In a nutshell
- High-quality mathematical problems for young children should connect to what they already understand and promote genuine engagement.
- Teachers should select rich tasks that allow children to use and compare their own strategies and solutions, supporting deeper conceptual understanding through productive struggle.
- Through repeated exposure to varied contexts, children gradually learn to recognise underlying structures in problem-solving situations, such as change, comparison between quantities, and part–whole relationships.
High‑quality mathematical problems for young children should connect to what they already understand and spark genuine engagement. When teaching mathematics through problem-solving, the learners develop conceptual understanding and see the relations between numbers (Van de Walle, 2014). The engaging element must rely on mathematical ideas that the child is ready to learn, and the task should allow children to show their thinking by explaining, acting, or drawing. Problems are most compelling when written in the present tense and when they relate to children’s everyday experiences within their cultural context (Van de Walle, 2014).
Teachers should select rich tasks that allow children to choose their own strategies and solutions, thereby supporting deeper engagement with mathematical ideas through productive struggle (Van de Walle, 2014). The same problem type can be revisited across different contexts over time to strengthen understanding. High-quality questioning supports children in reflecting on their strategies and provides teachers with opportunities to extend thinking through formative assessment. Early mathematical competence is also associated with stronger academic achievement in later grades (Chen, 2025).
A well-designed problem incorporates productive struggle: children should have the necessary prior knowledge and tools to work towards a solution, even when the task is challenging (Van de Walle, 2014; Kuzu, 2021). Young learners may use manipulatives, drawings, or informal representations to support their reasoning, which is developmentally appropriate given that fine motor and writing skills are still developing (Acevedo, 2020). The teacher or “educarer” does not demonstrate how to solve the problem; instead, children explore, test ideas, and discover solutions through acting, drawing, guessing, and checking. However, the teacher still plays an important role in scaffolding mathematical thinking and problem solving by prompting and guiding this exploration. Teachers ask high-quality questions that allow children to relate to their strategies and allow the teacher to extend their thinking. Children benefit from opportunities for discovery and exploration (Ortiz, 2016). Well‑designed problems help them develop both mathematical concepts and procedures, as well as the relationships between them (Van de Walle, 2014).
Where possible, food-based contexts should be used with caution, as some children may experience hunger during learning activities. If items such as cookies or bread are included in problem scenarios, paper representations can be used instead. Word problems should also incorporate culturally relevant names, contexts, and materials so that children can see their own experiences reflected in mathematical tasks (based on teaching experience).
Early arithmetic problem solving typically begins with familiar situations involving combining and separating quantities, long before learners are introduced to formal terminology such as “addition” and “subtraction”. At this stage, children encounter everyday situations in which quantities are gained, lost, compared, or shared. These experiences provide a conceptual foundation for understanding operations without relying on symbolic notation; for example, learners can solve problems without yet using the symbols + or − when working within meaningful contexts.
Early word problems commonly involve three quantities: an initial quantity, a change (increase or decrease), and a final quantity. Through repeated exposure to such situations, children gradually learn to recognise underlying structures in problems. For example, they begin to identify when a situation involves change, comparison between two quantities, or part–whole relationships. These structures form the basis of later problem-solving schemas (Van de Walle, Karp, & Bay-Williams, 2013).
Because these schemas develop through experience rather than direct instruction, children need exposure to a wide variety of rich, meaningful problem situations. Variation is essential, as it helps learners understand that the same underlying structure can appear in different forms. For instance, the unknown quantity in a problem may be the starting amount, the change, or the result, and shifting the location of the unknown supports flexible reasoning rather than reliance on memorised procedures.
In this way, early arithmetic learning is not primarily about performing calculations, but about developing intuitive models of how quantities behave. Symbolic operations introduced later in schooling, such as written addition and subtraction, are then grounded on a strong conceptual base that learners have developed through experience, language, and problem-solving (Naude & Meier, 2015).
