The four mentioned directions in mathematics education can be distinguished according to the presence or absence of the components of horizontal and vertical mathematisation
Horizontal mathematising is the modeling of problem situations thus that these can be approached with mathematical means. Or in other words: it leads from the perceived world to the world of symbols. And in the description of long division we have already observed that this ‘concrete’ perceived world is not an absolute level indication but a relative one. I.e that also parts of the world of symbols can become part of the perceived world, the personal reality. Vertical mathematising is directed at the perceived building and expansion of knowledge and skills within the subject system, the world of symbols.
Generally speaking both components of mathematising are missing in mechanistic education. Which is to say that neither the source nor the application area of arithmetic are sought in the perceived world. The start is made at the formal level of the world of symbols. Vertical mathematising is impeded and for many pupils even blocked because the perception principle is missing – with the result that the instruction becomes the presentation and drill of rules and regulations, in short, becomes algorithmic mathematics education. The formal straitjacket leaves room only rule – directed education.
Characteristic for the structuralistic approach of education is that there insight is pursued. This is done in a specific manner. Here also the start is on the formal arithmetic level. Only, the structuralists concretize the operations, structures and such with the aid of structured material in order to represent the subject system concretely and perceptibly. For instance with structure material in the form of MAB – or Dienes blocks as the embodiment of the decimal place value system. Vertical mathematising takes place with this structural material, by means of visual representations there of, to operations with symbols – ‘enactive, iconic, symbolic’ as Bruner calls the three, and Resnick refers to ‘mapping instruction’. Real problems play no essential part in the learning of arithmetic in the initial phase.
Informal solution methods of children from the pretext here, to arrive a arithmetic procedures via a gradual process of schematization, abbreviation and generalization. Context-bound mathematics is made subservient to formal arithmetic; models act as intermediaries. These models should however meet very special specifications in order to be able to fulfill the bridging function between informal context-bound arithmetic and formal arithmetic. They must be ‘models of’ context problems which will serve as ‘models for’ the pure subject restricted and applied arithmetic in the pertaining area – multifaceted, in other words. They must moreover be strong in the sense that they can be deployed in all phases of the learning-instruction process in the intended area, therefore in the first phase close to informal, context-restricted arithmetic and in the last phase close to formal standardized operating, as well as in the extensive area in between. And thirdly, these models must connect naturally to the working methods of children and naturally, or with some stimulation, lead to schematizing and abbreviation of pure arithmetic and to the generalization of the applicability thereof. In short, model situations with corresponding models must be sought of a kind where the children who work on them and with them will themselves indicate and negotiate the learning route to formal arithmetic, be it under the proper guidance of the teacher and the group.
Described in the following are a number of examples of learning strands in primary school which fit into the outlined realistic structure. These are:
2. Memorizing of addition and substraction tables to ten and twenty
3. Addition and substracton to one hundred
4. Multiplication and division tables
5. Mental arithmetic
6. Column arithmetic
Where necessary and illustrative, a few remarks will be made about alternative courses of mechanistic, empiristic and structuralistic nature, notably also there where this can throw light on the genesis of the particular learning strands. Can realists substantiate their edicational claim, namely that, starting from the informal, context-bound level, children can in a constructive manner be brought to the formal level with the aid of suitable model situations, models, schemes and symbols? – that is the question at issue.
One could relate the described series of lessons which illustrates horizontal mathematising or the vertical learning-instruction structure which was outlined in the first part. One could also investigate how the (empty, double) number line fulfils the bridging function in these courses between the informal context-bound level and the formal arithmetic level. Or make a detailed analysis of the issues and indicate their function in the learning-instruction structure, namely with long division. What formalistic instruction can lead to.