Mathematical Anticipation and Representational Coordination Within Van Hiele Levels: Evidence From Pre‐Service Teachers Learning Linear Equations

Source abstract

ABSTRACT This research examines how pre‐service mathematics teachers coordinate algebraic and graphical representations when solving linear‐equation problems. Particular attention is given to mathematical anticipation, including predictions, intermediate results, and verification as participants move between representations. A qualitative interpretive design was used with 126 pre‐service mathematics teachers working in 15 heterogeneous groups. Each group included participants representing the five van Hiele levels of geometric thinking: visualization, analysis, informal deduction, deduction, and rigor. Written and graphical solution artifacts from all groups were examined, and retrospective interviews were used to clarify participants' reasoning. Three representative cases were selected to capture distinctive patterns in the artifacts. The analysis identified three patterns of reasoning: separation of algebraic and graphical representations, emerging but incomplete coordination, and integrated coordination supported by prediction and verification. Across the cases, mathematical anticipation ranged from limited expectations to predictions that guided subsequent actions and were checked against information from other representations. The results indicate that spatial reasoning in linear‐equation problem solving involves more than producing or interpreting graphs. It also involves coordinating representations and using anticipation to guide and evaluate mathematical reasoning. The study provides a process‐oriented account of reasoning within heterogeneous groups organized according to participants' van Hiele levels.