4th Grade - Gateway 2

The instructional materials reviewed for Grade 4 partially meet the expectations for developing conceptual understanding of key mathematical concepts. Overall, the instructional materials do not consistently offer opportunities to use manipulatives and models to develop conceptual understanding. Also, more emphasis should be placed on discussion of mathematical concepts.

• Modules 1-9 and 11-12 contain lessons that specifically address standards which are explicitly outlined as standards developing conceptual understanding. (4.NF.A, 4.NBT.A and 4.NBT.B).
• “Number Case” and “Fundamentals” provide resources for students to use models.
• Lesson 3.3 addresses multiplying whole numbers (4.NBT.5). Students use a rectangular array to demonstrate the double and half strategy for multiplication when multiplying a one-digit factor by a two-digit factor. Students also draw on place value understanding to use the distributive property. This opportunity is provided in both whole group instruction and independent practice.
• Lesson 3.6 addresses multiplication properties (4.NBT.5). Students construct and multiply the cubes on rectangular prisms to demonstrate understanding of the associative and commutative properties. Students are prompted to discuss the efficiency of strategies and are gradually released from models to equations.
• Lesson 3.7 addresses multiplication properties (4.NBT.5). Students reason about the most efficient way to decompose numbers in a rectangular prism to make multiplication calculations easier. Students are encouraged to describe their strategies in selecting the order to multiply the numbers. Emphasis is placed on manipulating numbers for easier computations.
• Lesson 3.8 addresses multiplication strategies (4.NBT.5). Students use strategies based on place value to solve problems involving multiplication.
• Module 3, Check-Up 2, Problems 1a-d encourage the use of the associative and commutative properties (4.NBT.5) and place value understanding to solve multiplication problems.
• Lessons comparing fractions (4.NF.2) focus on developing procedural fluency with creating common denominators. There is only one lesson (3.11) focused on using unit fractions and number line strategies for comparing fractions. Very few problems focused on this standard further student conceptual understanding of the size of fractions by using reasoning skills related to benchmark fractions, visual fraction models or the number line.
• Lessons addressing equivalent fractions (4.NF.1) use the area model to help connect and develop the concept of equivalent fractions and why the algorithm for creating equivalent fractions works. However, in these lessons the student practice pages focus on using the standard algorithm without more lengthy practice and exploration on why it works.
• Lessons addressing multiplication and division (4.NBT.A and 4.NBT.B) offer opportunities at the beginning of the lessons to use arrays and other visual models (3.3 and 10.8), and the rest of the lesson focuses on procedural fluency with a given strategy. Students are not asked to justify or explain their answers using conceptual understanding.

The Grade 4 materials partially meet the expectations for procedural skill and fluency. They give some attention to individual standards that set an expectation of procedural skill and fluency. Lessons contain multiple examples of fluency practice pages.

• “Fundamental Games” provides opportunities for students to practice fluency through games.
• Some lessons have ongoing practice that primarily address procedural skill and fluency (e.g., 3.2, 5.8 and 6.2).
• Several lessons offer opportunities to develop fluency to "add and subtract multi-digit whole numbers within 1 million" (4.NBT.4). (Modules 2 and 4.)
  • The lessons on using the standard algorithm for addition and subtraction (Lessons 2.4-2.8 for addition; Lessons 4.1-4.7 for subtraction) have student work that primarily addresses developing procedural skill and fluency for the aspect of rigor stated in the standard. Some lessons include a generic application context (e.g., making change with money to connect to needing to rename numbers (Lesson 4.1); and a few lessons also include Justification/Explanation type problems where student must catch and explain an error (Lesson 2.5, Lesson 4.2).
  • However, unlike Grade 3, there is little indication that the expectation of fluency is consistently addressed throughout the year. Of the interview assessments, none address fluency with the standard algorithm for addition/subtraction.

The instructional materials reviewed for Grade 4 partially meet the expectations for students spending sufficient time working with engaging applications of the mathematics. Overall, the instructional materials do not consistently offer opportunities for students to engage in application of learning to real-world situations.

• Modules 2 and 4-12 contain lessons that address solving real-world problems.
• Several lessons specifically address standards that involve application (4.OA.3, 4.NF.3.D and 4.NF.4.C).
• "Stepping Into Financial Literacy" offers lessons on Financial Literacy.
• Each module contains three investigations that require application.
• Some lessons have ongoing practice available that focuses on application (e.g., 3.2, 4.10 and 5.6).
• Lessons aligned to the 4.OA.3 do not reach the full intent of the standard. Few word problems in these lessons are 2-step and rarely do students encounter multi-step problems.
• Most problem-solving lessons focus on measurement and do not offer opportunities to solve 2-step problems.
• Students are not consistently exposed to non-routine problems. There are missed opportunities in each module's problem-solving lesson to expose students to non-routine problems.
• Students are rarely asked to use a letter to represent the unknown when solving problems (Lesson 2.9, 4.7 and 6.12).
• Lessons addressing word problems with multiplying fractions (4.NF.4.C) primarily focus on the development of conceptual understanding or procedural skill rather than application.
• Students do not have opportunities to solve application problems in each lesson. Additionally, students are not exposed to application problems on each type of assessment.

The materials reviewed for Grade 4 partially meet the expectations for balance. Overall, the three aspects of rigor are neither always treated together nor always treated separately within the materials, and a balance of the three aspects of rigor within the grade is lacking.

• Although all three aspects of rigor are present in the materials, there is not a balance among the three aspects of rigor. There is an under-emphasis on conceptual understanding and application work compared to the emphasis given to fluency.
• There are almost no lessons that bring multiple aspects of rigor together. For example, lesson 4.9 only requires students to use application to find whole-number quotients. There is a missed opportunity to show conceptual understanding.
• Conceptual development related problems are included in the curriculum, but typically as introductory questions where the majority of time spent is on student independent work on procedural responses.
• Each module contains lessons addressing procedural skill (4.1) conceptual understanding (4.4) and application (4.7).
• Assessments do not include a balance of questions from all three aspects of rigor. Fluency and application questions are often missing from assessments. For example, summative assessment 1 (checkup, interview and performance task) do not contain questions that address fluency or application.

The instructional materials reviewed for Grade 4 partially meet the expectations for identifying the MPs and using them to enrich mathematics content within and throughout Grade 4. Overall, the instructional materials identify the MPs but do not consistently use them to enrich the content. Also, some MPs are over identified, and some are under identified.

• MPs are identified in the “Steps” portion of each module lesson.
• MPs are identified in 123 of the 144 lessons. These lessons have at least one MP as the focus.
• MPs are embedded within lessons.
• A chart, for each module, under the Mathematics tab, identifies MPs by lesson
• Videos can be found under the Resources tab which explains the MPs and Habits of Mind.
• MPs are not specifically listed on assessments.
• Explanations of how the MPs are being used and what to expect from students to show growth or mastery is not provided in the “Steps” portion of the lesson.
• The following lessons do not contain MPs to enrich the lesson content: 1.1, 1.3, 2.4, 2.5, 3.4, 4.4, 5.3, 5.4, 7.4, 7.5, 7.6, 8.4, 9.1, 9.5, 10.3, 10.4, 10.5, 11.3, 11.5, 12.4 and 12.5.
• The MPs are not treated equally. For example MP7 is identified in 64 lessons, whereas MP4 is only identified in 5 lessons.