Problems with Fallibilism as a Philosophy of Mathematics Education

Much reference has been made to Paul Ernest??s ??philosophy of mathematics education?? to legitimise a strong fallibilist trend in mathematics education. This article presents the argument that: (1) This philosophy makes unwarranted assumptions that have been taken as ??given??. For example, that ??absolutist?? or ??Platonist?? views of mathematics necessarily imply the transmission model of teaching mathematics. (2) The very basis of this philosophy contains a contradiction: that mathematics cannot be separated from its social origins, yet mathematics has a logical necessity that is independent of its origin. (3) This philosophy downplays mathematics as a formal, academic system of knowledge in the attempt to promote a child-centred pedagogy or the mathematics of social practices. (4) Ernest??s attempt to semiotically reduce proof to calculation is flawed. This article explores what is meant by fallibilism in relation to the views of many educationalists who appear not to like mathematics as a formal, academic body of knowledge and draws out the educational implications of these views.     

Learning What It Takes to Teach Science: High School Students as Science Teachers for Middle School Students

This study examines the influence of St. John's University Summer Science Experience and Teacher Mentoring Program on African American and Hispanic high school students' interest in science and science teaching as career goals. In the first phase of the program, high school students from six school districts in Suffolk County, Long Island (a suburb of metropolitan New York City) engaged in investigative science experiences that emphasized environmental science, chemistry, and technology and learned about effective science pedagogy. The second phase of the program functioned as a teaching practicum for the high school students, where they planned for instruction and taught middle school students investigations similar to those that they had engaged in during the summer program. Various surveys were developed to assess high school students' attitudes about science and science teaching, knowledge of effective teaching approaches, knowledge of ways to motivate younger students, and the overall impact of the program on the high students' interest in science and/or science teaching as career goals. Program evaluations reveal that over 75% of the students expressed an interest in considering science or science teaching as career possibilities. Implications for minority teacher recruitment are discussed.     

The intersection of culture and institutional support for Latino students’ academic success: Remediation or empowerment?

The study is framed by critical race theory to explore the intersection of cultural and institutional factors that influence Latino students’ completion of high school. The purpose of this study is to determine the extent to which factors related to students’ background, culture, socioeconomic status, and institutional-support such as participation in mentoring and/or dropout-prevention programs, can predict Latino students’ successful completion of high school. The overarching research question is: To what extent do family background, students’ educational aspirations, and institutional support programs predict whether Latino students’ complete high school? Using data from the Education Longitudinal Study of 2002 (ELS: 2002), from the National Center for Education Statistics (NCES) with 2,217 Hispanic participants, the study used a logistic linear regression model for the analysis. The findings identified students’ gender, socioeconomic status, first language, educational aspirations as well as the aspirations of their parents, school poverty concentration, and school support programs to be significant predictors of high school completion. The logistic regression model correctly classified between 78%, 85%, and 81% of the cases included in the group for timely completion according to first-, second-, and third-generation respectively. A similar classification was found for high school completion-within-two-years. The discussion highlights marked differences between the effect of dropout-prevention programs and that of mentoring programs on Latinos’ high school completion. In addition, that the factors represented by individual and institutional variables might not operate in isolation but instead might intersect with socioeconomic and cultural factors that ultimately create barriers for this minority group.     

Teacher Talk During Whole‐Class Lessons: Engagement Strategies to Support the Verbal Participation of Students with Learning Disabilities

The analysis identified discursive strategies used by general education teachers in inclusion classrooms to orchestrate and scaffold the verbal participation of all students, including students with learning disabilities (LD). The context was writing instruction. A whole‐class lesson involving teacher–student collaboration to write a text was analyzed for each of two teachers in two urban elementary inclusion classrooms totaling 67 students; 23 students had LD. Analysis of teacher talk focused on procedural strategies (help the lesson run smoothly and make it easier to follow) and involvement strategies (elicit students' attention to and participation in the lesson). Results indicated that both teachers used a variety of similar strategies to provide spaces for student contributions and, at the same time, move the lessons along. However, they also used contrasting strategies unique to their contrasting pedagogical frames of reference (structural vs. interactional).     

Assessing children's collaborative skills by paired observation

Summary

In some ways it now becomes too easy to compare — boys with girls, Jonathan with Michael (and other leaders), older with younger. My colleague's girls were more egalitarian than her boys; my least successful group, however, consisted entirely of 3rd year girls. Jonathan's leadership was of a notably selfish kind, but there were no other boy leaders of his age to compare with. There was certainly a feeling of development, younger to older, in terms of ability to plan, to sustain an investigation over time and draw conclusions when working together. Nevertheless one older group (3rd years) spent all its time trying to arrive at a workable social structure. The older often chose to co-operate in pairs with friends and showed considerable respect for each other's observations and ideas and care over their modification. At the same time, throughout both classes, children worked intensely in all their appointed groups on set tasks, even when no direct collaboration or conversation took place at all.

These facts in themselves make the evaluation of process and outcomes difficult. How, in effect, did we first recognise and second assess group success? Was joint completion of a task the only criterion or could social interaction be considered too? How important, indeed, was interaction to the final result?

If we say that achievement of a result of some kind through talk and mutual help was the main aim, then most of the older children actually brought that about. All but two groups came to a point of conclusion which they were able to discuss and summarise and those groups most closely monitored, either by tape-recording or adult eavesdropping, showed an ability to work very closely together and move forward through contributive conversation which seemed to exclude no-one. If I had the space to quote more copiously from tapes and notes the intensity of some of these sessions would become more apparent — the best of the 4th year girls' groups sometimes approaching an hour during which the quality of listening and concentration is clearly as important as the round of contribution. The boys' groups and some of the pairs were often more laconic, sometimes lapsing into silence for pondering or writing or moving around, but their overall persistence and their ability to plan and co-ordinate their efforts were notable.

My colleague claims less in terms of planning and outcome. Interactions did take place in her groups, sometimes with a tangible result, like a piece of floating plasticine, but children often worked alongside rather than together — in the group rather than with the group — with the marked exception of the girls' water experiments, where they were not only contributing in close canon, but narrating for the tape at the same time.

If one includes a social value in group work it is possible that even the group in my class which most obviously failed even to set up a single agreed investigation went through some kind of social learning. They tempt one to believe that smaller groups must have been better than larger and they certainly did help some of the shyer to participate more fully. On the other hand, large groups which had good cohesion actually seemed to be able to refine ideas better, more effectively eliminate less workable ones and arrive at more practicable and communicable solutions.

Children self-monitoring through taping seems to have been immensely successful; with the older as an aid to continuity and a means of reflection; with all ages in providing an awareness of audience, — the need to shape and present the whole process and its outcome. It seems not to have affected spontaneity of contribution, even where children imposed structures for taping (such as members reporting in turn), and the whole thing was handled in a very matter-of-fact way.

For the teachers the tapes were revelatory and entertaining — providing a means of study more complete and far reaching than any other devised. Adult intervention by the visiting teacher could be equally thorough, but rarely equally unobtrusive. However, children who knew each other well and seemed to be working perhaps too intuitively together were actually forced to define things more precisely for themselves in the process of reporting to an adult. The shortcomings of groups, older or younger, may ultimately return to this fact — an inability to see the need to share surmises, either with each other or with the community beyond the group. This leads me to believe that independent learning needs, nonetheless, to be carried out with eventual accountability in mind.

The benefits accruing to the two teachers are an echo of these thoughts. The very presence of another adult, planning, observing and discussing, raised the level of interest for both of us. We each had an audience and a means of sharing current problems, discussion on a number of occasions bringing about developments like my colleague's use of music to encourage greater mutual awareness and team-work in her younger children. Our perceptions of each other's children at work were often more objective and useful because they were less involved and, as my colleague pointed out, we tended to look at each other's groups much more as groups and much less as familiar individuals in groups. In practical terms it was also much easier for a visitor to a room to give entire attention to one group for a long period than it was for the class teacher who had overall responsibility for the activity of every group and individual in the room.

Some of the tasks performed by the visiting teacher — listening, talking, guiding — could equally have been (and often were) performed by the class teacher, but the greater value came in the later sharing of impressions — a sharing which led to modification, to a better assessment of success and failure, to useful comparison between age groups and to a wider broadcasting of these awarenesses to other teachers.

In our opinion, group tasks can be made more effective by careful setting up, the provision of awareness-raising frameworks, supporting collaborative exercises, questions to be considered (‘What do you think you have found out so far?’) and precise aims laid down for each session. But these do not preclude failure or actually deliver interactive skills. They are aids to learning — learning which may continue given the opportunity (a) to go on working together and (b) to reflect on the positives and negatives of each piece of work through writing and/or open class discussion.

All that one can say of the children applies equally to the teachers.

Finally, it is now clear that under the national curriculum we shall be dealing directly with matters such as how children cope with group tasks and whether or not they can talk and listen effectively in groups. Consequently teachers will be called on to increase their awareness of various aspects of collaboration, so that they can foster and then assess them. We hope this article goes some way towards raising the level of that awareness.     

Towards Quality Systems Development in NSW Public Schools

While the notion of “quality systems” is firmly established in many public and private sectors, this approach to quality management is only beginning to emerge in schools. After an initial discussion of the notion “systems thinking” and “quality systems,” this article suggests a model for the essential criteria for quality systems in schools. This model is then utilised to evaluate 3 widely accepted approaches to quality management within Australia for their applicability to the school situation – ISO 9001/4 2000, Investors in People and the Australian Business Excellence Framework. While it is suggested that all 3 approaches could have some relevance to quality systems development in schools, the unique culture of schools would present significant challenges to their utilisation.     

THE OLDER HEARING‐IMPAIRED STUDENT: MANAGING OBSTACLES TO LEARNING

With the number of older Americans increasing, many colleges and universities are attempting to serve the needs of this population by offering tuition reductions, alternative scheduling, and special courses to older learners. Consequently, more and more older persons appear to be returning to colleges and universities on a part‐time or full‐time basis. However, many of these older students may not be receiving maximum benefit from their educational experiences, owing to hearing impairments. This article discusses obstacles and problems that confront the older hearing‐impaired student in the classroom and presents possible management solutions.