The gains and the pitfalls of reification — The case of algebra

Algebraic symbols do not speak for themselves. What one actually sees in them depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice. It is this last statement which becomes the leading theme of this article. The main focus is on the versatility and adaptability of student's algebraic knowledge.The analysis is carried out within the framework of the theory of reification according to which there is an inherent process-object duality in the majority of mathematical concepts. It is the basic tenet of our theory that the operational (process-oriented) conception emerges first and that the mathematical objects (structural conceptions) develop afterward through reification of the processes. There is much evidence showing that reification is difficult to achieve.The nature and the growth of algebraic thinking is first analyzed from an epistemological perspective supported by historical observations. Eventually, its development is presented as a sequence of ever more advanced transitions from operational to structural outlook. This model is subsequently applied to the individual learning. The focus is on two crucial transitions: from the purely operational algebra to the structural algebra of a fixed value (of an unknown) and then from here to the functional algebra (of a variable). The special difficulties experienced by the learner at both these junctions are illustrated with much empirical data coming from a broad range of sources.     

Luck,Choice, and Educational Equality

Abstract

Harry Brighouse discusses two conceptions of educational equality. The first is a type of equality of opportunity, heavily influenced by the work of John Rawls, which he calls the meritocratic conception. According to this conception, an individual’s educational prospects should not be influenced by factors such as their social class background. The other, radical conception, suggests a person’s natural talents should not influence their educational prospects either. Brighouse favors the meritocratic conception, but this article argues that it is flawed and that the radical conception ought to be preferred. Although a superior conception of educational equality, the radical conception is still not quite right, so this article develops a luck egalitarian conception of educational equality. It is argued that this conception reflects much current thinking about equality and avoids some of the difficulties with Brighouse’s two conceptions. Finally, two objections to a luck egalitarian conception are considered.     

Exploring the relationship between K-8 prospective teachers’ algebraic thinking proficiency and the questions they pose during diagnostic algebraic thinking interviews

In this study, we explored the relationship between prospective teachers’ algebraic thinking and the questions they posed during one-on-one diagnostic interviews that focused on investigating the algebraic thinking of middle school students. To do so, we evaluated prospective teachers’ algebraic thinking proficiency across 125 algebra-based tasks and we analyzed the characteristics of questions they posed during the interviews. We found that prospective teachers with lower algebraic thinking proficiency did not ask any probing questions. Instead, they either posed questions that simply accepted and affirmed student responses or posed questions that guided the students toward an answer without probing student thinking. In contrast, prospective teachers with higher algebraic thinking proficiency were able to pose probing questions to investigate student thinking or help students clarify their thinking. However, less than half of their questions were of this probing type. These results suggest that prospective teachers’ algebraic thinking proficiency is related to the types of questions they ask to explore the algebraic thinking of students. Implications for mathematics teacher education are discussed.     

The promises of numeracy

The aim of this study is to better understand the notion of early algebraic thinking by describing differences in grade 4–7 students’ thinking about basic algebraic concepts. To achieve this goal, one test that involved generalized arithmetic, functional thinking, and modeling tasks, was administered to 684 students from these grades. Quantitative analysis of the data yielded four distinct groups of students demonstrating a wide range of performance in these tasks. Qualitative analysis of students’ solutions provided further insight into their understanding of basic algebraic concepts, and the nature of the processes and forms of reasoning they utilized. The results showed that students in each group were able to solve different number and types of tasks, using different strategies. Results also indicated that students from all grades were present in each group. These findings suggest the presence of a consistent trend in the difficulty level across early algebraic tasks which may support the existence of a specific developmental trend from more intuitive types of early algebraic thinking to more sophisticated ones.     

Computers and the language arts: Theory and practice

In this paper, the second of two, we set out a conception of critical thinking that critical thinking is a normative enterprise in which, to a greater or lesser degree, we apply appropriate criteria and standards to what we or others say, do, or write. The expression 'critical thinking' is a normative term. Those who become critical thinkers acquire such intellectual resources as background knowledge, operational knowledge of appropriate standards, knowledge of key concepts, possession of effective heuristics, and of certain vital habits of mind. We explain why these intellectual resources are needed and suggest that we can best teach critical thinking by infusing it within any curricular practice in which our students are involved.     

高等代数概念课的特征及教学原则

高等代数课程问题是高等代数教学改革的中心问题。作者在教改实践中探索了高等代数概念课的特征及教学原则,关键是使学生尽快提高数学思维水平,才能掌握高等代数概念的本质和来龙去脉。     

基于多元正义原则的教育公平观

教育公平是现代教育发展的基本诉求之一。但是,有关讨论并未意识到,不同教育公平观在基本原则上存在分歧,并且秉持不同原则的教育公平观对教育部门的设计、对教育与社会的关系定位都十分不同。为了澄清我们的教育公平理想,很有必要深入检讨其基本原则。为此,文章区分出基于平等原则的教育公平和基于多元正义原则的教育公平两个观念。随后,在批判分析的基础上,我们认为基于平等原则的教育公平观较难满足当前教育公平的核心诉求;由于多元正义原则与教育的个性化原则之间相互匹配,所以基于多元正义原则的教育公平观更为可取。     

“Is there an equal (amount of) juice?” Exploring the repeated question effect in conservation through conversation

The aim of this paper is to highlight and discuss advantages and constraints of different methods applied within the field of children's thinking studies, through the test of the repeated question hypothesis validity, using the conservation of liquid task. In our perspective, the Piagetian interview is an ecologically valid context for externalization and modification of children's thinking. We used an experimental procedure organized in standard and modified tasks, involving primary school children in Serbia. The results of quantitative and qualitative analyses show that the repeated question is not the unique cause of children's misleading in demonstrating to understand conservation. Other dimensions explain why children change their answers when they are tested by the two tasks we used, which offers an insight into the influence of research procedures on children's answers.     

Educational Equality: Luck Egalitarian,Pluralist and Complex

The basic principle of educational equality is that each child should receive an equally good education. This sounds appealing, but is rather vague and needs substantial working out. Also, educational equality faces all the objections to equality per se, plus others specific to its subject matter. Together these have eroded confidence in the viability of equality as an educational ideal. This article argues that equality of educational opportunity is not the best way of understanding educational equality. It focuses on Brighouse and Swift's well worked out meritocratic conception and finds it irretrievably flawed; they should, instead, have pursued a radical conception they only mention. This conception is used as a starting point for developing a luck egalitarian conception, pluralistic and complex in nature. It is argued that such a conception accounts for the appeal of equality of opportunity, fits with other values in education and meets many of the objections. Thus, equality is reasserted as what morally matters most in education.     

The impact of a thinking skills intervention on children's concepts of intelligence

The study reported was part of a large thinking skills intervention for 11–12-year-old children. This paper focuses on the impact of a thinking skills intervention on children's understandings of intelligence. A total of 178 children (n = 86 girls and n = 92 boys) across six schools participated in the study. Children were individually pre-tested in the classroom using written tasks designed to tap concepts of intelligence (definitions, characteristics, causes of intelligence, and the stability of intelligence: entity versus incremental concepts) and a variety of thinking skills. Schools were allocated into one of three intervention conditions: control condition; individual condition; collaborative learning condition. Children in the individual and collaborative learning conditions participated in an 8-week thinking skills intervention. Children in the individual condition worked individually on tasks to apply the thinking skills whereas learners in the collaborative condition applied the thinking skills on tasks in groups of four. Following the thinking skills intervention all children were individually post-tested using the pre-test measures. The results showed that the intervention had an impact on children's understanding of intelligence. In particular, the collaborative learning intervention led to most improvement in concepts of intelligence. The results are discussed with reference to theories of intelligence concepts and thinking skills interventions.     

Signs and meanings in students' emergent algebraic thinking: A semiotic analysis

The purpose of this article, which is part of a longitudinal classroom research about students' algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and(2) to provide accounts about the students' emergent algebraic thinking. The research draws from Vygotsky's historical-cultural school of psychology, on the one hand, and from Bakhtin and Voloshinov's theory of discourse on the other, and is grounded in a semiotic-cultural theoretical framework in which algebraic thinking is considered as a sign-mediated cognitive praxis. Within this theoretical framework, the students' algebraic activity is investigated in the interaction of the individual's subjectivity and the social means of semiotic objectification. An ethnographic qualitative methodology, supported by historic, epistemological research, ensured the design and interpretation of a set of teaching activities. The paper focuses on the discussion held by a small group of students of which an interpretative, situated discourse analysis is provided. The results shed some light on the students' production of (oral and written) signs and their meanings as they engage in the construction of expressions of mathematical generality and on the social nature of their emergent algebraic thinking. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: interiorization, condensation, and reification. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.     

油画创造力是创造性思维与富有想象力的技法操作的统一

油画创造力是创造性的思维和富有想象力的操作的结合。油画创作是艺术活动 ,不局囿于科学性的概念。通过思维灵感在操作技法上的互动方能得到有价值的画面。艺术创造力在艺术思维方向上具有多种预期发展 ,不同于一般的逻辑思维 ,创造性思维是一种归纳思维基础上的发散形式 ,是艺术灵感在操作技法上的发散     

Equivalence and relational thinking: preservice elementary teachers’ awareness of opportunities and misconceptions

This study examined awareness of equivalence and relational thinking exhibited by 30 preservice elementary teachers in order to assess their initial preparedness to engage students in these two important aspects of early algebraic reasoning. Findings indicated that preservice teachers collectively demonstrated an awareness of relational thinking both in identifying opportunities offered by tasks to engage students in this thinking and in identifying this thinking in samples of student work. However, in proposing difficulties students might have with selected tasks, few participants demonstrated the understanding that many elementary school students hold misconceptions about the meaning of the equal sign. Implications of these findings for preservice and inservice teacher education are discussed.     

Analogical reasoning and the development of algebraic abstraction

This paper presents a theory of the development of algebraic abstraction which extends Sfard's (1991, 1994a,b) and Mason's (1982, 1989) ideas on the learner's progress from operational or process-oriented thinking to the abstract or structural perspective. The theory incorporates a process of analogical reasoning to account for the means by which the learner might construct expressions of generality and subsequently manipulate them as mathematical objects. Such reasoning entails similarity comparisons in which a mapping is made between the corresponding relational properties of algebraic examples. These comparisons may firstly entail unpacking the relations in the examples in order to highlight the structural commonalities. The common relational structure is subsequently extracted to form a knowledge basis, namely, the construction of a mental model or representation that expresses the observed generalisation. The theory is applied to an analysis of secondary school students' approaches to classifying a set of complex equations. A student who appeared capable of algebraic abstraction within the domain of the task is contrasted with two students who were at a pseudostructural stage, where their focus on syntactic surface structures prevented them from forming the relational mappings needed for the construction of generalised models of the equation types.     

The role of analogical thinking in designing tasks for mathematics teacher education: An example of a pedagogical ad hoc task

This article discusses the design of tasks for teacher education. It focuses on tasks that are used in a university course for pre-service secondary school mathematics teachers. Special attention is given to tasks that use analogical thinking in their construction or implementation. These tasks are categorized by type of teacher education goal and analyzed with respect to the use of analogical thinking. Short examples are presented for three of the goal categories, while an elaborated example is given for the fourth one. The detailed example describes the goals and design of a task sequence following an emergent pedagogical need. Using the ad hoc constructed task-sequence the teacher educator avoids “telling” while demonstrating an alternative instructional approach, and seizing the opportunity to bring up additional pedagogical issues.     

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 Meritocratic Conception of Educational Equality: Ideal Theory Run Amuck

The dominant conception of educational equality in the United States is meritocratic: an individual's chances of educational achievements should track only (natural) talent and effort, not social class or other morally irrelevant factors. The meritocratic conception must presuppose that natural talent and effort can be isolated from social class — and environmental factors in general — if it is to provide guidance in the world of educational policy and practice. In this article Kenneth R. Howe challenges that presupposition and related elements of the meritocratic conception having to do with the role of competition and education as a positional good. Howe argues that, in use, the meritocratic conception unavoidably distributes education based on developed talent and motivation that cannot be isolated from the effects of social class, thus masking the real basis of the distribution and contributing to the perpetuation of illicit group privilege. Howe then entertains and rejects several rejoinders to his critique, including that the meritocratic conception might eliminate the presupposition of isolatable natural talent. Finally, Howe sketches an alternative conception of educational equality based on Elizabeth Anderson's adaptation of Rawls's “democratic equality.”     

Exploring the characteristics of an optimal design of non-programming plugged learning for developing primary school students’ computational thinking in mathematics

Existing computational thinking (CT) research focuses on programming in K-12 education; however, there are challenges in introducing it into the formal disciplines. Therefore, we propose the introduction of non-programming plugged learning in mathematics to develop students’ CT. The research and teaching teams collaborated to develop an instructional design for primary school students. The participants were 112 third- and fourth-grade students (aged 9–10) who took part in three rounds of experiments. In this paper, we present an iterative problem-solving process in design-based implementation research, focusing on the implementation issues that lead to the design principles in the mathematics classroom. The computational tasks, environment, tools, and practices were iteratively improved over three rounds to incorporate CT effectively into mathematics. Results from the CT questionnaire demonstrated that the new program could significantly improve students’ CT abilities and compound thinking. The results of the post-test revealed that CT, including the sub-dimensions of decomposition, algorithmic thinking, and problem-solving improved threefold compared to the pre-test between the three rounds, indicating that strengthened CT design enhanced CT perceptions. Similarly, the students’ and teacher’ interviews confirmed their positive experiences with CT. Based on empirical research, we summarize design characteristics from computational tasks, computational environment and tools, and computational practices and propose design principles. We demonstrate the potential of non-programming plugged learning for developing primary school students’ CT in mathematics.

     

Children's understanding of the commutativity and complement principles: A latent profile analysis

This study examined patterns of individual differences in the acquisition of the knowledge of the commutativity and complement principles in 115 five-to six-year-old children and explored the role of concrete materials in helping children understand the prinicples. On the basis of latent profile analysis, four groups of children were identified: The first group succeeded in commutativity tasks with concrete materials but in no other tasks; the second succeeded in commutativity tasks in both concrete and abstract conditions, but not in complement tasks; the third group succeeded in all commutativity tasks and in complement tasks with concrete materials, and the final group succeeded in all the tasks. The four groups of children suggest a developmental trend – (1) Knowledge of the commutativity and of the complement principles seems to develop from thinking in the context of specific quantities to thinking about more abstract symbols; (2) There may be an order of understanding of the principles – from the commutativity to the complement principle; (3) Children may acquire the knowledge of the commutativity principle in the more abstract tasks before they start to acquire the knowledge of the complement principle. This study contributes to the literature by showing that assessing additive reasoning in different ways and identifying profiles with classification analyses may be useful for educators to understand more about the developmental stage where each child is placed. It appears that a more fine-grained assessment of additive reasoning can be achieved by incorporating both concrete materials and relatively abstract symbols in the assessment.