THE GAINS AND THE PITFALLS OF REIFICATION -THE CASE OF ALGEBRA

ANNA SFARD AND LIORA LINCHEVSKI

Problem 3 (Dina’s case)

Development of Algebra

Definition

153 x

String of symbolsDescription of a computational

process.

A function - a mapping which translates every number into another

Result of the process- product of

a computation

Function as an object

What one actually sees in algebraic symbols depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice.

Problem 3

של • ערך לכל פתרון ?kיש

kyx

yk 2

•: לינאריות משוואות של הבאה שלמערכת נכון זה האם

: הצפויה  התשובה

של, • ערך לכל כי מקביל y=k-2הישר Kכן הוא- ה הם y=k-xהישר, Xלציר ולכן משופע הואנחתכים.

xky

ky

kyx

yk 22

דינה של פתרון

22

22

xyxy

yxk

yk

kyx

yk

יאנה של פתרון

22

22

xxkk

xky

ky

kyx

yk

מה אזהבעיה?

Reification

• The theory of reification is introduced, according to which there is an inherent process-object duality in the majority of mathematical concepts.

• It is the basic tenet of our theory that the operational (process-oriented) conception emerges first and that the mathematical objects (structural conceptions) develop afterward through reification of the processes.

The case of algebra-Reification• Abstract objects, such as functions or sets,

play the role of links between the old and the new knowledge. In algebra, function is what ties together the arithmetical processes (primary processes) and the formal algebraic manipulations (secondary processes). Thus, reification of the primary processes, or, in the case of algebra, the acquisition of the structural functional outlook, is a warranty of relational understanding.

Illustration

? מסוימת תבנית או תכונה יש בטבלה לערכים האם

… 4 3 2 1 0 -1 -2 -3 -4 … X

… 16 9 4 1 0 1 4 9 16 … X2

. המספר: אותו נקבל באלכסון אותם ונחבר באלכסון מספרים נחסיר אם טל...

. , עובד: שזה לי נראה דוגמאות עוד בדקתי טל...

... , : כללי. משהו נוסחה לנו הייתה אם שירלי

a … 4 3 2 1 0 -1 -2 … X

a2 16 9 4 1 0 1 4 X2

aaaa 22 11

Historical/ Didactical Parallel

• The nature and the growth of algebraic thinking is presented as a sequence of ever more advanced transitions from operational to structural outlook.

Stages in the development of algebraHistorical highlights Representation New focus on Stage Type

Rhind papyrusc. 1650 B.C

Verbal (rhetoric) Numeric computations

Operational Generalized

ArithmeticDiophantus c. 250 A.D Mixed: verbal +

symbolic)syncopated(

16th century mainly Viete(1540-1600)

Symbolic (letter as an unknown)

(Numeric) product of computations ('algebra of a fixed value')

Structural

Viete, Leibnitz (1646-1716), Newton (1642-1727)

Symbolic (letter as a variable)

Numeric function (functional algebra)

Stages in the development of algebra

British formalist school (De Morgan, Peacock, Gregory), since 1830

Symbolic(no meaning to a letter)

Processes on symbols (combinations of operations)

Operational 1) Abstract Algebra

XIX and XX century: theories of groups, rings, fields, etc., linear algebra

Symbolic Abstract structures

Structural