Kindergarten - Gateway 2

The Stepping Stones instructional materials for Kindergarten partially meet expectations for developing conceptual understanding of key mathematical concepts. The program consistently devotes instructional time to the use of manipulatives and multiple representations to introduce and develop students’ understanding of numbers and counting, as well as addition and subtraction. There are missed opportunities to develop a deep understanding of teen numbers as a foundation for place value.

K.CC.A-B

• The publisher identifies three aspects that help students develop a full understanding of numbers (start - grade K - module 1 - mathematics - focus): a concrete or pictorial representation (visual), a sound or number name being said (auditory), and a written numeral or number word (written). At least two of these three aspects are included in lessons that focus on understanding numbers and counting. An example occurs in Lesson 1.4, where students are given cards with numerals, number words, or pictorial representations of the numbers 1 to 5 (visual/written). The teacher calls out a number (auditory), and any student with a matching card stands up; those students then clap as many times as the number called.
• The Kindergarten program includes opportunities for students to engage in subitizing, where students practice recognizing numbers as groups rather than as individual objects. This emphasis on subitizing allows students to build number sense and prepares them to explore addition and subtraction concepts.
• Visual representations are used to help students visualize numbers and quantities (five-frame and ten-frame), as well as to understand the relative position of numbers (number track) and explore concepts of one more and one less.

K.OA.A

• The Kindergarten program has a strong focus on building a conceptual understanding of addition and subtraction, by structuring lessons purposefully using a concrete-pictorial-abstract approach (start - grade K - module 1 - mathematics - mathematical practices - resource tab - KMP2 video). For example, students begin to explore addition in module 6 by acting out situations (Lesson 6.1) and using manipulatives (Lesson 6.2). As students continue exploring, they add by making jumps on a number track (Lesson 6.4), and also drawing and analyzing pictures that represent addition stories. As students progress, they write addition sentences using numbers and words (Lesson 7.4: "2 add 5 is 7"), and then finally students write equations with symbols (Lesson 8.1: "2 + 5 = 7").
• “Concrete materials and/or pictures are always used to assist children” in developing an understanding of addition and subtraction concepts (start - grade K - module 6 - mathematics - focus). “[Students] use stories and concrete/pictorial materials to identify the special feature of [addition] – the number in each of the parts that form the total.” For example, in Lesson 8.4, students use linking cubes to model situations in a story, as they explore pairs of numbers that equal 10.
• As students transition to pictorial representations of subtraction, lessons are built around the idea of active subtraction: “When [children] work with pictures and it is not possible to move the objects, new words need to be introduced. The lessons suggest ‘cross out’ and ‘cover’ [as alternatives to ‘take away’]” (start > grade K > module 9 > mathematics > focus), which supports students understanding of the meaning of subtraction.
• Whole class lessons conclude with reflection time, where students are encouraged to verbalize their developing understanding of addition and subtraction concepts. Some examples of conceptual reflection questions include: Lesson 6.3 “The example shows that 8 add 2 makes 10. What are some other ways that you could make 10 with two groups?” Lesson 8.1: “What is happening in each picture? What numbers do you know for each story? What numbers do you need to find out for each?” Lesson 12.2: “What is the difference between adding and subtracting?”

K.NBT.A

• In discussion of the mathematics of module 11 (start - grade K - module 11 - mathematics - research into practice), the publisher states: “Because the number[s] 11 through 20 are so important to future counting, basic facts, and mental computation, it is sensible to devote special attention to them.” However, only 6 of the 72 lessons in the Kindergarten program focus on exploring the numbers 11-19; 2 of the 6 lessons focus on using pennies and dimes to represent teen numbers.
• Students begin their exploration of teen numbers in Lessons 10.3 with the numbers 14, 16, and 17 because these number names are considered “easy.” Students begin by examining the letters in the number names, and writing the numerals, and then recall real world contexts where they have seen these numbers used. The student journal page calls for students to write the number words and then draw a matching quantity. Lesson 10.4 follows a similar structure using 19, 18, and 15. Each of these lessons ends with a brief activity with the teacher saying: “I’m thinking of a number that has one group of ten and (nine) more,” and then expects students to respond with the correct number, even though the structure of “ten ones and some more ones” wasn’t explored.
• Lessons 11.2 and 11.3 develop the understanding of teen numbers as a group of ten ones and some more ones, with pictorial representations and the use of ten-frames. These lessons explore K.NBT.1 in a way that strongly supports students’ beginning understanding of the place value system. Teachers using this program should supplement with more lessons like these to develop this concept more fully.

The Stepping Stones instructional materials for Kindergarten partially meet expectations for attention to procedural skill and fluency. Overall, the program materials appropriately develop procedural skill and fluency with rote counting, number writing and recognition, and fact fluency throughout the year. Students have daily opportunities to practice and develop these skills; however, a significant portion of the exercises for developing fact fluency extend beyond addition and subtraction within 5 (K.OA.5).

• Rote counting practice, number writing, and number recognition are targeted routinely over the course of the year. Many lessons launch with counting practice before addressing the day’s learning target. Counting to 100 (K.CC.1) is explicitly targeted within lessons in modules 1, 3, 4, 5, 6, 7, 8, 10 and 12. Counting forward from a given number (K.CC.2) is explicitly targeted within lessons in modules 4, 6, 9, 11 and 12. Writing numbers (K.CC.3) is explicitly targeted within lessons in modules 1, 2, 5, 10 and 11.
• Lessons 1, 3, and 5 in each of the first eight modules include supplemental whole-class “fluency practice” activities with a focus on developing counting and subitizing skills (K.CC.A-B). Lessons 2, 4 and 6 in each of the first eight modules include “ongoing practice” student pages for the development of counting skills, number writing, and number recognition (K.CC.A).
• Progress toward fluency and procedural skill is interwoven with students’ developing conceptual understanding of the properties of operations. Students’ practice with fact fluency doesn’t begin until module 9, after students have had ample time to explore addition and subtraction concepts.
• Although K.OA.5 is included in the learning targets for modules 9 and 12, the program does not have any whole-class or small-group lessons in these modules that specifically focus on developing addition and subtraction fact fluency within 5. All of students’ practice around fact fluency occurs in supplemental activities.
• Lessons 1, 3 and 5 in each of the last four modules include supplemental whole-class “fluency practice” activities with a focus on developing fact fluency (K.OA.5). Lessons 2, 4 and 6 in each of the last four modules include “ongoing practice” student pages for developing fact fluency (K.OA.5). It should be noted that a significant amount of this practice (approximately one-third of all exercises) extends beyond addition and subtraction within 5, and practice doesn’t strategically emphasize mastering facts within 5 (K.OA.5) before directing students to explore facts within 10 (1.OA.7).

The Stepping Stones instructional materials for Kindergarten partially meet expectations for attention to engaging applications of grade-level mathematics. Overall, the program devotes adequate time to exploring addition and subtraction concepts using actions, manipulatives, pictures, and equations; however, there needs to be more problems presented in real-world contexts.

• K.OA.2 specifically calls for students to apply their mathematical understanding as they “solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.” Only 4 of the 72 lessons in the Kindergarten program explicitly address this standard, according to the “Grade K and the CCSS” document. These 4 lessons include opportunities for students to apply counting skills as they act out addition and subtraction situations, using objects, pictures, and/or interactive software to write and solve equations.
• The K.OA.A cluster heading calls for students to understand putting together, adding to, taking apart, and taking from situations (see Table 1, CCSSM page 88). The K-2, Operations and Algebraic Thinking Progressions document further identifies four specific problem subtypes that Kindergarten students should explore. Much of the work in this program is limited to adding to and taking from situations with the result unknown.
• There is a missed opportunity for students to work with put-together/take-apart situations where both addends are unknown in application settings. Exploration of these types of situations in application settings would support students with the CCSSM required fluency of adding and subtracting within 5 (K.OA.5).
• Much of the work with word problems in the Kindergarten program happens in whole-class lessons. The teacher poses addition and subtraction situations orally, or students interpret pictorial representations of addition and subtraction situations. There are minimal opportunities for students to generate their own representations of a given situation.

The Stepping Stones instructional materials for Kindergarten partially meet expectations for identifying and using the MPs to enrich grade-level mathematics content. Overall, the MPs are clearly identified in the program materials at both the module and lesson levels, and the practice standards are connected to mathematical content. However, two of the eight practices are significantly underrepresented, and the team noted misidentification of MPs in some lesson materials.

• The MPs are clearly identified in the instructional materials in multiple places. Within each module, there is a chart that identifies the MPs targeted in each of the module’s six lessons (start - grade k - module # - mathematics - mathematical practices). Users can also click on the Resources Tab to access a “Mathematical Practice Overview” document, which identifies the MPs targeted in each of the 72 lessons in the Kindergarten program. Within each lesson, the targeted MPs are listed in the learning objective for each lesson, as well as on a lesson contents chart (start - grade k - module # - lessons - contents).
• There were some inaccuracies/inconsistencies in the materials when identifying targeted MPs. For example, the Overview chart indicates that MP3 and MP5 are taught in Lesson 10.6; however, the lesson contents chart and whole class lesson reference MP3 and MP6. Another inconsistency occurs in Lesson 12.2, where the Overview chart indicates that MP4 and MP8 are targeted; the lesson contents chart and whole class lesson identify MP3 and MP8.
• The MPs are interwoven within the curriculum to enrich the grade-level content and are not taught as separate entities. Each lesson contains a pop-up text box that describes how the MP enriches the lesson. For example, in Lesson 6.4, students use the number track (MP5) to add numbers in a concrete way (K.OA.1); the text box reads: “SMP5 Students are making the connection between the number track and finding sums of two quantities.” A second example occurs in Lesson 5.2, where students use their fingers to represent quantities from one to ten and identify how many more fingers would be needed to make the quantity of 10 (K.OA.4, MP7); the text box reads: “MP7 As students figure out the number necessary to make 10, they are beginning to make use of structure. Making ten is a foundational concept.”
• The Kindergarten program focuses more heavily on some of the MPs than others. MP1 and MP5 are significantly under-represented. MP1 is only targeted in 2 of the 72 lessons; this limited experience is not enough for students to explore and develop the skills and understandings needed to develop perseverance in problem solving. MP5 is only targeted in 7 of the 72 lessons; Kindergarten students need extensive experience working in the concrete stage with a variety of manipulatives. The program’s limited focus on this practice would not allow students to develop an understanding of how to select and use tools strategically.

The Stepping Stones instructional materials for Kindergarten partially meet expectations for attending to the full meaning of each practice standard. The materials attend to the full meaning of some, but not all, of the eight MPs.

• Most of the lessons in the Kindergarten program (68 out of 72) identify only one or two targeted MPs. This pointed focus allows teachers and students to build a solid understanding of the targeted practices as they relate to grade-level content.
• Each module of the program includes this relevant commentary about the MPs (start - grade k - module # - mathematics - mathematical practices): “The Standards for Mathematical Practice describe the actions and ‘habits of mind’ that mathematically proficient student develop over time.” The Resources Tab includes links to professional learning videos which specifically focus on individual MPs and Habits of Mind in order to assist teachers in understanding the full meaning of these practice standards.
• MP1 Make sense of problems and persevere in solving them is not fully attended to in the Kindergarten program. This MP is listed as a target in 2 of the 72 lessons within the program, which is not enough to reach the full depth of this standard. In Lesson 6.1, students act out stories to model addition, “looking to see if their stories make sense” (whole class, pop-up text box); as a first introduction to MP1, it is appropriate to focus only on sense-making in this lesson. In Lesson 7.3, the whole-class lesson focuses largely on reading and writing the equal sign, while students find the total number of two groups of objects. MP1 is listed as a target in this lesson, with the note that during Step 4 of the lesson, teachers should “look for those students that persist in finding more than one way of making 10” (pop-up text box). It is appropriate to focus only on perseverance here; however, there are no instructional moves included to help students understand the importance of building perseverance and how to do that.
• MP5 Use appropriate tools strategically is not fully attended to in the Kindergarten program. This MP is listed as a target in 7 of the 72 lessons within the program (“Mathematical Practice Overview”), which is not enough to reach the full depth of this standard. In addition, while students are given opportunities to work with tools in many lessons, students are told which tool to use and are not provided opportunities to select tools to use in order to solve problems. In Lesson 2.6, students are introduced to the use of a 2-column “graph” as a tool to analyze data. In Lessons 3.2 and 3.3, students are introduced to the number track as way to identify numbers that come before and after a given number. In Lesson 6.4, students are given a number track to represent and complete addition equations. In Lesson 6.5, students explore the ideas of heavy and light; Step 4 calls for teachers to question students about how they might check to see if objects are heavier or lighter. In Lesson 6.6, students are introduced to the pan balance as a tool to compare the weight of two objects. In Lesson 12.4, students are again given a number track to identify one more/one less than a given number. [Lesson 10.6 is incorrectly identified.]
• Each whole class lesson includes an opportunity for students to reflect on the day’s learning (step 4: reflecting on the work). During this component of each lesson, the teacher poses questions to students about the work they completed in their journals and how that work ties to the day’s objective. This part of the lessons offers opportunities for students to engage in mathematical reasoning (MP2), construct arguments (MP3), explain how they used a particular tool or strategy (MP5), and/or explain an understanding of repeated reasoning (MP8).

The Stepping Stones instructional materials for Kindergarten partially meet expectations for prompting students to construct viable arguments and critique the arguments of others, concerning key grade-level mathematics detailed in the content standards. Many of the lessons include opportunities for students to construct viable arguments; however, these opportunities are often posed as whole-group questions. The program materials include few instances that prompt students to analyze the arguments of others.

• A majority of the opportunities for children to engage in constructing and analyzing mathematical arguments are posed as whole-group questions during whole-class lessons. Students are only able and held accountable to share their thinking if called on by the teacher, which does not ensure that all students practice this important mathematical reasoning.
• A significant number of the lessons tagged with MP3 focus on constructing viable arguments; there are very few opportunities available for students to critique the reasoning of others in this program. In Lesson 4.2, for example, students analyze pictorial quantities and are asked: “Which number is greater? How do you know?” The lesson doesn’t prompt students to consider the rationale of their peers. In Lesson 11.3, the class generates a list of equations to represent teen numbers: 10 + 1 = 11, 10 + 2 = 12, 10 + 3 = 13; students are asked to predict the next number sentence, and work with a partner to prove that the number sentences are true using a strategy of their choice. Pairs are invited to show and explain their work, but there is no mention of critiquing peers’ thinking or strategy choice.
• MP3, "Construct viable arguments and critique the reasoning of others", is specifically tagged in 15 of the 72 Kindergarten lessons, according to the “Mathematical Practice Overview” document. This means that students are prompted to construct arguments about mathematical content in 1 of every 5 lessons. In addition, MP3 is not targeted in any lessons in Modules 2, 8 and 9. This translates to approximately 45 days of instruction where instruction doesn’t intentionally focus on constructing and analyzing mathematical arguments. Teachers using this program will want to embed additional opportunities for students to engage in constructing and analyzing arguments.
• Each whole-class lesson includes an opportunity for students to reflect on the day’s learning (step 4: reflecting on the work). During this component of each lesson, the teacher poses questions to students about the work completed in Student Journals and how that work ties to the day’s objective. For example, in Lesson 4.3, students compare pictorial quantities of 1-9 to determine which amount is less; the teacher asks: “How do you know the number is less than 4?” Another example occurs in Lesson 12.2, where students are working with the concept of subtraction; the teacher asks: “What is the difference between adding and subtracting?” These questions allow students to construct arguments about their mathematical thinking, even in lessons that aren’t specifically tagged with MP3.
• The Kindergarten program includes missed opportunities to engage students in constructing and analyzing mathematical arguments. For example, Lessons 1, 3 and 5 in Modules 1-8 include ongoing “fluency practice” that focuses on developing students’ counting and subitizing skills; however, the teacher materials only state: “As you advance each slide, have the students count the objects or subitize the quantities.” Another example of a missed opportunity occurs in Lesson 1.3, where students are connecting number to quantity. Students are prompted to compare their work with a partner, and “raise your hand if you and your partner have the same answer. If we move our fingerprint faces around, will we still have four faces. How do you know?” This could be an opportunity for students to engage in partner talk, but the materials do not explicitly call for this.

The Stepping Stones instructional materials for Kindergarten partially meet expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Materials assist teachers in engaging students in constructing and analyzing arguments, although this assistance is inconsistent and focuses largely on constructing arguments more than analyzing arguments.

• Teacher notes in some lessons are lacking in explicit instruction in how to engage students in sharing their mathematical thinking. Simply posing the question “How do you know?” is not explicit enough to develop Kindergarten students’ abilities to construct and analyze arguments. Simply hoping that students say certain things is also not enough to support teachers in developing this math practice. An example of this occurs in Lesson 11.3, where students represent and name teen numbers; the MP3 pop-up text box states: “Students demonstrate [MP3] with responses such as, ‘The sums are one bigger each time’ or ‘This is just like counting except now there is a ten there’.” There is no guidance for teachers in how to support students in crafting such responses.
• This program gives more time and attention to constructing arguments, and significantly less time critiquing the arguments of others. Teachers are rarely prompted to promote partner talk or plan opportunities for students to consider incorrect mathematical reasoning.
• Throughout each lesson, suggested questions for teachers to pose to students are highlighted in pink.
• The Kindergarten materials for each module include a “research into practice” and “focus” page that gives teachers a summary of the mathematics being taught, why the content is important, and how students are exploring the content in a developmentally appropriate way. This information is helpful as it also draws attention to questioning that could play into reasoning and constructing a mathematical argument. For example, in Module 5 (start - grade k - module 5 - mathematics - focus), the publisher states: “For numbers 5 and 10, students should be encouraged to make 2 statements, for example, ‘Seven is 2 more than 5, and seven is 3 less than 10.’
• Each module of the program includes relevant commentary about the MPs (start - grade k - module # - mathematics - mathematical practices). The Resources Tab includes links to professional learning videos, which specifically focus on MP3 and Habits of Mind, in order to assist teachers in understanding the full meaning of this practice standard. The KMP3 video discusses the importance of student discourse in the mathematics classroom. The DHM2 video discusses justifying claims, proving conjectures, and distinguishing between agreement and logical necessity when constructing and analyzing arguments.