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Shape is everywhere in the
world, though in mathematics, they are mainly two-dimensional and three-dimensional,
so the classroom’s block corner is ideal in helping children explore shapes (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children and teachers can look around their physical classroom and discover
that there are endless shapes. However, in mathematics, these are often the
ones that share similar attributes and not random blobs.
The Big Ideas for Shape are,
firstly, shapes are defined and classified by their attributes, secondly, the
flat shapes of three-dimensional shapes are two-dimensional shapes, and
thirdly, shapes can be composed or decomposed to create new shapes (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Firstly, shapes are defined
and classified by their attributes.
Shapes have rules that make
each shape, such as a triangle having three sides or a square with four equal
sides, so teachers should craft activities that highlight these important rules
(Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Using these rules, it is easy to identify or even create shapes from loose
parts. The rules also ensure correction because though a rectangle and a square
have the same number of sides, only the square has four equal sides.
Secondly, the flat shapes of
three-dimensional shapes are two-dimensional shapes.
Children can explore and
discover that two-dimensional shapes are found on the faces of
three-dimensional shapes (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Concrete materials are great at illustrating this rule, as children rotate
common household items like toilet roll cores or a box, they can discover there
are hidden shapes everywhere.
Thirdly, shapes can be
composed or decomposed to create new shapes.
As children gain opportunities
to rotate, combine, and compare shapes, they will realise how shapes have part
and whole relationships, where there are shapes within shapes (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
They can break apart a shape into different shapes, or even use shapes to
create a different shape. The possibilities are endless.
Teachers understand that
manipulatives help children to learn abstract mathematical concepts, but they do
not contain mathematics for children to learn and are only helpful in guiding
children to think in problem-solving, and one example to teach shape is using
tangrams for spatial reasoning though teachers should not help them too much
that causes them to lose opportunities to think, and if a child faces
frustration it is better to provide an easier activity (Kamii, Lewis, &
Kirkland, 2001).
While concrete materials are often used for illustrative purposes, the main benefit
of them is to allow children to self-discover and problem-solve on their own
terms. Teachers should think critically about the types of materials found in
the classroom, and never intervene when unnecessary.
Instruction and construction
differ in that instruction is classroom practices the teacher carries out to
provide knowledge with objectives and systematic systems, whereas construction
is about how children learn through a process to actively build their skills
and concepts, and in modern classrooms, both exist together (Chen, 2014). As teachers build the
learning environment, children can construct knowledge and problem-solve.
Thus, the topics and Big Ideas
of mathematics have been elaborated through this series of six articles.
References
Brownell, J., Chen, J.-Q., Ginet, L., &
Hynes-Berry, M. (2013). Big Ideas of Early Mathematics. US: Pearson
Education.
Chen, J.-Q. (2014). Intentional Teaching: Integrating the
Processes of Instruction and Construction to Promote Quality Early
Mathematics Education. Early Mathematics Learning, 257-274.
doi:10.1007/978-1-4614-4678-1_16
Kamii, C., Lewis, B. A., & Kirkland, L. (2001).
Manipulatives: when are they useful? Journal of Mathematical Behavior, 20,
21-31. doi:https://doi.org/10.1016/S0732-3123(01)00059-1
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In this article, Data Analysis
and Spatial Relationships will be discussed.
How do children do data analysis?
Data Analysis is about asking
questions and finding out the answers, so activities may include complicated
graphs and charts, but it can also be as simple as writing down a list of items
(Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Data can be collected through these tasks.
The Big Ideas for Data
Analysis are firstly that the purpose of gathering data is to answer questions
when answers are not available immediately, secondly, data needs to be
represented to be analysed, and the questions frame how data is gathered and
organised, and thirdly, parts of data should be compared, and data as a whole can
be concluded (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Big Idea #1 of Data Analysis
Firstly, the purpose of
gathering data is to answer questions when answers are not available immediately.
Children realise that data
analysis helps in answering questions and thus they are motivated to understand
it more, and teachers need to know that, for problem-solving to happen, a real
problem must be present for children to solve by guiding them step-by-step, and
children should do data analysis like how survey experts do, where the answers are
attained only after analysis (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
It may be tempting to give the answers to children right away, but doing so deprives
them of the opportunities to self-discover and problem-solve.
Big Idea #2 of Data Analysis
Secondly, data needs to be
represented to be analysed, and the questions will frame how data is gathered
and organised.
When children gain experience
and feel empowered, they can follow steps to gather and represent data, and the
teacher is present to guide them through (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
This allows them to learn and imitate the process when they are older.
Big Idea #3 of Data Analysis
Thirdly, parts of the data can
be compared, and the data as a whole can also be concluded.
Adults typically guide
children to understand that data can be compared in parts and concluded as a
whole, so a new concept can be learned as questions are answered through the
data (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
For instance, a huge bag of sweets can be counted, but also the different types
of sweets within it.
What is spatial reasoning?
Spatial reasoning is an early
phenomenon in children, where they have mental understanding and physically
transform objects, and these are the five key areas: firstly, symmetry,
secondly, transforming, thirdly, composing and decomposing 2D images and 3D
objects, fourthly, locating, orienting, mapping and coding, and lastly,
perspective-taking (Novakowski, 2018), and even from birth,
infants are already learning about spatial relationships, as they reach for objects
around them or move from place to place (Brownell, Chen, Ginet, & Hynes-Berry, 2013).
Symmetry is about when one of
two shapes matches the other shape, and transforming is understanding what an
object can look like after being flipped or rotated (Novakowski, 2018). These can be
achieved by using blocks or paper.
Composing and decomposing 2D
images and 3D objects is about identifying shapes within shapes or creating a
new shape from two or more smaller ones (Novakowski, 2018). Children can learn
to use 2D images and craft 3D figures, or use 3D figures to make 2D drawings.
Different shapes can also be used during play for children to form new shapes.
Locating, orienting, mapping,
and coding are about understanding the location of objects within a space to
learn the relationships between positions, and also includes how 2D objects
look in 3D, and the sequence of numbers and symbols to show an action or
instruction, while perspective-taking is learning to see things from a
different perspective or knowing the changes in perspectives (Novakowski, 2018).
The Big Ideas for Spatial
Relationships are, firstly, that the relationships between places and objects
are described with mathematical accuracy, secondly, that a person’s experiences
of space and two-dimensional representations of space only show a certain
perspective, and thirdly, spatial relationships are visualised and manipulated
mentally (Brownell, Chen, Ginet, & Hynes-Berry, 2013).
Big Idea #1 of Spatial Relationships
Firstly, the relationships between
places and objects are described with mathematical accuracy.
Children know that when they
talk, draw, write, or create models, they can show movement and direction, so
teachers can use photos to show spatial relationships and encourage
discussions, and use language that describes space or movement games to show
movement in certain directions (Brownell, Chen, Ginet, & Hynes-Berry, 2013). Common phrases used
can include “below the table” or “on top of the shelf” to symbolise spatial
relationships.
Big Idea #2 of Spatial Relationships
Secondly, a person’s
experiences of space and two-dimensional representations of space only show a
certain perspective.
Children learn that when seen
through other perspectives, spatial relationships look very different, so they
need to listen to how others are seeing something through organic
self-discovery (Brownell, Chen, Ginet, & Hynes-Berry, 2013). A garden can look
very different when seen from a bird’s eye view or from a taller angle than a
child’s eye level.
Big Idea #3 of Spatial Relationships
Thirdly, spatial relationships
are visualised and manipulated mentally.
Young children may struggle to
imagine spatial relationships, but they can learn through concrete or pictorial
experiences, and for children who have mastered mental images, they do not need
concrete materials to create solutions (Brownell, Chen, Ginet, & Hynes-Berry, 2013). Mathematics has to
be taught progressively and according to the developmental stages of children,
which teachers are capable of understanding as they observe children.
Therefore, data analysis and
spatial relationships are both complicated topics with challenging Big Ideas,
but these are not impossible for children to attain through guidance from
teachers and self-discovery.
References
Brownell, J., Chen, J.-Q., Ginet, L., &
Hynes-Berry, M. (2013). Big Ideas of Early Mathematics. US: Pearson
Education.
For children to use patterns,
they will repeat or increase and decrease patterns, whereby repeating patterns requires
them to identify the regularity in the pattern that repeats, whereas increasing
or decreasing the pattern, they will identify the regularity that affects each
part of the pattern (Novakowski, 2015). For instance, if a
child were to repeat a pattern of blocks, it would look like red-blue-red-blue.
Whereas if the pattern is increased by a regularity, the child would build a
staircase that increases in height with every two red blocks.
The Big Ideas for Pattern are
firstly that patterns are sequences that repeat or grow, patterns are
rule-governed that exist both in mathematics and reality, secondly, once the
rule has been identified, then it can be predicted and generalised, and
thirdly, one pattern can take many forms (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Big Idea #1 of Pattern
Firstly, patterns are
sequences that repeat or grow, and patterns are rule-governed that exist both
in mathematics and reality.
A repeating pattern has a unit
of repeat, which is a segment that repeats, that becomes the rule for the
pattern, and that helps in understanding predictability (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children can understand that is how a pattern is formed and can be continued.
Big Idea #2 of Pattern
Secondly, once the rule has
been identified, it can be predicted and generalised.
The established rule must be
followed to continue a pattern, so children can tell missing parts in a pattern,
or even to extend it (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
If the rule is not followed, then it is no longer a pattern.
Big Idea #3 of Pattern
Thirdly, one pattern can take
many forms.
This is a more abstract
concept in which representation is used in simple algebraic concepts, such as
using a clap to show orange, and this can occur only when children have
multiple learning opportunities (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children can then learn to understand that a pattern structure can be
represented in many ways, so this can be achieved through concrete materials,
body movements, or even verbal prompts.
What is measurement?
Measurement is about a concept
and a process, the comparison of the sizes of objects, and it has a unit
descriptor and numerical value, while also requiring many skills and concepts,
such as attribute, conservation, transitivity, point of origin or baseline,
direct comparison, indirect comparison, unit, size of unit, iteration, and
estimation using a referent (Novakowski, 2016).
The Big Ideas for Measurement
are, firstly, that many different attributes can be measured just from one
object, secondly, all measurements require fairness in comparison, and thirdly,
quantifying a measurement helps in more precise comparison and descriptions (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Big Idea #1 of Measurement
Firstly, many different
attributes can be measured just from one object.
Attribute refers to an
object’s dimensions being measured, conservation is about how an attribute of
an object remains the same regardless of the position or movement, and
transitivity is about having a third object to compare the lengths of two
objects (Novakowski, 2016).
Measurement is how objects can
be identified by attributes of weight, temperature, length, circumference,
volume, or number, and for young children, this is a complex process, so they
can also identify which attribute of the object to focus on, because an object
can be both bigger and smaller than another, depending on the attribute (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children may claim that a taller bottle holds more water, but actually, they are
describing the height instead of the capacity.
Big Idea #2 of Measurement
Secondly, all measurements
require fairness in comparison.
Accuracy is important, so
there must be fairness during measurement, such as lining up objects to use
direct comparison (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
A child would not be able to tell if two vases are the same height if they are
not next to each other.
Point of origin or baseline is
using a zero point to start measuring objects for comparison, direct comparison
is placing two objects next to each other to compare lengths, and indirect
comparison is using another object to compare the lengths of two objects (Novakowski, 2016). Indirect comparison
is using a string to check if two toys are the same length.
Big Idea #3 of Measurement
Thirdly, quantifying a
measurement helps in more precise comparison and descriptions.
As children develop in more meaningful
comparisons of objects, they learn that exact units of measurement help in
describing and comparing objects better, and they are consistent, unlike using
hands to measure (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children can describe exactly how much bigger one object is than another, and
they can also identify units that are inconsistent and thus not accurate to use
as units of measurement.
Unit is used to measure
objects and they include non-standard and standard units, so in non-standard
units they are uniform and non-uniform, size of unit is about how the chosen
unit affects the numerical value of measurement, iteration is the use of many
copies of the same unit or if there is only one then the unit is used
repeatedly, and lastly estimation using a referent is about estimation of a
larger quantity using a known measurement (Novakowski, 2016). Units that use
standard units follow metric systems like centimetres and metres, while uniform
units are objects that are consistently sized, so non-uniform units are not.
Therefore, pattern and measurement
are covered in this article, and they are relevant skills for children to
understand and develop further as they grow older.
References
Brownell, J., Chen, J.-Q., Ginet, L., &
Hynes-Berry, M. (2013). Big Ideas of Early Mathematics. US: Pearson
Education.
There are two types of
counting, namely rote counting, where number names are recited from memory, and
rational counting, where a number name is matched to an object in a collection (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
The second type of counting is more engaging and more practical in everyday
life.
These are the following
counting opportunities, namely knowing number name sequence, one-to-one
correspondence, cardinality, stability, relative size, making connections
between number names, quantities and symbols, counting forwards, backwards, or
middle, and base-ten structure (Novakowski, 2015). Hence, a wide variety of learning
opportunities can be provided in a classroom setting.
The Big Ideas for Counting are
firstly that counting is used to find out the quantity in a collection, and
secondly, it also has rules that apply to any collection (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Secondly, counting has rules
that apply to any collection. This Big Idea has the following four key
principles: stable order, one-to-one correspondence, order irrelevance, and
cardinality (Brownell, Chen, Ginet, & Hynes-Berry, 2013).
Stable order refers to always
starting from one when counting, so if a child has not mastered this principle,
he will use random numbers during counting, and this is not just important for
number sequence memorisation but each number is bigger than the one that
precedes it and smaller than the next, and because the number system is
base-10, every number that ends with 9 will go back to 0 (Brownell, Chen, Ginet, & Hynes-Berry, 2013). Children should be
able to count in the correct number sequence from 1 to 10, and 10 comes after
9.
Order irrelevance builds on
stable order and generalises one-to-one correspondence, because no matter which
object the counting starts from, the result will remain the same, and the fact that
number words do not change the objects themselves but are only used during
counting, but then this affects reality, so when objects are counted, they
should be set aside to avoid confusion (Brownell, Chen, Ginet, & Hynes-Berry, 2013). So if a child counts
the number of friends seated in a circle, it does not matter if the child
counts from the left or right. The child will also understand that calling a
friend by a number does not mean that it will be their name. Adults should also
guide children to set aside counted objects during counting.
Cardinality is knowing that
the last number used is the quantity of objects, and cardinality serves two
purposes: children use numbers to apply to objects, and children use the last
number to represent the quantity of objects (Brownell, Chen, Ginet, & Hynes-Berry, 2013). While this may be a
simple concept as well, it takes a while for young children to understand.
Teaching Number Operations
Number Operations are tools
that need a foundation of understanding that every operation tells a story (Brownell, Chen, Ginet, & Hynes-Berry, 2013). These operations
simply refer to the following mathematical concepts: addition, subtraction,
multiplication, and division.
The Big Ideas for Number
Operations are firstly that sets can be changed through joining or separating,
secondly, sets are compared using numerosity and are ordered by more than, less
than, and equal to, and thirdly, a quantity can be decomposed into equal or
unequal parts, and the parts can be composed to form a whole (Brownell, Chen, Ginet, & Hynes-Berry, 2013).
Big Idea #1 of Number Operations
Firstly, sets can be changed
through joining or separating. Sets can change when objects are added or
removed, so children need to know the definitions of adding and taking away, and
there are strategies like counting all, where a child counts a set of objects
from the first object whenever there is a change, and counting on, which is
when new objects are added, but the number continues from the last counted
object (Brownell, Chen, Ginet, & Hynes-Berry, 2013). As children will be
more inclined to use the counting all strategy, teachers can provoke thinking
and encourage them to use counting on instead.
Big Idea #2 of Number Operations
Secondly, sets are compared
using numerosity and are ordered by more than, less than, and equal to. Children
will naturally know to use visual comparison to tell if a set has more objects,
and they use matching where sets are placed next to each other with one-to-one
correspondence to determine the quantities, and even if the objects vary in
size, children know that they are only focusing on the number of objects (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
So, even if a larger quantity of small erasers is placed next to a small
quantity of big erasers, children understand that there are more small erasers.
Ordering is about ordinal
numbers of first to last or most to fewest, thus the order of number sequences
tells the relationship between sets, in which sets are then determined whether
which has more than, less than, or equal to each other (Brownell, Chen,
Ginet, & Hynes-Berry, 2013). Ordinal numbers are
also how children attribute numbers to quantity, as they use the terms first,
second, and third.
Big Idea #3 of Number Operations
Thirdly, a quantity can be
decomposed into equal or unequal parts, and the parts can be composed to form a
whole. The concept of subitising is required as children learn that there are multiple
ways to get a certain quantity, and they can add and subtract automatically,
but this differs from rote thinking, as it is an understanding of part/whole
relationships (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
So, ten can be five and five, but also eight and two.
Therefore, as simple as
counting and number operations may seem to adults, these ideas are more
relevant and enlightening for children to learn more about the world around
them.
References
Brownell, J., Chen, J.-Q., Ginet, L., &
Hynes-Berry, M. (2013). Big Ideas of Early Mathematics. US: Pearson
Education.
Previously, it has been
mentioned that these are the topics of mathematics: Sets, number sense,
counting, number operations, pattern, measurement, data analysis, spatial
relationships, and shape. The Big Ideas stem from each of these topics. For
part 2 of this series of articles, the topics of Sets and Number Sense will be
elaborated.
Many educators still believe
that mathematics should only be taught in formal schooling years rather than
during early childhood, though there has been research done to show that these
experiences are crucial for children’s later development, so educators must
understand that mathematics is for every early learner, and that it is beyond
shapes and numbers (Knaus, 2017).
Next, the topics of Sets and
Number Sense will be introduced.
Teaching Sets
Sets mean using attributes to
create collections, with the same collection able to be sorted in different
categories, and they can be compared and organised (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
For instance, a child may sort some beads by size or colour, and thus has
created sets.
Three Big Ideas on sets are
that attributes are used to group collections into sets, a single collection
can be sorted in a variety of ways, and that sets can be ordered and compared (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
The concept of sorting is defined as unique from matching, as it is about
reorganising an entire collection or set into at least two subsets (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
For instance, a box of marbles is sorted into blue or green.
Big Idea #1 of Sets
Find my match.
What's my rule?
Firstly, the use of attributes
to help children sort collections into sets will be explored. The teacher can
guide the child to use different attributes, like colour, shape, or similar
objects, or even increase the difficulty of the activity by adding more attributes
or objects to the collection, and inviting children to figure out which object
is taken away (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
There is a wide variety of methods to be used with just a simple box of small
manipulatives. In the first image, the teacher uses all stars in the activity, and gets the child to find the exact colour to match the star. The game "What's my rule?" requires children to take out objects that share an attribute that the rest of the objects do not have.
Big Idea #2 of Sets
People sort activity.
Secondly, a single collection
can be sorted in a variety of ways. This is a more difficult concept, but
children can learn it through self-discovery during play, where they understand
that there are many ways a collection can be sorted (Brownell, Chen, Ginet, & Hynes-Berry, 2013). If a child has
sorted a box of crayons by colour, the crayons can also be sorted by size. In the People Sort activity, children understand that there are so many ways to sort objects.
Big Idea #3 of Sets
Thirdly, sets are able to be
ordered and compared. This involves comparing sets to find out which is better,
though it is more often about quantity, so children need to explore more to
understand the concept (Brownell, Chen, Ginet, & Hynes-Berry, 2013). Children will
naturally count the objects of each set during play and conclude that one of
them has a higher quantity.
Teaching Number Sense
Moving on to Number Sense,
which is about developing a purposeful sense of quantity, and the Big Ideas
include learning that numbers are used in different mathematical or
non-mathematical ways, knowing that quantity symbolises an attribute for a set
of objects with numbers being used to name quantities, and lastly, the quantity
of a small collection can be understood without counting (Brownell, Chen, Ginet, & Hynes-Berry, 2013).
Big Idea #1 of Number Sense
Firstly, learning that numbers
are used in different mathematical or non-mathematical ways. Numbers are not
just used to describe quantity or order, as they can become identifiers like a
name, and people normally do not think about all the other numbers that precede
it (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Hence, just like finding the classroom 105 does not require children to start
from 1, numbers may sometimes have a different function.
Big Idea #2 of Number Sense
Secondly, knowing that
quantity symbolises an attribute for a set of objects, with numbers being used
to name quantities. Numbers are sometimes used as attributes, but other
attributes must be ignored to understand them, as quantity is a mental image
when a child understands the relationships between sets (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Usage of this Big Idea thus involves the child having prior knowledge of sets
in order to compare two collections of objects.
Big Idea #3 of Number Sense
Thirdly, the quantity of a
small collection can be understood without counting. Subitising is being able
to tell “how many” in collections of objects quickly, without counting (Brownell, Chen, Ginet, &
Hynes-Berry, 2013).
Children who have developed in their mathematical thinking can tell how many
dots a face of a die has without physically counting each of them.
References
Brownell, J., Chen, J.-Q., Ginet, L., &
Hynes-Berry, M. (2013). Big Ideas of Early Mathematics. US: Pearson
Education.
Knaus, M. (September, 2017). Supporting Early Mathematics
Learning in Early Childhood Settings. Australasian Journal of Early
Childhood, 42(3), 4-13.
doi:https://doi-org.suss.remotexs.co/10.23965/AJEC.42.3.01
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