Approaches to Algebra: Perspectives for Research and Teaching

Portada
N. 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

INTRODUCTION
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
Copyright

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