Abstract:

Algebra serves as a fundamental prerequisite for mastering diverse mathematical domains. Proficiency in mathematics hinges significantly on grasping algebraic principles. By Grade 9, learners are expected to perform operations such as simplifying, multiplying, dividing, and factorizing algebraic expressions. These expressions encompass both numerical values and variable symbols, embodying a blend of arithmetic and symbolic notation. For instance, the expression 5𝑥 − 2𝑦 + 10 epitomizes this amalgamation of variables and constants. The transition from concrete numerical operations to symbolic algebra often poses challenges for learners, rooted in conceptual difficulties. Early education traditionally emphasizes concrete manipulatives and real-world examples, aligning with Piaget's developmental stages that advocate for tangible learning experiences. Manipulatives, designed objects that embody mathematical concepts more transparently than real-world items, facilitate the introduction of mathematics in a developmentally appropriate manner. Piaget's theory of meaningful learning underscores the importance of aligning external teaching methods with a child's internal cognitive development. Assimilation, where new information is interpreted through existing knowledge, plays a crucial role in this process. In advocating for a pedagogical approach that bridges the concrete and abstract, we propose employing algebra tiles as manipulatives that engage learners deeply and prompt reflective thinking. These tiles serve as tangible models that resonate with learners' existing knowledge, facilitating a dual representation—both concrete and abstract—of algebraic concepts. Vygotsky's zone of proximal development further support this approach, emphasizing the pivotal role of manipulatives in scaffolding learners' learning within their proximal zones of development. Our study involved employing algebra tiles with a sample of 22 Grade 9 learners from one South African school to address common errors in algebraic problem-solving. Through detailed analysis of specific errors, we illustrate how these manipulatives aid in bridging the gap between the physical and abstract dimensions of algebra. This dual representation helps learners recognize the symbolic relevance of algebra tiles while actively engaging with both concrete and abstract representations. Advocating for the pedagogical integration of manipulatives, particularly algebra tiles, enhances learners' comprehension of abstract algebraic concepts within their developmental zones. This approach aligns with educational theories that emphasize the importance of hands-on learning and meaningful engagement with mathematical ideas, ultimately supporting learners in mastering foundational algebraic skills.

Keywords:

Algebra, algebra tiles, manipulatives, concrete, symbolic.