Robyn Pierce

A Scale for Monitoring Students' Attitudes to Learning Mathematics with Technology

Abstract The Mathematics and Technology Attitudes Scale (MTAS) is a simple scale for middle secondary years students that monitors five affective variables relevant to learning mathematics with technology. The subscales measure mathematics confidence, confidence with technology, attitude to learning mathematics with technology and two aspects of engagement in learning mathematics. The paper presents a model of how technology use can enhance mathematics achievement, a review of other instruments and a psychometric analysis of the MTAS. It also reports the responses of 350 students from 6 schools to demonstrate the power of the MTAS to provide useful insights for teachers and researchers. ‘Attitude to learning mathematics with technology’ had a wider range of scores than other variables studied. For boys, this attitude is correlated only with confidence in using technology, but for girls the only relationship found was a negative correlation with mathematics confidence. These differences need to be taken into account when planning instruction.

Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software.

This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its availability in classrooms also provides opportunities for positive changes to teaching and learning. Very many examples are documented in the professional and research literature, and in this paper we organize them into 10 types. These are displayed in the form of a ‘pedagogical map’, which further classifies them according to whether the opportunity arises from new opportunities for the mathematical tasks used, change to interpersonal aspects of the classroom or change to the point of view on mathematics as a subject. The map can be used in teacher professional development to draw attention to possibilities for lessons or as a catalyst for professional discussion. For research on teaching, it can be used to map current practice, or to track professional growth. The intention of the map is to summarise the potential benefits of teaching with technology in a form that may be useful for both teachers and researchers.

Computer Algebra Systems and Algebra: Curriculum, Assessment, Teaching, and Learning

Computer algebra systems (CAS) were originally designed for mathematicians, scientists, and engineers, and their implementation into education, and especially into schools, is still very much in its infancy. This situation is reflected in the research on CAS and hence our ability to describe the influence of CAS on student learning in algebra. This chapter synthesises key results from research related to using CAS in algebra from two broad perspectives: firstly, the issues of curriculum, assessment, and teaching; and secondly, that of student learning. Our analysis supports the view that CAS has much to offer the teaching and learning of algebra, but that real benefits may accrue only from thoughtful and structured approaches which take into account the perspectives of the student and teacher, and the intricacies of the relationships between student, teacher, and CAS. In writing the chapter we are mindful that many more questions than answers have emerged and we have included a significant number of these in the hope that they may give impetus and direction to research in the area.

Observations on students’ responses to learning in a CAS environment

This paper reports on the responses of students using DERIVE, a computer algebra system, in an introductory calculus course at undergraduate level. There was strong evidence that the use of this technology was a catalyst for students to use three learning strategies which the literature links to enhanced learning outcomes: regularly using multiple representations, discussing meaning with peers and teachers, and including the computer in the conference process. Students felt that using DERIVE in their classes aided but did not underpin their learning of mathematics.

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.

The Influence of a Computer Algebra Environment on Teachers’ Practice

Using a computer algebra system (CAS) in the classroom provides many opportunities for improving student learning. However, to take advantage of such a powerful instrument as CAS requires changes to many aspects of the classroom. The different ways in which three pioneering Australian teachers adapted their teaching to use CAS are described (see also Appendix 4-1, for a comparison with similar experiences in other countries). Two of the teachers taught an eight-week course in each of two consecutive years (the CAS-Calculus project) at secondary school level, using a symbolic calculator. They gave CAS different roles in the instruction and in defining their curriculum goals. One teacher used CAS in a restricted way with the primary goal of increasing understanding while the second teacher adopted CAS as another technique freely available for solving standard problems and emphasized efficient routines. Over several years (in a separate project, at university level, using the computer program DERIVE), the third teacher has evolved a method of teaching with CAS, moving from an early emphasis on teaching about CAS as a tool and using it for difficult problems to incorporating its use for primarily pedagogical aims. In reporting on these case studies we comment on different ways of organising the classroom; the variety in approaches to teaching the use of CAS; the increased range of methods for solving problems and for teaching; the contrast between using graphics and symbolic calculators; the place of paper-and-pencil skills; devoting time to mathematics or technology; and finally the curriculum and assessment changes required in schools.

The design of lessons using mathematics analysis software to support multiple representations in secondary school mathematics

Current technologies incorporating sophisticated mathematical analysis software (calculation, graphing, dynamic geometry, tables, and more) provide easy access to multiple representations of mathematical problems. Realising the affordances of such technology for students’ learning requires carefully designed lessons. This paper reports on design research conducted with nine teachers in nine classes across two schools. The lesson came at the end of their study of quadratic functions (Year 10). Technology offered the ability to link representations and the teaching challenge was to use this ability effectively. Four issues emerged: variable naming; reducing cognitive load; maintaining motivation; and retaining a clear learning focus.

Using Dynamic Geometry to Bring the Real World Into the Classroom

This chapter reports on the use of dynamic geometry to support the use of real world contexts to enhance the learning of mathematics in the middle secondary years. Dynamic geometry, either linked to real world images or used to create dynamic simulations, not only can provide opportunities for students to collect real or simulated data to make conjectures, but also can improve their understanding of mathematical concepts or relationships through exploration. Tasks, which access these features, can be valuable for both increasing students’ engagement and their depth of mathematical thinking. The colour, movement and interaction can create a halo effect valued by teachers for its impact on students’ general attitude towards studying mathematics.

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.