Increasing Confidence in Mathematics Ability Through Logo and Collaborative
Dr Anthony Jones
Graduate School of Education
La Trobe University
Melbourne, Australia 3083
tel: +613 9479 2484
fax: +613 479 3070
Pre-service courses for most Australian primary teachers are 3 or 4 year undergraduate education degrees. La Trobe University offers a 1 yearpre-service course for non-education graduates. The majority of studentsin this course are female (90% in 1996 and 85% in 1997) and few have taken a mathematics subject in their undergraduate degree. In addition to learning how to teach mathematics, these students also have to confront an increasing use of computers for mathematics in primary schools.
For many years Logo has been a major focus of the mathematics education component of this course. In 1997 Logo will be supplemented by increased emphases on collaborative learning and teaching approach designed to be more appropriate for the mainly female participants. This paper briefly describes these three components and the theoretical background that led to their use.
teacher training, mathematics education, female students
This paper describes that planning and the theoretical background for a pre-service primary teacher education mathematics subject. While data is continually being collected from the participants, both students and teachers, insufficient has elapsed to enable any form of detailed analysis. The teaching approaches being applied involve the use of Logo and follow many of the beliefs expressed by Papert ,.
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 in Melbourne, Australia, has offered graduates a one year primary teacher certification course based on the secondary model.
Prospective primary teachers have to undertake studies in all eight subject areas that they will be expected to teach. At La Trobe University 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 throughout the year.
A significant number of students who enrol in the primary course have a weak mathematics background. This weakness usually manifests itself as a dislike of the subject or a complete lack of self confidence. Some students display both characteristics. For 1997 two new staff will plan and teach the subject. In an attempt to take positive steps to overcome these problems, the staff have agreed to introduce a different approach based around a combination of Papert's philosophy of learning mathematics through Logo , emphasising social interaction in learning through collaborative learning practices, and adopting a 'feminist mathematics pedagogy' .
3 Theoretical background
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' [2 p.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" [2 p.52]. This is something that is not understood by teacher education students who fear mathematics. Many of them have learned algorithms and facts without any associated understanding of the mathematical concepts involved. When they are in front of a class they reproduce both the content and the manner of their own learning.
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 is the approach that is being employed in this project. Apart from their initial explorations of the Logo environment in the MicroWorlds Project Builder context, the teacher education students will mainly work within mathematical microworlds. Those who desire to learn Logo programming will be encouraged and assisted, but this will not be part of the subject.
While writing procedures will not be a focus of this subject, the teaching strategies used by staff, and consequent implications for the mathematics curriculum in primary schools, will be issues for regular discussion and investigation. It is hoped to establish a Logo learning environment that encapsulates parts of Papert's philosophy for learning, together with cooperative learning in small groups and regular discussion and reflection.
Despite all the precautions that are being taken, it is still possible that the teacher education students in this project might not make cognitive links between what they are doing 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. Later in the year the pre-service teacher education students will have the opportunity of visiting 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 La Trobe University 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'  'feminist mathematics pedagogy' offered a range of practices that would interconnect with other desired facets of this subject. While many of the practices advocated by Rogers were already part of normal practice for the staff involved in this subject, examining research in areas such as gender equity in mathematics, and women's moral development provided insights into methods that might be appropriate for all students.
Similarly, the use of cooperative learning has been built into strategies used for this subject. There is a substantial body of research in this area (e.g. , ) 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 cooperative 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 cooperative learning practices it is hoped that students will never feel that they are on their own struggling with mathematics.
It is too early in the academic year (in Australia) to be able to make objective judgements about the degree of success for students enrolled in this subject. However initial indications are positive. Toward the end of 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.
In May the students spend three weeks teaching in local primary schools. We will know more about the effect the subject might be having as we watch these beginning teachers grapple with the demands of teaching real primary students in real classrooms.