An aspect of mathematical understanding: the notion of 'connected knowing'

Much advice about teaching for understanding implies that teacher should help children to develop connections between aspects of their experience, knowledge, and skills. This paper outlines points from the literature about different types of connections and describes relevant points from four case study teachers said and did.

The development of young children's mathematical understanding

Early childhood mathematics education research is burgeoning in Australasia. This chapter highlights and critiques key research in the area that has been published between 2004 and 2007. In particular, it considers specific mathematical topics such as number and numeracy, space and measurement, and structure and patterning; contextual matters such as links among home, school, pri or·to·sch 001 settings and community, indigenous learners and mathematics learning as children start school; assessment of mathematics learning in early childhood settings and the professional development of early childhood teachers of mathematics. It concludes with some suggestions for fruitful areas of future research.

Developing mathematical understanding

The notion of scaffolding is used to introduce research on differentiated learning trajectories that make use of activities termed 'prompts'. The prompts are used to enable children to develop the necessary mathematical understanding and skills to keep up with the rest of the class.  They focus on aspects of teaching that teachers identified as important aspects of classroom interactivity that contribute to the understanding: the use of physical representation of concepts: actions aimed at building conceptual links; and the use of language based activity.

Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

What does mathematics understanding look like?

The concept of mathematical understanding is central to curriculum development, classroom interaction, and the training of mathematics teachers In this paper, some models of the growth of understanding that the literature presents are outlined Some results of a study that documented four primary teachers’ mental models of, and beliefs about, different forms of understanding are reported It is proposed that linear models may restrict ways that these teachers plan lessons Questions for further research are raised

Testing the effectiveness of mathematical games as a pedagogical tool for children's learning

In an effort to engage children in mathematics learning, many primary teachers use mathematical games and activities. Games have been employed for drill and practice, warm-up activities and rewards. The effectiveness of games as a pedagogical tool requires further examination if games are to be employed for the teaching of mathematical concepts. This paper reports research that compared the effectiveness of non-digital games with non-game but engaging activities as pedagogical tools for promoting mathematical learning. In the classrooms that played games, the effects of adding teacher-led whole class discussion was explored. The research was conducted with 10–12-year-old children in eight classrooms in three Australian primary schools, using differing instructional approaches to teach multiplication and division of decimals. A quasi-experimental design with pre-test, post-test and delayed post-test was employed, and the effects of the interventions were measured by the children’s written test performance. Test results indicated lesser gains in learning in game playing situations versus non-game activities and that teacher-led discussions during and following the game playing did not improve children’s learning. The finding that these games did not help children demonstrate a mathematical understanding of concepts under test conditions suggests that educators should carefully consider the application and appropriateness of games before employing them as a vehicle for introducing mathematical concepts.

Students’ understanding of the function-derivative relationship when learning economic concepts

The aim of this study is to characterise students’ understanding of the function-derivative relationship when learning economic concepts. To this end, we use a fuzzy metric (Chang 1968) to identify the development of economic concept understanding that is defined by the function-derivative relationship. The results indicate that the understanding of these economic concepts is linked to students’ capacity to perform conversions and treatments between the algebraic and graphic registers of the function-derivative relationship when extracting the economic meaning of concavity/convexity in graphs of functions using the second derivative.

Integrating concrete and virtual materials in an elementary mathematics classroom : a case study of success with fractions

This paper describes an approach to introducing fraction concepts using generic software tools such as Microsoft Office's PowerPoint to create "virtual" materials for mathematics teaching and learning. This approach replicates existing concrete materials and integrates virtual materials with current non-computer methods of teaching primary students about fractions. The paper reports a case study of a 12-year-old student, Frank, who had an extremely limited understanding of fractions. Frank also lacked motivation for learning mathematics in general and interacted with his peers in a negative way during mathematics lessons. In just one classroom session involving the seamless integration of off-computer and on-computer activities, Frank acquired a basic understanding of simple common equivalent fractions. Further, he was observed as the session progressed to be an enthusiastic learner who offered to share his learning with his peers. The study's "virtual replication" approach for fractions involves the manipulation of concrete materials (folding paper regions) alongside the manipulation of their virtual equivalent (shading screen regions). As researchers have pointed out, the emergence of new technologies does not mean old technologies become redundant. Learning technologies have not replaced print and oral language or basic mathematical understanding. Instead, they are modifying, reshaping, and blending the ways in which humankind speaks, reads, writes, and works mathematically. Constructivist theories of learning and teaching argue that mathematics understanding is developed from concrete to pictorial to abstract and that, ultimately, mathematics learning and teaching is about refinement and expression of ideas and concepts. Therefore, by seamlessly integrating the use of concrete materials and virtual materials generated by computer software applications, an opportunity arises to enhance the teaching and learning value of both materials.

Primary students’ interpretation of maps : gesture use and mapping knowledge

Maps are used to represent three-dimensional space and are integral to a range of everyday experiences. They are increasingly used in mathematics, being prominent both in school curricula and as a form of assessing students understanding of mathematics ideas. In order to successfully interpret maps, students need to be able to understand that maps: represent space, have their own perspective and scale, and their own set of symbols and texts. Despite the fact that maps have an increased prevalence in society and school, there is evidence to suggest that students have difficulty interpreting maps. This study investigated 43 primary-aged students’ (aged 9-12 years) verbal and gestural behaviours as they engaged with and solved map tasks. Within a multiliteracies framework that focuses on spatial, visual, linguistic, and gestural elements, the study investigated how students interpret map tasks. Specifically, the study sought to understand students’ skills and approaches used to solving map tasks and the gestural behaviours they utilised as they engaged with map tasks. The investigation was undertaken using the Knowledge Discovery in Data (KDD) design. The design of this study capitalised on existing research data to carry out a more detailed analysis of students’ interpretation of map tasks. Video data from an existing data set was reorganised according to two distinct episodes—Task Solution and Task Explanation—and analysed within the multiliteracies framework. Content Analysis was used with these data and through anticipatory data reduction techniques, patterns of behaviour were identified in relation to each specific map task by looking at task solution, task correctness and gesture use. The findings of this study revealed that students had a relatively sound understanding of general mapping knowledge such as identifying landmarks, using keys, compass points and coordinates. However, their understanding of mathematical concepts pertinent to map tasks including location, direction, and movement were less developed. Successful students were able to interpret the map tasks and apply relevant mathematical understanding to navigate the spatial demands of the map tasks while the unsuccessful students were only able to interpret and understand basic map conventions. In terms of their gesture use, the more difficult the task, the more likely students were to exhibit gestural behaviours to solve the task. The most common form of gestural behaviour was deictic, that is a pointing gesture. Deictic gestures not only aided the students capacity to explain how they solved the map tasks but they were also a tool which assisted them to navigate and monitor their spatial movements when solving the tasks. There were a number of implications for theory, learning and teaching, and test and curriculum design arising from the study. From a theoretical perspective, the findings of the study suggest that gesturing is an important element of multimodal engagement in mapping tasks. In terms of teaching and learning, implications include the need for students to utilise gesturing techniques when first faced with new or novel map tasks. As students become more proficient in solving such tasks, they should be encouraged to move beyond a reliance on such gesture use in order to progress to more sophisticated understandings of map tasks. Additionally, teachers need to provide students with opportunities to interpret and attend to multiple modes of information when interpreting map tasks.

Prognostic modelling options for remaining useful life estimation by industry

Over recent years a significant amount of research has been undertaken to develop prognostic models that can be used to predict the remaining useful life of engineering assets. Implementations by industry have only had limited success. By design, models are subject to specific assumptions and approximations, some of which are mathematical, while others relate to practical implementation issues such as the amount of data required to validate and verify a proposed model. Therefore, appropriate model selection for successful practical implementation requires not only a mathematical understanding of each model type, but also an appreciation of how a particular business intends to utilise a model and its outputs. This paper discusses business issues that need to be considered when selecting an appropriate modelling approach for trial. It also presents classification tables and process flow diagrams to assist industry and research personnel select appropriate prognostic models for predicting the remaining useful life of engineering assets within their specific business environment. The paper then explores the strengths and weaknesses of the main prognostics model classes to establish what makes them better suited to certain applications than to others and summarises how each have been applied to engineering prognostics. Consequently, this paper should provide a starting point for young researchers first considering options for remaining useful life prediction. The models described in this paper are Knowledge-based (expert and fuzzy), Life expectancy (stochastic and statistical), Artificial Neural Networks, and Physical models.

Canards in advection-reaction-diffusion systems in one spatial dimension

This thesis contains a mathematical investigation of the existence of travelling wave solutions to singularly perturbed advection-reaction-diffusion models of biological processes. An enhanced mathematical understanding of these solutions and models is gained via the identification of canards (special solutions of fast/slow dynamical systems) and their role in the existence of the most biologically relevant, shock-like solutions. The analysis focuses on two existing models. A new proof of existence of a whole family of travelling waves is provided for a model describing malignant tumour invasion, while new solutions are identified for a model describing wound healing angiogenesis.

Shaping practice: worksheets as social artefacts

Part of a broader research project on the development of mathematical understanding in primary classrooms, the research reported in this paper focused on how opportunities to learn about percentage in two classrooms were shaped by worksheets. The words in the text, the way that the teachers replicated these orally, a grid used to illustrate the concept of percentage, the genre of both mathematical and non-mathematical aspects of the worksheet's presentation, and the calculation methods presented were all potentially influential aspects. It is argued that the "knowledge" conveyed by the texts of the worksheets was distributed over a much wider field than the classrooms involved.

A matemática do sensível pelas mãos do artesão: marcas da aprendizagem matemática e da cultura material dos ceramistas de Icoaraci

Esta tese aborda a discussão a respeito do raciocínio matemático manifestado no saber/ fazer dos artesãos ceramistas do Distrito Municipal de Icoaraci (Belém/ PA), visando o entendimento cognitivo e cultural desta prática, para abstrair contribuições à educação matemática – área de conhecimento na qual se inscreve, especialmente no âmbito da educação matemática. Trabalhado essa última, a tese analisa a realidade dos sujeitos mediante a Teoria dos Campos Conceituais, do educador matemático Gérard Vergnaud, que desenvolve estudos na linha construtivista, do psicólogo da educação Jean Piaget, possibilitando abordar na prática cotidiana do artesão, seus Campos Conceituais, a possibilidade ou não da existência de teoremas e conceitos-em-ato, fato esse que irá constatar ou não a essência ou „matematicidade‟ dos estudos educacionais matemáticos trabalhados por etnomatemáticos, pedagogos, especialistas de modelagem matemática, sociólogos e arqueólogos matemáticos. A epistemologia da educação matemática, disciplina filosófica, surge norteando esse entendimento sobre o raciocínio matemático, através da matemática do sensível, que acha origens na antiguidade grega, através dos ideários pitagórico, platônico e aristotélico, estendendo essa visão à matemática do mundo presente. Assim, a tese procura explicitar a manifestação de um raciocínio matemático por parte do artesão, que no seu fazer predominantemente não conhece e/ ou não utiliza a matemática acadêmica ou formal, como comprovado em outros estudos. Essa presença ou não de entendimentos matemáticos será constatada através de abordagem etnográfica e qualitativa, sob o enfoque fenomenológico, utilizando técnicas de observação, anotações de campo, inventário cultural e entrevistas, no intuito de analisar as representações existentes em suas obras e o fazer/ pensar manifestados nessa produção.