Introduction: The achievement goals framework has been researched and used to explain and account for individuals' learning and academic achievements. Over the past three decades, progress has been made in the conceptualizations and research development of different possible theoretical models of achievement goals. Notably, in this study, we explored the 2 x 2 achievement goal model. Method: Expanding this area of inquiry, and unlike previous research studies, we used an open-ended survey to explore the definition (mastery-performance) and valence (approach-avoidance) dimensions of achievement goals. Furthermore, differing from previous methodological approaches, we situated the items for students' responses in the context of secondary school mathematics learning. One hundred and sixty-nine (75 girls, 94 boys) Year 12 students participated in this research study. Open-ended responses were coded and analyzed using the software package, SPSS Text Analysis for Surveys 2.1. Results: Thematic analyses resulted in the categorization and identification of key words and themes. Frequency count arising from content analyses also indicated the potency of themes such as future development, external influences, and personal gratification. Discussion: By means of extrapolation and interpretation, we derived five refined facets for discussion: future development and aspirations, relevance and uniqueness of subject, extraneous forces, personal attributes, and attribution/motivation. (Contains 3 footnotes, 2 tables and 1 figure.)