The
observation in the grade 2 classroom suggested several ways in which the PRISM
model could be improved, and some features of mathematical activities that
seem to support deductive reasoning.
The
1995 version of the PRISM model distinguishes between types of deductive
reasoning only in the degree of formulation of the reasoning, with one
exceptions. Mechanical
deduction describes reasoning that is automatic.
The reasoner is unaware of the reasoning, but not out of lack of
self-awareness, but rather because the details of the reasoning have been
submerged in a process that is known to have the same result but which does
not require conscious attention. Much
of mathematics education is devoted to developing mechanical deduction, such
as the practice of arithmetic facts and algorithms to make them automatic, and
the practice of algebraic manipulations for the same purpose.
Other forms of mechanical deduction include the use of calculators and
software to perform calculations.
In the Grade 2 class the following sub-types of deductive reasoning were observed: Simple one step deductive reasoning, Simple multi-step deductive reasoning, Specialisation, Specialisation with multiple attributes, Hypothetical one step deductive reasoning, and Hypothetical multi-step deductive reasoning. These categories will now form part of the scheme under which the reasoning of older children are analysed. It should be said, however, that in the case of older children this level of detail may not be described, simply because the complexity of the reasoning of older children
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Psychology of Students' Reasoning in School Mathematics
David A Reid
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means that detailed descriptions might be difficult to understand and
so undermine their own usefulness.
A further addition to the model suggested by this study is a distinction between Surface and Structural Analogies. In previous work (e.g., Reid 1995) a distinction was made between strong and weak analogies, based on the number of common features shared by the source and target. IN this case the distinction is between analogies whose shared features are immediately apparent, without understanding, and those that are based on an understand of the structures of the two situations.
Several
features were found that seemed to support the deductive reasoning of the
grade 2 students. Teacher
questioning, especially questioning demanding explanation, was the main
feature, however the nature of the activity itself mattered a great deal. For activities that were focussed on pattern noticing the
frequency of inductive reasoning was higher, meaning that deductive reasoning
was less frequent. For activities
that involved complex (by Grade 2 standards) calculations or basing
conclusions on multiple pieces of information, multi-step deductions were more
common. This suggests that to
encourage deductive reasoning in early elementary grades, the activities in
which students are invited to engage should involve interpreting patterns
rather than noticing them in the first place.
An emphasis on pattern noticing in the curriculum may in fact interfere
with the development of deductive reasoning (for a similar conclusion arrived
at in a very different context see Healy & Hoyles, 2000, p. 409).
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Psychology of Students' Reasoning in School Mathematics
David A Reid
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Work remains to be done on the first research question of the PRISM project, which will involve integrating material from this study with others being done at the grade ten, seven and five levels.
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Psychology of Students' Reasoning in School Mathematics
David A Reid
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