3rd Grade - Gateway 2

The instructional materials reviewed for Grade 3 meet the expectations by attending to conceptual understanding. Overall, the instructional materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

• Problem strings are used throughout the year to provide a conceptual understanding approach to teaching procedural skills and computational fluency with an emphasis on making connections across representations, including number lines, arrays, and equations.
• A mathematics forum structure is used throughout the units, which allows students to share their thinking, ask questions, and explore key concepts. For example, Unit 5, Module 2, Session 2 is a forum where students work on their knowledge of the differences between sharing and grouping in division (3.OA.A.2).
• The Bridges Introduction pages of the Teacher Manual outline a variety of models that students access throughout the school year in order to demonstrate their understanding. In Unit 2, in the teacher directions and the introduction to the unit (pages ii, iii, iv, v, vi, vii and viii), there is information about the conceptual development of multiplication. Information on concepts, the properties of multiplication, strategies and models can be found and are explained for the teacher.
• Many representations are used throughout the sessions. For example, Unit 2, Module 1, Session 3 has students using arrays of stamps to determine the total cost of the stamps. The materials continue to focus on grouping to solve. In Unit 2, Module 1, session 4, in the Problems & Investigations, the students work with number line puzzles to solidify their efficiency and use of strategy to solve multiplication problems (3.OA.A.1).
• In Unit 4, Module 3, Session 1, students explore fractions with paper folding. Teacher direction 6 says, “Work with the class to compare non-congruent halves, and help them understand that pieces don’t need to be congruent to be equivalent.” (page 5.)
• In Unit 4, Module 3, Sessions 1 and 3, students use paper and pattern blocks to explore fractions, and then in sessions 4 and 5, they use a number line to explore “Fractions as Distances.”

The instructional materials reviewed for Grade 3 meet the expectations by attending to procedural skill and fluency. Overall, the instructional materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

• Problem Strings within the sessions provide fluency practice built from conceptual understanding. In Unit 2, Module 1, Session 2, the assessment is a problem string involving rows/arrays of stamps where students work with doubling strategies to find the totals of the arrays (3.OA.C). In Module 1, Session 3, students work again with groups of stamps to determine the total cost of the stamps. They are laid out in arrays using the digits 4, 5 and 2 (3.OA.C.7). Module 1, session 3 also has a problem string for practice connected to the work during the session (3.OA.C.7).
• Students build fluency with multiplication facts in Unit 2, Module 3, Sessions 3 and 4, “Multiplication Strategies, Parts 1 and 2.” Students look at the multiplication table and explore strategies such as double facts and double-plus-one facts.
• Students participate in “Work Place” activities throughout the sessions, which are “engaging, developmentally appropriate math stations that offer ongoing practice with key skills.” (Introduction to Bridges). In Unit 4, Module 3, Session 5, the Work Place activity has students engage with a spinner game called “2D Doubles Help” where they take turns spinning to generate a multiplication problem involving 3 or 4, and then they record the doubles fact that they can use to solve the problem.
• The Number Corner component of the Bridges curriculum "engages students and contributes to a math-rich classroom environment that promotes both procedural fluency and conceptual understanding." (Bridges Introduction).
  • The Computational Fluency component of Number Corner focuses on "activities, games, and practice pages designed to develop and maintain fluency," (page v, Teacher Manual), like Frog Jump Multiplication in October and Array Race in November. In December through March, students are focusing on multiplication facts starting with zero, one, and two, going to five and ten the next month, and then moving up through the other single-digit numbers.
  • In Number Corners, each month contains a number line component that focuses on procedural skills, for example: Rounding to the Nearest Ten in November, Comparing Fractions in February, and Put it on the Line in April.

The instructional materials reviewed for Grade 3 meet the expectations by attending to application. Overall, the instructional 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.

• In the Solving Problems component of the Bridges Number Corner, students spend time working on application problems. "Often the problems connect to another workout in the same month, which enables students to apply skills they learned elsewhere to a problem-solving context." (page vi, Number Corner Volume 1). In Number Corners, each month contains a problem solving component that focuses on application, for example: Multi-Step Problems and Equations in January, Data Problems in February, and Area and Perimeter Problems in March.
• Teachers pose contextual Problem Strings and Problems & Investigations throughout the Bridges curriculum that are grounded in real-world application in which students model, discuss, reason, and defend their thinking.
  
  • Within the materials, there are multiple sessions where a problem and investigation of a real-world scenario is paired with a forum for discussion and exploration of the problem solving strategies. For example, Unit 5, Module 3, session 1 has students writing their own story problems for given equations with and unknown quantity (7x5=m). Students are asked to create more than one operation in their problems. Then in Session 2, they have a forum to discuss their work (3.OA.D.8).
• Unit 4, Module 4 is an application-based exploration of fractions with the context of creating and measuring bean stalks. Students gather data with fractions and create a line plot of their beanstalk data.
• Unit 8 is a complete project-based application unit around the task of bridge design and construction. Standards involving fractions, measurement and data, and geometry are applied throughout the unit in real-world, problem-solving tasks.
• Reviewers note that many student pages and sessions throughout the materials contain application problems aligned to major work of the grade, however most of the multi-step, non-routine problems are at the end of the pages and labeled "Challenge."

The instructional materials reviewed for Grade 3 meet the expectations for balancing the three aspects of rigor. Overall, within the instructional materials the three aspects of rigor are not always treated together and are not always treated separately.

• The Problems & Investigations within the sessions call for students to apply procedural skill and fluency and conceptual understanding to solve application problems.
• Application problems often call for students to model their thinking through the use of area models, number lines, ratio tables, etc.
• Procedural skill and fluency is often noted side-by-side as students are working in conceptual models.
• Problem strings target procedural skill and fluency by targeting opportunistic strategies. Teachers represent student thinking with a variety of conceptual models. "Each time, students solve the problem independently using any strategy they like, and then the teacher uses a specific model (an array or a number line, for example) to represent students' strategies." (Teachers Manual Unit 1, Introducing Bridges Mathematics).
• Application is the focus in Unit 8, when students are designing bridges.
• Procedural skill and fluency is attended to separately in the Number Corner component "Computational Fluency."

The instructional materials reviewed for Grade 3 meet the expectations for the MPs being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials identify two to four MPs for each Bridges session and Number Corner activity. There are a few Bridges sessions that only identify MPs and no content Standards within those sessions. However, students using the materials as intended will engage in the MPs along with the content standards for the grade.

• The Introduction to Bridges Grade 3 includes a table describing what the MPs look like for students in Grade 3 for each practice.
• All eight MPs are identified throughout the curricular materials.
• The Bridges overall Scope & Sequence for Units does not note the practice standards, however, between two and four MPs are identified in the “Skills and Concepts” section at the beginning of every Bridges session.
• There are several Bridges sessions that identify only MPs and no content standards:
  • Unit 1, Module 1, Sessions 1 and 2 list 4 MPs in each session, with no content Standards.
  • Unit 8, Module 1, Session 1 lists 4 MPs, with no content Standards.
  • Unit 8, Module 1, Session 3 lists 2 MPs with no content Standards.
  • Unit 8, Module 4, Session 1 lists 4 MPs, with no content Standards.
• "Math Practices in Action" is located in the margin of the teacher notes within the Bridges sessions. They identify how the students engage with the MPs along with the content standards. For example:
  • In Unit 2, Module 1, Session 6, "Math Practices in Action" explains that teachers facilitate students engaging in MP2: "During this forum, you'll connect the symbolic notation (abstract) to the context of the problems (quantitative)."
  • In Unit 4, Module 3, Session 2, "Math Practices in Action" explains that students engage in MP8, "While repeating the process of dividing the rectangle into equal pieces and removing one of them, students will make use of the regularity they see in their reasoning to make sense of unit fractions. The repetition, and the patterns that emerge, help them develop a stronger understanding of unit fractions."
• The MPs are identified in the “Target Skills” section at the beginning of every Number Corner month and within the "Skills and Concepts" section at the beginning of the Number Corner activity types. There is a "Math Practices & the Number Corner Learning Community" section at the beginning of the first Number Corner binder, which describes how students engage with the MPs during Number Corners.

The instructional materials reviewed for Grade 3 meet the expectations for prompting students to reason by constructing viable arguments and analyzing the arguments of others. The students’ materials in both Bridges and Number Corner provided opportunities throughout the year for students to reason by both constructing viable arguments and analyzing the reasoning of others, however, while the students materials often prompt students to reason by constructing viable arguments, there is less consistency and opportunities for students to analyze the arguments of others, which leads to a lack of balance. More opportunities could be provided in the student materials, other than formative and summative assessments, to engage in analyzing the arguments of others.

• In Unit 1, Module 1, Session 2, question 4 says, "How many people did Sophia and Samir survey? How do you know?"
• In Unit 1, Module 1, Session 4, question 5 says, "Is the sum of 0 and any number always even, always odd, or sometimes even and sometimes odd? Explain."
• In the Unit 1 Post-Assessment, question 11 says, "Josh says that all the Doubles facts (6 + 6) have even sums. Do you agree with Josh? Why or why not?" Question 12 says "Explain how you can use a Make Ten Fact to solve 9 + 6. Include the answer to 9 + 6 in your explanation."
• In the Unit 4 Post-Assessment, question 13 says, "Is the statement true or false? Make a sketch to prove you're correct. You can use your pattern blocks to help if you like."
• In the Unit 6, Polygons & Quadrilaterals Checkpoint, question 4 says, "Damon says that this figure is not a polygon. Do you agree with him? Why or why not? Give two different reasons."
• "Math Forums, which occur a few times in most units, are a more formal and structured time for students to share and discuss their work. Students who are not sharing their own work are expected to listen carefully, compare their classmates’ work to their own, and ask questions to better understand each students’ ideas.” For example, in Unit 3, Module 2, Session 5, “Subtraction Strategies Forum,” the teacher directions say, “Invite students to present their work one student or student pair at a time. After each presentation, invite the rest of the class to ask questions, and have the presenters respond to those questions. After each student or student pair finishes, ask the students if they understood what the students did and whether anyone else used the same or similar approach. If a student shares something similar that elevates the level of discussion, model what that student did with sketches, numbers, and words." The focus in the forum seems to be on the strategies used, however that misses an opportunity to justify the strategy or even critique each others' reasoning.
• Within Bridges sessions, there are many prompts for students to explain how they got their answers or show their work, but there are missed opportunities to evaluate the thinking of others. In most of the problems or assignments, the directions say, "Show your work" or "Use numbers, labels, models or words to show your thinking." For example:
  • In Unit 2, Module 3, Session 2, question 2, says "If you paid $16.50 for rabbit food, how many pounds did you buy? Show your thinking."
  • In Unit 5, Module 1, Session 4, questions 1, 2, and 3 say, "Show all your work."
  • In Unit 1, Module 2, Session 1, question 5 says, "Show all your work using numbers, words, or labeled sketches."
  • In Unit 1, Module 2, Session 2, student page question 4 says, "Show all your work using numbers, words, or labeled sketches."
• In Number Corner activities, students are asked to "Use the space to solve the problem and record your thinking with numbers, words, equations, or models."

The instructional materials reviewed for Grade 3 meets the expectations for the materials explicitly attending to the specialized language of mathematics. Overall, the materials provide some instruction in how to communicate mathematical reasoning using words, diagrams and symbols, however more explicit instruction related to precise communication is needed. There are instances in the materials that introduce vocabulary that is not grade appropriate and in basic, incomplete ways.

• In the introduction to the series, “The curriculum includes a set of Word Resource Cards for every classroom. Each card features a mathematical term accompanied by illustrations, with a definition on the back. The cards are integrated into lessons and displayed in the classroom to support students’ acquisition and use of precise mathematical language.”
• In Section 3 of the Assessment Guide, Assessing Math Practices, an app called Math Vocabulary Cards, which included the same illustrated terms and illustrations as the Word Resource Cards, is also available to serve as a compact and convenient math dictionary.
• At the beginning of the sessions, a sidebar lists the vocabulary for the lesson, with an asterisk that identifies "those terms for which Word Resource Cards are available." Sessions contain directions for use of the cards. For example, in Unit 6, Module 4, Session 3, teacher direction 6 says, "Display the Word Resource Card for equivalent fractions, and review the meaning of fraction equivalency."
• Students are often supported to show their mathematical reasoning using words, diagrams and symbols. For example:
  • Unit 3, Module 1, Session 5, in the Problems & Investigations, the teacher directions say, "Remind students that they will have a chance to use some of the strategies that they generated a couple of months ago, or come up with new ideas, as they work with a partner on a new collection of story problem." and "Remind students to check the reasonableness of their work; ask them to make sure their strategies and their answers make sense." Also, the teacher directions say, "Make sure students have access to base ten pieces or other manipulatives, but don't suggest their use as sometimes students will employ a less sophisticated method if they think that is what you want." and "Let students know that both students in each partnership need to show their work in their journals."
  • In Unit 1, Module 2, Session 3, the student page, problems 2a and 2b, says, "Show all your work using numbers, words, or labeled sketches." Also, the student page, problem 3, says, "Record your work in your math journal, using numbers, words, or labeled sketches."
  • In the December Number Corner Problem String, the Key Questions in the sidebar prompt teachers to ask, "What strategy could you use?" "How can you show your thinking?" and "What model could you use to show your thinking?" Also, the "Big Idea" listed in the first string is, "When multiplying, you can think about equal groups or repeated jumps on a number line."
• "Math Practices in Action" that identify MP6 nearly all give direction to the teacher about accuracy with calculation or measurement, not precise communication. For example, in Unit 3, Module 3, session 3, the "Math Practices in Action - MP6" says, "It is important for students to decide when they need to attend to precision and when a mathematical question can be answered without an exact calculation. In this case, an exact answer is necessary, but the inexact estimates students have made can help them confirm whether their calculations make good sense."
• Unit 2, Module 3, Session 2 lists "ratio table" as a vocabulary word, and there is a Word Resource Card for it. Within the lesson, the teacher directions only refer to them as "tables". Teacher direction 1 says, "and letting them know they will be doing more work with tables." However, providing a word resource card for a concept beyond Grade 3 content may prompt teachers to teach this word to students.
• Unit 6, Module 4, Session 1 lists the vocabulary word "congruent", and there is a Word Resource Card for it. Teacher direction 7 says, "Work with the class to compare non-congruent halves, and help them understand that pieces do not need to be congruent (exactly the same shape and size) to be equivalent (the same amount)." This concept is beyond Grade 3 and a an imprecise definition.