Visualizing Fractions: Enhancing Problem-Solving Performance Through Diagrammatic Reasoning in Elementary Mathematics

In mathematics education, representations are used in place of mathematical structures, ideas, or relationships to concretize, transform, and represent them. When students interact with these representations, they engage in various cognitive activities such as thinking, reasoning, understanding, remembering, problem-solving, attention, and decision-making, which are difficult to observe. Therefore, uncovering these cognitive activities is very significant for mathematics education. However, they are not easy to uncover as they cannot be directly observed. Eye tracking is an important approach that can be used to reveal cognitive activities that cannot be directly observed. This study investigated how middle school students examine representations by examining their eye movements. Eighty-five (40 girls and 45 boys) 7th-grade middle school students participated in the study. In the study, gaze durations, fixation count, and fixation duration on four different representation types: verbal representation, symbolic representation, number line representation, and counters representation were compared. The findings showed that students fixated more on the verbal representation and gazed at it for longer. However, fixation durations on the verbal representation were quite short compared to the other representations. In contrast, when examining the counters, there were fewer fixations and shorter gaze durations, but fixation durations were longer. Gazes on the number line and symbolic representation did not differ across all three variables. The findings indicated that gaze on verbal and non-verbal representations differed to some extent, but not entirely. Finally, the findings are discussed in the context of mathematical representation and eye-tracking literature.

This study was conducted to develop a local wisdom-based E-comic as a learning medium that aimed at improving the understanding of regional economic concepts among fifth-grade elementary school students. The study was motivated by the low level of students’ comprehension regarding regional economic material and the limited availability of engaging and contextual learning media. The ADDIE development model was employed in this research. The product was designed in the form of a digital comic that features interactive visual narratives, with content focused on regional economic themes integrated with the local wisdom of Tulungagung Regency—such as economic activities, leading local products, and the potential of the creative economy. The research was carried out with fifth-grade students from State Elementary School 2 of Pucung Kidul, Tulungagung as participants. Data were collected through observations, questionnaires, interviews, document analysis, and concept comprehension tests. The e-comic was validated by media and content experts, and it was found to be highly valid and appropriate for use. Positive responses were obtained from both teachers and students, indicating that the e-comic was considered feasible and practical for learning purposes. Small-scale and large-scale trials were conducted, and the results showed that students’ conceptual understanding was significantly improved through the use of the local wisdom-based e-comic, as supported by the analysis of a paired sample t-test. To sum up, the developed e-comic was proven to be valid, practical, and effective in enhancing the understanding of regional economic concepts among fifth-grade elementary students.

The contribution of social attitudes, interest in learning, and teacher competence toward the fifth-grade elementary students’ adaptive skills in semester II at cluster 3 is discussed in this paper. The research is ex-post-facto research. The population is 137 students. The data collection method is a questionnaire. Based on the research that has been conducted, it is concluded that 1) there is a contribution of social attitudes toward the fifth-grade elementary students’ adaptive skills in Semester II at cluster 3 in Tabanan with a contribution of 23.50% and an effective contribution of 8.52%. Hereafter, 2) there is a contribution of learning interest toward the fifth-grade elementary students’ adaptive skills in Semester II Cluster 3 in Tabanan with a contribution of 32.9% and an effective contribution of 15.60%. In addition, 3) there is a contribution of teacher competence toward the fifth-grade elementary students’ adaptive skills in semester II at Cluster 3 in Tabanan with a contribution of 37.50% and an effective contribution of 24.97%. Lastly, 4) there is a significant contribution of social attitudes, learning interest, and teacher competence simultaneously toward the fifth-grade elementary students’ adaptive skills with a contribution of 49.10%.

In this study, it is aimed to examine the problem solving and problem posing skills of seventh grade students with rational numbers in the context of multiple representations. When the literature is examined, although there is a study examining the relationship between multiple representation and rational numbers, there is no research examining the relationship between students' problem posing skills in addition to this relationship. It has been considered important to examine problem posing in the context of multiple representation, which helps students to look at the subject from a new perspective and allows them to understand it more deeply. The research was carried out with 25 students attending a public secondary school in Istanbul in the 2022-2023 academic year. The case study method, which is one of the interactive designs included in the qualitative research approach, was used in the research. The "Rational Numbers Multiple Representation Test" developed by the researchers was used as a data collection tool. In line with the data obtained, the representation preferences of the students and the interpretations of the success in these preferences were given using descriptive statistics. In addition, the problems posed by the students were evaluated using the problem posing assessment rubric introduced by Katrancı (2014). The reason for choosing this rubric is that it allows grading students' problem posing performance. According to the research findings, the most preferred representation types in problem posing questions were verbal (textual), table, number line, algebraic (numerical) and model representation types, respectively. The problems posed by the students were found to be 64 points out of 100 according to the problem posing evaluation rubric. When the transition between representations were evaluated, it was found that the transition from the verbal representation type to the table representation type was the most successful, and the transition from the number line to the model type was the least successful. The type of representation that the students liked to solve the most was the model type (40%), while the number line (36%) took the second place. The types of representations that the students liked least were expressed as algebraic and verbal. Considering the reasons for liking the model representation type more, the reasons such as being visual, comprehensible and easy to construct were stated. On the other hand, reasons such as being easy and practical are shown in the reasons for liking the number line representation type.

Sixty (35 girls) ninth graders were assessed on measures of algebraic reasoning and usage of visual and symbolic representations (with a prompt for visual use) to solve equations and inequalities. The study grouped visual representations into two categories: arithmetic-visual, which entailed the use of real-world objects to represent specific values of variables, and algebraic-visual, which involved formal representations like the number line and the coordinate plane. Symbolic representations, on the other hand, encompassed the use of standard algorithms to solve equations, such as changing the place of terms in an equation. The results reveal that the use of algebraic visuals, as opposed to arithmetic visuals, was associated with enhanced algebraic reasoning. Further, although the students initially relied on standard algorithms to explain equations and inequalities, they could produce accurate algebraic-visual representations when prompted. These findings suggest that students have multiple representations of equations and inequalities but only express visual representations when asked to do so. In keeping with the general relationship between visuospatial abilities and mathematics, self-generated algebraic-visual representations partially mediated the relation between overall mathematics achievement and algebraic reasoning.

This study aims to describe the critical thinking skills of eighth-grade students attending public junior high schools with an A accreditation rating in Makassar City. The population of the study consisted of 4,167 students, and a sample of 365 students was selected using a cluster random sampling technique. The sample was drawn from 13 public junior high schools in Makassar City. The research instrument used was a critical thinking skills test in the form of 24 multiple-choice questions. Based on the results of the descriptive statistical analysis, the critical thinking skills of eighth-grade students in A-accredited public junior high schools in Makassar City were found to be in the moderate category, with an average percentage of 65.35%. Specifically, critical thinking skills on the elementary clarification indicator were in the high category (70.50%), basic support was in the low category (53.63%), inference was in the moderate category (60.32%), advanced clarification was in the high category (68.56%), and strategy and tactics was also in the high category (73.77%).

The study explored the elementary school student’s perception of improvement in the English language. The research was quantitative. The main objective is to investigate what techniques students use to improve their English. Moreover, find the difference of opinion between males and females regarding English improvement. Moreover, find the differences of opinion among public and private elementary school students. The study's population was elementary school students in Lahore, selected by a convenient sampling method. The study sample was 300 male and female elementary school students from Lahore. The researcher uses a self-developed questionnaire for students. For data collection, a questionnaire was used. The questionnaire was adapted to explore secondary school students' perception of improvement in the English language. The tool consisted of a six-point Likert scale. The questionnaire was divided into two parts; the 1st part consisted of demographic variables (gender, types of schools, and age). Data were analyzed through SPSS by applying descriptive frequency distribution, one-way ANOVA, and t-test. Based on data analysis, it was found that a significant percentage of students engage in various self-directed English learning activities, such as reviewing their work (29.3%), seeking opportunities to use English outside of class (22.8%), and practicing pronunciation through reading aloud (26.7%). Additionally, many students use strategies such as looking up words in dictionaries (24.4%) and paying attention to errors when writing or speaking (26.4%) to improve their language skills. Moreover, there was no significant difference between secondary students categorized on a gender basis and also not found a significant difference between students of private and public schools. However, significance was found between different groups of ages

Incorporating visual representations, rather than strictly relying on symbolic representations, is a research-based strategy for supporting fraction learning. However, students must also make transitions between visual and symbolic fraction representations to apply the conceptual understanding they gained from visual representations to symbolic fraction computation. Virtual manipulatives (VMs) offer opportunities for supporting students in making these transitions, as many fraction VMs integrate visual and symbolic representations into one manipulative. Some VMs also dynamically link the representations, so learners can observe how changes to one representation impact the other. For these features to support students in transitioning among representations, teachers must orchestrate opportunities for students to use and reflect on their use of the features. This study examined how six fourth- and fifth-grade teachers orchestrated opportunities in lesson plans for students to make transitions among the visual and symbolic representations within and beyond fraction VMs. Results showed that teachers used two strategies for orchestrating these transitions: VM choice and direct teacher intervention. Implications of teachers’ uses of these strategies are discussed in terms of what kinds of transition opportunities were made available to students and what professional learning experiences could be needed to support teachers in orchestrating transitions.

<p>This study aimed to determine the effectiveness of peer tutoring on the mathematical problem-solving performance of Grade 10 students in a public school in Cebu City during S.Y. 2021-2022. Utilizing a single-group pre-test and post-test research design, the study assessed the impact of peer tutoring on selected students' problem-solving performance in mathematics. Participants were chosen based on their math performance, and peer tutoring was conducted in a one-on-one tutor-tutee format. Two research tools were used: a Learner’s Questionnaire to gather students’ profiles and perceptions of the peer tutoring sessions and a Learners’ Achievement Test to measure their mathematical problem-solving performance. Both descriptive and inferential statistics were employed to analyze the data. Descriptive statistics, including weighted means and Likert scales, were used to summarize the students’ perceptions, while a z-test and paired t-test were used to compare pre-test and post-test scores. The results showed a significant improvement in the students' mathematical problem-solving abilities after the peer tutoring intervention, with a significant increase in their post-test scores. Furthermore, students perceived peer tutoring as highly beneficial in enhancing their understanding and ability to solve math problems. Based on these findings, it is recommended that educators utilize peer tutoring as an instructional strategy to improve students' mathematical problem-solving skills. This approach can also foster greater student participation and interest in learning mathematics.</p><p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0785/a.php" alt="Hit counter" /></p>

This study tries to discuss the central issue of the instilling plural values to learners of each level of Islamic schools in Makassar. This study focuses on Islamic schools, since the teaching of religious subjects were also offered at public schools. In order to limiting the scope of the research, this study highlights on the extent of religious teaching and how the teaching are synchronized with of the plurality values in schools that consists of diverse ethnicity and culture in one hand, and diverse religious beliefs outside the schools, in another hand. The main problem statement is how the plurality values are instilled at Islamic schools in Makassar. The instilling process of plural values is tilled at Islamic schools in Makassar. The instilling process of plural values and culture is qualitative with socio-cultural pedagogical approach. The finding is that the instilling of the plurality values had been implemented but is not yet an adequate process, because it is still sporadic. Also, it is not yet concretely reflected in the curriculum.

Teaching English vocabulary to children in the classroom should begin as early as possible because childhood is a golden age when everything may be absorbed simply and rapidly. Language holds the belief that the earlier children learn a foreign language, the easier it is for them to master it, and that children may learn a language more quickly and with less difficulty than adults. This research aims to develop a learning design instructional material model of basic English Vocabulary based on interactive multimedia for elementary students and to increase knowledge and skills of language, especially for elementary school students. Development of learning model and instructional material design followed a research design and development model designed. This research is an extensive process in designing the students' needs and develop the product or course material of basic English vocabulary to increase the ability or skills of elementary students in learning language through interactive multimedia. The sample of this research is the students at one of the elementary schools in Makassar. The research showed a learning model based on the interactive multimedia syllabus and instructional material design of basic English Vocabulary. The researchers suggested that formative assessment be undertaken each time the materials are delivered to determine which components should be modified to increase the materials' instructional and motivating value. Furthermore, summative evaluation should be carried out in order to make a policy decision on whether or not to use instructional materials based on real evidence of their success.

This study investigated the effect of training in Prior Knowledge Activation, Concept Mapping, and Think Aloud, on students’ problem solving performance. The relative effectiveness of the training among high and low achievers was also examined. Participants were 45 students in Grades 4 and 5 in two private schools in Lebanon (25 students experimental group; 20 as control). The experimental group received 3 weeks training, in addition to the regular methods while the control received the regular methods only. Curriculum-based assessments (reading comprehension and math word problem solving) and Raven’s Colored Progressive Matrices, were used to measure problem solving performance. ANCOVA results showed a significant difference in experimental and control groups’ posttest scores, in favor of the experimental group. ANOVA results showed that the mean differences between pre and post test scores of low and high achieving students in the experimental group differed significantly only on math word problem solving measure, in favor of the low achievers. Results are discussed and implications and future recommendations are presented.

How do kindergarteners solve different single-digit addition problem formats? We administered problems that differed solely on the basis of two dimensions: response type (approximate or exact), and stimulus type (nonsymbolic, i.e., dots, or symbolic, i.e., Arabic numbers). We examined how performance differs across these dimensions, and which cognitive mechanism (mental model, transcoding, or phonological storage) underlies performance in each problem format with respect to working memory (WM) resources and mental number line representations. As expected, nonsymbolic problem formats were easier than symbolic ones. The visuospatial sketchpad was the primary predictor of nonsymbolic addition. Symbolic problem formats were harder because they either required the storage and manipulation of quantitative symbols phonologically or taxed more WM resources than their nonsymbolic counterparts. In symbolic addition, WM and mental number line results showed that when an approximate response was needed, children transcoded the information to the nonsymbolic code. When an exact response was needed, however, they phonologically stored numerical information in the symbolic code. Lastly, we found that more accurate symbolic mental number line representations were related to better performance in exact addition problem formats, not the approximate ones. This study extends our understanding of the cognitive processes underlying children's simple addition skills.

BackgroundThe generic concept of number line, which maps numbers to unidimensional space, is a fundamental concept in mathematics, but its cognitive origins are uncertain. Two defining criteria of the number line are that (i) there is a mapping of each individual number (or numerosity) under consideration onto a specific location on the line, and (ii) that the mapping defines a unidimensional space representing numbers with a metric — a distance function. It has been proposed that the number line is based on a spontaneous universal human intuition, rooted directly in brain evolution, that maps number magnitude to linear space with a metric. To date, no culture lacking this intuition has been documented.Methodology/Principal FindingsBy means of a number line task, we investigated the universality proposal with the Yupno of Papua New Guinea. Unschooled adults did exhibit a number-to-space mapping (criterion i) but, strikingly, despite having precise cardinal number concepts, they located numbers only on the endpoints, thus failing to use the extent of the line. The produced mapping was bi-categorical and metric-free, in violation of criterion ii. In contrast, Yupnos with scholastic experience used the extent of the segment according to known standards, but they did so not as evenly as western controls, exhibiting a bias towards the endpoints.Conclusions/SignificanceResults suggest that cardinal number concepts can exist independently from number line representations. They also suggest that the number line mapping, although ubiquitous in the modern world, is not universally spontaneous, but rather seems to be learned through — and continually reinforced by — specific cultural practices.

Learning of the mathematical number line has been hypothesized to be dependent on an inherent sense of approximate quantity. Children's number line placements are predicted to conform to the underlying properties of this system; specifically, placements are exaggerated for small numerals and compressed for larger ones. Alternative hypotheses are based on proportional reasoning; specifically, numerals are placed relative to set anchors such as end points on the line. Traditional testing of these alternatives involves fitting group medians to corresponding regression models which assumes homogenous residuals and thus does not capture useful information from between- and within-child variation in placements across the number line. To more fully assess differential predictions, we developed a novel set of hierarchical statistical models that enable the simultaneous estimation of mean levels of and variation in performance, as well as developmental transitions. Using these techniques we fitted the number line placements of 224 children longitudinally assessed from first to fifth grade, inclusive. The compression pattern was evident in mean performance in first grade, but was the best fit for only 20% of first graders when the full range of variation in the data are modeled. Most first graders' placements suggested use of end points, consistent with proportional reasoning. Developmental transition involved incorporation of a mid-point anchor, consistent with a modified proportional reasoning strategy. The methodology introduced here enables a more nuanced assessment of children's number line representation and learning than any previous approaches and indicates that developmental improvement largely results from midpoint segmentation of the line.