15 research outputs found
Thomas Frederick Arndt ; Men in America : Photographs, 1973-1987
No full textTravis describes Arndt's photographs of stereotypical working-class heterosexual men. Biographical notes on both artist and author
Instructional strategies in explicating the discovery function of proof for lower secondary school students
No full textIn this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr
Geometrical reasoning in the primary school, the case of parallel lines
No full textDuring the primary school years, children are typically expected to develop ways of explaining their mathematical reasoning. This paper reports on ideas developed during an analysis of data from a project which involved young children (aged 5-7 years old) in a whole-class situation using dynamic geometry software (specifically Sketchpad). The focus is a classroom episode in which the children try to decide whether two lines that they know continue (but cannot see all of the continuation) will intersect, or not. The analysis illustrates how the children can move from an empirical, visual description of spatial relations to a more theoretical, abstract one. The arguments used by the children during the lesson transcend empirical arguments, providing evidence of how young children can be capable of engaging in aspects of deductive argumentation
Enacting Reasoning-and-Proving in Secondary Mathematics Classrooms through Tasks
No full textProof is the mathematical way of convincing oneself and others of the truth of a claim for all cases in the domain under consideration. As such, reasoning-and-proving is a crucial, formative practice for all students in kindergarten through twelfth grade, which is reflected in the Common Core State Standards in Mathematics. However, students and teachers exhibit many difficulties employing, writing, and understanding reasoning-and-proving. In particular, teachers are challenged by their knowledge base, insufficient resources, and unsupportive pedagogy.
The Cases of Reasoning and Proving (CORP) materials were designed to offer teachers opportunities to engage in reasoning-and-proving tasks, discuss samples of authentic practice, examine research-based frameworks, and develop criteria for evaluating reasoning-and-proving products based on the core elements of proof. A six-week graduate level course was taught with the CORP materials with the goal of developing teachers’ understanding of what constitutes reasoning-and-proving, how secondary students benefit from reasoning-and-proving, and how they can support the development of students’ capacities to reason-and-prove. Research was conducted on four participants of the course during either their first or second year of teaching. The purpose of the research was to study the extent to which the participants selected, implemented, and evaluated students’ work on reasoning-and-proving tasks. The participants’ abilities were examined through an analysis of answers to interview questions, tasks used in class, and samples of student work, and scoring criteria. The results suggest that: 1.) participants were able to overcome some of the limitations of their insufficient resource by modifying and creating some reasoning-and-proving exercises; 2.) participants were able to maintain the level of cognitive demand of proof tasks during implementation; and 3) participants included some if not all of the core elements of proof in their definition of proof and in their evaluation criteria for student products of reasoning-and-proving products
Fiji and New Zealand Pasifika students' perceptions of mathematics and their attitudes towards mathematics learning
No full textAbstract
This thesis explores the perceptions and attitudes towards mathematics learning of 36 students from Fiji and 12 Pasifika students living in New Zealand. The students were in Years 7 and 8, included a range of abilities in maths as assessed by their teachers. The New Zealand (NZ) students attended a school that had participated in the Numeracy Development Project (NDP) several years prior to the study. Data was collected using semi-structured and clinical interviews. Seven key questions were the main focus of this study. The students were asked about their views on: working collectively or individually, the importance of knowing and sharing solution strategies with others, the nature of mathematics, people who supported their maths learning, their attitudes towards mathematics, and how good they thought they were at maths. They solved problems involving subtraction, division and proportional problems.
The findings revealed that nearly all of the Fiji students but just over half of the NZ students supported group work. On their views about the value of knowing others' strategies, nearly seven tenths of the Fiji students and just under a half of the NZ students thought that it was important. However, all the children were unanimous in their view that explaining their solution strategies to others was important. The findings also revealed that nearly all the students from both countries thought that mathematics was about numbers and operations. Most of the Fiji children commented that it was also about problem solving, whereas the NZ students mentioned having alternative strategies as what they thought mathematics all about.
The students' views about a teacher's role in helping them learn mathematics greatly differed between the two countries. Slightly more than half of the Fiji students thought that the teacher's role was to show them strategies. The Fiji students also described their teachers as someone who gave them notes to copy and provided exercises from the textbook. In contrast the NZ students mentioned their teacher as someone who helped their mathematics learning by sharing clues, giving tasks that challenged them and grouped them by ability before helping them. The responses of all the students revealed that there was great support from friends, parents and relatives towards their mathematics learning. The students also rated their feelings towards mathematics on a three-point rating scale with happy, neutral and sad faces. Half of the students from both countries chose the happy face with the other half choosing the neutral face. None of them chose the sad face as matching how they felt about maths most of the time.
The students also assessed themselves on how good they were at maths. The majority of the students from both countries rated themselves as good and none of them chose the box showing very poor . The students were asked to do some tasks on subtraction, division and proportional problems. There was a major contrast between the two countries on how they worked out their answers. The Fiji students' responses showed their fluency with standard written algorithms and a high level of procedural knowledge. The NZ students on the other hand hardly used algorithms. Instead their responses showed the use of mental strategies for solving tasks ranging from stage 2, (counting from one on materials) to stage 7, (advanced multiplicative part-whole) on the NDP Number Framework
The principles of screen design for computer-based learning materials.
No full textThe critical interface between learners and computer-based learning materials is the
screen. If the display of learning is not effective then learning will be hindered.
Screen design is therefore an important element in the design of computer-based
learning.
This research investigated the three fundamental screen design elements of
text, colour and graphics. A review of literature, experimental design and a limited
survey of computer-based learning materials provided the background for this
research. The experimental materials reflected the results of the review and survey
by using representative subjects, providing a learning focus and employing computerbased materials. Two experiments were undertaken. The Colour and Graphics
experiments considered the effects of a number of variables on learners' behaviour
which included: the use of colour; the size and type of graphics; the learner's prior
knowledge of tutorial subject; and the complexity of the display.
The results of this research showed that colour is a powerful motivating force
as long as it is not used excessively. This was identified as the use of more than
seven colours. Graphics can be used more extensively in current computer-based
learning materials and users preferred representational graphics occupying a quarter
to a half of the screen. However, learners were not prepared to make the effort to
either use analogical graphics to make links with their prior knowledge or to extract
information contained in the structure and form of logical graphics. Subjects were
motivated by representational graphics.
Learners' behaviour in relation to the various screen displays they
encountered was affected by their prior knowledge of the tutorial content. This was
apparent in their choice of options (additional modules) within the tutorial, their
methods of interacting with the material and their responses to individual displays
Connecting the Points: An Investigation into Student Learning about Decimal Numbers
No full textThe purpose of this research project was to investigate the effects of a short-term teaching experiment on the learning of decimal numbers by primary students. The literature describes this area of mathematics as highly problematic for students.
The content first covered student understanding of decimal symbols, and how this impacted upon their ability to order decimal numbers and carry out additive operations. It was then extended to cover the density of number property, and the application of multiplicative operations to situations involving decimals. In doing so, three areas of cognitive conflict were encountered by students, the belief that longer decimal numbers are larger than shorter ones (irrespective of the actual digits), that multiplication always makes numbers bigger, and that division always makes numbers smaller.
The use of a microgenetic approach yielded data was able to be presented that provides details of the environment surrounding the moments where new learning was constructed. The characteristics of this environment include the use of physical artifacts and situational contexts involving measurement that precipitate student discussion and reflection.
The methodology allowed for the collection of evidence regarding the highly complex nature of the learning, with evidence of 'folding back' to earlier schema and the co-existence of competing schema. The discussion presents reasons as to why the pedagogical approach that was employed facilitated learning.
One of the main findings was that the use of challenging problems situated in measurement contexts that involved direct student participation promoted the extension and/or re-organization of student schema with regard to decimal numbers.
The study has important implications for teachers at the upper primary level wanting to support student learning about the decimal numbers system
Dual processes in mathematics: reasoning about conditionals
No full textThis thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students.
Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data.
In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised.
In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans & Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed
