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Cognitive Development, Mathematics, and Science Learnin
1. Cognitive Development,
Mathematics, and Science
Learning Objectives
After reading this chapter, you should be able to:
1. Define and explain the concept and components of cognitive
development.
2. Explain how early learning standards for cognitive
development relate to mathematical
and scientific thinking.
3. Describe early childhood curriculum activities that support
the development of math-
ematical concepts and processes.
4. Describe early childhood curriculum activities that support
development of scientific
concepts and processes.
10
Pretest
1. Infants are not capable of learning before
the age of 1. T/F
2. Bloom’s taxonomy is a tool teachers can
use to help promote higher-order
thinking. T/F
3. insecurity. Your children have enjoyed
meeting some of the nearby merchants, and they are starting to
play “store” in the dramatic
play center.
You enjoy listening to the children’s conversations and have
noticed that they have many
questions and theories about how things work. You’ve observed
that every day, Alonzo takes
out a bin of plastic animals and arranges them in different ways.
Yesterday, Monique and
Destiny came to you and asked how they could make paper
dresses that would be the same
size as the doll babies. You recognize that, in your role as an
early childhood educator, you
want to support the children’s interests, but you also need to
cultivate their cognitive develop-
ment, in part by providing intentional activities that teach
important concepts.
Cognitive development occurs as children acquire and process
different kinds of knowledge.
Mathematics and the sciences for children share a focus on
inquiry, problem solving, and
the development of critical thinking skills through processes
and practices that engage them
in hands-on explorations. This chapter focuses on the early
learning standards for cognitive
development and experiences that build a good foundation for
math and science standards
and curricula.
10.1 Cognitive Development and General Knowledge
Cognitive development is the process that occurs as thinking
and reasoning develop and
become more complex over time. Early learning standards for
4. cognitive development are
based on the broad assumption put forth by the National
Education Goals Panel (NEGP) that
“cognition and general knowledge represent the accumulation
and reorganization of expe-
riences that result from participating in a rich learning setting
with skilled and appropriate
adult intervention. From these experiences children construct
knowledge of patterns and rela-
tions, cause and effect, and methods of solving problems in
everyday life” (Kagan, Moore, &
Bredekamp, 1995, p. 4). In other words, cognition includes the
various ways in which humans
know and represent their understanding of the world.
According to cognitive psychologists, there are three different
kinds of interrelated knowledge:
1. Physical knowledge consists of concepts about physical
properties observed through
first-hand experience. Examples of how children might gain
physical knowledge include
learning about colors by mixing paints or using an ice cube tray
and freezer to learn that
water can change from a liquid to a solid and back again.
2. Logicomathematical knowledge consists of mentally
constructed relationships about
comparisons and associations between and among objects,
people, and events. This is
the least understood and most complicated cognitive process.
Examples include a child
sorting a group of small cars, who must apply criteria that make
sense to him, such as
color, to separate them into logical groups. He may then put
them back into a pile and
6. conforms to schema already formed. Using the ball example
above, if you give the child similar
balls that are smooth, round, and roll when pushed, the existing
concept is confirmed and the
child moves on to exploring other things.
If, however, you structure a discrepant event, by
giving the child a different kind of ball that he has
not previously encountered, disequilibrium (cog-
nitive conflict) arises owing to tension between the
child’s concept of “ball” and the new unfamiliar
ball. Because humans are wired to prefer equilib-
rium, the child will be motivated to expend men-
tal effort to make sense of the new balls. He will
thus accommodate the new information by modi-
fying or expanding the original schema to include
the characteristics of the new balls (e.g., whether
the ball is knobby or made of leather or wood, or
much larger or smaller than those he encountered
before).
Accommodation is a more complex process than
assimilation, affected by the quantity and kinds of
experiences a child encounters. As one concept
builds upon another, children develop more com-
plex thinking. This is one of the reasons early child-
hood experiences are considered so critical to future intellectual
and academic functioning
and one of the premises of early intervention programs such as
Head Start.
Early childhood educators foster accommodation as well as the
three kinds of knowledge, by
introducing a variety of familiar and new materials as children
are ready for them and using
language to help them expand and create new schema. In the
8. thus introduced an oppor-
tunity to develop new physical knowledge (a different kind of
ball that can be handled and
observed), logicomathematical knowledge (making mental
connections with other kinds of
balls), and social-conventional knowledge (giving names to the
characteristics of the new ball).
The Preoperational and Concrete Operations Stages of
Development
Children in the preoperational stage are beginning to expand
logicomathematical knowledge,
but the process is hampered by their tendency to center or focus
on one characteristic or fea-
ture of what they observe to the exclusion of others. For
example, if a child looks at a picture
of five dogs and two cats and the teacher asks, “are there more
animals or dogs?” the child
is likely to say, “more dogs.”
Further, they are egocentric, which means they tend to consider
the appearance of objects
from only their perspective. So if a teacher held a puppet with
its face toward a child, the child
would assume that the teacher also saw the puppet’s face rather
than its back. Third, preoper-
ational children are easily deceived by appearances and unable
to mentally conserve or retain
the idea of fixed quantities. For example, if eight ounces of
water is poured from a short, wide
container into a tall, narrow one, the child is not likely to
recognize that the amount of liquid
remains the same; instead, he may think that the taller glass
holds more (Figure 10.1).
During the concrete operations stage of cognitive development,
10. Cognitive Development and General Knowledge Chapter 10
Knowledge of Patterns and
Relationships
As young children encounter repeating
patterns in daily life, they begin to under-
stand that the natural world is organized.
As their thinking becomes more sophisti-
cated, they apply knowledge of concrete
patterns to more abstract concepts and
ideas—the essence of understanding the
predictability and rhythms of phenom-
ena, social interactions, and behavior. For
instance, the idea of taking turns repre-
sents a simple pattern—first I use the red
marker, then I give it to you, then you give
it back to me.
In the absence of an internalized sense
of predictability and patterns, the child
would not know what to expect next—a chaotic existence to be
sure. Decisions teachers
make about materials, routines, schedules, and how to organize
a classroom reinforce con-
cepts of patterns and relationships. Teachers promote an
understanding of patterns and rela-
tionships in many kinds of activities, as discussed later in this
chapter.
Cause and Effect
Young children frequently display magical thinking, proposing
11. preposterous or clearly unre-
alistic explanations (often humorous to adults) for why
something happens because they have
not yet discerned the relationship between cause and effect
(Catron & Allen, 2003; Hendrick
& Weissman, 2007). Determining why something happens and
predicting what might happen
when certain conditions are present or constructed represents a
complex hierarchy of increas-
ingly analytical concepts.
Logical reasoning develops slowly, gradually replacing magical
thinking and animism, children’s
tendency to attribute human qualities to inanimate objects or
animals (Copple & Bredekamp,
2009). Children’s explanations may be “intuitively reasonable”
and therefore hard to change;
thus the importance of a constructivist approach that aims to
facilitate reconstructing miscon-
ceptions through exploring, questioning, predicting, and testing
(Landry & Forman, 1999).
Understanding the relationship between cause and effect is also
fundamental to many aspects
of behavior—one of the reasons teachers and adults strive to be
clear about consequences.
Children begin learning about cause and effect intuitively from
birth: when I am wet someone
changes me, when I am hungry someone feeds me, when I smile,
my mommy smiles back,
etc. They learn intentionally through informal trial and error
during play when they exert force
on an object or mix colors of paint, for example.
After repeated trials with identical results, they begin to
understand causality and develop
13. and more on recall and reasoning (Campbell, 1999;
Charlesworth, 2005).
10.2 Promoting Cognitive
Development
To create an environment that promotes the three types
of cognitive knowledge—as well as learning about
cause and effect, patterns, and problem solving—teach-
ers should aim to:
• Provide a wide variety of interesting and challenging
materials and experiences for children.
• Foster cognitive conflict by introducing discrepant events or
information that motivate
children to experiment and test their theories.
• Document the way children solve problems to make their
thinking visible, and provide
opportunities for metacognition (thinking about their thinking).
• Promote conversation about problem solving; social debate
about ideas, theories, and
inferences leads to powerful learning (Landry & Forman, 1999).
Physical Knowledge
Mathematics and science are subject areas that help children
make connections between
concrete materials and abstract concepts. The preschool
classroom should have interest areas
designated for mathematics materials and science/discovery
investigations. The materials in
these centers provide children with opportunities to develop the
three kinds of cognitive
knowledge. They acquire physical knowledge through handling
17. making patterns,
non-standard
measuring
Plastic links,
paper clips, etc.
Counting,
sorting,
grouping,
making patterns,
non-standard
measuring
Interlocking
cubes
Counting,
sorting,
grouping,
making patterns,
modeling
operations, 3-D
data display
Cuisenaire rods Sorting,
grouping, sets,
base ten
operations
Abacus Counting, sets,
grouping,
modeling base
ten operations
Base ten blocks
24. Plastic links,
paper clips, etc.
Counting,
sorting,
grouping,
making patterns,
non-standard
measuring
Interlocking
cubes
Counting,
sorting,
grouping,
making patterns,
modeling
operations, 3-D
data display
Cuisenaire rods Sorting,
grouping, sets,
base ten
operations
Abacus Counting, sets,
groupong,
modeling base
ten operations
Base ten blocks
(Montessori
golden beads)
Counting, sets,
31. Counting,
sorting,
grouping,
making patterns,
modeling
operations, 3-D
data display
Cuisenaire rods Sorting,
grouping, sets,
base ten
operations
Abacus Counting, sets,
groupong,
modeling base
ten operations
Base ten blocks
(Montessori
golden beads)
Counting, sets,
grouping,
modeling base
ten operations
Dominoes Matching,
counting,
sorting,
grouping,
measuring
Dice Matching,
counting,
modeling
37. one-to-one
correspondence,
developing
recall
Counters: small
objects of
different colors
and shapes
(animals, boats,
planes, etc)
Counting,
sorting,
grouping,
making patterns,
non-standard
measuring
Plastic links,
paper clips, etc.
Counting,
sorting,
grouping,
making patterns,
non-standard
measuring
Interlocking
cubes
Counting,
sorting,
grouping,
making patterns,
38. modeling
operations, 3-D
data display
Cuisenaire rods Sorting,
grouping, sets,
base ten
operations
Abacus Counting, sets,
groupong,
modeling base
ten operations
Base ten blocks
(Montessori
golden beads)
Counting, sets,
grouping,
modeling base
ten operations
Dominoes Matching,
counting,
sorting,
grouping,
measuring
Dice Matching,
counting,
modeling
problems, shape
Tactile numbers
(rubber,
44. sorting,
grouping,
making patterns,
non-standard
measuring
Plastic links,
paper clips, etc.
Counting,
sorting,
grouping,
making patterns,
non-standard
measuring
Interlocking
cubes
Counting,
sorting,
grouping,
making patterns,
modeling
operations, 3-D
data display
Cuisenaire rods Sorting,
grouping, sets,
base ten
operations
Abacus Counting, sets,
groupong,
modeling base
ten operations
45. Base ten blocks
(Montessori
golden beads)
Counting, sets,
grouping,
modeling base
ten operations
Dominoes Matching,
counting,
sorting,
grouping,
measuring
Dice Matching,
counting,
modeling
problems, shape
Tactile numbers
(rubber,
Montessori
sandpaper,
magnetic)
Numeral
recognition,
writing numbers
Number puzzles Numeral
recognition,
matching, shape
Pegboards Numeral
49. areas, materials should be
organized and labeled for easy access and cleanup. Clear
storage containers or open baskets
make it easy for children to choose items and also provide
sorting and classification practice
when they are being put away.
Logicomathematical Knowledge
From infancy, children start to notice relationships, and any
time you introduce additional
complexity, unfamiliar materials, or a problem to solve, you
encourage children to construct
and refine concepts and discern relationships between materials
and ideas. For instance,
because infants are developing object permanence (knowing that
someone or something is
Observing:
Prisms, magnify-
ing glasses,
magni�er stand,
butter�y cage
Classi�cation:
Sea animals,
rocks, insects,
shells
Sensory Exploration
and Discrimination:
Montessori color tiles,
Montessori baric (weight)
tablets, feely box,
Montessori smelling jars
51. tate to encourage and extend their explorations (Geist, 2003, pp.
10–12).
Table 10.1: Facilitating Logicomathematical Knowledge with
Infants and Toddlers
Behavior/Activity Concept(s) Teacher Strategies
Sorting objects Discerning similarities
and differences among
objects
Offer a wide variety of toys in different shapes, colors, etc.,
such as large colored beads with containers or compart-
mentalized trays for sorting.
Shaking, striking,
beating instruments
Counting beats and
rhythm
Provide objects that make sounds and help children to use
in different ways; count out rhythms during use.
Nesting objects Comparing relative size Use “comparing” words
that describe what they are doing
(such as big, bigger, biggest).
Putting toys away Matching Provide picture or shape labels on
containers so children
can match an item they are holding with the corre-
sponding picture on a storage container.
Crawling, finding places
to sit/hide
52. Spatial relationships Set up a collapsible tunnel, large
cardboard boxes, or stack
of mats or pillows to climb on.
Filling/pouring Conservation Set up a sand/water table with
containers, funnels, etc., of
various sizes.
Stringing beads Patterns, shapes Point out and name patterns.
Social-Conventional Knowledge
Conversations are part of teachers’ daily interactions with
children. These exchanges provide
numerous opportunities to help them develop socio-
conventional knowledge by modeling the
language and vocabulary of mathematics and the sciences. As
you describe and label what
they do, you also ask them questions that prompt thinking. As
an educator, you will want to
formulate your questions so that they maximize the thinking
required to answer them.
One effective strategy is to use the categories provided in
Bloom’s taxonomy to guide your
questions. This is a model introduced in 1956 by cognitive
psychologist Benjamin Bloom that
illustrates the increasing complexity of intellectual behavior.
Revised during the 1990s, the
current model (Figure 10.4) provides guidance for teachers
about how to promote cognitive
development (Anderson & Cruikshank, 2001).
For instance, suppose you posed this question to older
preschoolers: “How many ways can we
54. the story of how you made it.”
Documenting Children’s Thinking
Using different strategies to document children’s work makes
their thinking visible.
Documentation gives teachers a means of reflecting on
children’s cognitive growth to inform
ongoing decisions about curricular materials, activities, and
instructional strategies. For exam-
ple, taking pictures and writing down children’s words or ideas
as they work on a prob-
lem or inquiry provides data that can later be used for
assessment, gives children a visible
record of their progress, and establishes concrete reference
points for conversations about
problem solving.
F10.04_ECE311
Creating
Evaluating
Analyzing
Applying
Understanding
Remembering
Figure 10.4: Bloom’s Taxonomy
Bloom’s Taxonomy can help teachers plan activities and use
interactions with children to promote
56. melt.
Wood Wood definitely floats. We can’t figure out how to attach
a sail.
Other Ideas
for Materials
Other Things
We Tried That Work
Other Problems
We’ve Had So Far
Foil
Bottle caps
Sticks
Clay
Cork
Trying the materials in the water first to
see if they float before we make the rest
of the boat.
Making the bottom of the boat bigger
than the sail keeps it from sinking.
If we put a toothpick and paper sail in a
cork, it falls over.
Opportunities for routine documentation of a work in progress
could include:
58. Mathematics and Science Standards Chapter 10
In the early childhood classroom, the rela-
tionship between mathematics and science
is evident in highly integrated activities and
investigations. Suppose that children are
helping the teacher make a snack mix. They
may look for recipes in cookbooks or on
the Internet to choose one they want to try.
They may hypothesize or predict how well
they think they will like it or what it might
taste like.
They will apply math concepts as they set
up and count their equipment (“We need
one big spoon, two bowls, three measur-
ing cups.”), sort/group ingredients (crack-
ers, dried fruits, cereal, seeds), and follow
ordinal (sequential) directions in determin-
ing what to do first, second, and so on and
in using different size cups to measure.
They apply operations to divide items (mix
in one big bowl and divide into smaller
bowls for serving) and one-to-one corre-
spondence for serving (one napkin/scoop
for each child).
They will also use science skills as they observe the process
(“Look at all the colors we have in
the bowl!”). They will decide when ingredients are fully mixed,
and ask questions (“Why are
the raisins and cranberries all wrinkly?”) that could lead to
further investigation (drying fruits).
They will evaluate the results of their recipe trial, perhaps
graphing the preferences of children
in the group.
61. 4. Measurement
5. Data analysis and probability
Emphasis on each of the strands varies over time, depend-
ing on where the children are developmentally and what
they’ve already learned. But one thing is sure: future suc-
cess in mathematics is based on sound foundations of
conceptual and procedural understanding in the early years
(Campbell, 1999; Linder, 2017;
NAEYC/NCTM, 2010; Seefeldt, 1999; Witzel, Ferguson, &
Mink, 2012).
Science Standards
The Next Generation Science Standards for K–12 reflect and are
guided by A Framework for
K-12 Science Education: Practices, crosscutting concepts, and
core ideas, published by the
National Research Council in July 2011. The framework
emphasizes an approach for stan-
dards development that integrates three dimensions:
science/engineering practices (the
methods used in science and engineering), cross-cutting
concepts (integrated understandings
across the science disciplines), and core ideas within each of the
four science disciplines—
physical science; life science; earth science; and engineering,
technology and applications.
Doing Math and Science
The process standards for mathematics and practices for science
focus on how children learn
and apply concepts. Common to the disciplines is an emphasis
on using concrete materials
63. Inquiry and problem solving
Reasoning and proof
Communication
Connections
Representations
Asking questions (for science) and defining problems (for
engineering)
Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Constructing explanations (for science) and designing solutions
(for
engineering)
Engaging in argument from evidence
Obtaining, evaluating, and communicating information
Processes focus on analysis of patterns, structures, and the
testing of hypotheses or pre-
dictions. Learning to make inferences and predictions and to
justify data-based conclusions
helps children begin to make sense of the physical world.
Through active listening to others’
64. explanations and communicating their own results and
reasoning, they begin to appreciate
multiple perspectives to problem solving and also learn to use
communication as an effective
means for sharing ideas.
Teachers apply knowledge of children’s general and individual
characteristics and learning
styles to encourage them to represent ideas in a variety of ways.
Mathematics and science
for young children are no longer primarily paper-and-pencil
activity, as our understanding of
children as concrete learners now stresses representation of
thinking with words, pictures,
materials, graphic organizers, and symbols (Campbell, 1999).
10.4 Mathematics Concepts and Curricular Activities
Content standards and mathematics concepts are heavily
emphasized in preschool and pri-
mary grades curricula and classrooms. It is important for early
educators to understand the
hierarchical nature of how mathematical thinking grows and
children’s need for concrete
materials and hands-on experiences to eventually develop
abstract reasoning. Each of the
following sections describes mathematical concepts, materials,
strategies, and activities that
support the goals and expectations expressed in the national
mathematics standards for
PreK–2 children.
Number and Operations
The primary goal for numbers and operations is developing
number sense, or “intuition”
about the meaning of numbers in relation to other numbers
66. Matching and one-to-one correspondence activities and
materials include:
• Lotto boards and matching games
(see Figure 10.2)
• Tossing rings of different colors
onto matching-colored posts
• Musical chairs
• Repeating hand-clap patterns or
clapping once for each word in a
rhyme
• Solving puzzles that have one space
for each matching piece (see Figure
10.2)
• Place mats with outlines for plate
and utensils that children use to set
their table
• Using tweezers or tongs to remove
one item at a time from a full bowl and transfer it to an empty
bowl
• Shadowing games—children repeat/mimic the motions of a
leader
• Making puzzle cards (such as the ones Ms. Phyllis made for
her insect unit in Chapter 6
(Figure 6.8)
Counting
68. through the use of rhymes/finger plays such as, “one, two
buckle my shoe, three four, shut
the door. . . ,” or counting songs, like “one little, two little,
three little monkeys, four little,
five little . . . .” These are reinforced by regular practice and
routines like “Let’s count to ten
before I open the door—repeat after me: one, two . . . .”
You promote rational counting to associate number with
discrete quantities by pointing to
each object as children count; also by asking children to count
groups of objects and then
saying, “How many did we count in this pile?” Daily
opportunities abound for rational count-
ing, including:
• Counting different numbers of sticks or straws and putting
them in a can with a cor-
responding number of dots.
• Counting the number of children in a group seated at a table
and then counting the
correct number of red crayons needed so that each child has
one.
• Counting the number of steps to “4” as they step on each
number of a number line
taped to the floor.
• Counting the number of fish in the aquarium.
• Counting off while standing in line waiting to go outside.
• Counting the number of stacking blocks needed for a
construction in groups of one,
two, three, etc.
69. Operations
Understanding numbers as parts of other numbers is the basis
for the operations of addition
and subtraction. Once children have achieved rational counting,
whether they can represent
numbers in writing or not, they can begin to perform operations.
The child who takes three
bears from a bin and puts them on the table, counting “one, two,
three, I have three bears,”
and then takes two more bears from the bin and counts, “four,
five—first I had three and
now I have five,” is beginning to perform the simple operation
of combining or adding sets of
objects. The child with five bears who announces separating
them into two separate piles of
3 and 2, is demonstrating understanding of the concept of
assigning referent numbers to sub-
sets (Campbell, 1999). There is some evidence to suggest that in
early education, emphasizing
parts and wholes over direct teaching of computation steps and
base 10 operations promotes
a more flexible understanding of algorithms (multiple-step
problem solving) (Campbell,
1999; Witzel et al., 2012). Thus if you use an open-ended
question to ask a child to partition
12 Unifix cubes in as many ways as possible, the child might
construct sets of 1 + 11, 2 + 10,
3 + 9, etc., but subsets of 1 + 4 + 5 + 2 would also be cor rect.
Later on, that approach may
lead to computation strategies that do not necessarily have to
match the “one right way” you
might remember from your own experiences with math
instruction.
Understanding the relationship between parts and wholes is al so
71. to count—such as an abacus, Unifix cubes, or domi-
noes—do not feature numerals. Conversely, being
able to trace or write a number may not mean that
the child understands the connection between the
numeral and the quantity it represents. In order for
that to happen, children need to recognize the sym-
bol, be able to identify and associate it with the cor-
rect number of objects in a set, and then represent
the number in writing legibly (Charlesworth, 2005,
p. 218; Witzel, Ferguson, & Mink, 2012).
Typical examples of materials that focus on one or
more of these three tasks are included in Figure 10.2;
they should be accessible and used in both preschool
and primary classrooms (Witzel, Ferguson, & Mink,
2012). Because of individual variations in the develop-
ment of this concept, teachers can best help children
to acquire symbolic representation through one-on-
one and small-group activities such as:
• Making number books with stickers or stamps
(Seefeldt & Galper, 2004)
• Writing numerals on sequence picture charts
• Using magnetic letters to represent numbers
• Using calendars with blank spaces and modeling writing in
the numbers
• Making numbers out of pipe cleaners, play dough, or wire
Algebra
The foundations for algebra begin with understanding
classification, ordering, and patterns
73. yellow bears; squares, circles, triangles). At first they
don’t necessarily know or express what the criteria
are, but they do demonstrate observation about how
the items are alike or different.
Gradually they begin to classify by less obvious cri-
teria such as material (hard/soft), pattern (striped/
checked), texture (rough/smooth), or function (moves/
doesn’t move) and more sophisticated characteristics
such as animal type (sea/land animals or reptiles/
birds). They also move to classifying objects by more
than one attribute, such as color and size (big blue
cars, small blue cars, big red cars, small red cars). A
child’s approach to classification reflects his or her
growing sense of logic. It is important to understand,
however, that children represent logical thinking in
different ways; what seems an obvious attribute to
one child might not be apparent to another. You can
learn a lot about how children think by asking them
to explain the way they sorted a particular group of
objects.
Teachers promote classification indirectly with activi -
ties such as lining up by shirt color or by boys vs.
girls or by listing foods that children like/dislike. They
also provide intentional sorting activities with materi-
als such as buttons or beads. Children can also be
encouraged to place different toys and props—such
as blocks, play foods, and cars—into appropriately
labeled bins during cleanup time.
Teachers facilitate classification skills by introducing and
modeling different ways to sort or
group and suggesting more challenging criteria. Using effective
language will also help. The
directions teachers give to children:
75. Mathematics Concepts and Curricular Activities Chapter 10
Ordering
At first children compare pairs of objects and later each
additional item to the selected crite-
ria. Ordering, or putting items into series—also called
seriation—represents the graduated
comparison of more than two things or sets in a larger group.
This procedure is more complex
than making comparisons in pairs (Charlesworth, 2005).
Seriation activities focus on ordering according to any
applicable attribute, by size in ascend-
ing or descending order, or temporal ordering of events from
first to last (Hendrick, 2007), or
ordering by graduated differences such as color shades, sound,
or weight.
Any activities that involve paired comparisons can be extended
to ordering and seriation sim-
ply by addition of materials such as:
• Graduated paper shapes
• Sticks, straws, or rods of different lengths (e.g., Cuisenaire
rods)
• Nesting items such as measuring cups and spoons
• Arranging different shades of a single color in order from
lightest to darkest
• Storing pots and pans in dramatic play on hooks from
76. smallest to largest
• Arranging pictures of children in order of birthday from
January to December
• Lining up by size
• Time lines
• Sequencing picture cards or flannel-board cutouts for
familiar stories such as
Goldilocks, The Three Little Pigs, or If You Give a Mouse a
Cookie
• Playing/singing a favorite song several times, varying the
volume from soft to loud
• Using pictures to represent the daily routine and having
children put them in order
Note that because of preschoolers’ egocentrism, they often
represent people or objects in
their drawings and paintings by relative importance rather than
by actual size/proportion.
Thus, in a picture of mom’s flower garden, mom and the flowers
might be bigger than the
house! Comparing, seriation, and ordering activities with real
objects help children gradually
move from psychological to more accurate concrete
representations.
Patterns and Patterning
In mathematics, a pattern represents a repeating series of any
kind. We want to help
children learn to recognize, replicate, represent, and extend
visual, sound, and motor pat-
78. tant not just for mathematics but for inter-
preting text in reading as well. Teachers
can foster learning about patterns by:
• Displaying photographs of patterns in
nature, such as a pine cone or nautilus
shell
• Pointing out a pattern in the brick-
work of a building
• Having a “pattern hunt” in the class-
room or on the playground
• Clapping out the rhythm patterns in
songs
• Making up motor games with actions that represent a pattern
• Decorating cupcakes with cutout patterns
• Filling in the days and weeks on a calendar
• Making patterns on paper with stamps or stickers
Geometry
“If mathematics is perceived as the search for order, pattern,
and relationships to characterize
ideas and experiences, then geometry and spatial sense should
be central topics in a math-
ematics curriculum for young children,” (Campbell, 1999, p.
124). Young children can begin
to learn about geometry as they encounter a variety of shapes in
play and daily life. Children
play with and often recognize objects and symbols (including
80. Mathematics Concepts and Curricular Activities Chapter 10
Activities that promote learning about shapes include:
• Holding and feeling the edges of cutouts or models of
different flat and three-
dimensional shapes
• Tracing shapes
• Matching shapes with cutouts
• Making shapes with their bodies
• Using shape cookie cutters with play dough and for baking
• Making and cutting foods like a tray of brownies or pancakes
into different shapes
• Folding paper to make simple origami
• Having a “shape hunt”
• Making mobiles with straight-sided shapes from objects like
straws or toothpicks or
making curved shapes with yarn or pipe cleaners
• Making silhouette cutouts of objects and matching
them or having guessing games
• Making collections based on different shapes
• Using geoboards to make shapes with rubber
bands
81. • Staking out “giant” shapes on the playground
with crepe paper or string
Spatial Relationships
Activities that promote spatial relationships focus on
encouraging children to locate bodies or objects in space,
use their knowledge of spatial relationships to describe
where something is located, interpret representations of
spatial relationships (mapping), and represent spatial rela-
tionships with symbols (mapping). Active games such as
hide and seek, duck-duck goose, or building an obstacle
course build spatial awareness.
Measurement
Children learn to measure first by nonstandard means
such as pacing off distances and later with uniform but
nonstandard measurement tools (e.g., measuring the
width of a tabletop with paper clips) and later still with
conventional measurement tools. As
they do so, they develop a sense of the kinds of things that are
measured. Concepts related to
measurement include measuring to represent comparisons, using
a variety of different tools
to measure, and seeing estimation as useful but not the same
thing as accurate measurement.
We encourage children to think about measurement with
questions that start with “How far,”
How much,” How long,” and so on.
Linear Measures, Weight, and Volume
Examples of standard measures—such as a growth chart, weight
scale, or masking tape
on the floor to mark off distances in inches or feet—should be
displayed and used to build
83. • Cooking activities
• Using a balance to compare the weights of different objects
and combinations of
objects
• Pacing off longer distances such as the length of a hallway,
sidewalk, or rows in the
garden
• Using standard measurement tools such as rulers, yardsticks,
or a tape measure
Temperature
Understanding that temperature is something that can be
measured is abstract and difficult
for young children other than in general terms such as hot, cold,
and warm. The classroom
should include different tools for measuring temperature with
displays in both analog and
digital format, including oral, candy, meat, and refrigerator
thermometers and outdoor digital
and clock-style thermometers. Children can be encouraged to
observe and record tempera-
tures and engage in activities that involve materials that they
can heat, melt, or freeze.
Children can also be encouraged to monitor the movement of
mercury or dials as temperature
changes. For example, many states’ licensing laws prohibit play
outdoors when the tempera-
ture exceeds 90 degrees Fahrenheit; in applicable climates, if
you mark the window thermom-
eter at the 90 degree mark and children can be on the lookout
for when the mercury or dial
handle reaches that mark to announce “It’s 90!”
85. Mathematics Concepts and Curricular Activities Chapter 10
The Twenty-Four Foot Python:
A Teachable Moment about Measurement
Ms. Deanna was working her way through Shel Silverstein’s
book Light in the Attic (1981, p. 44)
with her preschool/kindergarten class when she came to “Snake
Problem”:
It’s not that I don’t care for snakes,
But oh what do you do
When a 24-foot python says . . .
I love you.
The poem prompted an animated discussion about
how long a twenty-four-foot python would be.
Many ideas were suggested, but they could not
agree on a single answer. Ms. Deanna decided to
follow up, asking what they could do to find out.
The children said they wanted to make a twenty-
four-foot-long paper python model. It became evi-
dent that the focus of the investigation was going
to be accuracy—exactly twenty-four feet, not an
inch shorter or longer! Ms. Deanna produced a
ruler, introducing it as a standard unit of measure
for one foot. Using the ruler, the children quickly
realized the classroom floor tiles all measured
exactly one foot square. They spent several hours
measuring off distances in the classroom in floor
tiles but found that no matter how they measured,
there was no twenty-four-foot space in which they
could build their model.
88. • Installing a sundial outdoors
Currency
Children find American currency challenging because of
centration—they assume bigger
means more and that therefore a nickel should be worth more
than a dime or penny. They
also have trouble with paper vs. coin. As with the representation
of number as quantity in
general, it takes time for them to understand the symbol-
ism behind currency—that the nickel represents 5 cents,
the dime 10 cents, the dollar 100 cents, and so on.
Children do learn about the value of money and its con-
crete uses (buying things) and can be engaged in using
real money judiciously. For example, children in Mr. Dick’s
4-year-old class decided to use the outdoor playhouse to
set up a store for selling snacks. They made juice pop-
sicles, secured a “loan” to buy a big box of Goldfish crack-
ers, and determined that each item would cost a penny to
buy. They made signs for the store and dictated a note for
home, asking parents to send their friends with pennies
to spend in the store. They also “hired” children in the
2-year-old class to do jobs for them they didn’t want to do
(such as sweeping out the playhouse) for a penny!
While the value placed on work and their product was not
realistic in terms of the real world, it definitely showed
their understanding of how money is used and critical to
the exchange of goods and services. They carefully tracked
their revenues over a week and were able to determine
when they had enough pennies to pay back their loan.
They were also ecstatic to find, at the end of the week,
that they had made a profit of $3.34!
90. • Represent the same data set in multiple ways
• Lead to open-ended discussion questions
• Encourage children to name/title graphs
• Revisit data during/after discussions
• Model/demonstrate throughout process of data collection and
creating graphs (p. 39)
Graphs should represent data meaningful to children, such as
shoe colors, birthdays, tracking
the number of children present per day for a week, or
preferences. Very simple graphs can be
done with children as young as two or three. For example, Ms.
Stephanie conducted a unit
on babies with her older 2-year-old class, including sampling
baby foods. She made a picture
graph with the different jar labels across the top and each child
put their fingerprint under-
neath the picture of the food he or she liked the best.
Graphs can be effectively used to represent the cycle of
prediction, testing, and results in an
investigation of any kind. For example, if you plan to plant
seeds, children can predict how
many days it will take for them to sprout and compare
predictions with observations. Any
activity with an either/or outcome, such as sinking/floating can
be graphed in terms of predic-
tions/outcomes. Likewise, any unknown future activity can be
graphed by possible outcomes
children suggest. Suppose you are reading a new story with a
problem to solve. Before getting
92. “why” reflects the natural
human drive to make sense of the world.
Scientific thinking involves the application of curiosity and
reasoning to answering ques-
tions, and teachers promote systematic investigatio n by helping
children focus on ques-
tions like “What’s wrong here? What happened here?” and
“What proof do you have?”
(Campbell, 1999, p. 134). As documented in the
study of power, force, and motion in Chapter 6,
teachers support informal science inquiry during
exploratory play by choosing provocative materi-
als, posing questions, and furthering learning with
additional activities (Hamlin & Wisneski, 2012;
Stoll, Hamilton, Oxley, Eastman, & Brent, 2012).
Physical science describes and explains the prop-
erties of objects and phenomena. Life science is
the study of living things and their habitats. Earth
science focuses on learning about the forces of
nature and studying problems that affect the
health of our planet.
This part of the chapter will focus on general
ideas for planning activities and units and facilitat-
ing transformation of everyday concepts acquired
through play to science concepts that represent
structured thinking and logic using the language
of science (Hamlin & Wisneski, 2012, p. 85).
Physical Science
Everyday concepts about physical science are
acquired as children do such things as paint in the
94. • Light/shadow
• Color
• Magnetism
• Solids, liquids, and gases
• Weight, force, and motion
• Static electricity
Light and Shadow
Young children are highly intrigued
by the interplay of light and shadow
and the ways light can be manipu-
lated to achieve different kinds of
effects. Young children can under-
stand and use terms such as light,
shadow, reflection, filter, rainbow,
image, transparent, translucent, and
magnify.
Concepts that can be acquired by
young children include the ideas that:
• Light comes from the sun and
stars
• Light appears invisible but contains colors
• Blocking light creates shadows
• Some materials allow light to pass through (transparency)
• Light bounces off of shiny objects (reflection)
96. can also make shadow prints
by placing objects on photo-sensitive or construction paper and
exposing it to light.
Observing prisms in different locations and at different times of
day, using a water hose in the
sunshine to make rainbows, or adding oil to a water puddle
outdoors in the sunshine allow
children to see the light spectrum as a rainbow. They are
naturally intrigued by images in mir-
rors, and setting up several mirrors so that images are reflected
in multiple ways provides a
fascinating challenge for them.
Color
Color is all around us, providing a context for informal learning
and intentional activities to
help children learn concepts such as that:
• There are many different colors.
• A single color can have different shades/tints.
• Colors have names.
• Color is not an object but a means to describe objects.
• Colors can be combined.
• Sometimes colors can change.
• Objects can be classified by color.
• An object of one color can be changed to another color.
As with the difference between rote counting and number sense
97. in mathematics, children
may be able to recite the names of colors without being able to
identify the corresponding
color correctly; likewise, they may match and sort colors before
being able to name them.
Also, just as there are variations in the ways numerals are
represented or written in different
fonts, color tints or shades such as lemon, light, or gold may be
difficult for a child to all cate-
gorize as being in the yellow family. Therefore as children play
informally with colored objects
and engage in activities such as drawing with crayons or
markers or using paints, teachers
can help them learn the names of colors and distinguish between
them. Children should learn
words such as shade, tint, dark, light, primary colors, and
secondary colors.
Many materials are useful for explorations with color, including
food coloring, water, different
colors and kinds of paints, eggs, crayons, markers, colored
pencils, colored cellophane, con-
tainers, eyedroppers, ice cube trays, paint-chip sample cards,
and color sticker dots.
Activities that promote learning about colors include these:
• Sorting and/or matching paint-chip cards (from hardware or
paint store) within color
groups
• Having a color scavenger hunt
• Making a set of colored water bottles that represent the three
primary (red, blue, yel-
low) and three secondary (orange, green, purple) colors
99. • Only metal is magnetic
• Only certain kinds of metal are magnetic
• Magnets both attract and repel
• The earth has magnetic force
Discovery activities focus on providing assorted objects and
different kinds of magnets; chil-
dren can subsequently classify, graph, or label
magnetic/nonmagnetic materials. Informal play
with magnets can be set up, for example, by attaching magnets
to the fronts of small metal
cars and pulling them along a premade or improvised racetrack
or attaching paper clips to cut-
out paper fish and fishing with a magnet attached to the end of a
string or line. Children could
also use a magnet wand to move floating corks with an inserted
straight pin in the water
table. Although a bit on the abstract side as far as making
symbolic geographic connections,
young children can easily learn how to tell where “north” is
with a compass and understand
that magnetism is what makes the compass work.
Solids, Liquids, and Gases
Basic concepts related to states of matter appropriate for young
children include the following:
• Liquids assume the shape of their container
• Solids retain their shape
• Some solids dissolve in liquid; some do not
101. bubbles with wands fasci-
nates them, and the teacher can help them to understand that the
air inside the bubble is
trapped by a liquid “shell.” Children can also blow bubbles
through a straw into milk, water,
and syrup or pour these kinds of liquids back and forth to
explore viscosity.
Making fruit smoothies demonstrates the principle of a
suspension. Children can classify and
label objects in the classroom as liquids or solids or have a
liquid lunch, and they will certainly
spend a great deal of time at the water table exploring the
concepts of sinking and floating.
Weight, Force, and Motion
Applying force to an object makes it move—a concept children
employ every day as they push
cars or blocks across the floor, draw with a crayon, or pedal a
tricycle on the playground.
Simple machines such as pulleys, gears, ramps, and levers are
endlessly fascinating to children
in their efforts to figure out what makes things work, as
illustrated by the emergent investiga-
tion example in Chapter 6.
Other important ideas are the concepts that moving air is called
wind, friction occurs when one
object moves over another and produces
heat, objects fall down, moving water has
force, and machines help people work in
different ways. Vocabulary for young chil-
dren includes pulley, cause/effect, push/
pull, force, motion (and names of motions
such as roll, glide, fly, bounce), weight,
balance, friction, and incline.
102. Many materials for learning about this ele-
ment of physics are easily found in pre-
school classrooms, such as wood planks
in the block area, train track, marbles,
wheeled toys, straws, and heavy paper.
Other materials can include a balance,
pulleys, pendulum, scooters, oscillating
fan, rope, plunger or suction cups, clear
plastic tubing, and cove molding.
Activities that help children learn about weight, force, and
motion include:
• Using blocks and different types of materials to build
ramps of different heights/
lengths; timing the speed of different kinds of objects rolled
down and categorizing/
graphing them as fast/slow
• Setting up a clothesline pulley on the playground and using it
to move objects
• Going on a ramp hunt throughout a building or neighborhood
• Building a marble maze with tubing
• Setting up an obstacle course for scooter races
• Using magnets to move objects underneath paper
• Tracking shadows over the course of a day on the playground
• Blowing objects across a flat surface with straws
• Tying crepe paper streamers to a fan
104. putting an object into an electrical
outlet, learning about static electricity can be both fun and
harmless.
Life Science
Life science investigations focus on the study of living things
and their habitats. It makes the
most sense, in terms of concrete learning and consideration of
prior experience, to begin the
study of living things with those that are most relevant and in
closest proximity to your set-
ting. So, for example, if you live in a rural area, you might
study farm animals and local crops;
if you live on the coast, you might study ocean animals,
reptiles, and beach grasses.
Important life science concepts for young children to learn
include the facts that:
• All living things grow and change over time.
• Living things need food.
• When living things die, they decompose.
• Fossils are the remains of living things.
• Living organisms have systems that make them work.
• Living things inhabit and interact with different kinds of
environments.
Plants
Children learn about plants of different kinds with first-hand
experience by growing, examin-
106. different kinds to train their
(pole) bean plants, eventually growing a “bean house” big
enough to put a small table and
chairs inside.
At the end of the school year, the children wondered aloud how
big the weeds would become
over the summer, so they were left unattended; to the children’s
delight, when they returned
in the fall they had a “forest” of weeds, through which they
trampled paths. They made
“houses” and “forts” in this forest and enjoyed it for several
weeks before pulling it out
to begin a new garden. Finally, these children
wondered what would happen to a pumpkin
they had carved if they left it in the garden.
Over the entire winter they documented its
decomposition until it eventually hardened
into a petrified, shrunken shell.
Children love flowers and can collect, press,
dissect, and classify them by color, petal type,
etc. They can sort and classify seeds, pods,
and leaves; make collages or rubbings of dif-
ferent kinds of plants; cut or slice vegetables
and fruits; and make prints to compare shapes
and characteristics.
Similarly, to learn about trees, a “tree cookie”
(cross-sectional slice of a tree trunk) provides
opportunities to measure circumference and
count rings. Planting or adopting a tree in a
city park or finding and photographing the
oldest tree in the community helps children
begin to understand the long-term invest-