Mathematics in Early Childhood Part 5 (Data Analysis and Spatial Relationships)

Data Analysis and Spatial Relationships.

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.

The role of teachers

To understand if children are learning mathematics, teachers use observation, documentation, and formal assessments (Chaillé, 2021). Observations are how teachers observe children during activities or play, and analyse their behaviour. Documentation can include work samples, written observations, or photographs of children engaged with activities. Formal assessments include portfolios or checklists to gauge the development of 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.

Chaillé, C. (2021). ECE314 Facilitating children's mathematical thinking (study guide). Singapore: Singapore University of Social Sciences.

Novakowski. (2018). Spatial Reasoning.

 

 

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Mathematics in Early Childhood Part 4 (Pattern and Measurement)

Pattern and Measurement.

In this article, Pattern and Measurement will be discussed.

What are patterns?

Patterning is important as children can see connections and relationships between visual-spatial, addition, or multiplication elements, and it is about understanding, describing, creating, and extending patterns with a predicted repetition, and children should be able to articulate the pattern rule (Novakowski, 2015). Repeating patterns, spatial structure patterns, and growing patterns are three of some of the many patterns, and they all help children to predict, organise, and make connections (Chaillé, 2021).

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.

Role of the teacher

As a teacher, small manipulatives that allow for patterns can be provided in the learning environment, as children form patterns such as ABAB, AAB, AAB, and so on. Teachers can also encourage children to use mathematical language, which involves concepts like measurement, size, counting, numbers, and shape during daily routine, ask them math questions, and talk about their thinking (Chaillé, 2021). The learning environment is crucial in helping children develop mathematical concepts.

Books can also be provided that teach these concepts: Numbers, spatial relations, patterns, measurement and data, while using strategies like following children’s interest, stimulating mathematical thinking, involving parents, and using books as resources in projects (Chaillé, 2021).

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.

Chaillé, C. (2021). ECE314 Facilitating children's mathematical thinking (study guide). Singapore: Singapore University of Social Sciences.

Novakowski. (2015). Patterning.

Novakowski. (2016). Linear Measurement.

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Mathematics in Early Childhood Part 3 (Counting and Number Operations)

Counting and Number Operations.

In this article, Counting and Number Operations will be discussed.

Teaching Counting

Counting is vital because it helps children understand the base-ten system, which includes the numbers from 0 to 9, do number operations, and problem-solve, so children learn to count easily and accurately while figuring out the quantity of objects (Novakowski, 2015). While counting is important, it is not enough to only learn number concepts; children need to understand comparison, cardinality, and one-to-one correspondence (Chaillé, 2021).

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).

Big Idea #1 of Counting

Firstly, counting is used to find out the quantity in a collection. This Big Idea recognises counting as related to number sense, where the main purpose of counting is the cardinal use of numbers, which then allows children to use number operation activities or even to do subitising (Brownell, Chen, Ginet, & Hynes-Berry, 2013). Cardinality is understanding that the last number counted represents the group (Chaillé, 2021). Children can then move on to using sets during play with the foundational understanding of quantity.

Big Idea #2 of Counting

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.

One-to-one correspondence happens when children say each number that corresponds to an object in the group (Chaillé, 2021). This is a skill that often takes time for children to develop because they may point to different objects during counting. Adults can guide children to point to objects slowly and articulate the number words.

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.

Role of the teacher

So what does the role of a teacher look like? A classroom should have mathematically rich resources like tools, materials, and manipulatives, and also utilise space well (Chaillé, 2021). A teacher can look at the walls or outdoor areas beyond just the work tables or shelves that typically contain the mathematical activities. Children can use measuring tools to learn math concepts and use materials to count.

Teachers can also implement strategies that include intentional lessons and activities, learning centres, or provocations where children’s thinking is triggered as the teacher encourages a new direction of play, though the children do not necessarily have to follow the teacher’s guidance (Chaillé, 2021). These can include the teacher adding mathematical tools for children to use.

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.

Chaillé, C. (2021). ECE314 Facilitating children's mathematical thinking (study guide). Singapore: Singapore University of Social Sciences.

Novakowski. (2015). Counting.

 

 

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Mathematics in Early Childhood Part 2 (Sets and Number Sense)

Sets and number sense.

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.

Traditional classrooms use a behaviourist approach towards teaching numbers, with the main teaching strategy being direct instruction, whereby questions asked only have a correct answer, and the teacher gives the information and rewards children for answering accurately (Chaillé, 2021). While this method is better for children with special needs, it is better to use a constructivist approach for mainstream children. Mathematics can also be integrated into the classroom throughout the daily routine, during children’s play, and in the curriculum (Chaillé, 2021).

Whereas with a constructivist approach, the teacher creates a rich mathematical environment but gives little direct instruction and instead allows children to explore mathematics in their own ways (Chaillé, 2021). This requires a higher level of skill from the teacher, as the environment must be constructed to allow children to acquire math skills organically, and the teacher does not interfere with their learning.

The teacher is also observant of children’s mathematical explorations and caters for materials in response to that, such as by placing images of buildings of different heights when children are using blocks to create buildings (Chaillé, 2021). Intentional teaching directly contrasts rote learning and occurs in a supportive play environment, because children learn mathematical concepts when play and intentional teaching are combined, so for educators, they have to overcome their fear or lack of confidence in teaching mathematics (Knaus, 2017). The role of the teacher is to be a facilitator and observe children during play.

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.

Chaillé, C. (2021). ECE314 Facilitating children's mathematical thinking (study guide). Singapore: Singapore University of Social Sciences.

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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