Full Paper

Anthony Jones

T.Jones@latrobe.edu.au

Graduate School of Education

La Trobe University

This paper considers the planning, some theoretical background, and the implementation of learning technology applications in a pre-service primary teacher education mathematics subject. In response to the major employer of teachers in the state, the application of computer-based material was a major focus in the subject. The teaching approaches used included some aspects of Papert's Logo philosophy concerning the use of computers in learning mathematics (Paper 1980), an emphasis on co-operative learning practices, and an attempt to cater for learning styles preferred by females who represent the majority of students.

For many years the majority of Australian secondary school teachers have completed a three year undergraduate degree and then a one year teacher certification course. In contrast, primary teachers have completed their secondary education and then attended a teachers' college for three years. For the past twenty years La Trobe University (LTU) has offered graduates a one year primary teaching certification course based on the secondary model.

Prospective primary school teachers have to undertake studies in all eight subject areas that they will be expected to teach. At LTU these learning areas are organised into subjects titled Language Studies, Mathematics, Science, Technology Studies, Physical Education, Visual and Performing Arts, and SOSE (Study of Society and the Environment). In addition to the subjects they will teach, students also study educational psychology and general teaching methodologies. Within this crowded academic curriculum mathematics is allocated four hours per week and technology two hours per week.

A significant number of students who enrol in the primary pre-service teacher education course have not studied mathematics at tertiary level, and also have a weak background in computers and related technologies. These weaknesses usually manifest themselves as a dislike for the content or a complete lack of self confidence. Some students display both characteristics. In an attempt to take positive steps to overcome these problems, in 1997 staff have introduced an approach based around a combination of Papert's philosophy of learning mathematics through Logo (Papert 1980), emphasising social interaction in learning through collaborative learning practices (Webb and Farivar 1994), and adopting a 'feminist mathematics pedagogy' (Rogers 1995).

Included in the section that follows are several references to Seymour Papert and Logo. This is partly because Logo and an associated "Logo philosophy" were designed into the teaching and learning activities developed for this subject. Another reason for referring to Logo is the substantial body of research into Logo that has been published. However other software, especially spreadsheets and data bases, can be used in a "Logo-like" way, but there is not yet the same quantity of research evidence as there is for Logo. Although the focus of this paper might appear to be on Logo, in actual fact a number of other pieces of software were used in conjunction with Logo.

Throughout much of his writing Papert has made constructive comments on how mathematics teaching might be improved through the use of computers. He decries 'school math' (Papert 1980:51) and the way it is taught. He also makes it very clear that just providing access to computer hardware and software will not necessarily improve mathematics teaching. However it is the attitude of teachers, and society in general, to mathematics that Papert believes must change before school mathematics will change.

Papert also notes that ".. what could change most profoundly in a computer-rich world [is that the] range of easily produced mathematical constructs will be vastly expanded" (Papert 1980:52). This is something that is not understood by teacher education students who are uncertain about both technology and mathematics. In mathematics many of them have learned algorithms and facts without any associated understanding of the concepts involved. When they are in front of a class they reproduce both the content and the manner of their own learning, and this completely devoid of computer applications.

There are many different approaches to using Logo in the classroom. In a mathematics classroom Logo can be used as an intermediary, a medium between the teaching and the content. In this case Logo is a means to an end, rather than an end in itself. When using Logo as an intermediary the teacher can make use of Logo in different ways. First Logo can 'curtail' the learning of mathematics. Logo programming that is taught as a part of mathematics is a common example of Logo curtailing the learning of mathematics. In itself Logo programming is not mathematical, and so time spent learning to program must reduce the amount of time available for students to learn mathematics.

Using Logo to 'camouflage' mathematics is another problematic approach used in some classrooms. Unless a teacher makes overt links in students' minds between Logo activities and associated mathematical concepts, little mathematical learning is likely to occur because the mathematics has been camouflaged by the Logo. For example, many mathematics teachers encourage students to construct various regular polygons. In some cases students continue to enter a series of forward and turn commands for each polygon. If all the happens in the lesson is that students produce a drawings of regular polygons and the basic code necessary for the drawings, then it is unlikely that much mathematics has been learned. However both the process and learning outcomes might be different if, after drawing a couple of regular polygons in a 'long-hand' manner, students are introduced to the REPEAT command and are encouraged to look for a relationship between the number of repetitions and the size of the angle turned by the turtle.

Many teachers use Logo as a 'catalyst' for the learning of mathematics. Logo conferences and books contain many examples of Logo microworlds and other activities that enable students to learn mathematics through the use of Logo. This type of approach, guided exploration, was employed in this subject with a range of software. Apart from their initial explorations of the Logo environment in the MicroWorlds Project Builder context, the teacher education students worked mostly within mathematical microworlds. Those who desired to learn Logo programming were encouraged and assisted, but this was not an integral part of the subject.

While writing procedures was not a focus of this subject, the teaching strategies used by staff, and consequent implications for the mathematics curriculum in primary schools, were issues for regular discussion and investigation. The plan was to establish a Logo learning environment that encapsulated parts of Papert's philosophy for learning, together with co-operative learning in small groups, and regular discussion and reflection on what was being learned.

Despite all the precautions that were taken, it is still possible that the teacher education students might not make cognitive links between what they have done and how Logo can be an integral part of primary school mathematics. In order to supply a rich source of anecdotal material, and also to provide classroom based credibility for the subject, one of the staff is teaching with Logo in two local primary schools. This involves spending a half day in each school every week, working with six teachers and their classes in grades 1/2 and 5/6. The pre-service teacher education students have opportunities to visit these classes.

The fear or dislike that many teacher education students bring to their mathematics classes can not be overcome simply by placing the students in front of computers. In fact doing this on its own can compound the problem if there is also some degree of technophobia. Because almost all of the students involved in the LTU primary pre-service course are female (7 out of 48 in 1997), an approach that research suggests would not alienate female students was sought. Although developed in a different context, Rogers' (1995) 'feminist mathematics pedagogy' offered a range of practices that would interconnect with other desired facets of this subject. In particular, Rogers claims that goals for a subject need to provided in written form and discussed at the first meeting. Goals should be sub-divided into content specific, process and communication.

The approaches to teaching and learning are also important when students are uncomfortable or unsure about a subject. Lectures, and the use of transmission mode teaching in general, were used sparingly. Instead most sessions began and concluded with whole group dialogue, which is student centred unlike the teacher centred lecture. Activities such as brainstorming and reflecting on what had been learned were part of the whole group dialogue process. For most of each session students worked in pairs, particularly when using computers, or in small groups of four when problem solving, investigating or presenting.

In a similar fashion, the use of co-operative learning has been built into strategies used for this subject. There is a substantial body of research in this area (e.g. Good, Mulryan, & McCaslin 1992; Webb & Farivar 1994) and a number of positive effects have been identified, including increased achievement, greater self esteem, and more time spent on task. However there are still gaps in our understanding of all the dynamics and effects of small group co-operative learning in mathematics. This is also true of mathematics education in pre-service teacher education subjects. Throughout the year students will work in a variety of small group contexts, including student selected and teacher selected groupings. By encouraging co-operative learning practices it is hoped that students will never feel that they are on their own struggling with either the technology or the mathematics.

At this stage no formal evaluation has been completed. However initial indications are positive. During the first class students completed a questionnaire. In several responses many of the students expressed stereo-typical ideas about teaching mathematics. Classrooms consisted of straight and rigid rows of individual desks, and the teacher was usually depicted as being at the board and talking. These responses are different in nature to what was observed in some student-generated role plays that were an activity in week four. In the role plays teachers sat among their students, and students worked together at solving problems.

Initially technology was seen as an integral part of primary school mathematics. As students have become familiar with the primary school curriculum, and have developed a range of personal computing skills, this attitude has begun to change. However the general perception is still that technology is an add-on to the curriculum rather being part of the core.

At the end of the academic year students will complete a formal, open-ended evaluation of the subject. When these evaluations have been analysed it will be possible to make comparisons with evaluations from previous years and to gauge the impact of the features introduced in 1997.

(c) Anthony Jones

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