Assessment 1: Children as Mathematical Learners

Part A – Instrumental and Conceptual Understanding

To understand mathematics, multiple steps are required, which involve making sense of mathematical ideas and applying them flexibly (Haylock & Cockburn, 2017). Van de Walle et al. (2024), in their chapter, discuss the two contrasting concepts for mathematics that are instrumental and conceptual understandings. Instrumental understanding refers to the execution of rules or procedures without knowing why they work, e.g. a student might know that multiplying a number by 10 means putting a zero with that number, but does not understand the place value and base-ten numeration system that explains the rule. Whereas, conceptual understanding is the knowledge of mathematical principles and relationships that enables the students to explain and reason their method applied to novel problems (Andamon & Tan, 2018).

Van de Walle et al. (2024) stated that instrumental understanding enables students to achieve short-term results in the assessment due to rote learning. It leads to a superficial understanding of the concepts; thus, when these students are confronted with unfamiliar problems or non-routine tasks, they struggle with the concept and solution. However, according to Smith et al. (2018), the students who have conceptual understanding have the framework that enables number sense, fosters flexibility, and is critical for problem-solving. The place value lesson in primary school is an example to promote conceptual understanding. For instance, a student learning the number 204 can write 200+4 wrong if they do not have a concept of the zero's function as a placeholder. This misconception prevails when the instrumental method is applied within the classroom.  The teacher must make the student understand that each digit in a number has a value determined by its position. Activities such as base-ten blocks, place value charts, and number expanders can help the student grasp these concepts. The researchers have highlighted a few relevant terms within the context of learning, which include;

Productive struggle:

It is defined as the cognitive effort learners engage in that enables them to work through increasingly challenging tasks that require deep thinking. Students are made to solve new problems they have never seen before, intentionally making the problems harder such that student stretch their brains, and rather than providing immediate answers, teachers should create conditions for students to reason through problems. There are many studies supporting the significance of productive struggle in developing resilient, independent mathematical thinkers (Young et al., 2024).

Scaffolding:

Vygotsky presented a theory of the Zone of Proximal Development (ZPD). Based on that theory, scaffolding in mathematics refers to the temporary support and guidance provided to students by teachers to help students such that as they learn new concepts, the support gradually decreases as the students gain understanding and independence. Schukajlow et al. (2015) state that it enables students to perform on their own, as they can receive guidance. Effective scaffolding in mathematics is done by asking guiding questions, incorporating visual aids, and modelling strategies, which are then gradually withdrawn as learners develop confidence. It is like building a house on temporary support, which is scaffolding, that is removed afterwards as the structure becomes stable (Leggett, 2017).

Differentiation:

Differentiation is an instructional approach that requires a focus on all students, It refers to delivering instruction to address students’ varied learning styles, readiness levels, and interests. Simanjuntak et al. (2023) state that it is crucial because children do not learn the same things at the same rate and in the same way. In mathematics, differentiation might involve offering tasks at varying levels of complexity, providing different representations, be it concrete, pictorial, symbolic, or allowing multiple solution strategies, as class will always consist of students with varying abilities, no matter what level. Differentiated instruction is essential to ensure all learners access and understand place value concepts meaningfully, though as differentiation does not require a teacher to create individualised lessons, it requires three basic steps. The first is to plan lessons incorporating meaningful and authentic content, then secondly recognise students as learners, and lastly connect the content with the students (Prast et al., 2025).