Kindergarten - Gateway 1

The instructional materials for Connecting Math Concepts Level A, Kindergarten, do not meet expectations for assessing material at grade level. More than half of the program’s mastery tests and the two cumulative tests include items beyond the scope of expectations for the Kindergarten level, including adding and subtracting 2-digit numbers, identifying and counting money, and working with 3-digit numbers. It would not be possible for teachers or districts using this program to remove or amend the misaligned items without disrupting the overall structure and design of the grade-level program.

Review Team note: The Level A (Kindergarten) teacher guide identifies the cumulative tests as “optional” in two places (pages 3-4); however, the publisher orientation session identified these assessments as a required component of the grade level program. Therefore, the cumulative tests are included in this review.

Mastery Test 1: All items in parts 1-4 align fully with grade level standards, focusing on counting and cardinality.

Mastery Test 2: All items in parts 1-7 align with grade level standards, focusing on counting and cardinality. A minor exception occurs in Part 4, when students are asked to identify the plus symbol. While identifying symbols is not specifically called for in CCSSM, this is an acceptable expectation for Kindergarten students.

Mastery Test 3: Items in Parts 1-2 and 4-7 fully align with grade level standards, focusing on counting and cardinality and operations and algebraic thinking. Part 8 is partially aligned to CCSSM for Kindergarten. Items in Part 3 assess skills and/or content from future grades.

• Part 3 calls for students to understand the meaning of the equal sign, as they determine if a group of lines is equal to a given numeral; this is a Grade 1 expectation (1.OA.D.7).
• Part 8 calls for students to solve given expressions with lines underneath one of the addends as a visual representation. The given equations are within the scope of K.OA; however, students are prompted to “get going” (count on) from the first addend and count the lines to find the total. Counting on is a Level 2 method for addition, according to the K-2, operations and algebraic thinking progressions document (page 6). Direct modeling (drawing lines for both numbers) is a grade-level appropriate Level 1 strategy; teachers using these materials should consider accepting any solution strategies.

Mastery Test 4: Items in parts 1, 3, 4, 7, 9 and 10 align fully with grade-level standards, focusing on counting and cardinality and operations and algebraic thinking. Parts 2 and 8 are partially aligned to CCSSM for Kindergarten. Parts 5 and 6 assess skills and/or content from future grades.

• Part 2 items call for students to find sums within ten, but students are expected to use a counting on strategy. Direct modeling (drawing lines for both numbers) is a grade-level appropriate Level 1 strategy; teachers using these materials should consider accepting any solution strategies.
• Parts 5 and 6 call for students to complete an equation by drawing a quantity of lines to match a given numeral, and vice versa. These items are questionable, as they assess the meaning of the equal sign, which is a Grade 1 expectation (1.OA.D.7).
• Items in Parts 7-9 focus on counting and cardinality; an exception occurs in part 8, when students are asked to identify the “take away” symbol. While identifying symbols is not specifically called for in CCSSM, this is an acceptable expectation for Kindergarten students.
• Part 10 expects students to listen to a story problem and tell the teacher what symbols to write in order to represent the problem with an equation, which aligns with K.CC.A.3 and K.OA.A.1.

Mastery Test 5: Items in parts 1, 2, 5, 6,and 7 align fully with grade-level standards. Part 4 is partially aligned to CCSSM for Kindergarten. Part 3 assesses skills and/or content from future grades.

• Part 1 calls for students to listen to a story problem and tell the teacher what symbols to write in order to represent the problem with an equation, which aligns with K.CC.A.3 and K.OA.A.1.
• Part 2 aligns to K.OA.A.2, as students solve subtraction equations within 10 using drawings and equations.
• Part 3 calls for students to add 13 + 5 by drawing lines, which is beyond the grade-level expectation of sums within 10 (K.OA.A.2). The teacher presentation notes for Parts 3 and 4 call for students to find sums by drawing lines for only one of the addends, and counting on from the other addend to find the sum. Counting on is a Level 2 method for addition, according to the K-2, operations and algebraic thinking progressions document (page 6). Direct modeling (drawing lines for both numbers) is a grade-level appropriate Level 1 strategy; teachers using these materials should consider accepting any solution strategies.
• Items in Parts 5-7 are appropriate for Kindergarten students, focusing on counting and cardinality.

Mastery Test 6: Parts 1, 3, 4, 6 and 7 are aligned to Kindergarten standards. Part 5 is partially aligned to CCSSM for Kindergarten. Part 2 assesses skills and/or content from future grades.

• Part 2 of this assessment includes the equations 10 + 1, 13 + 1, and 17 + 1, all beyond the expectation of sums within 10 (K.OA.A.2). Also, the teacher presentation notes for these items state that students should “write answers to each of the plus-1 problems without making any lines” (Presentation Book 2, page 217). This indicates a focus on developing fluency; K.OA.A.5 calls for students to fluently add and subtract numbers within 5. All six items in part 2 have sums greater than 5.
• Part 5 calls for students to solve addition and subtraction equations. While all of these equations are within the appropriate scope of K.OA.A.2, the teacher presentation notes call for the use of a Level 2 counting on strategy as the only acceptable method for students to solve the given equations (Presentation Book 2, pages 217-218); teachers using these materials should consider accepting any solution strategies.
• Parts 8-9 of this mastery test do not have an explicit alignment to the standards, but they do not seem beyond the scope of grade-level expectations.

Cumulative Test 1: Items in Parts 1, 2, 7, 8, 10, 11 and 13 align fully with Kindergarten standards. Parts 5 and 12 are partially aligned to CCSSM for Kindergarten. Parts 3, 4, 6, 9 and 14 assess skills and/or content from future grades.

• Parts 1-2 assess counting and cardinality appropriately for the Kindergarten level.
• Parts 3-4 call for students to look at a numeral and represent it with the appropriate number of lines, and then look at a group of lines and write the appropriate numeral. The teacher is directed to say: “So you’ll make 3 lines on the other side to make the sides equal” (Presentation Book, page 223). Understanding the meaning of the equal sign is a Grade 1 expectation (1.OA.D.7).
• Items in Parts 5 and 12 call for students to solve addition and subtraction equations. While all of these equations are within the appropriate scope of K.OA.A.2, the teacher presentation notes call for the use of a Level 2 counting on strategy as the only acceptable method for students to solve the given equations; teachers using these materials should consider accepting any solution strategies.
• Part 6 requires students to cross out a given number of lines and then count what’s left “to make the sides equal” (Presentation Book, page 224), making this a Grade 1 expectation.
• Parts 7, 8, 10 and 11 align to K.OA.A work, although the teacher notes prescribe very specific solution strategies.
• Part 9 seems to assess fluency, as students are expected to complete +1 equations mentally, and all of these equations are beyond the “fluency within 5” expectation stated in K.OA.A.5.
• Part 13 assess counting skills appropriately (K.CC.A.1-2).
• Part 14 calls for students to state whether or not given “two-part” (two-digit) numbers (including 21, 31, 81) are teen numbers, which is beyond the expectation for K.CC.A.3.

Mastery Test 7: Part 3 is fully aligned with Kindergarten standards. Parts 1, 2, 4, 5, 6 and 7 assess skills and/or content from future grades.

• Part 3 calls for students to write and solve orally given addition and subtraction equations (K.OA.A.1-2).
• Part 1 calls for addition beyond 10 and involves missing addends (41 + ? = 45, 8 + ? = 13), resulting in items more appropriate for Grade 1 (1.OA.C.5-6, 1.OA.D.8).
• Parts 2, 4 and 5 call for an understanding of tens and ones (1.NBT.B.2).
• Parts 6-7 call for students to mentally “plus 10” to a given number, which is an expectation for 1.NBT.C.5.

Mastery Test 8: Items in Part 1 are partially aligned to CCSSM for Kindergarten. Items in Parts 2-9 assess skills and/or content from future grades.

• Items in Part 1 call for students to solve given addition equations, which is appropriate for Kindergarten students (K.OA.A.1-2); however, two of the four items include equations with missing addends, which isn’t an expectation until Grade 1 (1.OA.D.8).
• Part 2 items expect students to count coins to find the total value (2.MD.C.8).
• Part 3 expects students to write the expanded form for given 2-digit numbers (2.NBT.A.3).
• Parts 4-5 call for students to represent 2-digit numbers with symbols for tens and ones, and vice versa (1.NBT.B.2).
• Part 6 calls for students to count forward from given numbers beyond 120 (2.NBT.A.2).
• Part 7 calls for students to identify the value of underlined digits in 3-digit numbers (2.NBT.A.1).
• Part 8 calls for students to name the “turn-around” for a given addition equation; students aren’t required to apply properties of operations until Grade 1 (1.OA.B.3).
• Part 9 requires students add four tens to 25 mentally (2.NBT.B.5).

Mastery Test 9: Items in Part 7 align fully with Kindergarten standards. Part 1 is partially aligned to CCSSM for Kindergarten. Parts 2, 3, 4, 5, 8 and 9 assess skills and/or content from future grades.

• Items in Part 1 call for students to solve given addition and subtraction equations, which is appropriate for Kindergarten students (K.OA.A.1-2); however, two of the four items have sums and differences beyond 10 (1.OA.C.6), and 1 equation has a missing addend, which isn’t an expectation until Grade 1 (1.OA.D.8).
• Part 2 calls for students to compare numbers greater than 10 (1.NBT.B.3).
• Parts 3 and 5 call for students to add using 2-digit numbers (1.NBT.C.4).
• Part 4 calls for students to count coins to find the total value (2.MD.C.8).
• Part 6 expects students to write the expanded form for given 2-digit numbers (2.NBT.A.3).
• In Part 7, students identify 2D shapes (K.G.A.2).
• Part 8 calls for students to read 3-digit numbers (2.NBT.A.3).
• Part 9 calls for students to name the “turn-around” for a given addition equation; students aren’t required to apply properties of operations until Grade 1 (1.OA.B.3).

Mastery Test 10: Items in Part 9 fully align with Kindergarten standards. Parts 1, 2, 3, 4, and 5 assess skills and/or content from future grades. Parts 6, 7 and 8 are not aligned to CCSSM.

• Part 1 requires students to recognize that they can only write “turn-arounds” for addition equations (1.OA.B).
• Part 2 has students adding two 2-digit numbers using place value drawings (1.NBT.C.4).
• Part 3 calls for students to complete equations using given place value drawings with crossouts; this is beyond Kindergarten expectations.
• Part 4 has students completing +1 and +2 equations mentally, all beyond 10 (2.OA.A.2).
• Parts 5 and 10 call for students to count coins to find the value (2.MD.C.8).
• Parts 6 and 7 assess students’ ability to count backward from 10 to 1 and from a given number (not in CCSSM).
• Part 8 assesses students’ understanding of ordinal numbers (not in CCSSM).
• In Part 9 students identify 2D shapes (K.G.A.2).

Mastery Test 11: Items in Part 8 align with Kindergarten standards. Parts 1, 2, 3, 5, 6, 7 and 9 assess skills and/or content from future grades. Part 4 is not aligned to CCSSM.

• Part 1 calls for students to add with 2-digit numbers using “turn-around” equations to find the sums mentally, including 6 + 40, 1 + 27, 3 + 70, 1 + 35, 7 + 50 (numbers are beyond 2.OA.A.2 fluency expectations).
• In Part 2 students are supposed to subtract with two 2-digit numbers (2.NBT.B.5).
• Parts 3 and 7 call for students to “take away 1” and “plus 2” from numbers beyond 5 mentally (1.OA.C.5-6).
• In Part 4 students count paper money (not specifically required by the CCSSM).
• Part 5 calls for students to mentally add 10 to 2-digit numbers (1.NBT.C.5).
• Part 6 calls for addition and subtraction of two 2-digit numbers using “counters” (place value drawings) (1.NBT.C.4, 2.NBT.B.5).
• Part 8 includes questions about the attributes of 3D shapes (K.G.A.2-3).
• Part 9 calls for students to say equations for teen numbers (expanded form) (1.NBT.B.2, 2.NBT.A.3).

Mastery Test 12: Items in Part 1, 5, 7 and 8 align with Kindergarten standards. Part 6 is partially aligned to CCSSM for Kindergarten. Parts 2, 3 and 4 assess skills and/or content from future grades.

• Part 1 calls for students to recognize, count, and tally shapes, and then use that information to complete a table. While this activity does align with K.MD.B.3 and K.G.A.2, the team found it confusing; teachers using these materials should make modifications or find an alternate activity to better assess students’ understanding.
• Part 2 calls for students to write addition and subtraction facts for numbers that are connected on a given number line. This requires knowledge of the relationship between addition and subtraction, which is explored in 1.OA.B.
• Part 3 calls for students to write addition and subtraction equations from orally given story situations, using 2-digit numbers, and then to solve these equations using place value drawings (1.NBT.C.4, 2.NBT.A.5).
• Part 4 assesses understanding of expanded form (2.NBT.A.3).
• Part 5 displays an incorrectly ordered number line, and students are expected to correct it (K.CC.A-B).
• Part 6 calls for students to add and subtract 1 or 2 from given numbers mentally; 1 of the 6 items in Part 6 is within K.OA.A.5 expectations.
• Part 7 calls for students to look at two sets of items that are visually arranged to determine which group is “more” and “less” (K.MD.A.2, K.CC.C.6).
• Part 8 calls for students to identify 3D shapes (K.G.A.2-3).

Cumulative Test 2: Items in Part 20, 33, 35 and 36 align with Kindergarten mathematics standards. Parts 1, 6, 7, 9, 10 and 22 are partially aligned to CCSSM for Kindergarten. Parts 1-7, 9-18, 21-32 and 37 assess skills and/or content from future grades. Parts 8 and 34 are not aligned to CCSSM.

• Parts 1-2 call for students to use given place value drawings to identify numbers; 1 of the 8 items is a number within 10 (1.NBT.B.2).
• Part 3 calls for students to write orally given 2-digit numbers (1.NBT.B.2).
• Parts 4-5 call for students to use place value drawings to represent numbers (1.NBT.B.2).
• Parts 6, 7, 9, and 10 call for students to add and subtract 1 or 2 from given numbers mentally; 4 of these 20 items are within K.OA.A.5 expectations.
• Part 8 calls for students to count backward from a given number (not specifically required by the CCSSM).
• Parts 11-12 calls for students to “plus 10” mentally (1.NBT.C.5).
• Parts 13-14 assess understanding of expanded form (2.NBT.A.3).
• Parts 15, 16, 17 and 19 call for students to know the value of and count coins (2.MD.C.8) and paper money (not specifically required by the CCSSM).
• Part 18 calls for students to compare 2- and 3-digit numbers (1.NBT.B.3, 2.NBT.A.4).
• Part 20 calls for students to correctly complete a number line (K.CC.A-B).
• Parts 21-22 call for students to use drawings to solve addition and subtraction equations, including some with missing addends; 1 of the 6 items is appropriate for Kindergarten students.
• Parts 23-29 call for students to add and subtract with 1- and 2-digit numbers using place value drawings (1.NBT.C.4, 2.NBT.B.5).
• Part 30 calls for students to write addition and subtraction equations from orally given story situations, using 2-digit numbers, and then to solve these equations using place value drawings (1.NBT.C.4, 2.NBT.A.5).
• Part 31 requires students to recognize that they can only write “turn-arounds” for addition equations (1.OA.B).
• Part 32 calls for students to write addition and subtraction facts for numbers that are connected on a given number line. This requires knowledge of the relationship between addition and subtraction, which is explored in Grade 1 (1.OA.B).
• Part 33 calls for students to recognize, count and tally letters, and then use that information to complete a table. While this activity does align with K.MD.B.3 and K.G.A.2, the team found it confusing; teachers using these materials should consider making modifications or find an alternate activity to better assess students’ understanding.
• Part 34 assesses understanding of ordinal numbers (not in CCSSM).
• Part 35 calls for students to look at 2 sets of items that are visually arranged to determine which group is “more” and “less” (K.MD.A.2, K.CC.C.6).
• Part 36 calls for students to identify 2D and 3D shapes (K.G.A.2-3).
• Part 37 calls for students to read 3-digit numbers (2.NBT.A.3).

The instructional materials for Connecting Math Concepts Level A (Kindergarten) do not meet expectations for a viable amount of grade-level content for one year. The amount of content, including lessons and assessments, is viable for one school year; however, due to the large amount of content that is not aligned to the grade level, there is not enough instruction at the depth required by Kindergarten standards to prepare students for learning in future grades.

• This grade level program contains 120 core lessons, 11 optional parallel (reinforcement) lessons, 12 Mastery Tests, and 2 Cumulative Tests, spanning approximately 145 days, which is a sufficient number for a standard school year.
• The program is designed for use with small homogeneous groups; a 45-minute instructional period is suggested each day for each group, with 30-40 minutes of direct instruction and 5-15 minutes of independent work (Teacher Guide, page 6). The material allocated for instruction each day is reasonable for this suggested amount of time. Teachers using this program with the recommended three instructional groups would need to block 1 hour and 45 minutes of each day for mathematics instruction, which may be problematic; alternatives are suggested for decreasing the number of groups and required instructional minutes.
• Lessons 1-60 are mostly aligned to Kindergarten CCSSM. This first half of the instructional program contains a total of 640 exercises, with approximately 90 percent of the exercises focused on grade-level expectations. Some of these lessons may require slight modifications and/or omissions, but these changes would not affect the integrity of the program.
• Lessons 61-120 are not well-aligned to Kindergarten CCSSM. This second half of the instructional program contains a total of 651 exercises, with approximately 30 percent of these exercises focused on grade level objectives. The majority of exercises are focused on mastery of above level standards from Grades 1-2 (see report for Indicator 1b). This focus on above grade level objectives takes time and focus away from foundational grade level understandings. These lessons would require significant modifications and/or omissions that would affect the integrity of the grade level program.

The instructional materials for Connecting Math Concepts Level A (Kindergarten) do not meet expectations for following the learning progressions outlined in the CCSSM. These materials include a substantial amount of unidentified above grade-level content, which does not allow for students to have extensive work with grade-level problems. In addition, connections between concepts are not clearly articulated within or across lessons. The tendency of the lessons to move quickly beyond grade-level expectations and to force students to use a level of thinking they may not be developmentally ready for contradicts the “focused and rich” learning necessary to meet the full depth of Kindergarten standards.

i. Materials do not develop according to the grade-by-grade progressions. Prior or future content is not clearly identified.

• The publisher states: “The content is revised so that the children learn not only the basics but also the higher-order concepts that children sometimes do not master by third grade" (Teacher Guide, page 5). Similarly, “CMC Level A includes not only the basics but also concepts often not taught until third grade” (Teacher Guide, page 2). These are the only references the review team noted to above-grade level work. Although above grade-level content is introduced, explored, and assessed throughout the 120 lessons, this content is not clearly identified as beyond the grade level. For example, students are given multiple opportunities to identify and count coins and paper money (Lessons 71-75, 77-84, 89-97, and 101-111); however, money is not introduced in the CCSSM until Grade 2, when students solve word problems involving dollar bills and coins. The Planning Pages for these lessons make no mention of Grade 2 standards. The program also devotes a significant amount of time to developing 2- and 3-digit place value understanding (Lessons 52-95, 99-101, 110, 112, 113, 115 and 117). This understanding of place value aligns to Grades 1 and 2 (1.NBT.B.2 and 2.NBT.A.1), but there is no mention of these above grade-level standards on the Planning Pages or in the Tracks commentary.
• As explained for previous indicators (see 1b and 1d), CMC Level A Lessons 61-120 include significant work above grade level and therefore do not contribute to a deep understanding of the K.CC.C, K.OA and K.NBT domains, which all include clusters and standards that are major work and require substantial instructional time.

ii. Materials do not give students extensive work with grade-level problems.

• Careful attention is paid to developing the skill of counting in the CMC Level A materials, as “Counting is the foundation skill for every math operation children are to learn” (Teacher Guide, page 44). Students engage in practice with various types of counting activities, including rote counting, counting objects and events, counting on, counting from a given number to a given number, counting by tens, and counting backward, in each of the 120 lessons in the program. This emphasis on counting development aligns K.CC.A and K.CC.B, as outlined in the Kindergarten, Counting and Cardinality Progressions document. It should be noted, however, that students are expected to count up to 199, which is well beyond the expectation for Kindergarten (K.CC.A.1 Count to 100) and Grade 1 (1.NBT.A.1 Count to 120). There is also a focus on counting backward and counting by tens from non-decade numbers, which are not not specifically required by the CCSSM.
• Time and attention are given to understanding the operations of addition and subtraction, including learning about the symbols (+, =, -) used to assist with equations used when exploring Operations and Algebraic Thinking. “CMC Level A designs the operations of addition, subtraction, and algebra addition so that any child with the skills needed to enter the program will learn these operations” (Teacher Guide, page 119). Teachers using this program should note that the materials include an overemphasis on understanding the concept of equality and the meaning of the equal sign, which is a Grade 1 expectation (1.OA.D.7).
• These instructional materials include an emphasis on word problems: both in representing word problems with expressions and equations, and with solving word problems. K.OA.A.2 calls for adding and subtracting within 10; however, lessons in this program quickly move students beyond this expectation. By the end of Lesson 45, students are representing and solving problems within 20 (1.OA.A.1); by the end of Lesson 75 students have worked beyond 20 (2.OA.A.1), sometimes solving problems with two 2-digit numbers.
• For much of the word problem work in the lessons, the teacher presentation notes call for students to find sums by drawing lines for only one of the addends, and counting on from the other addend to find the sum. Counting on is a Level 2 method for addition, according to the K-2, Operations and Algebraic Thinking Progressions document (page 6). Direct modeling (using objects or drawings for both numbers) is a grade-level appropriate Level 1 strategy that isn’t given ample time and attention in this program. The K-2, OA Progressions document calls for “a great deal of focused and rich interactions in the classroom” for Kindergarten students “to enable all students to understand all of the numbers and concepts involved” (p. 11). The tendency of the lessons to move quickly beyond grade-level expectations and to force students to use a level of thinking they may not be developmentally ready for contradicts the “focused and rich” learning necessary to meet the full depth of the K.OA standards.
• There is an emphasis in the CMC Level A program on developing addition and subtraction fluency, which aligns with CCSSM expectations. However, K.OA.A.5 limits this work to within 5 for students at this level. These materials expect students to add and subtract quantities of 1, 2, and in some cases 10 to 1- and 2-digit numbers, which is well beyond expectations for Kindergarten students.
• The CMC Level A materials include 11 optional lessons for reteaching of concepts, based on student performance. These parallel lessons “present the same series of exercises that were presented in the corresponding regularly numbered lesson” (Teacher’s Guide, p. 18). This indicates that all students are expected to engage with and explore all tasks including the unmarked above grade level content.
• The CMC Level A program includes administration of a placement test for all students before beginning the program, “to measure children’s abilities to follow oral directions” (Teacher Guide, page 22). “If a child does not pass the test, the child should not be placed in Level A. The child needs more language instruction before beginning the program.” This information suggests that English Language Learners and other students with language deficits would not have their mathematical needs met when using this series. Note: During the Publisher Orientation, representatives stated that there were no prerequisites for Level A, and that all learners should use the program as designed.

iii. Most materials do not explicitly relate grade-level concepts explicitly to prior knowledge from earlier grades.

• An expectation to relate grade-level concepts explicitly to prior knowledge from earlier grades does not apply for the Level A instructional materials. The Teacher Guide outlines prerequisite skills for addition and subtraction (page 43), including counting skills and symbol identification and writing. Teachers using this program may wish to do additional research on the mathematical development of young learners, as the program materials do not include any background information or research on developmental stages of learning.

The instructional materials for Connecting Math Concepts Level A (Kindergarten) do not meet expectations for fostering coherence within the grade.

The materials do not include learning objectives that are visibly shaped by the CCSSM cluster headings.

• Both the teacher and student materials lack explicit connections to support student learning, due to the overall structure of the program and the exclusion of language from CCSSM cluster headings.
• The lesson materials do not emphasize references to or language from domain or cluster headings. The Common Core State Standards Chart (Teacher Guide, page 10-11) and elaborations (Teacher Guide, page 120-134) list the domains and cluster headings, but do not give attention to how the cluster headings relate to the lesson content. The Scope and Sequence of the Level A program (Teacher Guide, page 8-9) lists the different tracks, organized by skill(s), included in the 120 lessons. This chart lists skills such as: Reciting Counting Numbers; Count On—Next Number; Symbol Writing; Equality & Equations; Plus One-Digit Numbers, Column Problems, Word Problems, Shapes and 3-D Objects. The Planning Pages for each series of five lessons (Presentation Books) identify only the individual standards that align with each lesson, not the cluster headings.
• The exercises within lessons are presented as Student Learning Objectives on the Planning Pages in the Presentation Books. These objectives do not meaningfully relate the exercises to the standards; rather, the learning objectives simply name what students are doing in each exercise. For example, Student Learning Objectives for Lesson 101 include: count backward; identify numbers, shapes, and coins; add or subtract two-part numbers; compare relative size of shapes; use the take-away one rule; Use Ts and lines to complete a subtraction equation; determine when the turn-around rule applies; take away two-part numbers; learn plus-2 facts; work addition column problems with two-part numbers; complete problems independently.
• Lessons are not aligned at the cluster level, and do not focus on larger ideas. “Each lesson is composed of exercises from 6 to 9 tracks that present various (sometimes unrelated) skills” (Teacher Guide, page 44). Lessons are comprised of a series of 8-12 disparate exercises, with each exercise taught separately from the previous one. Problems and activities sometimes connect two or more clusters within a lesson; however, these are randomly placed and do not serve as an important or natural connection. For example, in Lesson 56 (Presentation Book 2, page 170-177), the topics covered throughout this lesson are placed in a random, unnatural order: Identify numbers in the 30s and 40s; Say the next number in a sequence; Determine when to add or subtract; Distinguish teen numbers versus other 2-digit numbers; Count by 10s; Use the plus-1 rule; Read non-teen 2-digit numbers; Use symbols for addition problems and solve using objects and equations. This lesson spans over the domains of: Counting and Cardinality, Operations and Algebraic Thinking, and Numbers and Base Ten without any listed reason by the publisher for doing so.

Materials rarely connect two or more clusters in a domain or two or more domains in a grade when appropriate.

• The program includes examples of exercises that connect two or more clusters, although these do not appear consistently throughout the year. For example, Lesson 56 Exercise 9 calls for students to solve word problems (K.OA.A) by writing an equation and then drawing lines to represent one of the addends (K.CC.B). A second example occurs in Lesson 97, Exercise 4, where students compare the sizes of various triangles (K.G.A) using superlative words, such as tallest or narrowest (K.MD.A). Also in Lesson 97 Exercise 2, students compare and discuss a rectangle and a square (K.G.A). By counting the sides (K.CC.B), they recognize that although the shapes have the same number of sides, they are not the same shape. Teachers using this program should take time when planning instruction to create opportunities to make explicit connections like these on a regular basis.