3rd Grade - Gateway 2

The instructional materials reviewed for Grade 3 partially meet the expectations for balance of the three aspects of rigor within a grade. Although the instructional materials meet expectations for each aspect of rigor, these aspects of rigor are often addressed in separate parts of the Sessions. Materials targeting application are often scaffolded, detracting from the balance of rigor. Overall, the three aspects of rigor are most commonly treated separately.

In general, conceptual understanding, procedural skill and fluency, and application are all adequately addressed in the Sessions; however, for the most part they are addressed in separate sections of the instructional materials. Conceptual understanding is typically addressed in the Discussion and Math Workshop portions of Sessions. Procedural skill and fluency is typically introduced in separate Sessions and then practiced in the Daily Practice portion of sessions. Application consists of routine word problems in the instructional materials. As a result, all aspects of rigor are almost always treated separately within the curriculum including within and during Sessions, Practice, and Homework.

The instructional materials reviewed for Grade 3 partially meet expectations that materials carefully attend to the full meaning of each practice standard (MP). Although the instructional materials attend to the full meaning of some of the MPs, there are some MPs for which the full meaning is not developed.

At times, the instructional materials only attend superficially to MPs. The following are examples:

• Unit 1, Session 1.1 in the Math Practice Note it lists MP1 and has students asking multiplication questions. In the Math Practice Note it just reminds teachers to notice how students are entering problems, choosing strategies, and how they are making use of drawings or cubes to solve problems. This session poses questions such as, “We agree that there are usually five toes on a person’s foot. How many toes would there be on four feet?” These questions do not lend themselves to students having to persevere in solving them.
• Unit 1, Session 3.1 in the Math Practice Note it lists MP5 and has students represent multiplication situations with arrays. The Math Practice Note talks specifically about what a rectangular array is and how it provides images students use to understand key properties. This activity only has students using arrays and does not allow them to choose any other tool.
• Unit 4, Session 2.2 in the Math Practice Note it lists MP4 and has students compare fractions on a number line. This is a literal model not attending to the full meaning of MP4 because it does not solve a real world problem. In the Math Practice Note it states that each time students place a fraction on the number line, they are modeling where that number fits in our system, and its relationship to whole numbers and to other fractions.
• Unit 5, Session 2.2 in the Math Practice Note it lists MP5 and has students using arrays to break up multiplication facts. The Math Practice Note talks specifically about using arrays and the story context to solve problems. This session does not allow them to choose any other tool.
• Unit 5, Session 3.3 in the Math Practice Note it lists MP1 and has students working with multi-step word problems. This is a new kind of problem structure for these students and they are not asked to solve anything. The Math Practice Note just informs teachers to help students focus on making sense of each problem, does not have them persevere through any of this session’s work.
• Unit 6, Session 1.2 in the Math Practice Note it lists MP5 and has students making fraction sets. In the Math Practice Note it comments about students using tools such as brownies and fraction sets to visualize an area model for fractions, however, it then specifically states that in this session they will be given pattern blocks to represent fractions and fraction relationships. This session does not allow students to choose their tool.
• Unit 6, Session 1.3 in the Math Practice Note it lists MP4 and has students create and label fraction pieces. This is a literal model not attending to the full meaning of MP4 because it does not solve a real world problem nor are students modeling to solve a mathematical problem.
• Unit 8, Session 3.4, in the Student Activity Book page 527 students write problems using a letter to represent the unknown, but it is not tied to word problems. The problem reads: “This equation is about Zupin’s marbles: z = 20 + (18 x 4). What does Z mean? What does the 20 represent? What does the 18 represent?” This does not attend to the full meaning of MP4 because it does not relate the mathematics needed to solving a real world problem.

At times, the instructional materials fully attend to a specific MP. The following is an example:

• Unit 1, Session 4.6 in the Math Practice Note it lists MP1 and has students making sense of problems, sharing problems, and understanding different solutions. This session has students solving division problems and then discussing questions such as, “How did you start solving this division problem? Is there a multiplication fact you know that might help you? What part of the problem have you solved? What is left over?” In the Math Practice Note it states that these questions communicate to students that they are not expected to immediately know the answer, but they are expected to think about what knowledge and tools they have to help them get started to figure it out.

The instructional materials reviewed for Grade 3 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

When MP3 is referenced, students are often asked to solve and share solutions. The independent work of the student is most often about finding the solution to a problem without creating a viable argument. Students often listen to peer solutions without being asked to critique the reasoning of the other student. Much of the student engagement in the class discussion is teacher prompted without giving students the opportunity to create their own authentic inquiry in the thinking of others.

• In Unit 4 Session 2.3 students are asked to discuss the area of a shape in their Student Activity Book with a partner. “How do you know that it’s 4 square inches? Who can explain how these triangles and squares go to together. Discuss with a partner.” There is no evidence that students are being guided to construct or critique mathematical reasoning.
• In Unit 6 Session 2.2 the teacher asks the students about the placement of a fraction on a numberline. “If I wanted to mark 2/4 on the number line, where would I mark it?” The students are neither constructing arguments nor analyzing the arguments of others.

At times, the materials prompt students to construct viable arguments and analyze the arguments of others.

• In Unit 8 Session 2.5 students are asked to explain why their solution to a problem makes sense while other students are encouraged to ask questions in order to help other students state their justifications clearly. By focusing questions that students will ask each other in this way, students are more clearly guided to critique the arguments of others and provide peer feedback.

The instructional materials reviewed for Grade 3 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.

Most of the time when MP3 is referenced, teachers are asked to have students share or explain their solutions. Teachers are also directed to have students ask questions but are not supported in focusing those questions toward critiquing the arguments of others.

• In Unit 2 Session 1.2 the teacher is prompted to comment on how different organizations of the data allowed students to answer different questions. No question or comments allow for students to construct and/or analyze viable arguments.
• In Unit 2 Session 2.4 the teacher prompts include: “What can we say about our class as a whole? What can you say about our data? Talk to a neighbor about the things you notice.” No question or comments allow for students to construct and/or analyze viable arguments.
• In Unit 4 Session 3.3 teacher prompts include: "What is the same about the shapes that are in this row labeled 'Quadrilaterals'? What is true of all quadrilaterals?" Then the teacher is told to post a chart and record student’s responses. The students are never prompted to argue or analyze with this line of questioning. The Math Practice Note for the teacher states, “you might want to talk explicitly with students about how using definitions of shapes in mathematics is not about opinion, but is about looking carefully at what properties the shape you’re considering does and does not have.”
• In Unit 7 Session 3.3 The Math Practice Note states, “some students may be able to use the context to explain what is happening…” and provides no support for students in critiquing the explanations of others.

The materials assist teachers, at times, in engaging students in constructing viable and analyzing the argument of others.

• In Unit 4 Session 2.2 teacher prompts include: "Which shapes worked to completely cover the rectangle? Does everyone agree? Anyone disagree? Can you talk about why you think these two shapes didn't work?"
• In Unit 4 Session 2.5 teacher prompts include: " Ask students how they solved the the problem. Does everyone agree the area of this shape is 55 square centimeters? Then, I noticed that Kim made a 4 x 10 and a 3 x 5 rectangle and Dwayne made a 7 x 5 and 4 x 5 rectangle. Why do these two different ways of finding area of 55 sq. cm both work? Take a minute to talk to a neighbor."