Procedural and conceptual changes in young children’s problem solving

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5–6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the “iterative model” hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving.     

Two non-traditional measures of chemistry learning: Word association and idea association

Conclusion The word association test and idea association test are open-ended tests which assess students' linking ability and also their accuracy in retrieving the relevant knowledge. These tests break through the “spoon-fed” and “drilled” types of assessment. The earlier study has shown that linkage of knowledge is one of the influential factors in determining problem solving performance (Lee, 1986). A replication study has confirmed the earlier findings that these two non-traditional measures assess learning performance as do the traditional methods. Apart from providing the numerical scores, for the learning performance, the two non-traditional tests also provide teachers with useful qualitative evidence of how students proceed with their learning and with their understanding of concepts. The technical procedures of setting, carrying out and scoring the tests are reasonably easy to handle. On the basis of the evidence shown, it is suggested that the word association and idea association tests should become part of the regular testing of chemistry learning in schools.     

Why a Special Issue on Collaborative Learning in Postsecondary and Professional Settings?

To introduce the special issue on collaborative learning in postsecondary and professional education, the editors describe their reasons for assembling it. First, most of the existing collaborative learning research focuses on K-12 settings; second, some forms of collaborative learning are used predominantly or exclusively in postsecondary or professional education; and finally, increasing numbers of adults are getting some or all of their college education online. The introduction concludes with the editors explaining their decision not to take a position on the nomenclature question (“collaborative” vs. “cooperative” vs. “peer” vs. “small group” learning) which pervades the literature.     

Learning and teaching: Differentiation and relation

In this paper we discuss the articulation between teaching and learning, how to differentiate them and how to establish relations between them, limiting ourselves to aspects dealing with knowledge. The aim is to allow the design of teaching situations more relevant for learning. The characteristics differentiating teaching and learning are used to analyse research studies relative to two time scales, one of the order of months or years and the other of the order of hours. The comparison shows the importance of the grain size chosen to analyse the knowledge involved both in teaching and in learning. On the first scale, the analysis of the students' knowledge and that of the knowledge to be taught are done independently to the extent that students' knowledge is not analysed in reference to the knowledge to be taught (in terms of error or missing aspect) but on the basis of the student's coherency. The decomposition of these two types of knowledge into similar components allows us to compare them and leads us to propose “intermediate notions” between the usual physics knowledge to be taught and the students prior knowledge. These intermediate notions can be rather far from complete correct physics knowledge but are learnable by the students. On the second scale, detailed analysis of a single teaching session and the students' processes during this session needs a fine level of knowledge granularity. Such a level allows us to make hypotheses based on the elements of students' prior knowledge from which they can construct new knowledge and not only on the prior knowledge which has to be modified. This granularity level allows an emphasis on the positive aspects of students' prior knowledge and enables us to construct hypothesis in order to design teaching situations. Making explicit “intermediate notions” in the knowledge to be taught at a rather large level of granularity of knowledge and the positive aspects of students' prior knowledge at a fine level of granularity, are proposed as ways to improve teaching for fruitful learning.     

Beyond rules and propositions: Reflections on Bereiter's concept of problem-centred knowledge

Bereiter's concept of problem-centred knowledge is criticized on two grounds. First, associating knowledge with particular problem types limits the use of that knowledge to those problem types, causing rather than curing mindlessness. Second, interesting problems are not independent of our knowledge but arise within particular knowledge structures. Hence, problems must be indexed by the knowledge structures they are relevant to, rather than the other way around. It is suggested that a more traditional concept of theoretical knowledge carries some of the same pedagogical potential as the concept of problem-centred knowledge without sharing its inherent difficulties.     

Student attitudes toward learning,level of pre-knowledge and instruction type in a computer-simulation: effects on flow experiences and perceived learning outcomes

Attitudes toward learning (ATL) have been shown to influence students’ learning outcomes. However, there is a lack of knowledge about the ways in which the interaction between ATL, the learning situation, and the level of students’ prior knowledge influence affective reactions and conceptual change. In this study, a simulation of acid-base titrations was examined to assess the impact of instruction format, level of prior knowledge and students’ ATL on university-level students, with respect to flow experiences (Csikszentmihalyi, 1990) and perceived conceptual change. Results show that the use of guiding instructions was correlated with a perceived conceptual change and high levels of “Challenge,” “Enjoyment,” and “Concentration,” but low sense of control during the exercise. Students who used the open instructions scored highly on the “Control flow” component, but their perceived learning score was lower than that for the students who used the guiding instructions. In neither case did students’ ATL or their pre-test results contribute strongly to students’ flow experiences or their perceived learning in the two different learning situations.     

Professional Development of Physics Teachers in an Evidence-Based Blended Learning Program

The present study examined continuity of learning between face-to-face and online environments in a “blended” professional development program designed for 16 physics teachers. The program had nine face-to-face meetings as well as continuous online exchanges between them through a website. The program focused on “knowledge integration” (KI) innovative activities in physics classes using an “evidence-based” approach: The teachers implemented the activities, collected and analyzed data about their practice and their students’ learning, and reflected on the evidence with their peers. Five reflective tools were used to promote continuity: Your Comments, Hot Polls, Smashing Sentences, Hot Reports, and Mini Research. Continuity was assessed with regard to the ideas discussed by the teachers and the reasoning patterns that they employed. Analysis of the online exchanges in relation to teachers’ face-to-face discourse revealed that the teachers discussed the same ideas (KI, evidence and learner-centered pedagogies), employed the same reasoning patterns (e.g., forming generalizations), and extended ideas in re-visitation. The online and face-to-face environments played different and complementary roles in the teachers’ learning. This study shows that appropriate use of an online environment in a blended program can lead to a continuous course of learning and can transform a “9 once-a-month-meetings” workshop into a “9-month” workshop.     

Knowledge building community: Keys for using online forums

Conclusion “The need for learning in a knowledge-based society is more important than ever, including traditional classroom teaching, online learning, and/or blended learning” (Levitch & Milheim, 2003). Accordingly, learning in a technology-supported collaborative knowledge building community is more desirable than ever. Whether in distance or blended learning, online forums can provide either an exclusive discussion or a supplement to in-class discussion. Through thoughtful planning and careful implementation, online forums can be used in “creative ways to help students internalize knowledge and share ideas in enjoyable and exchange environments” (Raleigh, 2000). Successful employment of online forums can foster knowledge building community in which desired student qualities are cultivated.     

What about social value?

We focus on three aspects of the articles of Reyna, of Perry, Stupnisky, Daniels and Haynes, and of Murdock, Beauchamp and Hinton. The first aspect is the logic of causal chain, a logic that we differentiate from a more deterministic approach. The second one is the mode of corrective action (attribution retraining) that is planned for students, whether cheaters or lower achievers. We differentiate this mode of action from the one that is based on the idea of normative awareness (or “clear-sightedness”): normative awareness does not imply that the speaker believes in what he says. Finally, we discuss the concept of social value (of a person) which seems to underlie the empirical results but which does not appear in the authors’ formulations. But this concept stems from a “sociologizing” meta-theory that these authors probably do not share and which makes the norm of enternality theory difficult to integrate within Weiner’s attribution theory.     

What Is a “Semiotic Perspective”, and What Could It Be? Some Comments on the Contributions to This Special Issue

This comment attempts to identify different “semiotic perspectives” proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to “semiotics in mathematics education”.     

Mathematics-for-Teaching: an Ongoing Investigation of the Mathematics that Teachers (Need to) Know

In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”. Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.     

IT Entry knowledge and skills of university teachers vis-à-vis their interest levels: the case of the oldest university in Asia

Technology, considered as the new language of teaching and learning, is an irrevocable reality. Any institution of higher learning is expected to look into the future without losing sight of its core capabilities-the malleability of its faculty members. Today, more than ever, teachers are expected to operate in an environment where “hiteach”, “hi-touch” and “hi-tech” govern their instructional behaviors. It is in this light that this study was conducted to situate teachers coming from a time-tested institution, such as the University of Santo Tomas, the oldest university in Asia, in terms of their IT knowledge, skills, and interests with a view to identifying implications on how faculty development programs may be made more responsive to the present-day educational structure.     

Improving mathematics instruction through lesson study: a theoretical model and North American case

This article presents a theoretical model of lesson study, an approach to instructional improvement that originated in Japan. The theoretical model includes four lesson study features (investigation, planning, research lesson, and reflection) and three pathways through which lesson study improves instruction: changes in teachers’ knowledge and beliefs; changes in professional community; and changes in teaching–learning resources. The model thus suggests that development of teachers’ knowledge and professional community (not just improved lesson plans) are instructional improvement mechanisms within lesson study. The theoretical model is used to examine the “auditable trail” of data from a North American lesson study case, yielding evidence that the lesson study work affected each of the three pathways. We argue that the case provides an “existence proof” of the potential effectiveness of lesson study outside Japan. Limitations of the case are discussed, including (1) the nature of data available from the “auditable trail” and (2) generalizability to other lesson study efforts.

The use of mathematical symbolism in problem solving: An empirical study carried out in grade one in the French community of Belgium

This article relates to an empirical study based on the use of mathematical symbolism in problem solving. Twenty-five pupils were interviewed individually at the end of grade one; each of them was asked to solve and symbolize 14 different problems. In their classical curriculum, these pupils have received a traditional education based on a “top-down” approach (an approach that is still applied within the French Community of Belgium): conventional symbols are presented to the pupils immediately with an explanation of what they represent and how they should be used. Teaching then focuses on calculation techniques (considered as a pre-requisite for solving problems). The results presented here show the abilities (and difficulties) demonstrated by the children in making connections between the conventional symbolism taught in class and the informal approaches they develop when faced with the problems that are put to them. The limits of the “top-down” approach are then discussed as opposed to the more innovative “bottom-up” type approaches, such as those developed by supporters of Realistic Mathematics Educations in particular.     

Ready...Fire...Aim! Toward meaningful technology standards for educators and students

Conclusion I applaud ISTE, AASL, AECT, and the other organizations involved for tackling the “messy work” of developing standards for the use of technology and information resources in schools. And, at the same time, I call for a “second generation” of standards that define realistic expectations for teachers based on the subjects and levels they are called upon to teach. I propose that professional organizations from each subject work with ISTE and AECT to complete this huge task, and I propose that we consider as a “next step” the creation of a set of on-line learning experiences through which teachers can gain the identified skills and knowledge by using the very technologies we’re hoping they’ll embrace in their own teaching. There’s an old saying, “If you don’t know where you’re going, any road will do.” As far as educational technologies are concerned, this is also true. For many, the goal seems to have been simply to “get more computers into the schools,” without much thought about purpose. To return to Phil Schlechty’s metaphor, It’s generally been a brief and misguided “Ready” stage (occupied with questions like “How many do we need?” “What type?” “Where?” and “How shall we connect them?”), followed by “Fire!” (the acquisition and installation of equipment). What we need is: “Ready” (the creation of appropriate teams of people who will combine their insights to plan for the district)... “Aim” (a series of discussions about what technologies can accomplish for schools and the students they serve)... “Fire” (acquisition, installation, and professional development according to plan)... “Aim” (an assessment of how well the technologies and related programs met the intended goals, and a new planning effort designed to close the gap)... “Fire” (acquisition and implementation designed to eliminate the gap)... “Aim” (another gap assessment)... “Fire” (another attempt to close gaps)..., And so on.     

Using educational software to support collective thinking and test hypotheses in the computer science curriculum

This study analyses the discourse among the teacher and the students, members of three (3) small groups, who learn in the environment of a stand-alone computer. Two educational environments are examined: the first one, a “virtual laboratory” (Virtual scale-DELYS) and the second one, a computer modeling environment (ModelsCreator). The ‘Virtual Scale’ environment provides users with curriculum focused feedback and in that sense it can be categorized as directive. The ModelsCreator environment provides users merely with a representation of their own conception of curriculum concepts, so it can be categorized as an open-ended environment. The goal of this research is to exemplify the way the two educational software environments support (a) the development of collective thinking in peer— and teacher-led discussion and (b) students’ autonomy. The software tools of the “Virtual scale” along with the resources provided for the problem solving created an educational framework of hypothesis testing. This framework did not limit the students’ contributions by directing them to give short answers. Moreover, it supported the students’ initiatives by providing tools, representations and procedures that offered educationally meaningful feedback. Based on the above results, we discuss a new educationally important structure of software mediation and describe the way the two software activities resourced collective thinking and students’ initiatives. Finally, for each type of software environment, we propose certain hypotheses for future research regarding the support of collaborative problem solving.     

中小学"数学情境与提出问题"教学的实验研究

This research tends to make the experimental study on the mathematics teaching model of “situated creation and problem-based instruction” (SCPBI), namely, the teaching process of “creating situations—posing problems—solving problems—applying mathematics”. It is aimed at changing the situation where students generally lack problem-based learning experience and problem awareness. Result shows that this teaching model plays a vital role in arousing students’ interest in mathematics, improving their ability to pose problems and upgrading their mathematics learning ability as well.      

How Children Regulate their Own Collaborative Learning

In this article we analyze the dialogic learning of one pair of students in order to investigate how these students cope with a collaborative learning situation in the classroom. Our aim is to substantiate the claims that not only are young students (8 year olds) capable of solving mathematical problems collaboratively, but that they also take an active role in regulating their collaborative learning activities. More specifically, our claim is that children appear to apply constructs of “mathematical level raising”, “social interaction” and “division of time” to steer their own collaborative learning and that they are rather successful in balancing these three aspects. The analysis is exploratory, but this new perspective on collaborative learning is relevant theoretically and consequential for classroom practice.     

Four dialogues and metalogues about the nature of science

We analyse and explore, in the form of dialogues and metalogues questions about the dialogic nature of beliefs and students belief talk about the nature of science and scientific knowledge. Following recent advances in discursive psychology, this study focuses not on students' claims but on the discursive resources and dialogical practices that support the particular claims they make. We argue that students' discourse is better understood as a textual bricolage that is sensitive to conversational context, common sense, interpretive repertoires, and textual resources available in the conversational situation. Our text is reflexive as it embodies the discursive construction of knowledge and undercuts any claims to authoritative knowledge. The very conception of “belief” is itself an expression or construction from within the mundane idiom.... We learn to use “belief” in conditions when the “objective facts” are unknown or problematic and we want to indicate the tenuous character of our claim.... The notion of “real world” or “objective reality” is embedded in an extensive, pervasive language game which includes as an intelligible move or possibility the use of the very concept of “belief” itself. (Pollner, 1987, p. 21)