Approaches to Algebra: Perspectives for Research and TeachingN. Bednarz, C. Kieran, L. Lee Springer Science & Business Media, 6 de des. 2012 - 348 pàgines In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an "arithmetic" of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano. |
Continguts
1 | |
PART I | 11 |
X | 19 |
Geometric proofs of algebraic rules | 26 |
Toward a homogeneous algebra | 32 |
THE ROLES OF GEOMETRY AND ARITHMETIC IN | 39 |
Two different conceptualizations | 47 |
THE ROLE OF PROBLEMS AND PROBLEM SOLVING | 54 |
What needs to be learned in algebra? | 174 |
Conclusion | 184 |
Algebra in the curriculum | 191 |
A MODELING PERSPECTIVE ON | 194 |
Mathematical narratives and early algebra | 216 |
How do students adjust their narratives when they note | 222 |
PART V | 237 |
The pedagogical assumptions of ComputerIntensive Algebra | 245 |
Lessons from history | 61 |
Algebra itself | 73 |
From action to expression | 81 |
AN INITIATION INTO ALGEBRAIC CULTURE THROUGH | 87 |
Generalization and the introduction of algebra | 102 |
What are the kinds and the characteristics of generalizations | 108 |
Arithmetic reasoning profiles used by students faced with problems | 124 |
DEVELOPING ALGEBRAIC ASPECTS OF PROBLEM | 137 |
Some remarks on the AngloMexican spreadsheet study | 144 |
PLACEMENT AND FUNCTION OF PROBLEMS | 155 |
Conclusion | 163 |
How does ComputerIntensive Algebra relate to the other three | 253 |
INTRODUCING ALGEBRA BY MEANS OF A TECHNOLOGY | 256 |
problems | 268 |
An initiation into algebra in the CARAPACE | 278 |
environment | 288 |
TWO ISSUES | 295 |
Real world problems | 301 |
SYNTHESIS AND DIRECTIONS FOR FUTURE | 314 |
Big ideas | 322 |
AUTHOR AFFILIATIONS | 343 |
Altres edicions - Mostra-ho tot
Approaches to Algebra: Perspectives for Research and Teaching N. Bednarz,C. Kieran,L. Lee Previsualització no disponible - 1996 |
Approaches to Algebra: Perspectives for Research and Teaching N. Bednarz,C. Kieran,L. Lee Previsualització no disponible - 1996 |
Frases i termes més freqüents
activities Al-Khwarizmi algebraic language algebraic reasoning algebraic syntax algebraic thinking algorithm analysis Analytic Art analytical Approaches to Algebra arithme arithmetic Arithmetica aspects Babylonian Babylonian mathematics Bednarz calculations CARAPACE Cardano chapter classroom Computer-Intensive Algebra concept consecutive numbers construction context curriculum Descartes development of algebra difficulties Diophantus discussion dots Eecke elementary algebra environment equal equations Euclid Euclid's Elements example experience Figure formula François Viète functional approach given Graph high school historical history of algebra interpretation interview introduction of algebra involved Janvier learning letters magnitudes mathematical narratives Mathematics Education meaning methods modeling multiplication notation operations particular pattern polygonal number problem solving procedures pupils quantities question rectangle Regiomontanus relations relationships representations Rojano role rules school algebra segments sequence situation solution spreadsheet square strategies structure symbolic algebra teacher teaching traditional algebra translation unknown values variable Viète word problems Zetetics