Adolescents' understanding of limits and infinity
Monaghan, J. (1986) Adolescents' understanding of limits and infinity. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1447624~S15
AIM To investigate mathematically able adolescents' conceptions of
the basic notions behind the Calculus: infinity (including the
infinitely large, the infinitely small and infinite aggregates);
limits (of sequences, series and functions); and real numbers. To
observe the effect, if any, on these conceptions, of a one year
EXPERIMENTS Pilot interviews and questionnaires helped identify
areas on which to focus the study. A questionnaire was administered to
Lower Sixth Form students with 0-level mathematics passes. The
questionnaire was administered twice, once in September and again the
following May. The A-level mathematicians had received instruction in
most of the techniques of the Calculus by May.
Interviews, to clarify ambiguities, elicit reasoning behind the
responses and probe typicality and atypicality, were conducted in the
month following each administration.
A second questionnaire, an amended version of the first, was
administered to a larger but similar audience. The responses were
analysed in the light of hypotheses formulated in the analysis of data
from the first 5ample.
PRINCIPAL FINDINGS Subjects have a concept of infinity. It exists
mainly as a process, anything that goes on and on. It may exist as an
object, as a large number or the cardinality of a set, but in these
forms it is a vague and indeterminate form. The concept of infinity is
inherently contradictory and labile.
Recurring decimals are perceived as dynamic, not static, entities
and are not proper numbers. Similar attitudes exist towards
infinitesimals when they are seen to exist. Subjects' conception of
the continuum do not conform to classical or nonstandard paradigms.
Convergence / divergence properties are generally noted with
infinite sequences and functions. With infinite series, however,
convergence / divergence properties, when observed, are seen as
secondary to the fact that any infinite series goes on indefinitely
and is thus similar to any other infinite series.
The concept that the hut is the saue type of entitiy as the
finite tens is strong in subjects' thoughts. We coin the term
generic hiuit for this phenomenon. The generic limit of 0.9, 0.99,
is 0.9, not 1. Similarly the reasoning scheme that whatever holds for
the finite holds for the infinite has widespread application. We coin
the term generic law for this scheme.
Many of the phrases used in calculus courses (in particular hut,
tends to, approaches and converges) have everyday meanings that
conflict with their mathematical definitions.
Numeric/geometric, counting/measuring and static/dynamic contextual
influences were observed in some areas.
The first year of a calculus course has a negligible effect on
students conceptions of limits, infinity and real numbers.
IMPLICATIONS FOR TEACHING On introducing limits teachers should
encourage full class discussion to ensure that potential cognitive
obstacles are brought out into the open. Teachers should take great
care that their use of language is understood. A-level courses should
devote more of their time to studying the continuum. Nonstandard
analysis is an unsuitable tool for introducing elementary calculus.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||L Education > LB Theory and practice of education
Q Science > QA Mathematics
|Library of Congress Subject Headings (LCSH):||Infinite -- Study and teaching, Calculus -- Study and teaching, Numbers, Real -- Study and teaching|
|Official Date:||July 1986|
|Institution:||University of Warwick|
|Theses Department:||Institute of Education|
|Supervisor(s)/Advisor:||Tall, David Orme ; Desforges, Charles|
|Extent:||v, 382 leaves|
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