2. EXPLORING THE ORIGIN OF MATHEMATICS
How is mathematics developed before
learning formal and well organized body of
mathematics knowledge?
3. COGNITIVE SCIENCE
Result in cognitive science->most of or
thought is unconscious.
Ordinary ideas from not mathematical
cognitive mechanism to characterize
mathematical ideas: such as basic spatial
relations, grouping, small quantities, motion,
distribution in space, basic manipulation and
so on
(Lakoff&Nunez,
2000)
4. WHERE MATHEMATICS COME FROM (LAKOFF &
NUNEZ, 2000)
Mathematical cognition is the extension of
ordinary cognitive behaviour rooted in daily
life experience.
The concepts and schemes derived from
ordinary cognitive behaviors (e.g., the spatial
relations used in everyday language) are the
ones used in learning advanced mathematics
by means of the mappings based on
metaphors.
5. CONCEPTUAL METAPHORS
Metaphor is not a matter of words, but of
conceptual structure
One of the principal results in cognitive
science is that abstract concept are typically
understood, via metaphor, in terms of more
concrete concepts.
Conceptual metaphor are part of our system
of thought and many arise from correlation in
our commonplace experience.
6. MAPPING BASED ON METAPHOR
Conceptual metaphors established from life
experiences (central cognitive mechanism)
Eg. Extending students mathematics from
innate basic arithmetic to more sophisticated
application of number
7. HOW DO WE GO FROM SIMPLE CAPACITIES TO SOPHISTICATED
FORMS OF MATHEMATICS?
Example 1 (embodied arithmetic from its
innate)
At least 2 capacities of innates arithmetic:
(1) capacity of subitizing
(2) capacity for the simplest form of adding
and subtracting small numbers relate to
counting
8. CHARACTERIZE ARITHMETIC OPERATION AND
ITS PROPERTY
Metaphorizing capacity: conceptualize
cardinal numbers and arithmetic operations
in terms of experience s of various kinds.
Conceptual blending capacity: need to form
correspondences across conceptual domains
(eg.combining subtizing and counting).
9. EXAMPLE 2 (RATIO DERIVED FROM REALISTIC
EXPERIENCES)
Ratio derived from sensory perception (Lin,
Hsu, Chen, Yang, ...).
Example of series of tasks for experiencing
the origin of mathematics @
10. CHARACTERISTICS OF THE TASK
Involve realistic context
Good entries for student to explore the origin
of ratio concept
Attain mathematics meaning and common
sense
11. REVISITING MATHEMATICS EDUCATION
(FREUDENTHAL ,1991)
Certainty as the most characteristic property of
mathematics, how certain is “certain”?
“ common sense takes things for granted, for
good reasons or for bad ones”
Mathematics as an activity (leading to ever
improved versions of common sense)
Common sense reveals in action –physical and
mental- which are common to people who share
common „realities‟ to the mere experience of
sensual impressions
12. MATHEMATICAL ENCULTURATION (BISHOP,
1991)
Mathematics as cultural phenomena
Mathematical enculturation process is a way
of encouraging individuals to experience & to
reflect on certain kinds of ideational contrast
in order to develop a particular way of
knowing.
13. COGNITIVE DEVELOPMENT CULTURALLY
Cognition that much to do with culture and
environment and less to do with genetics
(Lancy, 1983)
Eg. On cross cultural studies: cultures
studied do count and use numbers, do
measure, do develop geometric concepts, do
play rule-bound games, and do develop
explanation.
14. HISTORY OF MATHEMATICS FOR EXPLORING THE
ORIGIN OF CONCEPT
Integration of history of mathematics into
mathematics education addressed on (Goal):
- Epistemological status of mathematics
- Integration of history mathematics as way to
teach student about evolution & context
dependency of human knowledge
15. RATIO AND PROPORTION IN HISTORY (AS
EXAMPLE)
Nature of topics.
Ideas :
- One tribe is as twice as large as another.
- One leather strap is only half as long as another.
Both are such as would develop early in the history of
race, yet one working on the ratio of numbers and
other working on the ratio of geometric magnitudes.
(Smith,
1953)
16. GREEK WRITERS ON RATIO & PROPORTIONS
In Book VII of Euclid‟s elements, ratio is not defined
at all
In Book V, ratio is given the vague characterization of
„...a sort of relation in respect of size between two
magnitudes of the same kinds‟
Then, Smyrna writes „ratio in the sense of proportion
is a sort of relation of two terms to one another, as for
example double, triple‟
Elements, VII, definition 20, reads ‘Numbers are
proportional when the first is the same multiple, or
the same part, or same parts, of the second that the
third is the fourth‟
17. SUMMARY OF RATIO AND PROPORTION THEORY
(Rusnock &
Tagard, 1995)
19. TEACHING FOR ORIGIN OF MATHEMATICS
CONCEPT
Learning Goals:
- Enable students to derive mathematics idea
and meaning from their mathematics innate
- Enable students to derive mathematics idea
and meaning from reality (humanistic approach :
RME)
- Develop common sense for problem solving in
and out of mathematics
- Enable students experience the ideational
contrast of developing knowledge
