How to combine collaboration scripts and heuristic worked examples to foster mathematical argumentation – when working memory matters

Mathematical argumentation skills (MAS) are considered an important outcome of mathematics learning, particularly in secondary and tertiary education. As MAS are complex, an effective way of supporting their acquisition may require combining different scaffolds. However, how to combine different scaffolds is a delicate issue, as providing learners with more than one scaffold may be overwhelming, especially when these scaffolds are presented at the same time in the learning process and when learners’ individual learning prerequisites are suboptimal. The present study therefore investigated the effects of the presentation sequence of introducing two scaffolds (collaboration script first vs. heuristic worked examples first) and the fading of the primarily presented scaffold (fading vs. no fading) on the acquisition of dialogic and dialectic MAS of participants of a preparatory mathematics course at university. In addition, we explored how prior knowledge and working memory capacity moderated the effects. Overall, 108 university freshmen worked in dyads on mathematical proof tasks in four treatment sessions. Results showed no effects of the presentation sequence of the collaboration script and heuristic worked examples on dialogic and dialectic MAS. Yet, fading of the initially introduced scaffold had a positive main effect on dialogic MAS. Concerning dialectic MAS, fading the collaboration script when it was presented first was most effective for learners with low working memory capacity. The collaboration script might be appropriate to initially support dialectic MAS, but might be overwhelming for learners with lower working memory capacity when combined with heuristic worked examples later on

Effects of collaboration scripts and heuristic worked examples on the acquisition of mathematical argumentation skills of teacher students with different levels of prior achievement

A challenging demand for mathematics teacher students is to produce acceptable scientific mathematical argumentations. We investigated to what extent mathematics teacher students with different levels of prior achievement who collaborated in dyads can be supported in their development of mathematical argumentation skills by two different instructional approaches that were systematically varied in a 2 × 2-factorial design: collaboration scripts (with vs. without) and heuristic worked examples vs. problem solving. An experimental study was run in the context of a two-weeks preparatory course for beginning mathematics teacher students (N = 101). Mathematical argumentation skills were conceptualized as consisting of an individual-mathematical and a social-discursive component. Results indicated positive effects of both scaffolds on the social-discursive component. Moreover, the effects of both scaffolds on both components were dependent on learners' prior achievement (high-school GPA). Heuristic worked examples and collaboration scripts were particularly effective in the facilitation of mathematical argumentation skills for teacher students with higher general learning prerequisites. Possible process-based explanations for this pattern of results as well as ways to more specifically address the needs of teacher students with lower prior achievement are discussed

Good for learning, bad for motivation? A meta-analysis on the effects of computer-supported collaboration scripts

Scripting computer-supported collaborative learning has been shown to greatly enhance learning, but is often criticized for hindering learners’ agency and thus undermining learners’ motivation. Beyond that, what makes some CSCL scripts particularly effective for learning is still a conundrum. This meta-analysis synthesizes the results of 53 primary studies that experimentally compared the effect of learning with a CSCL script to unguided collaborative learning on at least one of the variables motivation, domain learning, and collaboration skills. Overall, 5616 learners enrolled in K-12, higher education, or professional development participated in the included studies. The results of a random-effects meta-analysis show that learning with CSCL scripts leads to a non-significant positive effect on motivation (Hedges’ g = 0.13), a small positive effect (Hedges’ g = 0.24) on domain learning and a medium positive effect (Hedges’ g = 0.72) on collaboration skills. Additionally, the meta-analysis shows how scaffolding single particular collaborative activities and scaffolding a combination of collaborative activities affects the effectiveness of CSCL scripts and that synergistic or differentiated scaffolding is hard to achieve. This meta-analysis offers the first counterevidence against the widespread criticism that CSCL scripts have negative motivational effects. Furthermore, the findings can be taken as evidence for the robustness of the positive effects on domain learning and collaboration skills

Learning to Diagnose with Simulations

This open access book presents 8 novel approaches to measure and improve diagnostic competences with simulation. The book compares the effects of interventions on these diagnostic competences in both teacher and medical education. It includes analyses showing that important aspects of diagnostic competences and effects of instructional interventions aiming to facilitate them are comparable for teachers and doctors. Through closely analyzing projects from medical education, mathematics education, biology education, and psychology, the reader is presented with multiple options for interventions that may be used in each of the subject areas and the improvements in diagnostic skills that could be expected from each simulation. The book concludes with an outline of promising future research on the use of simulations to facilitate professional competences in higher education in general, and for the advancement of diagnostic competencies in particular. This is an open access book

Learning to diagnose collaboratively – Effects of adaptive collaboration scripts in agent-based medical simulations

We investigated how medical students' collaborative diagnostic reasoning, particularly evidence elicitation and sharing, can be facilitated effectively using agent-based simulations. Providing adaptive collaboration scripts has been suggested to increase effectiveness, but existing evidence is diverse and could be affected by unsystematic group constellations. Collaboration scripts have been criticized for undermining learners' agency. We investigate the effect of adaptive and static scripts on collaborative diagnostic reasoning and basic psychological needs. We randomly allocated 160 medical students to one of three groups: adaptive, static, or no collaboration script. We found that learning with adaptive collaboration scripts enhanced evidence sharing performance and transfer performance. Scripting did not affect learners’ perceived autonomy and social relatedness. Yet, compared to static scripts, adaptive scripts had positive effects on perceived competence. We conclude that for complex skills complementing agent-based simulations with adaptive scripts seems beneficial to help learners internalize collaboration scripts without negatively affecting basic psychological needs

Learning to Diagnose with Simulations

This open access book presents 8 novel approaches to measure and improve diagnostic competences with simulation. The book compares the effects of interventions on these diagnostic competences in both teacher and medical education. It includes analyses showing that important aspects of diagnostic competences and effects of instructional interventions aiming to facilitate them are comparable for teachers and doctors. Through closely analyzing projects from medical education, mathematics education, biology education, and psychology, the reader is presented with multiple options for interventions that may be used in each of the subject areas and the improvements in diagnostic skills that could be expected from each simulation. The book concludes with an outline of promising future research on the use of simulations to facilitate professional competences in higher education in general, and for the advancement of diagnostic competencies in particular. This is an open access book

Mathematical conjecturing and proving

Most university courses in mathematics programs are characterized by a strong focus on the axiomatic nature of mathematics, and thus also on proof as the central scientific method of mathematics (Selden, A. & Selden, 2008). Lecturers write proofs on the blackboard, students attempt to demonstrate their understanding and skills by proving theorems on their own or in collaboration with others. However, there is often little systematic discussion in these courses on how new mathematical conjectures can be generated and on how proofs are constructed (Alcock, 2010). Students’ experiences with conjecturing and proving in schools or in university mathematics courses often lead them to “consider proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas” (Otten, Bleiler-Baxter, & Engledowl, 2017, p. 112). Yet, several authors (e.g., Epp, 2003; Savic, 2015a; Selden, A. & Selden, 2008) have hypothesized that often only little time can be devoted to illustrate students which strategies and processes may help to step through the proof construction process and to recover from proving impasses. Furthermore, the knowledge about what characterizes proof processes that lead to a successful outcome (i.e., an acceptable mathematical proof [according to local acceptance criteria]) is rare. To approach this issue, an extensive systematic literature search was conducted to summarize common claims and empirical findings about promising conjecturing and proving processes. 126 articles that focussed on conjecturing and proving were clustered using a topic modeling method. The algorithm identified 17 different topics. The most representative papers for each topic, in total 45 papers, were qualitatively analysed with regard to their research perspectives on which they were based and their claims and findings about the processes that are needed to successfully generate conjectures and construct proofs. This combination of statistical clustering and qualitative analyses allowed a systematic categorization of claims and empirical findings about successful conjecturing and proving processes in the literature. Based on this review, a set of characteristics of conjecturing and proving processes, that are assumed or reported to be crucial for success, is proposed. For the further analysis of such process characteristics, we started from a model differentiating students’ prerequisites they bring to bear on the proving situation, the conjecturing and proving processes they engage in, and the quality of the resulting product. The main question of the empirical work in this dissertation was, which process characteristics influence the quality of the final product (the formulated conjecture and constructed proof), and in which way they mediate the impact of students’ prerequisites on this product. Specifically, we distinguished between individual-mathematical and social-discursive process characteristics of conjecturing and proving. These process characteristics were extracted from prior research in mathematics education or in educational psychology or in the Learning Sciences. The central aim of this dissertation was to develop an instrument for assessing (prospective undergraduate) mathematics students’ conjecturing and proving processes in collaborative situations. A high-inference rating scheme with seven scales, based on theoretical considerations and on rating guidelines adapted from educational research was designed. The rating scheme was evaluated in a study with N=98 prospective undergraduate students working in dyads on an open-ended conjecturing and proving task. The results of the empirical study with regard to the basic analyses showed that collaborative conjecturing and proving processes could be rated with sufficient reliability and that the structure of the data corresponded to the underlying theoretical assumption that two dimensions, one related to individual-mathematical and one related to social-discursive process characteristics can be distinguished. The in-depth analyses pointed out that individual-mathematical process characteristics were predictive for the quality of the resulting product and mediated the relation between prerequisites (students’ prior knowledge on proof) and the quality of the product. In this way, the dissertation contributes to the scientific debate on how to assess (mathematical argumentation) skills (e.g., Blömeke, Gustafsson, & Shavelson, 2015; Koeppen, Hartig, Klieme, & Leutner, 2008) and provides theoretical and empirical insights on individual-mathematical and social-discursive process characteristics that describe the quality of collaborative conjecturing and proving processes

Analysis of Sociomathematical Norms in Mathematics Learning at 113 Junior High School

This study aims to describe the sociomatematical norms that are owned by students when learning mathematics in class. The subjects of this study were students of class VII-A Jakarta 113 Junior High School. The research method used is descriptive qualitative method. This research uses sociomatematical norms questionnaire and interview. Based on the results of the study, there were 9 students who had very good sociomatematical norms categories with a total percentage of indicators of 81% and there were 14 students who had good sociomatematical norms categories with a total percentage of indicators of 72%. The results showed that very good sociomatematical norms owned by students would affect the high results obtained by students in mathematics learning and good sociomatematical norms would affect the moderate and low results obtained by students in learning mathematics

Incorporating Scripts with Cooperative Learning to Promote Critical Thinking Skills in Secondary Science

A drastic growth of scientific and technological advancements in the 21st century have allowed for new jobs with innovative processes that require individuals who possess the ability to think deductively, reason through problems, and obtain information that can support the potential solutions to these problems. Many of the technological advancements have reduced the necessity to only memorize rote facts; rather, much of this information can be found through a quick internet search. What is needed, therefore, is education which requires students to think deeper than before – to examine new information through a more critical lens. The purpose of this research study is to investigate how the introduction of collaborative scripts into the cooperative learning of students in a secondary science classroom impacts critical thinking skills. A quasi-experimental non-equivalent control-group design was implemented. The sample was drawn from eight sections of ninth grade science at a secondary public school in a northeastern state. Students engaged in project-based learning with cooperation with peers on an inquiry-based science lesson with phenomena. The experimental group was presented with scripts to begin asking thoughtful questions of peers about the phenomena being studied. The control group was instructed to engage in peer discourse as they normally would. The CCT-X was administered to all participants as a pretest and posttest. The data was analyzed via ANCOVA testing. Although a greater improvement in scores can be seen in the group that was exposed to the cooperative scripts, the results were not statistically significant. Future recommendations were identified, such as recruiting a larger sample size, implementing a longer duration for the intervention of collaborative scripts, and considering a new instrument for measuring critical thinking skills