Basic ideas in mathematics

Basic concept in mathematics is one of the main subject areas in didactics . Intuitive ideas play an important role here, since all mathematical problem-solving processes , even at a higher level, are connected with ideas and accompanying assumptions. Thinking would not be possible without any ideas . Mathematical thinking can be influenced by ideas. Ideas can have either a positive or negative effect on thinking.

Basic concept

The basic concept was developed by Rudolf vom Hofe . It describes an approach that reinforces existing, correct ideas of the students and makes new ideas tangible (through experiments, possibly by handling, i.e. haptically ). The aim is an understanding-oriented acquisition of mathematical terms and mathematical procedures. It is based on a few ideas, the "basic ideas". In the further expansion of mathematics, the students should be able to form as specific ideas as possible, that is, they should not only "parrot" the content on a misunderstood symbolic or verbal level (understanding as opposed to know-how ). New basic ideas about the individual contents should be formed.

The basic mathematical understanding can be divided into three main groups (the concretization describes a possible learning objective control ):

Constitution of the meaning of a term by linking it to known factual and action contexts or ideas of action. - Specifically: The pupil thinks of a related example for a new task that she already understands.
Development of corresponding “ visual ” representations or “internalizations” that enable operative action on the level of imagination. - Concretely: The pupil can make a sketch with which an approach and at least parts of a solution path become clear.
Mathematical modeling ability . - Specifically: The pupil can translate a factual problem into an arithmetic problem and interpret its result as an approximate solution and compare it with other approximate solutions that were found in other ways.

The first step towards a usable basic mathematical understanding is the constitution of the meaning of a term by linking it to known factual and action contexts or ideas of action ("extraction from environmental references"). Applicability in everyday life is important. Mathematical content is learned or acquired and developed as soon as the mathematical content is also applied / used in everyday life. Without any application situations or everyday applications, mathematical content is like an "empty shell". Schoolchildren cannot understand the content and do not even think of using it in everyday life. Real-life application examples are therefore necessary so that the student makes the content their own .

The second step is the construction of corresponding “visual” representations or “internalizations” that enable operative action on the level of ideas. The last and third step of the mathematical understanding is the mathematical modeling ability: learned mathematical contents can be applied independently of individual experiences and ideas also in unknown factual situations and new problematic situations. One example is banking and there, for example, the interest calculation .

When solving factual and text problems, the basic ideas must be used.

Training of basic ideas

Schoolchildren (individuals) should consciously grasp subjective areas of experience, ideas for action and explanatory models for a factual context. The pupils should clarify their existing basic ideas from the factual contexts and build new ones on them. With the help of the basic ideas, mathematics should be understood on the level of terms, processes and results. Mathematics (terms, procedures, results) determine the basic ideas regarding the content. The basic ideas are didactically implemented in factual contexts. This didactic implementation is important because the factual contexts activate the individual to combine subjective areas of experience, ideas of action and explanatory models.

Different aspects of basic ideas

Just like basic mathematical understanding, basic ideas have three different aspects:

One aspect of the basic ideas is the normative aspect. The normative aspect describes what ideas students should have about mathematical content. The basic question is: "Which basic ideas are adequate for solving the problem from the learner's point of view?"
The descriptive aspect is another point of the basic ideas that describes the individual cognitive structures that are activated. The question here is: "What ideas can be identified in the student's attempt at a solution?"
A third aspect is the diagnostic aspect, which deals with the question "What can any divergences be due to, and how can they be remedied?"

Basic ideas based on examples

Every basic idea can be represented on different levels.

Proportional allocation
Functions

Basic ideas about the inverse proportional allocation

Allocation concept: One size is clearly assigned to another size ${\ displaystyle n}$ ${\ displaystyle p (n)}$
Change idea: Change the size , modified to the different size , so that the product off and remains the same. ${\ displaystyle n}$ ${\ displaystyle p (n)}$ ${\ displaystyle n}$ ${\ displaystyle p (n)}$
Object imagination: An anti-proportional assignment is viewed as a whole, as an independent object, its graph runs on a hyperbola.

Sample task for the proportional allocation: Task: A coach is booked for a fixed price for a school trip. If all 30 students come along, each has to pay 20 euros. How much does everyone have to pay if there are only 25 students traveling?

Basic ideas about functions

Allocation concept: One size is clearly assigned another size . ${\ displaystyle x}$ ${\ displaystyle f (x)}$
Change idea: If one size changes, the other size changes in a certain way. ${\ displaystyle x}$ ${\ displaystyle f (x)}$
Object concept: A function is viewed as a whole, as an independent object.

Basic ideas about the term fraction

Share concept ( break as part of a whole, as part of several wholes): of a pizza or 3 pizzas. ${\ displaystyle \ textstyle {\ frac {3} {4}}}$ ${\ displaystyle \ textstyle {\ frac {1} {4}}}$
Operator conception (fraction as a multiplicative arithmetic instruction): The profit is 120 euros (arithmetic operation is applied to the fraction). ${\ displaystyle \ textstyle {\ frac {3} {4}}}$
Ratio concept (fraction as (mixing) ratio): Apple juice and water are mixed in a ratio of 3: 4 to make apple spritzer.

When it comes to basic ideas about fractions, there can also be limits to basic ideas. One limit is the question of the number of fractions between and . Students have different ideas about this question and therefore different answers. In the models, the basic conception of a fraction is expressed as part of a whole and, closely related to the content, the basic conception of a fraction as a quasi-cardinal number in the verbal description. ${\ displaystyle \ textstyle {\ frac {1} {3}}}$ ${\ displaystyle \ textstyle {\ frac {2} {3}}}$

{\ displaystyle \ textstyle {\ frac {1} {5}} + {\ frac {3} {5}} = 1 \; {\ text {fifth}} + 3 \; {\ text {fifth}} = 4 \; {\ text {fifth}} = {\ frac {4} {5}}}

The basic idea of a fraction as part of a whole must be supplemented by the basic idea of expanding as a refinement of the division so that the task can be successfully solved by the students. In the case of the basic ideas, it should be noted that they can be punctual and fragmentary and therefore only have a limited load-bearing capacity.

Basic concept of zero

The number zero is regarded as a cardinal number and thus as an empty set ("nothing"). One problem with the basic idea of the number 0 is that students make mistakes with division , multiplication, and written arithmetic procedures. Mistakes can be avoided if the children develop a good basic idea of the number zero. The children easily add / subtract with the number zero.
Examples:

${\ displaystyle 2 + 0 = 2}$
${\ displaystyle 3-0 = 3}$

However, they also assume that they can multiply by the number 0 and divide by it.
Example:

${\ displaystyle 7 \ cdot 0 = 7}$ is a common student performance that needs to be corrected.

An example to explain multiplication appropriately for children: "If I don't even have 7 bananas, I don't have any bananas."

${\ displaystyle 7 \ cdot 0 = 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0}$
("Seven times nothing is still nothing.")

For divisions with the number 0 there are three cases that must be distinguished.

{\ displaystyle 0: 5 = 0}

("If I have nothing and it is distributed to 5 people, each person has nothing.")

${\ displaystyle 5: 0 = {\ text {impossible}}}$
By multiplication we get: This problem has no solution, because you couldn't insert a number so that the calculation would be correct.
${\ displaystyle 5 = x \ cdot 0}$

${\ displaystyle 0: 0 = {\ text {ambiguous}}}$
Multiplication results in: You could insert any number here and the calculation would always work out. So there is no one-stop solution. Therefore the rule is formulated: "You must not divide by the number zero!"
${\ displaystyle 0 = x \ cdot 0}$

literature

Günter Graumann : Mathematics lessons in school .
R. vom Hofe: Proposals for opening up normative basic concepts for descriptive working methods in mathematics didactics . In: H.-G. Steiner, H.-J. Vollrath (Ed.): New problem and practice-related research approaches. Cologne 1995, pp. 42-50.