The teaching-learning process is considered as a social interaction. In this microethnographical case study an elementary teacher and first graders are observed when they ascribe mathematical meanings of numbers and of numerical operations to empirical phenomena. Because of the differences of their ascriptions, the teacher and the students negotiate mathematical meanings. Also interactional regularities help the participants to cope with ambiguity. According to different theoretical approaches, the text discusses some indirect relations between social interaction and mathematics learning. Several classrooms episodes are interpreted to illustrate specific theoretical concepts.

https://link.springer.com/content/pdf/10.1007/BF01273665.pdf

Negotiation of mathematical meaning and learning mathematics

Preface.- Introduction to the Edition.- Introduction to the Contents of the Volume.- Chapter 1: Lectures on Projective Geometry (1891).- Chapter 2: Lectures on the Foundations of Geometry (1894).- Chapter 3: Holiday Courses, 1896 and 1898.- Chapter 4: Lectures on Euclidean Geometry (1898-99).- Chapter 5: The Foundations of Geometry: The Festschrift of 1899.- Chapter 6: Lectures on the Foundations of Geometry (1902).

REVIEWS-David Hilbert's lectures on the foundations of geometry 1891-1902

The 19 century witnessed a dramatic freeing of the human intellect from its naive assumptions. This development is particularly striking in mathematics. New geometries were discovered which challenged and expanded the familiar space of euclid, notably projective geometry and non-euclidean metric geometries. Developments in algebra led to similarly earth-shaking discoveries. This expansion of consciousness was accompanied by a desire to find a higher unity in the growing diversity. Hermann Grassmann made a bold, early attempt to bring the two domains together in his ”Science of extension” of 1844. Later in the century, Cayley and Klein contributed to this process by their discovery that metric geometry could be constructed atop projective geometry. William Clifford introduced a geometric product combining the metric inner product with Grassmann’s exterior product; the resulting structure has become known as a Clifford or geometric algebra. A completely satisfactory formulation of geometric algebra had to await the rigorous clarity of 20 century mathematics. The work presented here (the author’s Ph. D. thesis from the Technical University Berlin, 2011) represents an attempt to bring this stream of 19 century mathematics into the aforementioned rigorous, modern form, and to apply the resulting thought forms to understand kinematics and rigid body mechanics in the spaces under consideration (including euclidean, hyperbolic and elliptic space). One guiding light in the treatment is the reliance on a projective geometric foundation throughout, particularly a consequent application of the principle of duality. This projective approach guarantees that the result is metric-neutral – the results are stated in such a way that they apply equally well to the two non-euclidean geometries as well as euclidean geometry. And a consequent consideration of duality leads to the recognition that this classical triad must be extended by a fourth geometry, dual euclidean geometry. Readers of the MPK will not be surprised to learn that dual euclidean geometry has a close affinity with counter-space [in German, Gegenraum], a concept which Rudolf Steiner introduced and developed, particularly in his natural science lectures. That such a connection exists in the work presented here is not surprising, as the initial impulse for this thesis was provided by the later works of George Adams (for example, see [Ada59]), to whom the present publication is dedicated. One of Adams’ great gifts was to show how the results of modern science, particularly mathematical, when grasped with enlivened thinking, reveal deep, often unexpected, connections to the world and the human being – in a manner reminiscent of Goethe’s reading the “open secrets” of nature. Such insights can indicate fruitful directions for further research. In this spirit is the current publication offered, with special thanks to Peter Gschwind for making it possible.

Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries

PART 1: THE ORIGINS AND DEVELOPMENT OF GEOMETRICAL KNOWLEDGE PART 2: ABSTRACT PROJECTIVE GEOMETRY

Algebraic Projective Geometry. By J. G. Semple and G. T. Kneebone. Pp. viii, 404. 35s. 1952. (Oxford University Press)

Social functions of routines and consequences for subject matter learning

A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begrundeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces over arbitrary fields and of arbitrary dimensions requires two steps, the introduction of a metric in a pappian projective space and the definition of substructures as Cayley-Klein geometries. While the first step is taken in H. Struve and R. Struve (J Geom 81:155–167, 2004), the second step is made in this article. We show that the concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of non-euclidean geometries including the main geometries studied in the foundations of geometry by D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al.

Non-euclidean geometries: the Cayley-Klein approach

Projective metrics were first introduced by A. Cayley and F. Klein who constructed analytical models over the field of complex numbers. The aim of this paper is to give for the first time a purely synthetic definition of all projective spaces with Cayley-Klein metrics and to develop the synthetic foundation of projective-metric geometry to a level of generality including metrics over arbitrary fields of characteristic ≠2.

Projective spaces with Cayley-Klein metrics

Endliche Cayley-Kleinsche Geometrien

Zum Begriff Der Projektiv-Metrischen Ebene

We present a common axiomatic characterization of Cayley-Klein geometries over fields of characteristic \({\neq 2}\). To this end the axiom system of Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg 1973) for plane absolute geometry, which allows a common axiomatization of Euclidean, hyperbolic and elliptic geometry, is generalized. The notion of plane absolute geometry is broadened in several aspects. The most important one is that the principle of duality holds: the dual of a Cayley-Klein geometry is also a Cayley-Klein geometry. The various Cayley-Klein geometries are singled out by additional axioms like the Euclidean or hyperbolic parallel axiom or their dual statements.

Rolf Struve

Papers

Non-euclidean geometries: the Cayley-Klein approach

Projective spaces with Cayley-Klein metrics

Endliche Cayley-Kleinsche Geometrien

Zum Begriff Der Projektiv-Metrischen Ebene

An axiomatic foundation of Cayley-Klein geometries