The history of mathematics
Synonyms in a broader sense
Changes in math lessons, arithmetic lessons, arithmetic methodology, new mathematics, dyscalculia, arithmetic weaknesses
The term mathematics comes from the Greek word “mathema” and stands for science. Science is more extensive these days, however, and so the word mathematics stands for the science of counting, measuring and calculating as well as geometry.
Mathematics lessons therefore have the task of teaching counting, measuring, arithmetic and the geometric basics in such a way that an understanding of the content is achieved. Mathematics lessons always have to do with demanding and promoting performance. Special approaches and support are necessary, especially when there is a weakness in numeracy or even dyscalculia.
Historically, what is taught in math classes today has been further developed and defined over centuries. The origins of all arithmetic can already be found in the 3rd century BC, both among the ancient ones Egyptians as well as the Babylonians. In the beginning, computing was strictly following rules without questioning a specific why.
The questioning and proving were components that actually only existed in the times of the Greeks became important. During this time, the first attempts to simplify arithmetic were made. The calculating machine “ABAKUS” was developed.
It took a long time until arithmetic became generally accessible and while initially only a select few were allowed to learn to read, write and arithmetic, they formed with them Johann Amos Comenius and his demand for an overall education for young people of both sexes in the 17th century, the first signs of an education for all were gradually emerging. "Omnes, omnia, omnino: Allen, everything, all-embracing" were his slogans.
Due to the historical influences, the implementation of his demands was not initially possible. Here, however, it becomes clear what consequences such a requirement entails. Demanding education for everyone also meant enabling education for everyone. Associated with this was a change with regard to the teaching of (mathematical) knowledge, the so-called didactics. True to the motto: “What does my teacher's knowledge do for me if he can't convey it?”, It took a long time to realize that you can only gain insight and understanding of the facts if you work on different emotional levels Levels that treat the circumstances in a didactically meaningful way.
In addition to the transfer of knowledge, slide rules have already been used by Kern and Cuisenaire Illustration of numbers and their calculation methods invented. Jacob Heer also invented the in the 30's of the 19th century for illustration purposes Hundred table to illustrate the number ranges and their operations, other means of visualization followed.
In particular Johann Heinrich Pestalozzi (1746-1827) further developed modern arithmetic lessons. For Pestalozzi, mathematics lessons were more than just the simple application of various calculation methods. The ability to think should be encouraged and challenged through mathematics lessons. Six essential elements determined Pestalozzi's arithmetic lessons and his idea of a good arithmetic lesson. These goods:
- The math class is the focus, i.e. the most important part of the entire class.
- Concrete visual aids from everyday life (e.g. peas, stones, marbles, ...) to clarify the number concept and the operations (remove = subtraction; add = addition, distribute = division, bundling of the same value (e.g. 3 packs of six = 3 times 6)
- Thinking through instead of simply applying rules that are not understood.
- Mental arithmetic to automate and promote thinking skills.
- Class instruction
- Teaching mathematical content according to the motto: from easy to difficult.
In the 20th century developed what is known in pedagogy as reform pedagogy. The planned changes were tagged with “The Century of the Child”, or. "Pedagogy from the Child" driven forward. In particular Maria Montessori and Ellen Kay are to be mentioned by name in this regard. The weaker children were also given special consideration.
Similar to the development of various reading methods see reading and spelling weaknesses There were also two main methods of calculating, which were only comprehensively implemented in the classroom after the Second World War, i.e. especially in the 50s to the mid-60s. These goods:
- The synthetic process
- The holistic process
Johannes Kühnel's synthetic method assumes that different mathematical understandings are possible depending on the age of the child and that this sequence builds on one another. He felt the view as a particularly essential moment in mathematical knowledge transfer and the promotion of arithmetic weaknesses. Memorization alone did not necessarily imply an understanding of the knowledge to be learned. An essential visual aid was the hundreds sheet, which already resembled the hundreds sheet our children used in the second year of school.
The holistic procedure of Johannes Wittmann on the other hand, initially the numerals (1, 2, ...) “banished” from the classroom and sees the handling of the sets and the development of the set concept as an essential factor and a basic requirement for the ability to develop the number concept. Ordering (lining up), grouping (according to colors, according to objects, ...) and structuring (e.g. defining sequences of unordered quantities) were part of dealing with quantities.
Unlike Kühnel, who dictated the understanding of individual mathematical content for the age of the child, Wittmann assumes more understanding. In Wittmann's holistic process, a child can only count when the concept of quantity is established. The mathematical learning works here step by step, a total of 23 levels of arithmetic lessons are available.
While one was busy with the implementation of these procedures in the schools, pedagogical and didactic innovations were already developing, in particular through the research results of the Swiss psychologist Jean Piagets (1896-1980) were coined.
Jean Piagets (1896-1980) worked at the Jean Jacques Rousseau Institute in Geneva with questions from the field of child and adolescent psychology as well as the field of education. Numerous publications (see right banner bar) followed. In relation to math classes, Piaget's results can be summarized as follows:
- The development of logical thinking goes through different phases, so-called stages.
- The phases build on one another and can sometimes interact with one another, since one stage is not ended overnight and the next started.
- Building on one another implies that the goals of the phase taking place must first be achieved before a new phase can be started.
- The age information can vary individually, a time shift of approx. 4 years is conceivable. The reason for this is that a logical structure cannot be solved (adequately) by all children of the same age.
- At each level, the two mutually dependent functional processes of cognitive adaptation to the environment become noticeable: assimilation (= absorbing new content) and accommodation (= adapting behavior through exercise, internalization and mental penetration).
The stages of cognitive development according to Jean Piaget (1896-1980)
- The sensorimotor stage
from 0 to 24 months
Immediately after birth, the child only masters the simple reflexes, from which arbitrarily controlled actions develop.
Gradually, the child begins to combine the reflexes with others. Only at the age of around six months does the child consciously react to external stimuli.
At around eight to 12 months of age, the child begins to act purposefully. It can, for example, push objects away to grab another object that it wants. At this age, children also begin to distinguish between people. Strangers are viewed with suspicion and rejected (“strangers”).
In the further course, the child begins to develop and to get involved more and more with society.
- The preoperational stage
from 2 to 7 years
The training of intellectual activities is becoming more and more important. However, the child cannot put himself in other people's shoes but sees himself as the center and focus of all interests. One speaks of egocentric (ego-related) thinking, which is not based on logic. If ..., then ... - As a rule, it is not possible to mentally penetrate the consequences.
- The stage of concrete operations
from 7 to 11 years
At this stage the child develops the ability to penetrate the first logical connections with concrete perception. In contrast to egocentrism, decentration develops. This means that the child no longer only sees himself as the focus, but is also able to see and correct errors or wrong behavior.
In relation to math lessons, the ability to perform mental operations on concrete objects is very important. But this also includes the ability to look back at everything in your mind (reversibility). From a mathematical point of view, this means, for example: the child can perform an operation (e.g. addition) and reverse it using a counter-operation (inversion task, subtraction).
In his investigations to establish the side effects of the individual operations, Piaget carried out experiments that were intended to confirm his theories. An important attempt - related to this stage - was the transferring of equal amounts of liquids into vessels of different sizes. If a liquid is filled, say 200 ml, into a wide glass, the filling rim is deeper than in a narrow, high glass. While an adult knows that the amount of water remains the same despite everything, a child decides in the preoperational stage that there is more water in the tall glass. At the end of the stage of the specific operations, it should be clear that there is an equal amount of water in both glasses.
- The stage of formal operations
from 11 to 16 years
At this stage abstract thinking is enabled. In addition, at this stage children become increasingly better at thinking about thoughts and drawing conclusions from a wealth of information.
Each stage includes a development phase and therefore reflects a period of time. These time periods can vary by up to four years, so they are not rigid. Each stage reflects the spiritual foundations that have been reached and is in turn the starting point for the next phase of development.
With regard to the further development and design of child-centered mathematics lessons and the child-friendly promotion of learning problems, Piaget's results had some effects. They were integrated into Wittmann's teachings and so the so-called “operational - holistic method” developed from the holistic approach. In addition, there were also didacticians who tried to implement Piaget's findings without integrating them into other ideas. From this the "operative method" developed.
After the 2nd World War
The years after World War II were marked by the Cold War and the arms race between the then USSR and the USA. For example, the western-oriented countries perceived the fact that the USSR was able to launch a satellite into space before the USA as a shock, the so-called Sputnik shock. As a result, the OECD decided to modernize mathematics teaching, which was then passed on to schools by the Conference of Ministers of Education and Cultural Affairs in 1968: set theory was introduced into mathematics teaching. But that wasn't all. The modernization included:
- The introduction of set theory
- Increased integration of the geometry
- Insight into mathematical facts should come before the simple application of rules
- Brain teasers and brain teasers to emphasize so-called “creative” mathematics.
- Arithmetic in different place value systems (dual system)
- Equations and inequalities in advanced math lessons
- Probability theory, logic
- Solution of issues by means of calculation trees and arrow diagrams
These innovations were also unable to assert themselves over the long term. The “mathematics of set theory”, as it was called colloquially, was repeatedly criticized.The main point of criticism was the view that the application of arithmetic techniques and practicing were neglected, but that things were trained that sometimes had little relevance to everyday life. The “new mathematics” was considered too abstract. A fact that did not at all suit poor numeracy children.
nowadays one can find different approaches from the individual developments in mathematics lessons. So are for example Piagets Basic knowledge in mathematics didactics too still of great importance today. In addition to all the facts to be conveyed in terms of content, which are required by the school curriculum or framework plan, it is important to adhere to the sequence of newly learned mathematical content. Primary school children, for example, are at the stage of the concrete operations, sometimes even at the stage of the preoperational stage. Here is the Intuition for understanding is of great importance. New content to be learned should always be based on the E-I-S principle be penetrated in order to offer every child the possibility of understanding.
The E - I - S principle stands for Enactive penetration (acting with visual materials), iconic (= pictorial representation) and symbolic penetration.
This should now be clarified here - based on the addition. The understanding of addition can be achieved actively by using placement tiles, Muggle stones or the like. The child understands that something needs to be added. To the starting amount 3 (tiles, cars, Muggle stones, ...) 5 more objects of the same amount are added. It can see that there are now 8 (placement tiles, cars, Muggle stones, ...) and confirm this by counting them.
The iconic penetration would now be transferred to the visual level. So it now draws the task in circles in the exercise book:
0 0 0 + 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 (0 = placement plate, ...)
Images of the active penetration used (images of cars, etc.) can also be used. A transfer takes place when the numbers are added: 3 + 5 = 8
The systematic structure and the gradual reduction of the view, is particularly helpful for children who have problems capturing new content. In addition, is a Intuition As a general rule for all children to internalize mathematical content essential.
There may be children (with arithmetic weaknesses or even dyslexia) who immediately make the transition from the enactive to the symbolic level. It is also conceivable that children are able to think formally operational right from the start. One of the reasons for this is that the Development stages by no means rigid but that shifts of up to four years can occur. It is the task of the teacher to find out what level individual children are at and accordingly to orient the lessons accordingly.
Further information on partial performance weaknesses can be found at:
- Causes of Dyslexia
- Symptoms of dyslexia
- Diagnosing dyslexia
- Early detection of dyslexia
- Therapy for dyslexia
For more information on learning problems, see:
- Poor concentration
- Speech disorders
- Educational games
A list of all topics that we have published under our "Problems with Learning" page can be found under: Problems with Learning A-Z