Math and life are two different things
Math - more than one, two, three
People encounter mathematics in everyday life: when paying at the cash register at the bakery, when measuring ingredients for cooking, in traffic with various symbols, when using cell phones and remote controls. Even in nature, for example, we describe the shapes of leaves or plants using mathematical (geometric) terms. Find out for yourself where you encounter mathematical content in your everyday life!
When does math understanding begin?
Now the day-care centers in Germany have an educational mandate that is expressed in the various educational plans of the federal states based on guiding principles and goals. What is often lacking, however, is knowledge of how to develop a mathematical understanding. How are the relevant skills built up? In the research literature, the individual sequences of mathematical competence acquisition are given different names and focuses. Therefore, only a generalized outline of the acquisition of mathematical competence will now be given.
The acquisition of numerical skills occurs at the beginning of a person's life. As research in developmental psychology has shown, even infants have mental notions of quantities. Shortly after birth, infants are able to compare quantities of a maximum of four elements with one another. At the age of four months they show amazement when, after a familiar picture with eight points, they suddenly see one with sixteen points. Although they cannot estimate the exact number yet, they seem to notice the difference in quantity, presumably based on the total size of the elements. This is only possible if the number is as small as possible and the difference in quantity is presented in a certain ratio. Even infants seem to have an innate system for numerosities. This natural sense of quantity represents an important basis for the later understanding of the terms of comparison (more, less) and is therefore at the same time the basis for future arithmetic operations. At the same time, research is divided on whether these competencies are more likely to be assigned to the first mathematical or visual skills.
Towards the end of their first year of life, toddlers can recognize the sequence of larger or smaller quantities or areas. With the beginning of speaking in the second year of life, children learn the systematic use of numerals. These are babbled to themselves, the series of numbers is recited like a kind of rhyme - but this usually has nothing to do with counting. By the age of three, children are able to count up to five and also count elements in this area by tapping or pointing. The knowledge of the number range is now increasing steadily. At three and a half years old, children know numerical words up to ten, and by the age of four and a half they can count up to twenty, although some children are still relatively unsure about this. Up to the age of three, children still use the principle of “subitizing” (also known as simultaneous recording) when counting, which means estimating small amounts (maximum of four elements) through spontaneous visual recording. This ability is also used by adults to capture small amounts (check yourself out!). “Subitizing” seems to be fundamentally anchored in the human nervous system. The actual counting is only necessary for adults with a number of five or more objects. A structured arrangement of the individual elements is beneficial for the principle of spontaneous recording of a set.
Mathematical understanding in children from the age of three
From the age of three, the children begin to actually speak quantities, referring to the series of numbers that they have mostly learned by heart and mostly forget to actually count things. In the course of their fourth year of life they internalize certain counting principles, such as the cardinality principle (numbers are used to describe numbers, the last number mentioned also denotes the quantity), the one-to-one assignment (an element becomes a number assigned) and the principle of stable sequence (the arrangement or sequence of the quantity to be counted is not decisive for the counting result). The children seem to first apply and consolidate these principles to a smaller number range before applying them further to a larger number range. Understanding such principles enables children from the age of four to solve small addition and subtraction tasks, but they are still dependent on visual means of representation and tapping. In preschool age, the children then have the ability to recognize structures in ordered sets of objects (cube images), to count up from a certain starting number and to solve simple addition problems. For the latter, preschoolers use different strategies:
- Recall from memory (of already known tasks or representations, most children use this, but often get the wrong result),
- Representation by finger (high success rate),
- Counting (very few children use this, the success rate is 50%),
- Finger counting (displaying the cumulative lines with one hand and final counting, very time-consuming, but mostly with the correct result, a very complex method).
The basis for a solid mathematical understanding in the area of quantity and number knowledge is not only the knowledge of the numerical word, but also the reference to the corresponding quantity, also as a graphic representation (points, cube images). Furthermore, the cultural context and the language play an important role. Number symbols and set terms are culture-dependent, for example, some cultures only have very rough names for sets.
Now go back to yourself: In which contexts is mathematics already practiced in your children's home? Where else can quantities be represented and linked with numerical words? Where can children learn quantities and numbers?
Daseking, M., Lemcke, J. & Petermann, F. (2006): Precursory disorders of school skills: recording of cognitive performance in kindergarten age. In: Petermann, U. & Petermann, F. (Ed.): Diagnostics for special educational needs. Göttingen: Hogrefe. Pp. 211-237.
Oerter, R: & Dreher, M. (2002): Development of Problem Solving. In: Oerter, R. & Montada, L .: Developmental Psychology (5th edition). Weinheim & Basel: Beltz PVU. Pp. 469-494.
Pahnke, J. & Pauen, S. (2009): Development of Mathematical Thinking. In: Pauen, S. & Herber, V. (Ed.): From being small to Einstein. Berlin: Cornelsen Scriptor. Pp. 22-40.
Quaiser-Pohl, C. (2008): Promotion of mathematical precursor skills in kindergarten with the program "Playing Math". In: Hellmich, F. & Köster, H. (Ed.): Pre-school educational processes in mathematics and in the natural sciences. Bad Heilbrunn: Klinkhardt. Pp. 103-125.
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