Transcript Trapezoidal representations for an=5 2n

EXPLORING THE HIDDEN
CONTEXT OF PRE-SERVICE
TEACHERS’ INTUITIVE IDEAS IN
MATHEMATICS
Sergei Abramovich
Peter Brouwer
SUNY Potsdam, USA
TSG 29 issues
 How can one establish a link between
the kind of mathematics and the role of
mathematical experiences of pre-service
teachers?
 What are innovative approaches of
developing mathematical content
knowledge of pre-service mathematics
teachers?
Conference Board of the Mathematical Sciences (CBMS). 2001.
The Mathematical Education of Teachers.
Washington, D. C.: MAA.
“College courses … should make
connections between the mathematics
being studied and mathematics
prospective teachers will teach” (p. 7).
CBMS, 2001
Capstone course idea:
helping prospective teachers “make
insightful connections between the
advanced
mathematics
they
are
learning and high school mathematics
they will be teaching” (p. 39).
Connections across the
curriculum
By listening to prospective
teachers’ ideas, mathematics
educators can develop learning
environments that help the
teachers make connections across
the K-16 curriculum
How can elementary
mathematics concepts
motivate learning
environments for the
secondary classroom?
An answer is in using one’s
knowledge of hidden concepts
and structures of mathematics to
make connections across the
curriculum.
Abramovich, S. and P. Brouwer. (2007). How to
show one-fourth? Uncovering hidden context
through reciprocal learning. International Journal
of Mathematical Education in Science and
Technology, 38(6), 779-795.
HIDDEN MATHEMATICS CURRICULUM
A didactic space for the learning of mathematics
where seemingly unrelated concepts emerge to
become intrinsically connected by a common
underlying thread
Technological tools allow for the development of
entries into this space for prospective teachers of
mathematics
DIDACTICAL PHENOMENOLOGY OF
MATHEMATICS:
“A way to show the teacher the places where
the learner might step into the learning process
of mankind”
H. Freudenthal. (1983).
Learning by Transaction
Learning is a transactional process of
developing informed entrants into a culture
with the assistance of more advanced agents
of the culture
J. Bruner (1985)
Learning by transaction creates the Zone
of Proximal Development
L. S. Vygotsky (1978)
Theoretical framework for reciprocal
learning
Mathematical knowledge is a combination of
action, operation, and reflection (von Glasersfeld,
1995)
Mathematical teaching is a process through which
both students and teachers learn
In particular, it is possible for teachers to learn
from students (Steffe, 1991)
Pre-service elementary class:
How to show one-fourth?
One elementary pre-service
teacher’s representation
Representation of 1/n
Possible learning environments (PLE)
Possible learning environment – “a conceptual
generalization a teacher can use in the
creation of learning environments”
Steffe, L.P. 1991. The constructivist teaching
experiment. In E. von Glasersfeld (ed.),
Radical Constructivism in Mathematics
Education.
PLE 1: Iterated pentagons
A GSP
construction
n
t 1 2 + 3  0.649 2
Area D'DD'''' D'''D''
Area D'DD'''' D'''D''
0
2.75
9.01 cm 2
9.01 cm 2
1
3.37
18.03 cm 2
18.03 cm 2
2
3.89
27.04 cm 2
27.04 cm 2
3
4.35
36.05 cm 2
36.05 cm 2
4
4.77
45.07 cm 2
45.07 cm 2
Illustrating “the way in which software can embody
a mathematical definition” (CBMS, 2001, p. 132).
PLE 2: Making connections through
measurement
The Golden Ratio and pentagons
PLE 3: Explaining connections using
complex numbers
A pedagogical implication:
A capstone course for
pre-service secondary
teachers can be built by
uncovering the hidden
context of the
fundamental ideas of
elementary mathematics
Concluding remarks
Mathematics educators should listen to
pre- teachers and take their ideas seriously
PLEs can be developed to help pre-teachers
make connections across the curriculum
Technology use can motivate mathematical
learning
Each day, try to teach something that you
didn’t know the day before
The unity of history, mathematics and
technology addresses the CBMS
recommendations for the preparation of
teachers