Mathematical warrants, objects and actions in higher school mathematics.
'Higher school mathematics' connotes typical upper secondary school and early college
mathematics. The mathematics at this level is characterised by moves to (1) rigour in
justification,(2) abstraction in content and (3) fluency in symbolic manipulation.
This thesis investigates these three transitions - towards rigour, abstraction, and tluencyusing
philosophical method: for each of the three transitions a proposition is presented and
arguments are given in favour of that proposition. These arguments employ concepts and
results from contemporary English language-medium philosophy and also rely crucially on
classroom issues or accounts of mathematical experience both to elucidate meaning and
for the domain of application. These three propositions, with their arguments, are the three
sub-theses at the centre of the thesis as a whole.
The first of these sub-theses (1) argues that logical deduction, quasi-empiricism and
visualisation are mathematical warrants, while authoritatively based justification is
essentially non-mathematical. The second sub-thesis (2) argues that the reality of
mathematical entities of the sort encountered in the higher school mathematics curriculum
is actual not metaphoric. The third sub-thesis (3) claims that certain 'mathematical action'
can be construed as non-propositional mathematical knowledge. The application of these
general propositions to mathematics in education yields the following: 'coming to know
mathematics' involves:(1) using mathematical warrants for justification and self conviction;
(2) ontological commitment to mathematical objects; and (3)developing a
capability to execute some mathematical procedures automatically.