# Can Early Algebra lead non-proficient students to a better arithmetical understanding?

Sandra Gerhard Goethe-University of Frankfurt am Main, Germany gerhard@math.uni-frankfurt.de Abstract In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student's access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student's previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems. Introduction In the German national curricular standards (“Bildungsstandards”), the guideline for the curricula of the German federal states you can read the following: “Der Mathematikunterricht der Grundschule greift die frühen mathematischen Alltagserfahrungen der Kinder auf, vertieft und erweitert sie und entwickelt aus ihnen grundlegende mathematische Kompetenzen. Auf diese Weise wird die Grundlage für das Mathematiklernen in den weiterführenden Schulen und für die lebenslange Auseinandersetzung mit mathematischen Anforderungen des täglichen Lebens geschaffen.”(KMK, p. 6) “The mathematical education in primary school takes up, deepens and extends early mathematical everyday life experiences and develops basic mathematical competencies from those experiences. Thus the foundation is laid for learning mathematics in higher classes and for lifelong examination of the mathematical requirements of everyday life.”(translation by the author) Everyday life in mathematical education There are two contrasting ways to combine everyday life with mathematics: Looking at everyday life and trying to find mathematical content or learning mathematical concepts and applying those to ones everyday life. The former, which seem to be more in line with the quotation above, you can easily find in primary school textbooks. The German textbook “Das Zahlenbuch” for 4^{th }graders for example shows a map of Germany to motivate distances (p. 10), a handicraftsman to motivate calculating with money (p. 22) and a recreational lake to motivate calculating with decimal numbers (p. 71). There is also a double page about Christmas (pp. 122/123) and Easter (pp. 124/125) and a page about the benefits of mathematics (p.126) showing among others a doctor, a retiree and a consumer advisor, all talking about why they need mathematics. In the textbook you also can find a lot of word problems which are linked to the alleged everyday life of children, like car inspections (p.66) or buying lentils (p.73). Looking at the textbook brings up some questions: Is this everyday life of all children in our multisocial and multicultural society? Can you really find everyday life that all children have in common? Is it necessary to base mathematical education upon everyday life at all? There is no doubt that the mathematical background of children, the similarities and differences which arise by reason of children growing up in different quarters of a town to the point of totally different cultural backgrounds should definitely be part of mathematical education. But if you look at the background of children in a today classroom, one can easily see that there are a lot of differences and that it is hard to find a similarity for all of the children. The one everyday life which fits for all children in classroom does exist. Instead there is to find a way to look at the everyday life of every child in the classroom. A way of doing this can be the latter mentioned above, teaching mathematical concepts and letting the child apply those to its everyday life. But the question “How can you use this in your everyday life?” is hardly to find in textbooks and classrooms. Some reasons for putting everyday life on hold Teaching mathematical concepts without coming back to the student's previous knowledge of everyday mathematics and applying those concepts to everyday life later can be a way to cope with the