The transition from arithmetic to algebra
The transition from arithmetic to algebra
- Diagnosis-based experiment use of resources from Japan
Doctoral candiate
Abstract
The starting point for this project was a longstanding public concern about the state of school mathematics in Denmark, compared with results from other countries, in particular in East Asia, and not least when focusing on arithmetic and algebra. Our first step was to develop a method for diagnosing the current “state” in a more comprehensive way than to simply look at students’ results on tests and the obvious difference between school mathematics curricula in Denmark and in East-Asian countries like Singapore and Japan. The diagnosis is based on praxeological analysis in the sense of ATD (the anthropological theory of the didactic). It shows a gap between the official goals and the current teaching where algebra is not extensively taught as a modeling tool and students’ success with associated tasks is similarly modest. The didactic transposition of school algebra appears rather fragmented, distributed over several years and without a core progression that is actually taught to, and learnt by, all students.
The diagnosis provides further motivation and context to the main goal of this thesis: investigating to what extent a research-based textbook material can support teachers (who do not habitually have access to such material) in the teaching of introductory school algebra. Concretely, we have investigated how Danish lower secondary school teachers use a translation of the relevant chapter from a Japanese textbook, with almost no instructions for how to use the text; and to what extent this use supports the students’ transition from arithmetic to algebra.
Both the differences in mathematical progression in the two curricula, and in habitual organization of didactic processes, lead to expect a number of obstacles to the implementation of the Japanese text in a Danish context. While these were also to some extent observed, we found a number of unexpected or at least non-trivial potentials of the use of the Japanese material.
Teachers successfully used the chapters’ challenging “launch problem” to demonstrate school algebra as a modelling tool and to furnish a common example for several points in the chapter, and they were able to implement the focus on explaining and justifying important notational conventions (like the suppression of the multiplication symbol in most algebraic expressions). Concrete tasks and explanations in the text were generally used in the teaching without substantial obstacles. It appeared to be much less straightforward to realize the textbook’s aims concerning theory (general principles and definitions). Despite these challenges, our data suggest that the teacher’ didactical and mathematical profit from using the Japanese textbook went beyond what appeared directly in their first attempt to use it in teaching, so that later use could conceivably realize the potential of the text more fully.
Assessment Committee
Associate Professor Marianne Achiam, University of Copenhagen
Professor Heidi Strømskab, Norwegian University of Science and Technology, Norway
Professor Takeshi Miyakawa, Waseda University, Japan
Supervisor
Professor Carl Winsløw