2nd Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 83% |
|---|---|
Criterion 2.1: Rigor | 7 / 8 |
Criterion 2.2: Math Practices | 8 / 10 |
The instructional materials reviewed for Achievement First Mathematics Grade 2 partially meet the expectations for rigor and the Mathematical Practices. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The instructional materials partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics.
Criterion 2.1: Rigor
Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
The instructional materials reviewed for Achievement First Mathematics Grade 2 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The materials partially meet the expectations for application due to a lack of independent practice with non-routine problems.
Indicator 2a
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for developing 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 7, Introduction and Workshop, students engage with 2.NBT.5, fluently add and subtract within 10 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve problems about tens and ones by using a variety of representations (stick and dots, expanded form). Students roll two number cubes, record the two-digit number, represent the number using sticks and dots, and represent the number using expanded form. The teacher asks, “How will you figure out how to represent 2-digit numbers using sticks and dots and expanded form?” The students may reply with, “I will look at the digits in each place and think about the value of each digit.”
- In Unit 6, Lesson 2, Introduction, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they model three-digit numbers with place value blocks, then read and write the numbers. The materials state, “Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?”
- In Unit 6, Lesson 14, Introduction, students engage with 2.NBT.4, compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the result of the comparison, as they compare two three-digit numbers written in different forms using <, >, and =. The teacher poses the following comparison problem to students, “562 __ 5 hundreds, 2 tens, 6 ones.” A sample student response states, “We wrote 562 > 5 hundreds, 2 tens, 6 ones. We figured it out by showing both numbers in flats, sticks, and dots. For 562 we drew 5 flats, 6 tens, 2 dots. Then for 5 hundreds, 2 tens, 6 ones, we drew 5 flats, 2 sticks, 6 dots. We looked at the hundreds place and saw that they had an equal number of hundreds, so then we looked at the tens and saw that 562 has more tens than 5 hundreds, 2 tens, 6 ones, so 562 is greater than 5 hundreds, 2 tens, 6 ones.”
- In Unit 7, Lesson 2, Aim, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings, as they complete 2-digit addition problems using flats, sticks, and dots. The materials state, “SWBAT add 2-digit numbers with regrouping in one place by using flats, sticks, and dots.” Workshop Worksheet example, Problem 2A states, “$$550 + 268 =$$ ______.”
- In Unit 7, Lesson 7, Workshop Worksheet, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of why their strategy worked. The materials state, “Solve. ______ $$- 246 = 568$$. Explain how you solved the problem above. What strategy did you use? What steps did you take? Why did your strategy work?”
Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:
- In Unit 2, Lesson 10, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they add two-digit numbers by using a strategy that makes sense to them (sticks and dots, expanded notation/use known facts). Problem 1 states, “$$62 + 27 =$$ ___.” Students are directed to use sticks and dots or expanded notation to solve.
- In Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, as they identify the proper model for a given problem. Problem 34 states, “Circle which set of sticks and dots will help to find the total? $$62 + 24 =$$ ______.”
- In Unit 6, Lesson 2, Independent Practice Worksheet, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they independently model three-digit numbers with place value blocks, then read and write the numbers. Problem 1 states, “Draw flats, sticks, and dots to represent each number. 258. How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”
- In Unit 7 Lesson 7, Exit Ticket, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of how they solved the problem. Problem 2 states, “_____ $$- 567 = 293$$. Explain how you solved.”
- In Unit 7, Lesson 10, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction and relate the strategy to a written method. Problem 2 states, “Solve using flats, sticks, and dots. $$531 - 258 =$$ ____.”
Indicator 2b
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for attending 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:
- In Unit 2, Lesson 2, Introduction and Workshop, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they determine the missing part to make 10 using a strategy that works for them (count up, count back, just know). Students use a dot cube to roll for a number to subtract from 10 in a number bond. Potential strategy examples state, “Count up: You can start at 6 because that’s the first part and count up until 10 because that’s the whole. Like this… 6 -- 7, 8, 9, 10. So the missing part is 4. Subtract: You can start with the whole -- 10 and subtract 6 because that’s the part we know. The answer is 4, so the missing part is 4. Count back: I started at 10 because that is the whole and then I counted back 6 because that’s the part we know. Like this 10 -- 9, 8, 7, 6, 5, 4. So the missing part is 4. Just know: I just know that 6 and 4 make 10 because they’re number pairs. So the missing part must be 4.”
- In Unit 3, Practice Workbook B, Activity: Building Toward Fluency, students engage with 2.OA.2, fluently adding and subtracting within 20 using mental strategies, as they use various strategies to complete and discuss addition problems. The materials state, “Write the expression on the board or chart paper. Start with 4 + 10. Ask students to describe their strategy for solving the problem. Choose one or more students to explain their strategy to the class. Represent each strategy on the board using the number line or magnetic cubes. Once the student’s strategy is understood by the class, continue with the next sum.”
- In Unit 5, Practice Workbook B, Ten Plus Number Sentences, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice proficiency with their ten plus facts. The teacher says, “I will flash two ten-frame cards, ten and another card. Wait for the signal. Then tell me the addition sentence that combines the numbers.” The teacher flashes a 10 and 5. Students respond with, “$$10 + 5 = 15$$”
- In Unit 6, Lesson 9, Workshop Worksheet, students engage with 2.NBT.2, skip-count by 5s, 10s, and 100s, as they use skip counting by 10s and 100s to count up. Problem 1 states, “Count from 90 to 300.”
The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:
- In Unit 2, Lesson 3, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they independently complete a number bond with one unknown number and write 2 addition and two subtraction problems to match. Problem 1 states, “Finish the number bond and write number sentences to match.” Students are provided with a number bond diagram with 11 and 6 as addends and an unknown sum.
- In Unit 3, Practice Workbook B Pairs To Make Ten With Number Sentences, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve doubles +2 facts. Problem 3 states, “$$2 + 4 =$$ ___.”
- In Unit 4, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction, as they independently solve two-digit addition and subtraction problems. Problem 21 states, “$$53 -$$ ______ $$= 28$$.”
- In Unit 8, Practice Workbook D, students engage with 2.NBT.8, mentally add 10 or 100 to a given number 100-900, as they independently add 10 or 100 to given numbers betwembers under 1000. Problem 1a states, “Solve each problem using mental math, $$678 + 100 =$$ ____.”
Indicator 2c
Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
The instructional materials for Achievement First Mathematics Grade 2 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 2, “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, there are not any “Task Based Lessons” labeled in the Guide to Implementing AF Grade 2.
Routine problems are found in the Independent Practice and Exit Tickets. For example:
- In Unit 3, Lesson 2, Exit Ticket, students engage with 2.OA.1, using addition and subtraction within 100 to solve one- and two-step word problems, as they calculate take apart problems with the addend unknown. Problem 2 states, “Mr. Cruz has 19 basketballs. Mr. Hogan gave him some more basketballs. Now Mr. Cruz has 63 basketballs. How many basketballs did Mr. Hogan give to Mr. Cruz?”
- In Unit 3, Lesson 7, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 7 states, “Enmaries has a jump rope that is 68 inches long. Giada’s is 33 inches shorter than Enmaries’s jump rope. What is the length of Giada’s jump rope?”
- In Unit 5, Lesson 3, Independent Practice, students engage with 2.MD.8, solve word problems involving money, as they independently solve word problems with money. Problem 5 states, “King Jamonie has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?”
- In Unit 5, Lesson 9, Exit Ticket, students engage with 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, as they solve all types of story problems using the most efficient strategy. Problem 2 states, “Carlos bought a pack of gum using exactly $1. The cashier gave him 3 nickels, 2 pennies, and a quarter back. How much did the gum cost?”
Math Stories provide opportunities for students to engage in routine applications of grade-level mathematics. Students engage with Math Stories for 25 minutes, four days per week. The Guide to Implementing AF Grade 2 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 2 Math Stories:
- In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 2.OA.1, use addition and subtraction to solve one- and two-step word problems with unknowns in all positions, as they solve add to/change unknown word problems. Sample Problem 2 states, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?”
- In Unit 3, Guide to Implementing AF Math, Math Stories, December, students engage with 2.OA.1, using addition and subtraction within 100 to solve one- and two-step word problems, as they calculate take from problems with results unknown. Sample Problem 2 states, “Antonio gave 27 tomatoes to his neighbor and 15 to his brother. He had 72 tomatoes before giving some away. How many tomatoes does Antonio have remaining?”
- In Unit 4, Guide to Implementing AF Math, Math Stories, January, engages with 2.MD.5, use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, as they use a variety of strategies to solve measurement word problems. Sample Problem 6 states, There are 98 frogs on a log. 63 frogs jump into the water. 29 frogs hop back onto the log. How many frogs are on the log now?”
- In Unit 8, Guide to Implementing AF Math, Math Stories, May, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they complete math story problems. Sample Problem 1 states, “There were 45 students on the playground. 33 Students joined them. Then 37 students went inside. How many students are now on the playground?”
Non-routine examples could only be found in Unit 3, not across the yearly curriculum. For example:
- In Unit 3, Lesson 10, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if the thinking and process complete by another is correct. Problem 5 states, “Grace represented the problem below. What mistake did she make? Explain what mistake she made and what she can do to fix it. Mr. Cruz has 42 orange basketballs and 29 black basketballs. He gives away 32 basketballs. How many basketballs does Mr. Cruz have now? $$42 + 29 + 32 = N$$”
- In Unit 3, Lesson 11, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if the thinking and process complete by another is correct. Problem 7 states, “Lisa represented the problem below. Is she correct? If not, explain her mistake and how she could fix it. There are 28 lions at the zoo. There are 14 fewer lions than tigers at the zoo. 18 tigers come to the zoo. Now how many tigers are at the zoo?” (Two diagrams follow the question that represent her thinking towards the problem.)
It is important to note that some of the recommended Math Stories go beyond the expectations of Grade 2 mathematics standards and some are considered extensions. The Guide to Implementing AF Grade 2 states, “The problem types covered within each unit are marked with an O (on grade level) or an E (enrichment) to reflect their alignment to grade level standards.” For example:
- In Unit 7, Math Stories, April, Extension Problem 4 states, “(AA-UF) There are 32 stickers in a sticker book page. They are arranged so that there are 8 equal rows of stickers. How many stickers are in each road?” This word problem goes beyond Grade 2 Mathematics Standards, as it is a multiplication word problem, and better aligned with 3.OA.8, solve two-step word problems using the four operations.
- In Unit 7, Math Stories, April Extension Problem 12 states, “(C-DU with AA-UP) Nithi has 3 rows of 7 brownies on her tray. Raj has 3 rows of 5 brownies on his tray. How many more brownies does Nithi have than Raj? (Note: As an extension, look for students that see that Nithi has 3 rows of 2 brownies extra instead of calculating both and subtracting!)” This word problem is better aligned with 3.OA.8, solve two-step word problems using the four operations.
- In Unit 8, Math Stories, May, Problem 9 states, “(AT-SU with AA-UP) There are some desks in a classroom. Then, Ms. Colville brings in 3 rows of 4 desks. Now there are 30 desks in the classroom. How many desks were in the classroom originally?” This word problem goes beyond Grade 2 Mathematics Standards, as it is a two-step word problem including multiplication, and is better aligned with 3.OA.8, solve two-step word problems using the four operations.
Indicator 2d
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
The instructional materials for Achievement First Mathematics Grade 2 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 6, Lesson 2, Exit Ticket, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they read and write numbers within 1,000 after modeling with place value blocks (flats, sticks, and dots). Problem 2 states, “Draw models of ones, tens, and hundreds.” Students are given the number 508 and asked to answer the following questions, “How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”
- In Unit 6, Lesson 3, Independent Practice, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, and 2.NBT.3, read and write numbers to 1000, as they represent a three-digit numbers in a variety or forms and models. Problem 10 states, “Alexander has 529 M&Ms. Write the amount of M&Ms Alexander has in three different ways by filling in the blanks. (Unit Form, Base Ten Numeral Form, Place Value Models)”
- In Unit 7, Lesson 1, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit subtraction. The third item states, “Solve using flats, sticks, and dots ______ $$- 348 = 650$$. Explain how you solved ______ $$- 348 = 650$$.”
- In Unit 9, Practice Workbook E, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit addition problem using a number line. Problem 4 states, “Use the number line to solve. Show your work. $$578 + 237 =$$ ___.” Students are provided a blank number line.
Procedural skills (K-8) and fluency (K-6)
- In Unit 2, Lesson 4, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve addition problems. Problem 1 states, “Solve. $$7 + 8 =$$ _____.”
- In Unit 2, Lesson 24, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition/subtraction, as they solve two-digit subtraction problems with missing minuends by relating addition and subtraction. Problem 1 states, “Solve. $$93 -$$ ____ $$= 62$$.” Students are provided with a blank number bond model.
- In Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve put together problems with an unknown addend. Problem 5 states, “$$35 +$$ ___ $$= 50$$.”
- In Unit 5, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice fluently adding and subtracting. Problem 4 states, “$$45 +$$ ___ $$= 100$$”
Application
- In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve addition problems with the change unknown. Sample Problem 2 states, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?”
- In Unit 3, Lesson 4, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 4 states, “Ms. Reinhardt has 42 books. Ms. Gomez has 18 fewer books than Ms. Reinhardt. How many books does Ms. Gomez have?”
- In Unit 3, Lesson 4, Exit Ticket, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems, as they independently solve compare/smaller unknown word problems. Problem 1 states, “There are 59 girls on the bus. There are 26 more girls than boys on the bus. How many boys are on the bus?”
- In Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels and pennies, using $ and $$\cancel{C}$$, as they solve one-step story problems of all types that involve bills and coins by using the most efficient strategy. Problem 2 states, “Kevin has 75 cents. He spent 3 dimes, 3 nickels, and 4 pennies on a slice of cake. How much money does he have left?”
Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:
- In Unit 2, Lesson 6, Exit Ticket, students engage with 2.NBT.9, explaining why addition and subtraction strategies work, using place value and the properties of operations, and 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve take from problems with the result unknown (application) and show their thinking (conceptual understanding). Problem 1 states, “Represent and solve. Amya has 17 pencils. 13 are red and the rest are green. How many green pencils does Amya have? Describe how you solved.”
- In Unit 3, Lesson 2, Independent Practice, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, and 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if a representation is correct (application) and how they know (conceptual understanding). Problem 10 states, “Mr. Johnson has 46 pens. 24 are blue and the rest are black. How many of Mr. Johnson’s pens are black? Charlie and Henry represented the problem below. (Charlie $$46 + 24 =$$ ? represented/ Henry $$46 - 24 =$$ ? represented)”
- In Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving money, as they represent (conceptual understanding) and solve story problems (application) using the most efficient strategy (procedural skill). Problem 1 states, “Jacob bought a piece of gum for 26 cents and a newspaper for 61 cents. He gave the cashier $1. How much money did he get back?”
- In Unit 8, Lesson 2, Independent Practice, students engage with 2.OA.4, use addition to find the total number of object arranged in a rectangular array, as they draw a rectangular array and write addition equations (conceptual understanding) to represent and solve word problems (application) involving equal groups of objects. Problem 6 states, “Ja-yier put 5 toys into 4 different baskets. How many toys does Ja-yier have in all?”
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for Achievement First Mathematics Grade 2 partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics.
Indicator 2e
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials reviewed for Achievement First Mathematics Grade 2 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.
- Unit 2, Unit Overview, Standards for Mathematical Practice identifies MP8, look for and express regularity in repeated reasoning, as embedded in addition and subtraction lessons within 100.
- In Unit 5, Unit Overview, Standards for Mathematical Practice identifies MP6, attend to precision, as embedded in the measurement lessons of Unit 5.
- The MPs are listed at the beginning of each lesson in the Standards section. For example, in Unit 4, Lesson 3, the following MPs are identified as in the lesson: MP 2 and MP 6.
- The Mathematical Practices are not always identified accurately. For example:
- In Unit 3, MP 6 is not bolded as a focus MP. However, it is identified in all 14 lessons. MP 7 is identified as a focus MP but is not identified in any of the 14 lessons.
- At the unit level for Unit 7, MP 6 is not identified as a focus MP. However, at the lesson level, all 18 lessons identify it as connected. At the unit level, MP 3 is listed as a focus, but it is only connected to 3 of the 18 lessons.
- All MPs are represented throughout the materials, though lacking balance. For example, MP 8 is the focus of none of the second grade units, while MP 5 is the focus of all ten units.
- There are no stated connections to the MPs to the Math Stories component, Math Practice component, Cumulative Review 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:
- In Unit 2, Unit Overview, Standards for Mathematical Practice identifies MP1, make sense of problems and persevere in solving them, as embedded in the lessons of Unit 2. The materials state, “Students make math drawings and use composition and decomposition strategies to reason through the relationships in word problems. They write representations, equations, and word sentences to explain their solutions.”
- In Unit 5, Unit Overview, Standards for Mathematical Practice identifies MP5, use appropriate tools strategically, as embedded in the measurement lessons of Unit 5. The materials state, “Students apply their measurement skills and knowledge of the ruler to measure a variety of objects using appropriate measurement tools, such as inch rulers, centimeter rulers, meter sticks, and yardsticks.”
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The materials reviewed for Achievement First Mathematics Grade 2 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 2, Lesson 6, Narrative, What is new/or hard about the lesson? states, “This lesson is challenging because it’s pushing students to apply their understanding of part-part-whole relationships and pushing them to become fluent within 20. Students may have difficulty understanding the context of the story problem. They may also have difficulty calculating fluently.” Introduction, Pose the Problem states, “Carla baked 15 desserts. 9 of them were chocolate chip cookies and the rest were brownies. How many brownies did Carla bake?”
- In Unit 4, Assessment, students solve two-step story problems from data presented in a graph. Problem 4 states, “19 of the scholars who like fruit are girls. How many of the scholars are boys?” Students are provided a bar graph showing survey data regarding favorite fruits.
- In Unit 7, Lesson 17, Introduction, Check for Understanding states, “I’m going to solve ___ $$- 315 - 545$$ using expanded notation. (Set up number bond correctly and label parts and wholes correctly, but set up problem to subtract 545 - 315.) (EV) Am I right? No. (TT) Why not? What mistake did I make? Your number bond is correct, but if the whole is missing and you know that both parts are 315 and 545 you need to add the 2 parts to find the whole, not subtract. (TT) What strategies will you use to solve problems today? First we need to figure out what’s missing - a part or the whole. then we can use expanded notation, flats, sticks, and dots, or a number line to solve.”
The materials do not attend to the full meaning of MP5 because students do not choose their own tools. Examples include:
- Unit 2, Lesson 1, Step 1, “says, Roll, so I’m going to roll my dot cube gently. How many? SMS: 6!” The students are given the dot cube to roll, that is the only tool they are given and use during the lesson.
- Unit 4, Lesson 4, Students create survey questions with a partner and then use tally marks to organize the results. Students are told to use tally marks and no other tools are available or used during the lesson.
- Unit 9, Lesson 1, Materials, “pattern blocks” Students are given pattern blocks to build larger shapes to explore fraction concepts. Students do not have a choice in the tool they use.
Examples of the materials attending to the full intent of specific MPs include:
- MP2: In Unit 5, Lesson 9, Independent Practice Worksheet, students solve story problems involving coins, requiring them to abstract the value of a set of coins, to find the total value. Problem 8 states, “Anna gave Sean a dime, a nickel, and some more coins. Now Sean has $1.00 in coins. Draw and label two possible pictures of Sean’s coins.”
- MP4: In Unit 3, Lesson 2, Narrative, How does the learning connect to previous lessons? What do students have to get better at today? states, “In this lesson, scholars will focus on accurately representing the problem and then using that representation to choose the correct operation to solve. They will work with Add To/Take From - Change Unknown and Put Together/Take Apart - Addend Unknown story problems.” Introduction task states, “There are some birds on a fence. 19 birds flew away. Now there are 52 birds on the fence. How many birds on the fence were there to start?"
- MP6: In Unit 4, Lesson 3, Narrative, What is new and/or hard about the lessons? states, “Students may still be struggling to categorize data, and may still be struggling to create accurate, organized representations of data.” Introduction states, “(TT) How do I know where to stop on my scale? You need to find the category with the largest number, then you need to make sure your scale goes up to that number so that all your data fits. BPQ: Would it make sense to have my scale go to 100 if my largest category had 5 things? Why?”
- MP7: In Unit 8, Lesson 7, Workshop Worksheet, students decompose numbers and apply repeated addition to the structure of arrays to both the rows and columns. Problem 1 states, “Jeremiah drew an array with 20 squares. Draw 3 different arrays that have 20 squares in all. Write a repeated addition sentence to match each array you drew.”
- MP8: In Unit 9, Lesson 7, Narrative, How does the learning connect to previous lessons? What do students have to get better at today? states, “In the previous lesson, students partitioned shapes into the same fraction in more than one way and came to the understanding that the same fraction can have a different shape. Students also named and wrote unit fractions. Today, for the first time, students will partition rectangles into fractions in more than one way and prove the fractions are the same by cutting the parts and manipulating the pieces.”
Indicator 2g
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Indicator 2g.i
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
The instructional materials reviewed for Achievement First Mathematics Grade 2 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 2, Lesson 21, Exit Ticket, students explain the strategies they used to solve two problems, and why they chose each strategy. Problem C states, “How did you solve $$68 - 21$$? Explain the steps you took to solve question A. How is this different from how you solved $$68 + 21$$ in question B? Why?”
- In Unit 5, Lesson 3, Workshop Worksheet, Problem 3 states, “Megan has 4 dimes and 15 pennies. Joshua has the same amount of money as Megan but none of the same coins. What coins does Joshua have? How do you know?”
- In Unit 7, Lesson 19, Assessment, Problem 4 states, “Find the missing numbers to make each statement true. Show your strategy to solve. a. ___ $$= 407 - 159$$. Explain how you solved this using what you know about place value.”
- In Unit 9, Lesson 4, Intro, Problem 2 states, “Chase offers to share his pie with Jariah and Luke. They want to have the largest pieces possible. Should Chase cut it into thirds or fourths? Why?”
Examples of analyzing the arguments of others include:
- In Unit 2, Lesson 24, Workshop Worksheet, students use knowledge of place value and expanded notation to critique the work of other students, and show how their work can be fixed. Problem 2 states, “Cat solved the problem below using expanded notation. $$95 -$$ _____ $$= 27$$, $$90 + 5$$, - (subtraction symbol) $$20 + 7$$, _______, $$70 + 2 = 72$$, Is she correct? If not, how can she fix her work?” (Commas separate different lines of example student work.)
- In Unit 3, Lesson 2, Introduction, Problem 2 states, “Khaleel and Mauricia represented the problem below. There were 36 kids on the playground. Some more kids came over to join them. Now there are 62 kids on the playground. How many kids came to join them? Look at Khaleel and Mauricia’s representations. Who is correct? How do you know?”
- In Unit 4, Lesson 4.2, Cumulative Review, Problem 6 states, “Kimberly was solving the problem below using expanded notation. What mistake did she make? Fix Kimberly’s work in the box and then explain her mistake on the lines below. 67 - 39 = ___, $$60 + 7$$, - (subtraction symbol) $$30 + 9$$, ______, $$30+2=32$$.” (Commas separate different lines of example student work.)
- In Unit 5, Lesson 9, Independent Practice, Problem 7 states, “Calvin has 72 cents. Michelle says he needs 1 quarter and 5 pennies to make a dollar. Sarah says he needs 3 pennies, 2 dimes, and a nickel to make a dollar. a) How much money does Calvin need to make a dollar? b) Who is right and how do you know?”
- In Unit 8 Assessment, students use their knowledge of arrays to analyze the mathematical thinking of a fictitious student. Item 1 states, “Angela wants to make 3 pins. Angela wants to put 5 beads on each pin. Angela has a bead box with three rows in it. Each row has five sections. Angela has one bead in each section. Angela says that she has enough beads to make three pins. Is Angela correct? Show all of your mathematical thinking.”
Indicator 2g.ii
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
The instructional materials reviewed for Achievement First Mathematics Grade 2 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 3, Introduction, students are prompted to analyze the thinking of others as they determine the mistakes made by the teacher as he/she models how to find the unknown number in a number bond. The materials state, “(Model an intentional mistake--add when you should subtract.) (Show number bond with 9 in the center/whole and 5 in the part--label 9 red and blue crayons, 5 red crayons, missing # blue crayons.) Oh! I can find the missing number by adding 5 and 9. (EV) What do you think? No. (TT) What mistake did I make? How can I fix it? You added the 5 red crayons to the 9 red and blue crayons and we know that 5 of those crayons are red. We need to figure out how many of those crayons are blue. We need to subtract to find the missing part, not add.”
- In Unit 2, Lesson 21, Workshop, students solve two-digit addition and subtraction problems on a number line. While circulating during Workshop, the teacher may ask the following questions to check for understanding, “How did you solve the problem? Why did you make ___ jumps of ___? Which direction did you hop and why? (Where did you start with and where did you hop to and why?) Why did you start at XX and make jumps of XX?”
- In Unit 3, Lesson 6, Independent Practice, Check for Understanding, teachers are provided with a list of prompts designed to engage students in constructing viable arguments as they represent and solve story problems. The materials state, “How did you know that $$x$$? What strategy did you use? Why are you adding $$x$$ and x/subtracting $$x$$ and $$x$$? What in the story made you think that? Why did you have to regroup?”
- In Unit 5, Lesson 5, Independent Practice, Check for Understanding states, “Why did you trade ___ for ___? What strategy did you use to make trades?”
- In Unit 6, Lesson 9, Introduction, students count up between 90 and 1,000 by using skip counting. The materials state, “Pose the Problem: I have 286 gumballs. I want to see how many more I need to get to 500.” Students are given three minutes to represent and solve with a partner. The teacher then asks, “How did you find out how many gumballs I need to make 500 (call on someone with correct representation who solved on a number line)?” After the student shares, the teacher follows up with the following questions, “Agree/Disagree/Clarification; Why does that work? Why do you think we counted by ones first? Why do we count by tens next?”
- In Unit 7, Lesson 7, Introduction, students work to answer three-digit addition problems. Questioning guidance is provided to assist the teacher for two different scenarios, $$>\frac{2}{3}$$ correct and $$<\frac{2}{3}$$correct. The materials state, “Agree/Disagree/Clarification (TT) Why did they have to regroup? (CC) How did they show regrouping?”
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Achievement First Mathematics Grade 2 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 4, Lesson 2, Introduction, students are provided explicit instruction in the meaning of a pictograph as they learn to use them to represent data. The materials state, “Yesterday we worked so hard to categorize data and represent it in tally charts. Today we are going to represent using a new kind of graph--a pictograph! A pictograph represents data using pictures. There are important elements that must be part of a pictograph so people can understand it. (Reveal top quality graph VA as you go.) We need a title, category labels, and a KEY!”
- In Unit 5, Overview, Major Misconceptions & Clarifications states, “Misconception: Students struggle with the terms ‘quarter after’ and ‘quarter of’ since the coin ‘quarter’ represents 25 cents and is taught during this unit. Clarification: Have students shade their paper clocks into 4 quarters. Call attention to the fact that 4 quarters equal 1 dollar - and we name it a ‘quarter’ because it’s a quarter of a dollar.”
- In Unit 8, Lesson 3, Introduction, students are provided explicit instruction in the definition of an array, horizontally, and vertically, as they organize objects of equal groups into rows and columns. The materials state, “An array has rows that go horizontally, or side to side, and it has columns that go vertically, or up and down (fill in on VA). In an array, all of the rows are equal and all of the columns are equal. We can think of the rows as our groups (teacher rows and labels groups) and the columns help us see how many we have in each group (label on VA).”
- In Unit 9, Lesson 3, Introduction states, “Yesterday you used pattern blocks to divide shapes into equal parts-halves, thirds, and fourths. (TT) Teach your partner what you know about halves, thirds, and fourths. Halves means 2, the whole is divided into 2 equal parts, thirds means the whole is divided into 3 equal parts, fourths means the whole is divided into 4 equal parts.”
Examples of the materials using precise and accurate terminology and definitions:
- In Unit 2, Lesson 5, Introduction, accurate terminology is used as students use doubles to solve addition and subtraction problems. The materials state, “Skeleton VA: (remember that strategies section should be added/co-created with students during the intro and should include visual representations of strategies at work) 2. Solve. Addition: - Count on -Known doubles facts; Subtraction: -Count up -Count back -Known doubles addition facts.”
- In Unit 3, Lesson 4, $$>\frac{2}{3}$$ Correct representation, students use accurate terminology as they discuss solving a word problem. The materials state, “How did you represent the problem? I represented with a tape diagram. I put 29 in the long box because Anthony has 29 cars and he has more cars than Jason. I put 16 in one small box because Anthony has 16 more cars than Jason. Then I put J in the other small box because we don’t know how many cars Jason has.”
- In Unit 4, Lesson 3, Introduction, accurate terminology is used as students learn to create bar graphs to represent data. The materials state, “You know how to sort objects into categories to create tally charts and pictographs. Today we are going to use those categories to make bar graphs. (VH) Before we can make a bar graph what are some things we need in our graph? (Reveal top quality graph VA as you go) We need a title, categories, category labels, scale, scale labels, and bars! The scale is the number on the side of the bar graph that tells us how many in each category. (TT) How do I know where to stop on my scale? You need to find the category with the largest number, then you need to make sure the scale goes up to that number so that all of your data fits.”
- Unit 6, Lesson 2, Introduction, accurate terminology is used on a skeleton visual aid as students learn to represent three-digit numbers using multiple forms. Teachers fill in the different forms during instruction and leave the VA posted in the classroom. The materials state, “Writing 3-Digit Numbers, Standard Form 235, Place Value Models (Flats, Sticks, and Dots), Base Ten Numeral Form, Unit Form, Word Form, Expanded Form.”
- In Unit 9, Overview, Major Misconceptions & Clarifications states, “Misconception: Students confuse numerator and denominator. Clarification: Have students label their fraction with words. The numerator as the part and the denominator as all of the parts. Have VA posted and labeled and help students reference it until it becomes second nature.”