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Can Early Algebra lead non-proficient students to a better arithmetical understanding? - page 2 / 3

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different social and cultural backgrounds of primary school children. This is all the more important because, at least in Germany, children with migrational background are disadvantaged in the educational system (see Auernheimer 2003). But there are some other reasons, why it can be a good way not to build on children’s previous knowledge while teaching mathematical concepts. McNeil (2004) observed, that “the activation of existing knowledge can interfere with the acquisition of new information”. She explicitly refers to pre- school knowledge as well. El'konin (1975) differentiates between theoretical scientific and empirical knowledge. Empirical knowledge designates knowledge children extract from their everyday experiences, while theoretical, scientific knowledge is knowledge on a higher level. “The adult – the teacher – is the key figure and helps the child to develop ways of operating with objects through which he can discover their essential properties – those which constitute genuine concepts.” (El'konin 1975, p. 48) Hasemann & Stern (2002) addressed their research to the question how to foster mathematical understanding of lower achievers. They worked with 2nd graders on different programmes and drew following conclusion: “Die Auswertung der Tests ergab, dass bei schwächeren Kindern das alltagsnahe Programm eindeutig am wenigsten bewirkte, während bei den Kindern, die das abstrakte Programm durchlaufen haben, der größte Leistungszuwachs zu beobachten war.“ (Hasemann & Stern 2002, p. 222) „The evaluation of the tests shows, that the program close to everyday life definitely had the lowest effect on low-achievers while children who worked on the abstract program showed the biggest learning progress.”(translation by the author) Thus we are looking for an abstract teaching program that gives children tools that can aid them with solving mathematical problems of everyday life and also with solving the word problems in their text books. Thereby it is important that abstract does not mean doing it without concrete materials. If young children shall cope with abstract knowledge this knowledge has to be taught in an action-oriented way. An unconventional way of teaching Early Algebra In the first years of school children usually spend a lot of time with calculating with natural numbers. They also adopt strategies that cannot be transferred to calculations with decimal numbers or fractions. With a teaching experiment in the 60s Davydov (1975) chose a different approach to mathematical education. His idea was teaching the properties of numbers while already using the common algebraic symbolizations and before introducing numbers at all. Therefore he chose an action-oriented way by using direct comparison of magnitudes like length, area, volume, mass, time and so on. The children used concrete material like water containers for comparing volume and balance scales for comparing mass and learned to write down their findings with inequations. The question, how big the difference between the compared magnitudes was, lead to equations. Aided by the concrete material, the children learned to manipulate and to interpret different linear equations. After the children have learned dealing with the equations properly numbers are implemented by introducing a unit. This way of implementing numbers not only works for natural numbers but for the whole real numbers. Davydov’s idea was taken on by the MeasureUp-Program (see Dougherty & Venenciano 2007), which showed that children can successful deal with abstract equations, achieve a deep understanding of properties of numbers and use them effectually for solving word problems. Early Algebra as a guideline for word problems Certainly starting mathematical education without using numbers would be a big change for the German school system and would hardly become accepted by teachers and parents. But the idea of teaching the abstract properties of numbers by the aid of concrete comparison of magnitudes while firstly excluding numbers deserves a closer look in terms of its usefulness for helping children deal with word problems and mathematical problems of everyday life. The main questions are: Will the MeasureUp-Program work for school children of different grades although the already have been introduced to numbers and arithmetical operations? Can they transfer the knowledge about abstract equations to mathematical problems of everyday life? And can this program lead non- proficient students to a better arithmetical understanding? In a first project we modified the MeasureUp-Program for the use in a few weeks lasting teaching-experiment in grade three and fife. After the children have been introduced to the comparison of length, area and volume and the use of letters they learned how to set up and manipulate equations. To connect the abstract equations without numbers with word problems we gave the children word problems that contained letters instead of numbers. We the asked the children to make up word problems that are appropriate to given letter equations. Therewith we keep up our intention to firstly teach mathematical concepts and applying those to everyday life not till the children can handle the concept properly.

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