The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for assessing grade-level content. Above-grade-level assessment questions are present but could be modified or omitted without a significant impact on the underlying structure of the instructional materials.

The series is divided into nine units, and each unit contains a Unit Assessment available online in the Unit Overview document and can also be printed for students. The Unit Assessments contain written and interview questions. Some units contain suggestions for use of Post-Unit Assessment questions as Pre-Unit Assessment questions. Teachers are directed to adjust instruction according to the Pre-Assessment results. 

Examples of assessment questions aligned to grade-level standards include:

• In Unit 2, Geometry Interview Questions, Task 1 states, “Put a triangle above a square and ask, ‘Where is the triangle in relation to the square?’” (K.G.1)
• In Unit 5, Counting & Comparing Unit 5 Assessment, Question 2 states, “Look at the slices of pizza, (picture of five slices of pizza). Circle the group of ice cream cones that has more than the slices of pizza, (picture of one group of eight ice cream cones and another picture of six ice cream cones).” (K.CC.6)
• In Unit 6, Counting Unit 6 Assessment, Question 9 states, “There were 10 cupcakes on the table. Jamaine ate 4 cupcakes. How many cupcakes are on the table now?” (K.OA.2)
• In Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 3 states, “Draw a picture and write a number sentence to show 17 as tens and ones.” (K.NBT.1)

There are examples of above-grade-level assessment questions. In Unit 8, four of the seven questions assess above-grade-level content. The Guide to Implementing AF Math: Grade K states, “Teachers should remove these items or use them for extension purposes only.” For example:

• In Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 4 states, “How many tens are in the number 37? A. 3 B. 37 C. 7 D. 73.” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 
• In Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 5 states, “Keisha drew sticks and dots to show how many blocks she had. How many blocks does Keisha have? A. 34 B. 7 C. 43 D. 44.” A picture representing 43 is between the question and answers. The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 
• In Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 6 states, “Gloria wants to draw a picture to represent the number 26 as tens and ones. What could she draw to show the number 26 as tens and ones?” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value. 
• In Unit 8, Two-Digit Numbers Unit 8 Assessment, Question 7 states, “Use sticks and dots to show 42.” The Unit 8 Scoring Guide identifies this as a Grade 1 standard, 1.NBT.2. However, K.NBT.1 requires students to work with numbers between 11-19 to gain foundations for place value.

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for spending a majority of instructional time on major work of the grade. 

• The approximate number of units devoted to major work of the grade, including assessments and supporting work connected to the major work, is 6 out of 9, which is approximately 67%.
• The number of lessons devoted to major work of the grade, including assessments and supporting work connected to the major work, is approximately 124 out of 163, which is approximately 76%. 
• The instructional block includes a math lesson, math stories, and math practice components. The non-major component minutes were deducted from the total instructional minutes resulting in 9,420 major work minutes out of 13,855 total instructional minutes. As a result of dividing the major work minutes by the total minutes, approximately 68% of the instructional materials focus on major work of the grade. 

A minute-level analysis is most representative of the instructional materials because the minutes consider all components included during math instructional time. As a result, approximately 68% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The publishers identify connections between supporting content and major work within the lesson plan in the “Standards in Lesson” section, as well as in the Guide to Implementing AF Math: Grade K. Additional connections exist within the materials, although not always stated by the publisher. In addition, the publisher identifies the CCSSM clusters at the top of each lesson plan. However, in some cases, supporting clusters are misidentified as major clusters. For example, in Unit 2, Lesson 3, the publisher incorrectly identifies the Geometry Clusters as the major work of the grade. Examples of the connections between supporting work and major work include:

• In Unit 1, Lesson 5, Introduction, students engage with the supporting work of K.MD.3, classify objects into categories, and the major work of K.CC.5, count to answer “how many” questions. In Step 4, students sort pattern blocks by shape, then answer, “How Many? We need to figure out how many are in each group.”
• In Unit 2, Lesson 3, Exit Ticket, students engage with the supporting work of K.G.2, correctly name shapes, K.G.5, model shapes in the world by building shapes from components, and the major work of K.CC.5, count to answer “how many” questions. Students count sides and vertices and build them using geoboards. Problem 3 states, “Circle the shape that has 4 corners.” Students are shown pictures of a hexagon, a circle, a triangle, and a square. 
• In Unit 2, Lesson 12, Understand: Introduce the Problem, students engage with the supporting work of K.G.4, analyze and compare two-dimensional shapes, and the major work of K.OA.2, solve addition word problems. Students are asked to visualize the shapes being mentioned. The teacher says, “Get ready to make a mind movie! Close your eyes and turn on your ears!” The teacher poses the problem by reading it 2-3 times, “Noah has three shapes. Noah has one square. Noah has one rectangle. Noah has one triangle. Noah counts all the corners of each shape. How many corners does Noah count all together? Show and tell how you know.” After the problem is read, students create a drawing of the three shapes based on their knowledge of their attributes. They then count the corners to add them, and represent the addition with an equation. In this lesson K.G.4 is the only standard identified, not K.OA.2.
• In Unit 4, Lesson 5, Introduction, students engage with supporting work of K.MD.2, compare two objects to see which holds “more of”/”less of” the attribute, and the major work of K.CC.6, count to determine which group holds more. Students play a game called “Which holds more?” where they compare two objects and determine which holds more scoops of rice. The teacher asks, “How can we figure out which object has a larger capacity or holds more?” Students might say, “We can put in scoops of rice and count each to compare.” 
• In Unit 9, Lesson 5, Workshop engages with the supporting work of K.MD.3, classify objects into categories, count the number of objects in each category, and sort the categories by count; and the major work of K.OA.3, compose and decompose numbers less than or equal to 10 in more than one way while finding multiple combinations of 10 pink and blue beads to make groups of 10. The Workshop worksheet states, “Introduction: Linda has pink beads and blue beads. Linda has some bags. Linda wants to put 10 beads in each bag. Some must be pink and some must be blue. How many different ways can Linda put pink beads and blue beads in bags? Show all of your mathematical thinking.”
• Kindergarten Practice Workbook A, students engage with the supporting work of K.MD.3, classify objects into given categories, and the major work of K.CC.5, count the number of objects to answer “How many?” Problem 1 states, “Color each group of 3.” The directions are followed by a picture of three rectangles, five triangles, three circles, and four squares.

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations that the amount of content designated for one grade-level is viable for one school year. The Guide to Implementing AF, Grade K includes a scope and sequence which states, “Not every lesson is entirely focused on grade level standards, and, therefore, some lessons can be used for either remediation or enrichment.” As designed, the instructional materials can be completed in 163 days. One day is provided for each lesson and one day is allotted for each unit assessment.

• Nine units with 155 lessons in total.
• The Guide to Implementing identifies lessons as either R (remediation), O (on grade level), or E (enrichment). There are 22 lessons identified as E (enrichment), 1 identified as R (remediation) and 132 identified as O (on grade level).
• Eight days for unit assessments. Unit 9 does not have a unit assessment. 

The publisher recommends 85 minutes of mathematics instruction daily.

• There are two lesson types, Game Introduction Lesson or Task Based Lesson. Each lesson is designed for 45 minutes. 
• Math stories are designed for 25 minutes. 
• Calendar/practice is designed for 15 minutes.

The instructional materials reviewed for Achievement First Mathematics Kindergarten partially meet expectations for being consistent with the progressions in the Standards. Overall, the materials do not provide all students with extensive work with grade-level problems. The instructional materials develop according to the grade-by-grade progressions in the Standards. Content from future grades is clearly identified and relates to the grade-level work. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier units, when appropriate, given that Kindergarten does not have a previous grade. Within the overview for each unit, there is an “Identify the Narrative” component, which provides a description of connections to concepts in prior units and future grade levels.

The lessons follow a workshop model, including a math lesson, math stories, and calendar/fluency. Most lessons do not provide enough opportunity or resources for students to independently demonstrate mastery. The lessons include teacher-directed problems that the class solves together. Math stories are intended to occur every day there is a lesson, however there are insufficient number of math stories for each lesson day. In addition, many practice workbook pages are repeated across multiple units.

The materials develop according to the grade-by-grade progressions in the Standards. The Unit Overview documents contain an Identify the Narrative component that looks back at previous content or grade level standards and looks ahead to content taught in future grades. In addition, the Linking section includes connections taught in future grades, units, or lessons. Evidence of prior and future grade-level work supporting the progressions in the Standards is identified. Examples include: 

• In Unit 1, Sorting and Counting Unit Overview, Identify The Narrative, Linking states, ”Looking ahead to the remainder of kindergarten, students will continue the counting sequence beyond 10 and up to 100. They will use the counting skills developed in this unit to develop strategies for addition and subtraction and to compose and decompose numbers within ten (K.OA.3) and into tens and ones, beginning with teen numbers (K.NBT.1) and then with all two-digit numbers (1.NBT.2). They will use their understanding of counting to compare sets (K.CC.6) and their place value understanding to compare two digit numbers. (1.NBT.3)”
• In Unit 2, Geometry Unit Overview, Identify The Narrative, Linking states, “Most importantly, it will help them access more complex geometrical standards in first grade in regard to distinguishing between defining attributes versus non-defining attributes, composing two-dimensional shapes to create a composite shape, creating new shapes from the composite shapes, and partitioning circles and rectangles. In second grade, students will need to draw shapes based on a given set of attributes; in third grade, students will focus on quadrilaterals and understand that a quadrilateral can also be categorized in a number of different ways; in fourth grade, students focus on points, line, ray, and parallel versus perpendicular lines.”
• In Unit 5, Counting and Comparing Unit Overview, Identify the Narrative, Linking states, “Students expand the counting sequence beyond 100 in first grade and begin to relate the way we say and write numbers to place value understanding.”
• In Unit 8, Two-Digit Numbers Unit Overview, Identify The Narrative, Linking states, “In first grade, students will use their understanding of place value to represent two-digit numbers in expanded notation and begin to add and subtract two-digit numbers.” 
• In Unit 9, Exemplars Unit Overview, Identify the Narrative, Linking states, “In second grade, students must understand the place value of three-digit numbers and use that understanding to add, subtract, and compare three-digit numbers, as they did in first grade with two-digit numbers. They will work to add and subtract within 1000 and fluently add and subtract within 100. In third grade, scholars are expected to fluently add and subtract within 1000 using place value strategies. Finally, in 4th grade, scholars are introduced to the standard algorithm for addition and subtraction.” A structural overview is under the Linking narrative. The image shows the progression of addition and subtraction through 4th grade.  

Overall, the materials do not provide all students with extensive work on grade-level problems, nor do the materials address the full intent of some standards. The majority of the lessons implement 45 minutes of math workshop with a whole group introduction, workshop in pairs or small groups, mid-workshop interruption, whole group discussion, and closing with an exit slip. As it is unclear if students are working together or individually, workshop lessons may not provide enough opportunity for students to independently demonstrate mastery. The Guide to Implementing AF, Grade K, describes the workshop component as, “Collaborative processing time to continue to develop understanding of prioritized concept and strategy.” The lessons include a teacher-directed introduction to the workshop “game” and follows up with students tasked to participate in the “game.” Most lessons include an Exit Ticket with one or two questions for the students to complete individually. 

Beyond the lesson component of the math time, the Guide to Implementing AF Math, Grade K suggests 15 minutes of daily calendar and practice. Each unit indicates the Grade K Practice Workbook pages to be implemented during this time. However, the Practice Workbook pages contain a limited number of practice items and are recommended to be used repeatedly in different units. As a result of the limited number of opportunities to practice grade-level standards, the materials do not give students extensive work with grade-level problems.

Examples where the full intent of a standard is not met and/or extensive work is not provided include:

• In Unit 2, Lesson 4, Introduction and Workshop Resources, students engage with K.G.1 as they describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms. Students play a game called “Where is my Shape?” where they pick a card that tells them to draw a shape above, below, beside, and next to a picture of a Ninja Turtle. However, the full intent of the standard is not met as the positional words “behind” and “in front of” are not included in this lesson. Students are also not provided with extensive work as Lesson 4 and 5 Practice Workbook problems are the only opportunities out of 155 lessons that address K.G.1. 
• In Unit 2, Lesson 6, Exit Ticket, students engage with K.G.6 as they compose 2D shapes out of other 2D shapes by noticing their attributes. Students are provided with only two exit tickets to independently practice the standard. For example, “Jordan filled his hexagon using all triangles yesterday. Today he started using triangles but then ran out. How can he finish his puzzle? Circle the shape that would fit.” Students are given a picture of a hexagon with three triangle inside forming a trapezoid. They are then provided with three multiple choice items with shape pictures: A. trapezoid B. square C. oval.
• In Unit 3, Lesson 8, Introduction and Workshop, students engage with K.CC.2 as they use the strategy of counting on from a given number. This is the third lesson out of four addressing this standard, and students are not expected to use the counting on strategy. As a result, the full intent of the standard is not met for all students as they may choose one of the other strategies to find the total. Students are also not provided with extensive work on K.CC.2, as it is addressed in four lessons out of 155 lessons and in two games to be implemented during a ten minute practice time. In Roll and Record, students are directed to roll the dot cubes and find the total. The directions for the game are to “Find how many dots: Count all - Touch and Count, Just see one group and Count on, and Just see the total.”
• In Unit 3, Lesson 30, Student Task Page, students engage with K.CC.1 as they count to 100 by ones and tens. It is a task-based lesson that asks students to count by tens. However, they are not required to count higher than 30. Problem 1 states, “Hector shows Maria his penny collection. Hector has 3 jars in his collection. Hector has 10 pennies in each jar, Mario says that Hector has 40 pennies. Is Mario correct? Show and tell how you know.” The materials do not ask students to count to 100 by tens. 
• In Unit 5, Lesson 2, Exit Slip, students engage with K.CC.6 as they compare greater than, less than, and equal to using objects as required by the standard. Students are provided with three problems on exit tickets (Lessons 2, 3, and 5). As a result, students are not provided extensive work independently practicing comparing groups of objects. In addition, the full intent of the standard is not met since students are not provided with any independent problems practicing finding “equal groups” of objects in the exit slips. Problem 1 states, “Circle the tower that has more cubes.” Students are provided with an image of towers of seven blue cubes and five red cubes.

The Unit Overview supports the progression of Kindergarten standards by explicitly stating connections between prior grades and current grade level work. Each Unit Overview contains an Identify the Narrative component that identifies connections to what students learned before entering school and concepts previously learned in Kindergarten. Each Unit Overview also contains an Identify Desired Results: Identify the Standards section that makes connections to supporting standards learned prior to the unit. Examples include: 

• In Unit 2, Geometry Unit Overview, Identify the Narrative states, “Coming into this unit, students use the informal language of their everyday world to name and describe flat shapes (rectangle, triangle, square, circle, hexagon) without yet using mathematical concepts and the vocabulary of geometry. At this stage, a figure is a square because it looks like a book; another figure is a circle because it is round like the wheel of a car. Students make these observations without explicitly thinking about the attributes or properties of squares and circles.”
• In Unit 3, Counting Unit Overview, Identify the Narrative states, “Up to this point in K, students have worked intensively within 10. They have counted sets of objects and pictures and written numerals up to 10. This unit will help students build on their knowledge of numbers within 10 and extend it to larger quantities.”
• In Unit 4, Measurement Unit Overview, Identify the Narrative states, “After two units of counting (students can now count groups and record numbers to 20), and one unit of Geometry where students observed, analyzed, composed, decomposed and classified objects by shape, students now compare and analyze length, weight, capacity. This unit supports students’ understanding of amounts and their developing number sense.”
• In Unit 5, Counting and Comparing Unit Overview, Identify The Narrative states, “Lessons 10-12 ask students to write out hundreds charts to 100. This relates back to the pattern between and within the decades that students discovered in Unit 3. The kindergarten standard is to orally count to 100, while writing numbers to 100 is a first grade standard. In Kindergarten students must be able to orally count to 100 and write numerals up to 20. When doing the hundreds chart activities, point students toward appropriate resources in order to master the writing of numbers to 100 while allowing ample time for students to practice counting their strips and hundreds charts aloud.”
• In Unit 6, Addition and Subtraction Unit Overview, Identify Desired Results: Identify the Standards, K.OA.1, K.OA.2, and K.OA.3 (all are addition and subtraction standards) are identified as the standards to be learned in Unit 6. The previous kindergarten standards identified as foundational are counting standards, K.CC.1, K.CC.2, K.CC.4, K.CC.5, K.CC.6, and K.CC.7. 
• In Unit 8, Two-Digit Numbers Unit Overview, Identify Desired Results: Identify the Standards, K.NBT.1 (Compose and decompose numbers from 11 to 19 into ten ones and some further ones) is identified as one of the standards to be learned in Unit 8. The previous kindergarten standards identified as foundational are the counting standards (K.CC.1, K.CC.2, K.CC.3, K.CC.4) and K.OA.1 (Represent addition and subtraction).

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. The publisher identifies the CCSSM clusters at the top of each lesson plan. However, in some cases, supporting clusters are misidentified as major clusters. For example, in Unit 2, Lesson 3, the publisher incorrectly identifies the Geometry Clusters as the major work of the grade. 

The materials include learning objectives, or Aims, that are visibly shaped by CCSSM cluster headings. Examples include: 

• In Unit 2, Lesson 2, Aim is shaped by K.G.A, identify and describe shapes and K.G.B, analyze, compare, compose, and create shapes. The materials state, “SWBAT identify 2D shapes (triangle, square, rectangle, circle, hexagon) by noticing their attributes (sides and corners). SWBAT create 2D shapes (triangle, square, rectangle, hexagon) by noticing their attributes.”
• In Unit 3, Lesson 5, Aim is shaped by K.CC.B, count to tell the number of objects. The materials state, “SWBAT represent a quantity 5-15 pictorially by using a strategy to keep track of the count.”
• In Unit 5, Lesson 3, Aim is shaped by K.CC.C, compare numbers. The materials state, “SWBAT determine if groups are equal or which group has more and which has less by arranging the cubes (building towers or matching 1:1) and comparing directly.”
• In Unit 6, Lesson 2, Aim is shaped by K.OA.A, understand addition as putting together and adding to, and subtraction as taking apart and taking from. The materials state, “SWBAT find the total of two groups of objects and represent with an addition equation.”
• In Unit 8, Lesson 6, Aim is shaped by K.NBT.A, work with numbers 11-19 to gain foundations for place value. The materials state, “SWBAT compose teen numbers by looking at a group of ten ones and some more ones and using a strategy that works for them (count all, count on, just know).” 

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. The publisher identifies the CCSSM Clusters at the top of each lesson plan. However, in some cases, supporting clusters are misidentified as major clusters. For example, in Unit 2, Lesson 8, the publisher incorrectly identifies the Geometry clusters as the major work of the grade. Examples of connections include:

• In Unit 3, Lesson 24, Exit Ticket, students engage with K.CC.B, count to tell the number of objects, and K.NBT.A, work with numbers 11-19 to gain foundations for place value, as they count a number represented in two ten frames. Problem 1 shows two tens frames with ten and six ones and states, “How many are there?”
• In Unit 2, Lesson 8, Workshop, students engage with K.G.A, identify and describe shapes, and K.G.B, analyze, compare, create, and compose shapes, as they match the faces of 3D solids to 2D shapes. The Student Workshop Worksheet states, “Makkelle had a can of soup. She wanted to put a label on it that would cover the whole top face. What shape would the label be? A. (insert a picture of a rectangle) B. (insert a picture of a circle) C. (insert a picture of a triangle).” Students are provided with a cylinder to represent the can of soup.
• In Unit 8, Lesson 7, Workshop, Exit Ticket, students engage with K.NBT.A, work with numbers 11-19 to gain foundations for place value, and K.OA.A, understand addition as putting together and adding to, and understand subtraction as taking apart and taking from, as they make drawings to decompose teen numbers into ten ones, and some more ones. Problem 1 states, “Write a number sentence to show the number 13 as a group of ten ones and some more ones.”
• Practice Workbook B, students engage with K.G.A, identify and describe shapes, and, although not stated, K.G.B, analyze, compare, create, and compose shapes, as they draw shapes in relation to one another. Problem 2 states, “Look at the star. Draw a circle below the star. Draw a triangle above the star. Draw a rectangle next to the star.”

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include: 

• In Unit 2, Lesson 5, Introduction, students engage with K.G.6, compose simple shapes to form larger shapes, as they complete puzzles using geometric shapes. The materials state, “Step 1 says I’m going to pick a puzzle. Step 2 says I need to Decide what shape might fit. T & T: How can I make sure that happens? Strategy 1: Keep trying shapes until one fits the space. Strategy 2: Look at the space you are trying to fill. What shape might fit because of its attributes? Then find the shape that has the same attributes. Remember! You can flip and turn the pattern blocks. What shape do you think would fit? Why did you pick that shape?”
• In Unit 3, Lesson 2, Introduction and Workshop, students engage with K.CC.4, demonstrate understanding of the relationship between numbers and quantities, as they play “Counting Bags/Jars.” Students count the number of pattern blocks in the bag and then show the same amount using cubes. During the Workshop the teacher asks students, “How do you know this is the same amount / how are you showing the same amount?”
• In Unit 5, Lesson 6, Introduction, students engage with K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, as they play a game called Compare. During the Introduction, two cards are drawn (example, 7 and 9) and students are asked to pictorially show which is more or less by drawing circles on their whiteboards. The teacher asks, “How do you know from the picture?” A sample student response might be, “I know because in the picture you can see that there are extra circles in the row of 9 and the row of 7 is missing some.” 
• In Unit 6, Lesson 3, Introduction, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as the students complete a dice game while the teacher checks for understanding with questions leading the students to describe their thinking. Workshop states, “Step 1: Roll 2 cubes; record their amounts. (“roll” a 4 and a 2) Show first cube: How many? 4 (record) Show second cube: How many? 4 (reccord) Show second cube: How many? (give time to count as needed) 2 (record) Lap 2: Conceptual: Which strategies are kids using? What misconceptions are arising? Check for Understanding: How did you solve? Why does that work? How does your equation match what you did?”
• In Unit 8, Lesson 2, Workshop, students engage in K.NBT.1, compose and decompose numbers 11 to 19 into ten ones and some further ones, as students bundle objects into a group of ten and count on to determine the number of objects in a bag. The teacher is given suggestions for guiding the students to develop the concept of teen numbers. The materials state, “What did you notice about the group of ten ones, loose ones and the way we write the numbers?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: 

• In Unit 6, Lesson 10, Exit Slip, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situation, verbal explanations, expression, or equations, as they find the difference between two numbers using manipulatives and represent with an equation. Students are directed, “Use your counters and tens frames to find the difference. Fill in the equation to show what you did.” Students are provided with the digits, 8 and 5, and given a blank equation to fill in. 
• In Unit 7, Lesson 6, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they look at a picture of a rekenrek and find another equation that is equal/the same. Problem 1 states, “Look at the picture and equation in box 1. Use your teddies to write another equation that is equal/the same.” In box 1, there is a picture of a rekenrek with 4 on the top and 1 on the bottom and the matching equation, $$4 + 1 = 5$$. 
• In Unit 8, Practice Workbook G, students engage with K.NBT.1, compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, as they independently draw pictures to show the decomposition of the number 18 into ten ones and 8 more ones. Problem 6 states, “Draw a picture to show 18 as ten ones and some more ones. Write a number sentence to match.”
• In Unit 9, Practice Workbook F, students engage with K.OA.4, by finding the number that makes 10 for any number 1 to 9 by using objects or drawings and recording the answer. Problem 3 states, “Draw circles and write a number to show how many more are needed to make 10.” Students are given 2, 4, 7, 6, 3, 1, 8, 9, and 5.

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The instructional materials develop procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• In Unit 3, Lesson 12, Workshop Worksheet, students engage with K.CC.5, count to answer “How many?” questions about 20 things arranged in a line, a rectangular array, or a circle, as they count 18 pieces of silverware arranged in a rectangular array. The materials state, “Mr. Lohela was having a dinner party. He set out the silverware. How many pieces of silverware did he set out?”
• In Unit 6, Lesson 19, Introduction, students engage with K.OA.5, fluently add and subtract within 5, as they represent a story problem. The materials state, “Step 1: Visualize. Make a mind movie while I read. There were 7 carrot sticks on Hubina’s plate. She ate 3 of them. How many carrot sticks are on her plate? Step 2: Represent and Retell. Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.”
• In Unit 7, Practice Workbook E, Making 3, 4, and 5 finger Combinations, students engage with K.OA.5, fluently add and subtract within 5, as they play a game to develop fluency within 5. The materials state, “The teacher uses different finger flashes and students determine how many fingers are needed to make a target sum.” Once students understand the game, they play with a partner. 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they use fingers to calculate the missing addend. For example, “Activity: Making 3, 4, and 5 Finger Combinations. T: I’ll show you some fingers. I want to make 3. Show me what is needed to make 3. (Show 2 fingers.) S: (Show 1 finger.)”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include but are not limited to: 

• In Unit 3, Practice Workbook C, students engage with K.CC.3, write numbers from 2 to 20, as they independently complete a number sequence filling in missing numbers from 10 -15. Problem 5 states, “Fill in the missing numbers, “10, 11, ____, ____, ____, ____.”
• In Unit 6, Lesson 22, Assessment, students engage with K.OA.5, fluently adding and subtracting within 5, as they complete equations. Problem 5 states, “Solve. $$2 + 3 =$$ ___.”
• In Unit 7, Lesson 8, Exit Slip, students engage with K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, as they independently create equations with a sum of 10. The materials state, “Show all of the ways you could make 10. (You may not need to fill in every equation.)” Blank equations equalling 10 follow the directions. 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently add and subtract within 5, as they independently solve a series of addition problems with a sum of 2-5. Problem 1 states, “$$3 + 2 =$$____”

The instructional materials for Achievement First Mathematics Kindergarten partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The series includes limited opportunities for students to independently engage in the application of routine and non-routine problems due to lack of independent work during Workshop, and lack of non-routine problems. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher led questions and partner work. According to the Guide to Implementing AF Grade K, “Task Based Lesson Purpose: Students make sense of the mathematics they’ve learned by  working on a problem solving task and leveraging the knowledge they bring to math class to apply their math flexibly to non-routine, unstructured problems, both from pure math and from the real world. To shift the heavy lifting to scholars.” However, most of the task based lessons are considered enrichment and teachers may opt to not incorporate these non-routine opportunities into their math lessons. In addition, the task based lessons are not independent as they “encourage discussions between students about alternate methods or possibly incorrect solution paths.”

Routine problems are found in the Independent Practice and Exit Tickets/Slips. For example:

• In Unit 3, Lesson 6, Exit Ticket, students engage with K.CC.2, count forward beginning from a given number within the known sequence, as they independently use a counting on strategy to add numbers on two dice. Problem 1 states, “Use a strategy to find the total. Write the total on the line.” Students are shown dice with six and three dots, respectively.
• In Unit 6, Lesson 19, Exit Slip, students engage with K.OA.2, solve addition and subtraction word problems and add and subtract within 10, as they solve a story problem. The materials state, “There were 10 scholars in the lunch line. 7 scholars got their food and sat down. How many scholars are in the lunch line now? Represent, solve, and write a number sentence.” 
• In Unit 7, Lesson 5, Exit Slip, students engage with K.OA.2, solve addition and subtraction word problems within 10, as students calculate take apart problems with both addends unknown. The materials state, “There are 8 kids on the bunk bed. Show as many ways they can be arranged on the top and bottom as you can.” Nine blank equations are provided for students, “___ $$+$$ ___ $$= 8$$.”
• In Unit 7, Lesson 9, Understand: Introduce the Problem, students engage with K.OA.2, solve addition and subtraction word problems, and K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, by following the story problem protocol and using an efficient strategy to find all of the solutions. The materials state, “The grocer got another size box! He now has a box that holds exactly 9 apples. He has red and green apples that he needs to put into the box. What are all of the ways he could put red and green apples into his box?” 

Math Stories provide opportunities for students to engage in routine applications of grade-level mathematics. Students engage with Math Stories for 25 minutes, five days per week. The Guide to Implementing AF Kindergarten page four states the purpose of Math Stories, “Purpose: 

• To enable students to make connections, identify and practice representation and calculation strategies, and develop deep conceptual understanding through the introduction of a specific story problem type in a clear and focused fashion with deliberate questioning and independent work time.
• To reveal and develop students’ interpretations of significant mathematical ideas and how these connect to their other knowledge.
• To shift the heavy lifting to scholars.”

Examples of routine Grade K Math Stories:

• In Unit 2, Guide to Implementing AF Math, Math Stories, November, students engage with K.OA.2, solve addition and subtraction word problems, as they solve routine put-together/take apart-total unknown word problems. Sample Problem 1 states, “5 red crayons and 5 green crayons were in the basket. How many crayons were in the basket?” 
• In Unit 3, Guide to Implementing AF Math, Math Stories, January, students engage with K.OA.2, solve addition and subtraction word problems and add and subtract within 10, as they complete math story problems. Sample Problem 2 states, “There were 4 peas and carrots on the spoon. How many of each could be on the spoon? (0 + 4 is not a solution.)”
• In Unit 4, Guide to Implementing AF Math, Math Stories, February, students engage with K.OA.2, solve addition and subtraction word problems within 10, as they calculate take apart problems with both addends unknown. Sample Problem 6 states, “Ms. Smith has 7 yellow and purple markers. How many markers could be purple and how many could be yellow?”
• In Unit 5, Guide to Implementing AF Math, Math Stories, March, students engage with K.OA.2, solve addition and subtraction word problems, as they solve a math story. Sample Problem 8 states, “There were 9 clovers in a field. Some were big and some were small. How many of each could there be? ($$0 + 9$$ is not a solution)”

Examples of Math Stories that go beyond the standard, K.OA.2, as they incorporate addition and subtraction beyond 10, and include problem types beyond Kindergarten expectations:

• In Unit 3, Math Stories, December, Problem 3 states, “(PT/TA-TU) 8 green and 11 red lights were on the outside of the house. How many lights were on the outside of the house? This word problem is above grade level as it goes beyond 10. 
• In Unit 5, Math Stories, March, Problem 11 states, “18 pieces of gold were in a leprechaun’s pocket. 9 pieces of his gold fell out. How many pieces of gold are still in the leprechaun’s pocket?” This word problem is above grade level as it goes beyond 10. 
• In Unit 7, Math Stories, May, Problem 9 states, “(C-DU-F) Ms. Smith has 10 erasers. Jose has 6 erasers. How many fewer erasers does Jose have than Ms. Smith?” This word problem goes beyond Kindergarten problem types as it is a compare problem.

The instructional materials for Achievement First Mathematics Kindergarten meet expectations that the materials reflect the balance in the standards and help students meet the standards’ rigorous expectations by helping students develop conceptual understanding, procedural skill and fluency, and application. The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the program materials. For example:

Conceptual understanding

• In Unit 3, Lesson 16, Exit Slip, students engage with K.CC.5, count to answer “how many” questions about as many as 20 things, as they represent a quantity 10 -20 pictorially by using a strategy to keep track of the count. The Exit Slip shows the number 16 with two blank ten frames. Students are expected to draw circles on the ten frame to represent 16.
• In Unit 6, Lesson 11, Exit Ticket, students engage with K.OA.1, represent addition and subtraction with objects, fingers, mental images, drawings, sound, acting out situations, verbal or equations, as they represent and solve subtraction problems while using counters and tens frames. The materials state, “Use your counters and tens frames to solve. $$7 - 3 =$$____ and $$9 - 5 =$$ _____” 
• In Unit 9, Practice Workbook F, students engage with K.OA.3, decomposing numbers less than or equal to 10 into pairs in more than one way, as they draw pictures to show more than one way to make each number. Problem 4 states, “Draw a picture to show 2 ways to make each number. 6; 4; 7; 6; 3; 1; 8; 9; 5.”

Procedural skills (K-8) and fluency (K-6)

• In Unit 6, Lesson 5, Exit Slip, students engage with K.OA.5, fluently add and subtract within 5, as they are given an image with two numbers to add them together, and a spot for an equation. The materials state, “Cube 1 (6) Cube 2 (3) Equation _____ + _____= _____.” 
• In Unit 7, Practice Workbook E, Shake and Spill, students engage with K.OA.5, fluently add and subtract within 5, as they spill five two-sided counters in a cup to find combinations of 5. The materials state, “The students determine how many of each color is showing and record the sum using drawings or equations. The students should ‘shake and spill’ several times to show different pairs of numbers that sum to 5.” 
• In Unit 8, Practice Workbook E, students engage with K.OA.5, fluently adding and subtracting within 5, as they solve put together and take apart problems with the result unknown within 5. Problem 8 states, “$$2 + 2 =$$ ___.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level. All MPs are clearly identified throughout the materials, with few or no exceptions. However, there are inconsistencies between the identified MPs in the Unit Overview and the MPs identified in the Lesson Plans. The materials provide little direction as to how the MPs enrich the content and do not make connections to enhance student learning. The MPs are not treated separately from the content.

Evidence that all eight MPs are clearly identified throughout the materials, with few or no exceptions, though they are not always accurate. For example:

• In the Unit Overviews, the bolded MPs are the Focal MPs for the unit.
• In Unit 3, Unit Overview, Standards for Mathematical Practice identifies MP8, look for and express regularity in repeated reasoning, as embedded in the counting lessons of Unit 3. 
• In Unit 7, Unit Overview, Standards for Mathematical Practice identifies MP4, model with mathematics, as embedded in the compose and decomposing lessons of Unit 7. 
• The MPs are listed at the beginning of each lesson in the Standards section. For example, in Unit 2, Lesson 3, the following MPs are identified as in the lesson: MP 1, MP 3, MP 5, MP 6 and MP 7. 
• The Mathematical Practices are not always identified accurately. For example: 
• At the unit level for Unit 2, MP 3 is not identified as a focus MP. However, at the lesson level, 10 out of 12 lessons identify it as connected. At the unit level, MP 5 is listed as a focus, but it is only connected to five of the twelve lessons. 
• In Unit 5, MPs 2 and 8 are bolded in the Unit Overview. However, MP 3 is connected to all 12 unit lessons. MP 8 is only connected to 3 of 12 lessons. 
• In the Unit 6 Overview, MPs 1, 2, 3, 4, 5, and 6 are bolded as the focus MPs for the unit. However, at the lesson level MP 1, MP 3, MP 4 and MP 6 are not connected to any of the lessons. MP 5 is identified as connected to 19 out of 21 lessons. 
• All MPs are represented throughout the materials, though lacking balance. For example, MP 8 is the focus of two units, while MP 5 is the focus of seven units. 
• There are no stated connections to the MPs to the Math Stories component, Math Practice component, or Assessments. 

There are instances where the MPs are addressed in the content. However, these connections are not clear to the teacher.

• It is left to the teacher to determine where and how to connect the emphasized mathematical practices within each lesson. 
• There are connections to the content described in the Unit Overview. However, if a teacher is not familiar with the MPs, the connection may be overlooked as there are no connections within the specific lesson content to any MPs. Examples include: 
• Unit 6, Unit Overview, Standards for Mathematical Practice, identifies MP 6, attend to precision, as embedded in the addition and subtraction lessons of Unit 6. “Students utilize one-to-one-correspondence when counting, regardless of the orientation of objects/pictures. Students must also communicate precisely when they describe their representations and calculation strategies.” Note: Although MP 6 is identified as a focal practice standard in the Unit Overview, none of the Unit 6 lessons identify MP 6 as connected to lesson content. 
• Unit 8, Unit Overview, Standards for Mathematical Practice, identifies MP 7, look for and make use of structures, as embedded in the two-digit numbers lessons of Unit 8. “Students are introduced to the place value chart, deepening their understanding of the structure within our number system.”

The materials reviewed for Achievement First Mathematics Kindergarten partially meet expectation for meeting the full intent of the math practice standards. The Mathematical Practices (MPs) are represented in each of the nine units in the curriculum and labeled on each lesson. Math Practices are represented throughout the year and not limited to specific units or lessons. The materials do not attend to the full meaning of MPs 1 and 5.

The materials do not attend to the full meaning of MP1 because students primarily engage with tasks that replicate problems completed during instructional time. Examples include:

• In Unit 3, Lesson 8, Narrative states, “In this lesson, students continue to figure out the total of two groups that they get from rolling dot cubes/dice and write the numeral in the corresponding space on their game board.” Introduction states, “How are you going to figure out how many today? SMS: I’m going to figure out how many dots by touching and counting all the dots, pointing and counting, looking and counting, counting on.”
• In Unit 7, Lesson 5, students are finding all the ways that eight kids could be arranged onto the top and bottom bunk. Narrative states, “In the previous lesson, students visualize, represent, and find some but not all solutions. They understand that they need to maintain the total of 8 and began to discuss the strategy of compensating to do so. Today students solidify their understanding of maintaining a total through compensation. This is challenging because students need to reason about the amounts to recognize that the total stays the same without recounting.” 
• In Unit 9, Lesson 8, Understand: Introduce the Problem, Pose the Problem states, “Mom has 50 pennies. Mom has 4 cups. Mom asks Lydia to put 10 pennies in each cup. When Lydia is done she looks in the bowl. She sees some pennies are still in the bowl. How many pennies are still in the bowl? Show and tell how you know.”

The materials do not attend to the full meaning of MP5 because students do not choose their own tools. Examples include:

• Unit 1, Lesson 15, Materials, The materials include dot cubes and recording sheets. Students are not given tools to choose from to play the game rather they use the tools given to them.  
• Unit 2, Lesson 1, Materials, “Attribute blocks, Geoboards and rubber bands” In the activity students choose an attribute block and use the geoboard and rubber band to build the shape. No choice in tools is given. 
• Unit 4, Lesson 4, students are told to use either their hands or a balance to determine if objects are heavier, lighter, or the same. However, because the only way to determine the same is with the balance there is not a choice in the tools used during the lesson. 
• In Unit 7, Lesson 3, Exit Ticket, “Circle all of the expressions that make a total of 9. Use your counters to help you.” Students do not have a choice in the tool they use, they are given counters to use in the lesson and on the exit ticket. 

Examples of the materials attending to the full intent of specific MPs include:

• MP2: In Unit 2, Lesson 6, Narrative, students have been composing and decomposing flat geometric pictures. The materials state, “They will have to use their knowledge of shape attributes previously attained as well as analyze the remaining space to fill in the empty space with a shape that they know will fit based on the attributes.” In Exit Ticket, students are shown a picture of a hexagon made up of triangle pieces, and one missing piece, then prompted to find the missing piece with three choices. The materials state, “Jordan filled his hexagon using all triangles yesterday. Today he started using triangles but then ran out. How can he finish his puzzle? Circle the shape that would fit.”
• MP4: In Unit 6, Lesson 7, Introduction, Step 1 states, “There were 3 horses in the field. 4 more horses came out of the barn and into the field. How many horses are in the field now?” Step 2 states, “Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.”
• MP6: In Unit 6, Lesson 5, Narrative, students are finding the total of two groups by counting all or counting on. The materials state, “Students use number cubes today to encourage them to consider how to represent each addend and/or to think of strategies besides counting all. Students should recognize that counting all can be done when two groups are shown numerically (by representing concretely or pictorially, including with fingers) and that counting on can be done by saying the first amount and counting on the second as it is represented concretely or pictorially, including on fingers.” 
• MP7: In Unit 8, Lesson 2, Introduction, Step 4 states, “T&T: What do you notice about the group of ten ones and loose ones and how we write the number? SMS: I notice that there is 1 group of ten ones and so there’s a one right there. Then there’s 4 loose ones so there’s a 4 right here. Yes, this is called the tens place. There is the digit 1 here to show 1 group of ten ones. This is called the ones place. There’s the digit 4 here to show 4 loose ones.”
• MP8: In Unit 7, Lesson 8, Introduction states, “(Show representation on recording sheet) How does this representation match the story? It shows that there are 10 apples in each box and that some are red and some are green. It shows that we need to figure out all of the ways we could fill the boxes with some red and some green. (Make sure students understand that each ‘Row’ or ‘rectangle’ represents a box.)” Mid-Workshop Interruption states, “CC: Which starting combination helped us find more solutions? $$1 + 9$$. TT: Why does that help us find all of the solutions? It is the smallest possible amount of [red or green] apples and the largest possible amount of [opposite color] apples. Then we add one [red or green] apple at a time and take away one [opposite color] apple at a time until we have the largest possible amount of [red or green apples] and the smallest possible amount of [opposite color] apples, so we know we have found all of the solutions.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The student materials prompt students to both construct viable arguments and analyze the arguments of others even though mathematical dialogue is mainly between the teacher and individual students.

Examples of constructing viable arguments include: 

• In Unit 3, Lesson 26, Introduction, students transition from counting by ones to counting by tens and discuss the transition. T&T states, “How could I figure out how many? Counting by ones took a LONG time. Is there a way we can use our tens frames without having to count every dot?” 
• In Unit 4 Assessment, students compare the relative capacity of two items and explain how they know. Item 3 states, “(Give the student the cup and the basket). Ask, which item holds less? Use the words holds more/holds less. How did you figure that out?/How did you know that?” 
• In Unit 5, Lesson 3, Introduction, every pair of students have five yellow cubes and eight green cubes. The materials state, “Step 3 says Compare: We have gotten really good at telling which is more, but what if I ask Which is LESS? T&T: Figure it out with your partner- which color is less? How do you and your partner know?” Example response states, “Strategy 1:1: Matching 1:1: I know because I matched the cubes and some ___ cubes didn’t have a partner.”
• In Unit 6, Lesson 8, Introduction, students justify their thinking while solving a story problem using a strategy chosen by them. The materials state, “Step 2: Represent and Retell. Now you need to show the story. You can use your cubes or your whiteboard and marker; it’s up to you. Remember to include what we know and what we need to figure out. When you are done, put your whiteboard and cubes flat and be ready to explain how you represented and how it matches the story.” 

Examples of analyzing the arguments of others include:

• In Unit 3, Lesson 12, Share/Discussion, students determine and write how many objects in a set (10-20 objects) by using a strategy to keep track. The teacher asks 2-3 students to share their work/strategies. The materials state, “How did ____ count? How did ____ count? What is the same about these strategies? What is different? Why do they both work?” 
• In Unit 6, Lesson 6, Workshop, Problem 1 states, “Gia had 4 apples in her basket. She picked 4 more and put them in her basket. How many apples does she have now?” Share/Discussion states, “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: How did ____ represent? How did ____ represent? How do both of these strategies work to represent the story?”
• In Unit 7, Lesson 5, Share/Discussion, students are engaged in a class discussion sharing their strategies after finding all of the possible combinations of 7. The materials state, “2 - 3 students share their work/strategies involving compensation: How did ______ find another solution? How did ______ find another solution? How/why do both of these strategies work?”
• In Unit 8, Lesson 8, Lesson Task states, “Introduction: Tanya has ten beads and five beads. Marie has ten beads and three beads. Tanya says she has more beads than Marie. Is Tanya correct? Show and tell how you know.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Examples of the materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include:

• In Unit 2, Lesson 6, Introduction, Step 2 states, “What shape do you think would fit? Why did you pick that shape? SMS: It had a side that was the same length as the side of the puzzle (or some other attribute reference). If kids are struggling to articulate this idea, have a student come up and try it and the teacher can narrate the ideas to give kids the language - ‘Oh I see Danny is lining up the side of square with the side of the picture. Noticing the side helped here!’”
• In Unit 3, Lesson 18, Introduction, teachers are provided guidance in helping students to construct viable arguments demonstrating how to find the total of three numbers rolled on dot cubes. The materials state, “Step 2 is for us to find out the total. When we did this before, we only had to figure out the total for two dot cubes. T&T: How would we do it for three dot cubes? SMS: It’s the same! It’s just more dots, so I can count them all. If a student says this, have a student come up and demonstrate touching each dot and counting and a student showing the amounts on fingers and counting all. SMS: I can just see (subitize) and say the number on one dot and count on from there. Have a student demonstrate.”
• In Unit 4, Lesson 4, Share/Discussion, during Workshop, students are picking two objects and determining which is heavier or lighter using either a balance or hefting. The materials state, “Facilitate a discussion around a major misconception (i.e. an object that is longer/taller doesn’t always have to be heavier). Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR, 2-3 students share their work/strategies: How did ___ compare their objects? How did ___ compare their objects? What is the same about these strategies? What is different? Why do both work?”
• In Unit 5, Lesson 3, Mid-Workshop Interruption, students determine which number is more and which is less by building towers or matching one to one. The materials state, “If $$>\frac{2}{3}$$ of students are successful, ask students to describe the relationship between 2 towers (green 8 and blue 3) in a turn and talk. Hunt for a student who says one tower is more and another who says the other tower is less. Share their answers and ask who is right; students should see that both students are right- the green tower is more and the blue tower is less. Discuss how this is true; students should articulate that they are opposites and that if one tower is more the other will always be less and vice versa. Challenge students to circle the amount that is more as well one the recording sheets moving forward. If $$>\frac{2}{3}$$ of students are successful, call students back together to clarify expectations through a misconception protocol or role play.”
• In Unit 7, Lesson 1, Introduction, students name and record (with equations) various ways to decompose the totals 4 and 5. During a demonstration, the teacher tosses 3 red chips and 1 yellow chip. The materials state, “Step 2: says what do you see? TT: What do you see? I see 3 red counters and 1 yellow counter and 4 counters altogether. (Record guiding questions on VA.) -The purpose of this question is to get students to generate the numbers they will use in their number sentences; feel free to use the questions below to help: - If students say, ‘I see 4counters/chips,’ ask, ‘What colors do you see?’ -If students say, ‘I see red and yellow,’ ask, ‘How many red do you see? How many yellow?’ -If students say, ‘If students say, ‘I see red and yellow,’ but don’t notice the total, ask, ‘How many do you see altogether/How many does that make altogether?’”
• In Unit 7, Lesson 11, Introduction, Play Again and Check for Understanding, teachers are instructed to pose a fictional problem for the students to analyze. The materials state, “Rather than playing a full game, pose this problem: Mr. Lynch was playing the game, and he drew a 3, so he recorded like this: $$3 +$$ ____ $$= 10$$ (show). Then when he went to show that many on his tens frame, he realized that he didn’t have any! They had all been cut in half for an art project. So he used just the top half, like this (Show the top row of the tens frame with 3 counters on it.). Then, he said, ‘How many to ten?’ and he counted the empty squares. 2! He wrote $$3 + 2 = 10$$. EV: Does this work? No. TT: Why not? That is how many to 5, not ten. The tens frame works because it has 10 squares in all, so if we show how many we have we can count the empty squares to figure out how many to ten. There are not 10 squares in all.”

The instructional materials reviewed for Achievement First Mathematics Kindergarten meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials also use precise and accurate terminology and definitions when describing mathematics, and support students in using them. 

Examples of explicit instruction on the use of mathematical language include:

• In Unit 2, Overview, Identify The Narrative states, “Students are formally introduced to solid, 3D shapes in the lesson titled, ‘Stack/Roll/Slide.’ In the lesson, students observe solid shapes and begin to develop the vocabulary needed to describe their attributes, including vertices, faces, and edges. They begin to think about how these attributes distinguish them from 2D shapes.”
• In Unit 4, Lesson 1, Introduction, Introduce the Math states, “Today we’ll figure out how long (kinesthetic: make arms wide horizontally) or tall (kinesthetic: make arms wide vertically) things are by comparing two objects. That’s called length. CR: Length is (how long or how tall things are). Do again with kinesthetic movements. Show Measurable Attributes VA. When we are talking about how long or how tall things are, they can be LONGER (longer-choral response and motion: start with hands together and move apart) or SHORTER (shorter-choral response and motion: start with hands apart and move together).”
• In Unit 6, Lesson 1, Introduction, provides students with explicit instruction on the meaning of the + and = as they learn to write addition equations. Introduce the math states, “First, let’s learn the 2 math symbols we will be using today. This sign is called ‘plus’. It looks like the letter t. It means put together (put two hands together, interlocking fingers). We read it as ‘and’. (point to it 2 times and have kids say, ‘and’ and put their fingers together). This sign is called ‘equals’. It means is. (point to it 2 times and have kids say ‘is’ or ‘makes’.)”
• In Unit 7, Lesson 11, Introduction, students learn the name of the tool, tens frame. The materials state, “Today we are going to work with a special tool: The tens frame. (show tens frame) Why is this called a tens frame? What is special about it? It has 10 squares. Count them with me. (count the ten squares together) Today, we are going to play a game called ‘How many to Ten,’ let’s see how we can use our tens frames to help us today!”

Examples of the materials using precise and accurate terminology and definitions:

• In Unit 4, Lesson 7, Introduction, students are prompted to use accurate language to compare the capacity of two containers. The materials state, “Which one held more rice? The _______ held more rice than the _______. I know because it held _____ scoops of rice and the _____ held ______ scoops of rice. Be sure to prompt for accurate comparative language.”
• In Unit 5, Lesson 1, Assessment and Criteria for Success, students are expected to use the terms more, greater, the same, and equal to describe sets of objects. Questions are provided for teachers to support students in the use of these terms. The materials state, “Teachers should circulate during workshop to gather data on student mastery. All students should be able to use the words, ‘more,’ ’greater,’ ‘the same,’ and ‘equal’ to describe their sets. Teachers should ask: 1. Which is more/has a greater amount of cubes? How do you know? 2. How can you describe this tower? (pointing to a tower that is more). 3. How can you describe these towers? (showing two towers that are the same).”
• In Unit 6, Lesson 2, Introduction, Introduce the Math, the teacher adds a symbol to the visual aid to assist in students’ understanding. The materials state, “The symbol for addition is the ‘+’ sign that you all already know how to use. (add to picture part of va). It means we put the red and green together. (Draw the equals sign and the red and green together).”  
• In Unit 7, Overview, Identify the Narrative states, “They conclude that there are many different ways to make a total. They record using addition equations, with a continued emphasis on the language ‘and’ for ‘+’ and ‘makes’ or ‘is’ for ‘=.’ (If they have 3 red and 2 yellow, they record ‘$$3 + 2 = 5$$’ or ‘$$5 = 3 + 2$$’ and say ‘3 and 2 is five’ or ‘3 and 2 makes 5’ or ‘5 is 3 and 2.’) Teachers intentionally use this language when writing and reading equations and expressions in order to build deep conceptual understanding of the symbols.”