Working with digital textbooks or printed materials: A study with boys and girls on conditional probability

Büchter T., Steib N., Böcherer-Linder K., Eichler A., Krauss S., Binder K., Vogel M.

Questions involving Bayesian Reasoning often arise in events of everyday life, such as assessing the results of a breathalyser test or a medical diagnostic test. Bayesian Reasoning is perceived to be difficult, but visualisations are known to support it. However, prior research on visualisations for Bayesian Reasoning has only rarely addressed the issue on how to design such visualisations in the most effective way according to research on multimedia learning. In this article, we present a concise overview on subject-didactical considerations, together with the most fundamental research of both Bayesian Reasoning and multimedia learning. Building on these aspects, we provide a step-by-step development of the design of visualisations which support Bayesian problems, particularly for so-called double-trees and unit squares.

Wijaya T.T., Zhou Y., Houghton T., Weinhandl R., Lavicza Z., Yusop F.D.

Digital mathematics textbooks differ from traditional printed textbooks in, among other things, their dynamic structural elements, representing a potential that traditional textbooks cannot fulfil. Notably, dynamic structural elements, i.e., multimodal representations of mathematics, could be of particular importance for learning, which is why the scientific interest in digital mathematics textbooks has increased in recent years and many digital textbooks have been developed. However, research related to predicting teacher usage behavior of digital textbooks is still limited. Therefore, this research aims to analyze the predictors that may influence the intentions of mathematics teachers and the actual usage of digital textbooks by applying the Unified Theory of Acceptance and Use of Technology (UTAUT). Data were collected from 277 teachers in West Java Province, Indonesia, and analyzed using structural equation modeling (SEM). The results indicated that Performance Expectancy (PE) is the biggest significant factor, followed by Social Influence (SI), that influences the Behavioral Intention (BI) of mathematics teachers to use digital textbooks in Indonesia. Effort Expectancy (EE) does not affect the intention to use a digital textbook. In turn, BI has the largest and most significant effect on teachers’ actual usage of digital textbooks. This result contributes to the understanding of the predictors that can increase the use of digital textbooks by mathematics teachers.

Büchter T., Eichler A., Steib N., Binder K., Böcherer-Linder K., Krauss S., Vogel M.

Bayesian Reasoning is both a fundamental idea of probability and a key model in applied sciences for evaluating situations of uncertainty. Bayesian Reasoning may be defined as the dealing with, and understanding of, Bayesian situations. This includes various aspects such as calculating a conditional probability (performance), assessing the effects of changes to the parameters of a formula on the result (covariation) and adequately interpreting and explaining the results of a formula (communication). Bayesian Reasoning is crucial in several non-mathematical disciplines such as medicine and law. However, even experts from these domains struggle to reason in a Bayesian manner. Therefore, it is desirable to develop a training course for this specific audience regarding the different aspects of Bayesian Reasoning. In this paper, we present an evidence-based development of such training courses by considering relevant prior research on successful strategies for Bayesian Reasoning (e.g., natural frequencies and adequate visualizations) and on the 4C/ID model as a promising instructional approach. The results of a formative evaluation are described, which show that students from the target audience (i.e., medicine or law) increased their Bayesian Reasoning skills and found taking part in the training courses to be relevant and fruitful for their professional expertise.

Kaiser G., Zhu Y.

As mathematics has been seen for decades as a stereotyped male domain, gender differences in mathematics learning have received strong attention from the public and academia. In China, the issue of gender equity in education is a particularly interesting topic to most families with the implementation of the one-child policy since the late 1970s. This study aims to study in more depth the role of gender on Shanghai students’ mathematics attainment from a perspective of three societal factors (i.e., one-child status at home, socioeconomic status, and school types) via a secondary analysis of the Programme for International Student Assessment 2012 Shanghai–China mathematics data. In contrast to the official report by Programme for International Student Assessment on Shanghai–China and in line with own previous studies, the current analyses reveal that 15-year-old students in Shanghai performed significantly different on two content-related subscales (i.e., change and relationships and quantity) and two processes-related subscales (i.e., formulate and interpret). Furthermore, significant gender differences were found with students from one-child families but not multi-children families. Among schools of different types in terms of academic tracks and performance levels, the gender differences were largest in the more selective model (general) schools and then vocational schools followed by ordinary (general) schools. Given the nested structure of the Programme for International Student Assessment data, this study found that, on average, a Shanghai boy would achieve significantly higher marks than a girl in Programme for International Student Assessment 2012 mathematics test. The paper closes with discussions on societal and educational implications about these gender disparities, which are still apparent in the current school system of Shanghai.

Panadero E., Lipnevich A.A.

A number of models has been proposed to describe various types of feedback along with mechanisms through which feedback may improve student performance and learning. We selected fourteen most prominent models, which we discussed in two complementary reviews. In the first part (Lipnevich & Panadero, 2021) we described the models, feedback definitions, and the empirical evidence supporting them, whereas in the present publication, we analyzed and compared the fourteen models with the goal to classify and integrate shared elements into a new comprehensive model. As a result of our synthesis, we offered an expanded typology of feedback and a classification of models into five thematic areas: descriptive, internal processing, interactional, pedagogical, and students characteristics. We concluded with an Integrative Model of Feedback Elements that includes five components: Message, Implementation, Student, Context, and Agents (MISCA). We described each element and relations among them, offering future directions for theory and practice. Finally, feedback occurs within the context of a task that can be more specific –e.g. a mathematical exercise- or general –e.g. presenting to the public. Therefore, we included this idea in our model, below the students’ characteristics. It is important to consider how the task characteristics might influence the whole ecosystem while keeping in mind the larger goal – that feedback should improve the learner, not just the work. • Fourteen prominent feedback models were selected, analyzed and integrated. • An integrative typology is proposed that includes feedback content, function, presentation and source. • Models were thematically classified into descriptive, internal processing, interactional, pedagogical, and student's characteristics. • An Integrative Model of Feedback Elements is proposed: the MISCA (Message, Implementation, Student, Context and Agents).

Rezat S., Fan L., Pepin B.

In this survey paper we aim to provide an overview of research on mathematics textbooks and, more broadly, curriculum resources as instruments for change related to mathematical content, instructional goals and practices, and student learning of mathematics. In particular, we elaborate on the following themes: (1) The role of curriculum resources as instruments for change from a theoretical perspective; (2) The design of curriculum resources to mediate the implementation of reform ideas and innovative practice; (3) Teachers’ influence on the implementation of change through curriculum resources; (4) Students’ influence on the implementation of change through curriculum resources; and (5) Evidence of curriculum resources yielding changes in student-related factors or variables. We claim that, whilst textbooks and curriculum resources are influential, they alone cannot change teachers’ teaching nor students’ learning practices in times of curricular change. Moreover, more knowledge is needed about features of curriculum resources that support the implementation of change. We contend that curriculum innovations are likely to be successful, if teachers and students are supported to co- and re-design the relevant curriculum trajectories and materials in line with the reform efforts and their own individual needs.

Kaplar M., Radović S., Veljković K., Simić-Muller K., Marić M.

The purpose of this study is to analyse the effects of the Interactive Learning Materials Triangle (iLMT) on the learning and knowledge retention of 12-year-old students. The iLMT is a digital version of the standard school learning materials in Serbia, and is characterized by a high degree of interactivity and immediate feedback during the learning process. We conducted an experiment to explore whether iLMT influences student success in solving mathematical tasks that require different types of mathematical reasoning. Based on previous extensive research by Lithner, 4 types of tasks are discussed: high relatedness answer, high relatedness algorithm, local low relatedness, and global low relatedness. The study involved 633 students and 13 teachers of mathematics, equally distributed in control and test groups. The main findings indicate that student success on a knowledge test for high relatedness answer and local low relatedness tasks for the test group was significantly higher than for the control group. On the knowledge retention test, students in the test group outperformed students in the control group at high relatedness algorithm and local low relatedness tasks. Our results also suggest that, even when learning materials are carefully digitalized with the use of available technological advantages, student success in global low relatedness tasks may still be lacking.

Reinhold F., Strohmaier A., Hoch S., Reiss K., Böheim R., Seidel T.

Electronic learning environments used in mathematics lessons offer new ways to assess and analyze students' classroom engagement during authentic learning settings. In this study, we investigated students' electronic textbook-use as a measure for their individual engagement during mathematics instruction. To this end, we combined quantity measures (i.e., time on task, text length) and quality measures (i.e., on topic, mathematically valid, mathematical language used). Cluster analysis based on process data of 253 six-graders—who worked on three writing-to-learn exercises during fraction instruction—revealed four different clusters that we ordered hierarchically in terms of engagement, revealing gender differences in favor of girls. Analyses showed negligible differences in prior knowledge between the clusters, yet significant achievement differences in a posttest—with higher engaged clusters reaching higher outcomes. Our approach offers a viable way to unobtrusively measure students' classroom engagement utilizing process data from electronic textbooks.

Van der Kleij F.M., Lipnevich A.A.

The potential of feedback to enhance students’ performance on a task, strategies, or learning has long been recognized in the literature. However, feedback needs to be utilized by a learner to realize its potential. Hence, examining student perceptions of feedback and their links to effective uptake of feedback has been the focus of much recent feedback research. This paper presents a critical scoping review of the feedback perceptions literature. The review discusses the methods employed by 164 studies published between 1987 and 2018 and synthesizes the main findings across this body of literature. Lacking theoretical frameworks, repetitiveness (not replicability) of studies, and methodological problems observed among the reviewed have resulted in somewhat disappointing conclusions. Based on the findings, we present a framework for future investigations into student perceptions of feedback and suggest several avenues for the future of the field.

Hillmayr D., Ziernwald L., Reinhold F., Hofer S.I., Reiss K.M.

Based on systematic research of studies published since the year 2000, this comprehensive meta-analysis investigated how the use of technology can enhance learning in secondary school mathematics and science (grade levels 5–13). All studies (k = 92) compared learning outcomes of students using digital tools to those of a control group taught without the use of digital tools. Overall, digital tool use had a positive effect on student learning outcomes (g = 0.65, p

Leuders T., Loibl K.

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many systematic deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings.

Binder K., Krauss S., Wiesner P.

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2×2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch-structures, these two visualizations (compared to the 2×2 table) can even simultaneously display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2×2 tables and (double-) trees have a decisive disadvantage: While joint probabilities (e.g., P(AB)) are represented in 2×2 tables but no conditional probabilities (e.g., P(A|B)), it is exactly the other way around with (double-) trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2×2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2×2 table, double-tree, or net diagram with natural frequencies) are consistently superior to probability visualizations. The frequency net performs as well as the frequency double-tree. Only the 2×2 table with frequencies—the only visualization that participants were already familiar with —led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, and no uniform format effect becomes visible. Participants reached the highest performance in the versions with probability 2×2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability.

Rolfes T., Roth J., Schnotz W.

In this paper we present a laboratory experiment in which 157 secondary-school students learned the concept of function with either static representations or dynamic visualizations. We used two different versions of dynamic visualization in order to evaluate whether interactivity had an impact on learning outcome. In the group learning with a linear dynamic visualization, the students could only start an animation and run it from the beginning to the end. In the group using an interactive dynamic visualization, the students controlled the flow of the dynamic visualization with their mouse actions. The result of the experiment was that students learned significantly better with dynamic visualizations than with static representations. However, there was no significant difference in learning with linear or interactive dynamic visualizations. Nor did we observe an aptitude–treatment interaction between visual-spatial ability and learning with either dynamic visualizations or static representations.

Fraillon J., Ainley J., Schulz W., Friedman T., Duckworth D.

Weigand H., Trgalova J., Tabach M.

AbstractThe role of teaching, learning, and assessment with digital technology has become increasingly prominent in mathematics education. This survey paper provides an overview of how technology has been transforming teaching, learning, and assessment in mathematics education in the digital age and suggests how the field will evolve in the coming years. Based on several decades of research and educational practices, we discuss and anticipate the multifaceted impact of technology on mathematics education, thus laying the groundwork for the other papers in this issue. After a brief introduction discussing the motivations for this issue, we focus our attention on three lines of research: teaching mathematics with technology, learning mathematics with technology, and assessment with technology. We point to new research orientations that address the issue of teaching with technology, specifically describing attempts to conceptualise teachers’ mathematical and digital competencies, perspectives that view teachers as designers of digital resources, and the design and evaluation of long-term initiatives to support teachers as they develop innovative teaching practices enhanced by digital technologies. Our examination shows that learning with technology is still marked by new conceptualizations raised by researchers that can further our understanding of this complex issue. These conceptualizations support the recognition that multiple resources, ranging from paper and pencil to augmented reality, participate in the learning process. Finally, assessment with technology, especially in the formative sense, offers new possibilities for offering individualised support for learners that can benefit from adaptive systems, though more tasks for conceptual understanding need to be developed.