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: 4 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

1

Mathematics Educator, New Mexico State University,

MSC 3CUR, Las Cruces, NM 88003-8001, USA

2

New York University, New York, NY, USA

3

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 do 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 do 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 9 can be taken apart into the

addends or parts of 7 and 2 or 8 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 do not involve using

manipulatives or, worse, that students do 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 1 and 5. Kindergarten teachers often have these practices displayed as

anchor charts in the classroom:

Eight Mathematical Practices

1

I can make sense of problems and solve them (persistent problem

solver)

2

I can use numbers, words, and objects to understand problems

3

I can explain my mathematical thinking to someone else and I

listen to understand others math ideas

4.

I can show/model mathematical problems in different ways

5

I can use math tools to solve problems and know why I chose

them

6

I can figure things out in math so I am accurate. (Mistakes are

opportunities to learn)

7

I can use what I know to solve new problems

8

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 do 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 5 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 do 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.

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: 4 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

1

Mathematics Educator, New Mexico State University,

MSC 3CUR, Las Cruces, NM 88003-8001, USA

2

New York University, New York, NY, USA

3

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 do 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 do 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 9 can be taken apart into the

addends or parts of 7 and 2 or 8 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 do not involve using

manipulatives or, worse, that students do 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 1 and 5. Kindergarten teachers often have these practices displayed as

anchor charts in the classroom:

Eight Mathematical Practices

1

I can make sense of problems and solve them (persistent problem

solver)

2

I can use numbers, words, and objects to understand problems

3

I can explain my mathematical thinking to someone else and I

listen to understand others math ideas

4.

I can show/model mathematical problems in different ways

5

I can use math tools to solve problems and know why I chose

them

6

I can figure things out in math so I am accurate. (Mistakes are

opportunities to learn)

7

I can use what I know to solve new problems

8

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 do 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 5 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 do 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.

## Tài liệu A Case for Midspan Power-over-Ethernet Controlers ppt

## A CASE STUDY ON COMMON PROBLEMS IN LEARNING BUSINESS ENGLISH VOCABUALRY IN THE BOOK “BUSINESS BASICS” FACED BY THE 1ST YEAR STUDENTS AT VIETNAM UNIVERSITY OF COMMERCE, AND SOME SUGGESTED SOLUTIONS

## Tài liệu Báo cáo khoa học: "Man* vs. Machine: A Case Study in Base Noun Phrase Learning" pdf

## Tài liệu Reaching All Students - A Resource for Teaching in Science, Technology, Engineering & Mathematics pdf

## Building a Future for Women and Children The 2012 Report ppt

## A Case for Staged Database Systems ppt

## Building Ireland’s Smart Economy: A Framework for Sustainable Economic Renewal potx

## Civil liability resulting from transfrontier environmental damage: a case for the Hague Conference? pot

## building a pc for dummies 5th

## ASSET BUILDING FOR OLD AGE SECURITY - A CASE FOR HYBRID LONG-TERM SAVINGS MICROPENSION PRODUCTS potx

Tài liệu liên quan