1st Grade - Gateway 3

The materials reviewed for Achievement First Mathematics Grade 1 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 6, Two-Digit Numbers, Lesson 9 include:

• “What is new and/or hard about the lesson? This is challenging for students for two main reasons. They may have difficulty building/representing the two quantities and make mistakes when counting by tens. Teachers should encourage precision both when building/drawing quantities and counting by tens.”

• “Exemplar Student Response: Students should be able to explain their work by saying, “I drew 5 sticks because there are 5 tens and 0 dots because there are 0 ones. Then I drew 3 sticks because there are 3 more tens and 0 dots because there are 0 ones. I’m finding the total so I counted all of the sticks. I know each stick is worth 10 so I counted by tens: 10, 20, 30, 40, 50, 60, 70, 80.”

• “Potential Misconception: Student makes representing errors. Student makes calculation errors.” 

• “Mid-Workshop Interruption: If > \frac{2}{3} of students are successfully solving and using a range of strategies, ask students to discuss what they notice about the digits and how they change. Which digit changes when we combine tens? Why do you think that happens? (Why doesn’t the ones place change?) 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 path: Facilitate a discussion around a major misconception, 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. Possible Extension Problem: Solve 15 + 30.”

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 6, Two-Digit Numbers, Lesson 9 include:

• “What Key Points: Two digit numbers are made up of tens and ones. Tens are made up of tens and thus can be counted by tens. A multiple of ten is a group of tens with no extra ones. 

• “How Key Points: We can represent two-digit numbers by building them with cubes and creating tens sticks. We can represent two-digit numbers by drawing literal pictures. We can represent two-digit numbers by showing the tens with sticks and the ones with dots. We can combine two multiples of tens by counting all by tens. We can combine two multiples of ten by counting on by tens.”

The materials reviewed for Achievement First Mathematics Grade 1 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. Examples include:

• Unit Overviews provide thorough information about the unit's content, which often includes definitions of terminology, explanations of strategies, and the rationale for incorporating a process. Unit 5 Overview, Identify the Narrative, “Make 10 is a valuable strategy in the base-ten system because it allows students to work flexibly with numbers to solve more challenging problems by breaking them down into easier problems that they can solve fluently. The building blocks for the make ten strategy are built in Kindergarten, as students become familiar with number partners for numbers 1-10, decompose teen numbers into a group of ten and some more ones.”

• The Unit Overview includes an Appendix titled “Teacher Background Knowledge,” which includes a copy of the relevant pages from the Common Core Math Progression documents which includes on grade-level information.

Materials contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. Examples include:

• Unit 1 Overview, Counting, Linking, “Continuing through the rest of elementary school, students will use the counting sequence in all grades. In 2nd grade they’ll be using the counting and place value patterns to count to 1,000 and add and subtract within 1,000. This becomes fluent in 3rd grade. By fourth grade, they’ve generalized the counting and place value patterns to all numbers and can add and subtract any size of number.… In second grade, students will expand the counting sequence to 1,000. Many students will be able to count to 999 by the end of this unit even if they have not actually done it, based on their understanding of the repeating 0-9 pattern within 100 and the repeating 1-99 pattern after 100; therefore, this will be an easy next step for them. They will also add and subtract within 1000, applying the strategies acquired in this unit and Unit 3, with an emphasis on using a number line. Addition and subtraction within 1000 becomes fluent in third grade and by fourth grade, students have generalized the counting and place value patterns to all numbers and can add and subtract a number of any magnitude.”

• Unit 6 Overview, Two Digit Numbers, Linking, “By third grade, students use their understanding of place value to round numbers to the nearest ten or hundred and to add and subtract fluently within 1000. They will work to understand how 10 hundreds becomes a thousand. Third graders also begin to multiply one digit whole numbers by multiples of 10. In fourth grade, students come to recognize that in any multi-digit whole number, a digit in one place represents ten times what it represents in the digit to its right. (For example, in 888, the 8 in the hundreds place is worth ten times the 8 in the tens place.) They read and write multi-digit whole numbers using base ten numerals, number names, and expanded form (building on the expanded notation they learn in first grade), and they compare any two multi-digit numbers based on values of the digits in each place (as they did in first grade with two-digit numbers). Fourth graders also use place value understanding to fluently add and subtract using the standard algorithm and to multiply and divide using equations, arrays, and/or area models.”

• Unit 8 Overview, Measurement, Linking, “Moving into second grade, students begin to use standard units of measurement such as rulers, yardsticks, meter sticks, and measuring tapes to measure and estimate length. They relate the length of a unit of measurement to the length of the object being measured with that unit. (For example, students recognize that a table would be more inches long than feet because inches are shorter than feet.) Second graders also build on the compare work they did in first grade to determine how much longer one object is than another, expressing the difference in terms of a standard length unit. By third grade, students use rulers marked with halves and fourths of an inch to gather measurement data. They also begin to measure area and relate that measurement to multiplication and division, and estimate and measure volume and mass. In third grade, students solve problems related to these attributes for the first time. In fourth grade, students know the relative sizes of measurements within a system of units (i.e.- kilometers are larger than meters, which are larger than centimeters). They can convert units within a single measurement system (i.e. – meters to centimeters) and measure to the nearest eighth of an inch.”

The materials reviewed for Achievement First Mathematics Grade 1 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 1, 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. Examples include:

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 8, Lesson 7, Narrative, “What do students have to get better at today? As a result of this lesson, students can order 3 objects that cannot be directly compared by applying what they have learned in the previous lessons about measuring with precision to measure each object, and comparing the measurements. What is new and/or hard about that? Students are familiar with ordering objects by length when they are able to compare them directly (lesson 1). They also are familiar with comparing numerals to see which is greater or less for Kindergarten (K.CC.7). Today, students combine the strategies they use to order objects (asking which is longest and shortest and in between).”

• 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 8, standards addressed are 1.MD.1 and 1.MD.2. Previous Grade Level Standards/Previously Taught & Related Standards include 1.OA.1, 1.OA.2, K.MD.2, K.MD.1. An example grade level enduring understanding is, “The whole numbers are in a particular order that represents their magnitude. There are patterns in the way we say and write the numbers.” An example for what it looks like in this unit is, “Students apply the counting sequence when measuring by counting the number of units to determine length. Students also relate counting to addition when they use count up as a strategy for finding the difference in lengths.”

The materials reviewed for Achievement First Mathematics Grade 1 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 1 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 4, Lesson 4, Lesson Overview: “Materials: Graph packets (1 per student, double-sided), Data vocab poster, Intro bar graph, and Rapid Feedback tracker.”

• Unit 8, Lesson 5, Lesson Overview: “Materials: 9cm strip of paper, worksheet packets, measurement VA, rapid feedback tracker.”

The materials reviewed for Achievement First Mathematics Grade 1 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 3 Overview, Unit 3 Assessment: Story Problems 1, denotes the aligned grade-level standards and mathematical practices. Question 2, “Mr. Alese has 6 ties. 2 are red and the rest are blue. How many blue ties does he have?” (1.OA.1, MP1, MP2, MP4, MP5)

• Unit 6 Overview, Unit 6 Assessment: Two-Digit Numbers 1, denotes the aligned grade-level standards and mathematical practices. Question 14, “Write the symbol to show which number is greater/less than/equal to. 73 ____ 98” (1.NBT.3, MP2, MP4, MP7)

• Unit 8 Overview, Unit 8 Assessment: Measurement, denotes the aligned grade-level standards and mathematical practices. Question 6, “How many paperclips long is the toy car? ____ paper clips long?” A picture of a car above five paper clips is provided. The car spans from the second to the fourth paper clip. (1.MD.2, MP5, MP6)

The materials reviewed for Achievement First Mathematics Grade 1 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 1, 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, 1.OA.2, “If student does not understand the addition equation: Review the meaning of the symbols. See K, Unit 6, Lessons 1, 2, and 10. Review p/p/w relationships. See lessons 1-8 of this unit.”

• Unit 6 Overview, Unit 6 Assessment: Two-Digit Numbers 1, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 1.NBT.4, “If student does not understand how to apply the strategy they chose: See lessons 9-11 and lessons 15-16. Move back on CPA continuum of strategies. Encourage students to use concrete models and/or pictures of the tens.”

• Unit 8 Overview, Unit 8 Assessment: Measurement, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 1.MD.1, “If student is not attending to precision: Lead students in misconception protocol. Show student two different answers (one showing longest to shortest and other showing shortest to longest) to same question. Ask students to figure out which is correct and how they know. Once students identify the correct solution, look back at the error and discuss: What error did the incorrect student make? How can they avoid that error in the future? What did this mistake teach us about ordering objects by length?”

The materials reviewed for Achievement First Mathematics Grade 1 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 1.OA.1 (use addition and subtraction within 20 to solve word problems) is met. Item 1, “Maya had 3 books. Sean had 5 books. How many books did they have in all?” There are 12 available items, varied addition and subtraction situations, and space is provided for students to use objects, drawings, and equations to solve.

• In the Unit 6 assessment, the full-intent of standard 1.NBT.2 (understand that the digits in a 2-digit number represents tens and ones) is met. Item 3, “Which is not the same as 50 + 2? (multiple choice item with answer choices: 5 ones, 2 tens; quick tens image of 5 tens and 2 ones; 5 tens, 2 ones; 52).”

• In the Unit 7 assessment, the full-intent of standard 1.G.3 (partition circles and rectangles into two and four equal shares) is met. Item 2, “Which circle is partitioned into fourths? (MC item with four possible choices to choose from).” Item 7, “Show all of the ways that you could shade in half of the rectangles.” Eight blank rectangles are provided.

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

• Unit 3 Assessment, Item 11, supports the full development of MP2: Reason abstractly and quantitatively. “Which equation could you use to represent the problem: My teacher gave me 6 star stickers, 5 hear stickers, and 2 smiley face stickers. How many stickers do I have all together? 3 + 2 + __ = 5; 5 + 3 + 1 __; 3 + 5 + 2 = __; __ + 3 + 5 = 2.”

• Unit 5 Assessment, Item 9, supports the full development of MP3: Construct viable arguments and critique the reasoning of others. “Students were writing number sentences to figure out how many balloons and how many hats there were. (image of 3 balloons and 6 hats) Monica wrote 3 + 6 = 9. Terry wrote 3 + 6 = 9. a) Who is correct” (Answer choices - Monica only; Terry only; Both Monica and Terry; Neither Monica nor Terry) b) Why?”

• Unit 6 Assessment, Item 3, supports the full development of MP7: Look for and make use of structure. Assessment Item 3, “Which is not the same as 50+2? (Answer choices-5 ones, 2 tens; image of 5 lines to represent tens and 2 dots to represent ones; 5 tens, 2 ones; 52).”

The materials reviewed for Achievement First Mathematics Grade 1 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 1, 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.”

Examples of supports for special populations include: 

• Unit 1, Lesson 2, Workshop, Suggested intervention(s), “Students who are having difficulty recording numbers can use number lines or hundreds charts for reference.”

• Unit 4 Overview, Data, Differentiating for Learning Needs, “As children engage with making and interpreting graphs and other data representations for the first time in this unit, it is likely that they will bring a variety of experiences from kindergarten and preschool. While most students will be proficient in sorting into categories (K.MD.3), some students will have extensive experience with graphs and charts, while others will have none. Regardless of the experiences that children enter first grade with, teachers must meet their students where they are and ensure that all students are learning and deepening their understanding of the data concepts introduced in this unit. Teachers will need to know their students’ background knowledge and use that to differentiate both up and down while ensuring that students are all engaging in grade-level learning.” Suggested Interventions, “When students work on representing data sets, provide them with concrete data sets that they can sort into categories by moving the objects. 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, Fractions & Time, Differentiating for Learning Needs, “As children develop conceptual understanding of fractions and time in this unit, it is likely that they will bring a variety of experiences and knowledge from prior instruction. Some students will already be familiar with the language of fractions, especially halves and quarters and may intuitively know what they mean (though it is unlikely they will know their actual definitions). For others, halves and quarters/ fourths will be entirely new language. Similarly, some students will be familiar with clocks and telling time from home or math meeting, while others will have little to no experience with analog clocks. Regardless of the experiences that children enter the unit with, teachers must meet their students where they are and ensure that all students are learning and deepening their understanding of the mathematical concepts introduced in this unit. Teachers will need to know their students’ background knowledge and use that to differentiate both up and down while ensuring that students are all engaging in grade-level learning.” Suggested Interventions, “When showing the time on analog clocks by drawing in the hands, allow students to show it on a Judy clock or on their segmented paper clocks first and then copy that onto their papers. With a small group, have students fold shapes to partition or use scissors to actually cut shapes into halves. This will allow students to count their pieces and compare their sizes directly to ensure the correct number of parts and equal size.”

• 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 1 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 1, 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 5, Lesson 4, Workshop, Suggested Extension(s), “Push for use of known facts – students should always look for known facts first before attempting counting on. Some students may be able to use the associative property to derive known facts (ie – 5 + 5 + 6 is the same as

5 + 5 + 5 + 1 which is the same as 15 + 1).”

• Unit 8, Lesson 9, Workshop, Suggested Extension(s), “Challenge students to find the difference between the numerical measurements using strategies for comparing numbers. If they do this accurately and can explain how the strategies work, ask if they could represent it with an equation and why/how this works. (if the fish is 2 inches long and the dog is 5 inches long, I can find the difference by counting up from 2 to 5 because will tell me how many units longer the fish would have to be to be as long as the dog. I could write the equation 2 + 3 = 5 because if I start with a fish that is 2 inches long, and I added 3 more inches to its measurement, it would be as long as the dog which is 5 inches long.)”

• 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 1 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 1, 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 1, 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 6 Overview, Two-Digit Numbers 1, Differentiating for Learning Needs, Supporting MLLs/ELLs, Mathematical Language Routines, “8 mathematical language routines are outlined in detail in the Implementation Guide for Grade 1. These routines are worked into the lesson plans throughout the unit and explicitly highlighted for teachers in lessons 1, 8, 10, 13, 15, 18, and 21.. 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, MWI, and discussion. Over the course of the lesson, students refine their understanding of the concept and the language they use to articulate that understanding as they engage in these successive turn and talks. Turn and talks in the introduction of most lessons are often broad and mutl-part. For example, they often ask students to articulate the answer, how they came to it, and why that works. Teachers can and should feel comfortable modifying these turn and talks to align with the Stronger and Clearer Routine by breaking the question into 3 separate turn and talks, using the what/ how/ why framework: First, ask students to articulate what the answer to the problem is, then how they figured it out, and finally, why that works. Scaffolds are also provided for teachers as well and can be used to break questions into smaller parts. If students need support answering the larger question, teachers can pose the scaffolds (which are intentionally sequenced) as turn and talks to help students refine their thinking and language. 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. All lessons include an error analysis option as a potential focus of the mid-workshop interruption and 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 can continue to work this routine into the math stories block. They may also incorporate it into discussions of the core lessons when extensions include story problems by providing pairs of students with opposite parts of the story problem; the pairs will have to work together, communicating important information, to solve.

  • 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 work this into lesson extensions in the discussion by asking students to apply the learning in a new way by creating their own problems for partners to solve. Teachers may also continue 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.

  • 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 continue to work this routine into the math stories block and any other time MLLs/ ELLs work with story problems, including during the discussion when students engage in application problems together and during task-based lessons.

  • 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 each lesson as they connect the different focal strategies of the lesson. Several questions are scripted into each lesson’s introduction and then again in the second bullet of the MWI and 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 continue to build habits of discussion in this unit. Continue to prompt for students to engage in discourse by agreeing/disagreeing, building off one another’s thinking, and asking clarifying questions. Teachers should reinforce key vocabulary through the use of choral response. Teachers also support language development through this routine when they show place value concepts in different ways to build understanding, especially with concrete objects and pictures.”

• 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: “Place value: the value of the digit based on where the digit is in the numeral (1 is worth 1 unit if it is in the ones place, but it is worth 1 ten if it is in the tens place); Digit: a special symbol used to make a number; 0, 1, 2, 3, 4, 5, 7, 6, 7, 8, and 9 are all digits; 2-Digit Number: a numeral made up of 2 digits.”

• Unit Sentence Frames/Starters: “Providing sentence frames and starters is helpful for cultivating conversation, particularly in lower elementary. 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 4+4,’ the teacher would give the cue for students to turn and then say, ‘I solved 4+4 by…’ before students begin talking.” Examples include: “Sentence Frames for lessons 7-17: Ten more/ less than ______ is ______. I know because ______. That works because ______. The total is ______. I know because ______. That works because ______. The difference is ______. I know because ______. That works because ______. I solved by ______. That works because ______. I solved by ______. I chose that strategy because ______.”

The materials reviewed for Achievement First Mathematics Grade 1 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 1, Lesson 2, to address standard 1.NBT.1(count to 120), the materials include bags of loose objects and a sorting mat. In Overview, Identify the Narrative, “Lesson 1 focuses on reviewing the kindergarten strategies of keeping track of the count, and then students quickly move into strategically grouping objects into sets of tens and extra ones in lesson 2.” Introduce the Math, “Yesterday we counted large groups of objects to figure out how many we had. Today, we are going to play Counting Jar again and look for even more efficient ways to figure out how many when we have a lot of something. Model the game but not the strategies. Step 1 says Take one bag. Step 2 says how many? How could we figure out how many?” Students share review strategies from the previous day: move and count and organize and count (line up/arrange in an array). The materials state, “We need a strategy to keep track. We can move them as we count them or we can line them up as we count them.”

• Unit 3, Lesson 1, to address standard 1.OA.5 (relate counting to addition and subtraction), the materials include bags of cubes with two colors. In the Overview, Identify the Narrative, “For all of these story problem types, students should be able to represent with cubes or another manipulative, a 1:1 picture, a 1:1 tape diagram and/or number bond, a numerical tape diagram and/or number bond, and an equation. Further, the student should be able to relate all of these representations to each other, articulating how they all represent the story.” Introduce the Math, “Students should have bags on carpet that match the teachers’ bags 1 and 2. Step 1: Pick a bag (Pick bag 1). Step 2: Represent parts and wholes.”