Description of Main Innovative and Alternative Methodologies for Mathematical Learning of Written Algorithms in Primary Education

The traditional way of teaching mathematics generates significant learning difficulties in students that are reflected in their academic performance. In recent years, the number of teachers and researchers interested in finding innovative, flexible and comprehensible methodologies has increased. The main objective of this article has been to carry out a theoretical review of the methodologies for learning/teaching mathematics at school level. Central elements of international and national method initiatives have been highlighted. The empirical evidence on the Singapore method and the Algorithm Based on Numbers (ABN) method confirms the need to include innovative and manipulative strategies in the classroom. The Singapore method has been shown as a source that improves students' problem solving skill, mathematical competence, boosting reasoning, and a higher motivation. Some studies focused on the effectiveness of the ABN method and its influence on mathematical cognition. The use of this methodology and learning in flipped classroom have obtained significant results in teacher training. These results could provide guidance about how to improve pre-service education in Primary Bachelor's Degree. The findings presented in the manuscript could be a basis for opening new lines of quantitative research, with the aim of analysing problem solving and the use of manipulative materials in mathematics. Thus, future research should focus on analysing the cognitive processes involved in mathematical learning, carrying out empirical studies in schools. In addition, it is necessary to improve future teachers' training, so that they can learn about new alternatives for mathematical teaching and the available resources to be able to put it into practice. Moreover, alternative methodologies are a necessary driver for the improvement of mathematical performance both inside and outside the classroom, and also for the technological and economic development of countries.