested Interventions, “Students should work with cubes and/or place value models. They may use counting strategies including counting on / back or removing ( counting all can be used in high need situations). Teachers should explicitly model adding hundreds to hundreds, tens to tens, and ones to ones and removing hundreds from hundreds, tens from tens, and ones from ones. Provide sentence starters for the student and access to a visual anchor or checklist of the components of a written response.”

• K-4 Math Stories Guide, Differentiating Math Stories Instruction, “As noted in the Implementation Guides for each grade level, supporting all learners, including those with disabilities and special needs, English and Multilingual learners and advanced students, is a commitment of the Achievement First program, and Math Stories, like the other program components, is designed to meet all students where they are and to move them to grade level proficiency and deeper understanding of the Common Core Math standards through research-based best practices for differentiation.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

According to the Guide to Implementing AF Math: Grade 2, Differentiation, Supporting Advanced Students, “Part of supporting all learners is ensuring that advanced students also have opportunities to learn and grow by engaging with the grade level content at higher levels of complexity.” Daily lessons provide “suggested extension activities for students in the Workshop Section of the lesson plan so that teachers can encourage students to engage with the content at a higher level of complexity if they are not doing so naturally but are ready to. These extension suggestions include variations of the game that encourage more sophisticated strategies in Game Intro Lessons (K-2) and variations of the tasks or suggested strategies or tools students may use in Exercise Based Lesson (2-4). The independent practice for grades Exercise Based Lessons also includes problems labeled by difficulty. Teachers should differentiate for student needs by assigning the most challenging problems to advanced students while allowing them to skip some of the simpler ones, so that they can engage with the same number of problems, but at the appropriate difficulty level. Additionally, the Discussion section of the daily lesson plans always include a potential whole class extension/ application problem. These are often additional problems or tasks that ask students to apply the mathematical concepts taught that day, and like the focal problem of the day, students should be encouraged to use the strategy that makes sense to them in order to solve, once again allowing students to engage with the grade level content at a level that is appropriate to them.” Examples Include:

• Unit 2, Lesson 2, Workshop, Suggested Extension(s), “Have students start with 20 as the whole, give 2 number cubes instead of 1. Students can also skip ahead to the extension/ application problem (They should skip ahead and should not complete more problems). Extension Problem: “There were 17 cartons of milk in the refrigerator. 9 of them were chocolate milk. How many of them were regular milk? Represent with an addition AND subtraction equation and solve.”

• Unit 6, Lesson 15, Workshop, Suggested Extension(s), “Challenge students to solve problems where the numbers’ places are written out of order (ex: 50 + 8 + 200) and problems with more instances of regrouping.”

• K-4 Math Stories Guide, Differentiating Math Stories Instruction, “In the Math Stories block, heterogeneous groups of students are expected to work with a variety of tools and strategies as they work through the same set of problems; this ensures that all students access the content and build conceptual understanding while allowing advanced students to engage with the content at higher levels of complexity.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

According to the Program Overview, Guide to Implementing AF Math: Grade K, Differentiation, Supporting Multilingual and English Language Learners, “Both the Game Introduction Lessons in lower elementary and the Exercise Based Lessons in upper elementary along with the Math Stories Protocols used in Math Stories at all grade levels build on the four design principles for promoting mathematical language use and development in curriculum and instructions outlined by Stanford’s Graduate School of Education (Zwiers et al., 2017), Understanding Language/SCALE and recommended by the English Language Success Forum…” The series provides the following design principles that promote mathematical language use and development: 

• “Design Principle 1: Support sense-making, Principle in Action - Daily lesson plan scripts and the math stories protocols intentionally amplify rather than simplify student language by anticipating where students may have difficulty accessing the concepts and language and providing multiple ways for them to come to understanding. Every lesson includes multiple opportunities for students to engage in discussion with one another, often through turn and talks, as they make sense of the content, and this sense-making is also supported through the use of concrete and pictorial models and a lesson visual anchor that captures student thinking and mathematical concepts and key vocabulary… Additionally, teachers are provided with student-friendly vocabulary definitions for all new vocabulary terms in the unit plan that can support MLLs/ELLs further.”

• “Design Principle 2: Optimize output, Principle in Action - Lessons and the math stories protocols are strategically built to focus on student thinking. Students engage in each new task individually or with partners, have opportunities to discuss with one another, and then analyze student work samples as a whole class…All students benefit from the focus on the mathematical discourse and revising their own thinking, but this is especially true of MLLs/ELLs who will benefit from hearing other students thinking and reasoning on the concepts and/or different methods of solving.” 

• “Design Principle 3: Cultivate conversation, Principle in Action - A key element of all lesson types is student discussion. Daily lesson plans and the math stories protocol rely heavily on the use of individual or partner think time, turn-and-talks with partners, and whole class discussion to answer key questions throughout the lesson script. The rationale for this is that all learners, but especially MLLs/ELLs benefit from multiple opportunities to engage with the content. Students that are building their mastery of the language may struggle more with following a whole-class discussion; however, having an opportunity to ask questions and discuss with a strategic partner beforehand can help deepen their understanding and empower them to engage further in the class discussion…” 

• “Design Principle 4: Maximize linguistic and cognitive meta-awareness, Principle in Action - Every daily lesson and math stories lesson is structured so that students have many opportunities to get ‘meta’ about the mathematical processes they engage in. Students explain how they model and solve problems to the teacher and one another throughout the lesson, often through turn and talks in which they also evaluate their peers’ strategies and thinking. Lesson scripts also encourage students to draw connections between new content and previous learning as well as between different strategies....”

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Program Overview, Guide to Implementing AF Math: Grade 2, Differentiation, Supporting Multilingual and English Language Learners, “The Achievement First Mathematics Program appreciates the importance of creating a classroom environment in which Multilingual and English language learners (MLLs/ ELLs) can thrive socially, emotionally, and academically. We have strategically included several mathematical language routines (MLRs) to support the language and content development of MLLs/ELLs in all lesson plans and called them out explicitly for teachers in a third of lesson plans.” The Mathematical Language Routines, Vocabulary, and Sentence Frames are present throughout the materials. Examples include:

• Unit 2 Overview, Addition and Subtraction to 100, Differentiating for Learning Needs, Supporting MLLs/ELLs, Mathematical Language Routines, “8 mathematical language routines are outlined in detail in the Implementation Guide for Grade 2. These routines are worked into the lesson plans throughout the unit and explicitly highlighted for teachers in lessons 1, 3, 4, 6, 7, 11, 23, and 24. Teachers should use these lessons as a model for recognizing when routines occur in the remaining lessons and thinking about how they might incorporate additional routines into the remaining lessons if they feel their students need more language development support. A brief overview of each of the math language routines along with general guidance about how to implement them in the context of this unit are outlined below:

  • MLR 1 Stronger and Clearer Each Time: Teachers provide students with multiple opportunities to articulate their mathematical thinking, with the opportunity to refine their language with each successive share.This routine is often incorporated into lessons as students have multiple opportunities to articulate the key understanding/ key points of the lesson through turn and talks in the intro, workshop, and discussion. Over the course of the lesson, students refine their understanding of the concepts and the language they use to articulate that understanding as they engage in these successive turn and talks. Exemplar responses to turn and talks in the introduction of most lessons are often broad and mutl-part. The expectation is that students answer the question at hand, explain how they came to it, and why that works. Students who are not in the habit of giving such thorough answers or who are struggling with oral language may need more support than is scripted into the plans in order to give complete, exemplar responses. Teachers can and should feel comfortable modifying these turn and talks questions to align with the Stronger and Clearer Routine by breaking the question into 2-3 separate turn and talks, using scaffolds to break the question into more manageable parts without reducing the rigor of the question. For example, when asking students to discuss how they solved a problem, teachers may ask first what the solution is, then how they figured that out, and finally why that works. All turn and talks can also be posed as successive questions in which students engage in the same turn and talk several times in a row with different partners. As students practice articulating their ideas multiple times and hear different peers explain the concepts using different language and vocabulary, they will refine their language each time. 

  • MLR 2 Collect and Display: The teacher captures student thinking and/or strategies visually and leads the class in a discussion. In all lessons, teachers co-create a visual anchor with students. This visual anchor should include illustrations of the strategies at work, and teachers should reference them and encourage students to reference them in whole group discussion.

  • MLR 3 Critique, Correct, and Clarify: Teachers present students with a statement, an argument, an explanation, or a solution, and prompt them to analyze and discuss. Nearly all lessons include an error analysis option as a potential focus either of the introduction, MWI, or discussion. When following a misconception protocol, teachers should give students plenty of think time and allow them time to discuss the error and misconception with partners.

  • MLR 4 Info Gap: Students are put into pairs; each student in the pair is given partial information that when combined with their partner’s information provides the full context needed to solve the problem. Students must communicate effectively in order to solve the problem. Teachers may wish to work this routine into the math stories block by providing pairs of students with opposite parts of the story problem; the pairs will have to work together to communicate the important information needed to solve. This routine can be incorporated into lesson 6; further guidance is included in the lesson plan.

  • MLR 5 Co-Craft Questions and Problems: Teachers guide students to work with one another to create questions or situations for math problems or to create entire problems and then solve them. Teachers may wish to incorporate this routine into Math Stories by having students work in pairs to create story problems to exchange with one another, particularly on days when the class finishes the protocol early. Teachers may also work this routine into lesson 6; further guidance is included in the lesson.

  • MLR 6 Three Reads: Teachers support students in making sense of a situation or problem by reading three times, each time with a particular focus. Teachers should work this routine into the math stories block and any other time MLLs/ ELLs work with story problems, including when the problem of the day or try one more problem are contextual. When reading a story problem, prompt students to do a particular task for each read. For example, for the first read, teachers might direct students to focus on visualizing only. Then they might prompt students to represent during the second read and to check their representation against the story during the third read.

  • MLR 7 Compare and Connect: Teachers prompt students to understand one another’s strategies by comparing and connecting other students’ approaches to their own. Students engage in this routine multiple times in most lessons as they connect the different focal strategies of the lesson. Several questions are scripted into each lesson’s introduction and often in the Discussion that ask students to consider how strategies relate to one another. These questions should be posed as turn and talks with think time to best support language development.

  • MLR 8: Discussion Supports: Teachers use a number of moves to help facilitate student discussion including revoicing, encouraging students to agree, disagree, build on, or ask questions of their peers, incorporating choral response to build vocabulary, showing concepts multi-modally, and modeling clear explanations/ think alouds. Teachers build vocabulary including parts, wholes, regrouping, and more through explicit instruction and repetition. Teachers show the concept of regrouping multi-modally with concrete objects, pictures, and expanded notation. Teachers continue to encourage and build habits of discussion in this unit. Continue to prompt for students to engage in discourse by agreeing/ disagreeing with one another and introduce/ teach building off one another through explicit modeling, provision of sentence frames, and feedback and praise.”

• Vocabulary: “When introducing new vocabulary, words and their meanings should be explicitly taught with the use of concrete objects and/or visual models. Kinesthetic motions and choral response also are helpful for introducing new vocabulary, and when it is possible, it is often useful to pre-teach vocabulary for MLLs/ ELLs. To support sense-making, make sure that vocabulary is posted throughout the unit with visual illustrations of meaning.” Examples include: “Addend- a number that is added to another number; used to get the sum or the total; Subtrahend – the number which we subtract from another number in a subtraction sentence; Equal – having the same amount or value.” 

• Unit Sentence Frames/ Starters: “Providing sentence frames and starters is helpful for cultivating conversation, particularly for students who are developing oral language skills in new or multiple languages. Teachers should have these sentence frames posted in the classroom to assist students in engaging in discourse. Additionally, teachers can provide sentence starters at the start of each turn and talk by posing the question and then providing the starter. For example, if the turn and talk is ‘Turn and tell your partner how you solved 42 x 40,’ the teacher would give the cue for students to turn and then say, ‘I solved 42 x 40 by…’ before students begin talking.” Examples include: “Sentence Frames for Explaining Mathematical Thinking (all lessons): I represented by/with ______. I showed ______ because ______. I chose this representation/ model because ______. The solution is ______. I solved it by ______. This works because ______. I chose this strategy because _______.”

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are accurate representations of mathematical objects and are connected to written methods. Examples include:

• Unit 2, Lesson 7, to address standard 2.NBT.5 (fluently add and subtract within 100 using strategies based on place value…), the materials include place value blocks/cubes. In the Overview, Identify the Narrative, “Throughout the unit, manipulatives and math drawings  allow students to see the numbers in terms of place value units and serve as a reminder that they must add like units (e.g. knowing that 74 + 38 is 7 tens plus 3 tens and 4 ones plus 8 ones.)” Introduce the Math, “You learned about two-digit numbers in first grade. Today, you get to show what you know about two-digit numbers and work to represent them with sticks and dots and expanded notation.” 

• Unit 6, Lesson 2, to address standard 2.NBT.1(understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones), the materials include place value blocks. In Overview, Identify the Narrative, “They represent three-digit numbers  using flats, sticks, and dots, and they understand that ten ones can be grouped into a ten, ten tens can be made into a hundred, and ten hundreds are equal to a thousand. They work with hundreds, tens, and ones on a place value chart and then write numbers in standard form and base-ten numeral form.” Introduction, “A Flat is worth 100. Let’s try representing a 3-digit number using flats, sticks, and dots: Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?”