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.

The authors of this volume claim that mathematics can be usefully re-conceptualized as a special form of communication. As a result, the familiar discussion of mental schemes, misconceptions, and cognitive conflict is transformed into a consideration of activity, patterns of interaction, and communication failure. By equating thinking with communicating, the discursive approach also deconstructs the problematic dichotomy between individual and social research perspectives. Although each author applies his or her own analyses to the discourse generated by students and teachers grappling with mathematical problems, their joint aim is to put discursive research into the limelight and to spur thinking about its nature and its possible advantages and pitfalls. This volume is therefore addressed both to those interested in specific questions regarding classroom communication, and to those who are looking for a general conceptual lens with which to tackle the complexity of mathematical teaching and learning.

The authors of this volume claim that mathematics can be usefully re-conceptualized as a special form of communication. As a result, the familiar discussion of mental schemes, misconceptions, and cognitive conflict is transformed into a consideration of activity, patterns of interaction, and communication failure. By equating thinking with communicating, the discursive approach also deconstructs the problematic dichotomy between "individual" and "social" research perspectives.