Kindergarten - Gateway 2

The instructional materials reviewed for Kindergarten partially meet the expectations for giving attention to conceptual understanding. Overall, the materials identify many opportunities for students to develop conceptual understanding of mathematical concepts where called for in specific content standards or cluster headings, but some of these identified opportunities do not actually allow students to develop conceptual understanding.

• Each chapter includes a “Teaching for Depth” page. This page identifies strategies, research, mathematical practices, and professional development videos to help teachers attempt to achieve a deep understanding of content among students. For example, chapter 1, in the Teacher Guide on page 9E.
• The beginning of each lesson has an “About the Math” section to give teachers direction on teaching for depth and reasoning for teaching the specific skill. For example, chapter 1, on pages 13A and 19A in the teacher guide.
• In chapter 5, lesson 1, there are follow-up questions included that require conceptual thinking and are aligned to K.OA.A.1. The questions are: how many children are playing with the soccer ball? How many children are being added to the group? And, what will you do to find how many children are there now?
• Many of the sections marked as conceptual understanding do not have students understanding concepts; instead many are very procedural. For example, chapter 1, lesson 6, on page 44 of the teacher edition; chapter 2, lesson 1, on page 82 of the teacher edition; and chapter 3, lesson 7, on page 156 of the teacher edition are more procedural than conceptual.

There are instances where the lesson for the advanced learners is conceptual even though the on target lesson is not. For example, chapter 3, lesson 7, on page 156 of the teacher edition, and chapter 5, lesson 1, on page 232 of the teacher edition are excellent examples of a conceptual lesson. Unfortunately, the lessons in these materials are labeled for advanced learners; this does not allow all students access.

The instructional materials reviewed for Kindergarten partially meet the expectations for giving attention to applications. Overall, the materials identify many opportunities for students to engage in applications with Mathematical concepts, without losing time on the major work of the grade, but some of these identified opportunities do not actually allow students to engage in applications. 

• Real-world applications are identified in each lesson. An example can be found in chapter 12, lesson 4, on page 708 in the teacher’s guide. Kindergarten students are asked to use cubes to show what they know about making a graph.
• In the materials, each chapter and lesson include a “Think Smarter” and “Go Deeper” question which are intended to be application problems.
• Many of the problems marked as applications are not applications. For example, in Chapter 1, lesson 4, and chapter 3, lesson 4, the “Think Smarter” problems labeled as applications are conceptual understanding.
• In chapter 4, lesson 1, and chapter 5, lesson 9, the “Think Smarter” problems labeled as applications are conceptual understanding.

In chapter 2, lesson 3, the “Think Smarter” labeled as application is a procedural problem.

The instructional materials reviewed for Kindergarten partially meet the expectations for balance. Overall, there are many instances where problems or activities are labeled as conceptual understanding or application, and the problems or activities do not accurately align to those labels. These inaccurate labels disrupt the balance among the three aspects of Rigor within the materials.

• The materials claim all three are present in each lesson. However, many of the conceptual understanding problems are not truly conceptual problems and most of the application problems have the same issues. This creates an imbalance in the materials, with procedural skill and fluency being the strongest of the three aspects.
• The three aspects of rigor are identified in a balanced manner throughout the material. These aspects are identified in each lesson under Rigor: Understanding Concepts (Share & Show), Procedural Skills & Fluency (On Your Own), and Applications (Think Smarter & Go Deeper). An example can be found in chapter 2, lesson 1, on page 81A in the teacher’s guide.

In the materials, each element of rigor is identified. They are all touched upon in each lesson and sometimes overlap, but they are not always treated together or always separately. For example, chapter 1, lesson 1, on page 13a, the lesson identifies the three areas of rigor and where they are addressed (Share and Show, On Your Own, Think Smarter & Go Deeper).

The instructional materials reviewed for Kindergarten partially meet the expectations for carefully attending to the full meaning of each practice standard. Overall, the instructional materials carefully attend to the full meaning of some of the practice standards but not all of them. 

• The materials attend to the full meaning of most of the Mathematical practice standards. An exception is in regards to MP 5. In the materials for Kindergarten, there are many instances when MP 5 is listed, and students are given or directed to use a specific tool, which does not allow students to consider and choose the appropriate tool. An example of this can be found in chapter 3, lesson 3, on page 131 in the teacher’s guide. Teachers are directed to give students seven two-color counters for counting. Another example can be found in chapter 4, Lesson 1, on page 181 in the teacher’s guide. Teachers are directed to give students counters for counting work.
• On page 24 of the teacher’s guide, the materials identify the standards for Mathematical practice and where the lessons include these. For example, there are problem-solving lessons in lessons 1.9 and 2.4 for MP 1.
• Each chapter has MPs listed, but they do not always attend to the full meaning of the practice standard. For examples, in chapter 5, lesson 2, on page 237 of the teacher edition, MP 2 is referenced along with a question to ask the students. The question is very direct and does not require the students to reason abstractly or quantitatively. By asking the questions the way they are stated, the teacher is taking the thinking and reasoning away from the students.

In chapter 6, lesson 7, on page 347 of the teacher edition, makes a reference to MP 5; however, the students are given the tools to use, which means they are not selecting their own tool as the practice requires. This happens again, in chapter 7, lesson 5, on page 385 of the teacher edition, because students are not choosing their own tools as they are given counters and ten frames to use.

The instructional materials reviewed for Kindergarten partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level Mathematics detailed in the content standards. Overall, the materials consistently allow students to construct viable arguments, but they do not consistently prompt students to analyze other students' arguments.

• Examples can be found in the materials for opportunities to apply SMP 3. Two examples can be found in chapter 2, Lessons 2.2 and 2.3, on pages 88 and 94 in the teacher’s guide. Lesson 2.2 asks students, “How can you tell which set has a greater number of counters?” Lesson 2.3 asks students, “How can you tell which set has a number of counters that is less? Which number is less?”
• There are several places where MP 3 is identified in the student materials, but these identifications do not always meet the intent of the standard. For example, MP 3 is identified in chapter 11, lesson 2, page 649 of the student edition; lesson 3, page 655 of the student edition; and lesson 4, page 667 of the student edition, but the questions provided do not have students constructing viable arguments or analyzing the arguments of others.
• chapter 10, lesson 7, page 609 of the student edition; lesson 9, page 623 of the student edition; and lesson 10, page 627 of the student edition all identify SMP 3, but the prompts/questions do not have students engaging in the practice.

In chapter 11, lesson 3, page 661 of the student edition, students are asked to construct an argument, but they do not analyze another student’s argument.

The instructional materials reviewed for Kindergarten 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. Overall, the materials consistently assist teachers in having students construct viable arguments, but they do not consistently assist teachers in having students analyze other students' arguments.

• In the planning guide for the materials, page 26 is dedicated to identifying where examples of MP 3 are located throughout the materials. The planning guide also lists examples of MP 3 on page 98.
• In the materials, the “Think Smarter” problems and “Go Deeper” sections of each lesson prompt teachers to help engage students in constructing viable arguments. For instance, in chapter 5, lesson 5, page 257, teachers are prompted to ask students “What happens if you turn your counters over to the opposite color? Do you get another way to make 10? Explain your thinking.”
• In the “Go Deeper” section in chapter 10, Lesson 10.9, on page 623 of the teacher’s guide, teachers are prompted with the following: “Children should understand that the traffic cone they circled is beside an object shaped like a cube. The other cone inside the truck is not beside an object shaped like a cube. Help children see that they can also describe the position of the object shaped like a cube as being beside the traffic cone.”
• In the Math Journal in chapter 10, Lesson 10.7, on page 612 in the teacher’s guide, students are asked to make drawings of a square, discuss, and share their models. The teacher is then directed to ask students to explain how building models of solid real-world shapes helps them learn more about the shapes.
• On page 23 of the teacher planning guide, there are a list of prompts to ask students to help them construct viable arguments.

There are several places where MP 3 is identified; however, these identifications do not meet the meaning of the practice standard.  For example, chapter 3, lesson 9; chapter 7, lesson 1, on page 362 TE, and lesson 9, page 410, all identify MP 3. An explanation does not exist to show teachers how to get kids to participate in a conversation using viable arguments and analyzing the arguments of others. The questions often have right or wrong answers, and when they do not, they still are not asking students to explain how they went about solving a problem and then talking with others about how they solved the problem.