Kindergarten - Gateway 2

The materials reviewed in Kindergarten 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 K.CC.B:

• Unit 1, Module 1, Session 2: addresses and supports the developing understanding of cardinality and the conceptual understanding of K.CC.4 and K.CC.6 by sorting shoes in two lines then counting to identify which group is larger. The investigation uses concrete visual and verbal cues; there is correspondence across the mathematical representations as students are using verbal descriptions, concrete (actual shoes lined up on two different lines), and written value of each line.
• Unit 2, Module 1, Session 3: students reinforce their conceptual understanding of one-to-one correspondence (K.CC.4.A) as they are counting the number of boxes in the 10-frame and/or counting dots arranged on a ten-frame (K.CC.5). Students then use Unifix cubes to build a concrete representation of the 10-frame card, connecting the visual, verbal, and concrete representations.

The following are examples of attention to conceptual understanding of K.CC.6:

• Unit 2, Module 1, Sessions 4 and 5: conceptual understanding of comparing numbers is developed with a ten-frame. Strategies for determining which number has more or less are shared through discussion. In Session 5, the students play the game independently as the teacher observes and documents how students determine value of greater and less than.
• October Number Corner Calendar Collector: conceptual understanding is built with cubes and ten-frame representations. Discussion elicits evidence for which number is greater/less/equal by using multiple representations, including a simple array for comparison.

The following are examples of attention to conceptual understanding of K.OA.1:

• Unit 3, Module 2, Session 2: students develop conceptual understanding of addition and subtraction by acting out situations, using Unifix cubes, giving verbal explanations, and reading equations.
• Unit 6, Module 3, Session 3: students play the Work Place 6D Roll, Add & Compare game, roll 0-5 dice, build quantities to 10 with Unifix cubes, record the addition facts on a recording sheet, and then compare their total amount to their partners by snapping all their cubes together. Students are asked to justify their answer to who has more. Students connect the mathematical representations of dice, Unifix cubes/10-frames, written equations and Unifix trains to validate their comparison of who has more.
• April Number Corner Calendar Collector: writing addition equations is represented through direct modeling of frogs and represented as unit squares. Conceptual understanding of the addition equation sequence can be determined in multiple ways (example: 2 +1 + 1 + 1 is the same as 2 + 3).

The following are examples of attention to conceptual understanding of K.OA.3:

• Unit 1, Module 3, Session 1: students move from the 5-frame to the 10-frame in Terrific Tens. The 10-frame model helps develop students' understanding of part-part-whole relationship of 10 (K.OA.3). As students explore the 10-frame, they use their fingers to show the amount on various 10-frames.
• Unit 8, Module 4, Session 1: students compose and decompose numbers less than or equal to 10 and explore how they might see equations in the ten-frame. Students record their way of seeing various quantities within 10: 5 = 4 + 1, 2 + 3, etc. Students are then asked to think about what subtraction equation they can write or the same 10-frames: 5 - 1 = 4, 5 - 3 = 2. Students are asked to "show where they see the equation on the 10-frame" and "who has a different equation?"

The following are examples of attention to conceptual understanding of K.NBT.1:

• Unit 7 Module 2 Session 1 and Session 2: conceptual understanding of teen numbers is elicited from building numbers on a double 10-frame to see the unit of 10 as a whole with some more (10 and 3 is 13). Number line representations are also used to guide the counting sequence of more than 10.
• Unit 8, Module 3, Session 1: students develop conceptual understanding using place value mats of ones/tens to build numbers in the 10-20 range in Place Value Build and Win. They build the quantity with cubes on the mat, compare the numbers, and write inequality statements using the greater than and less than symbols. Cubes are pre-grouped into trains of 10. As students build the numbers, they are asked to explain how they used their cubes to build the number. Emphasis is placed on the "10 and some more" concept.
• February Number Corner Number Line: conceptual understanding of “ten and some more” is reinforced through multiple and concrete representations (double 10-frame and a manipulative number line). Connections are made between the concrete visual representation of the teen number and the written numeral representation.

The following are examples of attention to conceptual understanding of K.G:

• Unit 2, Module 4, Session 3, Pattern Block Puzzles: students observe and explore pattern blocks, identify the shape using characteristics and correct mathematical name (K.G.2), describe the positions (above, below, beside) (K.G.1) and develop understanding that shapes are the same regardless of orientation or size. Students also compose simple shapes to form larger shapes (K.G.6), as they cover various shapes with smaller pattern blocks. There is correspondence across mathematical representations as student give verbal descriptions of a shape's characteristics, practice using the correct word for the shape, use concrete pattern blocks to compose a larger shape. Conceptual discussions with high level questions occur (students quietly observe the shapes of various pattern blocks and then are asked, "Can you tell me about these shapes?") Students pair-share ways to build designs, have the opportunity to build, and then are invited to share design and finally asked, "Can you show me a different way to cover the shapes?"

The Kindergarten 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.

• Throughout the materials, computational fluency is elicited with both addition and subtraction equations. Students are able to use counters, 10-frames or drawings to assist them. Evidence is gathered to note if a student has moved to procedural fluency and no longer needs concrete materials to add and subtract within five or within ten by using a unit of five. The expectation within the last two units is that students will be able to decompose five into parts fluently without the support of concrete materials to show procedural understanding.
• Students spend a significant amount of time and have a variety of opportunities to fluently add and subtract throughout number corners activities. K.OA.5 is addressed in two areas of Number Corner. Although the publisher does not list the K.OA.5 standard in any of their Computational Fluency workouts, instead most often listing K.OA.4 in relation to adding and subtracting within five, Computational Fluency workouts use finger patterns, 5-frames, and the number line to help students develop fluency with addition and subtraction facts to the number five. Calendar Collector workouts have students collecting various items to count throughout the month.
  • In the March Calendar procedural fluency is guided by using subitizing images to state how many more to make a unit of 10. This builds from the conceptual understanding within an organized structure to see the parts of ten fluently without having to count (perceptual subitizing).
  • In the May Computational Fluency workout fact fluency to 5 is investigated by using multiple representations (number cards, 10-frames). Routines focus on looking at decomposing 5 into 2 or even 3 addends to build number flexibility.
• Fluency is developed throughout the sessions of the Kindergarten instructional materials.
  • In Unit 1, Module 2, Session 3 in “Fives with Fingers,” frames are flashed and students show number of dots with fingers of one hand and use their other hand to show how many empty boxes there are in the 5-frame and then add to find the total in all.
  • In the Unit 6, Module 4, Session 1 Work Place “Shake Those Beans,” students count how many red and how many white beans and how many in all to determine all combinations of five.
  • In Unit 7, Module 3, Session 5 in “Cubes in My Hand,” the teacher divides five cubes between her two hands. The teacher opens one hand to reveal cubes while keeping the other cubes hidden in other hand. Students determine how many cubes are hiding and then write the equations that represent the investigation.
  • In Unit 8, Module 1 students fluently subtract with minuends to 5 by using spinners and drawings to represent minuends and subtrahends.

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 each grade.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide single step contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. The majority of the application problems are done with guiding questions elicited from the teacher through whole group discussions that build conceptual understanding and show multiple representations of strategies. Materials could be supplemented to allow students more independent practice for application and real world contextual problems that are not teacher guided within discussions. This would provide students opportunities to show more evidence of their mathematical reasoning through common addition and subtraction situations as outlined in the CCSSM Glossary, Table 1.

The instructional materials include problems and activities aligned to K.OA.2 that provide multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. Examples of these applications include the following:

• In Unit 3, Module 3, Session 2, "Bicycle story problems," students are using their 10-frames to solve story problems given to them orally.
• In Unit 6, Module 4, students engage in application of addition skills to solve story problems.
• In Unit 7, Module 3, Sessions 1, 2 and 3, students solve frog addition/subtraction word problems using pictures. Students share out their strategies for solving. Students use Unifix cubes to model story problems and solve.
• In Unit 8, Module 1, Sessions 1-4, students use manipulatives, pictures, and 10-frame counting mats to demonstrate application of addition and subtraction skills for solving story problems.
• This module contains story problems set in the context of addition and subtraction. Student strategies are shared to elicit more sophisticated strategies over time within the unit. The unit also contains a checkpoint small group formative assessment to gather data to evaluate student strategies and misconceptions.
• In the Number Corner February Computational Fluency, students began to add to 10 in the context of themed story problems and application within the number corner computational fluency routine. Thinking within these contextual situations is extended toward building conceptual understanding of subtraction as a missing addend problem.

The materials reviewed in Kindergarten meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Kindergarten materials all there 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.

The instructional materials reviewed for Kindergarten 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, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Kindergarten 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 Kindergarten?" guide for teachers (AG, page 16).

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 for the teacher that shows the connection between the indicated MP and the content standard. Examples of the MPs in the instructional materials include the following:

• In Unit 1, Module 3 each of the six sessions list the same two Math Practices: MP6 and 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, four sessions (1, 2, 3, 6) list MP6, four sessions (1, 2, 5, 6) list MP7, three sessions list MP8 (3, 4, 5) and one session lists MP3 (4).
• In Unit 2, Module 3, Sessions 1, 4 and 6 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In Unit 4, Module 2 all five sessions list in the Skills & Concepts section two MPs: MP6 and MP7.
• In Unit 7, Module 1, sessions 2 and 5 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
• In the September Number Corner MP2 is referenced in the Calendar Collector; MP4 is referenced in Days in School; MP7 is addressed in Calendar Grid, Computational Fluency, and Number Line; and MP8 is addressed in Calendar Grid, Computational Fluency, Number Line, Days in School and Calendar Collector.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These lessons occur at the beginning and the end of the year. These sessions that focus entirely on MPs include the following:

• Unit 1, Module 4, Session 1
• Unit 1, Module 4, Session 2
• Unit 1, Module 4, Session 3
• Unit 1, Module 4, Session 4
• Unit 8, Module 4, Session 4
• Unit 8, Module 4, Session 5

The materials reviewed for Kindergarten 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 2, Module 1, Session 2, within the Problems and Investigations portion, students are introduced to the "think-pair-share" routine. They are asked to listen and explain their partners' thinking.
• In Unit 5, Module 4, Session 2 students are introduced to "There's a Shape in My Pocket." In this activity, students present arguments and critique the reasoning of their classmates to come to an agreement about which cards to remove.
• In Unit 7, Module 4, Session 1 students engage in a "think-pair-share" routine. As in other Sessions in the instructional materials, this activity allows students to share their thoughts, listen to the thoughts of classmates, and justify their own reasoning.
• In the March Calendar Grid and Number Line students share their thinking and justify their reasoning in developing their combinations of numbers to construct a ten.

The instructional materials reviewed for Kindergarten 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, difference, and ten-frame.

• 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 1, Section 5, in the Problems & Investigation section, the teacher is reminded to use the vocabulary for three-dimensional shapes, such as edge, face, vertex, surface.