Anne Watson. University of Southampton. January 2013. Working mathematically. For what values of . a. 0. . does this sequence have a limit?. a. n. +1. = (. a. n. 2. + 1. )/. 2. . . (i.e. is increasing and bounded above). ID: 576643
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The role of examples in mathematical reasoning
University of Southampton
For what values of
does this sequence have a limit?
(i.e. is increasing and bounded above)Slide3
an example is a particular case of any larger class about which students generalise and reason: concepts, representations, questions, methods etc.
(Watson & Mason)Slide4
How do we use examples?
Different kinds of relationship between examples and what they exemplify
specific instantiations of a previously defined class
genesis for identifying an uncharacterised class
Human agency: intention, dispositionSlide5
How does the
motivate definitions and build a sense of what is going on
are “standard cases” that link concepts and results, and are returned to again and again
indicate generic cases and can be copied or used to generate specific instances
sharpen distinctions and definitions of concepts
Examples – for: action or understanding
: counter-examples generate enquiry into new classes
Goldenberg and Mason: depends on attention and emphasisSlide7
Do different purposes indicate different dimensions of variation and ranges of change in examples?
How do people act on examples?Slide8
Analysing (Watson & Chick ZDM)
(an example of a sequence)
Generalising (Watson & Chick)
is 0, ½, ⅓
generalise to unit fractions?
Abstracting from examples (Watson & Chick)
converges when 0 <
≤ 1 and diverges when
treating it as a new entity makes it an
example – for,
e.g. for developing techniques
involves seeking plausible relations between elements of an example, from which conjectures might be
involves describing similarities among
: abstraction goes further and classifies similar examples, naming the similarity as a concept or class with its propertiesSlide12Slide13
Inductive generalisation (Bills & Rowland)
from patterns in sequential
expression of underlying structures or procedures, which could have arisen through analysis.Slide14
Zara’s use of examples (Watson & Chick)
a class beyond obvious
examples: construct new cases
Indicate a class – but subclass does not represent class
and sets of examples
classes - layout invites
used to generate others
Examples which express
emplates to deal
with other class
Sets of examples
Sets of examples
superficial (possibly incorrect)
as situations in which to develop
New example purposes
examples used to build other examples
examples that afford a shift of focus (new ways of thinking)
deductive (existence and counter-example)
abstract – a formative act (Harel)
new for whom?
deductive: symbolic manipulationequivalence
reasoning about properties
reasoning about structure
sets of examplesSlide17
, G. & Tall, D.
general, the abstract, and the generic in advanced mathematics,
For the Learning of Mathematics
, 11 (1), 38-42
Mason, J., &
, D. (1984). Generic examples: Seeing the general in the particular.
Educational studies in mathematics
Watson, A. & Chick, H. (2011). Qualities of examples in learning and teaching.
43 (2) p283-294.
Watson, A. & Mason, J. (2005).
Mathematics as a constructive activity: Learners generating examples.
Another situation ...
For what pairs of numbers can 48 be the LCM?
For what triples?
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