Home' Teacher : October 2010 Contents 24 TEACHER OCTOBER 2010
Technologies provide space within which
new mathematical knowledge and practices
emerge. Within this space students and teach-
ers may imagine their relationship with tech-
nology in different ways. To help illustrate
this, some colleagues and I have developed
four metaphors to describe the relationship
between user and technology. Technology
can be a master if students' and teachers'
knowledge and competence are limited to a
narrow range of operations. Students may
become dependent on the technology if they
are unable to evaluate the accuracy of the
output it generates. Technology is a serv-
ant if used by students or teachers only as a
fast, reliable replacement for pen and paper
calculations without changing the nature of
classroom activities. Technology is a partner
when it provides access to new kinds of tasks
or new ways of approaching existing tasks
to develop understanding, explore different
perspectives, or mediate mathematical dis-
cussion. Technology becomes an extension
of self when seamlessly integrated into the
practices of the mathematics classroom.
Another speaker at the recent ACER
conference, Professor Kaye Stacey, and her
colleague Robyn Pierce offer an alternative
representation of the ways in which tech-
nology can transform mathematical prac-
tices. Their pedagogical map classifies 10
types of pedagogical opportunities afforded
by a wide range of mathematical analysis
software. Opportunities to teach differ-
ently arise at three levels that represent the
teacher's thinking about:
the tasks they will set their students --
using technology to improve speed, accu-
racy and access to a variety of mathemat-
classroom inte ractions -- using technol-
ogy to improve the display of mathemati-
cal solution processes and support stu-
dents' collaborative work, and
the subject -- using technology to sup-
port new goals or teaching methods for
a mathematics course.
Sometimes the relationship between
student and technology can change dur-
ing a lesson. A favou rite example I have
for illustrating this fluid relationship comes
from a mathematics class conducted by my
colleague Vince Geiger a number of years
ago. In this particular classroom episode,
Vince asked his Year 11 students to use the
dynamic geometry facility on their graphi-
cal calculators to draw a line √45 units long.
His aim was to encourage students to think
about the geometric representation of irra-
tional numbers. The anticipated solution
involved using the Pythagorean relationship
62 + 32 = (√45 )2 to construct a right-angled
triangle with sides 6 and 3 units long and
hypotenuse √45 units long.
Table 1 summarises the flow of the epi-
sode and how technology was used. In this
episode, technology was initially used as a
servant to perform numerical calculations
that did not lead to the desired geometric
solution. It became a partner when students
passed their calculators around the group or
displayed their work to the whole class to
offer ideas for comment and critique, and
again as a partner when it gave the student
who found the solution the confidence he
needed to introduce his conjectured solution
into a heated small group debate.
If you look at this classroom episode in
terms of Pierce and Stacey's pedagogical
map, you start to see opportunities provided
by a task that links numerical and geometric
representations to support classroom inter-
actions where students share and discuss
their thinking. You also start to see how
technology can provide the kind of con-
ceptual construction kit that Olive, Makar
and co. refer to that can transform students'
mathematical knowledge and practices.
Technology and the curriculum
It's worth considering the extent to which
the Australian national curriculum supports
this transformative view of technology.
While I acknowledge that the curriculum
documents are still in draft form and are far
from finalised, it's been disappointing so far
to see how the role of technology was repre -
sented in the initial consultation versions of
the mathematics curriculum.
The shape paper that provided the ini-
tial outline of the Kindergarten to Year 12
Australian Curriculum -- Mathematics pro -
duced by the interim National Curriculum
Board in 2009 made it clear that technolo -
Table 1: Draw a line √45 units long
Role of technology
Students find the square roots of various numbers.
Students pass calculators back and forth to share and
critique each other's thinking.
Teacher invites student to present calculator work to whole
class. Audience identifies misconceptions about how
calculators display decimal versions of irrational numbers.
Master (prior group
work) then partner
(whole class display
Teacher hint: think about triangles. Students search for
Pythagorean formulation without geometric representation.
Teacher redirects students to consider geometry, not just
numbers. Student interrupts group discussion to propose
geometric solution; passes his calculator around group to
share and defend his solution.
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