Helen Chick

Solving the Problem with Algebra

This chapter draws together the major themes emerging from the 12th ICMI Study on The Future of the Teaching and Learning of Algebra and serves as an introduction to this book. The chapter begins with a short description of the major challenges that the teaching of algebra presents to researchers, curriculum writers and teachers. There follows a brief introduction to each chapter which surveys some key ideas presented, after which the significant suggestions for future algebra teaching and learning from that chapter are highlighted. The chapter finishes by drawing together the major themes that offer guidelines for making the future of the teaching and learning of algebra brighter than the past.

Data Representation and Interpretation by Primary School Students Working in Groups.

Twenty-seven Grade 5/6 students working in triads considered a supplied data set. They were asked to hypothesise about associations in the data and to represent these. Each student was classified according to the level of interpreting the information, the level of representing the chosen data, and the type of collaboration observed in the group. Levels of interpretation and representation skills were related and there was some indication of a possible association with the type of collaboration. There was no association of type of collaboration and students’ views on group work. Implications for future research and the classroom are considered.

Teachers' beliefs about statistics education

Beliefs have long been known to affect teaching and learning. In statistics education, little research has been conducted on the nature of teachers’ beliefs, despite the likely impact these beliefs have on teachers’ activities. This chapter first considers content-focused beliefs about statistics, its relationship with mathematics, and its place in the curriculum, before addressing beliefs associated with teaching and learning statistics. Influences on beliefs and the impact of beliefs on teaching are considered, and suggestions for further research are proposed.

Teachers of Mathematics as Problem-Solving Applied Mathematicians

Some of mathematics teaching is routine, like an exercise from a textbook for which you have received instruction and already know what to do. On other occasions, however, teaching mathematics is challenging, involving problems of teaching for which the solutions may not be readily apparent. These situations require the application of mathematical knowledge in concert with other types of knowledge for teaching. In this article, we explore the idea that teachers of mathematics act as applied mathematicians in applying mathematical knowledge to the resolution of teaching problems. This task involves the complex interplay of mathematical and teaching knowledge and processes of problem solving with success judged according to how well students learn. The article discusses these ideas through an examination of seven scenarios.RésuméTout comme certaines tâches mathématiques relèvent de la routine, par exemple la résolution d’un problème tiré d’un manuel, l’enseignement de certains aspects des mathématiques est également routinier. Mais il arrive parfois que l’enseignement des mathématiques présente un défi intellectuel et pédagogique, en raison de problèmes didactiques dont les solutions ne sont pas immédiatement apparentes. De telles situations exigent l’application de connaissances mathématiques de concert avec tous les autres types de connaissances nécessaires pour enseigner. Dans cet article, nous explorons l’hypothèse que les enseignants de mathématiques se comportent comme on le fait généralement en mathématiques appliquées lorsqu’ils appliquent leurs connaissances mathématiques à la résolution de problèmes pédagogiques. Ces tâches impliquent une interaction complexe entre des savoirs mathématiques et pédagogiques, ainsi que des processus de résolution de problèmes, dont le succès est déterminé par la qualité de l’apprentissage des étudiants. L’article se penche sur ces questions par le biais de l’analyse de sept scenarios dans lesquels figurent différents types de défis pédagogiques.

Collaborative influences on emergent statistical thinking — a case study

Abstract The purpose of this case study is to examine how collaboration affects the emergent statistical thinking of a group of three Grade 6 boys. Results of previous studies of students in Grades 3, 6, and 9 suggested that (a) when finding and justifying associations in data sets students working in groups may produce higher level outcomes than those working individually, and (b) there are numerous factors that influence the success or otherwise of collaborative activity. The current study, based on detailed analysis of video tape and transcripts of a group working collaboratively on a data handling task, documents various factors that affect collaboration and how these contribute to the attainment of desirable cognitive outcomes in terms of the task set. These outcomes are classified by emergent statistical themes and insight is gained into how naive statistical thinking begins to develop during the collaborative process. Implications for educators and researchers are considered.

Improving Teachers’ Professional Statistical Literacy

Given the deluge of data that school principals and teachers receive as a result of student assessment, it has become essential for them to have statistical literacy skills and understanding. Earlier work with primary and secondary teachers in Victoria revealed that, although most saw school statistical reports as valuable for planning and thought that they could adequately interpret them, their confidence was often not well founded, with some fundamental misconceptions evident in their statistical understanding. Based on these results, a workshop was developed to target key aspects of statistical literacy particularly relevant to the education context. The workshop incorporated simple hands-on activities to develop understanding of box plot representations, critiquing descriptions of distributions and applying the newly learned principles to participants’ own school reports. Although principals and teachers responded favourably to the activities, delayed post-testing indicated limited retention of the relevant aspects of statistical literacy. These results suggest that when teachers are dealing with data on only one or two occasions in a year, it may be important to provide timely and efficient access to reminders of basic concepts.

A statistical literacy hierarchy for interpreting educational system data

As a result of the growing use of state and national testing of literacy and numeracy among school students, there are increasing demands for teachers to interpret assessment data. In light of this, there is a need to provide benchmarks or a framework that identifies critical aspects of teachers’ understanding that are needed to interpret data effectively. In this study, 24 items from the Attitudes and Statistical Literacy Instrument are used with 704 teachers to provide a hierarchical scale of teacher ability to interpret these assessment data. Using an item response theory model for partial credit scoring, three levels of ability are identified, related to reading values, comparing values, and analyzing a data set as a single entity. Teacher ability is then compared across various demographic variables, such as number of years of teaching, main teaching grade levels, previous professional learning experience, last time statistics was studied, and gender. Implications are drawn for professional learning for teachers and for further research.

Teachers’ perceptions of the factors influencing their engagement with statistical reports on student achievement data

In Australia, as in other countries, school students participate in national literacy and numeracy testing with the resulting reports being sent to teachers and school administrators. In this study, the Theory of Planned Behaviour provides a framework for examining teachers’ perceptions of factors influencing their intention to engage with these data. Most teachers perceived the data to be useful, but there were some negatively held views. For both primary and secondary teachers, males were more positive and had weaker perceptions of barriers to their use of data from system reports compared to females. Teachers who had studied statistics at the post-secondary level and/or attended relevant professional learning generally felt more capable of using the data, and senior teachers and principals were more favourably disposed to using these kinds of statistical reports. Many teachers had concerns about the timeliness of the data’s release and the effort required to interpret them.

A Case Study of Graduate Professional Development in the TAFE Sector

There has been very little provision of professional development for teachers of mathematics in the technical and further education (TAFE) sector and even less evaluation and reporting on such programs. This paper presents a model for a professional development program in mathematics that was designed for TAFE teachers, as well as those from the K-12 sector. It is a case study that describes the implementation of the model with a group of TAFE teachers, considering such issues as the professional development needs perceived by the teachers, the components included in the program, and the evidence for the short- and long-term effectiveness of the program. The flexible structure of the program and the qualitative research base for the evaluation could be used as a model for professional development in other TAFE curriculum areas.

Commutative quasirecular rings with isomorphic additive and circle composition groups, ii: rational algebras

We show that any commutative nil ring which is an algebra over the rationals has its additive group isomorphic to its circle composition (or adjoint) group and demonstrate the importance of divisibility in establishing this result. We conclude by showing that if a ring has isomorphic additive and circle composition groups then this property need not be inherited by its quasiregular subrings.