Developing Purposeful Mathematical Thinking : a Curious Tale of Apple Trees. 'Desarrollo de un pensamiento matemático intencionado : un relato curioso de manzanos'.

Resumen basado en el de la publicación

Daily Problem-Solving Warm-Ups: Harboring Mathematical Thinking In The Middle School Classroom

In this action research study of my classroom of 8th grade mathematics, I investigated the use of daily warm-ups written in problem-solving format. Data was collected to determine if use of such warm-ups would have an effect on students’ abilities to problem solve, their overall attitudes regarding problem solving and whether such an activity could also enhance their readiness each day to learn new mathematics concepts. It was also my hope that this practice would have some positive impact on maximizing the amount of time I have with my students for math instruction. I discovered that daily exposure to problem-solving practices did impact the students’ overall abilities and achievement (though sometimes not positively) and similarly the students’ attitudes showed slight changes as well. It certainly seemed to improve their readiness for the day’s lesson as class started in a more timely manner and students were more actively involved in learning mathematics (or perhaps working on mathematics) than other classes not involved in the research. As a result of this study, I plan to continue using daily warm-ups and problem-solving (perhaps on a less formal or regimented level) and continue gathering data to further determine if this methodology can be useful in improving students’ overall mathematical skills, abilities and achievement.

Primary school teacher’s noticing of students’ mathematical thinking in problem solving

Professional noticing of students’ mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students’ mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers’ noticing of students’ mathematical thinking when students solve tasks that involve proportional and non-proportional reasoning. From the analysis of how pre-service primary school teachers notice students’ mathematical thinking, we have identified an initial framework with four levels of development. This framework indicates a possible trajectory in the development of primary teachers’ professional noticing.

Developing Students’ Mathematical Thinking and Algorithmic Culture in IT Classes

Report published in the Proceedings of the National Conference on "Education in the Information Society", Plovdiv, May, 2012

Scaffolding students’ thinking in mathematical investigations

This article describes some of the issues that teachers might encounter when scaffolding students’ thinking during mathematical investigations. It describes four episodes where a teacher’s scaffolding failed to support students’ mathematical thinking and explores the reasons why the scaffolding was ineffective. Understanding what is ineffective and why is one way to improve pedagogical practice. As a background to these episodes, we first provide an overview of the mathematical investigation. Our paper concludes with some recommendations for judicious scaffolding during investigations.

Mathematical prerequisites for learning statistics in Psychology: Assessing core skills of numeracy and mathematical reasoning among undergraduates.

This study sought to extend earlier work by Mulhern and Wylie (2004) to investigate a UK-wide sample of psychology undergraduates. A total of 890 participants from eight universities across the UK were tested on six broadly defined components of mathematical thinking relevant to the teaching of statistics in psychology - calculation, algebraic reasoning, graphical interpretation, proportionality and ratio, probability and sampling, and estimation. Results were consistent with Mulhern and Wylie's (2004) previously reported findings. Overall, participants across institutions exhibited marked deficiencies in many aspects of mathematical thinking. Results also revealed significant gender differences on calculation, proportionality and ratio, and estimation. Level of qualification in mathematics was found to predict overall performance. Analysis of the nature and content of errors revealed consistent patterns of misconceptions in core mathematical knowledge , likely to hamper the learning of statistics.

Using blogs to enhance the capacity of mathematical communication in High School

The aim of this investigation is to analyze the use of the blog as an educational resource for the development of the mathematical communication in secondary education. With this aim, four aspects are analyzed: organization of mathematical thinking through communication; communication of mathematical thinking; analysis and evaluation of the strategies and mathematical thought of others; and expression of mathematical ideas using mathematical language. The research was conducted from a qualitative approach on an exploratory level, with the case study method of 4 classrooms of second grade of secondary education in a private school in Lima. The observational technique of 20 publications in the blog of the math class was applied; a study of a focal group with a sample of 9 students with different levels of academic performance; and an interview with the academic coordinator of the school was conducted. The results show that the organization of mathematical thinking through communication is carried out in the blog in a written, graphical and oral way through explanations, schemes and videos. Regarding communication of mathematical thinking, the blog is used to describe concepts, arguments and mathematical procedures with words and examples of the students. The analysis and evaluation of the strategies and mathematical thinking is performed through comments and debates about the publications. It was also noted that the blog does not facilitate the use of mathematical language to express mathematical ideas, since it does not allow direct writing of symbols nor graphic representation.

Literacy, numeracy and learning in school-aged children identified as having speech and language impairment in early childhood

The progress of a nationally representative sample of 3632 children was followed from early childhood through to primary school, using data from the Longitudinal Study of Australian Children (LSAC). The aim was to examine the predictive effects of different aspects of communicative ability, and of early vs. sustained identification of speech and language impairment, on children's achievement and adjustment at school. Four indicators identified speech and language impairment: parent-rated expressive language concern; parent-rated receptive language concern; use of speech-language pathology services; below average scores on the adapted Peabody Picture Vocabulary Test-III. School outcomes were assessed by teachers' ratings of language/literacy ability, numeracy/mathematical thinking and approaches to learning. Comparison of group differences, using ANOVA, provided clear evidence that children who were identified as having speech and language impairment in their early childhood years did not perform as well at school, two years later, as their non-impaired peers on all three outcomes: Language and Literacy, Mathematical Thinking, and Approaches to Learning. The effects of early speech and language status on literacy, numeracy, and approaches to learning outcomes were similar in magnitude to the effect of family socio-economic factors, after controlling for child characteristics. Additionally, early identification of speech and language impairment (at age 4-5) was found to be a better predictor of school outcomes than sustained identification (at aged 4-5 and 6-7 years). Parent-reports of speech and language impairment in early childhood are useful in foreshadowing later difficulties with school and providing early intervention and targeted support from speech-language pathologists and specialist teachers.

How well are Australian children doing in the first year of school : a demographic analysis

Background: The transition to school is a sensitive period for children in relation to school success. In the early school years, children need to develop positive attitudes to school and have experiences that promote academic, behavioural and social competence. When children begin school there are higher expectations of responsibility and independence and in the year one class, there are more explicit academic goals for literacy and numeracy and more formal instruction. Most importantly, children’s early attitudes to learning and learning styles have an impact on later educational outcomes. Method: Data were drawn from The Longitudinal Study of Australian Children (LSAC). LSAC is a cross-sequential cohort study funded by the Australian Government. In these analyses, Wave 2 (2006) data for 2499 children in the Kindergarten Cohort were used. Children, at Wave 2, were in the first year of formal school. They had a mean age of 6.9 years (SD= 0.26). Measures included a 6-item measure of Approaches to Learning (task persistence, independence) and the Academic Rating Scales for language and literacy and mathematical thinking. Teachers rated their relationships with children on the short form of the STRS. Results: Girls were rated by their teachers as doing better than boys on Language and literacy, Approaches to learning; and they had a better relationship with their teacher. Children from an Aboriginal or Torres Strait Island (ATSI) background were rated as doing less well on Language and Literacy and Mathematical thinking and on their Approaches to learning. Children from high Socio Economic Position families are doing better on teacher rated Language and Literacy, Mathematical thinking, Approaches to learning and they had a better relationship with their teacher. Conclusions: Findings highlight the importance of key demographic variables in understanding children’s early school success.

How well are Australian children doing in the first year of school?

Background: The transition to school is a sensitive period for children in relation to school success. In the early school years, children need to develop positive attitudes to school and have experiences that promote academic, behavioural and social competence. When children begin school there are higher expectations of responsibility and independence and in the year one class, there are more explicit academic goals for literacy and numeracy and more formal instruction. Most importantly, children’s early attitudes to learning and learning styles have an impact on later educational outcomes. Method: Data were drawn from The Longitudinal Study of Australian Children (LSAC). LSAC is a cross-sequential cohort study funded by the Australian Government. In these analyses, Wave 2 (2006) data for 2499 children in the Kindergarten Cohort were used. Children, at Wave 2, were in the first year of formal school. They had a mean age of 6.9 years (SD= 0.26). Measures included a 6-item measure of Approaches to Learning (task persistence, independence) and the Academic Rating Scales for language and literacy and mathematical thinking. Teachers rated their relationships with children on the short form of the STRS. Results: Girls were rated by their teachers as doing better than boys on Language and literacy, Approaches to learning; and they had a better relationship with their teacher. Children from an Aboriginal or Torres Strait Island (ATSI) background were rated as doing less well on Language and Literacy and Mathematical thinking and on their Approaches to learning. Children from high Socio Economic Position families are doing better on teacher rated Language and Literacy, Mathematical thinking, Approaches to learning and they had a better relationship with their teacher. Conclusions: Findings highlight the importance of key demographic variables in understanding children’s early school success.

Promoting student understanding through complex learning

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s learning potential, as increasingly complex learners capable of dealing with cognitively demanding tasks.

Primary students' performance on map tasks : the role of context

This study investigated the longitudinal performance of 583 students on six map items that were represented in various graphic forms. Specifically, this study compared the performance of 7-9-year-olds (across Grades 2 and 3) from metropolitan and non-metropolitan locations. The results of the study revealed significant performance differences in favour of metropolitan students on two of six map tasks. Implications include the need for teachers in non-metropolitan locations to ensure that their students do not overly fixate on landmarks represented on maps but rather consider the arrangement of all elements encompassed within the graphic.