International Electronic Journal of Mathematics Education

Online ISSN: 1306-3030
Publications
Level descriptors for student reasoning from comparison of box plots including summary of student overall attainment level 
Year 11 (15-year-old) students are not exposed to formal statistical inferential methods. When drawing conclusions from data, their reasoning must be based mainly on looking at graph representations. Therefore, a challenge for research is to understand the nature and type of informal inferential reasoning used by students. In this paper two studies are reported. The first study reports on the development of a model for a teacher’s reasoning when drawing informal inferences from the comparison of box plots. Using this model, the second study investigates the type of reasoning her students displayed in response to an assessment task. The resultant analysis produced a conjectured hierarchical model for students’ reasoning. The implications of the findings for instruction are discussed.
 
Using a sample of 4th and 5th graders, this study investigated whether students’ mathematics anxiety differed significantly according to a group of variables. A total of 249 students participated in the study. “The Mathematics Anxiety Scale for Elementary School Students” and “The Personal Information Form” were used for data collection. Independent samples t-tests, Oneway Anova and Schefee test were used to analyze the data. Results showed that students’ mathematics anxiety differed significantly according to gender, whether they liked mathematics class or not, whether they liked their mathematics teachers or not and the achievement level in mathematics. Female students reported significantly higher mathematics anxiety than males. Students who liked mathematics class and those who liked their mathematics teachers had lower anxiety. Students with higher achievement in mathematics reported lower degrees of mathematics anxiety. However, results did not show any significant difference in students’ mathematics anxiety with respect to their grade level and gender-stereotypes about success in mathematics.
 
Quite often a mathematical object may be introduced by a set of equivalent definitions. A fundamental question is determining the “didactic effectiveness” of the techniques for solving a kind of problem associated with these definitions; this effectiveness is evaluated by taking into account the epistemic, cognitive and instructional dimensions of the study processes. So as to provide an example of this process, in this article we study the didactic effectiveness of techniques associated with different definitions of absolute value notion. The teaching and learning of absolute value are problematic; this is proven by the amount and heterogeneity of research papers that have been published. We propose a “global” study from an ontological and semiotic point of view.
 
Eastern Asian students repeatedly outperform U.S. students in mathematics. Some suggest that number-naming languages consistent with the base-10 number system found in many Eastern Asian countries presumably contribute to their students’ better understanding of the base-10 system and consequential performance. Such language features do not exist in English or other Western languages. The current study tests this assumption by comparing base-10 knowledge of students in kindergarten and first-grade from China, Romania, and U.S. who have developed number-naming language abilities but received relatively little formal school instruction. It is expected that since Chinese number-naming is linguistically more transparent and consistent with the base-10 system, Chinese students should outperform both their Romanian and U.S. peers. Romanians should show intermediate performance between Chinese and U.S. students since Romanian language is somewhat transparent and consistent with a base-10 system while English number-naming language is least consistent. However, the analysis of this study revealed that although Chinese children outperformed both Romanian and U.S. counterparts in accomplishing base-10 tasks, there were no significant differences between Romanian and U.S. children. This finding suggests that the extent to which number-naming language is linguistically transparent and consistent with the base-10 system may not necessarily align with the level of children’s understanding of the base-10 system and relevant mathematics performances.
 
This article considers one class of high school students as they worked on a task given to them by a company director of a haulage firm. The article provides details of students’ transformation of the given task into a subtly different task. It is argued that this transformation is interrelated with students’ understandings of mathematics, of technology and of the real world and students’ emerging goals. It is argued that the students did not address the company director’s task. Educational implications with regard to student engagement with realistic tasks are considered.
 
Today, many universities in the United States use mathematics placement tests in combination with high school grades and SAT scores to place students in freshman mathematics courses. In an attempt to make this process more convenient for students and universities, these tests are beginning to be given online. This paper describes the history of a university mathematics placement test, originally given in 1992, which was converted to an online format in 2005. The placement method is described and a logistic regression is used to evaluate the accuracy of the online placement procedure in comparison to the placement with the paper test.
 
Mathematicians in general claim that the Computer Algebra Systems (CAS) provide an excellent tool for illustrating calculus concepts. They caution, however, against heavy dependency on the CAS for all computational purposes without the mastery of the procedures involved. This study examined the effect of using the graphical and numerical capabilities of Mathematica as a supplemental instructional tool in enhancing the conceptual knowledge and problem solving abilities of students in a differential calculus course. Topics of differential calculus were introduced by the traditional lecture method to both the control and experimental groups comprised of students enrolled in two sections of the Business and Life Sciences I course. Mathematica was used only by the students of the experimental group to reinforce and illustrate the concepts developed by the traditional method. A content analysis was conducted using the qualitative data obtained from students’ explanations of the derivative of a function. The quantitative data, the students’ test scores, were analyzed using ANCOVA. The results showed that students in the experimental group scored higher than students in the control group on both the conceptual and the computational parts of the examination. The qualitative analysis results revealed that, compared to the control group, a higher percentage of students in the experimental group had a better understanding of the derivative.
 
A number of different representational methods exist for presenting the theory of linear equations and associated solution spaces. Discussed in this paper are the findings of a case study where first year undergraduate students were exposed to a new (to the department) method of teaching linear systems which used visual, algebraic and data-based representations constructed using the computer algebra system Maple. Positive and negative impacts on the students are discussed as they apply to representational translation and perceived learning.
 
Mathematics researchers generally agree that algebra is a tool for problem solving, a method of expressing relationship, analyzing and representing patterns, and exploring mathematical properties in a variety of problem situations. Thus, several mathematics researchers and educators have focused on investigating the introduction and the development of algebraic solving abilities. However research works on assessing students' algebraic solving ability is sparse in literature. The purpose of this study was to use the SOLO model as a theoretical framework for assessing Form Four students' algebraic solving abilities in using linear equation. The content domains incorporated in this framework were linear pattern (pictorial), direct variations, concepts of function and arithmetic sequence. This study was divided into two phases. In the first phase, students were given a pencil-and-paper test. The test comprised of eight superitems of four items each. Results were analyzed using a Partial Credit model. In the second phase, clinical interviews were conducted to seek the clarification of the students' algebraic solving processes. Results of the study indicated that 62% of the students have less than 50% probability of success at relational level. The majority of the students in this study could be classified into unistructural and multistructural. Generally, most of the students encountered difficulties in generalizing their arithmetic thinking through the use of algebraic symbols. The qualitative data analysis found that the high ability students seemed to be more able to seek the recurring linear pattern and identify the linear relationship between variables. They were able to co-ordinate all the information given in the question to form the algebraic expression and linear equations. Whereas, the low ability students showed an ability more on drawing and counting method. They lacked understanding of algebraic concepts to express the relationship between the variables. The results of this study provided evidence on the significance of SOLO model in assessing algebraic solving ability in the upper secondary school level.
 
In this paper we first describe the process of building a questionnaire directed to globally assess formal understanding of conditional probability and the psychological biases related to this concept. We then present results from applying the questionnaire to a sample of 414 students, after they had been taught the topic. Finally, we use Factor Analysis to show that formal knowledge of conditional probability in these students was unrelated to the different biases in conditional probability reasoning. These biases also appeared unrelated in our sample. We conclude with some recommendations about how to improve the teaching of conditional probability.
 
The goal of this follow-up study was to generalize the findings of a previous inquiry into how assessing teachers’ mathematical knowledge within a professional development (PD) course impacted the teachers’ perspective of their learning and their learning experience. This quantitative research study examined whether the teachers’ attitudes about assessments found in the original study were generalizable to a similar population as well as whether factors involving their No Child Left Behind (NCLB) status and prior experience with the PD facilitators were factors affecting their perspectives. Results indicate that the teachers felt that they learned more mathematics, increased their learning efforts, and gained confidence in their understanding of and ability to teach mathematics because they were assessed. Additionally, the teachers’ NCLB status or prior experience in PD with the facilitators had virtually no impact on the teachers’ perceptions about assessment. Characteristics of the PD that led to these results are explained.
 
The purpose of this study was to investigate different factors (grade level, gender, and ethnicity) that might affect the attitudes and learning environment perceptions of high school mathematics students in Los Angeles County, California, USA. The study involved the administration of the administration of the What Is Happening In This Class? (WIHIC) questionnaire and an attitude questionnaire based on the Test of Mathematics-Related Attitude (TOMRA) to 600 Grades 9 and 10 mathematics students in 30 classes in one high school. Quantitative research method was used in collecting information from the sample. The quantitative data were statistically analyzed using ANOVA and MANOVA. The results showed that male consistently reported slightly more positive perceptions of classroom environment and attitudes than did females. Anglo students’ scores consistently are a little higher than Hispanic students’ scores. There is strong evidence of associations between students’ attitudes and the learning environment.
 
This study examined the effect of behavioral objective-based (BOBIS) and study question-based (SQBIS) instructional strategies on students’ attitude towards Senior Secondary Mathematics. The three hypotheses for the study were tested at 0.05 level of significance. The issue of attitudinal changes of student in mathematics classroom is an evergreen topic which cannot be wished away. It is therefore important to search for more and simple methods/ways by which teachers could continually inspire positive attitude in mathematics classroom. The research adopted a pre-test, post-test, control group quasi experimental design. There were three treatment groups which are - two experimental groups (behavioral objective-based (group1, N=117) and study question-based (group II, N=95) instructional strategies) and a control group (group III, N=100). A total of 312 students were involved in the study. The classrooms were randomly selected in each school and all the students in the selected classroom constitute the sample (intact class). Students’ Attitude Questionnaire (SAQ) has a reliability coefficient of r = 0.81. Findings revealed a significant effect of treatments (BOBIS & SQBIS) on students’ attitude towards Mathematics. The result was (F (2,311) = 72.95, P < 0.05). There was a significant difference in attitude between behavioural objective-based instructional strategy group and the control group with the BOBIS group having far better attitude to mathematics than the control group. Similarly, significant difference was found between the attitude of SQBIS group and the control group but no significant difference in attitude was found between BOBIS group and SQBIS group. Behavioral objective-based and Study-question-based groups were found to have similar attitude towards. In other words, there was significant differences between the attitudes of subjects exposed to behavioural objectives and control group and between those exposed to study question and the control group and no significant difference in attitude between the behavioural objective and study question groups. Both experimental groups (BOBIS and SQBIS) proved to be superior to the control group. Based on the findings, behavioral objective-based and study question-based instructional strategies were found to be viable instructional strategies that could promote positive attitude towards mathematics. The implication of the result is that teachers’ method of instruction in classroom is important in changing students’ attitude and habits towards mathematics.
 
This report focuses on in-service teachers’ planning of stochastic education. The theoretical and methodological settings of the research will be outlined in-depth. The methodological settings will be illustrated by research results concerning one teacher. A further main focus is to present some results concerning the planning of stochastic education conducted by 13 teachers.
 
Concerns about students' difficulties in statistical thinking led to a study which explored form five (14 to 16 year olds) students’ ideas in this area. The study focussed on probability, descriptive statistics and graphical representations. This paper presents and discusses the ways in which students made sense of probability concepts used in individual interviews. The findings revealed that many of the students used strategies based on beliefs, prior experiences (everyday and school) and intuitive strategies. From the analysis, I identified a four category rubric that could be considered for describing how students construct meanings for probability questions. While students showed competence with theoretical interpretation, they were less competent on tasks involving frequentist definition of probability. This could be due to instructional neglect of this viewpoint or linguistic problems. The paper concludes by suggesting some implications for further research.
 
This article provides a philosophical conceptualization of mathematics given the particular tasks of its teaching and learning. A central claim is that mathematics is a discipline that has been largely untouched by the Darwinian revolution; it is a last bastion of certainty. Consequently, mathematics educators are forced to draw on overly absolutist or constructivist accounts of the discipline. The resulting “math wars” often impede genuine reform. I suggest adopting an evolutionary metaphor to help explain the epistemology/nature of mathematics. In order to use this evolutionary metaphor to its fullest effect in overcoming the polarization of the math wars, mathematical empiricism is presented as a means of constraint on the development of mathematics. This article sketches what an evolutionary philosophy of mathematics might look like and provides a detailed descriptive account of mathematical empiricism and its potential role in this novel way of thinking about mathematical enterprises.
 
This study investigated mathematics teachers’ interpretation of higher-order thinking in Bloom’s Taxonomy. Thirty-two high school mathematics teachers from the southeast U.S. were asked to (a) define lower- and higher-order thinking, (b) identify which thinking skills in Bloom’s Taxonomy represented lower- and higher-order thinking, and (c) create an Algebra I final exam item representative of each thinking skill. Results indicate that mathematics teachers have difficulty interpreting the thinking skills in Bloom’s Taxonomy and creating test items for higher-order thinking. Alternatives to using Bloom’s Taxonomy to help mathematics teachers assess for higher-order thinking are discussed.
 
The review discusses “What’s worth fighting for in education?” by Andy Hargreaves and Michael Fullan. While the book is now 10 years old, the problems of determining the important issues in school (and higher) education and fostering collaborative links between schools and the communities which they serve are as critical now as they have ever been.
 
This article identifies challenges involved in teaching a mathematics class with 350 or more students. It discusses issues of preparation, organization, course administration, instruction, use of technology, and student management, while offering constructive help and useful techniques for teaching large mathematics classes. General reflections from three instructors on their large class teaching experiences are followed by a model of how large freshmen Calculus courses are conducted at Simon Fraser University in Burnaby, BC.
 
An initial proposed 5-factor number sense model and related test question numbers
Factor loadings for the 25-item factor analysis with oblique rotation 
This study was to develop a computerized number sense scale (CNST) to assess the performance of students who had already completed the 3rd-grade mathematics curriculum. In total, 808 students from representative elementary schools, including cities, country and rural areas of Taiwan, participated in this study. The results of statistical analyses and content analysis indicated that this computerized number sense scale demonstrates good reliability and validity. Cronbach’s α coefficient of the scale was .8526 and its construct reliability was .805. In addition, the 5-factor number sense model was empirically and theoretically supported via confirmatory factor analysis and literature review.
 
The split-box for marble drops.  
Student-generated games: (a) Kate and Tana's game (b) Jim and Brad's game.  
This research focuses on fourth-grade (9-year-old) students’ informal and intuitive conceptions of probability and distribution revealed as they worked through a sequence of tasks. These tasks were designed to study students’ spontaneous reasoning about distributions in different settings and their understanding of probability of various binomial random events that they explored with a set of physical chance mechanisms. The data were gathered from a pilot study with four students. We analyzed the interplay of reasoning about distribution and understanding of probability. The findings suggest that students’ qualitative descriptions of distributions could be developed into the quantification of probabilities through reasoning about data in chance situations.
 
Probability classrooms often fail to develop sustainable conceptions of probability as strategic tools that can be activated for decisions in everyday random situations. The article starts from the assumption that one important reason might be the often empirically reconstructed divergence be-tween individual conceptions of probabilistic phenomena and the normative conceptions taught in probability classrooms, especially concerning pattern in random. Since the process of dealing with these phenomena cannot sufficiently be explained by existing frameworks alone, an alternative – horizontal - view on conceptual change is proposed. Its use for research and development within the so-called Educational Reconstruction Program is presented. The empirical part of the paper is based on a qualitative study with 10 game interviews. Central results concern the oszillation between conceptions and cognitive layers and the situatedness of their activation. In particular, diverging perspectives seem to root in contrasting foci of attention, namely the mathematically suitable long-term perspective being in concurrence to the more natural short-term attention to single outcomes. The Educational Reconstruction Program offers an interesting possibility to specify roots of obstacles and to develop guidelines for designing learning environments which respect the horizontal view.
 
Construing a collection of values of a sample statistic as a distribution is central to developing a coherent understanding of statistical inference. This paper discusses key developments that unfolded over three consecutive lessons in a classroom teaching experiment designed to support a group of high school students in developing such a construal. Instruction began by engaging students in activities that focused their attention on the variability among values of a common sample statistic. There occurred a critical shift in students’ attention and discourse away from individual values of the statistic and toward a collection of such values as a basis for inferring the value of a population parameter. This was followed by their comparisons of such collections and by the emergence and application of a rule for deciding whether two such collections were similar. In the repeated application of their decision rule students structured these collections as distributions. We characterize aspects of these developments in relation to students’ classroom engagement, and we explore evidence in students’ written work that points to how instruction shaped their conceptions.
 
Excerpt I from the booklet (Hjersing et al., 2004 p 16) "Logistic growth of a population (…) If the population is small, the rate of growth is proportional with the size of the population. If the population is so big, that it may not be fed or kept in the area, then the population will decrease -the rate of growth turns negative. (…) Based on this, several different equations may be set up. We choose a relatively simple one:
Excerpt II from the booklet (Hjersing et al., 2004 p 18)
Excerpt from the booklet (Hjersing et al., 2004 p 20)
This article reports on a research project, which was part of the research- and development project World Class Math and Science. The objectives of this part of the project were to research the potentials of computer use in upper secondary school mathematics for the teaching of differential equations from a modeling point of view. The project involved small scale teaching experiments with changes at two levels from the traditional viewpoint on school mathematics: 1) Change of view at curriculum level on the subject differential equations and 2) Change of view at the level of didactical reflections on the intentions of modeling and using models. The article discusses how students’ use of laptops can serve as a means for both changes by replacing complex, time consuming expressive modeling with more controlled exploration of differential-equations models. Finally, the perspectives for the teaching and learning of mathematics of such changes are discussed.
 
In this paper, I examine the notion of ‘real life’ mathematical applications as possible sites for ethical reflection in school mathematics. I discuss problems with the ‘real’ in mathematics education, and show how these problems are often based on faulty cognitive theories of knowledge transfer. I then consider alternative visions of mathematical application and suggest that attention to classroom discourse and the craft of mathematics offer ways of introducing the ethical into school mathematics.
 
Longstanding efforts by different researchers who were pioneers in the field of statistics education has led today to the introduction of statistics in school mathematics in many countries. Simultaneously, the teaching and learning of statistics has turned into a research area of increasing interest for mathematics educators, as shown in the two recent survey chapters by Jones, Langrall and Money (2007) and Shaughnessy (2007) and also in the existence of journals such as Journal of Statistics Education, Teaching Statistics and, more recently, Statistics Education Research Journal, specifically focused on statistics education research. The increased research in this area might not have been noticed by mathematics educators, as statistics education receives contributions not only from them but from many other different disciplines. Research into stochastic thinking, teaching, and learning started during the 1950s with the pioneering work by Piaget and Inhelder (1951) on the growth and structure of children’s probabilistic thinking and has always had an interdisciplinary character. Because psychology is an experimental science that heavily relies on statistics, the efforts to justify the scientific character of this field led psychologists to examine the validity of their research paradigms, including the use of statistics in empirical research. An amazing observation was that statistical inference and particularly significance tests were found to be misunderstood and misused by psychologists and experimental researchers at large over 30 years ago, and that the situation still persists in spite of strong debates ever since (Morrison & Henkel, 1970; Harlow, Mulaik, & Steiger, 1997; Batanero, 2000). Moreover, researchers in the field of reasoning under uncertainty suggested more than 20 years ago that, even after statistical instruction, students and professionals tend to continue to make erroneous stochastic judgements and decisions (Kahneman, Slovic, & Tversky, 1982). Statistics is one of the most widely taught topics at university level, where many service course students meet advanced stochastic thinking without any prior or concurrent experience of advanced algebra or calculus, so that didactical problems still persist at University level (Artigue, Batanero, & Kent, 2007). The diffusion of psychological reseach results and the increasingly easy access to powerful and user-friendly computers and statistical software, which save teaching time previously devoted to laborious calculations and allow a more intuitive approach to statistics (more real data, active learning, problem solving, and use of technology to illustrate abstract concepts through simulation) led statistics lecturers to increase their attention towards didactical problems (see, e.g., Moore, 1997). Consequently, an increasing number of statisticians or lecturers of statistics with different professional backgrounds started to develop educational research with a specific interest in creating curricular materials and to evaluate the teaching and learning of statistics at university level. The influence of the International Association for Statistical Education (IASE), created in 1991, helped to establish links among the different communities interested in statistics education and to support a more systematic research program, particularly through its conferences and publications. I am consequently very pleased to introduce this special issue of the International Electronic Journal of Mathematics Education (IEJME) on Emerging Research in Statistics Education that can help expanding knowledge about the state of the art in this field among mathematics educators. The issue will present some of the Invited Papers dealing with research that were presented at the Seventh International Conference on Teaching Statistics (ICOTS-7, Salvador (Bahia), Brazil, 2006, http://www.maths.otago.ac.nz/icots7). The ICOTS conferences are the most important means of interchange that the IASE offers to the community of statistics educators and were started 24 years ago by the International Statistical Institute (ISI). ICOTS-7 was successfully held in Brazil, with over 550 participants representing about 50 different countries. The more than 220 invited papers, 120 contributed papers, 120 posters, keynote lectures, panels, and special sessions gave a synthesis of the main tendencies and developments in statistics education. Selecting ICOTS-7 papers for this special issue was not an easy task, given the enormous number of contributions. We then focused, as a first step, on those invited papers dealing specifically with empirical research in themes that could be of interest to the IEJME audience. We, secondly, considered only those papers that had passed the double blind refereeing process in ICOTS (since refereeing was optional for invited papers). Taking into account these restrictions we selected a sample of papers that include a variety of research topics and nationality of authors, as well as both young and experienced researchers. The papers in this issue analyse probability, distribution and conditional probability, variation, statistical graphs, statistical literacy, sampling distributions, informal inference, multivariate data and teachers’ views about teaching statistics. It combines qualitative and quantitative research with methods including interviews, open-ended tasks, paper-pencil and computer based questionnaires, Rasch or factor analysis. Students in the samples range from primary school level to University, including prospective in-service teachers. The studies focus on students or teachers’ conceptions, assessment, instruction, use of technology or research methods. I hope this variety reflects the state of research in statistics education and increases the readers’ interest to know more about what is going on in this area; hopefully some of them will decide to undertake new research. In closing, I want to thank the contributors into this issue for the interest they put in improving their papers as well as the many referees, who accepted the challenging role to help this special issue achieve high scholarly standards. Acknowledgment: Preliminary versions of papers in this special issue were first published in Rossman and Chance (2006). Permission for publication of these expanded versions has been granted from the IASE and ICOTS-7 organizers.
 
How teachers think about student thinking informs the ways in which teachers teach. By examining teachers’ anticipation of student thinking we can begin to unpack the assumptions teachers make about teaching and learning. Using a “mathematics for teaching” framework, this research examines and compares the sorts of assumptions teachers make in relation to “student content knowledge” versus actual “learning paths” taken by students. Groups of teachers, who have advanced degrees in mathematics, education, and mathematics education, and tenth grade students engaged in a common mathematical task. Teachers were asked to model, in their completion of the task, possible learning paths students might take. Our findings suggest that teachers, in general, had difficulty anticipating student learning paths. Furthermore, this difficulty might be attributed to their significant “specialized content knowledge” of mathematics. We propose, through this work, that examining student learning paths may be a fruitful locus of inquiry for developing both pre-service and in-service teachers’ knowledge about mathematics for teaching.
 
Numbers of the students according to their explanations' categories
This study investigated the factors that 12th grade students in the United Arab Emirates take into consideration when judging the validity of a given statistical generalization, particularly, in terms of the sample size and sample selection bias. The sample consisted of 360 students who had not studied sampling yet. Results show that a small percentage of the students take the sample size and selection bias into consideration properly. Many students based their judgment on their personal beliefs regardless of the properties of the selected sample. This study identified some pre-teaching misconceptions that students have with regard to sampling. Such misconceptions include ‘any sample represents the population’, and, ‘any sample does not represent the population’.
 
Examples of main aspect of the questions raised about the road accident graphs 
The official inclusion of the teaching of graphing in school curricula has motivated increasing research and innovative pedagogical strategies such as the use of media graphs in school contexts. However, only a few studies have investigated knowledge about graphing among those who will teach this curricular content. We discuss aspects of the interpretation of media graphs among primary school student teachers from Brazil and England. We focus on data which came from questionnaires and interviews which gives evidence of the mobilisation of several kinds of knowledge and experiences, in the interpretation of media statistical graphs. The discussion of results might contribute to an understanding of the complexity of the interpretation of such graphs, and to the development of pedagogical strategies which can help teachers think about the teaching and learning of statistics in ways that will support the balance of these kinds of knowledge.
 
The ability to analyse qualitative information from quantitative information, and/or to create new information from qualitative and quantitative information is the key task of statistical literacy in the 21st century. Although several studies have focussed on critical evaluation of statistical information, this aspect of research has not been clearly conceptualised as yet. This paper presents a hierarchy of the graphical interpretation component of statistical literacy. 175 participants from different educational levels (junior high school to graduate students) responded to a questionnaire and some of them were also interviewed. The SOLO Taxonomy was used for coding the students’ responses and the Rasch model was used to clarify the construction of the hierarchy. Five different levels of interpretations of graphs were identified: Idiosyncratic, Basic graph reading, Rational/Literal, Critical, and Hypothesising and Modelling. These results will provide guidelines for teaching statistical literacy.
 
Computer tasks
The seven paper-based tasks
Infit statistic from Rasch scaling of test items
Outfit statistic from Rasch scaling of test items
We report a study where 195 students aged 12 to 15 years were presented with computer-based tasks that require reasoning with multivariate data, together with paper-based tasks from a well established scale of statistical literacy. The computer tasks were cognitively more complex, but were only slightly more difficult than paper tasks. All the tasks fitted well onto a single Rasch scale. Implications for the curriculum, and public presentations of data are discussed.
 
Protocol for post-interview 
This paper reports one recent study that was part of a project investigating tertiary students’ understanding of variation. These students completed a questionnaire prior to, and at the end of, an introductory statistics course and this paper focuses on interviews of selected students designed to determine whether more information could have been gathered about the students’ reasoning. Clarification during interviews reinforced researcher interpretation of responses. Prompting assisted students to develop better quality responses but probing was mostly useful for assisting students to re-express reasoning already presented. Cognitive conflict situations proved challenging. The diversity of activities identified by students as assisting the development of their understanding provides a challenge for educators in planning teaching sequences. Both educators and researchers need to listen to students to better understand the development of reasoning.
 
Achieving proficiency in mathematics appears to be a particular area of challenge for students in the United States. The Trends in International Mathematics and Science Study (TIMSS) recently released results for 2003 testing, and revealed that eighth graders in the United States rank 15th among 46 participating countries (Snell, 2005). Although these results are a significant improvement from the 1995 performance, the United States students still rank near the bottom when compared to other students from industrialized nations. Research in the area of mathematics achievement has examined a number of explanations as to why some students will test proficient and many will not (e.g., Hyde, Fennema, & Lamon, 1990; Mason, & Scrivani, 2004; Mevarech, Silber, & Fine, 1991; Rangappa, 1993, 1994). Using data extracted from the Education Longitudinal Study (ELS, 2004), the present study investigated the impact of student reading ability, student math self-efficacy, teacher expectations, and the use of computers in the teaching of mathematics in predicting student math achievement. Findings reveal that 56% of the variance in student math achievement can be explained by students' reading ability. The results of the final regression model also revealed that higher levels of math-self-efficacy and higher levels of teachers' expectations were associated with higher math achievement scores. However, a negative association between computer-assisted instruction and student math achievement scores was found.
 
This study reports on two excerpts from a 6th grade Taiwanese class, describing how a teacher examined and promoted his students' development of number sense. It illustrates an effort to integrate number sense activities into the mathematics curriculum in ways that encourage exploration, discussion, thinking and reasoning. The results indicate that number sense can be developed through well-designed number sense activities, effective teaching, and a good learning environment. It also demonstrates that students' number sense and mathematical thinking can be promoted through the use of multiple representations.
 
Strategy used with fractions, decimals and percentages
Frequencies and success rates, for the first answer of students for each problem
Frequencies and corresponding rates of use of conceptual (CON) and procedural strategies (PR) (only for the first successful strategy)
One of the attributes of rational numbers that make them different from integers are the different symbolic modes (fraction, decimal and percentage) to which an identical number can be attributed (e.g. 14, 0.25 and 25%). Some research has identified students’ difficulty in mental calculations with rational numbers as has also the switching to different symbolic representations between fractions and decimals. However, pupils’ performance, and repertoire of strategies have not been systematically studied in mental calculations with rational numbers expressed in different symbolic representations. The principal question of this research: how is the ability of students to perform mental calculations with rational numbers affected when the same number changes in fraction, decimal and percentage? For the purpose of the study 62 8th grade students were interviewed to examine how this symbolic shift in the number of operations affects the success and type of strategies they use, and the ability to alternate the rotation of these symbolisms. The results of the research show that the symbolic change of the rational numbers affects the success and the type of strategies that students use in mental calculations. Another result of the study demonstrated that students are not flexible when switching between the different symbolic representations of rational numbers as benchmark while performing mental calculations.
 
Long have we known that reasoning abilities are linked to learning, and specifically to learning mathematics. Even intelligence, considered a controversial construct, plays a significant role in the research on the explanation of academic performance. This article intends to highlight some important cognitive abilities or dimensions relevant to learning mathematics, synthesizing some research that defines such constructs and relates them to mathematical learning and achievement. General considerations about designing and implementing meaningful learning experiences are presented.
 
The aim of this study is to examine the effects of GeoGebra on third grade primary students’ academic achievement in fractions concept. This study was conducted with 40 students in two intact classes in Ankara. One of the classes was randomly selected as an experimental group and other for control group. There were 19 students in the experimental group, while 21 students in control group. The matching- only posttest- only control group quasi-experimental design was employed. As a pretest, student’s first term mathematics scores were used. Data were collected with post-test about fractions. The post-test consisted of 22 short ended questions. Thanks to the scores weren’t violated the normality, independent t test was employed. The findings of the study showed that there were significant differences in favor of the experimental group. According to findings of this study, it was recommended that GeoGebra supporting teaching methods can be used on teaching fractions in third grade.
 
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