Mathematical tasks and learning goals: examples from Japanese lesson study

This paper explores how to select, or design, the best mathematical task for a given learning goal. Examples are taken from a recent project in Victorian primary schools that employed Japanese lesson study as the means to provide teachers with professional learning within their own classrooms. The discussions by participating teachers and researchers provide some insights into the difficulties and solutions facing teachers intending to improve the critical thinking skills of their pupils. Examples of tasks and goals are provided.

Adapting the task : what preservice teachers notice when adapting mathematical tasks

This paper reports on an in-depth study that explores preservice teachers’ pedagogical adaptations to a rich mathematical task. Data were collected from six elementary preservice teachers working in pairs to first solve a mathematics problem and then design adaptations to make the problem more accessible and more challenging for diverse learners. Results indicate that preservice teachers are able to draw upon a range of strategies to vary the mathematical content, the context, and the question asked. However, they also did not notice or attend to how their adaptations changed the mathematical structure of the problem. This study provides insights into what is involved in learning to adapt classroom mathematical tasks as an important pedagogical practice.

A longitudinal analysis of estimation, counting skills, and mathematical ability across the first school year

In response to claims that the quality (and in particular linearity) of children's mental representation of number acts as a constraint on number development, we carried out a longitudinal assessment of the relationships between number line estimation, counting, and mathematical abilities. Ninety-nine 5-year-olds were tested on 4 occasions at 3 monthly intervals. Correlations between the 3 types of ability were evident, but while the quality of children's estimations changed over time and performance on the mathematical tasks improved over the same period, changes in one were not associated with changes in the other. In contrast to the earlier claims that the linearity of number representation is potentially a unique contributor to children's mathematical development, the data suggest that this variable is not significantly privileged in its impact over and above simple procedural number skills. We propose that both early arithmetic success and estimating skill are bound closely to developments in counting ability.

Towards mathematical literacy in the 21st century : perspectives from Indonesia

The notion of mathematical literacy advocated by PISA (OECD, 2006) offers a broader conception for assessing mathematical competences and processes with the main focus on the relevant use of mathematics in life. This notion of mathematical literacy is closely connected to the notion of mathematical modelling whereby mathematics is put to solving real world problems. Indonesia has participated as a partner country in PISA since 2000. The PISA trends in mathematics from 2003 to 2009 revealed unsatisfactory mathematical literacy among 15-year-old students from Indonesia who lagged behind the average of OCED countries. In this paper, exemplary cases will be discussed to examine and promote mathematical literacy at teacher education level. Lesson ideas and instruments were adapted from PISA 2006 released items. The potential of such tasks will be discussed based on case studies of implementing these instruments with samples of pre-service teachers in Yogyakarta. 

The task of designing tasks for teacher education and development

Posing, adapting, and creating worthwhile tasks that support student learning are significant pedagogical practices of teachers. However in mathematics education, task design and curriculum development are not often the responsibility of practicing teachers. Textbooks can become the main source of mathematics problems and teachers often follow the text and its sequence of problems. Although mathematics teachers can benefit from access to research-based curriculum resources, limited opportunities to adapt and design tasks can make it challenging to meet the diverse needs and interests of their students. More opportunities for learning to create problems are needed for both practicing and beginning mathematics teachers. What then is involved in learning to pose, adapt and create worthwhile mathematics tasks? More specifically, how can teacher educators design tasks that support teacher candidates in learning to create mathematical tasks? As teacher educators we have, over the past few years, used our own courses as sites in Australia and Canada for investigating various contexts to support preservice teachers in their learning to adapt and create mathematical tasks. Our research includes both large scale and small qualitative studies to explore the perspectives of teacher candidates on learning to create and pose mathematical problems, the kinds of problems they pose, and the opportunities and challenges this offers us as teacher educators. Our work is inspired by Variation Theory that focuses on learning as the act of awareness and discernment of variation. We are exploring both how variation theory can be useful for preservice teachers in their designing, posing and adapting of mathematical tasks and for teacher educators in their design of such pedagogical tasks. The results of our work support the argument that tasks to design, adapt and pose mathematical problems enhance their pedagogical understanding and should be a feature of teacher education courses. This is a new area for research and practice; further exploration of the suitability of particular tasks in elementary mathematics teacher education is recommended.

Learning to think spatially : what do students 'see' in numeracy test items?

Learning to think spatially in mathematics involves developing proficiency with graphics. This paper reports on 2 investigations of spatial thinking and graphics. The first investigation explored the importance of graphics as 1 of 3 communication systems (i.e. text, symbols, graphics) used to provide information in numeracy test items. The results showed that graphics were embedded in at least 50 % of test items across 3 year levels. The second investigation examined 11 – 12-year-olds’ performance on 2 mathematical tasks which required substantial interpretation of graphics and spatial thinking. The outcomes revealed that many students lacked proficiency in the basic spatial skills of visual memory and spatial perception and the more advanced skills of spatial orientation and spatial visualisation. This paper concludes with a reaffirmation of the importance of spatial thinking in mathematics and proposes ways to capitalize on graphics in learning to think spatially.

Mathematics and reading difficulty subtypes: Minor phonological influences on mathematics for 5-7 year olds

Linguistic influences in mathematics have previously been explored throughsubtyping methodology and by taking advantage of the componential nature ofmathematics and variations in language requirements that exist across tasks. Thepresent longitudinal investigation aimed to examine the language requirements of mathematical tasks in young children aged 5-7 years. Initially, 256 children were screened for mathematics and reading difficulties using standardised measures. Those scoring at or below the 35th percentile on either dimension were classified as having difficulty. From this screening, 115 children were allocated to each of the MD (n=26), MDRD (n=32), reading difficulty (RD, n=22) and typically achieving (TA, n=35) subtypes. These children were tested at four time points, separated by six monthly intervals, on a battery of seven mathematical tasks. Growth curve analysis indicated that, in contrast to previous research on older children, young children with MD and MDRD had very similar patterns of development on all mathematical tasks. Overall, the subtype comparisons suggested that language played only a minor mediating role in most tasks, and this was secondary in importance to non-verbal skills. Correlational evidence suggested that children from the different subtypescould have been using different mixes of verbal and non-verbal strategies to solve the mathematical problems.

A aprendizagem dos números racionais através das conexões entre as suas representações:uma experiência de ensino no 2º ciclo do ensino básico

Tese de doutoramento, Educação (Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2014

A aprendizagem da função afim no 8º ano : uma proposta apoiada nas tecnologias e na educação financeira

Trabalho de projeto de mestrado, Educação (Área de especialidade em Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2015

Awareness of number in children with severe and profound learning difficulties: three exploratory case studies

This paper reports on exploratory work investigating how children with severe and profound learning difficulties register an awareness of small quantities and how they might use this information to inform their understanding. It draws on studies of typically developing children and investigates their application to pupils whose response to conventional mathematical tasks are often limited because they lack relevance and interest. The responses of the three pupils to individualized learning contexts mirror the progression suggested in the literature, namely from awareness of number to simple actions using number cues to problem-solving behaviour

Dean's great discovery : multiplication, division and fractions

Multiplication, division and fractions are 'hotspots' for students in the middle years with many students experiencing difficulty with these concepts (Siemon, Virgona & Cornielle, 2001). Arrays effectively model multiplication and help children develop multiplicative thinking and learn multiplication facts (Young-Loveridge, 2005). In this article we show how an open-ended array problem enabled a Grade 5/6 student to think about the relationship between multiplication, division and fractions. In the article we describe the project and 'hot spot' mathematical tasks that we used and provide some background on multiplicative thinking before presenting the case and a commentary (Western Melbourne Roundtable, 1997) of one student's exploration. This case was documented whilst we were working on a collaborative project with a team of upper primary teachers and a group of pre-service teachers at a local primary school.

The use of video recording in professional development of teachers and teacher educators in Indonesia

Design-based research has gained currency in educational research over the past decade due to its strength to bridge the divide between theoretical research and educational practice in naturalistic settings. Design based approaches involve a process of designing mathematical tasks, observing the enacted design in classrooms and reflecting on the process from analysing the classroom artifacts. Video plays a central role in supporting teachers and teacher educators to study and reflect on students’ mathematical thinking and in capturing the dynamic of classroom teaching and learning process.

This chapter will examine and analyse practitioner’s lenses in capturing the dynamic and complexity of classroom mathematical learning using video segments, and classroom artifacts including digital photos of classroom moments and students’ work. Practitioners’ lenses are taken as a window to capture key teaching and learning moments from the lessons. Analysis of this selection of these video segments along with other classroom artifacts based on practitioners’ lenses provide insights into practitioners’ views on key teaching and learning moments in mathematics lessons.

Becoming numerate with information and communications technologies in the twenty-first century

This article draws on data from a three-year Australian Research Council-funded study that examined the ways in which young children become numerate in the twenty-first century. We were interested in the authentic problem-solving contexts that we believe are required to create meaningful learning. This being so, our basic tenet was that such experiences should involve the use of information and communications technologies (ICT) where relevant, but not in tokenistic ways. This article highlights learning conditions in which young children can become numerate in contemporary times. We consider ‘academic’ or ‘school-based’ mathematical tasks in the context of a Mathematical Tasks Continuum. This continuum was conceptualised to enable focused and detailed thinking about the scope and range of mathematical tasks that young children are able to engage within contemporary school contexts. The data from this study show that most of the tasks the children experienced in early years mathematics classes were unidimensional in their make up. That is, they focus on the acquisition of specific skills and then they are practiced in disembedded contexts. We suggest that the framework created in the form of the Mathematical Tasks Continuum can facilitate teachers thinking about the possible ways in which they could extend children’s academic work in primary school mathematics, so that the process of becoming numerate becomes more easily related to authentic activities that they are likely to experience in everyday life.

The Use of Videos and Classroom Artefacts in Professional Development of Teachers and Teacher Educators in Indonesia

This study is grounded on adaptations of Realistic Mathematics Education, Lesson Study and design-based research in Indonesian classroom contexts. Design-based research has gained currency in educational research over the past decade due to its strength to bridge the divide between theoretical research and educational practice in naturalistic settings. Design-based approaches involve a process of designing mathematical tasks, observing the enacted design in classrooms and reflecting on the process from analysing the classroom artefacts. Video plays a central role in supporting teachers and teacher educators to study and reflect on students’ mathematical thinking and in capturing the dynamic of classroom teaching and learning process. This chapter will examine and analyse practitioners’ lenses in capturing the dynamic and complexity of classroom mathematical learning using video vignettes and classroom artefacts including digital photos of classroom moments and students’ work. Practitioners’ lenses are taken as a window to capture key teaching and learning moments from the lessons. Analysis of this selection of these video vignettes along with other classroom artefacts based on practitioners’lenses provides insights into practitioners’ views on key teaching and learning moments in mathematics lessons.