Parsing the notion of algebraic thinking within a cognitive perspective

There is a growing consensus that algebra is an important aspect of mathematics teaching and learning and several abilities are required in order students to have successful performance in algebra. The present study uses insights from the domain of psychology to enrich what is currently known in the domain of mathematics education about the relationship of algebraic thinking with abilities involved in fundamental cognitive processes. In total, 190 students between the ages of 13–17 years old were tested through two tests. The first test addressed four types of cognitive systems which are responsible for the representation and processing of different types of relations in the environment: the spatial-imaginal, the causal-experimental, the qualitative-analytic and the verbal-propositional. The second test addressed algebraic thinking. The results support the key role of the four types of cognitive processes in students’ algebraic thinking. The results also suggest that abilities involved in the four types of cognitive processes predict algebraic thinking abilities, irrespective of the age of the students.     

Under-threes’ mathematical learning – teachers’ perspectives

This project highlights preschool teachers’ views of toddlers’ learning in mathematics. The Swedish national curriculum covers even the youngest children who are 1–3?years old. Interesting questions are thus: what should mathematics be for this age group and how should preschool teachers work with maths to achieve the curriculum objectives? Data were collected through interviews with six preschool teachers working in four different preschools. The data show that the teachers emphasize the body as very important for the learning process, which means that for these children, it is not a matter of simply talking about mathematical concepts, but experiencing them bodily. The teachers also report that they now pay more attention than previously to what material the children use and how they interact with it. They are more aware of how they organize and offer the various materials in the preschool and how this influences the way children use them and, consequently, their learning processes.     

The Effectiveness of Workbooks in the Elementary School Reading Program

Abstract

The study examined relations between ethclass group membership, the achievement syndrome of families, and the cognitive performance of 850 11-year-old Australian children from six ethclass groups. A semi- structured interview schedule was constructed to assess parents’ achievement orientations, press for English, press for dependence, educational and occupational aspirations, and individualistic-collectivistic value orientations. Standardized tests were used to measure children’s intelligence and performance in mathematics, word knowledge, and word comprehension. The results support the proposition that if children from different cultural groups are found to be characterized by distinct patterns of cognitive scores, then the groups are characterized by distinct patterns of family learning environments. Also, the findings suggest that some of the achievement syndrome measures may act as threshold variables so that until certain levels of particular environment measures are attained, other family processes may have limited associations with children’s cognitive performance.     

Meaning and Money

This paper examines how the inclusion of everyday mathematics into classroom instruction can make mathematics more meaningful to students. The concept of mathematical meaningfulness is reviewed and then compared to the experiences of children learning about money at home and at school. The empirical study used interviews and observations to determine what activities Hawaiian children from preschool through second grade did with money at home, while shopping and during classroom lessons. The interview data are used to show what kinds of knowledge children derived from these experiences at different ages. This everyday knowledge is compared to what children were expected to learn about money in school. The data support the conclusion that certain kinds of differences between everyday and school mathematics can make the inclusion of everyday mathematical topics in classrooms problematic. The paper concludes with a discussion of how everyday mathematics can be more profitably included in the curriculum, with examples from several mathematics programs.     

A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper.     

The algorithmic origins of counting

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.     

An assessment-based model for exploring the solving of mathematical problems: Utilizing revised bloom’s taxonomy and facets of metacognition

This paper provides a method that can be used to review the teaching, learning, and/or assessment of mathematics at either (or both of) the senior secondary and undergraduate levels. In this paper, how this method could be enacted is exemplified by considering the case of integral calculus. The method uses Revised Bloom’s Taxonomy (RBT) (Anderson et al., 2001) in conjunction with Efklides’s metacognition framework (Efklides, 2006, 2008) to design questions to address the different RBT cognitive processes and knowledge types. Using these two frameworks can help develop questions that target broader student thinking and a range of cognitive processes, including constructive ones, than traditional questions reach. In doing so, this method can be a starting point for Faculties seeking to reform their delivery and assessment of mathematics.     

Investigating the development of prospective mathematics teachers’ pedagogical content knowledge of generalising number patterns through school practicum

The purpose of this study is to investigate the development of three prospective elementary mathematics teachers?? pedagogical content knowledge (PCK) of number patterns with its two components: (1) knowledge of students?? understanding and difficulties and (2) knowledge of topic-specific strategies and representations. For our investigation, we use Radford??s (ZDM Math Educ 40:83?C96, 2008) framework called ??architecture of algebraic pattern generalisation?? to redefine these two components of PCK in the context of patterns. A case study was conducted to explore the development of PCK through a school practicum course and to examine how observation of number pattern lessons in schools and discussions of these observation contribute to prospective teachers?? PCK. Analysis of the data indicated that observations and discussions of number pattern lessons resulted in considerable change in the way prospective teachers take into account students?? understanding of and difficulties with patterns, as well as in the way they use pattern-specific strategies such as algebraic pattern generalisation during their lessons.     

An Activity Analysis of Secretarial Duties as a Basis for an Office Practice Course

The cognitive processes used by children during problem solving have received considerable attention in recent years. The identification of the processes used by both successful and unsuccessful problem solvers has been hampered by the difficulties involved in making accurate observations of mental processes which may or may not have physical manifestations during problem solving. In addition, problem solvers themselves often are unaware of many of the thought processes they use while attempting to solve problems. The most popularly used methods for studying problem solving processes are not sufficiently reliable or valid. This study explored the potential effectiveness of a new procedure for identifying and studying certain of the cognitive processes used during problem solving. In addition, the procedure was used to categorize the types of conceptual thinking problem solvers employ, to study the use of trial and error behavior, and to investigate problem solvers’ abilities to coordinate multiple bits of information. It appears that the procedure has application for generating hypotheses about cognitive processes problem solvers use.     

The influence of English-Korean bilingualism in solving mathematics word problems

The purpose of this study was to investigate language related difficulties and the language of cognitive processes of English-Korean bilingual students in solving mathematics word problems. Qualitative case study research methodology was used to collect, analyze, and present data. The principle of purposeful sampling was used to select six English-Korean bilingual students. Different types of bilinguals revealed distinct patterns of difficulties and languages in solving mathematics word problems written in English and Korean. Children in the transition stage that is unstable and changing revealed more difficulties in solving the mathematics word problems overall.This article is a summary of doctoral dissertation under the direction of James W. Wilson at the University of Georgia.     

Making Sense of Space: Distributed Spatial Sensemaking in a Middle School Summer Engineering Camp

Spatial thinking is important for success in engineering. However, little is known about how students learn and apply spatial skills, particularly in kindergarten to Grade 12 engineering learning. The present study investigated the role of spatial thinking in engineering learning at a middle school summer camp. Participants were 26 students (13 female, 13 male), predominantly from underrepresented groups. We took a cognitive ethnographic approach, using observations of hands-on engineering learning activities to identify moments when spatial problems arose and how learners made sense of these problems. We describe these processes as distributed spatial sensemaking because they involved both internal (cognitive) processes and also interactions with other learners, materials, and representations. We identified 90 distributed spatial sensemaking episodes in our data set. These episodes facilitated important engineering practices such as hypothesis testing and design iteration. We also found that different activities elicited different types of distributed spatial sensemaking episodes. Our results demonstrate how spatial thinking matters in everyday engineering learning and speaks to the types of engineering learning activities that scaffold particular spatial processes and practices. Our research also shows how cognitive, situated, and distributed theories can be used in tandem to make sense of a complex phenomenon like engineering learning.     

Understanding Individual Patterns of Learning: implications for the well-being of students

This article analyses the link between learning and well-being from the perspective that it is important to take into account the individual patterns of how young people learn and to encourage ways in which they can learn how they learn. Consideration is first given to recent insights and research in education and the cognitive and natural sciences that offer possibilities for new ways of thinking about learning and 'learning how to learn'. Using these trends as a background to her own 30 years of action research, the author offers a simple proposal: even very young children can explore their own unique learning processes. Building this capacity for exploration and encouraging the ongoing engagement in this activity are presented as vital to how well-being, and well-becoming are perceived by the learner. An approach that focuses on exploring individual processes of learning is introduced and the conditions for using it are discussed, including the role of teachers and parents in helping children bring forth their needs for learning in specific ways. Implications of this approach for classroom management and teacher training are also considered.     

Dynamic testing in indigenous and ethnic minority children

Our study aim was to examine whether ethnic-minority children would show different change patterns to indigenous children when presented with a seriation task within a dynamic testing context based on graduated prompt-techniques. We hypothesized that both Dutch-indigenous and ethnic-minority children would employ more sophisticated strategies after graduated prompt-training compared with untrained control-group children, and that trained ethnicity groups would show different progression paths from pre- to posttest. We also hypothesized that ethnic-minority and Dutch-indigenous children would progress differently in types of strategies over the course of their training, and that they would differ in number and type of instructions needed; i.e., whether various learning patterns could be detected. We further examined which variables predicted strategy-level best. Dynamically tested children changed their strategy behavior into the direction of a more advanced strategy; this change was the largest for the initial weaker scoring ethnic-minority children. These children also initially needed more but progressively needed less cognitive hints than Dutch-indigenous children during training. The graduated prompts approach we made use of clearly unfolds possibilities to detect and describe strong and weaker points in each child's solving processes before, during and after training. With a short intervention it appeared possible to describe how many prompts when and what type of prompts a child needed during training, i.e. to describe “the learning while tested” process.     

Strategies to Encourage Mathematics Learning in Early Childhood: Discussions and Brainstorming Promote Stronger Performance

This longitudinal study examined the influence of prekindergarten teacher characteristics and classroom instructional processes during mathematical activities on the growth of mathematics learning scores in prekindergarten, kindergarten, and first grade. Participants attended state-funded and Head Start prekindergarten programs. Mathematical performance was measured in fall and spring in prekindergarten and spring in kindergarten and first grade using the Test of Early Mathematics Ability–3 (TEMA-3; Ginsburg & Baroody, 2003). Two dimensions of the Classroom Assessment Scoring System (CLASS; i.e., instructional learning formats and concept development; Pianta, La Paro, & Hamre, 2008) were scored based on observed classroom mathematics activities. Teachers provided information about their education and years of prekindergarten teaching experience. Research Findings: Instructional processes that included elements of the CLASS concept development dimension, such as discussions and brainstorming to encourage children’s understanding, were related to growth of mathematics scores. Neither teacher characteristics nor instructional processes of the CLASS instructional learning formats dimension, such as using different modalities and materials, and learning objectives, were related to growth of mathematics scores. Practice or Policy: The findings extend our understandings of how instructional processes impact children’s early mathematical performance. These findings may be helpful in increasing our understanding of the types of instructional processes that might be emphasized in teacher professional development specifically related mathematical activities. Professional development that focuses on the CLASS concept development dimension may be easier for teachers to remember and implement in their classrooms and, consequently, have a greater impact on mathematics learning.     

Knowledge based systems for teaching and learning maths

The main task of an educational system is to generate instructional situations which induce students to learn knowledge and problem solving abilities as applied to a cognitive domain.
To this end, an instructional dialogue must be seen as a process aimed at making student and teacher learn from each other: the student has to learn the subject matter from the teacher, while the teacher has in turn to learn from the student how to regulate instructional interaction. The process relies on different kinds of expert knowledge: experience of the domain, about the actual student, and of teaching methods and objectives. This means that an instructional dialogue cannot be realised without an explicit representation of all the kinds of knowledge involved.
Research has established that Knowledge Based Educational Systems (KBESs) can be the keystone in building effective learning tools, but because of the difficulty in realising a system of this kind, most existing systems are prototypes and are intended only as workbenches for the computational analysis of educational processes.
This paper analyses the main problems which underlie the realisation of such systems, with reference to research into knowledge based systems intended for use with teaching/learning mathematics.     

Learning Disabilities in Guatemala and Spain: A Cross‐National Study of the Prevalence and Cognitive Processes Associated with Reading and Spelling Disabilities

The main purposes of this research were twofold. We examined the samenesses about learning disabilities (LD) in Guatemala and Spain, two countries with the same language but cultural, political, and educational differences, first analyzing data about the prevalence of reading and spelling disabilities in Guatemala City and the Spanish region of the Canary Islands. The focus of the second study was to determine whether there are cross‐national patterns of significant differences in cognitive processes associated with reading and spelling disabilities from a developmental approach in these two cultural contexts. We found some differences in the prevalence of specific LD in reading between both countries but we did not find significant differences between Guatemalan and Spanish reading‐disabled children in cognitive processes that are involved in reading and spelling acquisition in spite of the cultural and educational differences between the two countries.     

Effects of consistency and adequacy of language information on understanding elementary mathematics word problems

Two types of elementary mathematics word problems involving different linguistic structures were devised to examine the understanding and solution of these problems by 91 Grade 3, 4, and 5 children divided into “more able” and “less able” subgroups. One task consisted of 12 consistent and 12 inconsistent language problems on the basic processes of addition, subtraction, multiplication and division. Another task consisted of a total of 36 word problems with 12 items each containing adequate, inadequate, and redundant information, respectively, for problem solution. Subsidiary tasks of general ability, vocabulary, reading comprehension, mathematics concepts, reflection on mathematics learning, and working memory were also administered to provide estimates of the contribution of these “nonmathematics” tasks to the solution of elementary mathematics problems. Analyses of variance and covariance of group data showed significant main effects of grade, consistency, and adequacy of linguistic information in problem solution. Word problems containing inconsistent information were more difficult than those with consistent information. Further, word problems containing inadequate and redundant information were more difficult to classify, and for the children to explain, than those items with just enough information. Interviews with 12 individual children provided further insight into their strategies for problem solutions. Both cognitive and developmental perspectives are important for mathematics learning and teaching for children with or without learning disabilities.     

What kinds of things cause children’s reading difficulties?

The first part of this paper explains the distinction between proximal and distal causes of developmental disorders of cognition, with special reference to developmental disorders of reading. A number of different proximal causes of developmental disorders of reading have been identified. These correspond to a number of different patterns of developmental reading impairments – a number of “types of developmental dyslexia”, if you like – which are described in the second part of the paper. These patterns are interpreted in relation to a specific information-processing model of the cognitive system for reading. How each pattern is diagnosed, and how each might be treated, is discussed.     

中英澳高中向量习题认知领域的比较研究

选取中国、英国和澳大利亚三国代表性的三套高中数学教材,借鉴修订后的布鲁姆认知领域分类理论,对教材中向量内容的习题进行比较研究.研究发现:中英澳三国教材在知识维度的差异显著,主要体现在"程序性知识"和"元认知知识"两种类型;在认知过程维度的差异并不明显.对中国课程教学改革的启示:注重各知识类型的生成与转化;重视高认知水平的数学习题对学生再创造能力的培养;注重知识与认知过程的有效融合等.