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possibility problem appears. It is concentrated on the issue of whether or not
it is in fact possible to give mathematics education to the students of the cat-
egories considered, while satisfying the purposes and goals expressed in the
answers to the "why" question. So, the possibility problem contains mathe-
matical components such as the aspects of mathematics that are actualized
by the arguments put forward to justify mathematics education, including
the specific aims and objectives of mathematics teaching and learning en-
MOGENS NISS
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MATHEMATICS IN SOCIETY
tailed by these reasons. On this basis, psychological components are evi-
dently particularly crucial – "who can learn what? On which conditions, and
under what circumstances?" The same is true with the boundary conditions
and prerequisites necessary for pursuing (and achieving) the overall pur-
poses and goals of mathematics education, as well as the specific aims and
objectives of mathematics teaching and learning.
Last, but not least, enters the implementation problem. Assuming that the
justification problem and the possibility problem have been tackled, the im-
plementation problem deals with establishing the structural and organiza-
tional framework within which mathematics education is to take place. It
further deals with providing the immaterial resources (e.g., content, curric-
ula, pedagogy, teaching methods, teacher education, working forms), the
human resources (teachers, consultants, mathematics educators) and the
material resources (classrooms, textbooks, technology) for the realization of
mathematics education. The implementation problem also includes issues
related to the philosophy and modes of assessment. In other words, the im-
plementation problem focuses on the questions of "how?" and "what?" As
there is a continuum of answers to these questions, varying with, and de-
pending strongly on, the concrete circumstances, this problem is of a less
universal nature than the other two.
It is important to note that these three problems represent an analytical
reconstruction. Society does not normally see, articulate or tackle them as
they are stated here. The point is that no educational system that provides
mathematics education can avoid dealing with these problems directly or
indirectly, and that explicit or implicit versions of them constitute the main
driving forces of reform in mathematics education.
When one considers how these three problems have been tackled as a
function of place and time, it appears that conflicting sets of answers exist,
not only as regards the implementation problem, where differences would
be expected, but also with respect to the more fundamental justification and
possibility problems. For instance, it is an often observed phenomenon that
the political and administrative authorities in a society give answers that dif-
fer considerably from those suggested by the majority of mathematics
teachers and educators in that society, who, in turn, may well be in dis-
agreement with the dominant views of research mathematicians, while many
people in the arts, humanities and "soft" social sciences share a fourth set of
views of mathematics education.
The dominant interest of society at large in relation to mathematics edu-
cation is to provide for the utilization, maintainance and development of
mathematics as an applied science and as an instrument for practice as
means for technological and socioeconomic development, with the ultimate
purpose of increasing the material wealth in society. Herein lies, in most
countries, the general answer to the problem of justifying mathematics edu-
cation for the general population. However, because it is recognized that all
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this presupposes that mathematics thrives as a pure science, society has a
derived interest in providing as advanced a mathematics education as can be
afforded to a smaller number of students.
So, society aims at supplying itself with mathematical expertise. On the
other hand, society does not consist only of system components and anony-
mous power centres but also of individual human beings on behalf of whom
society holds ideals, visions and beliefs. Therefore, societies further take an
interest – of a varying degree – in providing mathematical prerequisites to
the population at large to master their private and social lives as individuals
and citizens. This is often called numeracy, matheracy (D'Ambrosio, 1985),