Michela Maschietto and Luc Trouche have written an article called Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories. The article was published online in ZDM on Monday. Although they start with a glimpse from a babylonian clay tablet, their main focus is on the development of tools and use of tools in the last century. In the main part of their article, they have a strong focus on the so-called mathematics laboratories. Here is the abstract of their article:

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.

A new issue of International Journal of Science and Mathematics Education has been published, Volume 7, Number 5, October 2009. The issue contains nine articles, several of which are related to mathematics education:

• CREATING OPTIMAL MATHEMATICS LEARNING ENVIRONMENTS: COMBINING ARGUMENTATION AND WRITING TO ENHANCE ACHIEVEMENT, by Dionne I. Cross
• APPROACHES TO TEACHING MATHEMATICS IN LOWER-ACHIEVING CLASSES, by Ruhama Even and Tova Kvatinsky
• ANALYSIS OF THE LEARNING EXPECTATIONS RELATED TO GRADE 1–8 MEASUREMENT IN SOME COUNTRIES, by Jung Chih Chen, Barbara J. Reys and Robert E. Reys
• EPISTEMOLOGICAL OBSTACLES IN COMING TO UNDERSTAND THE LIMIT OF A FUNCTION AT UNDERGRADUATE LEVEL: A CASE FROM THE NATIONAL UNIVERSITY OF LESOTHO, by Eunice Kolitsoe Moru

Aiso Heinze, Franziska Marschick and Frank Lipowsky have written an article that was published in the recent issue of ZDM. The article is entitled Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade. Here is the abstract of their article:

Empirical findings show that many students do not achieve the level of a flexible and adaptive use of arithmetic computation strategies during the primary school years. Accordingly, educators suggest a reform-based instruction to improve students’ learning opportunities. In a study with 245 German third graders learning by textbooks with different instructional approaches, we investigate accuracy and adaptivity of students’ strategy use when adding and subtracting three-digit numbers. The findings indicate that students often choose efficient strategies provided they know any appropriate strategies for a given problem. The proportion of appropriate and efficient strategies students use differs with respect to the instructional approach of their textbooks. Learning with an investigative approach, more students use appropriate strategies, whereas children following a problem-solving approach show a higher competence in adaptive strategy choice. Based on these results, we hypothesize that different instructional approaches have different advantages and disadvantages regarding the teaching and learning of adaptive strategy use.

Aiso Heinze, Jon R. Star and Lieven Verschaffel have written an article entitled Flexible and adaptive use of strategies and representations in mathematics education. The article was published in ZDM, Volume 41, Number 5 on Wednesday. Here is the abstract of their article:

The flexible and adaptive use of strategies and representations is part of a cognitive variability, which enables individuals to solve problems quickly and accurately. The development of these abilities is not simply based on growing experience; instead, we can assume that their acquisition is based on complex cognitive processes. How these processes can be described and how these can be fostered through instructional environments are research questions, which are yet to be answered satisfactorily. This special issue on flexible and adaptive use of strategies and representations in mathematics education encompasses contributions of several authors working in this particular field. They present recent research on flexible and adaptive use of strategies or representations based on theoretical and empirical perspectives. Two commentary articles discuss the presented results against the background of existing theories.

Abstract

In this paper, we focus on Finnish pre-service elementary teachers’ (N = 269) and upper secondary students’ (N = 1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6 = 83. How could you conclude from this relationship (without using long-division algorithm) what 491:6 = ? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency.

Three new articles have been published online in ZDM lately. One of these articles is entitled The role of fluency in a mathematics item with an embedded graphic: interpreting a pie chart, and it is written by Carmel Mary Diezmann and Tom Lowrie. Here is the abstract of their article:

The purpose of this study was to identify the pedagogical knowledge relevant to the successful completion of a pie chart item. This purpose was achieved through the identification of the essential fluencies that 12–13-year-olds required for the successful solution of a pie chart item. Fluency relates to ease of solution and is particularly important in mathematics because it impacts on performance. Although the majority of students were successful on this multiple choice item, there was considerable divergence in the strategies they employed. Approximately two-thirds of the students employed efficient multiplicative strategies, which recognised and capitalised on the pie chart as a proportional representation. In contrast, the remaining one-third of students used a less efficient additive strategy that failed to capitalise on the representation of the pie chart. The results of our investigation of students’ performance on the pie chart item during individual interviews revealed that five distinct fluencies were involved in the solution process: conceptual (understanding the question), linguistic (keywords), retrieval (strategy selection), perceptual (orientation of a segment of the pie chart) and graphical (recognising the pie chart as a proportional representation). In addition, some students exhibited mild disfluencies corresponding to the five fluencies identified above. Three major outcomes emerged from the study. First, a model of knowledge of content and students for pie charts was developed. This model can be used to inform instruction about the pie chart and guide strategic support for students. Second, perceptual and graphical fluency were identified as two aspects of the curriculum, which should receive a greater emphasis in the primary years, due to their importance in interpreting pie charts. Finally, a working definition of fluency in mathematics was derived from students’ responses to the pie chart item.

The other is written by Alan T. Graham, Maxine Pfannkuch and Michael O.J. Thomas. Their article is called Versatile thinking and the learning of statistical concepts. In the abstract you learn more about the main ideas in this article:

Statistics was for a long time a domain where calculation dominated to the detriment of statistical thinking. In recent years, the latter concept has come much more to the fore, and is now being both researched and promoted in school and tertiary courses. In this study, we consider the application of the concept of flexible or versatile thinking to statistical inference, as a key attribute of statistical thinking. Whilst this versatility comprises process/object, visuo/analytic and representational versatility, we concentrate here on the last aspect, which includes the ability to work within a representation system (or semiotic register) and to transform seamlessly between the systems for given concepts, as well as to engage in procedural and conceptual interactions with specific representations. To exemplify the theoretical ideas, we consider two examples based on the concepts of relative comparison and sampling variability as cases where representational versatility may be crucial to understanding. We outline the qualitative thinking involved in representations of relative density and sample and population distributions, including mathematical models and their precursor, diagrammatic forms.

Finally, George Gadanidis and Vince Geiger have written an article about A social perspective on technology-enhanced mathematical learning: from collaboration to performance. Here is the abstract of their article:

This paper documents both developments in the technologies used to promote learning mathematics and the influence on research of social theories of learning, through reference to the activities of the International Commission on Mathematical Instruction (ICMI), and argues that these changes provide opportunity for the reconceptualization of our understanding of mathematical learning. Firstly, changes in technology are traced from discipline-specific computer-based software through to Web 2.0-based learning tools. Secondly, the increasing influence of social theories of learning on mathematics education research is reviewed by examining the prevalence of papers and presentations, which acknowledge the role of social interaction in learning, at ICMI conferences over the past 20 years. Finally, it is argued that the confluence of these developments means that it is necessary to re-examine what it means to learn and do mathematics and proposes that it is now possible to view learning mathematics as an activity that is performed rather than passively acquired.

Kemal Altiparmak and Ece Özdogan have written an article that was recently published online in International Journal of Mathematical Education in Science and Technology. The article is entitled A study on the teaching of the concept of negative numbers. Here is the abstract of their article.

This study mainly aims to develop an effective strategy to overcome the known difficulties in teaching negative numbers. Another aim is to measure the success of this teaching strategy among a group of elementary level pupils in Idotzmir, Turkey. Learning negative concepts are supported by computer animations. The academic achievement test developed by the researchers was administered to 150 sixth-grade pupils at the beginning of and following the learning period. The teaching strategy was applied to the experiment group (n = 75) as stated above, while the traditional teaching model most frequently used in Turkey was applied to the control group (n = 75). At the end of the study, a significant difference was found in favour of the experiment group (t = 17.51, df = 148, p = 0.000 < 0.05).

To date, a number of studies have demonstrated the existence of mismatches between children’s implicit and explicit knowledge at certain points in development that become manifest by their gestures and gaze orientation in different problem solving contexts. Stimulated by this research, we used eye movement measurement to investigate the development of basic knowledge about numerical magnitude in primary school children. Sixty-six children from grades one to three (i.e. 6-9 years) were presented with two parallel versions of a number line estimation task of which one was restricted to behavioural measures, whereas the other included the recording of eye movement data. The results of the eye movement experiment indicate a quantitative increase as well as a qualitative change in children’s implicit knowledge about numerical magnitudes in this age group that precedes the overt, that is, behavioural, demonstration of explicit numerical knowledge. The finding that children’s eye movements reveal substantially more about the presence of implicit precursors of later explicit knowledge in the numerical domain than classical approaches suggests further exploration of eye movement measurement as a potential early assessment tool of individual achievement levels in numerical processing.