id: 06649814
dt: j
an: 2016f.00795
au: Fray, R.L.; Sheikh, T.O.
ti: Definitions and their application in writing proofs on divisibility of
integers.
so: Far East J. Math. Educ. 16, No. 1, 111-133 (2016).
py: 2016
pu: Pushpa Publishing House, Allahabad, Uttar Pradesh, India
la: EN
cc: E50 F60 D70
ut: definitions; proofs; divisibility of integers; skeleton proofs; dialogue
ci:
li: http://www.pphmj.com/abstract/9640.htm
ab: Summary: Beginning students in algebra classes have little experience and
struggle to write proofs. The aim of this study was to look at
studentsâ€™ use of definitions in determining the validity of
statements and in writing proofs for given propositions on divisibility
of integers. The main objectives were to highlight the difficulties
students have with proofs and suggest strategies to help them in
writing proofs.{ }In the first part of the study the class was given
the definition of divisibility and problems were set to which written
responses were obtained. In the second part of the study, divisibility
problems were set and students were guided through their proofs using
dialogue. The final set of tasks involved skeleton proofs to support
the writing of proofs.{ }We found that all the students had
difficulties with problems involving generalised variables as opposed
to numerical tasks. There was conflict between the new definition and
preconceptions and prior knowledge. Students had language problems,
problems of division by zero, and were reluctant to use proof by
counter-example or contradiction when no direct proof was possible.{
}The implications are that new definitions must be introduced
carefully, using a wide range of examples and appropriate prompts.
Dialogue and skeleton proofs are two strategies that can be used to
improve studentsâ€™ understanding of proofs.
rv: