The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating and monitoring students' developing conceptual understanding of matrices and systems of linear equations in an undergraduate mathematics course. The study also explored the tools' role in scaffolding and facilitating students' critical and conceptual analyses of problems in order to identify potential methods of solutions. Data collected included students' progressive maps/diagrams, journals of reflections and justifications of revisions, and final reports and researchers' annotated comments on students' maps/diagrams and anecdotal notes from presentations. Findings showed that students developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. Findings suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these metacognitive tools into students' repertoire of effective learning strategies.
Current emphases in national and state curricular frameworks urge the promotion of deep knowledge and deep conceptual understanding of students as well as enhancing students' abilities and skills in working, thinking and communicating mathematically. To achieve these content and process outcomes, mathematics teachers are encouraged to be innovative, investigative and explorative in their pedagogical approaches to designing and developing learning activities (NCTM, 2000; NSW 2002). External examination reports (MANSW, 2005) indicate that a high proportion of students have difficulties comprehending the meanings of key concepts in the context of problems, justifying solutions, and presenting coherent mathematical arguments. Furthermore, first year university students' mathematical performances (Mays, 2005) in diagnostic tests show that most have mathematical misconceptions with fractions, percentages and multi-digit subtraction. Similarly, national surveys in Samoa confirm that learning by rote-memorization is quite prevalent in most schools (DOE, 1995). Such findings resonate with recurring comments in examiners' reports concerning students' obvious inabilities to effectively apply existing knowledge to successfully answer exam questions (Afamasaga-Fuata'I, 2001, 2002a, 2002b, 2002c, 2003, 2005a, 2005b).
In foundation and undergraduate mathematics classes in Samoa, students find it difficult to explain and justify their answers mathematically in terms of the conceptual structure of relevant topics. Instead their verifications are often in terms of sequences of steps of procedures. Whilst this may work for familiar problems, this procedural view constrains them when solving qualitatively and structurally different problems (i.e., novel problems). According to Richards (1991), this manifestation is typically a communication problem resulting from students' inability to understand the meaning of a language (i.e., concepts, principles, theorems and theories) used in mathematical discussions and dialogues of more mathematically literate others. Subsequently, less mathematically literate students are unable to make sense of such conversations, offer conjectures or evaluate mathematical assumptions. When doubtful, students tend to use any procedure to get an answer without really checking whether an algorithm is suitable to the problem (Schoenfeld, 1996). …