ABSTRACT
As mathematical thought matures, it is often convenient to map the advance by referring to transitional stages. These can be quite minor and local, for instance when hierarchical levels of understanding for a concept are identified in a relatively stable mathematical context, or can be more far-reaching when the concepts themselves are in flux and different definitions are crafted accordingly to the state reached in a mathematical theory. Transitional stages can be gradual or abrupt. An institutional change tends to accelerate the sophistication in how mathematics is treated, and this is especially pressing when one considers the difference between the character of mathematics taught at high school and that taught at the tertiary level. (In the United States, this institutional change might be best reflected when a student enters the program of senior study at university.) Many students find this transition difficult, finding the style of the presentation of mathematics at university very different to what they were accustomed before (see for example De Guzmán, Hodgson, Robert, & Villani, 1998). It is natural to question whether the institutional change by itself is responsible in creating these difficulties. Students are abruptly faced with the apparent detached character of formalization and new standards set for argumentation that are largely foreign to the school experience. Is there a way to direct students’ learning of mathematical ideas in a “continuous” manner avoiding such “ruptures”? Some educators claim that, indeed, there is a way. For example, Rasmussen, Zandieh, King, & Teppo (2005) advocate the construction of “advancing mathematical activity … to think about the transition from informal to more formal mathematical reasoning” (p. 71). From another study, Marrongelle & Rasmussen (2008), it is clear that this approach is regarded to be more or less universal in application, suggesting it has the potential to instruct mathematics as a “continuum.” From this stance, there cannot be a tight characterization for the term Advanced Mathematical Thinking, a topic espoused by many mathematics educators considering the special needs to cope with tertiary level mathematics.
