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Children who encounter questions about quantities or numbers are observed to use their fingers in different ways to aid their problem solving. This study aimed to contribute to the area of finger counting research with an inquiry of what children's finger use tells us about their knowledge of numbers. A basic argument is that it is not sufficient to observe the actual use of fingers; there is a need for interpretations of what the finger use means to the children, taking the children's perspective as the outset. This was done by analyzing 4- and 5-year-olds' finger use through the lens of phenomenography and variation theory of learning to describe the qualitatively different ways in which children use fingers as an expression of their ways of experiencing the meanings of numbers. Five categories of finger use that show a variety in the meanings the fingers represent emerged: Fingers represent individual items, quantities, countables, number relations, and number structure. The results show that children's finger use may give access to their ways of understanding numbers; some ways of using fingers indicate expressions of more or less advanced meanings of numbers. A conclusion from the results is that to develop number knowledge and skills, children are aided by learning to see and use their fingers as representing composed units. Some children need the structural support of fingers to solve number problems, and fingers should be used to explore number structures rather than used as countables., (Copyright © 2024 The Author(s). Published by Elsevier Inc. All rights reserved.)

More than 50 years ago, Bongard introduced 100 visual concept learning problems as a challenge for artificial vision systems. These problems are now known as Bongard problems. Although they are well known in cognitive science and artificial intelligence, only very little progress has been made toward building systems that can solve a substantial subset of them. In the system presented here, visual features are extracted through image processing and then translated into a symbolic visual vocabulary. We introduce a formal language that allows representing compositional visual concepts based on this vocabulary. Using this language and Bayesian inference, concepts can be induced from the examples that are provided in each problem. We find a reasonable agreement between the concepts with high posterior probability and the solutions formulated by Bongard himself for a subset of 35 problems. While this approach is far from solving Bongard problems like humans, it does considerably better than previous approaches. We discuss the issues we encountered while developing this system and their continuing relevance for understanding visual cognition. For instance, contrary to other concept learning problems, the examples are not random in Bongard problems; instead they are carefully chosen to ensure that the concept can be induced, and we found it helpful to take the resulting pragmatic constraints into account., (© 2024 The Authors. Cognitive Science published by Wiley Periodicals LLC on behalf of Cognitive Science Society (CSS).)

The prefrontal cortex (PFC) is associated with many cognitive functions, including planning. In the neuropsychology literature planning is reduced to "look ahead" ability and most extensively studied with the "tower" tasks. The most influential theoretical explanation is that planning is required in the absence of a routine solution and PFC patients have difficulty coping with novelty. There is an alternate view of planning that emphasizes the distinction between real world tasks and laboratory tower tasks. This account focuses on the structure of problem spaces and why patients with lesions to right PFC have difficulty navigating ill-structured problem spaces. To further explore these issues we administered two real world travel planning tasks to 56 Vietnam War veterans with penetrating brain lesions and 14 matched normal controls. One planning task involved familiar knowledge while the other involved knowledge unfamiliar to our participants. Participants also completed the D-KEFS tower task. A subset of 18 patients-with lesions to right anterior prefrontal cortex (BA 10)-were impaired in the travel planning task compared to normal controls. The task familiarity/novelty dimension affected performance across participant groups (familiar-task scores were higher than unfamiliar-task scores), but it did not differentially affect any group. An examination of cognitive strategies utilized by participants revealed that the impaired patient group had difficulty maintaining a sufficient level of abstraction and engaged the task at a much more concrete level than other participants. Interestingly, patients impaired in the real-world planning tasks were not impaired in the tower tasks. We conclude that patients with lesions to right BA 10 have difficulty in real-world planning tasks that can be attributed to difficulties in engaging problems at the appropriate level of abstraction., (Copyright © 2023 The Author(s). Published by Elsevier Ltd.. All rights reserved.)

Recent results demonstrate that inducing an abstract representation of target analogs at retrieval time aids access to analogous situations with mismatching surface features (i.e., the late abstraction principle). A limitation of current implementations of this principle is that they either require the external provision of target-specific information or demand very high intellectual effort. Experiment 1 demonstrated that constructing an idealized situation model of a target problem increases the rate of correct solutions compared with constructing either concrete simulations or no simulations. Experiment 2 confirmed that these results were based on an advantage for accessing the base analog, and not merely an advantage of idealized simulations for understanding the target problem in its own terms. This target idealization strategy has broader applicability than prior interventions based on the late abstraction principle because it can be achieved by a greater proportion of participants and without the need to receive target-specific information. We present a computational model, SampComp, that predicts successful retrieval of a stored situation to understand a target based on the overlap of a random, but potentially biased, sample of features from each. SampComp is able to account for the relative benefits of base and target idealization, and their interaction., (© 2023. The Psychonomic Society, Inc.)

Simon Fraser University in Vancouver, Canada offers two introductory calculus courses designed for students enrolled in science and engineering programs. Students identified as needing additional support based on their admission grades take the version of the course where students meet weekly for four hours instead of three. A new approach for the fourth hour was introduced in Fall 2019 splitting the class into smaller sections of 20-40 students to work in small groups on vertical whiteboards under the supervision of the instructor, complemented by peer tutors. This format introduces an active learning element that improves student performance and attitude. The project was first implemented in Fall 2019, although interrupted due to COVID-19. After returning to the in-person classroom, the whiteboard seminar program was reinstated. In Fall 2022 and Spring 2023, survey feedback was collected from both students and peer tutors for a more holistic perspective on the program. In this study, the results of these surveys are analyzed, focussing on four areas of measurement: students' satisfaction, learning calculus concepts and problem-solving techniques, the contribution of the instructor, and the contribution of the peer tutors. Exploratory factor analysis revealed three latent variables underlying student survey responses. The survey results suggest that interactions with the instructor, peer tutors, and active learning activities all contributed to students' enjoyment of the course and their perceptions of their own learning. We compared students' final course grades from Fall 2018, where the fourth hour was devoted to lecture before the program was implemented, to Fall 2019 and 2022. The data shows a decrease in the percentage of students failing the course and an increase in those earning A's and B's since the program's implementation. Additionally, survey results show that replacing a typical lecture-style hour with a whiteboard seminar improved students' attitudes and perceptions of the course.

Aims: The aim of this concept analysis was to develop a contemporary operational definition of Clinical Judgement in nursing., Design: Concept analysis., Methods: Rodgers' evolutionary method of concept analysis. Throughout the study, the authors followed the SRQR checklist., Results: This concept analysis guided the development of an operational definition of clinical judgement, within the context of nursing, articulated as Clinical judgement is a reflective and reasoning process that draws upon all available data, is informed by an extensive knowledge base and results in the formation of a clinical conclusion., Conclusion: The purpose of this concept analysis was to propose a detailed definition of clinical judgement in nursing, that reflected its theoretical base within today's contemporary healthcare system. Using Rodgers' evolutionary method of concept analysis, surrogate terms, attributes, antecedents and consequences were identified, and a contemporary definition was developed. No patient or public contribution was required to design or undertake this research., (© 2022 The Authors. Journal of Clinical Nursing published by John Wiley & Sons Ltd.)

The authors developed an instructional nudges as part of a larger research project. These instructional nudges are designed to be small but powerful changes to teachers' existing practices. Some instructional nudges focus on modifying tasks used in classrooms. In this article, the authors share Rate and Review. The goal of Rate and Review is to support students in understanding that there are alternative solution strategies to what they might be used to using, provide students the opportunity to become more comfortable creating mathematical arguments, and develop a deeper understanding of a concept. Rate and Review provides formative assessment for teachers by supporting their reflections on students' understanding of current content and how students are making sense of common errors.

This study examined the extent to which college algebra instructors employ pedagogical practices previously found to assist students master difficult STEM content material and address their own previous math deficits. Faculty classroom behaviors were assessed with a modified version of the Generalized Observation and Reflection Platform (GORP). While it was not surprising that math faculty frequently lectured and wrote on the board, results also revealed that recommended practices were infrequently employed by instructors teaching college algebra. More specifically, faculty teaching introductory college algebra rarely addressed students' misconceptions, made references to their students' prior knowledge, or employed metacognitive teaching strategies. Moreover, while being observed, faculty never applied a mathematical concept to real life. Moreover, mathematics instructors rarely praised their students. The implications of these findings for professional development and hiring requirements are discussed and emphasized for college math department chairs and administrators generally. Results also highlighted the usefulness of the GORP assessment tool for assessing faculty classroom behaviors.

This study examined the effects of embodied learning experiences on students' understanding of computational thinking (CT) concepts and their ability to solve CT problems. In a mixed-reality learning environment, students mapped CT concepts, such as sequencing and loops, onto their bodily movements. These movements were later applied to robot programming tasks, where students used the same CT concepts in a different modality. By explicitly connecting embodied actions with programming tasks, the intervention aimed to enhance students' comprehension and transfer of CT skills. Forty-four first- and second-grade students participated in the study. The results showed significant improvements in students' CT competency and positive attitudes toward CT. Additionally, an analysis of robot programming performance identified common errors and revealed how students employed embodied strategies to overcome challenges. The effects of embodied learning and the impact of embodied learning strategies were discussed.

Identifying abstract relations is essential for commonsense reasoning. Research suggests that even young children can infer relations such as "same" and "different," but often fail to apply these concepts. Might the process of explaining facilitate the recognition and application of relational concepts? Based on prior work suggesting that explanation can be a powerful tool to promote abstract reasoning, we predicted that children would be more likely to discover and use an abstract relational rule when they were prompted to explain observations instantiating that rule, compared to when they received demonstration alone. Five- and 6-year-olds were given a modified Relational Match to Sample (RMTS) task, with repeated demonstrations of relational (same) matches by an adult. Half of the children were prompted to explain these matches; the other half reported the match they observed. Children who were prompted to explain showed immediate, stable success, while those only asked to report the outcome of the pedagogical demonstration did not. Findings provide evidence that explanation facilitates early abstraction over and above demonstration alone., (© 2022 John Wiley & Sons Ltd.)

In this article, we explored Grade 11 learners' algebraic and geometric connections when solving Euclidean geometry riders. A qualitative interpretive case study design was followed. Thirty Grade 11 learners from a non-fee-paying secondary school in the Capricorn North district of South Africa were conveniently sampled to participate in this study. Data were collected through learners' responses to classwork, homework exercises, and task-based interviews. Data were analysed thematically. The findings revealed that to solve Euclidean geometry riders successfully, learners need to establish the feature connections embedded in the given figure or diagram. The ability to make feature connections provides a point of departure in the solution process of a geometric problem. Contribution: Once the feature connection is established, other connections will subsequently emerge. In addition, the reversibility connections become a form of feature connection when solving Euclidean geometry riders. Therefore, we recommend that mathematics teachers emphasise and use mathematical connections in their daily teaching of mathematics.

Multiple representations are widely recognized for their significant role in concept learning. This study aimed to investigate the multiple representation translation skills of high school students at different grade levels about the concept of one-dimensional motion. 239 9th, 10th, and 11th-grade students participated in the study using a developmental research model. The data collection tool consisted of questions that required translating figures, tables, graphs, verbal explanations, and algebraic representations into other representation types in a multiple-representation translation test focusing on one-dimensional motion. Data analysis involved evaluating the translation among representations for each category and analyzing the multiple representation translation skills across different grade levels using one-way analysis of variance (ANOVA). The results revealed that students successfully translated from figure, table, and graphical representations to other forms while encountering challenges in translation from verbal and algebraic representations. Furthermore, the ANOVA results indicated a significant difference between the 9th and 11th grades, favoring the 11th grade.

Redox reaction is a core chemical concept. However, its abstract nature makes it very difficult for students. Students' conceptual structure reflects their mastery of concepts, which helps teachers implement targeted educational strategies. This study aimed to explore the conceptual structures of redox reaction held by students (grades 10 to 12) by employing MDS and HCA. A total of 606 students participated, with 195 students in 10th grade, 202 in 11th grade, and 209 in 12th grade. The results indicated that three-dimensional solutions were appropriate for the conceptual structures of 10th and 12th graders, while 11th graders demonstrated two-dimensional solutions. All students grouped the 15 concepts related to redox reaction into two large clusters: metrology and the redox reaction process. Moreover, both 10th and 12th graders further subdivided the 15 concepts into four subclusters: metrology, oxidation process, reduction process, and chemical reaction. Students' conceptual structures were rational across all three grades. The conceptual structures of 10th and 12th graders were more refined than those of 11th graders, and there was no significant difference between the conceptual structures of 10th and 12th graders. 11th graders learned about electrochemistry and tended to confuse concepts related to redox reaction with those related to electrochemistry.

The understanding mathematical concept is an error that often occurs in classroom learning among students when solving mathematical problems. The most difficult part for students is solving problems, because it requires numeracy skills, high concept mastery, as well as the ability to use good language, and so on so that students don't make any more mistakes when working on math problems. Student errors in solving mathematics problems are (1) errors in connecting concepts, (2) errors in operations and (3) errors in constructing concepts. The problem is what is the structure of students' mathematical misconceptions in solving mathematical problems based on cognitive style. The method in this research, namely an exploratory descriptive approach, aims to determine the structure of students' errors based on cognitive style in solving mathematical problems. Analysis of research data, namely: (1) Data reduction, (2) Data exposure, (3) data triangulation and (4) drawing conclusions. The cognitive styles referred to are field dependent and independent. The conclusions are (1) the structure of conceptual errors with an applied field dependent cognitive style begins with disequilibrating, then solving by linking applicable concepts, and (2) the structure of conceptual errors with a field independent cognitive style begins with disequilibrating, then solving using analyse.

The study aimed to analyze the content validation of the STEM-integrated on thermal and transport concept inventory instrument used to measure the problem-solving abilities of high school students. The instrument questions developed amounted to nine description questions. This type of study is development research. The steps in this research are define, design, development, and dissemination. However, in this study, it was limited to the design stage (expert validation). Furthermore, the instrument was assessed qualitatively and quantitatively by a validator consisting of three experts and five users. Assessment aspects include material, language, and construction. The qualitative data collected was analyzed qualitatively, and the quantitative data were analyzed using the Content Validity Ratio (CVR). The results showed that nine items had a CVR value exceeding 0.75, which meant that all items had valid criteria for content validity. The CVR value obtained is between 0.88 and 0.96, which means that the instrument has high criterion validity.

This study investigated the noticing of teachers who teach mathematics at different grade levels in the context of lesson study. The study specifically focused on teachers' noticing during the planning phase of the lesson. For this, the lesson planning phase of a group of four teachers consisting of kindergarten and primary school 1st grade teachers (vertical teacher team) was investigated. Teachers' planning process for the 1st grade level mathematics lesson was the focus in the context of lesson study. We examined categories and subcategories that emerged in the planning phase of the lesson study and noticing levels dealt with those categories. We examined which categories and subcategories emerged in planning and at which noticing levels they dealt with those categories. The results of the study provide comprehensive data on categories and noticing levels of teachers teaching mathematics at different grade levels in the collaborative planning process of a lesson study focusing on improving problem-solving. The results of the study showed that the teachers' noticing was clustered under three categories: curriculum, teaching methods, and conceptual understanding. In terms of noticing levels, the study's results revealed that noticing is performed at the attending to level in all categories. Although less frequently than the attending to level, teachers noticed making sense of level in all the other categories except the curriculum category. Besides, noticing that the deciding to respond level did not occur at any level might be due to the cultural challenges of adaptation of lesson study and the amount of support, we researchers provided teachers as facilitators.

A cohort of pre-service mathematics students was exposed to a teaching strategy based on retrieval practice and test-potentiated learning. The aim of the study was to determine how high and low prior topic knowledge study participants compare in terms of their procedural fluency and conceptual understanding after exposure to the teaching strategy. A pre-test and post-test repeated measures design was employed in the study to compare within groups. A revised taxonomy table based on Bloom's taxonomy was utilised to categorise test items. Findings indicate significant differences between pre-test and post-test scores within groups. Results from the independent samples t-test show a significant difference between the two groups. Outcomes confirm that the benefits of retrieval practice are greatest for unfamiliar content. Findings indicate that for low prior topic knowledge students, procedural fluency is enhanced and retained more than conceptual understanding whereas for the high prior topic knowledge students it was the reverse. The strategy was not as effective for improving conceptual understanding.

The purpose of this article is to conduct a mathematical and phenomenological comparison of three concepts: (1) the finite limit of a function at a point, (2) the finite limit of a sequence, and (3) the infinite limit of a sequence. Additionally, we aim to analyse the presence of these concepts in Spanish textbooks. The methodology employed is exploratory and descriptive. Our mathematical comparison revealed differences in several areas, including the dependence between variables, the involved infinite processes, the types of infinity, the dimensioning for each variable, and the intuition of continuity in the interval. Our phenomenological comparison found a correspondence between phenomena using a formal approach, but differences in phenomena when using an intuitive approach. Finally, our analysis of textbooks revealed that all three limits are most commonly presented in the verbal representation system and definition format.

This study aimed to examine the effect of four types of assessment on overall student success in an online college-level climate change course. Quizzes, midterms, lab assignments, and a capstone project as well as knowledge check questions were used to assess different aspects of student learning, consistent with Bloom's taxonomy hierarchy. Quizzes and midterms assess basic knowledge, including remembering and understanding concepts, laboratory assignments require students to analyze and integrate concepts, and the capstone allows students to evaluate their understanding and create new content. Binary logistic regression, multiple regression analysis, continuous-by-continuous interaction modeling, and path analysis were used to investigate the moderating and mediating effects of these assessment types. We found both direct and indirect positive interactions as well as one negative interaction. Positive interactions were identified between quiz and lab assignment achievement and between capstone achievement and lab assignment achievement. The total score for correctly answered knowledge check questions positively affected quiz and lab assignment achievements. The interaction between capstone project achievement and total score for correctly answered knowledge check questions showed a negative interaction. Finally, the total score for correctly answered knowledge-check questions had an indirect positive effect on overall student success in the course. Results show that different types of assessment in an online course are complementary and amplify student learning.

Motivation: This research investigates the transformative impact of integrating AI into mathematics education, aiming to enhance students' conceptual understanding and self-efficacy. It addresses the crucial need for innovative teaching methods in response to contemporary challenges in education and aims to fill gaps in understanding the potential drawbacks and benefits of AI implementation. Objectives: The primary goal of this study is to investigate the effects of AI tools, such as GeoGebra and ChatGPT, on students' conceptual understanding and self-efficacy in a mathematics classroom setting, with a focus on collaborative learning, teacher-led discussions, and problem-solving. Methods: Employing a mixed-methods approach, the research involves pre- and post-implementation surveys, supplemented by qualitative data from interviews, focus groups, and classroom observations. The study emphasises the utilisation of AI in collaborative learning environments and explores the challenges and opportunities associated with its integration. Results and Conclusions: The study reveals a noteworthy enhancement in students' conceptual understanding and self-efficacy belief when AI tools are incorporated into mathematics education. However, it highlights concerns such as technological challenges, potential teacher replacement fears, biases in AI systems, and difficulties related to pacing and social interaction. To optimise the use of AI, the paper suggests strategies like adaptive learning platforms, real-world simulations, intelligent tutoring systems, and data-driven instruction, positioning AI as a complement rather than a substitute for traditional teaching methods. The study recommends addressing challenges through additional support resources and transparent communication, emphasising the need for a thoughtful integration of AI in education.

The aim of this paper is to review recently calculus curriculum reforms and research studies that document what types of understanding students develop in their precalculus courses. We argue that it is important to characterize what difficulties students experience to solve tasks that include the use of foundational calculus concepts and to look for possible ways for students to develop a way of reasoning to work on problems that involve variational phenomena. Thus, we identified tasks in which calculus students exhibit limited understanding of essential concepts to approach and solve those tasks. The purpose is to illustrate and discuss how the systematic use of a Dynamic Geometry System (DGS) could provide a set of affordances for students to develop ways of thinking to grasp calculus foundational ideas and to study the derivative concept. Here, we relied on Thurston's seminal work that emphasizes the relevance for learners to identify, connect, and coordinate different dimensions and meanings (intuitive, symbolic, algorithmic, geometric, physical, and formal) to construct, understand, and apply the concept of derivative.

This study examines the multiplicative problem-solving strategies used by a 14-year-old student with autism spectrum disorder and intellectual disabilities during an instructional process based on the Conceptual Model-based Problem Solving (COMPS) approach. The instruction aimed to enhance conceptual comprehension of problem-solving by the use of model diagrams representing mathematical relations involved in word problems. These diagrams aid in selecting the appropriate operation for each type of multiplicative problem, including equal-groups, multiplicative comparison, and combination problems. We further discuss how the instructional process promoted conceptual understanding for the three problem types, highlighting the development of strategies (modeling, counting, and operations) and the pivotal role of the COMPS methodology components in this progression. The results indicate that the student adeptly adopted the COMPS approach, facilitating the transition from informal modeling to written operations, with his choice of strategies varying depending on the type of problem. While there was a higher utilization of modeling strategies in equal-groups and combination problems, modeling was not as frequently employed in comparison problems. We speculate how this differentiated strategy preference could be associated with certain characteristics of the disorder.

Spatial thinking is ubiquitous in design. Design education across all age groups encompasses a range of spatially challenging activities, such as forming and modifying mental representations of ideas, and visualizing the scenarios of design prototypes being used. While extensive research has examined the cognitive processes of spatial thinking and their relationships to science, technology, engineering, and mathematics learning, there remains a knowledge gap regarding the specific spatial thinking processes needed for open-ended problems, which may differ from those assessed in close-ended, analytical spatial tasks. To address this gap, we used educational design-based research to develop a nature-inspired, design-by-analogy project and investigate the spatial thinking processes of young, novice designers. 16 children from an international school in the Netherlands participated in this five-week design project. Multimodal evidence from classroom recordings and children's design works were triangulated to offer insight into the key spatial thinking processes involved in their creation of nature-inspired, analogy-based design prototypes. Our results revealed spatial thinking processes that might not align with those assessed in conventional spatial tests and may be unique to design or open-ended problem-solving. These processes include abstracting spatial features to infer form-function relationships, retrieving a range of relevant visual information from memory, developing multiple possible analogical matches based on spatial features and relationships, elaborating and iterating on the design concepts and representations to make creative and suitable solutions for the design challenge, as well as visualizing design prototypes in practical usage scenarios. By highlighting the nuanced differences between spatial thinking in open-ended, divergent thinking tasks and conventional spatial tasks that demand single correct solutions, our research contributes to a deeper understanding of how children utilize spatial thinking in design and open-ended problem-solving contexts. Furthermore, this case study offers practical implications for scaffolding children's analogical reasoning and nurturing their spatial thinking in design education.

This article uses the concurrent mixed methods design to explore the errors made by 171 Grade Seven learners in algebraic problem-solving within the Assin Central Municipality in Ghana. The participants were categorized into low-achieving and high-achieving groups based on their performance in a pretest, to help provide a detailed examination of the discrepancies in error occurrence between these groups. The Newman error analysis framework was used to unveil distinct patterns of errors among learners when tackling algebraic tasks. Quantitative data from the test were complemented by qualitative insights employing the Think Aloud Protocols (TAP). The analysis revealed that low-achieving learners struggled with reading, comprehension, and transformation errors, while high-achieving learners mainly encountered transformation and process skill errors. The findings contribute valuable insights into learners' challenges in mastering algebraic concepts, offering implications for educational interventions and curriculum development in mathematics education. The study recommends implementing differentiated instruction strategies and providing additional support for comprehension, and problem-solving skills to improve algebraic proficiency among learners.

Mathematics is crucial in science, technology, and engineering career paths for all individuals, including Students with Visual Impairments. This study examined teachers' perceptions of experiences and challenges that primary and secondary school Students with Visual Impairments faced concerning the learning of mathematics in Botswana. A descriptive survey research design was used in this study and data was collected from 84 participants. Mann Whitney U and Kruskal-Wallis tests were used to determine differences in teachers' perceptions by position and school type respectively. Moreover, content analysis was used to summarize responses to open-ended questions. Teachers perceived that Students with Visual Impairments experienced barriers in key areas such as concept development and problem-solving, braille literacy, resources and support, instructions and modifications, and attitudes and the learning environment. Significant differences were noted among teachers' perceptions based on whether they taught at a primary or secondary school. It is recommended that mathematics interventions for SWVIs should include creating a supportive learning environment, as well as collaboration within schools, between schools, and with external entities.

Contribution: This study uses a qualitative research method to analyze interviews where participants simplified an electric circuit while explaining their thought process. Background: Rearranging circuit diagrams is a fundamental skill in electrical and electronic engineering, yet students can struggle with unfamiliar configurations. Current research in the discipline is often quantitative, centered on conceptual understanding. By using a qualitative method, the process of "How" students interact with circuit diagrams is investigated. Research Question: How do students approach circuit diagram simplifications? Methodology: 15-min individual discussions with ten participants (undergraduate years 1-4) simplifying an unconventionally presented circuit diagram were recorded. Reflexive thematic analysis was used to identify common themes. Findings: (1) Participants initially rely upon pattern recognition to solve circuit problems before applying other analysis techniques; (2) two rearrangement methods were identified: "component focused," where combinations of components are grouped and then connected together, and "ground focused" where components in the circuit are related to ground and then connected together; (3) students using a ground focused strategy were less hesitant in their circuit rearrangement process; and (4) students broadly used mechanicalistic methods of error checking, selecting software tools rather than applying conceptual understanding.

Relational reasoning is a cornerstone of human cognition. Extensive work, drawing on the Relational Match to Sample paradigm (RMTS), has established that humans, at least above the age of five, are much more proficient relational reasoners than younger children or non-human animals. While sometimes differences between populations derive from differences in capacity (the capacity to create representations in a certain format or of a certain complexity, information processing capacity), other times such differences derive from different learning histories alone. Here we distinguish between two types of learning history explanations on the example of four-year-olds' failure on Premack's (1983) RMTS task: (1) that children four-year-olds have not yet created representations of the relations same and different with the properties need to support success on RMTS and (2) that four-year-olds have different inductive biases than do adults. Experiment 1 established that four-year-olds are at chance on the RMTS task we deploy as a transfer task in Experiment 2. Experiments 2A-C each provide children with a mere 8 trials of training on of one three MTS tasks (Number, Size and Identity MTS, respectively), none of which involves making matches of same to same or different to different. The very brief training (eight trials) on two of these tasks (Number MTS, Size MTS) leads to spontaneous success on RMTS in four-year-olds. Identity MTS has no effect on subsequent performance on RMTS. Given the brevity and non-relational nature of the training the successes after Number and Size MTS training must have resulted from changing inductive biases alone. Furthermore, the same two training tasks increased relational responding by adults on a related task (Kroupin & Carey, in press), whereas Identity MTS training did not, suggesting that the mechanisms through which the training changed inductive biases are at least partially continuous between ages four and adulthood., (Copyright © 2021 Elsevier B.V. All rights reserved.)

Nonhuman animals and preschoolers struggle with Relational-Match-to-Sample (RMTS), a classic test of the capacity for second-order relational, analogical, and reasoning. These failures are often explained by limitations in representational or computational capacities. Drawing on recent evidence for robust spontaneous RMTS success (i.e., without error-feedback) in crows and parrots after minimal second-order training, we present five experiments with human adults consistent with the possibility that population differences sometimes instead derive from differences in inductive biases alone. Experiment 1 confirms human adults have the capacities and requisite representations to succeed spontaneously on RMTS. Experiments 2-5 utilize a modified RMTS task in which adults make relational matches only about half of the time. Experiment 3 tests whether eight trials of various MTS tasks, nonsecond-order training featured in the aforementioned comparative studies, can increase spontaneous second-order relational responding in human adults. Two of the MTS tasks (Number, Size MTS) do so, demonstrating that MTS training can, in fact, increase relational responding by changing inductive biases alone. The other MTS tasks (Identity, Color MTS) do not do so, evidence that the facilitating effect is not a result of matching involved in MTS per se. Experiments 4 and 5 test one hypothesized mechanism by which specifically Number/Size MTS tasks may have led to increased relational responding, that is, by inhibiting preexisting biases to match on shape and/or color, making relational matches relatively more likely. We close by discussing the importance of research into inductive biases to the project of understanding relational reasoning. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Background: All thinking occurs in some sort of context, rendering the relation between context and clinical reasoning a matter of significant interest. Context, however, has a notoriously vague and contested meaning. A profound disagreement exists between different research traditions studying clinical reasoning in how context is understood. However, empirical evidence examining the impact (or not) of context on clinical reasoning cannot be interpreted without reference to the meaning ascribed to context. Such meaning is invariably determined by assumptions concerning the nature of knowledge and knowing. The epistemology of clinical reasoning determines in essence how context is conceptualised., Aims: Our intention is to provide a sound epistemological framework of clinical reasoning that puts context into perspective and demonstrates how context is understood and researched in relation to clinical reasoning., Discussion: We identify three main epistemological dimensions of clinical reasoning research, each of them corresponding to fundamental patterns of knowing: the representational dimension views clinical reasoning as an act of categorisation, the interactional dimension as a cognitive state emergent from the interactions in a system, while the interpretative dimension as an act of intersubjectivity and socialisation. We discuss the main theories of clinical reasoning under each dimension and consider how the implicit epistemological assumptions of these theories determine the way context is conceptualised. These different conceptualisations of context carry important implications for the phenomenon of context specificity and for learning of clinical reasoning., Conclusion: The study of context may be viewed as the study of the epistemology of clinical reasoning. Making sense of 'what is going on with this patient' necessitates reading the context in which the encounter is unfolding and deliberating a path of response justified in that specific context. Mastery of the context in this respect becomes a core activity of medical practice., (© 2021 The Authors. Medical Education published by Association for the Study of Medical Education and John Wiley & Sons Ltd.)

Background: The COVID-19 pandemic and the shift to digital learning materials have dramatically reduced the use of paper-based textbooks in higher education. In the field of engineering, students need a comprehensive understanding of mathematical concepts, which can be achieved through the use of e-textbooks. These digital learning materials can provide a more comprehensive and effective learning experience by incorporating a variety of multimedia elements, such as audio, visual aids, and interactive features. Rationale/Relevance: Mathematics students are expected to possess critical problem-solving skills, yet they are rarely asked to elaborate on and explain their mathematical reasoning and concepts through written argument in the engineering mathematics classroom. While students may excel in formula manipulation, they may still possess misconceptions about mathematical principles and concepts. A reliance on mechanical and procedural approaches, such as formula application, without proper conceptual understanding can result in serious misunderstandings of mathematics and its real-world applications. Without the ability to reason about mathematical concepts, students may struggle to connect what they are learning in class with real-world scenarios, leading to difficulties in solving practical problems. It is, therefore, crucial to foster conceptual understanding and critical reasoning skills in mathematics education, rather than solely relying on memorisation, in order to equip students with the necessary skills to succeed in their careers. Research Problem/ Research Question: This study will answer the following research question:How effective are e-textbooks in promoting conceptual understanding of engineering mathematics? Aim of the Study/Focus: The purpose of this study is to investigate conceptual learning in engineering mathematics using an e-textbook as a learning tool. Methodology: This qualitative case study explores e-textbooks as adaptive technology, with functionalities that include artificial intelligence allowing students to develop their understanding by interacting with digital text, watching videos of real-world mathematics concepts, and responding to quick quizzes on concepts; as well as practising and mastering further mathematical principles and concepts. It focuses on first-year engineering students and lecturers at a University of Technology in South Africa. The analysis of interview recordings was done with ATLAS.ti. analytical software. Theoretical Framework: The data from this study was analyzed through the lenses of cultural historical activity theory (CHAT) and Vosniadou's notion of conceptual change, allowing the researchers to explain complex real-world situations that students experience when engaging with the e-textbook to solve mathematics problems. Recommendations: The work presented here has implications for future studies of conceptual learning in mathematics research and may provide opportunities around learning engineering mathematical concepts, particularly in the context of developing countries. This is because it offers novel instructional approaches that are tailored to the specific needs and challenges of these countries.

Whole number computations are a critical skill that serves as a foundation upon which higher-order concepts in mathematics are taught to children. To facilitate their instruction, educators often use multiple representations to support a child's cognition. Representations with physical manipulatives are widely studied through a graduated instructional sequence featuring concrete, representational, and abstract stages of learning. In contrast, research on representational sequences featuring virtual manipulatives is less robust. Thus, this study evaluated an instructional strategy with virtual manipulatives, static representational drawings, and abstract algorithms to teach multiplication to three U.S. elementary students with mathematics difficulty. A functional relation was established via a single-subject multiple probe design between the treatment and students' accuracy performance. Baseline-corrected Tau estimates confirmed a medium effect size for all three students, while student performance on measures assessing the number of errors committed and the duration of sessions also returned favorable findings.

Nowadays, mathematics teachers in K-12 strive to promote their students' mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students' views toward them. In this work, we present the results of a research study, which included the design of a six-lesson learning activity aimed at fostering 16- to 17-year-old secondary students' CT skills in calculus lessons using the dynamic mathematics software "GeoGebra." Our goal was to investigate how students experienced the CT-embedded calculus lessons with "GeoGebra" and what challenges they faced during their interaction with the learning material and software. We collected and analyzed data from students' code in "GeoGebra," workbooks, semi-structured interviews, and questionnaires. Our findings suggest that most students mastered using CT concepts in calculus activities to a satisfactory degree and could reason about their computational solutions using "GeoGebra" and the generated graphs. Students' understanding of the mathematical content knowledge introduced was essential to complete the lesson series successfully and unnoticed gaps in prior knowledge emerged. Our study shows that students appreciate the CT-embedded calculus lessons and "GeoGebra's" exploratory approach to mathematics problems when provided with appropriate support. We conclude that an integrated approach to mathematics education and CT is viable and can contribute not only to fostering CT but also to increasing interest in mathematics.

The aim of this study is to explore the possibility of introducing the general notions of function and inverse function through a mathematical activity on linear functions, focusing on the quantitative meaning associated to the connection between a relation and its inverse. We present a genetic decomposition, that is, a viable cognitive path for learning these introductory concepts, in terms of mental structures and mechanisms, from the viewpoint of APOS (Action-Process-Object-Schema) theory. One of the innovative aspects of this study consists in the design of a genetic decomposition that involves the construction of two concepts. Through interviews, we explore the conceptions that two high school students--who had not been introduced to the function concept before--developed in connection with the notions in question. Our results confirm that not only it is possible to study a relation and its inverse together, but also doing that can enhance the understanding of the meanings involved in the mathematical relation. This leads us to question the learning of mathematics being linear, and to open a discussion about the organization of mathematical content for instructional purposes. We offer suggestions for further research both involving other mathematical concepts, and in terms of theoretical constructs.

In 2011, Cohen and Blanc-Goldhammer introduced a new unbounded version of the number line estimation task which they argued to overcome limitations of its 'traditional' bounded counterpart. In line with this proposition, there is increasing evidence that the unbounded number line estimation task may indeed reflect a purer measure of the underlying representation of number magnitude. However, only few studies used this task version yet. In the present literature review, we aimed at evaluating all studies employing the unbounded task version so far to provide an overview of the current state of research. To identify all relevant articles and to evaluate the validity of the task, we conducted a systematic literature search in different databases following the PRISMA guidelines in May 2021. Methodological differences and commonalities of the 16 studies that met the inclusion criteria are discussed here. Our evaluation indicated considerable differences between studies with respect to the number range covered or methodological features such as display size. Additionally, five studies observed evidence for estimation biases in the unbounded task as well. Nevertheless, this review also substantiated the claim that the unbounded task version might indeed be a more valid and purer measure of the mental representation of number magnitude as the results of 14 studies confirmed this hypothesis., (Copyright © 2021 The Authors. Published by Elsevier B.V. All rights reserved.)

Individuals use diverse strategies to solve mathematical problems, which can reflect their knowledge of arithmetic principles and predict mathematical expertise. For example, '6 + 38 - 35' can be solved via '38 - 35 = 3' and then '3 + 6 = 9', which is a shortcut-strategy derived from the associativity principle. The shortcut may be critical for understanding algebra, however approximately 50% of adults fail to use it. We review the research to consider why the associativity principle is challenging and highlight an important distinction between shortcut identification and execution. We also discuss how domain-specific skills and domain-general skills might play an important role in shortcut identification and execution, and provide an agenda for future research., (Copyright © 2021 Elsevier GmbH. All rights reserved.)

Instruction in an introductory programming course is typically designed to introduce new concepts and to review and integrate the more recent concepts with what was previously learned in the course. Therefore, most exam questions in an introductory programming course require students to write lines of code that contain syntactic elements corresponding to the programming concepts covered during the instruction. This study investigates the number of concepts involved in the exam problems of an introductory Java programming course. In addition, this study compares how the increase in the number of concepts correlates with the ability of students to write error-free lines of code. The instructional method adopted in this study focuses on providing students with a problem-solving schema and a resultant programming plan that integrates many concepts to meet the problem's goal. Results from this study indicate that as the course progresses through the semester, students, on average, apply appropriate problem-solving schemas and programming plans to produce more error-free lines of code, despite an increase in the concept count in the problems. Furthermore, the exam problems later in the course repeat the application of cluster concepts that have appeared in the past exam. This paper illustrates how programming is a cumulative skill and that repeating and building upon the applications of these concept clusters several times through the course increases the likelihood that students will produce more correct lines of code as the semester progresses.

The empirical data in this study are from a series of two lessons on measurement implemented in seven classes with 119 six-year-old students in Sweden. Both problem solving and problem posing were shown to be important in early mathematics when students in this study worked on one problem-solving task and one problem-posing task on measurement. As there are few studies specifically on problem posing in early mathematics and on young children's understanding of measurement, this study adds knowledge of value for both teachers and researchers. In the study, paper-and-pen work from the students was analysed together with interviews conducted after the students had worked on the two tasks. When solving the task on measurement, the students discerned shape, size, distance, and number as mathematical aspects of measurement. When asked to pose a similar task, only size and number reoccurred as mathematical aspects of measurement. However, other features from the problem-solving task reoccurred in the posed tasks: similar drawings were used in combination with questions on measurement as the mathematical content.

In the context of a rapid digital transformation, digital competence is now regarded as a fourth cultural skill complementing reading, writing, and arithmetic. We argue that a well-structured and sound competence model is needed as a shared foundation for learning, teaching, pedagogical diagnostics and evaluative schemes in the school system. Every competence model should build upon a consistent, theoretically sound framework for teaching and learning. We consequently develop a competence model for digital competence by drawing on the concept of computational thinking as well as on general learning taxonomies. By combining different knowledge and process dimensions with essential facets of computational thinking a cube model of digital competence can be constructed. Hence, we develop and substantiate a structure model for digital competence building upon the concept of computational thinking that goes beyond the existing frameworks only focusing on the subject-related context and present this for discussion. The next step would then be to supplement the structure model with specific learning objectives, so that developing approaches to teaching and learning digital competence has a sound basis. [Note: The publication year (2022) shown on the PDF is incorrect. The correct publication year for this article is 2023.]

This is an exploratory study considering instructional methods in mathematics education. The study examined misconceptions arising from selected assignments undertaken by sixty-five students. The students were attending a calculus mathematical course unit on limits of functions, in different academic areas offered at selected tertiary institutions in Western Uganda. In this study, forms of expressing ideas on limits of functions are considered as far as they facilitate access to underlying mathematical principles. We explored the use and application of tools for representing mathematical ideas to enhance students' conceptual understanding and problem solving skills. A small-scale pilot of interventions using readily available GeoGebra dynamical computer software was applied. The diagnostic assignment on limits of functions was used for data collection. The main objective was to examine whether or not students preferred the application of multiple representations to the analytic approach. The semi-structured interview protocol was also conducted to probe further and correlate students' responses and their problem-solving abilities. The results showed that multiple representations-based instructions significantly changed students' understanding of the limits of functions. The results promise better access and understanding of more abstract mathematical concepts and may support students' problem-solving abilities. The semi-structured interviews conducted indicated that multiple representations supported and, therefore, enhanced students' understanding and solving of the limits of functions. This study highly recommends that mathematics educators should adapt multiple representations-based instructions to enhance students' critical thinking, and problem-solving.

In this paper, we explored the type of mathematical connections Grade 11 learners make when solving two-dimensional (2D) trigonometric problems in an Activity-Based Learning (ABL) environment. We followed a qualitative case study design within an interpretive paradigm. Convenience sampling was used to select a whole class of 45 Grade 11 learners from one of the public non-fee-paying secondary schools in Capricorn District, Limpopo Province of South Africa. Group work presentations and classroom interactions were used to collect data. Data were analyzed using deductive thematic analysis guided by the mathematical connections' framework. The findings indicated that learners managed to make procedural, meaning, reversibility, different representations, feature, and inclusion part whole as well as integrated connections as they worked on 2D trigonometric problems in an ABL environment. We established that learners did not make generalization part-whole connections. In addition, we found that some learners lacked mathematical connections skills and failed to solve the problems. Engaging learners in an ABL environment provided a fine-grained approach that allowed them to make mathematical connections. We, therefore, recommend that teachers should create an ABL environment to enable learners to make different types of mathematics connections during the teaching and learning of trigonometric concepts.

Several studies compared three different types of questions (conceptual, algorithmic, and graphical) across various topics, however, few focused specifically on gifted students. This study addressed this gap. The aim of the study, hence, was to determine whether there were notable differences in gifted students' performance in the three types of tests. The study involved 115 gifted students aged between 17 and 18 years old. They responded to the three different tests including 10 test items in each with the same content. Significant differences were observed in their performances between the conceptual versus algorithmic in favour of the conceptual test, the conceptual versus graphical in favour of the conceptual test, and the algorithmic versus graphical in favour of the algorithmic test. Moreover, the statistical analysis results revealed that the students performed markedly poorer performance on the graphical test in comparison to both the algorithmic and conceptual tests. These results suggest the need to enhance students' graphical skills to facilitate a better understanding of physics concepts. Proper steps should be taken to improve their proficiency in interpreting and analysing graphical representations.

Active participation in project-based learning (PBL) could develop students` knowledge and crucial learning skills across various disciplines. However, the implementation of PBL in the K-12 classroom is usually impeded by the stepby-step PBL cycle. Micro project-based learning (MPBL), which advocates and adheres to the learning principles and mechanisms of PBL with constraining the learning cycle in shorter periods, has been considered as a lightweight alternative to PBL process. As an exploration, this study explored the impact of MPBL on the conceptual understanding of sodium bicarbonate and the crucial learning skills in chemistry class at upper-secondary schools. The quasi-experimental research was implemented for 125 students, with an experimental group receiving MPBL teaching and a control group receiving conventional teaching. In the study, data including knowledge tests, crucial learning skills survey, and student interview were collected and analysed. The results indicated that MPBL was effective in the development of students' conceptual understanding and crucial learning skills (i.e., communication and collaboration, information integration, independent learning, and problem-solving). The study will inform the pedagogical innovation in chemistry education and teaching practices in chemistry class.

This study examines learners' issues in learning the preparation and qualitative analysis of salts topic from Malaysian chemistry teachers' perspectives. The researchers adopted a qualitative design and conducted semi-structured interviews with sixteen informants who have experience of teaching chemistry for more than five years. They were selected via purposive sampling. Document analysis was conducted to verify and supplement findings obtained from the interviews. The study reports five issues that emerged from the data analysis: negative perceptions, dealing with the number of information, imagining abstract concepts, mastery of basic knowledge and solving problems, and fulfilling the grading criteria. The study results confirm the unanimity of past studies on students' unfavorable views on these topics and their struggle in learning abstract concepts involved in the salts formation and analysis. Besides, students' issues are also significantly rooted in having weak prior knowledge of salts, affecting their ability to solve related problems. The study suggested that teachers must practice various approaches and strategies deemed effective in aiding learners in learning and be more selective in choosing approaches tailored to the individual students' issues in learning these topics in class.

This paper investigates profiles of high school students' mathematical problem-solving beliefs following the introduction of a new mathematics curriculum. In this quantitative study, 490 high school students responded to a mathematical problem-solving beliefs questionnaire to provide data about their beliefs related to mathematical problem solving. A cluster random sampling technique was used to select participants for the study. To accomplish the major goals of the study, a K-means clustering technique was conducted to analyze patterns that were discernible from their beliefs. In addition, a one-way ANOVA conducted to examine mean differences of their beliefs between clusters. Results revealed that, in general, students strongly believe that conceptual understanding is important in mathematics. In one of the clusters students hold strong beliefs about the usefulness of mathematics in their day-to-day lives while in another cluster it was strongly believed that effort was key for one to increase their mathematical ability. Results are important for students' confidence to solve mathematical problems and for implementation of a problem-solving approach in the new mathematics curriculum.

Most students practice abductive reasoning in solving mathematical problems that encourage creativity. This study analyses the process of making conjectures based on abductive reasoning. This study used a qualitative design and involved 106 undergraduate mathematics students enrolled in the mathematics course Introduction to Ring. We analyzed the students' conjectures on two mathematics problems. The study was completed by grouping the types of conjectures made by students and then investigating each student's explanation of each conjecture. The results suggested two types of conjectures practised by students, namely creativity in investigating the converse of the proposition and creativity in dividing into cases.

This study aims to describe metacognitive interventions for students who experience suspension of sense-making when solving integration problems based on the characteristics of students' thinking. This research applied a qualitative research approach and provided essay tests followed by task-based interviews to classify students who experience suspension of sense-making into two categories based on their thinking characteristics namely pure procedural and mixed conceptual-procedural. The metacognitive intervention in this study was carried out by giving metacognitive questions to the participants based on their thinking characteristics. The metacognitive questions given included comprehension, connection, strategy, and reflection questions. The results of this study indicated that mixed conceptual-procedural students were able to raise their sense-making earlier than pure procedural students. It can be concluded that pure procedural students need metacognitive intervention in the form of complete metacognitive intervention while mixed conceptual-procedural students only need partial metacognitive intervention.

Special Education teachers serve students who are struggling in mathematics and continue to fall further behind each year. Educators are faced with the challenge of selecting the most effective program for assessing and teaching mathematics that will promote mathematics achievement for students who are served in resource settings. The KeyMath-3 Essential Resources (KeyMath-3 ER) is a mathematics program aligned with National Council of Teachers of Mathematics standards, which incorporates research-based practices for teaching mathematics to students with disabilities. This study utilized the KeyMath-3 ER program as an intervention with five second-grade students with various disabilities receiving their mathematics instruction in a resource setting. The results demonstrated the potential usefulness of KeyMath-3 ER as an effective program for students in a resource setting. The one outlier was a student with autism who did not show significant progress demonstrating a need for further research to assess the effectiveness of using this program with like students.