1. THE GAINS AND THE PITFALLS OF REIFICATION - THE CASE OF ALGEBRA ANNA SFARD AND LIORA LINCHEVSKI

2. Problem 3 (Dina’s case) Development of Algebra Definition

3. String of symbols Description 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.

4. Problem 3 יש פתרון לכל ערך של k? האם זה נכון שלמערכת הבאה של משוואות לינאריות :

5. התשובה הצפויה : כן, כי לכל ערך של K הישר y=k-2 הוא מקביל לציר ה -X, הישר y=k-x הוא משופע ולכן הם נחתכים.

6. פתרון של דינה פתרון של יאנה אז מה הבעיה ?

7. 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.

8. 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

9. האם לערכים בטבלה יש תכונה או תבנית מסוימת ? …43210-2-3-4…X …169410149 …X2X2 טל : אם נחסיר מספרים באלכסון ונחבר אותם באלכסון נקבל אותו המספר.... טל : בדקתי עוד דוגמאות, נראה לי שזה עובד..... שירלי : אם הייתה לנו נוסחה, משהו כללי... a…43210-2…X a2a2 16941014 X2X2

10. 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.

11. Stages in the development of algebra Historical highlightsRepresentationNew focus onStageType Rhind papyrus c. 1650 B.C Verbal (rhetoric) Numeric computations Operational Generalized Arithmetic Diophantus 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)

12. 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) Operational1)Abstract Algebra XIX and XX century: theories of groups, rings, fields, etc., linear algebra SymbolicAbstract structures Structural