Kindergarten - Gateway 3

The materials reviewed for Achievement First Mathematics Kindergarten 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 3, Counting, Lesson 13 include:

• “What is new and/or hard about the lesson? This is the first lesson in which students create equivalent sets of greater than 15. Students who used matching one-to-one to create equivalent sets within 10 or 15 will find that this strategy is inefficient and prone to error with larger numbers. They will discover that counting out a set is a more reliable strategy. Today’s lesson requires students to keep track of information and the count – they must keep track as they count the first set, remember that total as they determine how to record it with a numeral, and then keep track of the count as they count out another set of that many, all the while applying the 1:1 and cardinal principles to ensure accuracy.”

• “Exemplar Student Response: “I used [move and count/ touch and count/ organize and count]. I [moved/ touched/ lined up and touched] each object as I counted it, and the last number I said was [x], so there are x objects. I created the same amount by [getting one cube to match every one of these cubes/ counting out cubes until I had x in all]. I know they are the same amount because they both are x objects.”

• “Note: Students may not suggest matching one-to-one, as they have been counting out sets in previous lessons. There is no need to push for matching as a strategy, as long as students are able to articulate that both sets are the same. (Counting out is the preferred strategy, as it is required by the standard.) If students want to do matching one to one, a student should be called up to the front to demonstrate as it will not be possible from rug.”

• “Potential Misconception: Students lose track of what has already been counted and end up double-counting or leaving some out (when counting to tell how many and/or when counting out an equivalent set).”

• “Mid-Workshop Interruption: What is the next level for the skill in the aim? What do you want most of your students to start doing? What is a major misconception that needs to be clarified? If > \frac{2}{3} of students are successfully counting, recording and creating equivalent sets, push students to all use counting out to build the equivalent set (as opposed to matching one-to-one). Students may count out as they match one-to-one to build understanding of equivalency.”

• “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: Lead a discussion around a major misconception OR students share work OR ask students to apply their learning in a new way. Use workshop data to determine the appropriate discussion path.”

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 3, Counting, Lesson 13 include:

• “What Key Points: We use a strategy to keep track when counting so that we only say one number for each object; the number we say last tells how many.”

• “How Key Points: Strategies for counting to answer how many: Move and Count: I can move each object as I count to keep track of which I have counted and which I still need to count. Touch and Count: I can touch each object as I count to keep track of which I have counted and which I still need to count. (Works best with small quantities or in conjunction with organize and count.) Organize and Count: I can arrange the objects into a line or array (or into strategic groups which unlikely at this point) and move or touch and count from left to right/ top to bottom to keep track of which I have counted and which I still need to count.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 content, often including definitions of terminology, explanations of strategies, and the rationale for incorporating a process. Unit 3 Overview, First Steps, “In everyday use, ‘to count’ has two meanings. It can mean to recite the whole number names in their right order, beginning at 1 (I can count to 20. One, two, three, four, …). It can also mean to check a collection one by one to say how many are in it (I counted and found there were 14 left). Key Understanding 1 focuses on the latter meaning. The former is an aspect of Key Understanding 4.”

• 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, Sorting & Counting, Linking, “Continuing through the rest of elementary school, students will use the counting sequence in all grades. In first grade, students will expand the counting sequence to 120 and beyond (1.NBT.1). Starting in 1st and definitely by 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 is fluent by 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.”

• Unit 6 Overview, Addition and Subtraction, Linking, “Looking ahead to the first grade, students will apply the skills acquired in this unit to solve addition and subtraction problems within a magnitude of 20. They will add and subtract multiples of ten by counting all, counting on, counting back, and possibly counting up by tens using concrete and pictorial representations of two digit-numbers. The foundations for relating counting to addition and subtraction (1.OA.5), understanding subtraction as an unknown addend problem (1.OA.4), and being able to determine unknowns in all 3 positions in addition and subtraction problems (1.OA.8) are all laid in this unit. This unit also introduces students to strategies (count all, count on, count back, count up) that they will continue to use throughout the rest of elementary school and tools they will also make use of over the course of the next several years (especially the number line). This unit also builds fluency with facts within 5, which will be expanded to facts within 10 in first grade (1.OA.6). By second grade students will be able to add and subtract numbers fluently within 100 (2.NBT.5), by third, within 1000 (3.NBT.2), and in fourth grade, students will be able to add and subtract any multi-digit numbers using the standard algorithm (4.NBT.4). This year and next, students will continue to use equations to represent addition and subtraction scenarios. Throughout the rest of elementary school, students will continue to work with story problems following the protocol taught and practiced in this unit. In first grade, they will apply their understanding of addition and subtraction to represent and solve add-to/ take-from change-unknown, put-together/take-apart addend-unknown and compare, difference-unknown story problems. In second grade, students will also master the start unknown, compare-bigger unknown-fewer, and compare-smaller unknown- more problem types, and they will begin to solve two-step story problems. They will continue to expand their bank of representation and solution strategies and work with larger magnitudes of numbers in all addition and subtraction story problems throughout both grade levels. In upper elementary, students continue to work with story problems, now including multiplication and division in third grade and multi-step with all four operations in fourth. While the language of the protocol changes slightly in the upper grades, the steps of visualizing, representing and retelling, and solving taught in first grade and introduced in kindergarten continue throughout elementary school.” 

• Unit 8 Overview, Measurement, Linking, “Looking ahead to the first grade, students will apply the skills acquired in this unit to solve addition and subtraction problems within a magnitude of 20. They will add and subtract multiples of ten by counting all, counting on, counting back, and possibly counting up by tens using concrete and pictorial representations of two digit- Numbers. The foundations for relating counting to addition and subtraction (1.OA.5), understanding subtraction as an unknown addend problem (1.OA.4), and being able to determine unknowns in all 3 positions in addition and subtraction problems (1.OA.8) are all laid in this unit. This unit also introduces students to strategies (count all, count on, count back, count up) that they will continue to use throughout the rest of elementary school and tools they will also make use of over the course of the next several years (especially the number line). This unit also builds fluency with facts within 5, which will be expanded to facts within 10 in first grade (1.OA.6). By second grade students will be able to add and subtract numbers fluently within 100 (2.NBT.5), by third, within 1000 (3.NBT.2), and in fourth grade, students will be able to add and subtract any multi-digit numbers using the standard algorithm (4.NBT.4). This year and next, students will continue to use equations to represent addition and subtraction scenarios. Throughout the rest of elementary school, students will continue to work with story problems following the protocol taught and practiced in this unit. In first grade, they will apply their understanding of addition and subtraction to represent and solve add-to/take-from change-unknown, put-together/take-apart addend-unknown and compare, difference-unknown story problems. In second grade, students will also master the start unknown, compare-bigger unknown-fewer, and compare-smaller unknown- more problem types, and they will begin to solve two-step story problems. They will continue to expand their bank of representation and solution strategies and work with larger magnitudes of numbers in all addition and subtraction story problems throughout both grade levels. In upper elementary, students continue to work with story problems, now including multiplication and division in third grade and multi-step with all four operations in fourth. While the language of the protocol changes slightly in the upper grades, the steps of visualizing, representing and retelling, and solving taught in first grade and introduced in kindergarten continue throughout elementary school.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 Kindergarten, 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 9, Lesson 1 Narrative, “How does the learning connect to previous lessons? What do students have to get better at today? This is the first lesson in the final Kindergarten math unit. This unit will require scholars to use all of the knowledge and skills they’ve acquired throughout the year to make sense of various story problems and solve them. In this problem, students will need to understand that a square has four sides, then add two numbers to find the total number of chairs, and then compare the total to 10 to be able to say whether there are enough chairs for the students.”

• 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 Grade K: Unit 9, the standards addressed are K.OA.2, K.OA, 1, K.OA.5. Previous Grade Level Standards/Previously Taught & Related Standards include K.CC.1, K.CC.2, K.CC.A.3, K.CC.4, K.CC.5, K.CC.6, K.CC.7, K.OA.3, and K.G.B.4. An example grade level enduring understanding is, “We can count a collection to find out how many are in it and use numbers to represent.” An example for what it looks like in this unit is, “Students will use counting to solve an assortment of story problems.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 Kindergarten 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 5, Lesson Overview: “Materials: pattern block puzzles, pattern blocks, puzzles! VA, math workshop rules VA, Attribute Blocks.”

• Unit 6, Lesson 9, Lesson Overview: “Materials: story problem steps poster, blown up intro problem, whiteboards/markers, and cubes/other manipulatives.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 1 Overview, Unit 1 Assessment: Counting, denotes the aligned grade-level standards and mathematical practices. Interview Item 3c, “T: (Arrange 10 cubes in a circle). How many cubes are there?” (K.CC.4, MP6)

• Unit 5 Overview, Unit 5 Assessment: Counting & Comparing, denotes the aligned grade-level standards and mathematical practices. Written Item 3, “Circle the number that is less.” The numerals 7 and 9 are written inside of two rectangles. (K.CC.7, MP2)

• Unit 7 Overview, Unit 7 Assessment: Counting, denotes the aligned grade-level standards and mathematical practices. Question 1, “Samuel had 7 markers. Some were red and some were green. How many red and green markers could he have? Show your work.” (K.OA.2, K.OA.3, MP1, MP2, MP5)

The materials reviewed for Achievement First Mathematics Kindergarten 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 K, 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: Counting, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, K.CC.2, “Student needs repeated practice with counting on from a given number. Consider using these counting routines at school (during math meeting/practice time or at other ‘down’ times such as during transitions or when students are waiting in line to use the bathroom, for example) and at home: - Count around the Circle: Students make a circle and count in sequence with one student saying each number at a time as they move around the circle. The first student says, ‘1,’ the next says, ‘2,’ and so on and so forth. Be sure to start with different students each time or start at different numbers so that students get practice counting on from different numbers. - Start with/ Get to: Have students pick a number card to ‘start with’ and another to ‘get to.’ Have them count from their ‘start with’ card to their ‘get to’ card. - Provide hundreds charts and/or number lines for students to point to as they count. Explicitly model how to use these tools.”

• Unit 6 Overview, Unit 6 Assessment: Addition and Subtraction, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, K.OA.2, “If the student is choosing the correct operation but making calculating errors: Present the problem again and see if student makes the same error. (This will help you determine if this is a fluency/counting gap error or an error of precision.) If student makes the same error, address that error specifically through counting practice and provision of additional tools/aids (ie- a number line if student is struggling with stable order when counting). If student does not make the same error, show them their previous answer. Ask: Which is correct? How do you know? What can we do to make sure we don’t make a mistake like this again? Student should come up with a strategy to check his/her work. This may be counting twice to make sure he/she gets the same answer both times, or it may be solving again with a different strategy. Whichever strategy to check work the student comes up with, have the student create a visual anchor of the steps they will take to check their work for themself and attach it to their desk for reference. Make sure student articulates why checking their work is important.”

• Unit 8 Overview, Unit 8 Assessment: Two-Digit Numbers, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, K.NBT.1, “Student does not understand that the number of ‘extra’ ones is the number of ones that need to be added to the ten to make the teen number (and is represented by the digit in the ones place): See lessons 3-4. Explicitly model building the amount and counting the ‘extra’ ones only to determine how many “‘extra’ ones are in the teen number. As students develop better understanding of teen numbers, transition to using count up as a strategy to determine the number of extra ones. Explicitly model this first concretely by starting with a full tens frame and then counting up the teen number as you add ones. Once you reach the teen number, ask students which objects you added to the ten to make the teen number and have them count them to determine the number of extra ones. Capture the thinking steps visually: 1. Start with 10. 2. Add more until you reach teen number total. 3. See how many you added. Engage student or small group of students in guided practice by taking them through these steps with decreased scaffolding/ support. Once students are able to move through the steps independently, consider moving to pictorial strategies for counting up using the same thinking steps. If students transition well to pictorial, you may wish to model counting up on fingers and discuss how/ why this works, though it is perfectly okay if students continue to work with concrete/ pictorial strategies. See guidance above about supporting students in developing understanding of the meaning of the digits; the suggested chart is particularly helpful as it helps students visually see the pattern in the ones place as it relates to the number of extra ones.”

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

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

• In the Unit 4 assessment, the full-intent of standard K.MD.2 (directly compare two objects with a measurable attribute in common) is met. Item 1, “(Give the student the book and the pencil) Ask the students which item is shorter. Use the words longer/shorter. How did you figure that out? How did you know that?”

• In the Unit 5 assessment, the full-intent of standard K.CC.6 (tell whether the number of objects in one group is greater/less than/or equal to the number of objects in another group) is met. Item 1 compares numbers and groups within 10. Students are provided six colored tiles, 10 colored tiles, the number 4, and the number 7 and must tell which group has more. In item 2, students are given an image of six pizza slices in a row and two groups of ice cream cones in an array configuration (eight in one group and six in another group) and they must tell which group of ice cream cones that has more than the pizza slices.

• In the Unit 8 assessment, the full-intent of standard K.NBT.1 (compose and decompose numbers from 11-19 into ten ones and some further ones) is met. Item 1, “Which number sentence shows 14 as tens and ones? (MC-A. 7 + 7, B. 8 + 6, C. 10 + 4, D. 10 + 1)).” Item 2, “How many (image of a filled ten-frame and 2 dots outside of the ten-frame; MC-A. 10, B. 12, C. 11, D. 22).” Item 3, “Draw a picture and write a number sentence to show 17 as tens and ones.”  

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

• In the Unit 1 Assessment, Task 5, students engage with MP1: Make sense of problems and persevere in solving them. “Can you sort these blocks? How did you sort them?” 

• In the Unt 6 Assessment, Item 9, students engage with MP2: Reason abstractly and quantitatively. “There were 10 cupcakes on the table. Jamaine ate 4 cupcakes. How many cupcakes are on the table now?”  

• In the Unit 8 Assessment, Item 1, students engage with MP7: Look for and make use of structure. “Which number sentence shows 14 as tens and ones? (a. 7 + 7; b. 8 + 6, c. 10 + 4 ; d. 10 + 1).”

The materials reviewed for Achievement First Mathematics Kindergarten 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 K, 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 2, Lesson 3, Workshop, Suggested intervention(s), “Explicitly model referencing visual anchors to help you identify shapes and their names. Consider giving students their own set of mini-visual anchors to reference. When working with geoboards or straws/ popsicle sticks to build shapes, model thinking aloud about the number of sides as you select that many elastic bands or straws/ popsicle sticks to build. Then build the shape and model checking it by counting sides and corners. Engage the group in this same thought process through guided practice before they work independently. Because building and drawing shapes often requires fine motor skills, it can be frustrating for some students who may understand conceptually but struggle to build or draw a given shape. Support students behaviorally by explicitly modeling trying again or using strategies to cope with frustration. If needed, support with goals and rewards. For example, teachers may say something along these lines to support a student who is having difficulty and becoming frustrated when attempting to draw straight sides: ‘It may be hard for you to make a shape with 3 straight sides, but I know you can try your best. How many times do you think you should try before you ask for help? Let’s see if you can meet the goal of trying 3 times on your own first and taking a deep breath if you get frustrated. Every time you do that, you’ll earn a star. If you get 4 stars today, we can call your mom to tell her how hard you tried even when it was frustrating for you!’”

• Unit 4 Overview, Measurement, Differentiating for Learning Needs, “As children engage with measurement for the first time in this unit, it is likely that they will bring a variety of experiences from preschool and home. Some students will enter kindergarten with experience measuring the length, weight, and potentially even capacity of objects with nonstandard units, while others will have little to no experience with these measurable attributes. Regardless of the experiences that children enter kindergarten 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’ data and use that to differentiate both up and down while ensuring that students are all engaging in grade-level geometry.” Suggested Interventions, “When determining which measurable attributes can be compared, explicitly model thinking aloud how you know which attributes can be measured; capture thinking steps and then take the small group through them in guided practice. To ensure that students grasp the understanding that size does not equate with weight, provide students with objects with large variation in size AND weight where the lighter object is bigger. Some suggestions include poster paper and a laptop, tissue and rock, paperback big book and stapler, etc.”

• Unit 7 Overview, Two-Digit Numbers, Differentiating for Learning Needs, “As children begin to develop conceptual understanding of place value in this unit, it is likely that teachers will need to strategically differentiate instruction based on prior learning to ensure that all students are learning and deepening their understanding of the math. For most students, the concept of place value will be new, but some may have experience with place value understanding from experiences outside of school and for many place value will be intuitive; some students may already have developed some understanding of place value from their work with counting and writing numerals in math meeting, as well. 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, “Support conceptual understanding through the use of concrete objects and/or pictorial representations of teen numbers arranged onto tens frames. To build conceptual understanding, explicitly model counting and/or counting out towers of 10 cubes and extra single cubes by tens and ones, thinking aloud about why you are counting the towers by tens and the single cubes by ones. Use strategically designed recording tables to draw attention to the number of towers of ten and single cubes and how they relate to the digits of the two-digit number they represent.”

• 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 Kindergarten 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 K, 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 3, Lesson 4, Workshop, Suggested Extension(s), 1. “Challenge students to use more complex strategies to keep track of the count. Students may organize and skip count, for example. Be sure they can articulate how/why this works, particularly the concept that each object is counted once, even when counting two or more at a time.” 2. “Have students work with larger quantities within the range of the standard.”

• Unit 7, Lesson 5, Workshop, Suggested Extension(s), “If students find all of the totals, ask them to articulate how they know they found all of them. What could they start with to be sure they have them all? How would they know they’d gotten them all? Why does this work?”

• 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 Kindergarten 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 K, 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, Geometry, Differentiating for Learning Needs, Supporting MLLs/ELLs, Mathematical Language Routines, “8 mathematical language routines are outlined in detail in the Implementation Guide for Kindergarten. These routines are worked into the lesson plans throughout the unit and explicitly highlighted for teachers in lessons 1, 4, 7, and 11. 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. 

  • 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 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.The game in lesson 4 is essentially a version of this routine as one student has the information about the location of a shape and must describe it to the other using precise language so that they can draw it in the correct place. 

  • 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. 

  • 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 task based lessons and any extension problems that are story problems. 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 then again in the second bullet of the MWI and Discussion that ask students to consider how strategies relate to one another. 

  • 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 introduce and reinforce key vocabulary in this unit through the use of movements and repetition with choral response. Teachers continue to build habits of discussion in this unit. Prompt for students to engage in discourse by agreeing/ disagreeing with one another.” 

• 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: “Positional Words/Phrases – words that tell where an object is in relation to another object; Above* - a positional word that means over; Below* - a positional word that means under/underneath.”

• Sentence Frames: “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 1-3, I notice that this shape has _______. It is a _______. If needed: I notice that this shape has _______ straight sides and _______ pointy corners. This shape is a _______. I know because _______. If needed: This shape is a _______. I know because it has _______ straight sides and _______ pointy corners. I built /drew a _______ by building _______. If needed: I built/ drew a _______ by building _______ straight sides connected at _______ pointy corners.”

The materials reviewed for Achievement First Mathematics Kindergarten 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 3, Lesson 2, Workshop, students count to tell how many by using bags of 5 to 15 manipulatives (cubes or counters). Students play “Counting Bags/Jars,” where they “move and count, touch and count, or organize and count” the manipulatives and then “record with a number.”

• Unit 5, Lesson 5, Workshop, students use number cards to identify whether the objects in one group are greater than, less than, or equal to the number of objects in another group (K.CC.6). Students draw two cards from a deck, record the numbers, and then circle which number is the greatest. They can use a variety of strategies to determine the larger number including matching “non-linking manipulatives (teddy bears, pennies, counters, anything that doesn’t connect)” to each number.