LOGO and Elementary Mathematics Education in Portugal

LOGO and Elementary Mathematics Education in Portugal

Paulo Carvalho (Cávado Sul Group of Schools, Portugal), Pedro Palhares (University of Minho, Portugal), António Osório (University of Minho, Portugal), Alexandra Gomes (University of Minho, Portugal) and Ema Mamede (University of Minho, Portugal)
DOI: 10.4018/978-1-4666-0068-3.ch018


In the second part, the authors describe an elementary teachers’ training programme on mathematics education that during four years has every year provided a small time for the exploration of LOGO and promoted its use to teachers in the classroom. They reflect on this program and offer insights on teachers’ resistance to change their practices. The authors conclude the chapter with their own ideas for the implementation of LOGO in a group of schools in the near future.
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Reflections on the Teaching and Learning of Mathematics

When we think about the teaching and learning of mathematics we can consider two contrasting views. One is the traditional view, which is the one dominant among teachers at the start of the initial training (Brown & Borko, 1992). The other is the constructivist view. We defend the constructivist view, however with some nuances.

The first distinction between the two visions of mathematics education concerns the acquisition of knowledge. Thorndyke’s connectionism theory, later reinforced by Skinner’s behaviourism, followed the tabulae rasae principle, implying the transmission of knowledge (Orton, 1992).

Constructivism, in all its variants, defends a different perspective: every learner constructs his/her knowledge. For some variants of constructivism, knowledge is socially originated leading theoretically to the conclusion that all knowledge can be constructed in the classroom (Ernest, 1991). For other variants, however, there is knowledge of a social origin and knowledge of a logical type (Nunes & Bryant, 1996). Hewitt (1999), in particular makes a distinction between arbitrary things (students must be informed) and necessary things (students should construct them).

This is also our view: mathematics curriculum should take into account those things that have to be transmitted by the teacher (number names, algorithms, etc) and those that have to be constructed by the students (notion of number, etc).

Another distinction we can perceive concerns mathematical tasks. In the beginning of the twentieth century Thorndyke proposed that exercises should be the core of mathematics teaching. Exercises and the practice of routines became so valued that even nowadays they are everywhere (Kilpatrick, 1992). However, a problem solving approach arose with Pólya (1957) and is now suggested in many curricula around the world. Nowadays, mathematics teaching and learning also includes problem posing, investigations and open problems. The trend is to enhance cognitive demand and shift the decision power to the learners (Ernest, 1991).

In the traditional view of teaching and learning Mathematics, exercise solving is the fundamental component. For us, however, adequate Mathematics teaching-learning includes problem solving and investigations as a fundamental component.

Watson and Mason (2007) point out that tasks do originate students’ activity, but this activity may not be what the teacher intended implicitly when planning their teaching. Learners’ activity is affected by personal dispositions, and some learners may jump to the first idea they have, while others will wait to be told what to do. Learners’ activity is also influenced by what Brousseau (1997) called the milieu, which include the established practices or learners’ expectations as influenced by the system, both in the present and in their past.

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