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Young Children’s Voices in Mathematical Problem Solving
21 Jan 2026
Contributed by Dr Ho Siew Yin and Sng Wei Qin Abbie, from NTUC First Campus, for SingTeach Virtual Staff Lounge
Why Children’s Voices Matter
In preschool years, mathematics learning is often framed as the learning and development of foundational numeracy concepts and skills, such as relationships and patterns, counting and number sense, and shapes and spatial concepts (MOE, 2023, p.12). Yet, at the core of meaningful learning of mathematics and problem-solving lies something more: Children’s voices. Listening to children express their mathematical ideas and reasoning offers profound insights into their mathematical sense-making and mathematical agency.
In his book, Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics, renowned mathematics education researcher, Alan Schoenfeld, highlighted that developing mathematical sense making is about developing a mindset that appreciates and uses mathematical concepts or situations by connecting them with prior knowledge, engaging meaningfully, reasoning logically and developing insights about them, rather than just memorizing procedures (Schoenfeld, 1992). Boaler (2000) highlighted that mathematical agency involves giving learners the opportunity to ask questions, the space to propose ideas, critique ideas and take charge of their own learning, thereby fostering deeper understanding and engagement in mathematical problem solving. This is in contrast with traditional classrooms where the authority lies with the teacher and students are passive receivers of instruction.
Both mathematical sense-making and mathematical agency emphasize empowering learners and encouraging their active participation in creating mathematical meaning for themselves. To foster this empowerment in young children, it is vital that teachers value children’s voices in mathematical problem-solving classrooms.
Listening to Children’s Voices: More Than Just Words
Children communicate their mathematical thinking and ideas in varied ways – not only through words (verbal) but also through gestures, drawings and even the use of physical objects such as blocks (non-verbal). This non-verbal mode provides young children a way to express their mathematical understanding, especially when they are too young to express complex ideas in words. Hence, these two modes, verbal and non-verbal provide insights into how young children construct mathematical knowledge as well as their understanding of mathematical concepts.
Further, when teachers value children’s voices, it can reveal how children understand their family, their community and the environment which they live in.
The Research Study
As part of a recent study by NTUC First Campus, preschool children were asked to solve mathematical story problems – the “Dogs-and-Balls Problem” and the “Children-and-Caps Problem”. These two problems were obtained from the Brigance Early Childhood assessment tool (Brigance, 1992).
Children’s Voices: Vignettes
The following vignettes show children’s voices as they solve the two mathematical story problems. In both story problems, children are asked whether each animal or person gets an object. These vignettes also demonstrate giving children the opportunity to engage in sense-making and exercise mathematical agency.
Both Vignette 1 and Vignette 2 show two young children, Eric and Adil’s thinking as they articulate how they solved two story problems. Both vignettes show what sharing means from these children’s perspective.
Vignette 1: Eric (4 years old)
Dogs-and-Balls Problem
Researcher: Each dog can have a ball. Are there enough balls?
Eric: Yes!
Researcher: Can you show me how?
Eric: [Pick up pencil and drew on the picture]
Researcher: [Pointed to Ball 2]: Are you giving this ball to 2 dogs?
Eric: Yes, share.
Researcher: But the other dogs have 1 ball each?
Eric: Share.
Researcher: The other dogs have 1 ball each. It is okay for these 2 dogs to share 1 ball?
Eric: Yes, share.
Vignette 2: Adil (4 ½ years old)
Children-and-Caps Problem
Researcher: Suppose each child wants to wear a cap. Will every child have a cap?
Adil: Yes!
Researcher: Yes [repeats after Adil]. Can you show me how?
Adil: [Picks up pencil and drew on the picture]
Researcher: [Pointed to Cap 1 and Children 5 and 6]: Are you giving this cap to these two children?
Adil: Yes, share.
Adil: How can 2 children wear a cap [Pointed to Children 5 & 6]?
Adil: Share.
Researcher: There is only 1 cap. How can 2 children wear a cap? [Pointed to Children 5 & 6]?
Adil: He wear [Pointed to Child 5], then after that she wear [Pointed to Child 6].
Reflection
If you were Eric or Adil’s teacher, how would you have responded to these two children? How would you have graded these two children’s solutions?
Both Eric and Adil used non-verbal ways through drawing lines to articulate their solutions to the problem. In addition, these two children may not have shown formal mathematical reasoning in their problem solving. However, their responses show sense-making within a real-world context. Both children connected the story problem to their everyday experiences. When Eric says “Yes, share”, he is indicating the idea that two dogs can play with one ball, which often happens in real-world in the early childhood classrooms where children do share things with each other, rather than applying the formal mathematical reasoning which involves one dog getting one ball. Similarly, when Adil indicated that two children can share a hat, he is connecting everyday experience that two people can share one hat through turn-taking.
Instead of focusing on the incorrectness of the answer, teachers can build on these children’s real-world understanding as a foundational starting point for developing their mathematical skills, such as probing for example, why one dog gets a ball, while another dog shares, then bridging these everyday ideas to formal mathematical reasoning.
Vignettes 3 and 4
In the following two vignettes, Vignette 3 and Vignette 4, two children were asked to create their own story problems. Here, mathematical agency emerges as the children are given the opportunity to create their own story problems.
In Vignette 3, five-year-old Yusuf created story problems which are full replication (type of objects and numerical relationships) of those given to him.
Vignette 3: Yusuf (5- years old)
Vignette 4 shows two story problems drawn by another young child, Ford. The first problem (above) replicated the Dog-And-Balls Problem’s numerical relationships, substituting Ninjas for Dogs and Weapons for Balls. The second problem (below) is a new story problem with new objects and numerical relationships (3 Ninjas, 3 Weapons).
Vignette 4: Ford (5- years old)
4 ninjas and 3 weapons
3 ninjas and 3 weapons
Reflection
Vignette 3 shows that Yusuf was able to translate his understanding of the story problems into his own story problem, mirroring the same mathematical structure of those story problems given to him.
Vignette 4 suggests that Ford was able to move beyond the given problems, connecting prior knowledge with his own ideas and engaging meaningfully in mathematical sense-making.
If you were Yusuf and Ford’s teacher, what further activities would you give to them to advance their mathematical development?
Further Reflection: Children’s Voices Matter!
Valuing children’s voices fosters meaningful mathematical communication, empowers them to articulate their mathematical ideas and supports the construction of their own mathematical understanding and problem-solving. These observations provide teachers with useful insights to design and tailor differentiated learning activities that advance children’s mathematical development.
Why Children’s Voices Matter in Early Childhood Classrooms
Valuing children’s voices and incorporating children’s perspectives into the early childhood classrooms firstly empower children to express themselves mathematically, taking ownership of their own learning, thereby transforming their mathematics problem-solving experiences positively. Secondly, this illuminates the direction for mathematics curriculum decision-making, as well as shape the effectiveness of education policies, ensuring such policies are relevant and executable in current real-world contexts.
References
Boaler, J. (2000). Identity, Agency, and Knowing in Mathematics Worlds. In J. Boaler (Ed.), Multiple Perspectives on Mathematics Teaching and Learning. Ablex Publishing.
Brigance A. (1992). Brigance K & 1 Screen – Revised. North Billerica, MA: Curriculum Associates, Inc.
Ministry of Education (2023). A curriculum framework for preschool education in Singapore: Educator’s guide for numeracy. Singapore.
Schoenfeld, A. H. (1992). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370). New York: Macmillan.
