3rd Grade - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Does Not Meet Expectations | 38% |
|---|---|
Criterion 2.1: Rigor | 6 / 8 |
Criterion 2.2: Math Practices | 1 / 10 |
The instructional materials reviewed for Grade 3 Stepping Stones do not meet expectations for rigor and the MPs. The instructional materials give attention to all three aspects of rigor, but there is not equal emphasis given to the three aspects. Materials do not consistently offer opportunities for conceptual understanding, which under-emphasizes that aspect of rigor. The lesson and assessment materials do not consistently provide opportunities for students to work with all three aspects of rigor in a balanced way. Overall, the instructional materials do not reflect the balance in the CCSSM which helps students meet rigorous expectations by developing conceptual understanding, procedural skill and fluency, and application. The instructional materials do not support the Standards’ emphasis on mathematical reasoning. Sometimes the materials prompt students to construct viable arguments, but they do not consistently prompt them to analyze other students' arguments. They do not assist teachers in engaging students in constructing viable arguments or analyzing the arguments of others. They do not explicitly teach or 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 Grade 3 Stepping Stones partially meet expectations for rigor and balance. The instructional materials give attention to all three aspects of rigor, but there is not equal emphasis given to the three aspects. Materials do not consistently offer opportunities for conceptual understanding, which under-emphasizes that aspect of rigor. The lesson and assessment materials do not consistently provide opportunities for students to work with all three aspects of rigor in a balanced way. Overall, the instructional materials do not reflect the balance in the CCSSM which helps students meet rigorous expectations by developing conceptual understanding, procedural skill and fluency, 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 instructional materials reviewed for Grade 3 partially meet the expectations for developing conceptual understanding of key mathematical concepts. Overall, the instructional materials do not consistently offer opportunities to use manipulatives and models to develop conceptual understanding. Also, more emphasis should be placed on discussion of mathematical concepts.
- Lessons 1.6, 3.1, 3.2, 3.6, 5.1, 5.2, 5.3, 6.7, 9.5, 9.7 and 9.8 specifically address standards which are explicitly outlined as conceptual standards (3.OA.1 and 3.OA.2).
- Lessons 4.8-4.12, 6.8-6.12, 11.1-11.5 and 12.1-12.8 specifically address standards which are explicitly outlined as conceptual standards (3.NF.A).
- The “Number Case” channel provides resources for students to use models.
- Students are given opportunities to develop multiplication concepts when asked to represent multiplication with cubes and other manipulatives in groups and arrays.
- Lesson 1.6 addresses multiplication concepts (3.OA.1) and includes visual models, discussions of equality of arrays, interpretations of array rows, and equations associated with visuals models.
- Lesson 3.1 develops the concept of multiplication (3.OA.1). Students model multiplication problems with number blocks, write multiplication equations and word problems, and draw a picture to match the story. However, sufficient time is not allowed for students to develop the concept of multiplication as defined by the standards. There is little opportunity for students to explain their thinking/reasoning. Opportunities for student explanations is not addressed in the following sections of the program: observation, discussion, journal, portfolio, or the interview.
- Lesson 3.6 addresses multiplication concepts (3.OA.1) using engaging problems with recipes and groups of guests. Students are asked to interpret multiplication using doubling and quadrupling.
- Lessons 9.5 and 9.7 develop multiplication concepts (3.OA.1). However, the major focus of these lessons is fluency, and therefore does not support the full intent of the standard. Less emphasis is placed on using manipulatives as tools to conceptualize.
- Lessons 5.1, 5.2 and 5.3 develop division concepts (3.OA.2). These lessons introduce the concept of whole-number quotients. There is some use of visual representations to help students understand the concept, but there is little opportunity for students to verbalize their thoughts or understanding and is not provided in the following sections of the program: observation, discussion, journal, portfolio, or the interview.
- Lesson 4.9 addresses fraction concepts (3.NF.A) by using paper squares folded in equal parts to model the same fraction in different ways. Squares are marked with dashed lines for folding or are represented in the student journal. Students are not asked to take squares and partition them into equal parts independently.
- Lesson 6.8 addresses fraction concepts (3.NF.A) by using fractions strips to model how folding certain fractions creates other fractions. Students are provided fraction strips and must individually partition them to show fractions with the denominators 2, 3, 4 and 6.
- Lesson 11.1 addresses fraction concepts (3.NF.A) through the use of area models to identify equivalent fractions. Students are not required to partition shapes on their own. The partitioning is always provided and students do not create the models themselves.
- Lesson 12.1 addresses fraction concepts (3.NF.A) through the use of number line models to identify equivalent fractions. Number lines are provided for students with the partitioning already done. Students do not have to create number lines and then identify fractions.
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 Grade 3 materials meet the expectations for procedural skill and fluency. Fluency is strategically developed, interwoven throughout the curriculum, and connected to conceptual development. In addition, it is very clear that this curriculum expects the development of fluency as described in the grade levels standards.
- “Fundamental Games” provides opportunities for students to practice fluency through games.
- Multiplication/division facts are taught in a designed sequence (e.g., 5s and 10s in Module 1; 2s and 4s facts in Module 3; connection of 5s, 10s, 2s, and 4s facts to division facts in Module 5; etc.) that show a focus on supporting development of fact fluency for multiplication and division.
- Several lessons offer opportunities to fluently multiply and divide within 100. (Modules 2, 3, 5, 7, 8 and 9)
- Fluency is built upon a series of steps in the materials. Students are introduced to a strategy with pictorial support. They then build fluency based on facts related to strategies they have been using.
- Every module includes interview-based assessments which include assessment of student fact fluency and often include recording of student strategy use.
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 reviewed for Grade 3 meet the expectations for students to spend sufficient time working with engaging applications of the mathematics. Overall, the instructional materials consistently offer opportunities for students to engage in application of learning to real-world situations.
- Lessons 3.6, 5.1, 5.4, 5.5, 6.7, 7.5, 8.9, 9.7 and 10.5 specifically address standards which are explicitly outlined as application standards (3.OA.3).
- Lessons 1.12, 2.12, 3.6, 6.7, 7.5, 8.6, 9.7, 9.10-9.12, 11.12 and 12.3 specifically address standards which are explicitly outlined as application standards (3.OA.8).
- Students are exposed to single and multi-step contextual problems and non-routine problems.
- Each module contains one lesson focused on “Solving Real World Problems.”
- "Stepping Into Financial Literacy" offers lessons in financial literacy.
- Each module contains three investigations that require application of learning.
- A variety of one- and two-step story problems are presented in the lessons and assessments.
- Lesson 3.6 extends student exploration with standard equal groups and rows using word problems that introduce the concept of making multiples of a recipe.
- Lessons 9.5 and 9.6 provide abstract, interesting game contexts to help students explore sets of factors that create the same product and the order of operations and its effect on the size of a total.
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 materials reviewed for Grade 3 partially meet the expectations for balance between the three aspects of rigor with the grade.
- Although all three aspects of rigor are present in the materials, there is not a balance among the three aspects of rigor. There is an under-emphasis on conceptual understanding compared to the emphasis given to procedural skill and fluency.
- Students assessments do not offer a balance of rigor and are often missing questions requiring application.
- Each module contains one lesson focused on “Solving Real World Problems” and three investigations that require application.
- There are a few areas where the three aspects of rigor are well balanced. For example, in lesson 3.6, students develop conceptual understanding of interpreting contexts where a recipe needs to be multiplied by a factor, work through a range of more difficult application problems and develop fluency with creating multiplication equations to model more familiar multiplication contexts, and work with strategies (double a double when multiplying by 4).
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 3 do not meet the expectation for supporting the Standards’ emphasis on mathematical reasoning. Sometimes the materials prompt students to construct viable arguments, but they do not consistently prompt them to analyze other students' arguments. They do not assist teachers in engaging students in constructing viable arguments or analyzing the arguments of others. They do not explicitly teach or 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 Grade 3 partially meet the expectations for identifying the MPs and using them to enrich mathematics content within and throughout Grade 3. Overall, the instructional materials identify the MPs but do not consistently use them to enrich the content. Also, some mathematical practices are over-identified and some are under-identified.
- MPs are identified in the “Steps” portion of each module lesson.
- MPs are identified throughout all 144 lessons. Each lesson has at least one MP as the focus.
- MPs are embedded within lessons.
- A chart for each module, under the Mathematics tab, identifies MPs by lesson.
- Videos can be found under the Resources tab which explains the MPs and Habits of Mind.
- MPs are not specifically listed on assessments.
- MP2 was over-identified in the materials with 70 lessons out of 144 addressing this practice.
- MP5 was greatly under-identified with only 6 lessons out of 144 and was found in 3 of the 12 modules.
- Module 4 does not address MP1, MP3, or MP5.
- Explanations of how the MPs are being used and what to expect from students to show growth or mastery is not provided in the “Steps” portion of the lesson.
Indicator 2f
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for Grade 3 do not meet the expectations for materials carefully attending to the full meaning of each MP. Overall, the instructional materials do not meet the full meaning of four or more MPs.
- MP1: Often routine problems are given and are quoted as MP1. MP1 is not fully addressed in lessons 3.10 and 5.1. Students are not asked to make meaning of problems and are asked to work in small groups, not independently. This MP is typically assigned in this curriculum when a problem type is being used in the whole class discussion or in a game that is more difficult due to being slightly out of the range/focus of the grade (i.e., rate contexts with multiplication/division or has multiple steps) or requires generating multiple answers. There is little evidence this MP is addressed regularly on student independent work or on assessments.
- MP2: Students are asked to reason abstractly when introducing vocabulary as part of lessons in Module 2, but they are not asked to use the same skills during the lesson or ongoing practice. In lessons 5.2 and 8.1, students are asked to represent their thinking symbolically but not required to manipulate the symbols.
- MP4: Modeling is not always tied to a real-life context. For example, in lesson 3.1, there is no real-life context provided when using multiplication or addition equations. Also, when creating mathematical models, students are told which models to use and are not provided opportunities to select for themselves. For example, in lesson 9.6, students are taught adding fractions using number lines and fraction models, but during independent practice, they are instructed to use number lines.
- MP5: The tools are usually chosen for the student and the teacher rather than allowing the student to choose the tool. For example, in lesson 1.3 students are told to use a number line and in lesson 2.1 students are told to use a pan balance.
- MP6: Students are given multiple opportunities to use a variety of mathematical symbols including the equal sign. However, students are often provided with equations and do not get to state the meaning of the symbols that they choose.
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 3 do not meet the expectation for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, the materials sometimes prompt students to construct viable arguments, but they do not consistently prompt them to analyze other students' arguments.
- There are some occasions where students are asked to present an argument when prompted by a teacher to tell how they know; however, opportunities for constructing arguments are rarely found in the student journal.
- An example where this is done well: Module 2, Lesson 5, where students explain their thinking. Student pages include these questions: “Which strategy on this page do you like the best? Why?” These questions prompt the critique of others’ thinking.
- There are many missed opportunities including:
- Module 1, Lesson 2: Students are not prompted by the materials to construct an argument or explain their thinking.
- Module 7, Lesson 5: Students explain why the letter t is chosen as the variable and create questions and number sentences relating to a menu, but there is no opportunity to analyze the thinking of others.
- MP3 is not identified in Modules 3, 4 or 5 and minimally so in the other Modules. Students occasionally construct viable arguments, but rarely critique the reasoning of others (i.e., lesson 1.7 and 2.3). Students are often encouraged to share their strategies, but are not provided guidance on critiquing others (i.e., lesson 7.5 and 11.6).
- No opportunity for students to reconsider their own argument in response to the critique of others.
- No opportunity for students to construct an argument and/or analyze the thinking of others on assessments.
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 Grade 3 do not meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others.
- Teacher materials sometimes prompt students to have discussions, but do not provide true opportunities for students to construct arguments or analyze the arguments of others. For example, in Module 8, Lesson 2, students are only prompted to discuss “How is the recording of the steps the same or different from other methods we have used?”
- The materials do not provide ample open-ended questions that allow students to grapple with concepts.
- Many questions are low depth of knowledge and do not promote deep understanding of the concepts. (e.g., Module 3, Lesson 1, "What amount do we have in each place value? What will we write in each place so we know the amount when we do not have base-10 blocks?")
- There are many missed opportunities for teachers to prompt students to analyze the arguments of others.
- There is no evidence of supporting teachers in helping students create viable arguments. There are instances in lessons (especially at the beginning) that do ask questions where arguments/responses would be created and discussed, but there is no guidance for the teacher on how to assist students in creating clear arguments.
- There is no support for teachers in supporting students critiquing each other.
Indicator 2g.iii
Materials explicitly attend to the specialized language of mathematics.
The materials reviewed for Grade 3 do not meet the expectation for attending to the specialized language of mathematics.
- There is little to no explicit instruction on how to use the language of mathematics.
- There are instances where using precise and accurate mathematical language is avoided.
- The commutative property is referred to as the "turn-around facts."
- In Module 1, lesson 6, the introduction of the array is not supported with precise language. “Introduce or remind students of the term array which is used to describe this type of arrangement of objects.” (rows of/groups of.)
- In Module 6, lesson 10, numerator and denominator are referred to without being defined for students. Fold is used instead of partition in reference to fractions. Students are not prompted to use the precise language of numerator and denominator when referring to numbers in a fraction.
- In Module 5, lesson 9, no specific definitions are used to introduce vertices. The Student Journal page refers to the vertices as corners. Students sort shapes into “shapes with dents” and “shapes without dents.”
- Students are not always prompted to use precise language when writing and speaking about mathematics.