Non-routine Problems Research Articles

Pictorial Model: The Secret of Singapore's Pupils' Excellence in Mathematics

The research aimed to introduce mathematics education researchers, mathematics teachers and developers of mathematics curricula in Egypt and the Arab world with the secret of excellence of Singapore students in mathematics and how they come at the top order in the international study of the trends of science and mathematics education every four years. This secret lies in a number of communities, school, family and classroom models, the most important of which is the pictorial model, which offers magic solutions to any mathematical problem. Problem solving is the center or focus of mathematics education as it involves acquiring and applying mathematical concepts and skills in a wide range of situations involving real-world problems, open-ended and non-routine problems. The Singaporean model is used to solve verbal problems where the student identifies the main information in the problem in a pictorial model that includes units in the whole figure of rectangles and the information and the unknown is indicated in the problem to be solved on the pictorial figure and through the figure the process or calculations to be used are clarified and the problem is resolved.

Challenges of Mathematics Learning with Heuristic Strategies

Problem solving is an essential aspect of students' mathematical activities. This ability could practice by using heuristic strategies in learning. Besides, these are assured to be able to promote metacognitive skills. In the implementation, teachers faced several challenges. This research aims to describe the challenges of teachers in implementing learning with heuristic strategies. This research is a descriptive qualitative. Participants of this research were 12 junior high school mathematics teachers from 12 high schools in the Special Region of Yogyakarta and Central Java. Data collection was taken by questionnaire and added with documentation. Data were analyzed using the Miles and Huberman stage-data reduction, data display, and drawing conclusion/verification. The data then validated using triangulation technique. The results revealed that some of teacher has already implement heuristic strategy in the learning process, but unable to define the heuristic strategy correctly. In its implementation, teachers experience several obstacles. These obstacles were found in providing non-routine problems to students, solving problems by students, and in discussions conducted to solve problems.

Self-efficacy in non-routine problem solving in STEM education

Self-efficacy in non-routine problem solving in STEM education

Gifted eighth, ninth, tenth and eleventh graders’ strategic flexibility in non-routine problem solving

This descriptive survey study aimed to examine gifted students’ success and strategic flexibility in non-routine problem solving. The study group consisted of 50 gifted students. A test consisting of seven problems was used to collect data. Answers were assessed in terms of correctness and strategy use. Flexibility was evaluated based on the use of appropriate strategies, intra-task flexibility, and inter-task flexibility. Descriptive statistics, Pearson’s correlation coefficient and one-way ANOVA were used for analysis of the data. Students displayed an above-average performance in solving problems and strategy use. Students’ intra-task flexibility was low, whereas their inter-task flexibility was high. There was a high correlation between flexibility and success. While success differed according to grade level, there was no significant difference between grades in terms of flexibility. Results are discussed in terms of their implications related to education, and recommendations aimed at mathematics education environments and curricula are made.

Students’ proportional reasoning in solving non-routine problems based on mathematical disposition

Proportional reasoning is the important process in solving social arithmetic problems. A students’ successes in solving mathematical problem is also determined by affective abilities, one of which is a mathematical disposition. This study aims to describe the students’ proportional reasoning in solving non-routine problems based on mathematical disposition. This research is a qualitative study by conducting an in-depth analysis of the proportional reasoning ability of students in students with high, medium, and low mathematical dispositions. The research instrument used was a problem solving ability test, a mathematical disposition scale, and an interview guide. The result shows that in solving non-routine problems, students with high mathematical disposition have mathematical reasoning abilities at level five (Functional and Scalar Relationships), students with moderate mathematical disposition at level four (Accomodating Covariance and Invariance), and students with low mathematical disposition at level five one (Qualitative). Thus it can be concluded that the higher the mathematical disposition of students, the proportional reasoning in solving non-routine problems will be better.

ON CREATIVITY THROUGH MATHEMATIZATION IN SOLVING NON-ROUTINE PROBLEMS

This study aimed to describe and compare the students’ fluency, flexibility, and originality in solving non-routine problems in the Palembang context. They were depicted from the student’s fluency, flexibility, and originality of solving the horizontal and vertical mathematization forms. This qualitative study employed. The subjects of this study were 30 students of grade nine of junior high schools in Palembang. The instruments used were tests and interviews. The tests were employed to investigate the written horizontal and vertical mathematizations forms. Meanwhile, the interviews were to explore the students’ ideas with inadequately detailed answers. Then, the test and interview data were reduced and grouped based on the indicators of creativity. The reduced data were presented in a descriptive form for conclusions. The results of the data analysis showed that the high-ability students were the most fluent and flexible in solving the problems. Still, the provided solutions were less original and tended to use formal mathematics in the forms of formulas, symbols, and operations. Meanwhile, the moderate-ability students tended to start to solve problems by simplifying them, then presenting them in visual images. The answer sheets of the moderate-ability students revealed their fluency in understanding the problems and solutions, flexibility, and originality of thinking. This study obtained different results from the low-ability students who tended to have difficulties understanding the problems and made many errors in solving them. Such a condition showed their inability to write the known data and relate the data to other facts they had already learned. As a result, their answers did not represent fluency, flexibility, and originality.

Metacognitive Skills and Problem-Solving

The purpose of this study is to investigate the metacognitive strategies that middle school students used in the process of solving problems individually. The study group consisted of 37 middle school students in the eighth grade. The students were asked one non-routine word problem, and their written answers were collected. After solving the problem, the students filled out a self-monitoring questionnaire that requested them to reflect retrospectively on the metacognitive strategies they employed for the problem. In order to obtain detailed data, semi-structured interviews were conducted with six students, three of whom gave correct responses and the other three gave wrong answers to the problem. The data was analyzed through the model of metacognitive activity during problem-solving. The results showed that metacognitive skills have a significant effect on students’ problem-solving success. The study found that students with high metacognitive skills tend to solve the problem correctly by using appropriate strategies, mathematical notations and logical reasons. The results also revealed that students with poor metacognitive skills have difficulties in understanding the problem, selecting appropriate strategies, and finding correct answer. Although the students thought the opposite, it was concluded that their habits of checking, detecting, and correcting mistakes in their solutions were poor.

Use of Open-Ended Student Worksheet to Improve Mathematics Communication Skills in Algebra

Communication is the foundation of every learning activity. Mathematical communication is a skill to represent ideas in mathematical language so that it is understood by others who are used in problem-solving activities. This study aims to improve mathematical communication skills by using open-ended student worksheets in algebra. The research sample involved 6 junior high schools in Purworejo Regency, Indonesia which is divided into 2 groups, the experimental group consists of 91 students and the control group consists of 96 students. The study design used a randomized static group comparison design. The research instrument used 5 item essay tests that contained non-routine problems. The data analysis technique used an independent sample T-test. The results showed that the communication skills of students who were subjected to learning using open-ended student worksheets were better than the communication skills of students who were subjected to learning using student worksheets that are commonly used by teachers. These results provide an important impact that teachers should always integrate the presentation of non-routine problems in the form of worksheets as an integral part of mathematics learning.

Prospective mathematics teachers’ thinking styles and problem-solving skills

The aim of this study is to examine problem-solving skills and thinking styles of prospective mathematics teachers and investigate whether the thinking styles of them are related to their mathematical problem-solving skills and the problem-solving strategies preferred by them differ depending on their thinking styles. The data of the study were collected from 32 prospective middle school mathematics teachers through five non-routine problems and “Thinking Styles Inventory”. Prospective mathematics teachers’ solutions to non-routine problems were analyzed qualitatively. For quantitative data, descriptive statistics, correlation and chi-square were used. The results showed that majority of the prospective teachers have difficulties in using problem-solving strategies appropriately and solving non-routine problems. It was found that prospective teachers mostly preferred legislative, hierarchical and liberal thinking styles respectively whereas they displayed least preferences for oligarchic, conservative and monarchic thinking styles respectively. The results revealed that problem-solving skills were significantly, negatively and moderately related to only monarchic thinking style. High level of problem-solving skills were found to be related to liberal and internal thinking styles significantly, negatively and moderately whereas there was no significant relationship between low level of problem-solving skills and thinking styles. It was also found that there were no significant differences between the thinking styles of prospective mathematics teachers and their problem-solving strategies. This finding was supported by qualitative data. More comprehensive results can be obtained through observing and monitoring prospective teachers for long period.

Adaptive Reasoning Based on Microsoft Mathematics

The demands of professional teachers to design technology-assisted learning that can facilitate all students to develop their potential. the fact that the quality of students' adaptive reasoning in solving math problems is still low which has an impact on other mathematical competencies. The purpose of this study was to determine the differences in the improvement of adaptive reasoning abilities between students who have implemented Search Solving Create and Share learning assisted with Microsoft Mathematics and conventional learning. The method used is quasi-experimental with the non-equivalent pre-test and post-test control group design. Through the Simple Random Sampling technique, two classes were selected, namely class X1 and X2 one of the SMA in Bandung Indonesia. Test instrument: story questions about adaptive reasoning abilities. Analysis using independent T-test data. Result: There is a difference in the improvement in adaptive reasoning abilities between students who using search-solve-create- share learning assisted by Microsoft mathematics compared to conventional learning. Students' difficulties in solving story problems with adaptive reasoning abilities can be minimized by getting used to solving non-routine story problems. Learning search-solve-create-share assisted by Microsoft mathematics accustoms students to convey assumptions, provide reasons for answers, draw conclusions from statements, and students can find patterns of various mathematical problems through multi-interactive media.

Exploration of self-regulated learning: Mathematical problem solving

Self-regulated learning is needed to regulate and direct itself, adjust, and control self-learning. This study aims to get an overview of students’ self-regulated learning in solving junior high school mathematics problems. This research is a qualitative descriptive study with three research subjects. The data collection techniques used math ability tests, problem-solving tests, self-regulated learning questionnaires, and interview guidelines. The results showed the profile of self-regulated learning raised by junior high school students when solving math problems, especially problems related to SPLDV, namely: 1) Planning, thinking and activation stages: (understanding the problem is expressed in using one’s own language or the language of the problem and there is an estimate of SPLDV completion through arithmetic procedures, logic and elimination substitution methods; 2) Monitoring Stage: monitoring is carried out on the correctness of the variables with or without involving the whole conversation; 3) Control stage: answer checking with reverse technique; and 4) Reaction and Reflection Stage: Explore the difficulties faced related to routine and non-routine problems.

طبیعة حل المشکلات الریاضیة اللفظیة فی مناهج الریاضیات للمرحلة الابتدائیة بالمملکة العربیة السعودیة وسنغافورة وبریطانیا والولایات المتحدة الأمریکیة (دراسة مقارنة)

هدفت الدراسة إلى التعرف على طبيعة حل المشکلات الرياضية اللفظية في مناهج الرياضيات للمرحلة الابتدائية بالمملکة العربية السعودية ومقارنتها بمناهج رياضيات سنغافورة وبريطانيا والولايات المتحدة الأمريکية، وذلک للتعرف على نوع المشکلات الرياضية وطريقة صياغتها وأماکن تمرکزها بمنهج رياضيات الصف الأول ابتدائي، والتعرف على عدد خطوات حلها واستراتيجياتها والمجالات الرياضية التي تکررت فيها، وتم استخدام المنهج الوصفي من خلال أسلوب تحليل المحتوى، وأظهرت النتائج تفوق کل من دولة سنغافورة وبريطانيا على المملکة العربية السعودية، و أظهرت النتائج بالنسبة لمناهج المملکة العربية السعودية وجود مشکلات رياضية روتينية فقط تم صياغتها بشکل لفظي في معظمها، وتباينت في أماکن تمرکزها بين وجودها بشکل مدمج بالدروس أو استقلالها في درس أو استخدامها کتقويم، واعتمادها على استراتيجيات حل مناسبة للصف الأول الابتدائي، إلا أنه توجد مشکلات تم استخدام عدة خطوات بحلها بدلا من خطوة واحدة أو خطوتين، کما أظهرت النتائج الترکيز في المشکلات الرياضية اللفظية على مجال الأعداد بنسبة تفوق مجال الهندسة، في عدم تواجد لمجال الإحصاء والبيانات. وتوصي الدراسة في ضوء هذه النتائج بالتنويع في المشکلات الرياضية في کتب الرياضيات بالصف الأول ابتدائي من خلال تضمين مشاکل غير روتينية لإثارة التفکير لدى التلاميذ، وصياغتها بطريقة مناسبة، والبعد عن الحشو، والترکيز على الخطوة الواحدة في حل المشکلات الرياضية، وتضمين مجال الإحصاء والبيانات کمشکلات رياضية بسيطة، کما تقترح الدراسة، تحليل محتوى کتب الرياضيات للتعرف على طبيعة المشکلات الرياضية بالمراحل الابتدائية الأخرى ومقارنتها مع الدول الأخرى. The study aims to expose the nature of solving verbal mathematical problems in mathematics curricula for the elementary stage in the Kingdom of Saudi Arabia and compare them with the mathematics curricula in Singapore, Britain and the United States of America; so as to determine the type of mathematical problems, their formulation methods and their scopes in the first-grade elementary mathematics curriculum; and to identify the number of steps needed to solve them, their strategies, and the mathematical areas in which they were repeated. The descriptive approach was applied through the content analysis method. The results reflected the superiority of both Singapore and Britain over the Kingdom of Saudi Arabia. The results also reflected that for the curricula of the Kingdom of Saudi Arabia there are only routine mathematical problems that were formulated verbally for the most part with varying scopes whether being integrated in lesson, independent as a lesson or used as an assessment; They relied on solving strategies suitable for the first grade of elementary school, nonetheless there are problems in which several steps were used to solve them instead of one step. The results also reflected that the focus in verbal mathematical problems on the area of numbers is more than in the area of geometry, in the absence of statistics and data areas. In light of these results, the study recommends the diversification of mathematical problems in mathematics books for the first grade of elementary school by including non-routine problems to stimulate thinking among pupils, appropriate formulation, keeping away from unnecessary prolongation, focus on one step in solving mathematical problems, and inclusion of the area of statistics and data as simple mathematical problems; the study also suggests, the analysis of the mathematics books content to identify the nature of mathematical problems in other elementary stages and compare them with other countries.

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.

The Relationship Between Students’ Reflective Thinking Skills and Levels of Solving Routine and Non-Routine Science Problems

Abstract: Reflective thinking skills are thought to be an important factor affecting the success of problem solving both routine and non-routine science problems. This research aimed to determine the relationship between eighth-grade students’ reflective thinking skills toward problem-solving and their level of solving routine and non-routine science problems. Determining this relationship can form a basis for the decisions teachers make in courses and contribute to increasing teaching quality. A correlational and regressional research design was used in this study. A total of 408 eighth-grade students from four secondary schools in Kocaeli, Turkey constituted the sample group. The correlational results showed that, as the students’ reflective thinking towards problem solving scores increased, which were identified as questioning, reasoning and, evaluating, their level of solving routine and non-routine problems also increased. But multiple regressional results showed only the evaluation skill was a significant predictor in solving routine problems and only the reasoning skill was a significant predictor in solving non-routine problems. Therefore, students should also be equipped with reasoning skill such as examining the cause-and-effect relationships by investigating the reason behind the decisions made while solving a problem, to contribute to the solution of non-routine problems.

About time: Syntactically-guided reasoning with analog and digital clocks

ABSTRACT Around the globe, young students are expected to learn about time, yet how is it that they themselves make sense of this topic? From a sociocultural perspective, sense-making about time emerges in relation to properties of available tools and representations, such as analog clocks or digital notation. Such interactions with the symbols and structural properties of clocks are examples of syntactically-guided reasoning, a key domain of early algebra. In this paper, I focus on how students’ syntactically-guided reasoning emerged when reasoning about non-routine time problems with different clocks, whether or not such reasoning was consistent with accepted conventions of time measure. I present three case studies of typical Grade 2 students describing time on a particular clock (analog or digital) as they solved tasks related to time identification and elapsed time. I describe how symbolic properties of clocks enabled pathways of thinking about time-related ideas in relation to that particular clock. Implications for the treatment of time in elementary mathematics are discussed.

An Instructional Design Process for Developing Students Higher Order Thinking Skills in Number Patterns

Higher order thinking skills in learning are the application of thinking processes for complex situations and of course have many variables. Every student can think, but most students need encouragement and guidance for higher order thinking processes. The research used a qualitative approach, namely literature review. This study aims to explain the outline of activities planned for designing learning on number pattern class VIII material, the development of learning objectives, expected results, and practice questions that require the use of high order skills. Several efforts must be made to improve students' higher order thinking skills in mathematics, namely: (1) involving students in non-routine problem solving activities; (2) facilitating students to develop their ability to analyze and evaluate (critical thinking) and creative ability (creative thinking); and (3) encourage students to construct their own knowledge, so that learning becomes meaningful for students.

Analysis of Creative Thinking Abilities of Prospective Mathematics Teachers in Solving and Posing Quadrilateral Open-ended Problems

Prospective teachers are required to develop their creative thinking abilities. The creative thinking abilities can help prospective teachers in designing the learning process, one of which is in designing non-routine problems as an evaluation of learning. This study aims to describe the creative thinking abilities of six sixth-semester students of Mathematics Education at Sanata Dharma University in solving and posing open-ended problems on the topic of quadrilateral. Data collection methods used are written tests to solve and pose open-ended problems and interviews. Data analysis adopted the framework of qualitative data analysis namely data reduction, data display, and conclusion drawing. The results showed that (1) the students' creative thinking abilities in solving open-ended quadrilateral problems were categorized into four categories namely very poor (1 student), poor (3 students), fair (2 students) and good (1 student), (2) the students' creative thinking abilities in posing open-ended quadrilateral problems were categorized into three categories of creative thinking abilities namely very poor (2 students), poor (1 student) and very good (3 students).

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.

A Preliminary Study: Why Develop High Order Thinking Skills Worksheets Based on the 3N

This qualitative research is a preliminary study of a 4-D model development research that aims to describe the basic problems, student characteristics, the formulation of learning outcomes, and specific learning goals as the initial stage of developing a high order thinking skills (HOTS) exercise bank based on 3N in Algebra. Research subjects at the analysis stage of student characteristics are all students taking Algebra courses in Mathematics Education Department Universitas Sarjanawiyata Tamansiswa for the 2019/2020 academic year. Data were collected using a literature study, observation, documentation. The data obtained were analyzed descriptively with the following steps: (1) preparing the data, (2) data coding, (3) searching for findings, (4) constructing the framework, and interpreting the data. The results of this study are: (1) Algebra learning centered on lecturers and students has difficulty solving non-routine problems, (2) there are only 1.59% of students who implemented to check the correctness of solutions in solving HOTS problems, (3) CPMK Algebra, and (4) formulation of specific learning goals. These results are used as a reference in the design stage of exercise bank development. 
 Keywords: high order thinking, exercise bank, problem-solving.