Proof in the curriculum: an overview
John Mason
Open University - UK
A meeting recently took place on the role of reasoning and proof in
mathematics classrooms, organised by the Qualifications and Curriculum
Authority (QCA) for England. This was in some ways a landmark meeting
in that a quasi-autonomous non-governmental organisation (QUANGO) which
was set up to advise the Minister of State for Education on qualifications
and curriculum, chose to invite experts to discuss a topic which the
QCA consider to be important but which has not yet been signalled by
the Minister as requiring attention. Their aim was to seek advice and
information form a range of countries concerning possible directions
for revisions to the U.K. National Curriculum. Since the QCA also tests
children at the ages of 7, 11 and 14, and provides optional tests for
all other years, they do more than just give advice: they actually formulate
the National Curriculum and chart learner progress.
So it was that a rather bemused group of people with interests and concerns
about teaching mathematical reasoning in schools, met in Cambridge over
two days. Fourteen countries were represented: Australia, Bulgaria,
Canada, Czechoslovakia, France, Italy, Israel, Japan, Mexico, Poland,
Russia, Singapore, UK, USA.
Some colleagues had been asked to describe what was done regarding reasoning
and proof in their school curricula. The overwhelming impression was
that despite differences of emphasis, of approach and of orientation,
success has not been universally great. Where reasoning has been deeply
embedded in the ethos of the classroom, learners do appear to be able
both to recognise and to construct reasoned justifications for assertions
and conjectures. But where concentration has been largely in one domain
(perhaps algebra, perhaps geometry) there is relatively little cross-over
into the other domain. Where reasoning has been bolted on or assigned
to a specific part of the curriculum, a majority of learners failed
to demonstrate significant grasp of the idea of mathematical reasoning.
Some colleagues were asked to address general issues concerning reasoning.
Emphasis was on psycho-social aspects of epistemological development,
and on the construction of tasks which promote mathematical thinking.
A number of questions arose which need to be addressed and for which
further research is necessary, including
What needs to be done do prompt children to develop from accepting
the authority of outside sources (adults, institutions), to acting
responsibly (literal meaning: to be prepared to justify one's actions
or assertions) by taking on the mantle of authority for themselves
through being able to justify their assertions-claims-conjectures;
to develop from "just because!" and "it just is"
as warrants, to reasoned justifications?
How is it that children can spontaneously use counter-examples in
social discourse, but rarely do in mathematics unless specifically
encouraged?
How can children be encouraged to make use of writing and software
as a means of expression through which they refine their explanations
and justifications; moving from convincing themselves, to convincing
a friend who is present so that alterations and further justifications
can be offered as needed, to convincing someone who is not present
and so not relying on responding to their questions and concerns but
rather envisaging what these might be and addressing them?
How can an ethos of scientific-debate, of mathematical thinking, of
a conjecturing and probing atmosphere become the norm in every classroom,
not just in a few classes?
How might learners be encouraged to incorporate if…then…
reasoning into their ordinary and mathematical language?
How can tasks be constructed which promote a sense of need to convince
others (through unexpected or non-intuitive results, among others)?
Distinctions made included
Pragmatic and Formal-Academic reasoning;
Proof as Explanation and proof as Justification, which relates to
the source of authority and warrants for conviction;
Empirical Induction (perhaps developing into exhaustion by cases);
Deduction based on firm foundations; Abduction and Insight requiring
further justification; Deduction based not on formal foundations but
on accepted 'facts';
Must, May, and May-not, as possible states, related to notions of
for-all and the nature of certainty and conviction;
Social reasoning and Mathematical reasoning (role of counter-example,
empirical evidence, provision of warrants, etc.)
It was noted that in the domain of number and algebra, the transition
from arithmetic thinking (working from the known towards the as-yet-unknown)
to algebraic thinking (using the as-yet-unknown or the general to express
structural relationships and then using symbol manipulation to find
values for unknowns) is a significant hurdle for many learners. A popular
strategy in the face of a problem is to use guess-and-test, which can
be refined into trial-and-improvement, with the improvements made more
and more systematically and with reference to structures in the problem,
but there is still a gap between this and the use of algebraic language
to express generality and-or structure. Spreadsheets offer promising
support for making such a transition.
It was noted that whereas some countries are drawing back from specific
attention to mathematical exploration and group work, others are embarking
on emphasising these components of learners' experiences.
Some colleagues were asked to describe particularly the role and impact
of technology with respect to reasoning and proof. Evidence was offered
that spreadsheets provide a tool in which aspects of reasoning can be
developed, especially in the transition between arithmetic and algebraic
thinking through the use of cell-references as a form of symbolic expression
and manipulation.
There was widespread agreement that we did not want to see curricula
specified which encouraged teachers to get learners to memorise proofs,
or otherwise mechanise the use of reasoning. It was noted that this
does run counter to the force from teachers and learners for ever-more
precisely specified problem types so that learners can succeed through
familiarity with similar worked examples.
Although not discussed at length, many were concerned with equity issues,
in that while access to mathematical reasoning must be provided for
all future citizens, this does not mean that everyone develops at the
pace of the slowest.
The meeting ended with participants discussing three questions chosen
by the QCA representatives:
1. How can pedagogic structures such as
scientific-debate;
personal-work-group-work-leading-to-a-poster-collective-work-around-posters-synthesis-and-overview;
group-work-sharing-synthesising-extending-or-varying
and many variants become standard pedagogical structures in all classrooms?
2. The intended curriculum is attenuated by the taught curriculum
which is in turn attenuated by the assessed curriculum. What forms
might assessment take which focuses on reasoning without promoting
rote memorisation and without promoting attempts to mechanise specific
reasoning structures?
3. (Most urgent for UK). The Government will be instituting a programme
for talented and gifted learners (at all ages) (say top 10%): what
could be offered in place of simple acceleration, particularly in
regard to reasoning?
Another way of thinking might be to ask
What could be sensibly and effectively implemented more
or less immediately with suitable materials?
What could be sensibly and effectively implemented with suitable professional
development for all mathematics teachers?
What could be sensibly and effectively implemented through curricular
change and different assessment over the longer term?
If you want to see what was suggested, visit the QCA website (http://www.qca.org.uk)
when the papers for and from the meeting are released.
©
John Mason
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