This is a summary from the other article in Mathematics written by A.Teffers** “Didactical Background of a Mathematics Program for Primary Education”**

** 1. Introduction**

This contribution outlines an instruction-theoretical framework of realistic mathematics education for primary school that was developed in The Nedherlands in the period between 1970 and 1990. At the outset a learning-instruction structure is described. Next it is illustrated how this structure is inter-twined in the various learning strands. Then the conclusion is how these strands of learning give concrete shape at lesson level.

** 2. Instruction – Theoretical Framework of Realistic Mathematics Education**

The instruction-theoretical framework is outlined on the basis of column on the basis of column arithmetic for division. A realistic course for ‘long division’ is reached through some steps, from the simple problem until the complex problem. Students will solve the problems using their own ways that might be various from the simple method until they can learn to carry out the long division procedure. The five major learning and teaching principles that lie at the basis of such “realistic” courses will be described in the context of the previously mentioned learning strands for long division.

**2.1 Construction and Concretising**

The first learning principle in learning mathematics is constructive activity. This is clearly visible in the outline coursed where students can discover the division procedure for themselves because a concrete orientation basis is laid for the skill to be learned. Finally, they come to realize what the arithmetic operations are leading up to.

**2.2 Levels and Models**

The learning mathematics is a process which moves at various levels of abstraction. This level-characteristic can be seen from the solutions methods that students used. To be able to achieve the raising in level from informal to formal arithmetic the students must have at his disposal the tools to help bridge the gap between the concrete and the abstract. Materials, visual models, model situations, schemes, diagrams and symbols serve this purpose.

**2.3 Reflection and Special Assignments**

The learning of mathematics and in particular the raising of the level of the learning process is promoted through reflection. In this third instruction principle the pupils must constantly have the opportunity and be stimulated at important junctions in the course, to reflect on learning strands that have already been encountered and to anticipate on what lies ahead. Important assignments through which one and other can be achieved are the previously mentioned free productions and the conflict problems.

**2.4 Social Context and Interaction**

Learning is not merely a solo activity but something that occurs in a society and is directed and stimulated by that socio-cultural context. Students can find solution for a problem through discussion among them. Consequence of this principle is that mathematics education should by nature be interactive, which besides room for individual work, it must also offer opportunity for the exchange of ideas, the rebuttal of arguments, and so forth.

**2.5 Structuring and Interviewing**

Learning Mathematics does not consist of absorbing a collection of unrelated knowledge and skill elements, but is the construction of knowledge and skills to a structured entity. This fifth principle means that learning strands must be intertwined with each other. Besides it, pure arithmetic and the making of applications must from the very start be connected with each other.

**2.6 The Structure of the Learning-Instruction Principle**

In the foregoing the learning principles and instruction principles that were connected are: the concept of learning as construction with the lying of a concrete orientation basis; the level-character of learning seen on the long term with the previous of models, schemes, and symbols; the reflection aspect learning with the assigning of special tasks, with as main categories free production items and conflict problems; learning as a social activity through interactive instruction; and the structural of learning with the intertwining of learning strands.

**3. Four Directions in Mathematics Education**

Beside the realistic direction in mathematics education, there are three others that can be distinguished, namely: the empiristic, the mechanic and the structuralistic.

**3.1 The Empiristic Approach to Long Division**

The empiric approach to long division because, in general, it is one that is not thought. Instead, the short cut of informal arithmetic methods is pursued, in part via mental arithmetic. At this approach what stands out especially is the difference in regard to the use of the model in order to break away from informal to formal arithmetic.

**3.2 The Mechanistic Approach to Long Division**

In tens of lessons the diffusion algorithm is practiced from simple case to complex case. The degree of complexity is especially determined by the magnitude of the numbers (dividend and divisors); regrouping actions for multiplications and subtraction; and ‘bothersome’ zeroes. Divisions with remainders do not appear until near the end of the course.

**3.3 The structuralistic approach to long division**

In the structuralistic approach the emphasis in the teaching long division lies very much on the place value system. The problem with the structuralistic set up of the learning strand for long division is that the algorithm is taught primarily at the formal arithmetic level. The sucject-systematic final algorithm is not pursued in a direct manner, but developed gradually via context problems and context restricted arithmetic.

**3.4 Direction in the 5 by 5 Learning-Instruction Structure**

The structuralistic method does not fit into the learning-instruction structure well either. For instance, the construction principle can only be applied to a limited degree because a solid concrete orientation basis is lacking. Observe the contracting background of the realistic 5 x 5 education-learning structure have more relief as in the example of long division. Therefore, the build up of elementary skill can take a place via a process of reinvent, yet, must imperatively take place if the mathematical rules and structures are to be widely applicable.

**3.5 Summary and Generalisation**

The four mentioned directions in mathematics education can be disguised according to the presence or absence of the components of horizontal and vertical mathematisation like below:

Horizontal | Vertical | |

Mechanistic | – | – |

Empiristic | + | – |

Structuralistic | – | + |

Realistic | + | + |

Horizontal mathematising is the modeling of problem situations thus that these can be approached with mathematical means (it leads from the perceived world to the world of symbols). Vertical mathematising is directed at the perceived building and expansion of knowledge and skills within the subject system, the world of symbols.

**4. Examples of realistic Learning Strands with an Emphasis on Vertical Mathematising**

**4.1 Counting**

Counting is a realistic approach as a reaction to the structural concept. The expense of counting activities is to develop number concept by way of practising logical forms of reasoning. Elementary arithmetic is founded on counting movement, in short synchronous counting. When young children are asked to count how many number, they are not yet able to count resultatively. There is a third educational path to help them understand about counting. That is playing the game of counting and doing arithmetic by rolling the dice, dominoes, and a dot card on which tokens are placed in circle.

**4.2 Automatiing and Memorizing of Addition and Subtraction up to Twenty**

Working with dice or a dot charts has helped many children automatise and even memorize ever so many elementary additions, subtractions and ‘split-ups’ to twelve. Expansion and completion of that process of automatising and memorizing to twenty demands a new model situation. The models that can be chosen are: a string of beads with a five-structure for arithmetic to ten; and the arithmetic rack with five-structure for arithmetic to ten and especially twenty, a cardinal model.

**4.3 Addition and Subtraction to a Hundred**

In the realistic viewpoint addition and subtraction to a hundred is thereof not immediately algorithmised, but there is room for all sorts of varied strategies of efficient (mental) arithmetic. So, students can use their own ways in various strategies from informal level until formal level. Place value or empty number line can be used in this case. When students use corresponding subtraction or addition strategies, it can facilitate memorization.

**4.4 Learning the Tables**

In general there are two methods by which to learn tables: reproduction and reconstruction didactics. These methods are useful for reproduction of memorized knowledge, reconstruction knowledge via skilled arithmetic and connecting to informal working method. When student use table, they also can learn about sums, multiplication and division.

**4.5 Mental Arithmetic**

Mental arithmetic is considered as doing arithmetic mentally where standard procedure is carried out mentally and the calculation is made in the head. Various forms of mental arithmetic are distinguished, namely: estimating, varied and standard mental arithmetic.

**4.6 Column Arithmetic**

Characteristic for column is that the arithmetic is done with the individual digits, while with mental arithmetic the numbers that are operated with retain their own ‘value’.

**4.7 Ratio**

The most realistic feature of the realistic approach of ratio is that it does not steer directly to the so called rule of three in working out the fourth term (a : b = c : ? , maka ? = bc/a). If numbers are used in ratios then the double number line can help in solving of ratio problems, namely: determining the ratio relationship, comparing equivalent ratios, making equivalent ratios, and determining the fourth proportional.

**4.8 Fractions**

The double number line can also be used for operation with fractions. The comparison with fraction can take place by switching to a different unit of measure where one another can be depicted on the double number line.

**5. An Example of Realistic Mathematics Education at Classroom Level with an Emphasis on Horizontal Mathematising**

Students are given a problem and they work in groups. There is discussion among them and the teacher guide them by asking some questions that will lead them get better understanding about the problem.

**6. Conclusion**

One could relate the described series of lessons which horizontal mathematising or the vertical learning strands to the learning-instruction structure. One could also investigate how the number line fulfils the bridging function in these courses between the formal context-bound level and the formal level, or make a detailed analysis of the issues and indicate their function in the learning-instruction structure.