In all these he is using a mathematical mode of description based upon, though somewhat changed from, the proportional geometry of Euclid, Book VI and Archimedes (for details on the change see Palmieri 2002).
Based on analyzing mathematical mode of the permanent magnet linear synchronous motor (PMLSM), three-closed-loop control system is presented in this paper.
Mathematical modes of enquiry, in contrast, tend to point to mathematical practices as fundamental to developing mathematical understanding.
The analysis suggests that the presence of, even multiple elements of, mathematical practices cannot necessarily be considered equivalent to take up of mathematical modes of enquiry.
The data suggest that the presence of elements of mathematical practices cannot be viewed as equivalent to the presence of mathematical modes of enquiry.
Watson and Barton (2011) might consider this outcome to be the consequence of failing to view mathematical modes of enquiry as fundamental to all mathematical working.
In this 'separate and integrative' frame, where mathematical practices are selectively, rather than fundamentally, drawn upon, it would appear that a possible outcome of sporadic absence of explicit attention to mathematical practices is the disruption of mathematical modes of enquiry.
Christopher's story suggests that our more 'separate and integrative' presentation of mathematical content and practices was not sufficient for eliciting broad-ranging moves towards mathematical modes of enquiry within the time frames that we had available.
The analysis suggests that while elements of mathematical practices can be developed, moving towards mathematical modes of enquiry may require more explicit, consistent, and longitudinal attention to problem-solving approaches to support orientational change.
As noted already, sporadic responses from other teachers across all three years indicate that presences of aspects of mathematical practices allied with absences in relation to an overarching orientation to mathematical modes of enquiry were more broadly prevalent.
Watson and Barton (ibid), however, tend to disagree with this view, arguing that curriculum lists are: 'hypothetical until mathematical modes of seeking, using, and exemplifying understanding are understood' (p. 66).
