Patterns of Metacognitive Behavior During Mathematics Problem-Solving in a Dynamic Geometry Environment

Currently, there is a strong political focus in Ethiopia on science, engineering, and mathematics. This to be efficient and practical strong mathematics knowledge, which is built through effective mathematics learning and problem solving, is momentous. For this in turn students’ beliefs about mathematics learning and problem solving play an important role; since students’ beliefs are vital forces in students’ mathematics learning and problem solving. Therefore, investigating students’ beliefs about mathematics learning and problem solving is essential in educational research. To address the problem quantitative approach using survey design was employed. The data was collected from four schools in West Arsi Zone using multistage sampling. The quantitative data obtained was analyzed using mean and independent samples t-test. Consequently, this study displayed that students’ beliefs about mathematics learning, and students’ beliefs about mathematics problem solving, were neutral; and there was statistically significant difference in students’ beliefs about mathematics learning and problem solving according to stream, and parents’ residence. However, even though the mean of students’ beliefs about mathematics learning and problem solving for male students was greater than that of female students, there was no statistically significant difference between male and female students in their beliefs about mathematics learning and problem solving.

The Analysis of Pre-service Teachers’ Beliefs about Mathematical Problem Solving

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.

The aimed of the research are: (1) to analyze the effect of Cooperative Learning of Type Think Pair Share Based on Mandailing Culture (TPSM) to mathematical problem solving ability (MPSA) , (2) to analyze the effect of mathematical initial ability (IMA) to MPSA of the students , (3) to analyze effect of interaction between TPSM and IMA to mathematical problem solving ability of the students. The research population was all students of middle secondary school (MSS) at Sidikalang which consisted of nine classes by taking classes VII - 1 and VII - 2 as the sample The instrument used to measure MPSA of the students is MPSA test. Type of the test is essays test . MPSA data is analyze through Analysis of Variance (ANOVA). Findings of the research are: (1) there is the effect of TPSM to MPSA of the students , (2) thre is the effect of IMA to to MPSA of the students , ( 3 ) there is significant effect of learning approach (TPSM, conventionl approach) and IMA (high, medium, low) to MPSA of the students. Another finding is th time available is not enough to conduct learning trough TPSM, but it can be overcome by good time management. Another one, learning through TPSM make the students become aware to their own culture.

Digital tools have a potential to change significantly the form of mathematical learning taking place in classrooms, with research pointing to various affordances in comparison with physical tools such as paper-and-pencil environments. Nevertheless, there is a scarcity of research that has examined in-depth the interrelated roles these two types of tools fulfil in mathematics learning. This issue of inter-related roles is important because, when digital tools are incorporated into classrooms, students usually also have notebooks and worksheets within which they carry out actions complementary to their use of digital tools. In this article, we focus on the use of dynamic geometry environments (DGEs) in conjecturing and proving, and, in particular, we examine the interplay between students’ paper-and-pencil activity and their use of a DGE during the producing and proving of a generalisation of a statement. We analyse a series of lessons involving secondary school students (aged 14–15, Grade 9) and show that, while DGE use supported the students in generalising a statement, they were initially unable to prove the generalisation while using the DGE, but subsequently succeeded through their paper-and-pencil activity. Our research illustrates the affordance of paper-and-pencil environments to support students in working on different representations, and thus highlights how the interplay between paper-and-pencil activity and DGE use can be important for the progress of conjecturing and proving. We also show the roles taken by the teacher in supporting the students’ work, and point to the need for further research into the back-and-forth use of digital and physical tools.

The ability of a student to obtain, analyze, measure and compare the many instances of a mathematical proposition in a dynamic geometry environment gives him/her opportunities to make conjectures and test a proposition. These roles for dynamic geometry software are widely acknowledged as having the potential to enrich the teaching of geometry. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by ‘‘what if’’ and ‘‘what if not’’ questions, has the potential to promote links between empirical and deductive reasoning. A classic set of problems in Euclidean plane geometry consists of finding the path of a point that is subject to given constraints. Its systematic study goes back to a lost work (in two books) of Apollonius of Perga, Plane loci (Botana and Valcarce 2003). Except for the simplest loci, such as lines, circles and perhaps the conics, this subject is usually avoided in most geometry texts. This is due to the common difficulties faced when mentally visualizing various objects with different movements. With the emergence of dynamic geometry software, the locus problems have attracted new interest from researchers. Using the locus generation features of these software one can easily obtain the locus of a point under some constraints. Although there are some minor differences, different dynamic geometry software, behave in a similar way when obtaining loci. In general, two objects must be selected: the first one, usually a point called the driving point or mover, is bound to a path, whereas the other, the locus point, must depend somehow on the first one. Since the element

PurposeThis paper aims to examine the self-study of three Koshi in analyzing lesson plans (LP) to prepare for the “final comments” for novice pre-service primary teachers in Australia. Analyzing LP is a critical step in preparing “final comments” by Koshi.Design/methodology/approachThrough self-study methodology, we reflected on our role as Koshi and examined our taken-for-granted assumptions. Drawing on Seino and Foster’s (2021) themes, we analyzed 12 LPs across a 2-year period where we acted as Koshi. Data analysis involved examining our feedback on the LP and analyzing our discussions and reflections to understand what we learned during the process.FindingsWhile some alignment with the Seino and Foster’s (2021) framework was found, context-specific sub-categories emerged. Our self-study highlighted three considerations: (1) the novelty of lesson study for novice pre-service teachers (PSTs) who were unfamiliar with structured problem-solving in mathematics, (2) the need to manage expectations we placed on ourselves as Koshi as well as our PSTs and (3) the need to temper the “critical” nature of our feedback for it to be constructive. We argue cultural nuances and our understanding of PSTs’ current mathematical concepts and pedagogies shaped these considerations.Research limitations/implicationsThe findings are based on a self-study of three Koshi, which inherently limits the generalizability of the results. The personal and contextual experiences of the participants may not fully capture the broader variability in Koshi’s practices or PSTs’ responses in other contexts. Additionally, the study focuses on feedback during a single lesson study cycle, offering a snapshot rather than a longitudinal perspective on the impact of Koshi’s comments on PSTs’ professional growth. Finally, the absence of direct input from PSTs regarding their perceptions of the feedback limits the ability to assess its effectiveness comprehensively. Addressing these limitations in future research could provide a more nuanced understanding of the role of Koshi and its adaptation to various educational settings.Practical implicationsBy reflecting on our own practices, we recognized the need to emphasize key aspects of structured problem-solving, including the selection of rich, open-ended tasks that align with curriculum goals and foster deep mathematical understanding. This approach emphasizes the importance of reflective practices for Koshi to continuously develop our learning stance (Watanabe, 2021).Social implicationsThis study highlights the complexities of being and becoming Koshi in a context where structured problem-solving and lesson study are not yet embedded in mainstream educational culture. As a result, this reflective practice not only enhances the Koshi’s capacity to deliver constructive comments but also strengthens the overall effectiveness of the lesson study process in diverse educational settings.Originality/valueThis paper contributes to further understanding of the role of the Koshi in non-Japanese contexts and wider educational setting particularly in initial teacher education.

Mathematical problem solving: What we know and where we are going

The research aims to analyze differences in students' mathematical problem solving abilities through the picture and picture learning model assisted by media and given regular learning, analyze differences in students' mathematics learning motivation through the picture and picture learning model assisted by video media and given regular learning, analyze the interaction between learning and ability students' initial abilities on students' mathematical problem solving abilities, and analyzing the interaction between learning and students' initial abilities on students' mathematics learning motivation at SDN 100312 Pargarutan Jae Tapanuli Selatan. The sample was determined as 62 people consisting of 2 (two) classes. The research design used in this research is an experimental design with a 2x3 factorial. The results of the research concluded that students' mathematical problem-solving abilities through the picture and picture learning model assisted by video media were better than if they were given regular learning at SD Negeri 100312 Pargarut.an Jae Tapanuli Selatan. Students' motivation to learn mathematics through the picture and picture learning model assisted by video media is better than being given regular learning at SD Negeri 100312 Pargarutan Jae Tapanuli Selatan. There is an interaction between learning and students' initial abilities on students' mathematical problem-solving abilities at SD Negeri 100312 Pargarutan Jae Tapanuli Selatan. There is an interaction between learning and students' initial abilities on students' mathematics learning motivation at SD Negeri 100312 Pargarutan Jae Tapanuli Selatan.


 
 
 
 This research is an experimental research, with the aim to find out: (1) how the students' mathematical problem solving ability before the application of the Power of Two method; (2) how is the students' mathematical problem solving ability after the application of the Power of Two method; (3) whether the Power of Two method is effective on students' mathematical problem solving abilities. The population in this study were all students of class XI MA Attarbiah Islamiah Kolaka in the 2020-2021 school year which consisted of 1 class with a total of 30 students. Sampling is done by using a census or total sampling. Sampling is done by using all the population. The analysis used is descriptive analysis and inferential analysis. Furthermore, the results are based on hypothesis testing: a statistical t-test (one sample t-test) was performed. It was obtained that tcount = -8.53 with ttable = -1.67. Accept H0 if -1.67≤ tcount≤ 1.67 because tcount = -8.53 is outside the H0 acceptance interval, then H0 is rejected and H1 is accepted. Thus there is a significant difference between students' mathematical problem solving ability scores before using the Power of Two method and after using the Power of Two method.
 
 
 

This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.

This study examines the effect of 5-year-old children’s mathematical problem solving ability and their self-esteem based on the Havruta method using math storybooks. The subjects of this study were 40 5-year-old students attending a kindergarten in the Incheon area: 20 students comprised the treatment group and 20 students comprised the control group. An instrument originally created by Ward (1993) but adapted by Hwang (1997) and later modified by Ryu (2003) was used to test the children’s mathematical problem solving abilities. A modified version (Kim, 1997) of an instrument developed by Harter and Pike (1984) was used to measure children’s self-esteem. Test results were analyzed using SPSS ver. 18.0 for Windows. The findings are as follows. First, the treatment group that had Havruta classes utilizing math story books was found to improve significantly more than the control group in their mathematical problem solving ability. Havruta classes had positive effects on children’s mathematical problem solving abilities. Second, there was no significant difference found between the two groups in terms of self-esteem when the children’s self-esteem was compared after Havruta classes that utilize math storybooks. It may not be possible to see immediate changes in children’s self-esteem because positive parent and teacher feedback had the strongest influence on 5-year-old children’s self-esteem, as opposed to self-learning. The results of this study provide meaningful basic data for Havruta classes that focus on questions and discussions through math story books to increase children’s mathematical problem solving abilities in the child education field. Keywords: math storybook, Havruta method, mathematical problem solving ability, self-esteem

Recent literature about educational reform has emphasized problem solving as a focus of school mathematics. In particular, problem solving has been characterized as a nonroutine activity in which students develop their own approaches to problems, select their own tools, and pursue investigations of their own design. Although there is abundant evidence that technology is used productively for mathematical problem solving, there are relatively few reports of spontaneous, learner-initiated use of technology as one of many options for approaching a problem. As with other problem-solving techniques, it is important for students to be able to (a) recognize when computer-based solution methods are appropriate, (b) select from a variety of software tools applicable to a given problem situation, and (c) combine a variety of appropriate technological and nontechnological tools to investigate different aspects of a problem. This article reports work with K–12 teachers to establish a learning environment that allows students to use technology spontaneously and appropriately for mathematical problem solving in a context in which problems were selected without regard to their suitability for solution with any particular technological or nontechnological tool. As participants gained experience judging when and how to take advantage of computer investigation methods during a 15-day institute, they showed an increasing number of learner-initiated applications of technology to mathematical investigation and problem solving, increasingly active involvement in those investigations, and increasingly effective computer use.

Students’ conceptions through the lens of a dynamic online geometry assessment platform

The article describes the development and analysis of a quantitative study investigating to what extent the potential of dynamic geometry environments (DGEs), in relation to mathematical reasoning competency, are utilized in lower secondary schools in Denmark (grades 7–9). The study entails a questionnaire, which was developed on the basis of an extensive review that uncovered four potentials of DGEs in relation to reasoning competency: feedback; dragging; measuring; tracing. 220 Danish lower-secondary mathematics teachers completed the questionnaire. Analysis indicates that the potentials of measuring and dragging are utilized to some degree, feedback to a lesser degree, while tracing is almost non-existent. Furthermore, there are signs that DGEs are used as a substitute for the paper-and-pencil environment to solve tasks that were originally designed for paper and pencil. Possible improvements of praxis are discussed and the integration of the results into praxis in further research is elaborated upon.