1st Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 66% |
|---|---|
Criterion 2.1: Rigor | 8 / 8 |
Criterion 2.2: Math Practices | 4 / 10 |
The instructional materials reviewed for Grade 1 partially meet the expectations for rigor and MPs. The instructional materials meet the expectations for the criterion on rigor and balance but do not meet the expectations of the criterion on practice-content connections. Overall, the instructional materials are stronger in regards to conceptual understanding and application.
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 1 meet the expectations for rigor. The instructional materials meet the expectations for the criterion for conceptual, fluency and procedure and application. Conceptual understanding gets much more emphasis than the other two aspects of rigor. Overall, the instructional materials are stronger in regards to conceptual understanding and application.
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 materials meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. There are good conceptual discussion pieces located throughout the materials. Some good conceptual Home-Link and practice problems exist; however, these come before the focus lessons, which, without the lesson to understand the concept, could present possible issues for the students.
Below are lessons where the full depth of conceptual understanding is addressed.
- Lesson 4-11: Students begin to compare numbers on a 100 chart in columns and rows and discuss what the numbers have in common.
- Lesson 5-1 and Lesson 5-2: Students use concrete representations (base-10 blocks) along with number grids, 100's charts, and calculators to explore place value for 10's and 1's.
- Lesson 5-6: Students fill in a 100's chart while the teacher asks about digits in the 10's and 1's places and relates this to ordering and comparing numbers.
- Lesson 6-6: Page 535 has students using double ten frames to make ten as a strategy.
- Lesson 7-1: Students explore fact families using dominoes.
- Lesson 8-2: Students explore halves by partitioning pancakes and crackers.
- Lesson 9-5: Students are asked to share their strategies and solutions for adding 55 and 35 including possible use of concrete models such as base-10 blocks, mentally counting up from the larger number, and changing to an easier number by making 10's. Students are similarly encouraged to use a strategy to subtract.
- Most lessons in the materials have a "Math Message" which targets conceptual understanding.
- Most lessons call for students to use concrete and/or visual representations when solving problems. For example, in lesson 7-5, students are introduced to the meter and are estimating and measuring lengths.
In addition, the following routines also build conceptual understanding:
- Routine 1, Number of the Day, engages students in counting concrete objects by ones, bundling and making 10's, and bundling to make 100.
- Routine 2 ,Calendar Routine, references 1.NBT.C and asks if students can "mentally find 10 less or 10 more than a date on the calendar."
Lessons which partially meets the requirements for conceptual understanding are listed below.
- Lesson 2-2 introduces 10-frames and 10-frame Top-It, which introduces students to the idea of grouping counters (dots) in groups of 10. The lesson still does not explicitly link the 10-frames to place value.
- In Lesson 5-8, students exchange base-10 blocks, exchanging 1's for 10's but are still referring to these blocks as flats, longs, and cubes, missing the connections with hundreds, tens, and ones.
- In Lesson 6-10, students work with base-10 blocks and place-value mats to solve riddles such as "show 4 longs and 6 cubes. Is this number larger or smaller than 64? How do you know?" This lesson is using the terms longs and cubes rather then 10's and one's.
Lessons which miss opportunities to develop conceptual understanding are listed below.
- Lesson 1-3 references 1.NBT.B in the Penny-Dice game. The game, if played as described in this lesson, misses an opportunity for students to group 10 counters (pennies) to show a "bundle" of 10. The "Observe" question asks "What strategies do children use to count their total pennies?" but strategies have not been explicitly taught in or before this lesson or in the game.
- Lesson 1-5 references 1.NBT.B in the Top-It Game, but it asks if students use a number line to compare numbers rather than using place value.
- In Lesson 4-5 on page 338, Building with base 10 blocks, "Children compose shapes with base-10 blocks and gain familiarity with the names, shapes, and sizes of these manipulatives." In this activity, children refer to base-10 blocks as flats, longs, and cubes rather than 100's, 10's, and 1's.
A concern does exist concerning the lack of lessons for some of the standards. Students may not be able to develop a deep conceptual understanding of the following standards:
- 1.OA.A.1: "Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem." Much of the work in this series focuses on addition and subtraction within 10.
- 1.NBT.B.2: "Understand the two digits of two-digit numbers represent 10's and 1's." Only lessons 5-1, 5-2 and 5-3 address this standard with a few more questions in math journal and/or homework.
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 Grade 1 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
According to the spiral tracker, there are 399 exposures to 1.OA.6 in the instructional materials. Approximately fifty-six different lessons address 1.OA.6 in the Focus portion of the lesson.
There are some places where fluency is given attention in the materials.
- Most lessons in the materials have a "Mental Math and Fluency" piece which allows for students to practice fluencies required in first grade.
- Several online games help students with the expectation of fluency, including Top It, Plus or Minus, Beat the Computer, and Bingo.
- Most lessons have a "Practice" section which has students practicing skills and building fluency. For example, lesson 7-3, page 635 is "Drawing a Picture Graph."
- Online is a reference sheet called "Do Anytime Activities" with suggestions to help students build fluencies at home.
Note: On page 157 lesson 2-4 states “Tell children that they should subtract the smaller number from the larger number to find the difference.” This procedure will set students up for procedural misconceptions later in mathematics.
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 materials 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.
Each unit contains a two-day "Open Response" lesson which engages students in application of mathematics. For example, lesson 4-4 has students engaging in application of the mathematics by measuring markers. Online in the resource section, several "Projects" are available to help students with application of mathematics.
Word-problem contexts are generally familiar to first grade students including children playing, classroom materials, books, fish and dogs. In addition to word problems provided within some of the daily warm ups (24 warm ups include word problems) and focus lessons, the student journals provide many opportunities to engage students in working with word problems. Add-to, take-from, and result unknown problems are the most frequently presented.
Home-links and math journal problems focus on one-step story problems. The number of one-step story problem lessons create a good sense of application, but only for numbers under 10. This is an area of concern.
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 Grade 1 Everyday Mathematics instructional materials meet the expectations for balance. Overall, the three aspects of rigor are neither always treated together nor always treated separately within the materials.
While some lessons target each individual aspect of rigor, some lessons focus on more than one aspect of rigor. For example, some lessons blend application and conceptual understanding, and some lessons use conceptual knowledge to help build procedural skill and fluency.
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 Grade 1 do not meet the expectations for practice-content connections. The materials only partially meet the expectations for attending to all of the indicators 2e through 2g, except for 2f which did not meet expectations. 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 every practice standard, especially MP3, in regards to students critiquing the reasoning of other students, and the use of correct vocabulary throughout the materials.
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 1 partially meet the expectations for identifying the MPs and using them to enrich the mathematics content. Each of the standards is identified in the Grade 1 materials. The practices are not under-identified. For example, Unit 5, page 393 discusses how MP2 and MP6 unfold within the unit and lesson. Within the lesson are spots where the MPs are identified. However, within the lessons, it does not give teachers guidance on how to help students with the MPs. Because there is limited guidance on implementation, it is difficult to determine how meaningful connections are made. MP3, MP4 and MP8 are the least identified in the Grade 1 materials. MP1 and MP6 are over-identified.
The Assessment Handbook includes "Mathematical Practices for Unit(s) Individual Profile of Progress" that can be used to assess practice standards. The Beginning-of-Year assessments do not identify any practice standards for assessment, and the Middle-of-Year assessment identifies MP1, MP2, MP3, MP4, MP5, MP6 and MP7 for assessment. The End Of Year assessment identifies MP1, MP2, MP3, MP4, MP6 and MP7 for assessment. MP8 is identified with only on unit assessment while MP1 is assessed in six unit assessments.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The First Grade Everyday Mathematics do not meet the expectation for carefully attending to the full meaning of each practice standard. They do not treat each MP in a complete, accurate, and meaningful way. The lessons give teachers limited guidance on how to implement the standards. Some lessons are attached to standards without having students actually attending to them.
Below are examples where the full intent of the MPs is not met.
- MP1: Lesson 1-2 cites MP1; however, simply having students know a faster way to count their fingers is not having students persevere in a problem. Lesson 2-3 cites MP1; however, simply having students count and model tally marks is not having students make sense of problems or persevere in problem solving. Lesson 3-6 cites MP1; however, telling students to make jumps on a number line does not have students making sense of problems or persevering in solving them. Lesson 6-3 cites MP1; however, simply having students determine if a equation is true or false does not have them making sense of problems or persevering in solving them.
- MP4: Lesson 1-8 cites MP4; however, telling students to use tally marks is not having students choose an appropriate mathematical model, which is the intent of the MP. Lesson 2-2 cites MP4; however, again students are told to use tally marks. Lesson 3-3 cites MP4, and again the lesson tells the students how to model the mathematics.
- MP5: Lesson 1-3 cites MP5; however, telling the students to use the pattern block template is not having students choose an appropriate tool, which is the intent of the MP. Lesson 2-7 cites MP5; however, telling students to use their calculators is not having students choose an appropriate tool. Lesson 3-5 cites MP5 but tells students to use number lines instead of having students choose an appropriate tool. Lesson 3-11 tells students to use calculators instead of letting students choose an appropriate tool. Lesson 6-7 cites MP5; however, telling students to use the Table of Contents is not having students choosing an appropriate tool.
- MP6: Lesson 1-9 cites MP6; however, simply asking students "how many of each type of base 10 blocks were used and how many in all" is not having students attend to precision. Lesson 2-4 cites MP6; however, simply asking students "which problems were the easiest to solve, which problems were hardest to solve, and can you think of any other strategies that might have helped with the hardest problems?" is not having students attending to precision. Lesson 7-2 cites MP6; however, having students race to find the answer on calculators is not having them attend to precision.
- MP8: Lesson 2-6 cites MP8; however, telling students to restate the turn-around rule and counting strategy in their own words is not having students look for and express regularity in repeated reasoning. Lesson 6-4, cites MP8, but telling students that a strategy is called near doubles is not having them look for and express regularity in repeated reasoning.
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 materials partially meet the expectation for prompting students to construct viable arguments and analyze the evidence of others. Although the materials at times prompt students to construct viable arguments, the materials miss opportunities for students to analyze the arguments of others, and the materials rarely have students do both together. There are some questions that do ask students to explain their thinking on assessments and in the materials. Sometimes there are questions asking them to look at other's work and tell whether the student is correct or incorrect and explain.
The following are examples of MP
3 in the assessments:
- Unit 1 Assessment: Students explain their thinking in items 3 and 6, or two of the eight problems.
- Unit 2 Assessment: Students explain their thinking in items 1, 2, 4 and 6, or four of the six problems. They explain their thinking in both problems on the Unit Challenge and items 10 and 11, two of the 11 problems on the Cumulative Assessment.
- Unit 3 Assessment: Students explain their thinking in items 1, 3, and 5, or three of the seven problems, in one of two problems in the unit challenge and in the open response problem.
- Unit 4 Assessment: Students explain their thinking in one of six problems, in one problem on the unit challenge, and three of seven problems on the cumulative assessment.
- Unit 5 Assessment: Students explain their thinking on items 8, 12 and 13, or three of 14 problems and on one of two problems on the unit challenge and on one of three problems on the open response assessment.
- Unit 6 Assessment: Students explain their thinking on items 3, 4 and 10, or three of 11 problems, and on two of 12 problems on the cumulative assessment.
- Unit 7 Assessment: Students explain their thinking on items 3 and 4, or two of 12 problems, and on one of two problems on the open response assessment.
- Unit 8 Assessment: Students explain their thinking or critique another person's thinking on item 9, one of 17 problems and items 3, 4, 7 and 8, four of eight problems on the cumulative assessment.
- Unit 9 Assessment: Students explain their thinking on items 4, 6, 7 and 9, four of nine problems, and on the open response assessment.
- The mid-year assessment includes 6 of 14 problems that ask students to justify, explain, show their thinking or critique reasoning of others.
- The end-of-year assessment includes 8 of 28 problems that ask students to justify, explain, show their thinking or critique reasoning of others.
Examples of opportunities to construct viable arguments: (All pages reference Student Journals)
- Lesson 2-2, page 4: "How did you figure out how many more?"
- Lesson 2-5, page 8: "Tell your partner how you know."
- Lesson 3-2, page 23: "How does solving 3+5 help you solve 5+3?"
- Lesson 3-9, page 36: "How did you know what numbers to write?"
- Lesson 4-1, page 43: "Tell your partner how you know;" page 45: "Can you count up to find the answer? How?"
- Lesson 4-4, page 52: "Explain how you can tell how many without counting."
- Lesson 7-7, page 147: "How do you know what to label the name-collection box?"
- Lesson 7-8, page 149: "How are these pictures alike?"
- Lesson 7-10, page 153: "Raoul wants to show [10's] in the box. Is that right or wrong? Explain."
- Lesson 8-1, page 159: "How do you know that they both show the same number?"
- Lesson 8-2, page 161: "How can you find the rule in problem 4?"
- Lesson 8-4, page 165: "Explain how you know;" page 166: "How can knowing 7+7 help you solve 8+6?"
Examples of opportunities to analyze the arguments of others:
- Lesson 7-6, page 145 of the Student Journal has students explaining if both Dan's and Pam's strategy will get the same answer.
- Lesson 7-10, page 153 of the Student Journal has students explaining if Raoul's mathematical model is right or wrong and why.
- Lesson 9-6, page 201 of the Student Journal has students sharing their answers with a partner and checking answers.
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 partially meet the expectation for 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. In the materials, usually only one right answer is available, and there is limited teacher guidance on how to lead the discussion besides a question to ask. Many missed opportunities to guide students in analyzing the arguments of others exist. Students spend time explaining their thinking but not always justifying their reasoning and creating an argument.
The following are examples of lessons aligned to MP3 that have missed opportunities to assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others:
- In routine number 4, the teacher is asked to have children support their arguments about weather trends based on data from the weather bar graph. No further direction is given.
- Lesson 1-8 cites MP3, but it only gives the teachers questions with right and wrong answers. Additionally, there is no direction for the teacher to help with facilitating a discussion.
- In lesson 4-1, there is a question for teachers to ask, and there is a little direction for the teacher. Additionally, there is no direction on how to get students to analyze the arguments of others.
- Lesson 4-2 does a good job providing questions so that teachers can help students guide constructing their own arguments; however, it has a missed opportunity for students to analyze the arguments of others.
- Lessons 4-7, 4-11, 5-1, and 5-4 cite MP3; however, they do not give the teacher enough direction.
- Lesson 6-7 cites MP3; however, simply asking students to tell what they find interesting about the reference book is not participating in constructing arguments or analyzing the arguments of others.
MP3 is well represented in Lesson 5-8. The teacher directions say to "have children work in groups to make arguments about which object is taller, citing evidence that goes beyond just looking at the objects" (page 439) and includes a sentence frame to help students prepare their arguments.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 1 partially meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics; however, often the correct vocabulary is not used.
- Each unit includes a list of important vocabulary in the unit organizer which can be found at the beginning of each unit.
- Vocabulary terms are bolded in the teacher guide as they are introduced and defined but are not bolded or stressed again in discussions where students might use the term in discussions or writing.
- Each regular lesson includes an online tool, "Differentiating Lesson Activities." This tool includes a component, "Meeting Language Demands," that includes vocabulary, general and specialized, as well as strategies for supporting beginning, intermediate, and advanced ELLs.
- The units do not have lessons or activities dedicated to developing mathematics vocabulary.
- Terms are introduced in the text of the lessons. For instance, when the term "vertices" first appears, the instructions to students are to put their finger on the shape that has exactly four vertices, or corners (1-3, page 65).
- Everyday Math comes with a Reference book that uses words, graphics, and symbols to support students in developing language.
- Correct vocabulary is often not used. For example, turn-around fact is used rather than the term commutative property, number sentence is used instead of equation, name-collection box instead of equivalent equations or equivalent expressions, and big cube instead of base-ten block.