To practitioners wrestling with the discussion of maths etc.
I think the discussion of pure maths & applied etc falls into the differences between Expert & Novice differences as follows:
Being uninitiated in the conversations and conventions of academic discourse, many students enter courses of study not knowing what actually transpires when one writes or thinks in the culture of the academic discipline. Although they are acquainted with some of the artifacts (e.g., textbooks, lectures, term papers, lab manuals), they are not familiar with the linguistic and cognitive processes used when producing such artifacts nor with the superordinate knowledge structures underlying the academic discipline.
Expert and Novice Differences
We know that specific knowledge domains represent particular ways of mapping out the world so that each general field of study has developed specific epistemologies, i.e., superordinate knowledge structures. In fact, related studies investigating the differences in mental processes between discipline-specific experts and novices indicate that experts acquire knowledge through internalized schema-dependent taxonomies and approaches, whereas novices do not (Larkin, 1980; Chi, 1981; Greeno, 1983; Smith, 1992; Tudor, 1992; Bookman, 1993; Zeitz, 1994; Byrnes, 1995). In other words, the difference between experts and novices is not just the amount of knowledge they possess but more importantly the way that knowledge and information is organized and processed.
Historians, for example, examine primary sources as they arrive at coherent interpretations of events. These interpretations involve deductive reasoning, question formulating and high-level inferential conjecturing. Yet, their results are commonly summarized in texts written in a narrative style. These narrative summaries are, then, presented in lectures as declarative information. Needless to say, most students are hard pressed to "unpack" the superordinate knowledge structures understood by the experts or truly ascertain what Lyotard (1984) calls metanarratives, those underlying foundational stories or beliefs. Literature and the performing arts, also, rely on various depths of understanding and appreciation that emulate the cognitive processes of professionals. And, like the humanities, science and mathematics also have schema-dependent content requiring semiotic mediation to facilitate learning. Because novices lack the superordinate knowledge structures discipline experts use when writing and thinking, these newcomers need to learn how to learn within specific knowledge domains.
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Another important feature is that development repeats the stages already passed, but repeats them in a different way, on a higher plane; development advances, so to speak, in spirals, not in steps. These higher psychological processes form as new psychological systems and are not superimposed as a second story over the elementary processes, nor are they superimposed as disparate hierarchical steps as held by "mental discipline" learning theory advocates. More importantly, these new cognitive and affective systems transform the meanings of the lower ones.
It is here that the thought processes of the consummate expert and the uninitiated novice diverge. The subject-matter expert uses decontextualized reasoning, where the novice uses more contextualized thought processes. This variance is an important distinction all teaching professionals need to remember. They need to remember that because their students--at least in the initial stages of learning subject-matter--don't think in discipline-specific ways, the learning facilitators need to initiate them into the conversations and conventions of academic thought.
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Successful postsecondary pedagogy capitalizes on the interpersonal/ intrapersonal, sociohistorical, spiraling recursive nature of thought/language development as professors facilitate discipline-specific learning through the stages of ZPD. As their students interiorize more complex understandings and discipline-specific taxonomies, higher level decontextualized principles are presented. It is at this point that students move from mere performance to competence. As these newcomers leave the zone of proximal development and enter the domain of actual development, they move to a higher intrapsychological plane.
Thus a fact hitherto underrated by psychology becomes
apparent: sociohistorical shifts not only introduce new
content into the mental world of human beings; they also
create new forms of activity and new structures of cognitive functioning. They advance human consciousness
to new levels. (Luria, 1976, 163)
For what's it's worth.
Michel E.
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