An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed.     

Two kinds of B-basis of the algebraic hyperbolic space

In this paper, two new kinds of B-basis functions called algebraic hyperbolic (AH) Bezier basis and AH B-Spline basis are presented in the space (?)k=span{1,t,...,tk-3,sinht,cosht}, in which K is an arbitrary integer larger than or equal to 3. They share most optimal properties as those of the Bezier basis and B-Spline basis respectively and can represent exactly some remarkable curves and surfaces such as the hyperbola, catenary, hyperbolic spiral and the hyperbolic paraboloid. The generation of tensor product surfaces of the AH B-Spline basis have two forms: AH B-Spline surface and AH T-Spline surface.     

Uncovering students’ thinking about thinking using concept maps

A method for uncovering students’ thinking about thinking, specifically their meta-strategic knowledge, is explored within the context of an ongoing, multi-year intervention designed to promote the development of students’ thinking dispositions. The development of a concept-map instrument that classroom teachers can use and an analytic framework for interpreting students’ responses is presented. In a preliminary study, the concept map instrument is piloted to evaluate changes in students’ conceptions of thinking after a year’s participation in classrooms where their teachers actively sought to make thinking more visible by noticing and naming the thinking observed as well as introducing and using thinking routines (Ritchhart and Perkins. Educational Leadership, 65(5), 57–61 2008). Concept maps from 239 students from grades 3 through 11 were analyzed. Results suggest that students’ conceptions of thinking do improve with age but also can be substantially developed through a classroom culture where thinking is modeled and rich opportunities for thinking are present. The concept map instrument itself proved to be a robust instrument for uncovering students’ thinking about thinking.     

Exploring cultural differences in critical thinking: Is it about my thinking style or the language I speak?

Critical thinking is deemed as an ideal in academic settings, but cultural differences in critical thinking performance between Asian and Western students have been reported in the international education literature. We examined explanations for the observed differences in critical thinking between Asian and New Zealand (NZ) European students, and tested hypotheses derived from research in international education and cultural psychology. The results showed that NZ European students performed better on two objective measures of critical thinking skills than Asian students. English proficiency, but not dialectical thinking style, could at least partially if not fully explain these differences. This finding holds with both self-report (Study 1) and objectively measured (Study 2a) English proficiency. The results also indicated that Asian students tended to rely more on dialectical thinking to solve critical thinking problems than their Western counterparts. In a follow-up data analysis, students' critical thinking was found to predict their academic performance after controlling for the effects of English proficiency and general intellectual ability, but the relationship does not vary as a function of students' cultural backgrounds or cultural adoption (Study 2b). Altogether, these findings contribute to our understanding of the influence of culture on critical thinking in international education.     

Working with systems and thinking systemically – disentangling the crossed wires

This article explores two separate traditions that educational psychologists (EPs) in the UK have for working with systems. One of these is “systems work” with organisations such as schools. The other is “systemic thinking” for working with families. Over the years these two traditions, systems work and systemic thinking, have become crossed and intertwined. This article clarifies the connections between them by providing a historical overview, largely based on articles published by Educational Psychology in Practice and the thinking of R. Dallos and R. Draper on Family Therapy, of how they have developed. Opportunities for EPs working in Children Services to use systemic thinking and systems work are highlighted.     

Patterns and purposes in the uses and misuses of the term ‘critical thinking’ in the social sciences

This article analyses the uses and representations of the term “critical thinking” in the social science literature, based on a qualitative content analysis of titles, abstracts and keywords retrieved from the SCOPUS database for Germany, France and Russia over the last two decades. Our analysis focuses on how the use of the term “critical thinking” has increased over time, in which contexts the term is used and in which part of article texts it is used. Our findings are counterintuitive on several levels. First, the term “critical thinking” is seldom used in a pluri-disciplinary context. More commonly, it is used within specific discourses—notably education. Second, we found that it is mainly used instrumentally, rather than analytically. Third, most of the articles that use the term do not engage in actual critical analysis. There are also important geographic variation in the use of the term. In articles from Germany and Russia the term is used in similar ways—and differently in France. These nuances are difficult to analyse however, due to the different topics addressed, as well as author preferences. The use of the term “critical thinking” is diverse; indeed, at times weak and paradoxical. Finally, we discuss how editorial policy in academic journals possibly influences the discourse on critical thinking.     

From arithmetical thought to algebraic thought: The role of the “variable”

The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years.

On the application of primary process thinking to the naming of second‐order factors. 1

Several serious problems in the theoretical foundations of factor analysis are reviewed. The first, factor‐naming, is discussed as a determinant of factor fame. A second problem, that of rounding errors in the computerized extraction of factors, is treated from the perspective of psychoanalyticametric theory. Results of an empirical study of the causes of rounding errors are presented and discussed.     

Why teach critical thinking?

Abstract

The purpose of this article is to expose lack of clarity that surrounds the justification for teaching critical thinking and to argue for communally beneficial reasons for teaching this important skill. An explication of critical thinking is done and the prerequisite conditions for its optimal utilisation are indicated. The views of eight educators on why we teach critical thinking are offered. An argument for a semiotic understanding of culture is presented and this is followed by views on why critical thinking is taught.     

Combinatorial proofs and algebraic proofs — I

It is sometimes the case in mathematics that the same result can be proved in two or more essentially different ways (a luxury of mathematical riches, you might say). A recurrent theme in this context is the tension between combinatorial proofs and algebraic proofs. Inthis two-part article we describe some results which allow for proofs of both kinds and give us an opportunity to contrast them. We shall see some beautiful examples of both kinds of proofs. In Part I we dwell on results that deal with the binomial coefficients and with Fibonacci numbers. We will not be overly zealous in our usage of the term ‘combinatorial’: we will allow a proof to be so categorized if it is largely combinatorial, with only a ‘small’ component of algebra. However, that small part may well be an essential component of the proof. The simplicity of counting arguments can be deceptive. In recent times, Paul Erd?s has shown just how subtle and powerful such arguments can be, and how far they can reach.     

Child maltreatment and adult criminal behavior: Does criminal thinking explain the association?

Criminal thinking styles were examined as mediational links between different forms of child maltreatment (i.e., sexual abuse, physical abuse, and physical neglect) and adult criminal behaviors in 338 recently adjudicated men. Analyses revealed positive associations between child sexual abuse and sexual offenses as an adult, and between child physical abuse/neglect and endorsing proactive and reactive criminal thinking styles. Mediation analyses showed that associations between overall maltreatment history and adult criminal behaviors were accounted for by general criminal thinking styles and both proactive and reactive criminal thinking. These findings suggest a potential psychological pathway to criminal behavior associated with child maltreatment. Limitations of the study as well as research and clinical implications of the results are discussed.     

Combinatorial proofs and algebraic proofs — II

In Part I of this article we considered some binomial identities and also some identities involving the Fibonacci numbers, and proved them using methods which we described as ‘largely’ combinatorial. Now we shift our focus to number theory and to prime numbers in particular, and showcase some proofs having a strong combinatorial element. Throughout this article, p denotes an odd prime.     

Recurring patterns in the development of high school biology students’ system thinking over time

The goal of this study was to identify and understand the mental models developed by 67 high school biology students as they learn about the human body as a complex system. Using concept maps, it sought to find an external way of representing how students organize their ideas about the human body system in their minds. We conducted a qualitative analysis of four concept maps created by each student throughout the 3-year learning process, which allowed us to identify that student’s systems thinking skills and the development of those skills over time. The improvement trajectories of the students were defined according to three central characteristics of complex systems: (a) hierarchy, (b) homeostasis and (c) dynamism. A comparative analysis of all of our students’ individual trajectories together revealed four typical learning patterns, each of which reflects a different form of development for systems thinking: “from the structure to the process level”, “from macro to micro level”, “from the cellular level to the organism level,” and “development in complexity of homeostasis mechanisms”. Despite their differences, each of these models developed over time from simpler structures, which evolved as they connected with more complex system aspects, and each indicates advancement in the student’s systems thinking.     

Self‐regulation of student epistemic thinking in science: the role of metacognitive prompts

The purpose of the present study is to examine the effectiveness of a metacognitive prompts intervention‐science (MPI‐S), which is based on the nature of science with 162 eighth‐grade science students. It was hypothesised that students exposed to the intervention will show higher levels of content knowledge and knowledge about the nature of science than students in a comparison group. Attempts were also made to determine what cognitive processes are triggered during the use of metacognitive prompts. Findings showed significant improvements in students’ content knowledge and nature of science. In addition, qualitative findings revealed that the experimental group made choices based on evidence in the inquiry unit whereas the comparison group made decisions based on authority. Educational implications for practice into the classroom are discussed.     

Genetically modified food for the future: examining university students’ expressions of futures thinking

Futures thinking, or foresight, is increasingly recognised as a key element in science and environmental education. We integrated a futures thinking activity into an undergraduate general science course in Taiwan by using an online thinking and discussion tool that provides scaffolding questions to help students think systematically about future possibilities of the specific topic – genetically modified (GM) food. As a course assignment, the enrolled students were asked to work on and write their responses to this online scaffolding tool over five weeks. The purpose of this study was to examine students’ expressions of futures thinking on GM food as elicited through this assignment. We content analysed the written responses of 99 students who completed the online task. A particular emphasis of analysis was placed on examining students’ thoughts and ideas about ‘preferable’ or ‘desirable’ futures of GM food. Nine themes emerged from the analysis and were found be organised into three categories using factor analysis. These results provided insight into characteristics of students’ futures thinking and underlying perspectives, and can serve as a basis for future-focused curricular and instructional innovations.