Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7-16.
 
The author criticises studies which focus on students’ relationship with and understanding of proof independent of borader educational and social context. She argues that such an approach is locked into producing results which suggest a hierarchy of interaction, such as a proof heiracrchy “with empirical "proof" or procedural validation by action at the bottom, and rigorous deductive argument or relational validation based on premises and properties at the pinnacle” (p. 7). The author suggests that we need to look for more wide ranging influences on student responses as well as how the goals for incorporating proof into the curriculum are operationalised. A social dimension can be advantageously used in explain proofs by getting students to explain to a peer. (To link this idea to something currently trendy, whiteboard tutorials could incorporate validation or explanation of proofs.)
 
The author discusses the role of proof in the current (1997) UK national curriculum, which has an unfortunate implication that “it is now official that proof is very hard and only for the most able” (p. 9), due to the layered and hierarchical nature of the curriculum. Just a side note: all the papers I’ve read so far emphasis the intent student focus on “empirical validation” and the complexities around how that differs from formal symbolic proofs. Some take it as a big problem, some see it as a difference which is unavoidable and both sides have worth, some see it as a step on a developmental path. The author goes on to discuss a study on justification and proof, which looked at students conceptions of proofs and how they constructed proofs, rather than taking a deficit view. The author goes on to discuss interesting data which I don’t have space for here. Finally the author argues that, if we wish students to develop a robust conception of proof and skills in proof construction and validation, such skills development needs to be explicitly included in a scaffolded curriculum. We must also avoid thinking that development of skills related to proof are developed linearly.
 
Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.
 Muller, J. (2009). Forms of knowledge and curriculum coherence, Journal of Education and Work 22(3), 205-226.
 
I found this paper very detailed and rich with interesting information and discussion. My summary here will only touch on selected points of the paper. I was particularly interested in the links between disciplinary knowledge and identity. If you are teaching a “hard pure” subject (such as mathematics, theoretical physics, chemistry) to students who are studying for a “hard applied” disciplinary qualification (such as engineering), how does that difference affect their identity development in the context of the knowledge being valued in the classroom?
 
The author starts with a historical look at the origin of universities and the subjects studied in the early centuries of European universities. These were the liberal arts, divided into the dominant Trivium (grammar, logic and rhetoric) and the less dominant Quadrivium (arithmetic, astronomy, geometry and music). Three branches of philosophy were also taught and law and medicine “were frequently tolerated” (p. 206). There was substantial focus on understanding the world through inner cultivation and inner discipline. In the seventeenth century the precedence of the Trivium over the Quadrivium started to shift as empiricism and science gained in followers. Application, practicality, innovation and research became more greatly valued than previously.
 
The beginnings of what became modern calculus come from both Newton and Leibnitz. Newton’s development was problem-driven and Leibnitz’s was driven more by rationalism and abstract mathematical logic.
“The difference between these two giants lay in their intellectual approach, deductive in Leibnitz’ case, problem-driven in Newton’s. But it lay too with the internal organisation of science as a ‘pure’ discipline as opposed to that of ‘applied’ disciplines … Scientific knowledge grows by the evolution of ever more abstract and general propositions; this is its epistemic destiny, so to speak. Applied knowledge grows through an accretion of practical solutions to particular problems. Of course it can be, and is, retrospectively rationalised in terms of its scientific generalizability. But its raison d’etre is procedures that work; science’s is principles that are true.” (p. 208)
 
The author points to two “fault lines” in the evolution of the disciplines. The first is between that Arts and Humanities on the one hand (inner) and the Sciences on the other (outer). The second is between pure and applied.
 
The author draws on other work to describe a quadrant of discipline types. There is hard pure (natural sciences), soft pure (social sciences), hard applied (science-based professions) and soft applied (social professions). The different quadrants are discussed in the paper with regard to “paradigmicity”, research, supervision and codification. The characteristics of how knowledge is defined, what knowledge is valued and to what degree it is codified or integrated in a knowledge community impacts on how fast students can gain entry into the knowledge community and how quickly innovation can be recognised.
 
“Although these professions [law, medicine, engineering, accounting and architecture] exercise a strong contextual grip over professional training, the university-based trainers have over the time developed an impressive autonomy over their work, and they tend to present a united front to both the academy and the world. The traditional professions have thus evolved a powerful way to develop a robust professional habitus and identity in their practitioners, deep induction into the ‘values of the profession, its standards of professional integrity, judgement and loyalty’ (Beck and Young 2005, 188). Indeed, these professional identities, albeit ‘projected’ from the profession rather than solely ‘introjected’ from the discipline (Bernstein 2000), are similarly stable and robust.” (p. 214, emphasis in original)
 
The author continues by discussing newer professions of various types. In general the newer professions (he names teaching, clinical psychology, social work, tourism, business studies and information science) tend to be more fluidly defined than the older professions, labour divisions are possibly less defined, they are sometimes less well organised, and the knowledge base is less well-defined and does not yet have a core element which transcends context or location. Professional identities are weaker than in the traditional professions.
 
“To see why, it is useful briefly to consider the dual nature of academic identity. Identity is, like many social science objects, Janus-faced: the one is identification, induction into a community of practice, joining a club of those with similar values and competences; the other face is individuation, developing one’s unique niche or ‘voice’, becoming a recognised innovator in an established tradition. The first face points to identity as dependence, conformity to the community’s values and standards; the second points to identity as independence and novelty, setting new standards (Henkel 2000, 2005). A strong academic identity thus binds the social to the cognitive: it means both a strong, stable intellectual or professional community, and a robust means for recognising and generating innovation within it. The one depends on the other.” (p. 214)
“The curriculum planning message here is that disciplinary foundations are one key to strengthening both the identities of adepts and the research activity in the region.” (p. 215)
 
The author discusses how curricular knowledge and disciplinary knowledge are strongly linked but are not the same. The choices that have to go into designing a curriculum describe a body of knowledge which is related to but not the same as that practised by adepts in the discipline. The author goes on to discuss the difference between conceptual coherence and contextual coherence in curricula. Some linkages are:
Conceptual coherence – vertical curriculum – hierarchy of abstraction and conceptual difficulty – sequence matters – regulated by logic – adequacy is internally guaranteed.
Contextual coherence – segmentally connected – sequence matters less – context is a specialised form of practice – adequacy externally guaranteed (by professional body perhaps).
The form of disciplinary knowledge therefore does constrain a well-designed curriculum by insisting on either conceptual or contextual coherence.
 
Asking the question “What kind of graduate should this qualification produce?” has four possible answers in the form of four qualification routes, which the author calls:
Route 1: Academia; fourth generation professions; leads to PhD; e.g. researchers
Route 2: Traditional and some fourth generation professions; leads to Masters and possible PhD; e.g. engineers
Route 3: General occupations; leads to Bachelors via diplomas; e.g. engineering trades
Route 4: Particular occupations; FET qualifications; e.g. travel agents
[The paragraph above is a terse summary of a much longer discussion.]
 
I shall end my summary with this quote:
“… it is regarded as a matter of social justice [that] the curriculum ought, by moral right, to give access to mobility upwards. To do this, a curriculum ought, as a matter of fairness, include abstract knowledge of the kind that would, if desired, grant the academic conditions for access upwards.” (p. 222)
If anything, this view increases the burden on someone teaching a hard pure subject to students of a hard applied discipline. We want our (applied) students to develop disciplinary identities, yet we need to teach them this (pure) subject in such a way that the knowledge they understand to be valued is that valued by the pure discipline should they desire to take that subject further. I walk away from this paper better informed and even more troubled.
 
Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.

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