Algebra slaves and agents in a Logo-based mathematics curriculum

For many students the ideas and methods of algebra appear obscure and mysterious, their sense and purpose unclear, and their applicability to anything genuinely real or interesting very remote. Students often fail to acquire an understanding of the key concepts, despite their inherent simplicity. Even when they gain the notion of variables, expressions and equations, students often lack the strategic knowledge required to motivate and direct the global planning and detailed execution of an attack on a problem. These conceptual and strategic difficulties are compounded by the needs for precise performance of the arithmetic and symbolic operations required in manipulating expressions. Extended operations like subtracting an expression from both sides of an equation or expanding a product of three terms, are very difficult for beginning students. Their buggy performance in carrying out the detailed manipulative work greatly confounds and frustrates their acquisition and assimilation of the most important and central ideas.In an effort to confront these difficulties and show how they can be overcome, we are developing a Logo-based introductory algebra course for sixth graders. Our approach has three major components: work on Logo programming projects in algebraically rich contexts whose content is meaning ful and compelling to students, the use of algebra microworlds with concrete iconic representations of formal objects and operations, and the introduction of the algebra workbench, an expert instructional system to aid students in performing extended algebraic operations.The algebra workbench will employ a set of powerful symbolic manipulation tools for performing the standard manipulations of high school algebra. It will have two main modes of use: demonstration mode, which uses an expert tutor program to solve algebra problems incrementally, explaining its strategy and its step by step operations in straightforward terms along the way; and practice mode, in which the student tries to solve a problem with the assistance of the tutor, which performs the operations requested by the student at each step and which can be called at any point to advise the student of the correctness of a step, to perform or explain any step, to evaluate the student's solution, or to perform a problem that she poses.These powerful aids make it possible to effectively separate out the difficulties in performing the formal and manipulative aspects of algebra work from those encountered in learning the central conceptual and strategic content. Distinctly different kinds of instructional tools and activities-Logo programming, expert tutors, or algebra microworlds-can thus be brought to bear where each is most appropriate and effective.     

A Macro–Micro–Symbolic Teaching to Promote Relational Understanding of Chemical Reactions

The purpose of this research is threefold: (1) to identify the difficulties that Grade 10 students in a Lebanese school have that hinder their conceptual understanding at the micro–macro–symbolic interface in chemistry, (2) to investigate the effect of a macro–micro–symbolic teaching approach on students’ relational understanding of chemical reactions, and (3) to characterize students’ conceptual profiles regarding their understanding of chemical reactions in terms of macro, micro, symbolic levels and the relations among them, at the end of the teaching sequence. Forty six 10th graders from two sections participated in the study. A student-centered approach was followed in both sections based on constructivist pedagogy. Hence the teacher played the role of a facilitator who guided students in a meaning making inductive learning process, through questioning, monitoring, validating, and clarifying ideas. Instruction in the experimental group was characterized by macro–micro–symbolic teaching that focuses on the interplay between the levels, integrates various representations, and engages students in an epistemic discourse about the nature of knowing in chemistry. Data sources for the study included a pre-test and two post-intervention tasks: a post-test and a concept map task, in addition to interviews with selected students from both sections. Findings indicated that macro–micro–symbolic teaching enhanced students’ conceptual understanding and relational learning of chemical reactions. Besides, four assertions related to students’ conceptual and epistemological thinking in response to the different teaching approaches are presented. Implications for instruction and for teacher education programs, as well as recommendations for further research, are discussed in light of these findings.     

Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking

A common approach used for introducing algebra to young adolescents is an exploration of visual growth patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of these difficulties at an early age. A teaching experiment was conducted with two classes of students with an average age of eight years and six months. From the results it appears that young students are capable not only of thinking about the relationship between two data sets, but also of expressing this relationship in a very abstract form.     

Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving

In Singapore, 6–12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13–17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is correlated with the ability to inhibit intrusions from the earlier arithmetic strategies. Similar to typical school practice in Singapore, we asked 14 year-old students (N = 157) to use only letter-symbolic algebra to solve 9 algebra word problems. After having controlled for algebraic knowledge, working memory, and intelligence, better inhibitory ability still predicted fewer arithmetic intrusions and higher problem solving accuracy. Path analysis revealed 2 types of inhibition. Inhibition-of-reified-processes predicted accuracy through arithmetic intrusions. Inhibition-of-recently-learned-associations predicted accuracy through intelligence. Findings suggest establishing pedagogical links between arithmetic and algebraic methods may facilitate students' transition to letter-symbolic algebra.     

Predicting Middle School Profiles of Algebra Performance Using Fraction Knowledge

Fraction knowledge and algebraic skill are closely linked. Algebra is a gatekeeper for advanced courses (Booth & Newton, 2012; Brown & Quinn, 2007). This study uses the person-centered approach of latent profile analysis to examine individual differences in middle schoolers’ (N = 350) algebra performance at the end of the year (EOY). The relative importance of a range of fraction skills is considered in predicting the likelihood of displaying a particular profile of EOY algebra knowledge, measured by feature knowledge, equation encoding, equation solving, and word-problem skills. Notably, fraction number line estimation did not predict algebra performance on any of the four measures. Fraction arithmetic at the beginning of the year is most predictive of displaying the highest performing EOY algebra profile.     

Secondary School Students’ Errors in the Translation of Algebraic Statements

In this article, we present the results of a research study that explores secondary students’ capacity to perform translations of algebraic statements between the verbal and symbolic representation systems through the lens of errors. We classify and compare the errors made by 2 groups of students: 1 at the beginning of their studies in school algebra and another 1 completing their studies on algebra in compulsory education. This comparison allows us to detect errors which require specific attention in instruction due to its persistence and to identify errors that disappear as students advance in their study of algebra. The results and conclusions have a pedagogic value to inform instruction and also lead to backed conjectures and research questions to push forward research on student’s translation capacity and students’ knowledge of algebraic symbolism.     

从算术到代数

从算术向代数过渡,是学生数学学习过程中极为重要的转变阶段.符号是代数不同于算术的典型特征,学生从算术向代数的过渡,是从对数的思考向对符号的思考的转变,是从算术思维向代数思维的转变,是思维层次从个别到一般、具体到抽象的飞跃.     

The Adidactic Interaction with the Procedures of Peers in the Transition from Arithmetic to Algebra: A Milieu for the Emergence of New Questions

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of each student with the procedures generated by other students during stage 1. The two kinds of processes – adaptation to a milieu and social interactions – play a critical role in the change of “rationality” required for the move from arithmetic to algebra. Both the design of the initial mathematical problem given to the students and the organization of the interactions leading to the debate under study in this article are based on this hypothesis. The research presented in this article is set in a broader work of didactic engineering that aims at studying didactic conditions for making a connection between arithmetic practices and algebraic practices.     

Pizzas or no pizzas: An advantage of word problems in fraction arithmetic?

Fractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties by making fraction arithmetic less abstract. The current study therefore investigated to what extent students (106 sixth graders, 187 seventh graders, and 192 eighth graders) perform better on fraction arithmetic problems presented as word problems compared to these problems presented symbolically. Results showed that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts. However, in addition, subtraction, and multiplication of two fractions, symbolic problems were easier. There were no performance differences by students’ grade, but higher conceptual fraction knowledge was associated with higher fraction arithmetic performance. Taken together this study showed that situating fraction arithmetic in a realistic setting may support or hinder performance, dependent on the problem demands.     

Bridging the gap: a longitudinal study of the relationship between pedagogical continuity and early Chinese literacy acquisition

This longitudinal study examined the relationship between pedagogical continuity in literacy education and early literacy development by comparing Chinese children in Hong Kong and Shenzhen. Stratified random sampling was used to select 24 preschool and Primary 1 classes in four communities catering to middle‐class families in each city. The 24 teachers were interviewed about their teaching methods and views on pedagogical continuity in individual sessions. Further, their teaching activities were videotaped over a period of a week. Their students (n = 758) completed a Chinese literacy attainment test at the beginning and at the end of the same academic year. Analyses indicated that Hong Kong students outperformed their Shenzhen counterparts in Chinese literacy attainment, at both preschool and primary levels, concurrently and longitudinally. This suggests that the holistic approach followed in Hong Kong might have a more positive impact on children's literacy development than the approach followed in Shenzhen. Implications of the findings for early childhood curriculum reform are discussed.     

Liberation or Oppression?—Western TESOL Pedagogies in China

In this article, we examine power relations in College English teaching in China, focusing on the symbolic capital of English as a global language. Framing our discussion with Bourdieu's concept of symbolic capital and a review of literature, we problematize the importation of pedagogies from Western countries to China and argue that seemingly liberating pedagogies, such as the communicative language teaching approach, can be turned into a form of oppression of both the instructors and the students. Drawing on Freire's critical pedagogy, we propose that a truly liberating pedagogy should be based on a dialogical relationship between policy-makers, teachers, and students in consideration of the specific economic, social, and cultural backgrounds of the students and teachers.     

STUDENTS’ OPPORTUNITIES TO ENGAGE IN TRANSFORMATIONAL ALGEBRAIC ACTIVITY IN DIFFERENT BEGINNING ALGEBRA TOPICS AND CLASSES

This study compares students’ opportunities to engage in transformational (rule-based) algebraic activity between 2 classes taught by the same teacher and across 2 topics in beginning algebra: forming and investigating algebraic expressions and equivalence of algebraic expressions. It comprises 2 case studies; each involves a teacher teaching in two 7th grade classes. All 4 classes used the same textbook. Analysis of classroom videotapes (15–19 lessons in each class) revealed that the opportunities to engage in transformational algebraic activity related to forming and investigating algebraic expressions were similar in each teacher’s 2 classes. By contrast, substantial differences were found between 1 teacher’s classes with regard to the opportunities to engage in transformational algebraic activity related to equivalence of algebraic expressions. The discussion highlights the contribution of the interplay among the mathematical topic, the teacher, and the class to shaping students’ learning opportunities. Specifically, the mathematical topic appeared to play a prominent role in certain situations, with the topic involving deductive reasoning generating high variation in classes of 1 teacher but not in the other’s.

     

列方程解应用题的认识与思考

在列方程解应用题教学的起始阶段 ,要使学生的认识完成从“算术到代数”的转变     

Improving pre‐service teacher learning through peer teaching: process,people and product1

This research investigates the perceptions of first‐year Bachelor of Teaching students (primary and secondary) and Diploma of Education students (secondary) about their peer teaching experience in a postgraduate subject called Curriculum and Assessment. Peer teaching is a learner‐centred approach to teaching and learning that is intended to provide significant benefits for learners' knowledge, skills and metacognition. However, concerns have been raised over the quality of the learning and teaching and the risks associated with such a pedagogy. In the present study, student responses to questionnaires and semi‐structured interviews were analysed, using a mixed methods approach, with respect to three broad and somewhat interconnected categories: process, people and product. These responses suggested a wide range of reactions to peer teaching, but overall students feel they benefited from the experience. The findings of this study should be of interest to lecturers and students in pre‐service teacher education courses, especially. Knowledge about peer teaching, learning and assessment would be especially valuable for both education lecturers and beginning teachers seeking to design and manage learner‐centred pedagogy in their own primary, secondary and tertiary classrooms. However, the results of this research would have far‐reaching appeal for all teaching and learning contexts.     

Connecting past and present through case-comparison learning in history: views of teachers and students

History education frequently aims at developing active citizenship by using the past to orientate to the present and the future. A pedagogy for pursuing this aim is making connections between the past and the present by means of comparing cases of an enduring human issue. To examine the feasibility and desirability of this case-comparison teaching approach, students (n = 444) and teachers (n = 15) who participated in an implementation study conducted in the Netherlands were questioned about their experiences and views. Results show that both students and teachers felt that case-comparison in the context of an enduring human issue is feasible and not more complex than the usual history teaching in which topics are studied separately without explicitly making comparisons between past and present, even if some students thought that taking account of episodes from different historical periods concurrently required an extra learning effort. Both students and teachers believed that connecting past and present in history teaching enhances engagement and meaning making. They suggested a curriculum combining the case-comparison approach with the type of history teaching they were accustomed to. Mixed methods were used for data collection. Implications for further research on case-comparison learning in history are being discussed.     

STUDENTS’ OPPORTUNITIES TO ENGAGE IN TRANSFORMATIONAL ALGEBRAIC ACTIVITY IN DIFFERENT BEGINNING ALGEBRA TOPICS AND CLASSES

This study compares students’ opportunities to engage in transformational (rule-based) algebraic activity between 2 classes taught by the same teacher and across 2 topics in beginning algebra: forming and investigating algebraic expressions and equivalence of algebraic expressions. It comprises 2 case studies; each involves a teacher teaching in two 7th grade classes. All 4 classes used the same textbook. Analysis of classroom videotapes (15–19 lessons in each class) revealed that the opportunities to engage in transformational algebraic activity related to forming and investigating algebraic expressions were similar in each teacher’s 2 classes. By contrast, substantial differences were found between 1 teacher’s classes with regard to the opportunities to engage in transformational algebraic activity related to equivalence of algebraic expressions. The discussion highlights the contribution of the interplay among the mathematical topic, the teacher, and the class to shaping students’ learning opportunities. Specifically, the mathematical topic appeared to play a prominent role in certain situations, with the topic involving deductive reasoning generating high variation in classes of 1 teacher but not in the other’s.     

Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context

Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated multiple components (symbolic equations, symbols, categorization, hint) in the source, or target, or both, to address the four analogical tasks. Different combinations of the components were tested in a series of four experiments. Symbolic equations (main component) fostered a mental construction of the problem in its solution mode. Categorization enabled an identification of the problem category. A hint in the target directed the learners to the source problem. The interaction between these components facilitated the mapping of the symbolic equations in the source onto the target, resulting in the superiority of the MC approach in fostering analogical transfer. Neither the main component alone nor the main component plus one sub-component was sufficient for analogical transfer. Hence for analogical transfer to occur, at least the main component (symbolic equations) and two sub-components (categorization and hint) are required. However, symbols may not have additional effects for transfer to occur.     

《线性代数》教学探讨与实践

为了让学生学好《线性代数》,增加学生的学习兴趣,我们进行一系列的探索和实践：建立和谐的师生关系;尝试探究性教学,问题讨论式教学以及多媒体教学,开展数学建模竞赛等实践活动。     

Learning from students,inquiry into practice,and participation in professional communities: Beginning teachers' uneven progress toward equitable science teaching

In this research project, we investigated two beginning secondary science teachers' efforts to learn to teach science in ways that build from and celebrate the ethnic, gender, linguistic, and academic diversity of their students. To do so, we followed Troy and Brian from their preservice teacher education experiences through their first year of teaching 8th grade physical science at local junior high schools. We also conducted a follow‐up observation and interview with each participant after he had moved past the beginning stage of survival in the teaching profession—once in his fourth year of public school science teaching. Through qualitative analysis of interviews, classroom observations, and teachers' written work, we identified patterns and explored commonalities and differences in Troy and Brian's views and practices tied to equity over time. In particular, we examined successes and challenges they encountered in learning to teach science for all (a) from their students, (b) from inquiry into practice, and (c) from participation in professional communities. In our implications, we suggest ways teacher educators and induction professionals can better support beginning teachers in learning to teach science to all students. In particular, we highlight the central roles both individual colleagues and collective school cultures play in aiding or impeding beginning teachers' efforts to learn from students, from practice, and from professional communities. © 2006 Wiley Periodicals, Inc. J Res Sci Teach 44: 586–612, 2007.