Home' Teacher : November 2010 Contents PROFESSIONAL DEVELOPMENT 7
The polarities that often characterise educa-
tional discussion -- 'student centred' versus
'teacher centred,' for example, or 'abstract
thinking' versus 'concrete contexts,' tend
to freeze us into less than helpful positions.
That's the conclusion of David Clarke, who
in 2006 reviewed what we've learnt from
10 years of international research in maths
education, including his international video
study of well-taught mathematics in class-
rooms in 16 countries.
Clarke, the Director of the International
Centre for Classroom Research in the Mel-
bou rne Graduate School of Education at
the University of Melbourne, points out
that, in practice, the complexity of teach-
ing and learning readily runs across these
categories, and is less easily pigeonholed
than those terms suggest.
Another international comparison that
questions these categories is Lipang Ma's
1999 Knowing and Teaching Elementary
Mathematics, in which she compares prac-
tice and approach in classrooms in the United
States and China. While some US research-
ers have suggested that teaching algorithmic
processes interferes with children's natural
development of understanding, and should
be abandoned in favour of more immersive
and exploratory approaches, Ma fou nd
that many Chinese teachers foster a richer
understanding of elementary maths along
with their teaching of the algorithm.
The Chinese teachers in Ma's study pro -
mote a detailed examination of algorithmic
processes for, say, long multiplication or
column subtraction, as well as a strong con-
ceptual understanding of why these work.
Thus a close look at these classrooms refutes
the common idea that procedural or 'algo -
rithmic thinking' is the natural enemy of
conceptual understanding, as though they
are in competition. Similarly the simple
opposition of 'traditional' and 'progres-
sive' approaches -- though well cited in the
literature -- is more often than not used in
a rhetorical manner to lambast one side or
the other. In reality, things are often more
complex, and effective teaching and learn-
ing can draw on all approaches.
As Clarke observes, 'constructing such
dichotomies (as teacher centred or student
centred and so forth) as oppositional creates
a set of false choices, sanctifying one alterna-
tive, while demonising the other.' He argues
a more inclusive view is needed. In the US,
the tension in these debates developed into
such acrimony that the term 'Math Wars'
was coined to describe the furore. Yet both
Clarke and Ma's detailed studies of actual
classes refute the claim that one 'side' or
approach is universally more effective.
At the most recent conference of the
Mathematics Education Research Group
of Australasia, Peter Grootenboer and Julie
Ballantyne presented a paper that looked at
the practice of eight excellent maths teachers,
as rated by peers and students. They noted
that those teachers used a very diverse range
of styles and methodologies, from highly
teacher directed, with public ranking of stu-
dent performance, to predominately group-
work-based approaches where the teacher
gave only minimal feedback. Common
elements across these classes and teachers
existed less at the level of broad classroom
practice, such as 'they all used student-
directed approaches,' and more at the level
of teacher identity -- including the teacher's
strong sense of mathematical identity. Stu-
dents also reported on their teacher's unique
personality and relationship with their class.
All these studies indicate that identifying
the definitive 'best' classroom approaches
from educational research is difficult, and
that, given the diversity of practice in effec-
tive classrooms, we need a more integrated
model that goes beyond the oppositional
narratives that characterise much discussion
and indeed research.
An integrated model
A thumbnail sketch of a more integrated
model is found in Alfred Whitehead's clas-
sic Aims of Education. Whitehead -- him-
self a deep mathematician -- proposed that
learning passes through stages. First, there's
'romance,' where the general idea and outline
of the subject holds appeal, for example, the
child who is drawn into an appreciation of
music or science, though still largely unaware
of many of the technical details. Exposure
to the broad strokes and potential of the
domain is the goal in this stage. Then fol-
lows a necessary period of 'precision,' which
focuses on intensive development of techni-
cal competence and accuracy. This in turn is
followed by 'generalisation,' which yields a
more mature appreciation of the field, now
informed with technical skill. Whitehead
saw these three phases of romance, precision
and generalisation simultaneously apply-
ing to different time scales; they are both
large-scale phases a learner passes through
over several years, and are also a model for
smaller units of work; and ideally every les-
son should have something of this dynamic.
'Romance' and 'precision' are of course
somewhat opposed; Whitehead was realistic
enough to observe that it's a difficult chal-
lenge to take a whole class very far down the
path of precision without dulling some of the
'romance.' Yet reconciling all these dynamics
is a richer model of real-life classes than what
appears to be the default polarised model.
'And' not 'or'
'Do we learn best by instruction, or doing?'
'Is worked example or self discovery
SHOULD MATHS TEACHING BE 'STUDENT CENTRED' OR 'TEACHER CENTRED,'
FOCUSED ON 'ABSTRACT THINKING' OR 'CONCRETE CONTEXTS'? THE ANSWER,
SAYS ROB COSTELLO, IS SIMPLER THAN YOU MIGHT THINK, SO LONG AS YOU'RE
PREPARED TO QUESTION THE USUAL DICHOTOMIES.
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