1. EXPLORING THE HIDDEN CONTEXT OF PRE-SERVICE TEACHERS’ INTUITIVE IDEAS IN MATHEMATICS Sergei Abramovich Peter Brouwer SUNY Potsdam, USA

2. 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?

3. Conference Board of the Mathematical Sciences (CBMS). 2001
Conference Board of the Mathematical Sciences (CBMS) 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).

4. 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).

5. 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

6. 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.

7. Abramovich, S. and P. Brouwer. (2007). How to show one-fourth
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),

8. 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

9. 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).

10. 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)

11. 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)

12. Pre-service elementary class: How to show one-fourth?

13. One elementary pre-service teacher’s representation

14. Representation of 1/n

15. Possible learning environments (PLE)
Possible learning environment – “a conceptual generalization a teacher can use in the creation of learning environments”
Steffe, L.P The constructivist teaching experiment. In E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education.

16. PLE 1: Iterated pentagons
A GSP construction
Illustrating “the way in which software can embody a mathematical definition” (CBMS, 2001, p. 132).

17. PLE 2: Making connections through measurement
The Golden Ratio and pentagons

18. PLE 3: Explaining connections using complex numbers

19. 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

20. 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