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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Theories of Language Development

Language development.

How do we understand how a child learns language? Is it through the environment, the child himself, or a mix of both? In this article, language development is analysed through developmental theories.

There are three main theories in language development for young children: Behaviourist, Maturationist/Nativist, and Constructivist/Interactionist. In this article, these theories will be articulated and referenced with real-life examples.

Behaviourist

Firstly, the behaviourist understands that children learn language through the environment, be it through conditioning or reinforcements, and that children are mere learners of knowledge (Chen, 2020). John Watson and B. F. Skinner were two such proponents for this school of thought. A child learns language as he or she receives conditioning, reinforcement, and imitates others through the words and behaviours of adults in a repetitive manner (Lemetyinen, 2023). Rewards and punishments are often used in a behaviourist teacher’s classroom, so to teach writing or spelling, a teacher might give rewards such as stickers. This is a common occurrence in Singapore.

Maturationist or Nativist

Secondly, the maturationist or nativist believes that children are born with a Language Acquisition Device or LAD, that every child is capable of mastering any language and that certain milestones happen between 40 weeks and 5 years, so every child should be respected in terms of their individual growth and development (Chen, 2020). Arnold Gesell is one such psychologist who subscribed to the maturationist theory. A key difference between the maturationist and behaviourist educators is that the behaviourist puts more emphasis on the stimuli outside of the child. However, language could never be learned through external sources without internal elements, since the LAD predisposes children to learn language during a critical period, because even in poor learning environments, a child could still acquire languages (Lemetyinen, 2023). In a maturationist classroom, the teacher does not in any way force a child to do writing until the child is developmentally ready. Rarely does this happen in Singapore.

Interactionist or Constructivist

Lastly, the interactionist or constructivist theory believes that a child learns through his or her exploration and discovery within the learning environment, and as such, they are not mere passive learners but active discoverers of knowledge (Chen, 2020). Two of such well-known theorists are Vygotsky and Piaget. Constructivists understand that children create their own knowledge as they interact and experience the world around them, so the curriculum has to be more learner-centred with teachers acting as facilitators rather than givers of knowledge, while providing a rich learning environment with social interactions (McLeod, 2025). A constructivist classroom has the teacher not giving the answers but allowing children to explore and learn. Learning centres will be print-rich with writing tools, so children will explore writing on their own terms.

References

Chen, D. (2020). ECE108 Supporting communication and emergent literacy through play. Singapore: Singapore University of Social Sciences.

Lemetyinen, H. (2023, September 7). Language Acquisition Theory. Retrieved from Simply Psychology: https://www.simplypsychology.org/language.html

McLeod, S. (2025, March 31). Constructivism Learning Theory & Philosophy of Education. Retrieved from Simply Psychology: https://www.simplypsychology.org/constructivism.html

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2. The Construction of Knowledge

Construction of knowledge.

While behaviourists believe in nurture and maturationists believe in nature, constructivists advocate for a balance between the two.

Under the realm of constructivism, we have 3 theorists: Piaget, Vygotsky, and Gardner. We have to provide opportunities for children to have active engagement through the use of
1. Relevant, meaningful learning experiences
2. Concrete, open-ended materials
3. Unrushed time 
4. Differentiated learning.

Piaget’s Cognitive Development Theory entails 4 stages: Sensorimotor (0 to 2), preoperational (3 to 7), concrete operational (7 to 11), formal operational (11 onwards). Children may deviate from these stipulated age groups, according to their individual development.

Piaget also describes an ability known as adaptation, which involves assimilation and accommodation. Assimilation is when children transform new information into what they already know, while accommodation is about changing what is already known to make sense of new information. So if I know that tables have 4 legs, and I see a chair, I will still call it a table as I change it to fit what I currently know. But if a teacher tells me that it is not a table, but a chair, I change what I know to fit this new information. Hence, this is a back-and-forth process.

Equilibrium or disequilibrium is simply about whether a balance between new and current information is achieved.

Vygotsky defines the zone of proximal development as the gap between the most difficult independent task a child can do and the most difficult task the child can do with support. A more knowledgeable other has to guide the child, but ensure the level of support slowly reduces to enable the child to be capable of his or her own.

This process is defined as scaffolding. Scaffolding entails 5 components: Instruction (information on what to do), Modelling (showing behaviour for children to imitate), Feeding-back (offering information based on what the child has done concerning the task objective), Questioning (enabling children to think in new ways), Cognitive Structuring (structure for thinking and acting, like stating the steps to do). Vygotsky also believes in the importance of language. Unlike Piaget, Vygotsky believes in the role of social relationships.

As for Gardner, Multiple Intelligences are in every person, in various degrees. It is possible to have more than one. Learning can involve multiple intelligences too, for instance, a cookery lesson must have logical-mathematical competency, but it can also involve naturalistic intelligence, as they begin to discover and wonder how ingredients work with each other.

Children learn through play, and play is defined as an activity that is voluntary, requires active involvement, symbolic, free of external rules, focuses on action rather than outcomes, and pleasurable.

Play can also be divided into types, such as constructive play, or Parten’s Six Levels of Social Play: Unoccupied behaviour, solitary, onlooker, parallel, associative, and cooperative.

Smilansky defines Four Types of Cognitive Play, further enhancing Parten’s theory: Functional (Repetitive physical action), Constructive (Sensorimotor or creativity to build something), Games with rules (Understanding and accepting rules), and Dramatic (substituting an object for something imaginary).

Play is beneficial in teaching pre-academic skills. Using objects to represent supermarket groceries is one such example, and the use of symbols benefits them in learning to read. Using a toy phone to represent a real phone is no different from using a word to represent an idea or concept.

Finally, purposeful play, as defined by MOE, it must be enjoyable to children, requires the active involvement of children in exploring, developing and applying knowledge and skills, involves learning objectives that have been carefully thought through by the teacher while taking in consideration children’s interests and abilities, and it requires facilitation by teachers and this involves observing children at play to discover what they have learnt, and then designing learning activities to reinforce and/or extend their learning based on clearly identified learning objectives.

References
Chen, D. (2020). ECE102 Children as thinkers and meaning makers (study guide). Singapore: Singapore University of Social Sciences.

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