2nd Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Does Not Meet Expectations | 55% |
|---|---|
Criterion 2.1: Rigor | 5 / 8 |
Criterion 2.2: Math Practices | 5 / 10 |
The instructional materials reviewed for Grade 2 do not meet the expectations for rigor and mathematical practices. The instructional materials partially meet the expectations for rigor and do not meet the expectations for mathematical practices.
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 Grade 2 partially meet expectations for rigor and balance. The materials include specific attention to both conceptual understanding and procedural skill and fluency; however, there are limited opportunities for students to work with engaging applications. As a result, the materials do not exhibit a balance of the three aspects of rigor.
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 for Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
Clusters 2.MD.A, 2.NBT.A, and 2.NBT.B focus on measuring and estimating lengths in standard units and understanding and using place value and properties of operations to add and subtract.
- Teacher Resources Part 1 Unit 5 has students use charts, tens and ones blocks, and playing cards to develop place value. In Lesson 8 students use base ten blocks to add without regrouping (2.NBT.5).
- In Teacher Resources Part 1 Unit 7, students focus on measuring lengths in metric units. In Lesson 4 students work with centimeter cubes and centimeter rulers to measure objects. In Lesson 6 students estimate the length of an object by measuring with their finger (approximately 1 centimeter) (2.MD.A).
- Teacher Resources Part 2 Unit 2 focuses on three-digit numbers. Students use base tens blocks, charts, place value cards, and equations to use place value to understand three-digit numbers and to add and subtract three-digit numbers. Teacher Resources Part 2, Unit 3 continues by developing strategies for large numbers.
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 materials for Jump Math Grade 2 meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.
- The teacher's edition gives strategies for mental math in the teacher's edition page A-30. The strategies are not incorporated into the lesson plans for the teacher.
- There are games in the teacher's edition pages A45-A46 that help to build student fluency. These games focus on pairs of numbers that count to 10, but these games are not mentioned in the lessons.
Standard 2.OA.2 requires students to fluently add and subtract within 20.
- Much of the work in Grade 2 is around adding and subtracting. Teacher Resources Part 1 Units 1, 2, 3 and 4 and Teacher Resources Part 2 Units 1 and 3 focus on Operations and Algebraic Thinking standards.
- Teacher Resources Part 1 Unit 2 Lesson 19 has students developing fluency in their knowledge of pairs of numbers that add to 10 by using their fingers, tables, equations, and ten-frames. Students continue adding in Lessons 20 and 21, this time to more than 10. Students continue with working towards fluency in the remaining lessons of the unit and the following unit.
Standard 2.NBT.5 requires students to fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
- Students work with strategies based on place value, properties of operations and/or the relationship between addition and subtraction in Teacher Resources Part 1 Units 5 and 6 and Teacher Resources Part 2 Units 2 and 4, as well as throughout their work with 2.OA.2 (mentioned above).
- Teacher Resources Part 1 Unit 6 Lesson 18 has students subtracting from a multiple of ten to develop fluency in subtraction with 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 do not meet the expectation for being 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.
There is little evidence of the opportunity to work with engaging applications of the mathematics. There are very few non-routine problems throughout the year. Word problems are present in the materials, but the context has no bearing on the mathematics. Students are asked to use real-world items frequently, but these are used in place of counters and are more of a tool.
- Few problems regarding 2.OA.A, "Represent and solve problems involving addition and subtraction," include meaningful context.
- In Teacher Resources Part 1 Unit 3 Lesson 32, the teacher walks students through a word problem by writing the sentences on the board, talking about each sentence, and writing the number sentence that correlates. Then the teacher writes "There were five ladybugs. ___ more ladybugs came. Now there are seven ladybugs." The teacher is prompted to write the number sentence (in vertical form) on the board by asking students what to write from the first, second, and last sentence.
- In Teacher Resource Part 2 Unit 1 Lesson 47 students are deciphering comparing word problems. Students are told to circle what is more and underline what is fewer. Given Emma has 3 fewer pencils than David, students should underline Emma and circle David.
Most word problems are single step and routine.
- Teacher Resources Part 1 Unit 1 Lesson 8: "6 snails, 2 more snails come, how many altogether?"
- Teacher Resources Part 1 Unit 2 Lesson 25: The teacher is told to draw circles to show subtraction. "How many are left? 3 boys, 2 go away."
- Teacher Resources Part 2 Unit 1 Lesson 48: "Rani has 9 more tetras than goldfish. Rani has 17 tetras. How many goldfish does she have?"
- Teacher Resources Part 2 Unit 3 Lesson 54: "There were 18 apples on the table. Rick put some more apples on the table. Now there are 25 apples. How many more apples did he put on the table?"
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 partially meet the expectation that the materials balance all three aspects of rigor with the three aspects almost always treated separately within the curriculum including within and during lessons and practice. Overall, many of the lessons focus on procedural skills and fluency with few opportunities for students to apply procedures for themselves. There is a not a balance of the three aspects of rigor within the grade.
- The three aspects of rigor are not pursued with equal intensity in this program.
- Conceptual knowledge and procedural skill and fluency are evident in the instructional materials. There are multiple lessons where conceptual development is the clear focus.
- The instructional materials lack opportunities for students to engage in application and deep problem solving in real world situations.
- There are very few lessons that treat all three aspects together due to the relative weakness in application. However, there are several lessons that include conceptual development leading to procedural practice and fluency.
- There are minimal opportunities for students to engage in cognitively demanding tasks and applications that would call for them to use the math they know to solve problems and integrate their understanding into real-world applications.
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 Jump Math Grade 2 do not meet the expectations for practice-content connections. Although the materials meet expectations for identifying and using the MPs to enrich mathematics content, they do not attend to the full meaning of each practice standard. Overall, in order to meet the expectations for meaningfully connecting the Standards for Mathematical Content and the MPs, the instructional materials should carefully pay attention to the full meaning of each MP, especially MP3 in regards to students critiquing the reasoning of other students and teachers engaging students in constructing viable arguments and analyzing the arguments of others.
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 Grade 2 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
In Teacher Resources Part 1, a discussion of MPs is given starting on page A-21 . According to a statement in the materials, “We guide students to develop Mathematical Practice Standards by explicitly teaching the skills required.” The MPs are not listed in the beginning with the lesson goals but in parentheses in bold within the lesson at the part where they occur. As stated on page A-22 in Teacher Resources Part 1, "While the development of these practices occur in virtually every lesson, only some lessons have grade-level application to the standards. These grade-level applications are identified in the margin"
Overall, the materials clearly identify the MPs and incorporate them into the lessons. The MPs are incorporated into almost every lesson; they are not taught as separate lessons. All of the MPs are represented and attended to multiple times throughout the year, though not equally. In particular, MP5 receives the least attention.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 2 do not meet the expectations for carefully attending to the full meaning of each practice standard. The publisher rarely addresses the Mathematical Practice Standards in a meaningful way.
The materials only identify examples of the Standards for Mathematical Practice (MPs), so the teacher does not always know when a MP is being carefully attended to. MPs are marked throughout the curriculum, but sometimes the problems are routine problems that do not cover the depth of the Math Practices. Many times the MPs are marked where teachers are doing the work.
Examples where the material does not meet the expectation for the full meaning of the identified MP:
- MP1: In Teacher Resources Part 1 Unit 2 Lesson 15, the teacher is drawing circles on the board, asking students how many circles were drawn, and then writing the numeral and the number word. Students are not problem solving or persevering. In Teacher Resources Part 2 Unit 5 Lesson 17, the teacher draws a set of pattern blocks 7 units long and below it one of 8 units in length. Between the two pattern block lengths, the teacher draws a line. Students must explain if the line is closer to 7 or 8 units. This occurs during direct instruction, and students do not have to preserve in solving any problems.
- MP5: There are few opportunities for students to engage in MP5. In Teacher Resources Part 1 Unit 3 Lesson 27, students are given a Black Line Master and told to use the "hexagon to write the addend they start from, trace the number of blanks needed, and count on." Students are not choosing appropriate tools strategically. In Teacher Resources Part 1 Unit 8 Lesson 14, students are told to check measurements with string and a ruler.
- MP6: In Teacher Resources Part 1 Unit 2 Lesson 12, the teacher is drawing number lines and asking questions to the whole class. Precise language is being used, but the nature of direct instruction has the students repeating the correct vocabulary word rather than communicating with others.
- MP7: In Teacher Resources Part 1 Unit 1 Lesson 6, in Activity 8 students are supposed to say the next even or odd. In this activity students are not identifying the structure of odd and even; they are just identifying odd and even numbers after 7 activities to practice the skill. In Teacher Resources Part 2 Unit 5 Lesson 17, the teacher places pattern blocks next to a ruler and asks students "how long a pattern block is? (1 inch) How many inches long are five pattern block squares? (5 inches)." Students are not determining the structure on their own.
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 Grade 2 partially meet expectations that the materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. Materials occasionally prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards; however, there are very few opportunities for students to both construct arguments and analyze the arguments of others together.
In the lessons provided in the Teacher Resources Part 1 and 2, examples identified as MP3 are often in a whole group discussion, though there are occasional suggestions for students to work in groups. Students rarely have the opportunity to either construct viable arguments or to critique the reasoning of others in a meaningful way because of the heavy scaffolding of the program. For example, in the Teacher Resources Part 1 Unit 4 Lesson 41, the teacher walks students through word problems asking specific questions. "SAY: Let's try another pencil problem. Write on the board: You had 7 colored pencils. You gave some to a friend. Now you have 3 left. How many many did you give to your friend? Read what you have written and SAY: We can show this by drawing." The teacher then draws pictures and explains how to solve the problem. The next problem follows suit but asks "How many balloons do we need to draw? How many balloons were there to start? (6) How many balloons were left at the end? (2) How many balloons should we count? (2)" These questions lead to understanding but do not address MP3 by having students construct their own arguments and/or critiquing the reasoning of others. Another example is Teacher Resources Part 1 Unit 6 Lesson 20 page 29. In an activity, students are asked to determine the number of base ten blocks left in a bag after some have been taken out. Students are not asked to justify to prove their 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 materials reviewed for Grade 2 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
Within lessons, the teacher materials are not always clear about how teachers will engage and support students in constructing viable arguments or critiquing the reasoning of others. Materials identified with the MP3 standard often direct teachers to "chose a student to answer" or "have a volunteer fill in the blank." Questions are provided but often do not encourage students to deeply engage in MP3. In addition, although answers are provided, there are no follow up questions to help re-direct students who didn’t understand. Few problems or activities are labeled as MP3.
- Teacher Resources Part 1 Unit 7 Lesson 1 : Students are shown a picture of two different sized pencils. The materials prompt teachers with the following: "ASK: Which pencil will be longer when I line them up correctly? Explain how you know. ASK: Is there more extra red at one end or extra blue at the other end? (more extra blue at one end than extra red at the other end) Line up the pencils correctly to check the prediction. Repeat with pencils that gradually become more similar in length." The prompts do not assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others with this particular activity.
- Teacher Resources Part 1 Unit 1 Lesson 2: In the extension section the directions state, "The addition sentence has one number that does not match the picture. Write the correct number." There are no directions for the teacher in how to help students engage in constructing viable arguments or analyzing the arguments of others.
- Teacher Resources Part 1 Unit 6 Lesson 18: In the extension section, students are asked “Why do the underlined numbers go up by 10?” The given response is “The totals go up by 10." There is one response and no information or directions for the teacher to foster students ability to construct arguments and analyze the arguments of others.
Overall, some questions are provided for teachers to assist their students in engaging students in constructing viable arguments and analyzing the arguments of others; however, additional follow-up questions and direct support for teachers is needed.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Jump Math Grade 2 partially meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit, but there are times the materials do not attend to the specialized language of mathematics.
Although no glossary is provided in the materials, each unit introduction includes a list of important vocabulary, and each lesson includes a list of vocabulary that will be used in that lesson. The teacher is provided with explanations of the meanings of some words.
- In Teacher Resources Part 1, page A-21 states that “words being introduced and defined for the first time are presented in bold font in the list and in italics in the lesson plans.”
- Vocabulary words are listed at the beginning of each lesson plan in the Teacher’s Guide, but definitions, if any, are within the lesson.
While the materials attend to the specialized language of mathematics most of the time, there are instances where this is not the case.
- Often students are not required to provide explanations and justifications, especially in writing, which would allow them to attend to the specialized language of mathematics. For example, in Teacher Resources Part 1 Unit 7 Lesson 2 vocabulary includes the terms length, longer, measure, unit, and unit of measurement. Each time, however, that these words are used in the lesson, they are used by the teacher. The student is not required to provide an explanation or justification for their answers that would allow them to use the words in this lesson.
- Many of the discussion prompts provided are guided by the teacher so that the student is merely repeating the teacher's language. This limits student ability to actively use mathematical language.