Building a case for blocks as kindergarten mathematics learning tools

Abstract: Kindergarteners need access to blocks as thinking tools to develop, model, test, and articulate their mathematical ideas. In the current educational landscape, resources such as blocks are being pushed to the side and being replaced by procedural worksheets and academic ‘‘seat time’’ in order to address standards. Mathematics research provides a solid basis for advocating for hands on resources to explore geometry and number concepts. Through the use of blocks in standards based mathematical tasks, students have the opportunity to develop important mathematical concepts and reasoning strategies. Kindergarten teachers’ instructional actions can be grounded in history, research, personal wisdom, and professional knowledge regarding what is appropriate and meaningful for their students in learning mathematics with thinking tools such as blocks.

Early Childhood Educ J (2016) 44:389–402 DOI 10.1007/s10643-015-0717-2 Building a Case for Blocks as Kindergarten Mathematics Learning Tools Cathy Kinzer1 • Kacie Gerhardt2 • Nicole Coca3 Published online: July 2015 Ó Springer Science+Business Media New York 2015 Abstract Kindergarteners need access to blocks as thinking tools to develop, model, test, and articulate their mathematical ideas In the current educational landscape, resources such as blocks are being pushed to the side and being replaced by procedural worksheets and academic ‘‘seat time’’ in order to address standards Mathematics research provides a solid basis for advocating for hands on resources to explore geometry and number concepts Through the use of blocks in standards based mathematical tasks, students have the opportunity to develop important mathematical concepts and reasoning strategies Kindergarten teachers’ instructional actions can be grounded in history, research, personal wisdom, and professional knowledge regarding what is appropriate and meaningful for their students in learning mathematics with thinking tools such as blocks Keywords Mathematics Á Standards Á History of blocks Á Research informed instruction Á Practical Á Research and practice Visualize 5- and 6-year-olds in a kindergarten classroom discussing ideas, solving problems, representing objects, and observing the shapes, sizes, patterns, and qualities of a & Cathy Kinzer cakinzer@nmsu.edu Kacie Gerhardt kk1338@nyu.edu Nicole Coca nicklecoca@yahoo.com Mathematics Educator, New Mexico State University, MSC 3CUR, Las Cruces, NM 88003-8001, USA New York University, New York, NY, USA Las Cruces Public Schools, Las Cruces, NM, USA complex block structure that they have constructed collectively Then, look to the majority of kindergarten classrooms in the United States On a typical day, 5- and 6-year-old children spend less than 30 min—and often no time at all— in child-initiated exploratory play or other learning activities with resources such as blocks (Miller and Almon 2009) This article is the result of a university-school partnership in which university educators participate in early childhood classrooms, listening to teachers and learning with young children Through extensive interaction with educators, as well as visits to other kindergarten classrooms in the area, a common concern emerged about the lack of opportunity for kindergartens to use physical blocks in their curriculum Historically, blocks have been an integral part of kindergarten classrooms as resources for play, instruction, and learning However, as academic seat-time in kindergarten to address literacy and numeracy standards and carry out the required assessments has increased, the result has been fewer opportunities for children to develop visual, spatial, and fine motor skills by using blocks as mathematics thinking tools Many teachers in our partnership expressed concern that, while their mandated curriculum includes pictures of blocks on worksheets, there are currently not many standards-based lessons that used real blocks—such as geoblocks, pattern blocks, unit blocks, tree blocks—for actively learning mathematics In spite of the lack of support for utilizing physical blocks in contemporary standards-based curriculum, we observed how many teachers in our community continue to incorporate learning centers that include blocks whenever possible The learning centers are important because children can play, explore, and informally engage in mathematical ideas in ways that support their mathematical development; however, they are not enough Teachers in our partnership have advocated for ways to bring blocks 123 390 back from the margins of the classroom One teacher asked, ‘‘The opportunities to learn through using blocks are disappearing from our kindergarten school day, except occasionally in our centers! What can we do?’’ Kindergarten teachers hope to make a case for using blocks as learning tools to address mathematics standards and to be an integral part of the curriculum One kindergarten teacher in our district said, If we can articulate both the research and the connections to our standards, we will have a solid foundation for advocacy… We are teaching lessons aligned to the mathematics standards, but how that is done—how students experience and contribute to the learning—makes all the difference in the world Initially, children need concrete, hands-on tools for thinking about and representing mathematical ideas They have transitioned from their homes and preschools where they were interested in using different types of blocks in activities that stimulated language, creativity, math knowledge, and enjoyment Kindergarten educators recognize the need for relevant interactive learning activities that connect physical objects with abstract concepts, and they seek methods to use learning tools in ways that promote conceptual understanding of the standards that they are required to teach Another teacher noted, Blocks support my students’ learning and interest in doing math It is more tangible and real for children as they relate to blocks Children use 3-D blocks to compare sizes and shapes and see relationships between blocks They explore the features of shapes in developing spatial sense and connecting to number concepts like counting the number of sides or edges of a block Children value using blocks as learning tools The purpose of this article is to consider mathematics learning opportunities with blocks through research and the wisdom of teachers in kindergarten classrooms The hope is that kindergarten teachers will gain historical, research, and practice-oriented perspectives as well as instructional resources that will enable them to advocate for incorporation of blocks as learning tools in mathematics lessons while addressing required state or district mathematics standards The Context for Considering Blocks as Mathematics Learning Tools Blocks have been an integral part of many young children’s lives, whether through child-initiated block play, constructions, or guided block-learning experiences Research shows that children have powerful intuitive mathematical 123 Early Childhood Educ J (2016) 44:389–402 competence (Ginsberg 1983) They not see mathematics as a separate subject of study until they enter formal schooling Children naturally think mathematically as they compare, quantify, and explore space and shapes in the world around them The most powerful opportunities for learning mathematics in primary grades are those that seek to build from children’s cultures, languages, and pre-existing informal mathematical experiences Many kindergarten students can connect with blocks as tools for exploration and learning because blocks are often part of their background experiences at home or in preschool These prior ways of knowing are powerful resources for developing learning activities in the kindergarten classroom (Moll et al 1992) Young children’s early experiences in mathematizing through familiar objects such as blocks can contribute to collective negotiation not simply of mathematical knowledge but also social interactions and communication in the formal setting of kindergarten In the following section we discuss the historical landscapes of blocks as manipulatives that connect to students’ curiosities, ways of knowing, and developing mathematical ideas Historical Foundations for Blocks as Mathematics Learning Resources Throughout history, humans have utilized natural materials in the environment such as soap, wood pieces, rocks, and boxes to build and test their ideas and inventions (Hewitt 2001) The way that blocks became integrated into more formal educational environments is fundamental to understanding why they are important resources for kindergarten classrooms today Many prominent early childhood educators incorporated blocks into the curriculum because the structure and nature of blocks provide important opportunities for young learners to connect to, and further refine, their mathematical schemas That is, blocks and other manipulatives became foundational in educational contexts because they are a way of exploring and articulating the mathematical ideas that children are already beginning to develop In the following sections, we will highlight the history of block use within the educational practices of several seminal early childhood educators Fredrick Froebel (1772–1852), the originator of kindergarten (‘‘children’s garden’’), utilized blocks in school as learning objects based on mathematical relationships of size, shape, and geometric structures (Zuckerman 2006) Froebel focused on children’s learning from the natural environment through structured activities and wooden materials to develop geometric concepts and spatial reasoning skills in young children through hands-on design and construction Following Froebel, Maria Montessori, a physician in Italy (1870–1952) dedicated her Early Childhood Educ J (2016) 44:389–402 life to supporting students with special needs through sensory training and stimulation for deliberate use of ‘‘didactic materials’’ that taught abstract concepts For example, children constructed individual pieces of a ‘‘pink tower’’—a graduated building made of blocks The wooden blocks in the tower had specific qualities such as dimensions, surface, temperature, and sounds (Montessori 1916/ 1964) Froebel and Montessori shared numerous principles in designing sensory and concept-based modular learning objects for young learners to engage in three-dimensional exploration to develop mathematical and science concepts such as identifying attributes, materials, structures, and relationships including shape, size, and symmetry that are present in our geometric world Swiss psychologist Jean Piaget formalized many educational theories and built on the ideas of Froebel and Montessori Piaget developed the learning schema for children’s logico-mathematical knowledge that includes important ideas in both arithmetic and spatial knowledge Piaget supported learning through active experiences, utilizing concrete materials, interconnecting subject areas, and peer interactions According to Piaget, the principle goal of education is to develop people who are capable of doing new things, not simply repeating what other generations have done—people who are creative and inventive discoverers (Piaget 1976) Many early childhood educators have contended that children should be actively engaged in learning processes for constructing knowledge, social skills, and dispositions that engender curiosity and contribute to collective knowledge building Children’s spatial and geometric learning trajectory is dependent on their opportunities to develop relevant language while exploring concepts through spatial activities such as planning and building block cities, designing homes for animals, studying towers around the world, and building ramps to study movement of objects [National Research Council (NRC) 2009] Pratt (1948/1990) designed unit blocks with mathematical proportions of 1:2:4 These wooden unit blocks provide foundations in geometric properties and empower students as structural designers as they build, compare, describe, and analyze block construction Pratt’s unit blocks are utilized in home and school settings today (City and Country and School 2015) These blocks are powerful tools for creating a mathematical unit, or unitizing, which occurs in geometry, number, and measurement contexts in early childhood settings Children might combine three blue triangular pattern blocks to make a unit of one yellow hexagon or make a repeating pattern with wooden blocks that includes a cube, then a triangular prism, then another cube and triangular prism, as the unit of the ab pattern The activity of combining blocks to make a composite shape or knowing that ten ones is a unit of ten are very important math concepts and reasoning processes for young learners 391 in developing an understanding of the base ten number system (NRC 2009) Blocks are typically an integral part of the constructivist curriculum in Reggio Emilia schools that originated after World War II (North American Reggio Emilio Alliance 2014) This curriculum emanates from students’ interests, curiosity, and relationships with peers and materials in their learning environment Reggio-inspired schools typically view children as having impressive potential and curiosity Children are seen as capable of constructing their own learning and negotiating a sustained process of shared learning in their environment Media and materials such as blocks are utilized to promote play, discovery, and cognitive and social connections in the processes of learning (Gandini 2008) Children explore sizes and shapes of blocks to engage in visualization, problem solving, and development of collaborative social skills in an environment that connects their creations to reading, science, mathematics, storytelling, and art Children in Reggio-inspired settings often view learning as engaging, connected, and interdisciplinary This is a way for educators to utilize blocks in instructional activities or sequences of related activities that integrate content domains such as numeracy, literacy, art, history, and science Blocks as Mathematical Reasoning Tools While the preceding examples provide a historical perspective for blocks as resources for mathematics learning, blocks should continue to serve as powerful objects to externalize and advance children’s mathematical thinking in today’s classrooms Their attributes are particularly important for uniting concepts that are foundational for learning Mathematics Learning in Early Childhood (NRC 2009) research synthesis recommends two foci in mathematics for young children: (a) number, and (b) geometry/ measurement Individually, these domains are important for young learners, but the connections between number and geometry are equally significant, for example, dividing a rectangle into two equal parts or quantifying categories or attributes of 3–D shapes Through the use of blocks, these mathematical connections between numbers and geometry become tangible and observable In the area of geometry, children can move through succeeding levels of thought as they learn about geometric shapes in two and three dimensions (Clements and Battista 1992; van Hiele 1986) Initially, children recognize geometric shapes and form visual schemes for 2-D and 3-D shapes and spaces As they develop spatial capacity through experiences with tools such as blocks, they match 3-D shapes, name common geometric shapes, use relational language, categorize shapes based on properties, and 123 392 represent 2-D and 3-D relationships with objects Children use spatial structuring as they build in space with blocks They fill rectangular containers with layers of cube blocks They begin to understand the concepts of perspective, symmetry, and size through building block configurations They can describe why some blocks stack easily (or why they not), according to their attributes These block activities bolster students’ understanding of geometric shapes and mathematical reasoning According to the Mathematics Learning in Early Childhood recommendations (NRC 2009), children use four major ideas, or reasoning processes, in their study of mathematics content Blocks are explicitly named as tools for developing mathematical reasoning within these four ideas Children in kindergarten often compose and decompose numbers and geometric shapes For example, several smaller rectangular prisms are combined to make one large rectangular prism This idea of composing and decomposing is very important in learning about number or quantities and their relationships (e.g., knowing that the quantity or total of can be taken apart into the addends or parts of and or and 1) The second major idea is unitizing, or creating or discovering, a mathematical unit To create a repeating pattern, children have to know the parts that make up the unit (square rectangle square, repeated) and see it as a composite whole or unit Relating and ordering are major mathematics ideas that are developed with blocks This is investigated when children compare two stacks of blocks that have the same number of blocks but are different in height, or one stack has more than another stack Through this process they observe, compare, and describe differences in measureable attributes such as length The fourth major idea in mathematical reasoning for young learners is looking for patterns and structures and organizing or classifying information Blocks are resources for building, describing, and extending unit patterns For example, a unit of hexagon and a rhombus can be taken as the basis for understanding patterns when children are asked, ‘‘What would the pattern look like if we repeated this unit four times?’’ Or, kindergarten children can be asked to determine how groups or categories of blocks are similar or different These four main ideas in developing mathematical reasoning guide mathematics learning in kindergarten and build a strong foundation for mathematics studies in later grades Children’s geometric thinking is strengthened through well-designed activities, use of appropriate physical manipulatives (e.g., blocks, computer), and resource-rich learning opportunities that support their growing geometric and spatial skills The historical and research review presented above leads to the question, How might blocks be a typical resource to support mathematics learning in kindergarten classrooms today? In response to current accountability and high-stakes testing practices, many kindergarten educators have pushed blocks and other useful instructional resources to the side to meet curriculum requirements 123 Early Childhood Educ J (2016) 44:389–402 Kindergarten teachers are often part of the substantive educational accountability systems and focus on testing that is occurring in many schools In this realm, classroom activities and learning experiences are often narrowed to procedurally ‘‘cover’’ academic standards The standards are not always the prominent issue The high-stakes testing that is driving educational ‘‘reform’’ has an impact on the quality of learning experiences in early childhood classrooms The emphasis on developing academic skills quickly limits opportunities for creativity, negotiation, communication, and relational problem solving with mathematical tools Contemporary Perspectives on Blocks as Learning Tools There are contemporary examples of schools that integrate blocks in the kindergarten curriculum The City and Country School in New York City develops a range of intellectual, social, mathematics, problem solving, and research skills through creative block projects (City and Country School 2015) However, a growing number of kindergarten teachers have determined that their current Common Core State Standards (CCSS) ‘‘aligned’’ curriculum resources include more skills-based worksheets that not involve using manipulatives or, worse, that students not engage in rich problem solving or activities that promote mathematical reasoning, as they are often told step-by-step how to ‘‘solve the problem.’’ While the current curriculum presents a scarcity of mathematics tasks that are interesting on an individual basis, children are further alienated from opportunities for deep mathematical learning through limited peer interactions, including sharing individual or collective mathematics thinking strategies Children begin to believe that mathematics is about doing worksheets rather than engaging in rich activities that include resources for learning and require students’ mathematical reasoning and communication of important mathematical ideas Currently, there is a crisis in kindergarten as teachers report major factors that inhibit children’s opportunities to learn through block play or block activities (Miller and Almon 2009) Early childhood educators are often required to teach prescribed standards, evaluate student progress, and utilize most of the day’s schedule to focus on literacy and numeracy, the two content areas that are assessed by CCSS standardized tests in later grades This leaves little or no time for exploring, creating, or utilizing geometric objects as thinking tools to promote deeper understanding of number and geometry concepts Meeting academic standards should not come at the price of denying young children access to engaging and robust mathematics learning experiences Early Childhood Educ J (2016) 44:389–402 Advocacy Research Blocks provide opportunities for many forms of play and can support development of mathematics concepts and processes Through engaging with blocks, children classify, measure, count, and explore symmetry, shape, and space (Piaget and Inhelder 1967; Kamii et al 2004) Research conducted by Wolfgang et al (2001) determined that children who engaged in sophisticated block play during preschool years were more successful in junior high and high school and achieved higher mathematics grades and overall achievement scores Exploratory play by young children often reflects the logic of, and causal structure of, scientific inquiry (Cook et al 2011; Schulz and Bonawitz 2007) The inherent mathematical qualities of blocks support geometric reasoning and mathematical thinking as children explore their shape and combinatorial aspects (Ginsburg, Inoue, and Seo 1999) Young children use blocks to reason spatially in three dimensions—a skill that is necessary for future engagement in mathematics, science, and engineering disciplines Spatial thinking is important in many areas, such as measurement and geometry, and is predictive of achievement in mathematics and science (Clements and Sarama 2007; Shea et al 2001) Using blocks can develop mathematical and scientific thinking; young children who engaged in block learning experiences also scored significantly higher than peers without these experiences on language acquisition assessments (Christakis et al 2007) Based on this review, it is clear that blocks can support academic learning, innovative play, and achievement across subject domains In addition to cognitive development, blocks as learning tools promote a range of socioemotional skills and competencies and provide children with opportunities to interact, design, plan shared goals, negotiate, and develop persistence in solving problems together (Cartwright 1995) Professional Wisdom: A Vignette of a Kindergarten Classroom In light of current trends that eliminate such valuable handson learning materials, it is imperative that teachers and administrators understand and articulate the research and the implications of including thinking tools such as blocks in a child’s learning day Through professional knowledge, educators are empowered to make informed decisions in planning learning activities for young children They can take action based on historical perspectives, research, and professional wisdom regarding what is appropriate for their kindergarten students Young children need access to blocks 393 as thinking tools, particularly in mathematics, to develop, construct, test, and reflect on their learning One of the teachers in our partnership, who has a range of learners in her inclusion classroom, described this imperative: As a kindergarten teacher, it is important to provide young students with many opportunities to explore and manipulate blocks to deepen their geometry understanding By allowing students time to build with blocks while using guiding questions, they begin to make important connections between various shapes that can be composed and decomposed This teacher described how her use of blocks in the classroom arises out of the children’s own understandings and experiences of shape in the everyday world as this abstract understanding is concretized through block activities that are integrated across the academic year: At the beginning of kindergarten, students enter with their own conceptions about shapes, and through guided explorations they begin to develop a more concrete understanding of geometry Students have a general idea of shapes in the environment and some students with preschool experience know the correct names of shapes Through songs, literature, classroom discussions, activities, and videos, all students are exposed to shapes and their attributes By providing time for them to use blocks they begin to make a tangible connection to these attributes and are then able to gain a conceptual understanding of geometry rather than just an abstract understanding This teacher highlighted how pattern blocks and other 2-D resources not only provide an essential connection to mathematical ideas, but enable students to develop essential vocabulary and social competencies in the classroom: Throughout the first semester of kindergarten, 2-D shapes are the focus Students learn the proper names of these shapes, how they can be composed and decomposed, as well as how to describe their attributes, and how to sort and classify these shapes by their attributes While students are engaged in various tasks with blocks, they are able to verbalize their geometry connections while using correct vocabulary and mathematical reasoning When students are allowed to use blocks they are excited to share their creations with each other and their teacher This excitement provides a wonderful avenue to develop their vocabulary and geometry concepts as students describe, and draw or represent, what they have built By the second semester, this teacher’s class has made substantial progress in naming and recognizing shapes through their work with 3-D shapes, block activities, and 123 394 the use of supporting video and literature The teacher described how the second semester’s activities build from, and promote, further study of shapes and their properties: During the second semester of kindergarten, when 3-D shapes are introduced in our class, block activities help reinforce children’s knowledge of shape and the properties and relationships of shapes They begin to point out when they find cubes or cylinders in the environment In fact, students are also able to identify rectangular and triangular prisms and consider how to construct equivalent shapes by making connections to geometry videos, (like the Shape Name Game; Have Fun Teaching.org), that they have previously viewed in the classroom While this kindergarten teacher is addressing the required state standards, the integration of blocks and other manipulatives contributes significantly to student learning and confidence in geometry For this teacher, a resourcerich approach to geometry includes foundational experiences that are needed to progress to higher levels of geometric thinking: All students are capable of learning the names of shapes and can identify them in everyday situations However, students that are allowed to explore with various types of blocks have a deeper understanding of geometry and are able to verbalize their understanding more articulately These students have a greater understanding of spatial relationships and can see how shapes can be composed and decomposed, made into a unit or pattern that can repeat, or classified and ordered with more ease than students who have not had the opportunity to learn geometry through these interactions and experiences While a significant body of literature substantiates this teacher’s views, the practical implications of using blocks in ways that align with Common Core State Standards is worth further discussion Early Childhood Educ J (2016) 44:389–402 Another instructional strategy is to integrate literacy activities that include writing, representing mathematical concepts, graphing, and so forth There is a wealth of children’s educational books that focus on blocks, block constructions, and geometry to support these activities Books recommended by kindergarten teachers include: Bear in a Square (Blackstone 1998), The Shape of Things (Dodds 1996), Mouse Shapes (Walsh 2007), When a Line Bends a Shape Begins (Gowler 1997) and Shapes, Shapes Shapes (Hoban 1996) These literacy resources connect to geometry activities Several examples linking literacy and numeracy are incorporated in the block learning opportunities that follow Blocks provide many opportunities to integrate both the Common Core Content and Standards for Mathematical Practices (National Governors Association Center for Best Practices 2010) When children are solving problems, modeling, representing ideas, reasoning quantitatively, developing persistence, constructing, and using blocks as thinking tools in mathematics, they are experiencing the mathematics practice standards In addition to the mathematical concepts and big ideas, children need opportunities to develop habits of mind or ways of engaging in mathematics as described in the Standards for Mathematical Practices These eight practices in the CCSS are mechanisms for children to develop, refine, and extend their mathematical thinking They are the ways in which mathematicians make sense of complex ideas; for young children, they are avenues to reasoning and communication in problem solving Children engage in these mathematical practices when they solve mathematics problems using various types of blocks For example, using of mathematical tools such as blocks to think about mathematics concepts while solving problems could include Mathematical Practice Standards and Kindergarten teachers often have these practices displayed as anchor charts in the classroom: Eight Mathematical Practices I can make sense of problems and solve them (persistent problem solver) I can use numbers, words, and objects to understand problems I can explain my mathematical thinking to someone else and I listen to understand others math ideas I can show/model mathematical problems in different ways I can use math tools to solve problems and know why I chose them I can figure things out in math so I am accurate (Mistakes are opportunities to learn) I can use what I know to solve new problems I can look for patterns and organize information to help solve problems Block Activities Many types of blocks can be used in block activities in standards-based mathematics lessons When implementing such activities, the role of the teacher is critical for integrating learning with hands-on experience A kindergarten teacher in our partnership remarked, ‘‘I have the essential role of asking questions that connect the block activities, math concepts, and children’s thinking.’’ Effective questioning and listening to children’s ideas as they engage in thinking, reasoning, and making sense of mathematical ideas are critical to supporting learning 123 Early Childhood Educ J (2016) 44:389–402 Child-friendly versions of the Mathematical Practices are available online Standards-based lessons provide important opportunities for children to develop these practices and ways of learning mathematics while engaging in rich tasks utilizing blocks It is important for young students to have something tangible when learning about shapes and their attributes Tangible objects allow them to feel the sides and touch the corners that they are expected to describe in CCSS Through access to blocks, children begin to come to their own conclusions about how shapes are related or different To develop clear understanding of geometry, children need to use these materials extensively with their hands A kindergarten teacher noted, They cannot learn that a building is made of cubes from a picture of a building made of cubes unless they have hands-on experience with a ‘‘real cube.’’ They begin to see that shapes can be composed of other shapes and are enthusiastic in their discoveries as they connect tangible objects with abstract concepts Through structured activities, blocks can be a vital part of the primary mathematics curriculum The examples of lessons that follow provide explicit connections to the CCSS They are not entire lesson plans; rather, they present key ideas for early childhood educators to consider in providing opportunities for kindergarteners to learn through using blocks as thinking tools to address CCSS for mathematics Learning environments should provide opportunities for children to experience instructional activities that include blocks, as well as learning centers that honor children’s ways of making sense of geometric ideas This requires understanding the broader policy landscape and advocating for teaching and learning experiences that are informed through research and the wisdom of practice to ensure a viable engaging mathematics education that integrates blocks as learning resources for young children in kindergarten 395 also address Counting and Cardinality (K.CC.4.a.b.) Selected Mathematical Practice Standards Standards MP 1: I can make sense of problems and solve them MP 2: I can use numbers, words, and objects to understand problems Students are provided a small bag with an assortment of 8–10 pattern blocks Students utilize work mats or yarn tied to make a circle They study the shapes of the pattern blocks and organize or group them by attributes Attributes may be size, shape, color, and number of sides or corners They put their categories/groups on separate work mats or encircle them with yarn They then describe their categories and ways of thinking about their shapes to another student or to the class Math conversations: ‘‘How did you group the blocks? What did you notice about the shapes? How are the shapes alike or different? How many groups did you make?’’ Were students thinking about the attributes of the 2-D shapes? How did students describe the groups? Did students utilize the vocabulary word wall? What did students notice about the shapes? Take pictures or make a poster of several students’ representations for further study Connecting Blocks as Learning Tools to Common Core State Standards for Mathematics in Kindergarten: Lesson Learning Opportunities Learning Opportunity: Pattern Block 2-D Design and Count How Blocks Help me in Learning Geometry? What are the Names, Shapes, and Attributes of 2-D and 3-D Shapes? Learning Goals: I can make a design with to 15 pattern blocks and count the colors and/or geometric shapes (K.CC.4.A.B.C) and (K.G.B.5) Learning Opportunity: Pattern Block Sort Learning Goals: I can analyze and compare shapes (K.G.B.4.1) (Kindergarten Geometry Standards) This can MP 3: I can explain my mathematical thinking to someone else MP 4: I can model mathematics problems in different ways 123 396 Students select a specified number of pattern blocks from a tub or bag They design a shape with that number of pattern blocks They count and record on paper how many of each color and shape they used They share their strategy and thinking with a learning partner The teacher documents several student responses and asks the class to analyze and respectfully agree or disagree with the work Several of the students’ representations can be used the next day during a math talk for ten minute math activities Learning Opportunity: Pattern Block Pictures Learning Goals: I can correctly name shapes (regardless of the orientations/positions or size) (K.G.A.2) Early Childhood Educ J (2016) 44:389–402 MP 1: I can make sense of problems and solve them MP 6: I can figure things out in math so I am accurate Students are given a set of pattern and/or attribute blocks along with a folder or some sort of divider The divider will be used to shield blocks from the partner or small group in which the student is working One student asks the other student to cover his/her eyes and then selects a block and places it behind the divider The first student then gives the partner or group clues about the selected shape by giving statements about its attributes For example, if the student selected a triangle, the student could say, ‘‘This shape has three sides This shape has three corners This shape has straight edges This shape looks like a slice of pizza.’’ MP 3: I can explain my mathematical thinking to someone else MP 6: I can figure things out in math so I am accurate Students use pattern blocks either to create their own pictures or to complete pattern block pictures that the teacher has provided Once the pictures are completed, they students describe the picture to a partner by sharing the shapes that were used For example, ‘‘I used three squares and four triangles to make my picture.’’ Once the designing partner has shared the work, the listening partner asks a question, such as, ‘‘Did you use any hexagons?’’ This could also be done with wooden or foam blocks during a free-choice center This would address (K.G.A.3): Identify shapes as two-dimensional (lying in a plane, ‘‘flat’’) or three-dimensional (‘‘solid’’) as well Learning Opportunity: Pattern/Attribute Block Share and Ask Learning Goals: I can describe attributes of 2D or 3D shapes (K.G.B.4) MP 3: I can explain my mathematical thinking to someone else MP 6: I can figure things out in math so I am accurate Learning Opportunity: Guess My Shape Learning Goals: I can describe attributes of shapes by analyzing and comparing them (K.G.B.4) 123 Students are given pattern and/or attribute blocks to work in small groups They are also given the following sentence frames: ‘‘I have a shape with _sides Who has a shape with _ sides?’’ or ‘‘I have a shape with _ corners Who has a shape with corners?’’ They fill in the blanks with their own number of sides or corners, depending on the selected shape When asking the ‘‘Who has’’ portion of the question, they not have to use the same number of sides or corners as the selected shape Thus, they learn to identify and describe the attributes of shapes This can be done with other types of blocks, such as geoblocks and addresses (K.G.A.3) Early Childhood Educ J (2016) 44:389–402 397 Learning Opportunity: Making Shapes Learning Opportunity: Building Block Houses for Animals Learning Goals: I can use simple shapes to make a larger shape (K.G.6) Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) I can actively engage in groups with peers and in reading activities with purpose and understanding (RL.K.10) I can use a combination of drawing, dictating, and writing to compose an informative text (W.K.2) MP 4: I can show/model mathematical problems in different ways MP 5: I can use math tools to solve problems Students are given a variety of shapes of blocks and asked to use two or more blocks to compose larger shapes or shapes that have different faces and shapes (triangle, rectangle, square, hexagon), for example, ‘‘Find other unit blocks that can make a square prism.’’ Over time, students name the new shapes that kindergatrteners have formed, as well as the shapes that they used to compose the new shape Students construct a block wall or building with equivalent blocks (e.g., a rectangular prism that is equal to two triangular prisms) They compose and decompose physical block shapes to make sense of their attributes, shapes, and sizes in informal ways They can make equivalent shape blocks over time Kindergatrteners are asked to find all the possible ways to make this rectangular prism using other blocks How did students compose shapes? What did they discover? How did children approach this task? What did students notice about equivalency? MP 1: I can make sense of problems and solve them MP 4: I can show/model my work in many ways The teacher reads a book about animal houses, such as Too Tall Houses (Marino 2012) Students select a stuffed animal and build a house for the animal, including a door that fits the animal Once the animal house is complete, the student draws a diagram of the house and writes a description Students are developing informal measuring skills, representing 3-D buildings in their 2-D drawings and expressing their mathematical ideas in response to literature 123 398 Early Childhood Educ J (2016) 44:389–402 MP 1: I can make sense of problems and solve them MP 4: I can show/model my work in many ways Students use unit blocks to build towers or tall structures or buildings They research real-world towers and post pictures of these towers, such as the Empire State Building They engage in discussion about what defines a tower and the necessary components of towers, for example, ‘‘What is the best way to build a foundation that a tower could be built on?’’ Once the tower is built, each student draws a diagram of the tower and writes a description The block gallery includes students’ ‘‘towers’’ and diagrams and descriptions for discussion and inquiry Learning Opportunity: Building Towers Learning Opportunity: Building Bridges Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) I can participate in shared research and writing projects (W.K.7) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) I can use a combination of drawing, dictating, and writing to compose an informative text (W.K.2) Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) I can compare and contrast adventures and experiences of characters in familiar stories (RL.K.9) I can actively engage in group and reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) I 123 Early Childhood Educ J (2016) 44:389–402 can use a combination of drawing, dictating, and writing to compose an informative text (W.K.2) MP 1: I can make sense of problems and solve them MP 4: I can show my work in many ways Students use unit blocks to build bridges After the teacher has read two or more ‘‘Three Billy Goats Gruff’’ stories (e.g., Asbjornsen et al 1957; Carpenter 1998; Galdone 1981), the students compare and contrast the stories They build a bridge with unit blocks and then reenact or retell the story, using figurines They draw a diagram and write a description of their bridge 399 foundation of unit blocks and connect ramp sections to build a pathway for rolling balls They place ramps at different slopes and test results They experiment and determine the effect of rolling different sizes and weights of balls (e.g., wooden, plastic, golf balls) down ramps They are encouraged to try various strategies, experiment and discover principles for themselves If they form misconceptions, the teacher can ask questions to invoke experimentation and understanding Learning Opportunity: Constructing Ramps Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front, behind, and next to (K.G.1) I can actively engage in group and reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) MP 1: I can make sense of problems and solve them MP 3: I can explain my thinking and listen to understand others Students investigate constructing and rolling balls down elevated ramps (sections of wood cove molding) The teacher reads a book and facilitates discussion about constructing ramps (e.g., Roll, Slope, and Slide (Dahl 2006)) Students work with partners to build a 123 400 Early Childhood Educ J (2016) 44:389–402 Learning Opportunity: Using Slope and Speed to Knock Down Towers Learning Opportunity: Using Angles to Turn Corners on Ramps Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front, behind, and next to (K.G.1) I can actively engage in group reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) Learning Goals: I can model shapes in the world by building shapes from components (K.G.5) Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front, behind, and next to (K.G.1) I can actively engage in group and reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) MP 1: I can make sense of problems and solve them MP 3: I can explain my mathematical thinking to someone else and I listen to understand others MP 4: I can show/model mathematics problems in different ways Students investigate ways of knocking down towers placed at the end of ramps They can experiment in building various sizes of towers to study ways the ramp slope affects results, as well as the influence of various sizes and weights of balls (e.g., wooden, plastic, golf balls) They are encouraged to try various strategies, experiment, and discover principles for themselves The teacher can ask questions to invoke experimentation and understanding 123 MP 1: I can make sense of problems and solve them MP 4: I can show/model my work in many ways Students investigate strategies of getting balls to turn corners on ramps They try various ways of building corners on ramps, using various angles They can experiment with various slopes of ramps and diverse structures of walls that will keep the balls rolling on the ramps They are encouraged to try a range of strategies, experiment, and discover principles for themselves The teacher and students can ask questions to invoke experimentation and understanding Early Childhood Educ J (2016) 44:389–402 401 Acknowledgments The authors appreciate the proffesional contributions of kindergarten teachers Glenda McShannon and Julie Ormond Conflict of interest of interest The authors declare that they have no conflict References General Suggestions for Addressing Kindergarten Counting and Cardinality Standards with Blocks as Learning Tools Over time and through experiences, students will count a set of blocks, correctly naming each block by the number of objects that it represents For each block counted, the student should be able to match each object with the correct number name (cardinality, keeping track, sequencing, and one-to-one correspondence) The use of enlarged five frames and ten frames for counting blocks is helpful Create two separate groups of blocks One group should have more blocks (up to 10) and one group should have fewer blocks (up to 10 but fewer than those in the other group) Students are asked to determine which group has more blocks and which group has fewer blocks Create two separate groups of blocks with an equal number of blocks (each group should contain no more than 10 blocks) Ask students whether the two groups have a different number of blocks or are equal, then ask them to explain their response Students can generate block towers with equivalent shapes Discuss and ask questions about which is taller, shorter, or the same quantity of blocks or same height They can deconstruct and rebuild the tower, which helps in counting sequence and decomposing numbers Use blocks and categories of blocks to represent quantities Students can engage in role-play with blocks to represent the actions of addition and subtraction Students can enjoy making a block book to represent the combinations of ten or an appropriate number Asbjornsen, P C., Moes, J E., & Brown, M (1957) The three billy goats gruff Orlando, FL: Harcourt Brace and Company Blackstone, S (1998) Bear in a square Concord, MA: Barefoot Books Carpenter, S (1998) The three billy goats gruff New York, NY: Harper-Collins Cartwright, S (1995) Block play: Experiences in cooperative learning and living Retrieved from http://www.issa.nl/mem bers/articles/pdf/5010339.pdf Christakis, D., Zimmerman, F., & Garrison, M (2007) Effect of block play on language acquisition and attention in toddlers: A pilot randomized controlled trial Archives of Pediatrics and Adolescent Medicine, 161, 967–971 City and Country School (2015) Retrieved from http://www cityandcountry.org/page Clements, D., & Battista, M (1992) Geometry and spatial reasoning In D Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp 420–464) New York, NY: Macmillan Clements, D., & Sarama, J (2007) Early childhood mathematics learning In F K Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp 461–555) New York, NY: Information Age Cook, C., Goodman, N., & Schulz, L (2011) Where science starts: Spontaneous experiments in preschoolers’ exploratory play Cognition, 120, 341–349 Dahl, M (2006) Roll, slope and slide Minneapolis, MN: Picture Window Books Dodds, D (1996) The shape of things Somerville, MA: Candlewick Press Galdone, P (1981) The three billy goats gruff New York, NY: Clarion Books Gandini, L (2008) Introduction to the fundamental values of the education of young children in Reggio Emilia Retrieved from http://www.klaschoolsfranchise.com/reggioemilia.pdf Ginsberg, H P (1983) The development of mathematical thinking New York, NY: Academic Press Ginsburg, H., Inoue, N., & Seo, H (1999) Young children doing mathematics: Observations of everyday activities In V Cooper (Ed.), Mathematics in the early years (pp 88–89) Reston, VA: National Council of Teachers of Mathematics Gowler, R (1997) When a line bends a shape begins New York, NY: Houghton Mifflin Hewitt, K (2001) Blocks as a tool for learning: A historical and contemporary perspective Young Children, 56(1), 6–13 Hoban, T (1996) Shapes, shapes shapes New York, NY: HarperCollins Kamii, C., Miyakawa, Y., & Kato, Y (2004) The development of logico-mathematical knowledge in a block-building activity Journal of Research in Childhood Education, 19(1), 44–57 Marino, G (2012) Too tall houses New York, NY: Viking Books Miller, E., & Almon, J (2009) Crisis in the kindergarten: Why children need to play in school College Park, MD: Alliance for Childhood 123 402 Moll, L., Amanti, C., Neff, D., & Gonzalez, N (1992) Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms Theory Into Practice, 31(2), 132–141 Montessori, M (1916/1964) The Montessori method New York, NY: Schocken Books National Governors Association Center for Best Practices (2010) Common core state standards Washington, DC: Council of Chief State School Officers National Research Council (2009) Mathematics learning in early childhood: Paths toward excellence and equity Washington, DC: National Academies Press North American Reggio Emilio Alliance (2014) The child has a hundred languages Retrieved from http://www.reggioalliance org/reggio_emilia_italy/history.php Piaget, J (1976) To understand is to invent: The future of education New York, NY: Penguin Books Piaget, J., & Inhelder, B (1967) The child’s conception of space New York, NY: Norton 123 Early Childhood Educ J (2016) 44:389–402 Pratt, C (1948/1990) I learn from children New York, NY: Harper & Row Schulz, L E., & Bonawitz, E B (2007) Serious fun: Preschoolers engage in more exploratory play when evidence is confounded Developmental Psychology, 43, 1045–1050 Shea, D., Lubinski, D., & Benbow, C (2001) Importance of assessing spatial ability in intellectually talented young adolescents Journal of Educational Psychology, 93, 604–614 van Hiele, P (1986) Structure and insight: A theory of mathematics education Orlando, FL: Academic Press Walsh, E (2007) Mouse shapes San Diego CA: Harcourt Books Wolfgang, C H., Stannard, L L., & Jones, I (2001) Block play performance among preschoolers as a predictor of later school achievement in mathematics Journal of Research in Childhood Education, 15(2), 173–180 Zuckerman, O (2006) Historical overview and classification of traditional and digital learning objects Cambridge, MA: MIT Press [...]... doing mathematics: Observations of everyday activities In V Cooper (Ed.), Mathematics in the early years (pp 88–89) Reston, VA: National Council of Teachers of Mathematics Gowler, R (1997) When a line bends a shape begins New York, NY: Houghton Mifflin Hewitt, K (2001) Blocks as a tool for learning: A historical and contemporary perspective Young Children, 56(1), 6–13 Hoban, T (1996) Shapes, shapes shapes... intellectually talented young adolescents Journal of Educational Psychology, 93, 604–614 van Hiele, P (1986) Structure and insight: A theory of mathematics education Orlando, FL: Academic Press Walsh, E (2007) Mouse shapes San Diego CA: Harcourt Books Wolfgang, C H., Stannard, L L., & Jones, I (2001) Block play performance among preschoolers as a predictor of later school achievement in mathematics Journal of... National Research Council (2009) Mathematics learning in early childhood: Paths toward excellence and equity Washington, DC: National Academies Press North American Reggio Emilio Alliance (2014) The child has a hundred languages Retrieved from http://www.reggioalliance org/reggio_emilia_italy/history.php Piaget, J (1976) To understand is to invent: The future of education New York, NY: Penguin Books Piaget,... fewer blocks Create two separate groups of blocks with an equal number of blocks (each group should contain no more than 10 blocks) Ask students whether the two groups have a different number of blocks or are equal, then ask them to explain their response Students can generate block towers with equivalent shapes Discuss and ask questions about which is taller, shorter, or the same quantity of blocks. .. behind, and next to (K.G.1) I can actively engage in group and reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) MP 1: I can make sense of problems and solve them MP 3: I can explain my mathematical thinking to someone else and I listen to understand others MP 4: I can show/model mathematics. .. Effect of block play on language acquisition and attention in toddlers: A pilot randomized controlled trial Archives of Pediatrics and Adolescent Medicine, 161, 967–971 City and Country School (2015) Retrieved from http://www cityandcountry.org/page Clements, D., & Battista, M (1992) Geometry and spatial reasoning In D Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp 420–464)... different ways Students investigate ways of knocking down towers placed at the end of ramps They can experiment in building various sizes of towers to study ways the ramp slope affects results, as well as the influence of various sizes and weights of balls (e.g., wooden, plastic, golf balls) They are encouraged to try various strategies, experiment, and discover principles for themselves The teacher can ask... NY: Macmillan Clements, D., & Sarama, J (2007) Early childhood mathematics learning In F K Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp 461–555) New York, NY: Information Age Cook, C., Goodman, N., & Schulz, L (2011) Where science starts: Spontaneous experiments in preschoolers’ exploratory play Cognition, 120, 341–349 Dahl, M (2006) Roll, slope and slide Minneapolis,... declare that they have no conflict References General Suggestions for Addressing Kindergarten Counting and Cardinality Standards with Blocks as Learning Tools Over time and through experiences, students will count a set of blocks, correctly naming each block by the number of objects that it represents For each block counted, the student should be able to match each object with the correct number name... environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front, behind, and next to (K.G.1) I can actively engage in group and reading activities with purpose and understanding (RL.K.10) I can participate in collaborative conversations with diverse partners about kindergarten topics (SL.K.1) MP 1: I can make sense of problems and solve