Algebraic Insight Underpins the Use of CAS for Modelling

Computer Algebra Systems (CAS) performs algorithmic processes quickly and correctly. Concern is commonly expressed that students using CAS will merely be pushing buttons but this paper indicates that, while CAS may assist students, this facility impacts on only one section of the mathematical modeling process: CAS may be used to help find mathematical solutions to mathematically formulated problems. Controlling and monitoring the use of CAS to perform the necessary routine processes requires the mathematical thinking referred to as algebraic insight. This paper sets out a framework of the aspects, and elements of algebraic insight and illustrates the importance of students developing each of the two key aspects: algebraic expectation and ability to link representations. This framework may be used for both planning teaching and monitoring students’ progress

Potential of technology and a familiar context to enhance students\u27 concept of rate of change

Students\u27 concept image of rate of change may be incomplete or erroneous. This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students\u27 existing schema of informal understandings of rate of change to more formal mathematical representations. Students developed numerous \u27models of\u27 rate of change in a motion context which then transferred to serve as a \u27model for\u27 rate of change in other contexts.<br /

Video evidence : what gestures tell us about students\u27 understanding of rate of change

This paper reports on insights into students&rsquo; understanding of the concept of rate of change, provided by examining the gestures made, by 25 Year 10 students, in videorecorded interviews. Detailed analysis, of both the sound and images, illuminates the meaning of rate-related gestures. Findings indicate that students often use the symbols and metaphors of gesture to complement, supplement, or even contradict verbal descriptions. Many students demonstrated, by the combination of their words and gestures, a sound qualitative understanding of constant rate, with a few attempting to quantify rate. The interpretation of gestures may provide teachers with a better understanding of the progress in their students&rsquo; thinking.<br /

Gifted are lifted higher : an exploration of the development of higher order thinking skills of gifted students playing strategy games

Strategy games can provide an opportunity to develop higher order thinking skills in students gifted in mathematics. Extending and engaging gifted students is a demanding task. This paper reports on a twelve-week project undertaken with a group of nine gifted lower secondary school students. These students played and analysed five traditional strategy games. Following this experience, they were asked to create a challenging strategy game of their own. This paper discusses the rationale for the use of traditional strategy games, outlines the methodology employed, explains the selection of specific games and describes the observed improvement in students\u27 higher order thinking skills.<br /

CAS : student engagement requires unambiguous advantages

Encouraging students to develop effective use of Computer Algebra Systems (CAS) is not trivial. This paper reports on a group of undergraduate students who, despite carefully planned lectures and CAS availability for all learning and assessment tasks, failed to capitalize on its affordances. If students are to work within the technical constraints, and develop effective use of CAS, teachers need to provide assistance with technical difficulties, actively demonstrate CAS\u27 value and unambiguously reward its strategic use in assessment.<br /

Algebraic insight underpins the use of CAS for modelling

Computer Algebra Systems (CAS) performs algorithmic processes quickly and correctly. Concern is commonly expressed that students using CAS will merely be pushing buttons but this paper indicates that, while CAS may assist students, this facility impacts on only one section of the mathematical modeling process: CAS may be used to help find mathematical solutions to mathematically formulated problems. Controlling and monitoring the use of CAS to perform the necessary routine processes requires the mathematical thinking referred to as algebraic insight. This paper sets out a framework of the aspects, and elements of algebraic insight and illustrates the importance of students developing each of the two key aspects: algebraic expectation and ability to link representations. This framework may be used for both planning teaching and monitoring students’ progress.C

Consciousness embodied: language and the imagination in the communal world of William Blake

This dissertation examines the philosophical and spiritual beliefs that underpin William Blake’s account of the imagination, his objections to empiricism and his understanding of poetic language. It begins by considering these beliefs in relation to the idealist principles of George Berkeley as a means of illustrating Blake’s own objections to the empiricism of John Locke. The philosophies of Locke and Berkeley were popular in Blake’s society and their philosophical positions were well known to him. Blake and Berkeley are aligned against Locke’s belief in an objective world composed of matter, and his theory of abstract ideas. Both reject Locke’s principles by affirming the primacy of the perceiving subject. However, Blake disagrees with Berkeley’s theologically traditional understanding of God. He views perception as an act of artistic creation and believes that spiritual divinity is contained within and is intrinsic to man’s human form. This account of human perception as the creative act of an immanent divinity is further elucidated through a comparison with the twentieth-century existential phenomenology of Maurice Merleau-Ponty. In the Phenomenology of Perception (1945), Merleau-Ponty examines human experience as the functioning of an embodied consciousness in a shared life-world. While Merleau-Ponty does not make any reference to a spiritual deity, his understanding of experience offers a link between Berkeley’s criticisms of Locke and Blake’s own objections to empiricism. Through a comparative examination of Blake and Merleau-Ponty, the imagination is revealed to be the creative or formative consciousness that proceeds from the integrated mind-body complex of the “Divine Body” or “human form divine”. This embodied existence locates the perceiving self in a dynamic physical landscape that is shared with other embodied consciousnesses. It is this communal or intersubjective interaction between self and other that constitutes the experienced world. Merleau-Ponty’s account of the chiasm and his notion of flesh, discussed in The Visible and the Invisible, are applied to Blake in order to elucidate his belief in poetic vision and the constitutive power of language. The form and function of language are compared with that of the body, because both bring the individual experience of a perceiving subject into being in the world and facilitate the reciprocal exchange between the self and other. Ultimately, this dissertation argues that Blake characterises the body and language as the living media of the imagination, which facilitate a creative exchange between a perceiving self and a shared life-world

Using CAS to enrich the teaching and learning of mathematics

Computer Algebra Systems (CAS) are powerful tools for both doing and learning mathematics. They may be used to perform algorithmic routines both quickly and correctly but harnessing this power in a manner which is effective for promoting learning is not trivial. Research undertaken with both secondary school and undergraduate students clearly indicates that, while students quickly come to appreciate the availability of CAS to check their answers, several key factors influence the development of their use of the facility of CAS to extend both access to mathematics and support learning of mathematical concepts. First, the institutional value which the technology is afforded influences the degree to which students are willing to apply themselves to the task of learning technical skills necessary to work with CAS. Second, the use of multiple representations may both increase students’ conceptual understanding and provide them with alternative methods through which to progress solution of problems. Finally, students need to be guided in judicious use of CAS. This will involve teaching students to be discriminating in their use of technology for functional purposes, that is, to find solutions to difficult or time consuming problems, and strategic in their use of CAS to explore patterns and link representations in order to gain greater insight into mathematical processes and concepts.E

Linear functions and a triple influence of teaching on the development of students' algebraic expectation

The study of linear functions is important as it provides students with their first experience of identifying and interpreting the relationship between two dependent variables. This paper, which builds on previous research, reports a study undertaken with 64, year 9 students from two Australian schools. Linear functions were introduced to these students through a graphics calculator supported, functional approach to modelling contextual problems. The teaching was generally successful. Scrutiny of pre- and post-tests highlights the triple influence of the teaching on their progress in each element of Algebraic Expectation relevant to this stage.E

Ontogenetic Survey of Histone Modifications in an Annelid

Histone modifications are widely recognized for their fundamental importance in regulating gene expression in embryonic development in a wide range of eukaryotes, but they have received relatively little attention in the development of marine invertebrates. We surveyed histone modifications throughout the development of a marine annelid, Polydora cornuta, to determine if modifications could be detected immunohistochemically and if there were characteristic changes in modifications throughout ontogeny (surveyed at representative stages from oocyte to adult). We found a common time of onset for three histone modifications in early cleavage (H3K14ac, H3K9me, and H3K4me2), some differences in the distribution of modifications among germ layers, differences in epifluorescence intensity in specific cell lineages suggesting that hyperacetylation (H3K14ac) and hypermethylation (H3K9me) occur during differentiation, and an overall decrease in the distribution of modifications from larvae to adults. Although preliminary, these results suggest that histone modifications are involved in activating early development and differentiation in a marine invertebrate