EXPLORING THE HIDDEN CONTEXT OF PRE-SERVICE TEACHERS’ INTUITIVE IDEAS IN MATHEMATICS
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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?

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Tsg 29 issues
EXPLORING THE HIDDEN CONTEXT OF PRE-SERVICE TEACHERS’ INTUITIVE IDEAS IN MATHEMATICS Sergei AbramovichPeter BrouwerSUNY Potsdam, USA


Tsg 29 issues
TSG 29 issues INTUITIVE IDEAS IN MATHEMATICS

  • 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?


Tsg 29 issues
Conference Board of the Mathematical Sciences (CBMS). 2001. INTUITIVE IDEAS IN MATHEMATICS 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).


Tsg 29 issues

CBMS, 2001 INTUITIVE IDEAS IN MATHEMATICS

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).


Tsg 29 issues

Connections across the curriculum INTUITIVE IDEAS IN MATHEMATICS

By listening to prospective teachers’ ideas, mathematics educators can develop learning environments that help the teachers make connections across the K-16 curriculum


Tsg 29 issues

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.


Tsg 29 issues

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.


Tsg 29 issues

HIDDEN MATHEMATICS CURRICULUM one-fourth? Uncovering hidden context through reciprocal learning.

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


Tsg 29 issues

DIDACTICAL PHENOMENOLOGY OF MATHEMATICS: one-fourth? Uncovering hidden context through reciprocal learning.

“A way to show the teacher the places where the learner might step into the learning process of mankind”

H. Freudenthal. (1983).


Tsg 29 issues

Learning by Transaction one-fourth? Uncovering hidden context through reciprocal learning.

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)


Tsg 29 issues

Theoretical framework for reciprocal learning one-fourth? Uncovering hidden context through 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
Pre-service elementary class: one-fourth? Uncovering hidden context through reciprocal learning. How to show one-fourth?


One elementary pre service teacher s representation
One elementary pre-service teacher’s representation one-fourth? Uncovering hidden context through reciprocal learning.


Representation of 1 n
Representation of 1/ one-fourth? Uncovering hidden context through reciprocal learning. n


Possible learning environments ple
Possible learning environments (PLE) one-fourth? Uncovering hidden context through reciprocal learning.

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
PLE 1: Iterated pentagons one-fourth? Uncovering hidden context through reciprocal learning.

A GSP construction

Illustrating “the way in which software can embody a mathematical definition” (CBMS, 2001, p. 132).


Tsg 29 issues

PLE 2: Making connections through measurement one-fourth? Uncovering hidden context through reciprocal learning.

The Golden Ratio and pentagons


Tsg 29 issues

PLE 3: Explaining connections using complex numbers one-fourth? Uncovering hidden context through reciprocal learning.


A pedagogical implication

A pedagogical implication: one-fourth? Uncovering hidden context through reciprocal learning.

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

Concluding remarks one-fourth? Uncovering hidden context through reciprocal learning.

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