Educational innovation, learning technologies and Virtual culture potential’

Learning technologies are regularly associated with innovative teaching but will they contribute to profound innovations in education itself? This paper addresses the question by building upon Merlin Donald's co‐evolutionary theory of mind, cognition and culture. He claimed that the invention of technologies for storing and sharing external symbol systems, such as writing, gave rise to a ‘theoretic culture’ with rich symbolic representations and a resultant need for formal education. More recently, Shaffer and Kaput have claimed that the development of external and shared symbol‐processing technologies is giving rise to an emerging ‘virtual culture’. They argue that mathematics curricula are grounded in theoretic culture and should change to meet the novel demands of ‘virtual culture’ for symbol‐processing and representational fluency. The generic character of their cultural claim is noted in this paper and it is suggested that equivalent pedagogic arguments are applicable across the educational spectrum. Hence, four general characteristics of virtual culture are proposed, against which applications of learning technologies can be evaluated for their innovative potential. Two illustrative uses of learning technologies are evaluated in terms of their ‘virtual culture potential’ and some anticipated questions about this approach are discussed towards the end of the paper

The importance of being accessible: The graphics calculator in mathematics education

The first decade of the availability of graphics calculators in secondary schools has just concluded, although evidence for this is easier to find in some countries and schools than in others, since there are gross socio-economic differences in both cases. It is now almost the end of the second decade since the invention of microcomputers and their appearance in mathematics educational settings. Most of the interest in technology for mathematics education has been concerned with microcomputers. But there has been a steady increase in interest in graphics calculators by students, teachers, curriculum developers and examination authorities, in growing recognition that accessibility of technology at the level of the individual student is the key factor in responding appropriately to technological change; the experience of the last decade suggests very strongly that mathematics teachers are well advised to pay more attention to graphics calculators than to microcomputers. There are clear signs that the commercial marketplace, especially in the United States, is acutely aware of this trend. It was recently reported that current US sales of graphics calculators are around six million units per year, and rising. There are now four major corporations developing products aimed directly at the high school market, with all four producing graphics calculators of high quality and beginning to understand the educational needs of students and their teachers. To get some evidence of this interest, I scanned a recent issue (April 1995) of The Mathematics Teacher, the NCTM journal focussed on high school mathematics. The evidence was very strong: of almost 20 full pages devoted to paid advertising, nine featured graphics calculators, while only two featured computer products, with two more featuring both computers and graphics calculators. The main purposes of this paper are to explain and justify this heightened level of interest in graphics calculators at the secondary school level, and to identify some of the resulting implications for mathematics education, both generally, and in the South-East Asian region

Graphics calculators and assessment

Graphics calculators are powerful tools for learning mathematics and we want our students to learn to use them effectively. The use of these hand held personal computers provides opportunities for learning in interactive and dynamic ways. However, it is not until their use is totally integrated into all aspects of the curriculum that students regard them with due importance. This includes their use in all kinds of assessment tasks such as assignments, tests and examinations as well as in activities and explorations aimed at developing students’ understanding. The incorporation of graphics calculators into assessment tasks requires careful construction of these tasks. In this paper, discuss issues of equity relating to calculator models, levels of calculator use and the purpose and design of appropriate tasks. We also describe a typology we have developed to assist in the design and wording of assessment tasks which encourage appropriate use of graphics calculators, but which do not compromise important course objectives

Next steps in implementing Kaput's research programme

We explore some key constructs and research themes initiated by Jim Kaput, and attempt to illuminate them further with reference to our own research. These 'design principles' focus on the evolution of digital representations since the early nineties, and we attempt to take forward our collective understanding of the cognitive and cultural affordances they offer. There are two main organising ideas for the paper. The first centres around Kaput's notion of outsourcing of processing power, and explores the implications of this for mathematical learning. We argue that a key component for design is to create visible, transparent views of outsourcing, a transparency without which there may be as many pitfalls as opportunities for mathematical learning. The second organising idea is that of communication, a key notion for Kaput, and the importance of designing for communication in ways that recognise the mutual influence of tools for communication and for mathematical expression

Process over product: It\u27s more than an equation

Developing number and algebra together provide opportunities for searching for patterns, conjecturing, justifying, and generalising mathematical relationships. It allows the focus to be on the process of mathematics and noticing the structure of arithmetic, rather than the product of arriving at a correct answer. Two of the big ideas in mathematics are multiplicative thinking and algebraic reasoning. By noticing the structure of multiplicative situations, students will be in a position to reason algebraically, and the process of reasoning algebraically will allow students to appreciate the value of thinking multiplicatively rather than additively

Structural-Symbolic Translation Fluency: Reliability, Validity, and Usability

Standardized formative mathematics assessments typically fail to capture the depth of current standards and curricula. Consequently, these assessments demonstrate limited utility for informing the instructional implementation choices of teachers. This problem is particularly salient as it relates to the mathematical problem solving process. The purpose of this study was to develop and evaluate the psychometric characteristics of Structural-Symbolic Translation Fluency, a curriculum-based measure (CBM) of mathematical problem solving. The development of the assessment was based on previous research describing the cognitive process of translation (Mayer, 2002) as well as mathematical concept development at the quantitative, structural, and symbolic levels (Dehaene, 2011; Faulkner, 2009; Griffin, 2004). Data on the Structural-Symbolic Translation Fluency assessment were collected from 11 mathematics and psychometrics experts and 42 second grade students during the spring of 2016. Data were analyzed through descriptive statistics, frequencies, Spearman-Brown correlation, joint probability of agreement, Pearson correlation, and hierarchical multiple regression. Psychometric features of interest included internal consistency, inter-rater reliability, test-retest reliability, content validity, and criterion-related validity. Testing of the 9 research questions revealed 9 significant findings. Despite significant statistical findings, several coefficients did not meet pre-established criteria required for validation. Hypothesized modifications to improve the psychometric characteristics are suggested as the focus of future research. In addition, recommendations are made concerning the role of assessing the translation process of mathematical problem solving