The mutual influence of theory and practice in Mathematics Education:implications for research and teaching

Questions related on how to connect Theory and Practice in school mathematics are under debate from several years. Also, different forms of co-operation between academic researchers and school teachers are widely discussed. In the search of boundary conditions to mediate knowledge between the two poles, there is evidence that any conception which assigns to “theory” the place of instructing “practice” is doomed to fail and the necessity of developing the notion of cooperation comes as a consequence. According to this assumption, existing literature provides interesting contributions supporting the idea of blending mathematical content with pedagogical knowledge. This contribution focuses on the role that theoretical models, as emerged from the observation of students at work, can play on instructing practice. In particular we will approach algebraic thinking and refer to a theoretical model based on the distinction between sense and denotation of algebraic expressions. We will discuss how this theoretical model can shed light on students difficulties when solving equations and inequalities. Finally, we will point out how findings coming from research can suitably orient teachers and promote further development.