This master’s thesis investigates the learning of mathematical formulas with understanding, focusing on geometric formulas used for measuring quantities in geometric objects and the ways ninth-grade students perceive these formulas. Students frequently rely on rote memorization, which weakens cognitive connections and leads to lower learning outcomes. The purpose of the study is therefore to examine how different modes of perception of geometric formulas influence students’ understanding and their success in solving geometric tasks.
The theoretical part discusses the concept of learning with understanding through Bloom’s taxonomy and Gagné’s classification of knowledge. In the context of algebraic formulas, various interpretations of the variable are presented, while visualization is shown to play an essential role in understanding geometric formulas by supporting transitions between algebraic and geometric representations. As a central theoretical foundation, the thesis identifies seven modes of perceiving geometric formulas, which differ in the dynamics and direction of mental connections that learners establish: identity, shape, recipe, equation, plan, text, and coexistence.
The empirical part is based on a qualitative case study involving nine ninth-grade students. Data were collected through a written knowledge test and mathematical dialogue, and the analysis focused on interpreting students’ reasoning and explanations during problem solving. The findings reveal substantial differences in how students perceive geometric formulas, both among individuals and across situations. The results indicate that more complex and interconnected modes of perception lead to a deeper understanding of geometric formulas and enable more effective transitions between different representations. Students with stronger visualization skills move more easily between modes of perception, select more appropriate strategies, and achieve better results on complex tasks. Learners who approach formulas with understanding demonstrate greater flexibility, deeper geometric knowledge, and higher performance on the knowledge test.
The study confirms the significant impact of teaching approaches and learning methods on students’ use and understanding of mathematical formulas. It highlights the need for teaching that supports flexible thinking and emphasizes conceptual understanding and representation, supported by models that promote visualization. Such approaches enable students to achieve deeper comprehension and more durable knowledge.
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