Non-routine Mathematical Problems Research Articles

KAJIAN MENDALAM TENTANG KEMAMPUAN PEMECAHAN MASALAH MATEMATIKA NON-RUTIN SISWA

This study aims to describe non-routine mathematics problem solving in relation to students' self-efficacy and math anxiety. We collect data from journal databases both national and international. The eligibility criteria are used with the keyword "non-routine mathematical problem solving" with intervals of the last 10 years. Based on the results of the synthesis, it was found that several important variables related to the topic were self-efficacy, reading comprehension, mathematic attitudes, think-aloud processes, mathematics anxiety, and truth-seeking. From these variables it was found that self-efficacy (self-confidence) and mathematics anxiety (anxiety mathematics) have an important role in solving non-routine mathematics problems

Ways of thinking siswa dalam menyelesaikan masalah pola bilangan non rutin: Suatu penelitian fenomenologi hermeneutik

This study was conducted with the aim of investigating and exploring students' ways of thinking (WoT) in solving non-routine number pattern problems. This study used a qualitative method with a hermeneutic phenomenological approach with grade 8 students at a junior high school in Banda Aceh. To achieve the research objectives, data collection was carried out using a written test instrument with a number pattern on a 4-digit palindrome, structured documentation, and clinical interviews. The results of the study show that students' WoT in solving non-routine number pattern problems is that there are four approaches used in solving non-routine problems, namely: first, determining the special case; second, determining the pattern; third, using a mathematical model; and fourth, using a similar problem. The subject of critical reflection uses the three WoTs above except for using similar problems. The subject of explicit reflection uses the three approaches above except for using a mathematical model. While Subjects who cannot solve the problem only use the strategy of identifying special cases. Another finding is that the subject of critical reflection tends to use different strategies from those given by the teacher, is unique, and gives reasons for algebraic forms. In contrast, explicit reflection subjects tend to be less flexible in using strategies and tend to use inductive or arithmetic reasons. To support students in their ability to pattern and think algebraically, teachers must accustom students to solving non-routine mathematical problems in various contexts of learning by using number patterns.

Individual Differences in Mathematical Problem-Solving Skills Among 3- to 5-Year-Old Preschoolers

AbstractThe aim of this study was to investigate individual differences in mathematical problem-solving among 3- to 5-year-old children (N = 328; n3-year-olds = 115, n4-year-olds = 167, n5-year-olds = 46). First, we examined the extent to which children in this age group were able to solve open and closed non-routine mathematical problems representing a variety of mathematical domains. Second, we investigated the extent to which underlying academic and cognitive skills (i.e., expressive and receptive language, visuospatial, and early numeracy skills) were associated with individual differences in mathematical problem-solving concurrently and longitudinally (i.e., one year later). The results showed that 4- to 5-year-olds were able to solve a variety of non-routine mathematical problems. However, though 3-year-olds were also able to solve a variety of problems, the mathematical problem-solving measure did not meet the reliability criteria, resulting in excluding 3-year-olds from further analyses. Expressive and receptive language, visuospatial, and early numeracy skills were associated with mathematical problem-solving concurrently among 4-year-olds. Among 5-year-olds, only visuospatial and early numeracy skills were associated with mathematical problem-solving. Furthermore, only prior mathematical problem-solving skills and early numeracy skills predicted mathematical problem-solving skills longitudinally. These findings indicate that preschoolers are able to solve open and closed non-routine mathematical problems representing a variety of mathematical domains. Additionally, individual differences may stem not only from differences in mathematical problem-solving skills but also from early numeracy.

Students’ mathematics-related belief systems and their strategies for solving non-routine mathematical problems

This study explored students’ mathematics-related beliefs and the relationship between the beliefs and their strategies for solving non-routine mathematical problems. The study was guided by Daskalogianni and Simpson’s 2001 belief systems categories and strategies for non-routine mathematical problems. The participants were 625 grade 11 students from five high schools in Tshwane North District, Gauteng province of South Africa. Data were collected using a mathematics beliefs questionnaire, a mathematics problem-solving test and interview. Quantitative and qualitative research techniques were used for data analysis. It was found that the students held all the three belief systems (utilitarian, systematic and exploratory) at different degrees of intensity and the belief systems and strategies for problem-solving had a weak positive linear relationship, and there were no statistically significant differences among mean scores of the students holding systematic, exploratory and utilitarian beliefs. They apply unsystematic guess, check and revise; systematic guess, check and revise; systematic listing; looking for patterns; consider a simple case; modelling; logical reasoning; no logical reasoning; trial-and-error and use a formula in solving non-routine mathematical problems. Furthermore, it was found that the systematic belief system could explain the students’ behaviour in problem-solving more than the exploratory and utilitarian belief systems.

Pelatihan Strategi Pemecahan Masalah Matematika Nonrutin Sebagai Persiapan KSN Matematika Di SMP Al Irsyad Purwokerto

The purpose of this service activity is to introduce, prepare and increase students' knowledge, especially students of mathematics achievement at SMP Al Irsyad Al Islamiyah Purwokerto about effective and efficient as a provision to take part in the National Science Competition in the field of mathematics or other similar math competitions. So far, the selected students are given independent guidance by the teachers at the school. However, due to scientific limitations and the teacher's busy schedule, the activity of building mathematics achievement is still not optimal. The method of implementing this activity consists of three stages starting with the first stage, namely the introduction of mathematical problem solving strategies, followed by the second and third stages in the form of material deepening activities. The post-test results showed an increase in the average post-test score of 22% from the average pre-test score. This shows an increase in the non-routine mathematical problem solving ability of the participants after being given training. The results of observing participant activities for three meetings also showed that the percentage of activity during the training increased at each meeting. This shows the increasing ability and motivation of the trainees in solving non-routine math problems.
 Keywords: Problem Solving, Non-routine Mathematics Problems, National Science Competition

Students' Skills In Solving Non-Routine Mathematical Problems

Mental skills are required for individuals to think quickly and make correct decisions. It can be said that mental skills, which play an important role in the acquisition of thinking and questioning skills, are only possible with creative, critical, and other high-level thinking skills. In this regard, the PISA exam, which reveals the mathematics achievement levels of countries, is important. Due to the absence of such an exam in the Turkish Republic of Northern Cyprus (TRNC), the descriptive scanning method, one of the quantitative research methods, was used in order to determine the effects on students' abilities in solving non-routine mathematical problems with the methods applied in the mathematics course. Based on this research conducted to determine the skills of 9th-grade high school students in solving non-routine mathematical problems, the mathematics achievement levels of the schools that accept students with special exams in the TRNC have been revealed.

Teaching Mathematics Based on Graphic Management in Problem Solving Topics

Problem solving is a very important skill in mathematics that students need to master. These skills require logical thinking, deep understanding, mastery of concepts and making connections with existing knowledge. This study aimed to determine the effectiveness of graphic management -based mathematics teaching in problem-solving topics. This study uses a qualitative research approach with a case study design involving one study participant. Participants of the study were randomly selected among year 3 students with a moderate level of Mathematics proficiency. To collect data, graphic management of “Four Corners and A Diamond” and semi -structured interview sessions were conducted. Data collection from graphics management was analyzed descriptively and these interviews were analyzed inductively to obtain specific themes based on the transcripts. Findings of the study show that with this graphic management, students can organize information from questions in the form of appropriate graphics and facilitate their understanding. The implication of this study is that students can solve mathematical problems through the arrangement of information correctly based on the graphic management of "Four Corners and A Diamond". This study can also encourage teachers to use graphic management in helping students to solve non-routine mathematics problems more conceptually.

Creative Mathematical Reasoning: Does Need for Cognition Matter?

A large portion of mathematics education centers heavily around imitative reasoning and rote learning, raising concerns about students’ lack of deeper and conceptual understanding of mathematics. To address these concerns, there has been a growing focus on students learning and teachers teaching methods that aim to enhance conceptual understanding and problem-solving skills. One suggestion is allowing students to construct their own solution methods using creative mathematical reasoning (CMR), a method that in previous studies has been contrasted against algorithmic reasoning (AR) with positive effects on test tasks. Although previous studies have evaluated the effects of CMR, they have ignored if and to what extent intrinsic cognitive motivation play a role. This study investigated the effects of intrinsic cognitive motivation to engage in cognitive strenuous mathematical tasks, operationalized through Need for Cognition (NFC), and working memory capacity (WMC). Two independent groups, consisting of upper secondary students (N = 137, mean age 17.13, SD = 0.62, 63 boys and 74 girls), practiced non-routine mathematical problem solving with CMR and AR tasks and were tested 1 week later. An initial t-test confirmed that the CMR group outperformed the AR group. Structural equation modeling revealed that NFC was a significant predictor of math performance for the CMR group but not for the AR group. The results also showed that WMC was a strong predictor of math performance independent of group. These results are discussed in terms of allowing for time and opportunities for struggle with constructing own solution methods using CMR, thereby enhancing students conceptual understanding.

Metacognition and Performance in Mathematical Problem- Solving Among Bachelor of Elementary Education (BEEd) Pre-service Teachers

This study assessed the metacognition and performance in mathematical problem- solving of Bachelor of Elementary Education (BEEd) pre- service teachers. A survey questionnaire was used to explore the knowledge and regulation of cognition of the respondents based on the model of Schraw and Dennison (1994). Also, a test was administered to determine the performance of pre- service teachers in mathematical problem- solving. The study involved 151 respondents who already finished their content area courses that included strategies in mathematical problem- solving. Descriptive-correlation design was utilized in the study as it ascertained the association between the metacognition and performance of the pre- service teachers in mathematical problem- solving. The result of the study shows that pre- service teachers have gained the average level of metacognition and have satisfactory performance in mathematical problem- solving. Furthermore, pre- service teachers with high level of metacognition also have high level of performance while those with low level of metacognition also have low level of performance in mathematical problem- solving. The result of the study is an input to further explore metacognition as this is significant not only in the academic activities but also in the complex roles of daily lives. References Akyol, Z., & Garrison, D. R. (2011). Assessing metacognition in an online community of inquiry. Internet and Higher Education, 14(3), 183–190. https://doi.org/10.1016/ j.iheduc.2011.01.005 Allen, B. A., & Armour-Thomas, E. (1993). Construct validation of metacognition. The Journal of Psychology, 127, 203. Barana, A.; Boetti, G.; Marchisio, M.(2022). Self-Assessment in the Development of Mathematical Problem-Solving Skills. Educ. Sci. 2022, 12, 81. https://doi.org/ 10.3390/educsci12020081 Clark, D. (1999). Learning Domains or Bloom’s Taxonomy. 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Development of the Mathematical Problem-Solving ability using applied cooperative learning and Polya's Problem-Solving process for grade 9 students. Journal of Education and Learning, 11 (3), 40-46.

An Investigation Into Problem Solving Skills of Gifted Students' via Multi-Facet Rasch Analysis

Problem solving, which is an important skill for mathematics, has long been recognized as the focus of the curriculum. This study focuses on both the systematic of problem solving and the use of strategies for different solutions, so aim of this study was to examine gifted middle school students' mathematical problem-solving skills via multi-facet Rasch analysis. For this purpose, data were collected via 12 non-routine mathematical open-ended problems from a total of 276 students who attend Science and Art Centers. The data were scored by two independent researchers according to the criteria in the range of 1-5 points per problem. In the study, it was seen that Grader 1 behaved more tolerant than Grader 2 in terms of evaluation for all problems except 6th problem. In addition, the most difficult and the easiest problems were differed between graders. From the findings of the study, it can be said that even if the evaluation criteria were clearly defined the difference between graders was not resolved. It can be suggested that to take necessary precautions for making vital decisions in all grade levels via teacher made tests and to use more than one grader instead of just one grader for evaluation.

DIGITAL TOOLS AND PAPER-AND-PENCIL IN SOLVING-AND-EXPRESSING: HOW TECHNOLOGY EXPANDS A STUDENT’S CONCEPTUAL MODEL OF A COVARIATION PROBLEM

This study aims at understanding the role of the tools chosen throughout the processes of solving a non-routine mathematical problem and communicating its solution. In assuming that problem-solving is a synchronous activity of mathematization and expression of mathematical thinking we take our proposed Mathematical Problem Solving with Technology (MPST) model to analyze the processes of solving-and-expressing-problems. Resorting to qualitative methods for data collection and analysis, we report on the case of an 8th grader working on a covariation problem to examine the role that paper-and-pencil and digital tools play in the development of a conceptual model of the situation. We found that the resources used throughout the solving-and-expressing activity influenced the depth of the conceptual model developed, within a process of progressive mathematization. Whereas paper-and-pencil led to the emergence of a conceptual model based on exploring particular cases, the digital transformation of the solution was triggered by the process of communicating its mathematical justification and expanded the previous model. Moreover, the complexity of this activity is evidenced by its multiple sequences of processes. Finally, the integration process seems crucial as the concomitant use of technological and mathematical resources precedes major advancements in the expansion of the conceptual model.

Assessing impact of a Teacher professional development program on student problem-solving performance

In this paper we report on the impact of a year-long teacher professional development (PD) workshop on student problem-solving skills. The PD workshop was held in Chile and promoted the use of collaborative problem-solving activities in the classroom, through monthly sessions during the school year. Participant teachers met with a monitor to solve a new problem collaboratively, plan a problem-solving activity for their next lesson, and reflect on their activities from their previous lesson. The study documents the performance of two groups of students when solving non-routine mathematics problems in two similar and non-equal tests consisting of three problems each, before and after the year of application of the workshop. The first group of students was the experimental group with collaborative problem-solving activities, as proposed in the PD workshop. The second group was the control group, composed of students whose teachers followed traditional teaching. In the first part of the paper we present the results of pre- and post-tests in which the students in the experimental group improved their problem-solving performance in a significant way compared with the control group. In the second part, we present a sample of representation strategy used by successful students in both groups and we discuss, in general, student strategies in solving test problems.

Classification of Students’ Non-Routine Problem Solving Skills

Non-routine mathematical problems are complex problems that require a level of creativity and uniqueness in solving them. Classification students’ characteristics of solving non-routine problems have not been many researchers who provide an overview. This study aims to classify the characteristics of students’ problem solving skills. This type of research is a qualitative study involving five students of class VIII at SMP Negeri 1 Kintamani. Five students were selected through a purposive sampling technique on the condition that students of class VIII have high, medium and low mathematical skills. Data regarding non-routine mathematical problem solving students are collected through problem solving tests and supported by interview tests to obtain a picture of students’ ability to understand the questions along with an understanding of the answers that have been done. Data were analyzed with data reduction, data presentation, and data verification steps. Data that has been verified are analyzed using the constant comparison method. Characteristics of non-routine mathematical problem solving students can be classified into 4 levels, namely Level 1 (students’ understanding of problems along with mathematical problem solving is very lacking), Level 2 (Students’ understanding of problems less and students’ problem solving is lacking but able to plan problem solving), Level 3 (students ‘understanding of problems takes a long time and students’ problem solving is good), and Level 4 (students ‘understanding of problems is good and students’ problem solving is very good). Classification is obtained based on reference to problem solving according to Polya.

The mathematical problem-solving ability of junior high school students based on their mathematical resilience

This research aims to describe the mathematical problem-solving ability of grade VIII students at State Junior High School in Sragen Region and found the patterns of mathematical problem-solving abilities that were analysed based on students’ mathematical resilience. The method used in this research was descriptive quantitative research with 153 students as the samples. Data were collected using a questionnaire and a written test. The results of the research give an overview that mathematical problem-solving ability of students has high resilience level of mathematics which can deal with and be able to overcome obstacles and negative situations related to the problem-solving process because they can successfully coach themselves. Students who are given non-routine mathematical problem show differences in their mathematics problem-solving based on levels of mathematical resilience.

Evaluation of Preservice Teachers’ Skills in Solving Non-Routine Mathematical Problems through Various Strategies

This study aims to investigate how the different solutions, problem-solving strategies, and the number of strategies used by the preservice secondary school mathematics teachers on non-routine mathematical problems change according to grade levels. This study was conducted with 152 preservice teachers studying at the department of elementary mathematics education of a state university located in the Western Black Sea region in Turkey. The purposeful sampling method was used to select the participants. A 12-question test with non-routine problems was used as the data collection tool. Content analysis, one of the qualitative data analysis techniques, was used for analyzing the data. The results indicated that the preservice teachers at each grade level generally proposed single solution and more than half of them had difficulties in finding solutions to the problems. It has been observed that the most used strategies of the preservice teachers are the strategies of reasoning, prediction-control, finding a correlation, formation-testing, systematic listing, drawing, and forming equations in all grade levels.

Implementation of Snowball Throwing Type of Cooperative Learning in Calculus Course

Our students experience difficulties in working on non-routine mathematical problems. They cannot see the general through special things or instead see things that are special through common things when dealing with mathematical concepts and examples. They cannot see the essential features of the way or recognize it when presented in various forms. We needed to improved their learning motivation and thinking creativity through the appropriate learning form. For that purpose, the action research was carried out by applying the snowball throwing type of cooperative learning in two cycles by involving as many as 70 participants in our calculus course. We implemented the teaching techniques for finding a diffractive solution in period 1 and the methods of solving differential equations in period 2. The learning implementation of period two was the improvisation of the cycle one’ scenario by adding the learning resources, the Internet. The increase in learning motivation and thinking creativity had been able to meet the predetermined targets and concluded that the snowball throwing type of cooperative learning used in calculus lectures is useful.

Analyzing students’ character values in non-routine mathematics problems

This research is a descriptive qualitative research that aims to determine the character value of senior high school students which includes the value of honesty, hard work, creative, and curiosity in solving non-routine mathematics problems. Methods of data collection are done by written test consisting of three non-routine mathematics problems about systems of linear equations with three variables, observation of student character values during the test, then continued with the interview that aims to ensure the correctness of test sheet analysis results and observation analysis results. Based on the results of the analysis it is known that when solving non-routine problems, students show the character value of honesty, hard work, creative, and curiosity, with curiosity character values as the dominant character and honesty values as rarest characters.

On values in non-routine mathematical problems for senior high school students

This study aims to describe values in non-routine problems for senior high school students. This research used the qualitative method. The subjects of this study were the senior high school students. The data collection techniques used interviews and observations. An interview was used to get information about curriculum and the abilities of the students. The observation was used to find out the mathematical values contained in the non-routine problems. The results of the study show that in non-routine problems there are values that exist in mathematical values. These values are rationalism, objectivity, control, progress, openness, and mystery.

فاعلیة برنامج قائم على تکوین وحل المشکلات الریاضیة لتنمیة قدرة الطلبة مرتفعی التحصیل على تکوین المشکلة فی ضوء قدرتهم الریاضیة

هدفت الدراسة إلى الکشف عن مدى فاعلية برنامج قائم على تکوين المشکلات الرياضية وحلها في تنمية القدرة على تکوين المشکلات الرياضية لدى الطلبة مرتفعي التحصيل بالصف العاشر الأساسي في ضوء قدرتهم الرياضية، ولتحقيق هدف الدراسة تم تصميم برنامج رکز على تدريب الطلبة مرتفعي التحصيل على توظيف استراتيجيات حل المشکلات الرياضية، واستراتيجيات تکوين المشکلات الرياضية على مجموعة من الأنشطة والمشکلات الرياضية غير الروتينية، وتم تصميم اختبار لقياس الأداء القبلي والبعدي للطلبة في تکوين المشکلات الرياضية. کما تم تطبيق اختبار القدرة الرياضية (TOMA-3) لقياس القدرة الرياضية للطلبة. وتکونت عينة الدراسة من 64 طالبا وطالبة من الطلبة مرتفعي التحصيل بالصف العاشر الأساسي تم اختيارهم بطريقة عشوائية، وتم توزيعهم في مجموعتين ضابطة وتجريبية. وللإجابة على أسئلة الدراسة فقد تم استخدام اختبار تحليل التباين المشترک (ANCOVA)، وقد کشفت نتائج الدراسة عن: - وجود فرق دال إحصائيا (α = 05,0) بين متوسط درجات طلبة المجموعة التجريبية، ومتوسط درجات طلبة المجموعة الضابطة في التطبيق البعدي لاختبار تکوين المشکلات الرياضية لصالح المجموعة التجريبية. وبلغت قيمة مربع آيتا(η2) 0,244 في تنمية القدرة على تکوين المشکلات الرياضية. - عدم وجود فرق دال إحصائيا (α = 05,0) بين متوسط درجات طلبة المجموعة التجريبية ومتوسط درجات طلبة المجموعة الضابطة في التطبيق البعدي لاختبار تکوين المشکلات الرياضية، يعزى إلى التفاعل بين المجموعة (تجريبية، ضابطة) والقدرة الرياضية (مرتفع، منخفض) لدى طلبة الصف العاشر الأساسي. وفي ضوء النتائج السابقة تمت التوصية بالاستفادة من البرنامج المقترح في تنمية وإثراء قدرة الطلبة مرتفعي التحصيل على تکوين المشکلات الرياضية، وتضمين المناهج الرياضية المشکلات الرياضية المناسبة للطلبة مرتفعي التحصيل وتدريبهم على استراتيجيات متنوعة لحل وتکوين المشکلات الرياضية. This study aimed to reveal the effectiveness of a proposed enrichment program based on problem posing and problem-solving in the development of the ability of the mathematical problem-posing. To achieve the study aim, an enrichment training program was designed to focus on a set of mathematical ideas and skills in mathematics using problem-posing and problem-solving strategies to pose and solve non-routine mathematical problems, for students of high achievement to develop their problem-posing. The researcher also designed a test to measure the mathematical problem-posing. In addition, the Test of Mathematical Abilities (TOMA-3) was used to measure the students mathematical abilities before the experiment. The sample of the study consisted of 64 grade ten high achievers students, who were selected randomly from South Batinah governorate. The students were divided into control and experimental groups. For testing the study hypothesis data was analyzed by Analysis of Covariance (ANCOVA). The finding showed that: - There is a statistically significant difference at the level of significance (α= 0.05) between the mean scores of students in the experimental group and the mean scores of students in the control group in the post problem-posing test in favor of the experimental group. Eta square was calculated (η2 = 0.244) for problem posing. - There is no statistically significant difference at the level of significance(α= 0.05) between the mean scores of students in the experimental group and the mean scores of students in the control group in the problem-posing test due to the interaction between the group (control- experimental) and the mathematical ability (low-high). In light of the study findings, the study recommends to enhance the abilities of mathematical problem-posing of students of high achievement by including the contents of the proposed program as part of the mathematics curricula.

The Effectiveness of a Proposed Enrichment Program Based on Problem-Solving and Problem Posing in Developing the Ability of Mathematical Reasoning and Sense-Making among Grade Ten High Achievers

This study aimed to reveal the effectiveness of a proposed enrichment program based on problem-solving and problem posing in developing the ability of the mathematical reasoning and sense making among students of high achievement. To achieve this goal, an enrichment training program was designed to focus on a set of mathematical ideas and skills in mathematics using problem-solving and problem- posing strategies to solve and pose non-routine mathematical problems for students of high achievement to develop their mathematical reasoning and sense making ability. The researcher designed a test to measure the mathematical reasoning and sense making. In addition, the Test of Mathematical Abilities (TOMA-3) was used to measure the students' mathematical abilities before the experiment. The sample of the study consisted of 63 grade ten high achievers randomly selected from South Batinah governorate, which was divided into control and experimental groups. For testing the study hypothesis, data was analyzed by analysis of covariance (ANCOVA) in which eta squared was obtained. The results showed that the proposed program was effective and the mathematical reasoning and sense making skills were highly achieved.