Kaye Stacey

Finding and Using Patterns in Linear Generalising Problems.

Linear generalising problems are questions which require students to observe and use a linear pattern of the formf(n)=an+b withb≠0. This study reports responses of students aged between 9 and 13 to these questions, documenting the mathematical models that they select, the strategies used in implementing them and the explanations they give. Substantial inconsistency of choice of model is observed; students who began a question correctly frequently adopted a simpler but incorrect model for more difficult parts of the question. Students who had undertaken a course in problem solving implicitly used a linear model more frequently and consistently and their explanations more often related the spatial patterns and the number patterns. They seemed to understand the relationship between the data and the generalising rule more completely.

STUDENTS' UNDERSTANDING OF ALGEBRAIC NOTATION: 11–15

Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns. Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names. The principal explanation given in the literature has been a general link to levels of cognitive development. In this paper we present evidence for specific origins of misinterpretation that have been overlooked in the literature, and which may or may not be associated with cognitive level. These origins are: intuitive assumptions and pragmatic reasoning about a new notation, analogies with familiar symbol systems, interference from new learning in mathematics, and the effects of misleading teaching materials. Recognition of these origins of misunderstanding is necessary for improving the teaching of algebra.

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.

The Impact of Teacher Privileging on Learning Differentiation with Technology

This study examines how two teachers taught differentiation using a hand held computer algebra system, which made numerical,graphical and symbolic representations of the derivative readily available. The teachers planned the lessons together but taught their Year 11 classes in very different ways. They had fundamentally different conceptions of mathematics with associated teaching practices,innate ‘privileging’ of representations, and of technology use. This study links these instructional differences to the different differentiation competencies that the classes acquired. Students of the teacher who privileged conceptual understanding and student construction of meaning were more able to interpret derivatives. Students of the teacher who privileged performance of routines made better use of the CAS for solving routine problems. Comparison of the results with an earlier study showed that although each teachers teaching approach was stable over two years, each used technology differently with further experience of CAS. The teacher who stressed understanding moved away from using CAS, whilst the teacher who stressed rules,adopted it more. The study highlights that within similar overall attainment on student tests, there can be substantial variations of what students know. New technologies provide more approaches to teaching and so greater variations between teaching and the consequent learning may become evident.

Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS Video Study criteria to Australian eighth-grade mathematics textbooks

Australian eighth-grade mathematics lessons were shown by the 1999 TIMSS Video Study to use a high proportion of problems of low procedural complexity, with considerable repetition, and an absence of deductive reasoning. Using definitions from the Video Study, this study re-investigated this ‘shallow teaching syndrome’ by examining the problems on three topics in nine eighth-grade textbooks from four Australian states for procedural complexity, type of solving processes, degree of repetition, proportion of ‘application’ problems and proportion of problems requiring deductive reasoning. Overall, there was broad similarity between the characteristics of problems in the textbooks and in the Australian Video Study lessons. There were, however, considerable differences between textbooks and between topics within textbooks. In some books, including the best-selling textbooks in several states, the balance is too far towards repetitive problems of low procedural complexity.

The Effect of Different Approaches to Algebra on Students' Perceptions of Functional Relationships

National curriculum guidelines advise teachers to begin algebra with the study of patterns leading to their description as algebra rules relating two variables. Traditional approaches are based on the use of algebraic letters to stand for specific but unknown numbers. In this paper we show that a pattern-based approach does not automatically lead to better understanding of functional relationships and algebraic rules. Approximately 1200 students in Years 7 to 10 in 10 schools were tested on recognising, using and describing rules relating two variables. Fourteen students were interviewed. Students saw a variety of patterns, many of which were not helpful for algebra. The critical steps in moving from a table to an algebraic rule are outlined.

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.

THE EFFECT OF EPISTEMIC FIDELITY AND ACCESSIBILITY ON TEACHING WITH PHYSICAL MATERIALS: A COMPARISON OF TWO MODELS FOR TEACHING DECIMAL NUMERATION

Multi-base arithmetic blocks (MAB) are the most frequently used physical materials for teaching about decimal numbers, despite published reservations about their appropriateness. This paper presents an alternative, LAB (linear arithmetic blocks) and compares the two materials on the basis of epistemic fidelity and accessibility for students. Two teaching experiments involving 30 matched students indicated that LAB is considerably more accessible for students, and identify three contributing factors (LAB modeling number with length rather than volume, MAB incorporating an apparent dimensional shift and having prior use). Use of LAB was associated with more active engagement by students and deeper discussion. Epistemic fidelity is critical to facilitate teaching with the models, but we attribute the enhanced classroom environment to the greater accessibility of the LAB material. Further research is warranted, so that teaching of mathematics with physical materials can be improved.

Tracing learning of three representations with the differentiation competency framework

The support of technology for working with multiple representations of functions has substantial potential for teaching calculus. For teaching differentiation, these representations relate to finding difference quotients, finding gradients of curves and tangents, and using symbolic differentiation rules. For students to use them all and link them together requires a wide range of skills, which have been organised into aDifferentiation Competency Framework. This paper also describes a balancedDifferentiation Competency Test that was created from the Framework and used in two Year 11 classes to monitor students’ understanding of introductory differentiation. The Framework helps identify both student achievement and teaching focus.

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.