ROOSELYNA EKAWATI

OCT 1, 2012

ORIGIN OF MATHEMATICS

CONCEPT

EXPLORING THE ORIGIN OF MATHEMATICS

How is mathematics developed before

learning formal and well organized body of

mathematics knowledge?

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)

WHERE MATHEMATICS COME FROM (LAKOFF &

NUNEZ, 2000)

Mathematical cognition is the extension ofordinary cognitive behaviour rooted in dailylife experience.

The concepts and schemes derived fromordinary cognitive behaviors (e.g., the spatialrelations used in everyday language) are theones used in learning advanced mathematicsby means of the mappings based onmetaphors.

CONCEPTUAL METAPHORS

Metaphor is not a matter of words, but ofconceptual structure

One of the principal results in cognitivescience is that abstract concept are typicallyunderstood, via metaphor, in terms of moreconcrete concepts.

Conceptual metaphor are part of our systemof thought and many arise from correlation inour commonplace experience.

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

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

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).

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 @

CHARACTERISTICS OF THE TASK

Involve realistic context

Good entries for student to explore the origin

of ratio concept

Attain mathematics meaning and common

sense

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 andmental- which are common to people who sharecommon „realities‟ to the mere experience ofsensual impressions

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.

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.

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

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 ofrace, yet one working on the ratio of numbers andother working on the ratio of geometric magnitudes.

(Smith,1953)

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 twomagnitudes of the same kinds‟

Then, Smyrna writes „ratio in the sense of proportionis a sort of relation of two terms to one another, as forexample double, triple‟

Elements, VII, definition 20, reads ‘Numbers areproportional when the first is the same multiple, orthe same part, or same parts, of the second that thethird is the fourth‟

SUMMARY OF RATIO AND PROPORTION THEORY

(Rusnock & Tagard, 1995)

Lin et al (2012)

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