The instructional materials reviewed for Grade 2 are aligned to the CCSSM. Most of the assessments are focused on grade-level standards, and the materials spend the majority of the time on the major work of the grade. The materials are also coherent. The materials follow the progression of the standards and connect the mathematics within the grade level although at times off-grade level content is not identified. There is also coherence within units of each grade. The Grade 2 materials include all three aspects of rigor, and there is a balance of the aspects of rigor. The MPs are used to enrich the learning, but additional teacher assistance in engaging students in constructing viable arguments and analyzing the arguments of others is needed. Overall, the materials are aligned to the CCSSM.
Focus & Coherence
The materials reviewed for Grade 2 meet the expectations for Gateway 1. These materials do not assess above-grade level content, and they spend the majority of the time on the major clusters of each grade-level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are partially consistent with the mathematical progression in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Grade 2 materials are focused and follow a coherent plan.
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The instructional materials reviewed for Grade 2 meet the expectations for focus. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials. The instructional materials spend the majority of the time on major clusters of the grade. This includes 2.OA.A, 2.OA.B, 2.NBT.A, 2.NBT.B, 2.MD.A and 2.MD.B. Overall, the materials meet the expectations for focus.
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Materials do not assess topics before the grade level in which the topic should be introduced.
The instructional materials reviewed for Grade 2 meet the expectations for assessing grade-level content. Overall, the instructional materials can be modified without substantially affecting the integrity of the materials so that they do not assess content from future grades within the summative assessments provided. Summative assessments considered during the review for this indicator include unit post-assessments and Number Corner assessments that require mastery of a skill.
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.
The assessment materials reviewed for Grade 2 meet expectations for focus within assessment. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials.
For this indicator, several pieces in the Assessment Overview section of the Assessment Guide were used to identify summative assessments. On page 3 of the assessment overview, the authors state that the post assessments “examine students’ growth over each three- to four-week period of instruction.” Additionally, on page 4, the authors identify the Number Corner Checkups as a long range assessment which “reflects the Common Core Critical Areas of Focus; check for conceptual understanding, procedural fluency and application of coherent and rigorous standards; and provide a snapshot of each student’s skills near the end of each quarter of the school year." Lastly, the Grade 2 Assessment Map found on pages 11–15 indicates when mastery of each standard is expected and where the mastery standard is assessed. Based on these criteria, the following were considered to be the summative assessments and were reviewed for Indicator 1a:
- All Unit Post-Assessments
- Number Corner Checkups 1–4
- Comprehensive Growth Assessment
- Select Unit Checkpoints where mastery is indicated on Assessment Map:
- Unit 4 M2 S5: Inches, Feet and Yards Checkpoint
- Unit 5 M1 S5: Three-digit Numbers Checkpoint
- Unit 7 M2 S5: Metric Measuring and Fractions Checkpoint
The Unit Post-Assessments that contain above-grade level assessment items are noted in the following list:
- In the Unit 4 Post-Assessment on page 82, questions 6a-c are identified by the authors as assessing proportional reasoning and aligned to MP2 and MP4. For example, an envelope pictured on a grid is described as being 2 bricks tall and a dog is 3 bricks tall; question 6a asks, “How many envelopes would you have to stack up to reach the height of two dogs?” Proportional reasoning is not an expectation in the K – 5 grade band. There are three lessons in Unit 4 Module 3 which address this topic and are identified by the authors as “laying the foundation for the kind of multiplicative comparisons and proportional reasoning that will be expected in future grades.” Therefore, it should not be assessed, and its inclusion is not reasonable. Skipping the assessment question and the associated lessons will not impact the integrity of the grade level standards being taught.
- In the Unit 6 Post-Assessment on page 128, question 7a asks students to draw a line to cut a trapezoid in half and color one-half red. This task is aligned to 2.G.3 (Partition circles and rectangles into two, three, or four equal shares...). It is not intended that students partition any shapes other than circles and rectangles when exploring these fractional parts. All other questions in the assessment are grade-level appropriate, and including the trapezoid does not affect the integrity of the grade level standards taught.
- In the Unit 6 Post-Assessment on page 128, question 7b asks students to color half of a rectangle which has been divided into fourths. The authors have aligned this task to 2.G.3 and states that it supports 2.NF; however, there is no NF domain in Grade 2. This task goes beyond the indicated standard and is more appropriately aligned to 3.NF.3.A (Understand that fractions are equivalent if they are the same size…). In the Scoring Guide, it is stated that “Divisions of half other than that suggested by the lines are acceptable.” Therefore, students are permitted to draw a different line to cut the rectangle in half, and because of this, it is reasonable to leave the question in the assessment.
- In the Unit 7 Post-Assessment on page 146, question 4 asks students to color half of an array shown. This is more of an area fraction model, which is not the intent of standard 2.G.3. Since this occurs in the second-to-last unit of the year, the question and any corresponding lessons could be skipped without loss of integrity to the Unit.
The Comprehensive Growth Assessment (CGA) questions that contain above-grade level assessment items are noted in the following list:
- In the Comprehensive Growth Assessment (CGA) written portion on page 10, question 11 is assessing standard 2.OA.4 to total the number of objects arranged in a rectangular array up to 5 rows and 5 columns. This question is an array of 6 columns. The problem could be modified to fit the intent of the standard.
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Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.
The instructional materials reviewed for Grade 2 meet the expectations for focus on the major clusters of each grade. Students and teachers using the materials as designated will devote the majority of class time to major clusters of the grade.
Instructional material spends the majority of class time on the major cluster of each grade.
The instructional materials reviewed for Grade 2 meet the expectations for focus by spending the majority of class time on the major clusters of the grade. All sessions (lessons), except summative and pre-assessment sessions, were counted as 60 minutes of time. Number Corner activities were counted and assigned 20 minutes of time. When sessions or Number Corner activities focused on supporting clusters and clearly supported major clusters of the grade, they were counted. Reviewers looked individually at each session and Number Corner in order to determine alignment with major clusters and supporting clusters. Optional Daily Practice pages and Home Connection pages were not considered for this indicator because they did not appear to be a required component of the sessions.
When looking at the modules (chapters) and instructional time, when considering both sessions and Number Corners together, approximately 81 percent of the time is spent on major work of the grade.
- Units – Approximately seven out of eight units spend the majority of the unit on major clusters of the grade, which equals approximately 88 percent. Unit 6 did not focus on major work of the grade.
- Modules (chapters) – 24 out of 32 modules spend the majority of the time on major clusters of the grade, which equals approximately 75 percent. Units 1, 2 and 3 had three modules that focused on major work of the grade. Units 4 and 7 each had 3.5 modules that focused on major work of the grade. Units 5 and 8 had all four modules focused on major work of the grade.
- Bridges Sessions (lessons) – 123 out of 160 sessions focus on major clusters of the grade, which equals approximately 77 percent. Major work is not the focus of the following sessions:
- Unit 1, Module 1, Sessions 1-4
- Unit 1, Module 2, Session 1
- Unit 2, Module 4, Sessions 1 and 2
- Unit 3, Module 4, Session 2 and 3
- Unit 4, Module 3, Session 2
- Unit 4, Module 4, Sessions 2 and 4
- Unit 5, Module 2, Session 5
- Unit 6, Module 1, Sessions 1-5
- Unit 6, Module 2, Sessions 1-5
- Unit 6, Module 3, Sessions 1-6
- Unit 6, Module 4, Sessions 1-4
- Unit 7, Module 2, Sessions 1-3
- Unit 8, Module 4, Session 2
- Bridges sessions require 60 minutes. A total of 123 sessions are focused on major work grade work of the grade. Bridges sessions devote 7,380 minutes of 9,600 minutes to major work of the grade. Number Corner activities require 20 minutes. A total of 160 days of Number Corner activities address major work of the grade. Number Corner activities devote 3,200 minutes of 3,400 minutes to major work of the grade. In total 10,580 of 13,000 minutes, approximately 81 percent, is devoted to major work of the grade.
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The instructional materials reviewed for Grade 2 meet the expectations for coherence. The materials use supporting content as a way to continue working with the major work of the grade. For example, students count shapes in categories and then compare the quantities. The materials include a full program of study that is viable content for a school year, including 160 days of lessons and assessment. All students are given extensive work on grade-level problems, even students who are struggling, and this work progresses mathematically. However, off-grade-level content is not consistently identified. These instructional materials are visibly shaped by the cluster headings in the standards; for example, one module is called "Attributes of Two-Dimensional Shapes." Connections are made between domains and clusters within the grade-level. For instance, materials make connections between numbers and operations in base ten and measurement and data. Overall, the Grade 2 materials support coherence and are consistent with the progressions in the standards.
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Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The instructional materials reviewed for Grade 2 meet expectations that supporting content enhances focus and coherence by engaging students in the major work of the grade.
Supporting standard 2.OA.3 is connected to 2.OA.A, 2.OA.B and 2.NBT.B, major work of the grade, throughout the instructional materials. For example, in Unit 3, Module 4, Session 1, standards 2.NBT.6 and 2.NBT.9 support major work of 2.OA.3 by connecting even and odd numbers to place value and addition. Another example is found in the September Calendar Grid. In this Calendar students determine if a group of objects has an even or odd number of members and solve story problems.
Supporting standard 2.OA.4 is connected to 2.OA.B and 2.NBT, major work of the grade, throughout the instructional materials. For example, in Unit 1 Module 2, Session 3 in Workplace 1F "Count and Compare Fives" students are provided cards with various arrays (in rows or columns of 5) and practice counting by 5's with them and compare them to see which is greater; this supports both 2.NBT.2 and 2.NBT.5. Also, in Unit 2 Module 4, Session 1 "Thinking About Twos" and in Unit 4 Module 4, Session 2 "Thinking About Threes," students write equations to match dot arrays. This supports 2.OA.B and 2.NBT.5. The September, October, November, December, and January Number Corner Daily Rectangle activities also connect 2.OA.4 with 2.OA.2 and 2.NBT.5
Supporting standard 2.MD.7 is connected to 2.NBT.2 throughout the instructional materials. For example, these connections can be seen in the September, October, and February Number corner Calendar Collectors and the November Calendar Grid. These activities connect time with skip counting as students are always asked to “count by 5's” whenever determining the time on an analog clock.
Supporting standard 2.MD.8 is connected to 2.OA.1, 2.NBT.2, 2.NBT.5, and 2.NBT.7, major work of the grade, throughout the instructional materials. Work with money strongly supports place value understanding. For example, in the Unit 1 Module 2, Session 3 Workplace 1F "Count and Compare Fives" work with nickels provides a context for practicing counting by 5's. In Unit 5, Module 2, Sessions 1–6 use 5-frames and 10-frames to counting money and build place value understanding. Also, in Unit 7 Module 4, Sessions 1 – 2 students write their own word problems involving money and then solve each other’s problems. This supports 2.OA.1, 2.NBT.5 and 2.NBT.7. Also, the Number Corner March Calendar Collector "Two Quarters a Day" features word problems dealing with money which supports 2.OA.1.
Supporting standard 2.MD.9 strongly supports work with 2.MD.A, major work of the grade. For example, in Unit 8 Module 2, Sessions 4 - 5 in the "Marble Roll Experiment 1 Ramp Height" and in the Module 3, Sessions 1–5 "Marble Roll Experiment 2 Marble Mass," students are involved in an engaging project in which they collect and plot measurement data to see what factors will affect marble rolling distance. These lessons strongly support 2.MD.A as the students practice measuring skills while they measure distances to plot. Another example of the connection between 2.MD.9 and 2.MD.A are the Number Corner April and May Calendar Collector activities "Measuring and Plotting Plant Growth" and "Measuring and Plotting Student Heights."
Supporting standard 2.MD.10 is connected to 2.OA.1 in the instructional materials. Although these connections are seen in the materials, the graphs do not always use a single-unit scale as prescribed by this standard; instead many of the graphs are scaled beyond a single unit. For example, in Unit 3 Module 4, Session 3 a bar graph, not single-unit scale, is used to solve word problems involving addition and subtraction. Also, in the Number Corner December Calendar Collector "Student Surveys," students work with surveys and make pictographs and bar graphs and solve simple put together/take apart word problems.
The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
The instructional materials reviewed for Grade 2 meet the expectations for this indicator by providing a viable level of content for one school year. Overall, the materials have expectations for teachers and students that are reasonable.
- Materials provide for 160 days of instruction. Each Unit has 20 sessions = 20 days. There are eight units. (20x8=160)
- The prescribed daily instruction includes both unit session instruction and a Number Corners session. (170 days). There were no additional days built in for re-teaching.
- Assessments are incorporated into sessions and do not require an additional amount of time. Instead, they are embedded into module sessions one on one as a formative assessment.
- The Number Corner Assessments/Checkups (a total of 10 assessments, 1 interview and 1 written, in each of the following months: September, October, January, March and May) would require additional time to conduct a 7-10 minute interview with each student.
- A Comprehensive Growth Assessment is completed at the end of the year and will require additional number of days to administer. There are no additional time/days built in for additional support, intervention or enrichment in the pacing guide. The publisher recommends re-teaching of strategies, facts, and skills take place in small groups while the rest of the class is at Work Places (math stations) or doing some other independent task. There is a concern that if a particular session’s activities take up most of the 60 minutes allotted, there will be no time for the remediation and enrichment to take place.
- Based on the Bridges Publisher Orientation Video and Guide provided to the reviewers, unit sessions are approximately 60 minutes of each Instructional Day.
- Each Unit session contains: Problems & Investigations (whole group), Work Places (math stations), Assessments (*not found in each session), and Home Connections (homework assignments *not found in each session).
- Based on the introduction in the Number Corners Teacher Guide, as well as the Bridges Publisher Orientation Video, Number Corners sessions are approximately 20 to 25 minutes of each Instructional Day.
- Approximately 80-85 minutes is spent on the Bridges and Number Corner activities daily.
Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.
The instructional materials reviewed for Grade 2 are partially consistent with the progressions in the standards. Although students are given extensive grade-level problems and connections to future work are made, off-grade level content is not always clearly identified to the teacher or student.
At times, the session materials do not concentrate on the mathematics of the grade. Some of the sessions within each module concentrate on below-grade level concepts such as non-standard units of measure, sorting, using concrete objects to determine sums and differences to 20 or above-grade level concepts such as scaled bar graphs, multiplication and division notation, growing patterns, ratio and proportional reasoning, rotational symmetry and fractions of a set. The inclusion of off-grade level concepts takes away from the number of sessions that could be spent more fully developing the work on the mathematics of the grade.
In some cases, the below or above-grade level content is identified as such by the publishers, and in other cases it is not. On the first page of every session, the Skills & Concepts are listed along with the standard to which it has been aligned by the publisher. In some cases, this alerts the user to the inclusion of off-grade level concepts. Examples include:
- Unit 1, page 13: "Classify objects into categories" is identified as a focus and is aligned to K.MD.3, alerting teachers to the fact that this session involves below grade-level concepts.
- Unit 2, Module 2, Session 3, page 13: One of the skills is listed as “Find the perimeter of a polygon, given its side lengths" and is aligned to 3.MD.8, alerting teachers to the fact that this session involves above grade-level concepts.
- Unit 4, module 3, Session 3, page 13: The authors describe activities within the Giant's Door activities in the next three sessions which deal with proportional reasoning lay foundations for the kind of multiplicative comparison and proportional reasoning that will be expected in later grades.
In other cases, the below or above grade-level concepts are not identified as such within the sessions in the "Skills and Concepts" listing or at the beginning of the units in the "Skills Across the Grade Levels" sections. Examples of unidentified below or above-grade level content include:
- Unit 2 Module 2 Session 2, page 7: One of the skills listed is “Determine exactly how much longer one objects is than another,” and it is aligned to 2.MD.4. Within the actual lesson, students are determining the difference in length using non-standard units, while the full intent of 2.MD.4 is to “measure to determine how much longer one object is than another, expressing the length difference is terms of standard length unit.” In cases like this, below grade-level work is not explicitly made known to teachers.
- Unit 3 Module 4 Session 2, page 7: One of the skills listed is “Make a bar graph to represent a data set with up to 4 categories," and it is aligned to 2.MD.10. However, the bar graphs within the lesson are created with a scale of 2 and 2.MD.10 specifically indicates a single-unit scale in its description. In cases such as this, above grade-level work (scaled bar graphs are actually aligned to 3.MD.3) is not explicitly made known to teachers.
- Unit 6, Module 4, Session 2: students determine the number of lines of symmetry in a block. Symmetry is a Grade 4 standard (4.G.3).
- Unit 6, Module 4, Session 3: Students work with transformations including translations, reflections, and rotations. Transformations are not included in the standards until Grade 8.
- Unit 7, Module 2, Session 1: Students work with modeling division situations, a Grade 3 concept.
- Unit 7, Module 2, Session 5: Students make predictions about outcomes. Probability is a Grade 6 concept.
- Unit 8, Module 1: Students solve word problems with 3-digit numbers. This is a Grade 3 concept.
- The content includes non-grade level standards as evidenced by the inclusion of Domain 2.NF. The publisher lists this standard in the curriculum; however, there is no Grade 2 2.NF. For example, 2.NF is found in Unit 7, Module 2, Sessions 1, 2, and 5.
Materials provide students opportunities to work with grade-level problems. The majority of differentiation/support provided is on grade level. Extension activities are embedded within sessions and allow students to engage more deeply with grade-level work. Additional extension activities are also provided online.
Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.
The instructional materials reviewed for Grade 2 meet the expectations for fostering coherence through connections at a single grade, where appropriate and when the standards require. The standards are referred to throughout the materials. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings and include problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.
Instructional materials shaped by cluster headings include the following examples:
- Unit 1, "Figure the Facts," is shaped by 2.OA.B.
- Unit 2, Module 1, Session 5 “Base Ten Riddles” connects 2.NBT.1 and 2.NBT.3 (standards within the 2.NBT.A cluster heading) by having students demonstrate an understanding of a 3-digit number and then write those numbers.
- Unit 2, Module 3, "Adding on the Open Number Line," is shaped by 2.MD.6.
- The Unit 5, Module 1 session learning objective "Use strategies based on place value, properties of operations or the relationship between addition and subtraction to add with sums to 1,000 and subtract with minuends to 1,000" is shaped by to 2.NBT.B.
- Unit 6, Module 1, "Attributes of Two-Dimensional Shapes" is shaped by 2.G cluster heading, "Reason with shapes and their attributes."
Units, Modules, and Sessions that connect two or more clusters in a domain or two or more domains include the following examples:
- In Unit 1, Module 4, Session 1 students use bead strings to count by 5's connecting 2.OA to 2.NBT.
- Unit 1, Module 4, Session 2 makes a connection between three clusters, 2.MD, 2.NBT and 2.OA. Students use the number line to count by 5's and solve word problems.
- Unit 2, Module 1, Session 3 “How Many More? How Many Fewer?” connects 2.OA with 2.NBT by having students solving word problems and demonstrate their understanding that 100 can be thought of as a bundle of 10 tens.
- Unit 2, Module 2 "Measuring Jack's Giant Beans with Tens" connects with 2.NBT and 2.MD.
- Activities in Unit 5, Module 2 connect 2.MD with 2.NBT when working with money organized into 10-frames to make connection with place value.
- Activities in Unit 5, Module 3 connect 2.MD measurement with 2.NBT, understanding place value and using place value understanding to add when measuring with paper clips.
Rigor And Mathematical Practices
The materials reviewed for Grade 2 meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials meet the expectations for the connections between the MPs and the mathematical content. More teacher guidance about how to support students in analyzing the arguments of others is needed.
Rigor and Balance
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The materials reviewed for Grade 2 meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials meet the expectations for the connections between the MPs and the mathematical content. More teacher guidance about how to support students in analyzing the arguments of others is needed.
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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.
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 reviewed in Grade 2 for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.
The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open ended questions that require evidence to show their thinking and reasoning.
The following are examples of attention to conceptual understanding of 2.OA:
- Unit 1, Module 2, Session 2 investigates the number rack structure of units of 5 and 10 with sums to 20. Students build numbers through a chain of reasoning relating the representation of one sum (5+5=10 so 5+6=11) to another sum using the number rack.
The following are examples of attention to conceptual understanding of 2.NBT:
- In Unit 3, Module 4, Session 4 in Problems & Investigations, students complete a "Sticks and Bundles" activity where they discuss estimates and then count the number of sticks using double-digit adding.
- In Unit 5, Modules 1-3, students use hundreds charts, stick bundles, base 10 blocks, and Unifix cubes to represent place value and skip counting up to the hundreds place to solidify conceptual understanding of clusters 2.NBT.A and 2.NBT.B.
- In Unit 8, Module 1, students use number lines, place value squares, and expanded notation to represent place value and skip counting up to hundreds place to solidify conceptual understanding of clusters 2.NBT.A and 2.NBT.B.
- In the December Number Corners, in Activity 2, students are working with craft sticks in singles and bundles of 10 and 100, the number line, and a greater than/less than chart. Students are first asked to represent a number (214) on the number line with the craft sticks and explain their thinking/observations of the final amount built on the place value chart. Next, they explore jumping the number line , starting with jumps of 10 forward and backward then moving to jumps of 100 (214, 314, 414, 514...). Finally, students guess a secret number from the number line; with each guess they are told "greater than, or less than" and the numbers guessed are recorded on a greater than, less than chart. At the same time, students are moving clips on the number line to mark the numbers that have been guessed.
The following are examples of attention to conceptual understanding of 2.MD.A:
- In Unit 1, Module 4, Session 1, students are using the "Number Line to 20 Mat" to represent sums of various numbers within 20.
- In Unit 1, Module 4, Session 2, students are given the number line as one of three tools of choice (number line, bead string, number rack), to solve addition/subtraction word problems.
- In Unit 8, Modules 2-4, students create ramps and roll marbles from various heights and then measure distance that the marble rolled to represent real world measurement to solidify conceptual understanding of cluster 2.MD.A.
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 2 materials 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.
Fluency is developed throughout the sessions of the Grade 2 instructional materials.
- In Unit 3, Module 2, Session 4, the number line is used to subtract within 30, fluently leading to mental strategies involving place value.
- In Unit 4, Module 2, Session 5, students are introduced to the Work Place "Climb the Beanstalk." Fluency is practiced as students roll two dice and add to get the sum. Then they spin the spinner and subtract the amount on the spinner from the sum and move to that space on the game board.
- As students play the Work Place game "Star Power" in Unit 3, Module 1, Session 3, they are building their fluency while seeing who can reach 100 first. Students get to choose the order in which they add, promoting the strategy of anchoring on 10's. The game also allows students to explore the associative and commutative properties of addition.
- In the October Computational Fluency Number Corner, students review and practice combinations to 10 using 10-frames and the activities, "Make Tens" and "Break Tens."
- In the March Computational Fluency Number Corner, students use their "Quick Facts" forms to practice their fluency each day. Students work on various strategies: Count On/Make Ten, Quick Fact Doubles/Doubles Plus or Minus One, Quick Facts Add Nine/Add Ten.
- The April Number Line Number Corner routine reinforces procedural skill leading to fluency by adding and subtracting units of 1, 5 and 10 on a number line to reach a specific number.
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
Materials meet the expectations for having engaging applications of mathematics as they 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 the grade.
Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. Examples of these applications include the following:
- Unit 1, Module 3, Session 3: Students decompose numbers in a variety of ways using the context of a dinner party. Equations and visual representations are used to record combinations for various scenarios related to the context of the dinner party and how to seat the guests.
- Unit 1, Module 4, Session 2: Students engage in application of using number lines and number racks to assist in solving story problems.
- Unit 3, Module 2, Session 3: Students compare and contrast three different solutions to a story problem involving length.
- Unit 7, Module 3, Session 2: Students apply place value and addition knowledge and skills to story problems involving money.
- The Number Corner March Number Line on days 5, 10, 15 and 20: When students play "Put It on the Line," they are solving addition and subtraction word problems on the open number line. Examples of problem types include add to and start unknown.
- The Number Corner January Calendar Grid: As students complete various graphs that represent class data, the teacher creates word problems for students to solve.
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 in Grade 2 meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.
In the Grade 2 materials, all three aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the unit sessions and in Number Corner activities. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve understanding and conceptual understanding.
Although some sessions focus on individual aspects of rigor, some session do combine the three aspects of rigor. For example, in Unit 3, Module 1, Session 2, the session includes all aspects of rigor for domain 2.MD.B and 2.NBT.A. Students investigate measurement through a contextual problem leading to understanding and application of using a number line tool to represent their thinking. Another example is Unit 3, Module 3, Session 5. This lesson includes all aspects of rigor as students create and solve their own story problems. Strategies and solutions are shared as students use various tools and mathematical representations to show evidence of their thinking.
Mathematical Practice-Content Connections
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The materials reviewed for Grade 2 meet this criterion. The MPs are often identified and often used to enrich mathematics content. There are, however, several sessions that are aligned to MPs with no alignment to Standards of Mathematical Content. The materials usually attend to the full meaning of each practice. The materials reviewed for Grade 2 attend to the standards' emphasis on mathematical reasoning. Students are prompted to explain their thinking, listen to and verify the thinking of others, and justify their own reasoning. Although the materials often assist teachers in engaging students in constructing viable arguments, more guidance about how to guide students in analyzing the arguments of others is needed.
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Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
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 the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level Standards for Mathematical Content, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.
The Grade 2 Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The chart is broken down into four categories: Habits of Mind, Reasoning and Explaining, Modeling and Using Tools, and Seeing Structure and Generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Grade 2?" guide for teachers (AG, page 17).
Each Session clearly identifies the MPs used in the Skills & Concept section of the session. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation of the connection between the indicated MP and the Standards for Mathematical Content for the teacher. Examples of the MPs in the instructional materials include the following:
- In Unit 1, Module 2 each of the 6 Sessions address MP7. There is a "Math Practices In Action" reference in two of the six sessions.
- In Unit 2, Module 3 in the Skills & Concepts section, Session 5 lists MP1, Session 1 lists MP2, Session 6 lists MP3, Sessions 1, 2, 4, 5 and 7 list MP7, and Sessions 3 and 6 list MP8.
- Unit 3, Module 3, Session 2 references MP2 within the Problems & Investigations portion of the session in "Math Practices in Action." Students are engaged in the "Presents & Parcels" activity. The teacher helps transition students from quantitative to abstract reasoning when she labels the sketches as "10" instead of showing each of the 10 presents. The labels remind students of the quantities without actually seeing the presents.
- In Unit 7, Module 1, Sessions 2 and 3 reference the MPs within the Problems & Investigations portion of the session as, "Math Practices in Action."
- In the September Number Corner MP1 is referenced in the Calendar Grid; MP2 is referenced in the Computational Fluency and Number Line; MP4 is referenced in Calendar Grid, Daily Rectangle, and Number Line; MP6 is addressed in Calendar Collector; MP7 is addressed in Calendar Grid, Calendar Collector, and Daily Rectangle; and MP8 is addressed in Computational Fluency.
Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These sessions that focus entirely on MPs include the following:
- Unit 1, Module 1, Session 1
- Unit 7, Module 2, Session 1
- Unit 8, Module 4, Session 2
Materials carefully attend to the full meaning of each practice standard
The materials meet the expectations for attending to the full meaning of each practice standard. Each session clearly identifies the MPs used in the Skills & Concept section of the session. Typically there are two standards for MP listed for each session, so there is not an overabundance of identification.
Each Session clearly identifies the MPs used in the Skills & Concept section of the session Typically there are two MPs listed for each session, so there is not an overabundance of identification. Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, the MPs are not discussed or listed in unit overviews or introductions (Major Skills/Concepts Addressed); however, they are listed in section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.
In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Daily Rectangle, Computational Fluency, Number Line). The MPs are also listed in the assessment section of the Introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of the MPs within Number Corner.
The following are examples of times that the instructional materials attend to the full meaning of an MP:
- Unit 1 Module 2 Session 2 attends to MP7. Students are shown a number greater than 10 on a number rack for less than 3 seconds. This elicits the students to make sense of the structure of groups of 5 and 10 since they are unable to count unit by unit.
- Unit 4, Module 3, Session 1 attends to MP6. Students are comparing two different measuring tools: the student-made inchworm and footworm. As they measure things around the room, they first make estimates and then are asked to measure using each tool. After measuring they are asked to discuss the difference between using the footworm and inchworm rulers. They then work in their Student Books, choosing which tool, the inchworm or footworm to measure lengths of various objects, explaining their choices.
- Unit 8, Module 1, Session 3 attends to MP1. In the Problems & Investigations section of the session, students are solving story problems. The teacher models how to restate what the problem is asking, identify the information required to solve it, and make a reasonable estimate before starting to solve the problem. The teacher helps students get in the habit of orienting themselves before beginning their computations; making sense of problems and persevere in solving them.
- In the May Daily Rectangle Number Corner, MP7 and MP2 are attended to as students investigate the relationship between rectangles and area through rectangular arrays. Repeated addition equations are written to express the total as a sum of equal addends.
- At times MP4 was attended to fully in the sessions as students are posed contextual problems that require the use of mathematical understanding and application of strategies. Examples include Unit 3, Module 2, Session 1; Unit 4, Module 4, Session 1; Unit 4, Module 4, Session 3; and Unit 7, Module 3, Sessions 2 and 3.
However, at times the materials only partially attend to the meaning of MP4. Examples include Unit 6, Module 1, Session 2 and Unit 6, Module 2, Session 1.
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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 reviewed for Grade 2 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.
Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.
- In Unit 3, Module 2, Session 2, students reflect upon a variety of strategies and efficiency of jumping by larger numbers on a number line. The context of the problem involves a comparison with the difference unknown. The problem can be solved with both addition and subtraction. This leads to a discussion with students analyzing and critiquing each other’s approach to the solution.
- In Unit 3, Module 3, Session 5 strategies and solutions are shared as students use various tools and mathematical representations to show evidence of their thinking in solving student written word problems. Strategies are compared and contrasted as students investigate effective and efficient ways to solve the problem. Students listen to one another present their strategy and are encouraged to ask questions and add new ideas to their own paper based on the discussion.
- In Unit 4, Module 3, Session 1, students are given inchworms and a footworm to measure various lengths then they are asked their estimations, measurements, strategies, and why they would use inchworms or footworms to measure various sized objects.
- In Unit 5, Module 4, Session 1, students use Unifix cubes to search for patterns in various arrangements. Students are encouraged to share their own ideas about how the pattern will extend beyond what is shown while also considering their classmates' ideas. This provides the opportunity for students to construct viable arguments and critique the reasoning of others.
- In Unit 6, Module 2, Session 3, students work in pairs in their Student Books and with Geoboards to partition rectangles into rows and columns of the same size squares. They show/share their reasoning, expressing their arguments verbally as well as using models, sketches and symbolic notation (Math Practices in Action).
- In Unit 6, Module 2, Session 5 students engage in discussion with the teacher and critique strategies used to find the area of rectangles. Then students are directed to use efficient strategies in pairs and record the areas of the rectangles.
- In the Number Corner February Daily Rectangle during Activity 2, "Daily Deposit," students are adding the base ten blocks in the chart. They are asked to share their solutions and explain their thinking. Again, there is sample dialogue; however, none include samples encouraging students to analyze the thinking of others.
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 2 partially meet the expectations 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. Although the instructional materials often assist teachers in engaging students in constructing viable arguments, there is minimal assistance to teachers in how to guide their students in analyzing the arguments of others.
There are Sessions containing the "Math Practice In Action" sidebars that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. A few of the sessions contain direction to the teacher for prompts and sample questions and problems to pose to students.
Many lessons give examples of teacher/student discourse by providing teachers a snapshot of what questions could be used to generate conjectures and possible student thinking samples. The following are examples of sample discourse:
- Unit 4, Module 1, Session 2
- Unit 7, Module 2, Session 5
- Unit 8, Module 1, Session 5
Although teachers are provided guidance to help students construct arguments and students are provided many opportunities to share their arguments, more guidance is need to support teachers in guiding their students through the analysis of arguments once they are shared. For example, in Unit 5, Module 1, Session 3, students revisit base 10 area pieces. They arrange and count 10 hundreds pieces (mats) to discover that there are 1,000 units in 10 mats. The dialogue prompts teacher to ask: "How many units will be on the mat in all?" "How do you know?" "Which benchmark, 10 or 100 seems more useful in trying to estimate the total number?" "Why?" "Who'd like to share their thinking with the class?" The teacher is not provided enough support to encourage students to critique the thinking of others.
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for Grade 2 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 that is present throughout the materials.
The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.
The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon, and trapezoid and using operations and algebraic thinking terminology such as equation and difference.
- Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
- The online Teacher Materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
- At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
- In Unit 6, Module 2, Section 2, students are exploring area with pattern blocks and use terms such as area, half, triangle, quadrilateral, hexagon, and rhombus.
- In the October Number Corner Daily Rectangle, students are introduced to the terms rectangular array and column while describing units of one as they build conceptual understanding of early multiplication.
The materials reviewed meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning and the purpose of each is explained well. The visual design is not distracting or chaotic and supports learning. The materials support teachers in learning and understanding the standards. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole.
A baseline assessment allows teachers to gather information on student's prior knowledge, and teachers are offered support in identifying and addressing common student errors and misconceptions. Materials include opportunities for ongoing review and practice. All assessments include information on standards alignment and scoring rubrics. There are no systems or suggestions for students to monitor their own progress. Activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.
All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional technology-based, assessment opportunities, and the digital resources are not easily customized for individual learners. Overall, the materials meet the expectations for usability.
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Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
Materials are well-designed, and lessons are intentionally sequenced. Typically students learn new mathematics in the Problems & Investigations portion of sessions while they apply the mathematics and work towards mastery during the Work Station portion of Sessions and Number Corner. Students produce a variety of types of answers including both verbal and written answers. Manipulatives such as 10-frames, craft sticks, and Unifix cubes are used throughout the instructional materials as mathematical representations and to build conceptual understanding.
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
The Sessions within the units distinguish the problems and exercises clearly. In general, students are learning new mathematics in the Problems & Investigations portion of each Session. Students are provided the opportunity to apply the mathematics and work toward mastery during the Work Station portion of the session as well as in daily Number Corners.
For example, in Unit 2, Module 1, Session 3, during "How Many More? How Many Fewer?," students are estimating, counting, and comparing quantities of beans in order to develop place value counting skills. During the Problems & Investigation section of the lesson, students are scooping kidney beans into containers in groups of 10. Students pair-share their estimates, revising as they work, and then discuss as a whole class. In the Student Book page, students are counting out beans, recording results, and calculating how many more beans or fewer beans did they have than 50. In the Work Place, "Scoop, Count, & Compare," students are playing a game where they get three chances to scoop as close to 125 beans as possible. They first create a benchmark by counting 10 or 25 beans. Then they scoop, count, and record the number of beans they actually scoop. Finally, students write an expression to show whether the number of beans they scooped was less than, greater than, or equal to 125; the player who is closest, wins.
Design of assignments is not haphazard: exercises are given in intentional sequences.
The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continued to be reviewed, practiced and extended throughout the year.
The "Skills Across Grade Level" table is present at the beginning of each unit. This table shows the major skills and concepts addressed in the unit. The table also provides information about how these skills are addressed elsewhere in the grade, including Number Corner, and in the grade that follows. Finally, the table indicates if the skill is introduced (I), developed (D), expected to be mastered (M), or reviewed, practiced or extended to higher levels (R/E).
Concepts are developed and investigated in daily lessons and are reinforced through independent and guided activities in Work Places. Number Corner, which incorporates the same daily routines each month (not all on the same day) has a spiraling component that reinforces and builds on previous learning. Assignments, both in class and for homework, directly correlate to the lesson being investigated within the unit.
The sequence of the assignments is placed in an intentional manner. First, students complete tasks as a whole group in a teacher-directed setting. Then students are given opportunities to share their strategies used in the tasks completed in the Problems & Investigations. The Work Places activities are done in small groups or with partners to complete tasks that are based on the problems done as a whole group in the Problems & Investigations. The students then are given tasks that build on the session skills learned for the Home Connections.
For example, 2.MD.6 is Introduced in Unit 2, developed in Units 3 and 5, mastered in Unit 7, and is Reviewed/Practiced/Extended in Unit 8. The standard continues to be developed in Number Corners in Computational Fluency and Number Line sections in all months except March. The standard is continued in Grade 3 as a Reviewed/Practiced/Extended skill. Another example is 2.OA.2. This standard is Developed in Units 1, 2 and 3, and continues to be Developed in all months of Number Corners in Computational Fluency.
There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners. Often, working with concrete and moving to more abstract models as well as verbally explaining their strategies. Students are asked to produce written evidence using drawings, representations of tools or equations along with a verbal explanation to defend and make their thinking visible.
For example, in Unit 5, Module 1, Session 2, students are responding to a place-value task in a variety of ways by estimating, counting, recording, comparing, and adding total amounts. They are building on their place-value understanding to 1,000 as they count large amounts of craft sticks. They first estimate the total number and then work in groups to count and bundle the sticks into 10s and 100s. They then come together to record, compare, and order the number of sticks giving verbal explanations of their models and recording/comparing amounts using "greater than" and "less than" symbols. Finally, they find the total number of sticks and compare it to their original estimates.
Also, in the Number Corner December Number Line, students are responding to place value in a variety of ways: building, sharing, estimating, confirming as they work with craft sticks in singles and bundles of 10 and 100, the number line, and a greater than/less than chart. Students are first asked to represent a number (214) on the number line with the craft sticks and explain their thinking/observations of the final amount built on the place value chart. Next, they explore jumping on the number line, starting with jumps of 10 forward and backward then moving to jumps of 100 (214, 314, 414, 514...). They are asked to share with their neighbor on what number they think the next jump will land; then they confirm their thinking by actually jumping on the number line. Finally, students guess a secret number from the number line. With each guess, they are told "greater than or less than", and the numbers guessed are recorded on a greater than, less than chart. At the same time, students are moving clips on the number line to mark the numbers that have been guessed.
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. Manipulatives are used and provided to represent mathematical representations and provide opportunities to build conceptual understanding. Some examples are the 10-frames, number lines, Unifix cubes and craft sticks. When appropriate, they are connected to written representations.
For example, in Unit 4, Module 1, Sessions 1 and 2, students are exploring customary units of measure using traditional and invented tools (cut-outs of: student's foot - varies in size, teacher's foot - foot, giant's foot - using a yard stick 36"). They measure several things and distances around the room and outside. They compare measurements and discuss the differences in the tools chosen and the resulting measurements.
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
The material is not distracting and does support the students in engaging thoughtfully with the mathematical concepts presented. The visual design of the materials is organized and enables students to make sense of the task at hand. The font, size of print, amount of written directions and language used on student pages is appropriate for Grade 2. The visual design is used to enhance the mathematics problems and skills demonstrated on each page. The pictures match the concepts addressed such as having the characters that are in the story problems placed in picture format on the page as well. Some problems may even require students to use the pictures to solve the story problems.
For example, in Unit 4, Module 2, Session 4, in the Work Place, the design of the students' Record Sheet supports students engaging in thoughtful work with measurement. Based on the card drawn, students are measuring various items in inches. For example, a student may draw a card with "book" and another with "work place bin." They fill out the "Measure & Compare Record Sheet" with the name of the object, length in inches, and number line to show the difference between the two lengths, and they fill in a sentence frame that states the difference between the two objects. Each portion of the sheet is clearly marked with spaces for work. The line is present to scaffold students' number line diagram, and the sentence frame is there to scaffold the sentence.
Also, in the Number Corner December Calendar Collector "Student Surveys," students are collecting survey data. After several whole-class surveys, students create and conduct their own surveys. A graphing sheet is provided for students to organize their data. Data graphs are clear and provide appropriate scaffolding to support second graders in their understanding of graphing data.
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Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
The instructional materials support teachers' learning and understanding of the standards. The instructional materials provide questions and discourse that support teachers in providing quality instruction. The teacher's edition is easy to use and consistently organized and annotated. The teacher's edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Lessons provide teachers with guiding questions to elicit student understanding and conduct discourse that allows student thinking to be visible. Discussion questions provide a context for students to communicate generalizations, find patterns, and draw conclusions.
Each unit has a Sessions page, which is the Daily Lesson Plan. The materials have quality questions throughout most lessons. Most questions are open-ended and prompt students to higher-level thinking.
In Unit 2, Module 1, Session 3, teachers are prompted to ask the following questions:
- "Who would like to share how many dots there are total on the domino I picked-the one with 4 and 3? "
- "What did you all get for the total number of dots on your domino-the one with 5 dots on one-half, and 4 on the other? "
In Unit 4, Module 2, Session 2, as students are working with a number line, the teacher removes numbers so that only the zero and 100 are showing with one blank space in between. Teachers are prompted to ask the following questions:
- "What number goes in the box?"
- "Yes, why would 50 go in the box? Can you say more about why it should be 50?"
- "If we replaced the 50 with a 5 would the number in the empty box change? Why? What would the new number be - how do you know?"
- "The emperor penguin is 45 inches tall. Is that more or less than 3 feet?"
- "Can someone tell us in your own words how you know?"
- "Does anyone see another way to do that?"
- "Would someone like to tell us why that's a good strategy?"
In Unit 7, Module 4, Session 2, students are working to count dots on a double ten-frame, and teachers are prompted to ask the following:
- "What do you think is the same and what is different about these cards?"
- "What else do you notice?"
- "What do you mean they look different?"
- "Can you tell me a bit more? How do they look different?"
In Number Corner April Days In School, the following questions are provided to help students think about numbers on the hundreds grid:
- How many squares are marked? How did you count them? Is there another way?
- What number comes next? How do you know?
- How many tens have we made so far?
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials; however, additional teacher guidance for the use of embedded technology to support and enhance student learning is needed.
There is ample support within the Bridges material to assist teachers in presenting the materials. Teacher editions provide directions and sample scripts to guide conversations. Annotations in the margins offer connections to the MPs and additional information to build teacher understanding of the mathematical relevance of the lesson.
Each of the eight units also have an Introductory section that describes the mathematical content of the unit and includes charts for teacher planning. Teachers are given an overview of mathematical background, instructional sequence, and the ways that the materials relate to what the students have already learned and what they will learn in the future units and grade levels. There is a Unit Planner, Skills Across the Grade Levels Chart, Assessment Chart, Differentiation Chart, Module Planner, Materials Preparation Chart. Each unit has a Sessions page, which is the Daily Lesson Plan.
The Sessions contain:
- Sample Teacher/Student dialogue;
- Math Practices In Action sidebars that provide information on what MP is connected to the activity;
- Literature Connection sidebars that list suggested read-alouds that go with each session;
- ELL/Challenge/Support notations where applicable throughout the sessions;
- and a Vocabulary section that contains vocabulary pertinent to the lesson and indicators showing which words have available vocabulary cards online.
Technology is referenced in the margin notes within lessons and suggests teachers go to the online resource. Although there are no embedded technology links within the lessons, there are technology resources available on the Bridges Online Resource page such as videos, whiteboard files, apps, blogs, and online resource links (virtual manipulatives, images, teacher tip articles, games, references). However, teacher guidance on how to incorporate these resources are lacking within the materials. It would be very beneficial if the technology links were embedded within each session, where applicable, instead of only in the online teacher resource. For instance, the teacher materials would be enhanced if a teacher could click on the embedded link, (if using the online teacher manual) and get to the Whiteboard flipchart and/or the virtual manipulatives.
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.
Materials contain adult-level explanations of the mathematics concepts contained in each unit. The introduction to each unit provides the mathematical background for the unit concepts, the relevance of the models and representations within the unit, and teaching tips. When applicable to the unit content, the introduction will describe the algebra connection within the unit.
At the beginning of each unit, the teacher's edition contains a "Mathematical Background" section. This includes the mathematics concepts addressed in the unit. For example, Unit 3 states, "Unit 3 pushes students towards mastery of key number facts and fact strategies for single-digit addition and subtraction...The ability to subitize is central to a well-rounded sense of numbers and operational fluency in general. To subitize is to give up the need to count every object in a set in order to name the quantity...Unit 3 also emphasizes the concept of part-part-whole reasoning. A precursor to algebraic reasoning, knowledge of part-whole relationships is useful in problem contexts that involve either combining or separating numbers..."
The Mathematical Background also includes sample models with diagrams and explanations, strategies, and algebra connections. There is also a Teaching Tips section following the Mathematical Background that gives explanations of routines within the sessions such as think-pair-share, craft sticks, and choral counting. There are also explanations and samples of the various models used within the unit such as frames, number racks, tallies/bundles/sticks, and number line.
In the implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSSM Focus for Grade 2 Mathematics.
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
Materials contain a teacher’s edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.
In the Unit 1 binder, there is a section called "Introducing Bridges in Mathematics." In this section, there is an overview of the components in a day (Problems & Investigations, Work Places, Assessments, Number Corner). Then there is an explanation of the Mathematical Emphasis in the program. Content, Practices, and Models are explained with pictures, examples and explanations. There is a chart that breaks down the MPs and the characteristics of children in that grade level for each of the math practices. There is an explanation of the Skills Across the Grade Levels chart, the assessments chart, and the differentiation chart to assist teachers with the use of these resources. The same explanations are available on the website. There are explanations in the Assessment Guide that go into the Types of Assessments in Bridges Sessions and Number Corner.
The CCSSM Where to Focus Grade 2 Mathematics document is provided in the Implementation section of the Online Resources. This document lists the progression of the major work in grades K-8.
Each unit introduction outlines the standards within the unit. A “Skills Across the Grade Level” table provides information about the coherence of the math standards that are addressed in the previous grade as well as in the following grade. The "Skills Across the Grade Level" document at the beginning of each unit is a table that shows the major skills and concepts addressed in the Unit and where that skill and concept is addressed in the curriculum in the previous grade as well as in the following grade.
Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
The materials provide a list of lessons in the teacher's edition cross-referencing the standards covered and providing an estimated instructional time for each lesson and unit. The "Scope and Sequence" chart lists all modules and units, the CCSSM standards covered in each unit, and a time frame for each unit. There is a separate "Scope and Sequence" chart for Number Corners.
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
Home connection materials and games sometimes include a “Note to Families” to inform them of the mathematics being learned within the unit of study.
Additional family resources are found at the Bridges Educator's Site.
- Grade 2 Family Welcome letter in English and Spanish - This letter introduces families to Bridges in Mathematics, welcomes them back to school, and contains a broad overview of the year's mathematical study.
- Grade 2 Unit Overviews for Units 1-8 in English and Spanish.
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.
Materials contain explanations of the instructional approaches of the program. In the beginning of the Unit 1 binder, there is an overview of the philosophy of this curriculum and the components included in the curriculum. There is a correlation of the CCSSM and MPs as the core of the curriculum in the Mathematical Emphasis section. The assessment philosophy is given in the beginning of the Assessment binder. The types of assessments and their purpose is laid out for teachers. For example, informal observation is explained as "one of the best but perhaps undervalued methods of assessing students...Teachers develop intuitive understandings of students through careful observation, but not the sort where they carry a clipboard and sticky notes. These understandings develop over a period of months and involve many layers of relaxed attention and interaction."
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Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
The instructional materials offer teachers resources and tools to collect ongoing data about student progress. The September Number Corner Baseline Assessment allows teachers to gather information on student's prior knowledge, and the Comprehensive Growth Assessment can be used as a baseline, quarterly, and summative assessment. Checkpoint interviews and informal observation are included throughout the instructional materials. Throughout the materials, support sections provide common misconceptions and strategies for addressing common errors and misconceptions. Opportunities to review and practice are provided in both the sessions and Number Corner routines. Checkpoints, Check-ups, the Comprehensive Growth Assessment, and Baseline Assessments clearly indicate the standards being assessed and include rubrics and scoring guidelines. There are, however, limited opportunities for students to monitor their own progress.
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
The September Number Corner Baseline Assessment is a 4-page, written assessment designed to ascertain students' current levels of skills targeted for mastery in Grade 1, such as story problems within 20, numbers to 120, place value, measurement, 2-digit addition/subtraction, and fractions. The Comprehensive Growth Assessment contains interview items and written items and addresses every Common Core standard for Grade 2. This can be administered as a baseline assessment, an end-of-the-year summative assessment or quarterly assessment to monitor students' progress.
Informal observation is used to gather information. Many of the Sessions and Number Corner workouts open with a question prompt: a chart, visual display, a problem, or even a new game board. Students are asked to share comments and observations, first in pairs and then as a whole class. This gives the teacher an opportunity to check for prior knowledge, address misconceptions, as well as review and practice with teacher feedback. There are daily opportunities for observation of students during whole group and small group work as well as independent work when they work in Work Places.
Materials provide strategies for teachers to identify and address common student errors and misconceptions.
Materials provide strategies for teachers to identify and address common student errors and misconceptions.
Most Sessions have a Support section and ELL section that suggests common misconceptions and strategies for re-mediating those misconceptions that students may have with the skill being taught.
Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 6 Module 2 Session 3 “Support” section gives suggestions for struggling students. The materials suggest that if students are still unsure that two triangles make up one rectangle, invite them to trace the figure on a sheet of geoboard paper, cut out the triangles, and slide them together to make a rectangle.
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
Materials provide opportunities for ongoing review and practice, with feedback for students in learning both concepts and skills.
The scope and sequence document identifies the CCSSM that will be addressed in the sessions and in the Number Corner activities. Sessions build toward practicing the concepts and skills within independent Work Places. Opportunities to review and practice are provided throughout the materials. For example, in Unit 7, Module 3, Session 2, as students are working on toy store story problems, they are reviewing and practicing their ability to solve a two-step subtraction story problem with minuends to 100 (2.OA.1).
Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, in the Number Corner December Calendar Grid, as students are working with 2-dimensional shapes to find patterns, they are getting review and practice with recognizing that shapes have specific attributes (2.G.1).
Materials offer ongoing formative and summative assessments:
Assessments clearly denote which standards are being emphasized.
All assessments, both formative and summative, clearly outline the standards that are being assessed. In the assessment guide binder, the assessment map denotes the standards that are emphasized in each assessment throughout the year. Each assessment chart notes which CCSSM is addressed.
For example, the Unit 2, Module 2, Session 1 Unit checkpoint includes a Checkpoint Scoring Guide that lists each prompt, the correct answer, each standard, and the points possible. The Unit 5, Module 3, Session 5 Post-Assessment includes a Post-Assessment Scoring Guide that lists all items, correct answers, standards and the possible points. The Number Corner Checkup 1 includes a Number Corner Checkup 1 Scoring Guide for the written part of the assessment that contains the item, the CCSSM, and the possible points.
Another example is Unit 6, Module 3, Session 5; this Unit 6 Assessment includes a Unit Scoring Guide that lists all items, correct answers, standards, and the possible points. Another example is Number Corner Checkup 4; the Interview Response Sheet has a CCSSM Correlation for each of the questions at the top of the Response Sheet as well as a Number Corner Checkup 4 Scoring Guide for the written part of the assessment.
Also, each item on the Comprehensive Growth Assessment lists the standards being emphasized on the Skills & Concepts Addressed sheet, the Interview Materials List and the Interview and Written Scoring Guides.
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting students' performance and suggestions for follow-up.
All Checkpoints, Check-ups, the Comprehensive Growth Assessment, and Baseline Assessments are accompanied by a detailed rubric and scoring guideline that provide sufficient guidance to teachers for interpreting student performance. There is a percentage breakdown to indicate meeting, approaching, strategic, and intensive scores. Section 5 of the Assessment Guide is titled "Using the Results of Assessments to Inform Differentiation and Intervention.” This section provides detailed information on how Bridges supports RTI through teachers' continual use of assessments during the school year to guide their decisions about the level of intervention required to ensure the success of each student. There are cut scores and designations assigned to each range to help teachers identify students in need of Tier 2 and Tier 3 instruction. There is also a breakdown of Tier 1, 2 and 3 instruction suggestions.
Materials encourage students to monitor their own progress.
There is limited evidence in the instructional materials that students are self-monitoring their own progress.
Section 4 of the Assessment Guide is titled, "Assessment as a Learning Opportunity." This section provides information to teachers guiding them in: setting learning targets, communicating learning targets, encouraging student reflection, exit cards, and comparing work samples from earlier and later in the school year.
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Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
Session and Number Corner activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The Bridges and Number Corner materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
Units and modules are sequenced to support student understanding. Sessions build conceptual understanding with multiple representations that are connected. Procedural skills and fluency are grounded in reasoning that was introduced conceptually, when appropriate. An overview of each unit defines the progression of the four modules within each unit and how they are scaffolded and connected to a big idea.
In the Sessions and Number Corner activities, there are ELL strategies, support strategies, and challenge strategies to assist with scaffolding lessons and making content accessible to all learners.
For example, in Unit 5, Module 1, Session 4, students are playing the game, "Place Value Triple Roll." They are rolling dice and building various numbers with sticks and bundles. Support is offered: "...work with students to find the next smallest number on the list and record it on their sheets." Challenge is offered: "Ask students to complete the last section of the Record Sheet."
In the Unit 3, Module 3, Session 1 Work Place 3F "3-D Base Ten Triple Spin," the following suggestions are provided:
- Support: "If students are struggling to decide which denomination to take each spin in, have them take their three spins before they spin the Greater than/Less than spinner."
- ELL: "Review the idea of using sketches to record numbers. Draw the shorthand version for each base ten piece and, right next to the sketch, write the number it represents."
- Challenge: "If students are playing the game with confidence and ease and might benefit from a challenge, invite them to take six spins each, build two 3-digit numbers, and add the numbers to determine the winner."
Materials provide teachers with strategies for meeting the needs of a range of learners.
The instructional materials provide teachers with strategies for meeting the needs of a range of learners.
A chart at the beginning of each unit indicates which sessions contain explicit suggestions for differentiating instruction to support or challenge students. Suggestions to make instruction accessible to ELL students is also included in the chart. The same information is included within each session as it occurs within the teacher guided part of the lesson. Each Work Place Guide offers suggestions for differentiating the game or activity. The majority of activities are open-ended to allow opportunities for differentiation. Support and intervention materials are provided online and include practice pages, small-group activities and partner games.
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
The instructional materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations. Tasks are typically open-ended and allow for multiple entry-points in which students are representing their thinking with various strategies and representations (concrete tools as well as equations).
In the Problems and Investigations section, students often are given the opportunities to share strategies they used in solving problems that were presented by the teacher. Students are given multiple strategies for solving problems throughout a module. They are then given opportunities to use the strategies they are successful with to solve problems in Work Places, Number Corner, and homework.
For example, in Unit 2, Module 1, Session 4, students are working on showing "Tens & Ones" with cubes. The teacher begins with an intentional progression: 23 to 43; 43 to 73. As students share out their representations, the teacher continues to ask "Did someone have another way to make the change?" "How about a different way to show 45?" Students can represent the numbers by counting out single cubes, combinations of 10s, and combinations of 5s; some simply add two additional groups of ten to the original number (23-43). Any correct representation is accepted, and students are encouraged to solve using different representations.
Another example is found in the Number Corner February Computational Fluency. Students are working on their addition facts according to where they have demonstrated mastery. Each student has a different table where they are coloring in the facts that they've mastered, with non-mastered facts easily identified. The table also has a sidebar with various strategies that can be used to master facts to 20, for example: Add Zero Facts, Count on Facts, Doubles Facts, and Make Ten Facts. According to students' individual level of mastery and understanding of strategies, they identify the strategy that they will use in today's practice and the facts they will be working on. This fluency practice allows all students to enter the practice at their own point of understanding/mastery and allows for practice using different strategies.
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
The instructional materials suggest supports, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.
Online materials support students whose primary language is Spanish. The student book, home connections and component masters are all available online in Spanish. Materials have built in support in some of the lessons in which suggestions are given to make the content accessible to ELL students of any language.
There are ELL, Support, and Challenge accommodations throughout the Sessions and Number Corner activities to assist teachers with scaffolding instructions. Examples of these supports, accommodations, and modifications include the following:
- Unit 3, Module 3, Session 6 provides an ELL and a support suggestion. The ELL suggestion reads as follows: "If students have trouble reading a problem, encourage them to examine the talk bubble for clues as to what they're supposed to do. If that doesn't work, suggest that they ask the student who wrote the problem to read it to them." The Support suggestion is to "be prepared to have two or three problems in mind to suggest to students who may have trouble choosing, or may have trouble solving some (perhaps most) of the problems posed by their classmates. You may even want to have one problem in mind to work with a smaller group of students as you send other students out to shop."
- In Unit 4, Module 2, Session 1, students are working on measuring feet and yards. The following "ELL" suggestion is provided: "Give students a visual reminder of the relationship between feet and yards to help them better understand that there are three feet in every yard. If possible, display a yardstick in the classroom with three inchworm rulers lined up beside it. Post the statement 1 yard = 3 feet near the display."
- In the Number Corner January Daily Rectangle, students are working on creating arrays in four different quadrants of a hundreds grid. The "Support" suggestion is: "Use two 5x8 cards to mask all but the quadrant in question each time. Also remind the students that they only need to worry about the gray squares in each quadrant, not all the squares."
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
The instructional materials provide opportunities for advanced students to investigate mathematics content at greater depth. The Sessions, Work Places, and Number Corners include "Challenge" activities for students who are ready to engage deeper in the content.
Challenge activities found throughout the instructional materials include the following:
- In Unit 3, Module 2, Session 2, the challenge part of this session encourages students to add on the number line.
- In Unit 4, Module 2, Session 2, in the Problems & Investigations section, students are measuring in yards. The "Challenge" suggestion is as follows: "Finding items that are exactly 2, 3 or 4 yards long may be difficult. Some pairs may enjoy competing to see who can get the closest to these measures. Discuss how you would judge who is the closest using a smaller unit of measure such as inches."
- In the Number Corner October Calendar Collector, students are collecting time in a game called" How Much Time Did We Collect." The Challenge suggestion is as follows: "Challenge students to figure out how many hours and minutes they collected over the month and record at the bottom of their Rolling for Minutes Record Sheet."
Materials provide a balanced portrayal of various demographic and personal characteristics.
The materials provide a balanced portrayal of demographic and personal characteristics. Most of the contexts of problem solving involve objects and animals, such as frogs and penguins. When students are shown performing tasks, they are cartoons that appear to show a balance of demographic and personal characteristics.
Materials provide opportunities for teachers to use a variety of grouping strategies.
The instructional materials provide opportunities for teachers to use a variety of grouping strategies.
The instructional materials offer flexible grouping and pairing options. Throughout the Units, Work Places, and Number Corners, there are opportunities to group students in various ways such as whole group on the carpet, partners during pair-share, and small groups during Problem & Investigations and Work Places.
In Unit 6, Module 4, Session 1, students are grouped as a whole for the Read Aloud, A Cloak for the Dreamer, and the discussion about quilt blocks. Then, they work individually on creating their own quilt block. After the Problems & Investigations session, they move into Work Places where they work either independently or in small groups with the Work Place centers.
In the Number Corner January Computational Fluency, the teacher models the game, "Fact Strategy Game," and plays the game with the whole class. Students may then play the game in pairs.
Materials encourage teachers to draw upon home language and culture to facilitate learning.
There is limited evidence of the instructional materials encouraging teachers to draw upon home language and culture to facilitate learning. The materials provide parent welcome letters and unit overview letters that are available in English and Spanish.
Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional technology-based, assessment opportunities, and the digital resources are not easily customized for individual learners.
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
Materials integrate technology, such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software, in ways that engage students in the Mathematical Practices.
Each session within a module offers online resources that are in alignment with the session learning goals. Online materials offer an interactive whiteboard file as a tool for group discussion to facilitate discourse in the MPs. Resources online also include virtual manipulatives and games to reinforce skills that can be used at school and home. In the Bridges Online Resources, there are links to the following:
- Virtual Manipulatives - a link to virtual manipulatives such as number lines, geoboard, number pieces, number racks, number frames, and math vocabulary;
- Interactive Whiteboard Files - Whiteboard files that go with each Bridges Session and Number Corner;
- Online Games- online games such as 100 Hunt using the hundreds grid, 2-D Shape Pictures, Interactive math dictionary, Addition With Manipulatives, and Balloon Pop Comparisons (greater than/less than); and
- Images - for example, 1,000 M&M candies arranged on hundred grids by students.
Within the Teacher's Edition, there is no direct reference to online resources. If embedded within the Teacher's Edition, the resources would be more explicit and readily available to the teacher.
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
The digital materials are web-based and compatible with multiple Internet browsers. They appear to be platform neutral and can be accessed on tablets and mobile devices.
All grade level Teacher Editions are available online at bridges.mathlearningcenter.org. Within the Resources link (bridges.mathlearningcenter.org/resources) there is a sidebar that links teachers to the MLC, Math Learning Center, Virtual Manipulatives. These include games, Geoboards, Number Line, Number Pieces, Number Rack, Number Frames and Math Vocabulary. The resources are all free and available in platform neutral formats: Apple iOS, Microsoft and Apps from Apple App Store, Window Store, and Chrome Store. The apps can be used on iPhones and iPads. The Interactive Whiteboard files come in two different formats: SMART Notebook Files and IWB-Common Format. From the Resource page there are also many links to external sites such as ABCYA, Sheppard Software, Illuminations, Topmarks, and Youtube.
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
The instructional materials do not include opportunities to assess students' mathematical understanding and knowledge of procedural skills using technology.
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
The instructional materials are not easily customizable for individual learners or users. Suggestions and methods of customization are not provided.
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
The instructional materials provide opportunities for teachers to collaborate with other teachers and with students, but opportunities for students to collaborate with each other are not provided. For example, a Bridges Blog offers teacher resources and tools to develop and facilitate classroom implementation.