EXPLORING THE HIDDEN CONTEXT OF PRE-SERVICE

TEACHERS’ INTUITIVE IDEAS IN MATHEMATICS

Sergei AbramovichPeter Brouwer

SUNY Potsdam, USA

EXPLORING THE HIDDEN CONTEXT OF PRE-SERVICE

TEACHERS’ INTUITIVE IDEAS IN MATHEMATICS

Sergei AbramovichPeter Brouwer

SUNY Potsdam, USA

TSG 29 issuesTSG 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?

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.

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).

“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.

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.

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

HIDDEN MATHEMATICS CURRICULUM

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

of mankind”

H. Freudenthal. (1983).

DIDACTICAL PHENOMENOLOGY OF MATHEMATICS:

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

Learning by transaction creates the Zone of Proximal Development

L. S. Vygotsky (1978)

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

Theoretical framework for reciprocal learning

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:

How to show one-fourth?

One elementary pre-service teacher’s representation

One elementary pre-service teacher’s representation

Representation of 1/nRepresentation of 1/n

Possible learning environments (PLE)

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.

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.

A GSP construction

n t12+ 3⋅0.649( )2 Area D'DD''''D'''D'' Area D'DD''''D'''D''

0 2.75 9.01 cm2 9.01 cm2

1 3.37 18.03 cm2 18.03 cm2

2 3.89 27.04 cm2 27.04 cm2

3 4.35 36.05 cm2 36.05 cm2

4 4.77 45.07 cm2 45.07 cm2

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

The Golden Ratio and pentagons

The Golden Ratio and pentagons

PLE 2: Making connections through measurement

PLE 3: Explaining connections using complex numbers

A pedagogical implication:

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

A capstone course for pre-service secondary teachers can be built

by uncovering the hidden context of the fundamental ideas of

elementary mathematics

Concluding remarksConcluding remarksMathematics educators should listen to pre- teachers and take their ideas seriously PLEs can be developed to help pre-teachers make connections across the curriculumTechnology use can motivate mathematical learning Each day, try to teach something that you didn’t know the day beforeThe unity of history, mathematics and technology addresses the CBMS recommendations for the preparation of teachers

Mathematics educators should listen to pre- teachers and take their ideas seriously PLEs can be developed to help pre-teachers make connections across the curriculumTechnology use can motivate mathematical learning Each day, try to teach something that you didn’t know the day beforeThe unity of history, mathematics and technology addresses the CBMS recommendations for the preparation of teachers