Multiple Solution Tasks Research Articles

Exploring creativity in mathematics assessment: An analysis of standardized tests

This paper aims to investigate whether US standardized tests provide opportunities for students to demonstrate their creative thinking abilities through the inclusion of creativity-directed problems in their mathematics assessments. Our results indicated that two commonly used standardized national tests (i.e., PARCC and SBAC) do offer students some opportunities to exhibit their creative thinking skills by incorporating creativity-directed problems (e.g., multiple solution tasks) in their assessments. However, not all creativity-directed tasks are present at every grade level. The findings of this paper are significant not only because they reveal the potential of these tests to include creativity-directed tasks but also because they underscore the importance of assessment materials in fostering students’ creative thinking skills in mathematics, as assessments significantly influence teachers’ instructional practices and curriculum materials.

Students' Creative Thinking Process in Solving Multiple Solution Tasks on Geometry Material

Students' Creative Thinking Process in Solving Multiple Solution Tasks on Geometry Material

Reactive motion generation on learned Riemannian manifolds

In recent decades, advancements in motion learning have enabled robots to acquire new skills and adapt to unseen conditions in both structured and unstructured environments. In practice, motion learning methods capture relevant patterns and adjust them to new conditions such as dynamic obstacle avoidance or variable targets. In this paper, we investigate the robot motion learning paradigm from a Riemannian-manifold perspective. We argue that Riemannian manifolds may be learned via human demonstrations in which geodesics are natural motion skills. The geodesics are generated using a learned Riemannian metric produced by our novel variational autoencoder (VAE), which is intended to recover full-pose end-effector states and joint-space configurations. In addition, we propose a technique for facilitating on-the-fly end-effector/multiple-limb obstacle avoidance by reshaping the learned manifold using an obstacle-aware ambient metric. The motion generated using these geodesics may naturally result in multiple-solution tasks that have not been explicitly demonstrated previously. We extensively tested our approach in task-space and joint-space scenarios using a 7-DoF robotic manipulator. We demonstrate that our method is capable of learning and generating motion skills based on complicated motion patterns demonstrated by a human operator. Additionally, we assess several obstacle-avoidance strategies and generate trajectories in multiple-mode settings.

Analisis kemampuan berpikir kreatif siswa dalam menyelesaikan soal multiple solution task ditinjau dari gaya belajar

Pada penelitian ini implementasi berpikir kreatif akan diterapkan pada materi Sistem Persamaan Linier Dua Variabel (SPLDV). Penelitian ini bertujuan untuk mendeskripsikan kemampuan berpikir kreatif siswa SMP dalam menyelesaikan soal Multiple Solution Task (MST) ditinjau dari gaya belajar. Jenis penelitian ini adalah deskripsi kualitatif. Subjek penelitian ini adalah siswa kelas VII B SMPN 1 Sambi telah menempuh materi SPLDV 2, dipilih sebanyak 2 siswa masing-masing kategori gaya belajar visual, auditory dan kinestetik. Teknik pengumpulan data menggunakan angket, tes soal, dan wawancara. Teknik keabsahan data menggunakan triangulasi metode. Hasil dari penelitian ini menunjukkan subjek tipe belajar visual tersebut rata-rata telah memenuhi 4 indikator MST yaitu fluency, elaboration, flexibility, dan collaboration. Subjek tipe belajar auditory memenuhi 2 indikator MST yaitu fluency dan flexibility. Subjek tipe belajar kinestetik dari hasil penelitian tersebut didapat bahwa objek tersebut telah memenuhi 4 indikator MST yaitu fluency, elaboration, flexibility, dan originality.

Supporting creative problem solving in primary geometry education

This intervention study aimed to identify how creative thinking can be supported in geometry education. Fifth-graders received five geometry lessons that incorporated divergent and convergent thinking. Children were assigned to a condition with either no creative thinking support (n = 60), support for divergent thinking (partial support; n = 55) or support for divergent and convergent thinking (full support; n = 59). Divergent thinking was assessed by a multiple-solution task and scored in terms of fluency, flexibility and originality. Convergent thinking was assessed by an idea evaluation task. Repeated measures MANOVA showed that different aspects of divergent thinking (i.e., fluency, flexibility and originality) benefited from different types of support. Students’ fluency scores improved in the full support condition, but decreased in the other two conditions. A reverse pattern was found for flexibility and originality, which decreased in the full support condition but increased in the other conditions. No time and condition effects were found for convergent thinking. In all conditions, there was a non-significant increase in geometry scores. However, geometry performance differed between conditions, and was lower (across pretest and posttest) in the full support condition than in the no support and partial support conditions. These results provide some initial evidence that divergent thinking can be promoted in geometry education. Convergent thinking, on the other hand, might be more difficult to support effectively. As different types of support seem to affect different aspects of divergent thinking, teachers might consider adapting support to the specific task or individual needs of children.

EVALUATING THE APPROPRIATENESS OF TASKS AND THE ELABORATION OF MULTIPLE SOLUTIONS TO OCCASION FOURTH-GRADERS’ MATHEMATICAL CREATIVE THINKING

Creativity is not only for gifted students, but also for regular ones. This case study was aimed to analyze the appropriateness of tasks and the elaboration of multiple solutions to occasion fourth-graders’ mathematical creative thinking through a documentary multiple-solution counting task in a figurative setting. The data came from the written report of 48 fourth graders in two classes in Taiwan, China. The appropriateness of creativity was reflected in the appropriateness of tasks and solutions, particularly suitable for complex problem solving. Elaboration was detail-dependent, and visualization was beneficial to the analysis of elaboration. The regular students who had just entered the fourth grade could show their creative thinking through different angles (horizontally or vertically) and starting points (holistic or partial), but with slightly more partial and horizontal than holistic and longitudinal, more adaptation than transformation. These fourth-grade students have had the basic mathematical creative thinking capability of adaptation, combination, change, rearrangement, extension or going back by using counting, combining, adding and reducing, overlapping, moving, and diagonal division strategies. Keywords: creative thinking, mathematical creativity, multiple-solution task, primary school students

An empirical study of mathematical creativity and students’ opinions on multiple solution tasks

Creativity and problem solving are considered to be twenty-first-century competencies, therefore promoting mathematical creativity should be an important part of school mathematics. The study presented in this paper is inspired by the notion of mathematical creativity and the utilization of multiple solution tasks (MSTs) to investigate students’ creativity. Multiple solution tasks are mathematical tasks with an explicit requirement to be solved with different methods. For the purpose of the study, we used a textbook question which allows multiple solution pathways as our research instrument. This MST was administered to first-year mathematics students (18–20 years) along with additional questions related to their experience in school mathematics. The results of the study showed that students had difficulties complying with the demands of the task, but generally have a positive disposition toward MSTs. Moreover, the results underline the necessity of incorporating MSTs in mathematics classroom teaching to create a more coherent corpus of mathematical knowledge. The study also provides an example of how a regular textbook task can be used as an MST.

Relationships between general giftedness, expertise in mathematics, and mathematical creativity that associated with pattern generalization tasks in different representations

The current study investigated creativity-directed problem-solving processing that explicitly requires solving pattern generalization problems in multiple ways. To examine mathematical creativity, we employed a multiple solution tasks approach, asking participants explicitly to solve pattern generalization problems in multiple ways. The participants were engaged with pattern generalization-multiple solution tasks in which patterns were presented numerically, diagrammatically, or contextually. The study aimed to examine whether success and creativity associated with solving pattern generalization-multiple solution tasks are affected (I) by expertise in mathematics (EM) and general giftedness (G) and (II) by task complexity and representation. Two hundred and ninety-eight participants from four research groups that differed in levels of general giftedness (G) and mathematical expertise participated in the study. We found that G affected statistically significant all the components of creativity when solving pattern generalization-multiple solution tasks, independently of task representation. Surprisingly, the effect of EM was lower and dependent on the type of task representation. Also surprising, we found statistically significant interactions between G and EM factors when solving pattern generalization-multiple solution tasks. The findings demonstrate a significant effect of task representation on students’ levels of success and creativity when engaged in solving pattern generalization-multiple solution tasks: contextual representations were approached with the highest level of success and creativity, whereas numerical representations appeared to be tackled with the lowest level of success and creativity. We suggest that mathematical instruction should integrate pattern generalization-multiple solution tasks to promote students’ mathematical expertise and creativity.

Students\u2019 collaborative creative process and its phases in mathematics: an explorative study using dual eye tracking and stimulated recall interviews

In the age of artificial intelligence where standard problems are increasingly processed by computers, creative problem solving, the ability to think outside the box is in high demand. Collaboration is also increasingly significant, which makes creative collaboration an important twenty-first-century skill. In the research described in this paper, we investigated students’ collaborative creative process in mathematics and explored the collaborative creative process in its phases. Since little is known about the collaborative creative process, we conducted an explorative case study, where two students jointly worked on a multiple solution task. For in-depth insight into the dyad’s collaborative creative process, we used a novel research design in mathematics education, DUET SRI: both students wore eye-tracking glasses during their collaborative work for dual eye-tracking (DUET) and they each participated in a subsequent stimulated recall interview (SRI) where eye-tracking videos from their joint work served as stimulus. Using an inductive data analysis method, we then identified the phases of the students’ collaborative creative process. We found that the collaborative creative process and its phases had similarities to those previously found for solo creative work, yet the process was more complex and volatile and involved different branches. Based on our findings, we present a tentative model of the dyad’s collaborative process in its phases, which can help researchers and educators trace and foster the collaborative creative process more effectively.

How could the Presentation of a Geometrical Task Influence Student Creativity?

This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on Multiple-Solution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.

Multiple-Solution Tasks in Pre-Service Teachers Course on Combinatorics

In the paper, we present a study devoted to the utilization of multiple-solution tasks (MSTs) in combinatorics as a part of a pre-service teachers course on didactics of mathematics from the view of the mathematics teachers’ specialized knowledge (MTSK) theoretical framework. The study was carried out over the standard course of a summer semester in 2021. The course was attended by 13 pre-service teachers (PSTs). It was carried out online, due to COVID-19 restrictions. Ten combinatorial multiple-solution tasks were assigned to the PSTs. Analyzing pre-service teachers solutions to these tasks, we sought the description and better understanding of the combinatorial knowledge of the topic from the perspective of MSTK. The results revealed some critical aspects of mathematical knowledge in combinatorics that pre-service teachers education should focus on.

Creativity in mathematics performance: The role of divergent and convergent thinking.

BackgroundCreativity requires both divergent and convergent thinking. Previous research established that divergent thinking relates to mathematics performance, but generally ignored the role of convergent thinking and, hence, leaves it unclear how both might interact when children work on mathematical tasks. This study addressed this paucity in the research literature, with the goal of improving our understanding of the role of creative thinking in primary school mathematics.AimsThis study examined how divergent and convergent thinking contribute to mathematics performance, both directly and jointly, on single‐ and multiple‐solution tasks.SampleThe study was conducted with 229 Dutch fifth graders of 12 primary schools.MethodDivergent and convergent thinking were measured with a visual and verbal task. Path analysis was used including verbal and visual divergent and convergent thinking tasks in relation to single‐ and multiple‐solution mathematics task performance. Working memory was included as a covariate.ResultsVerbal convergent thinking positively predicted single‐ and multiple‐solution task performance. Verbal divergent and convergent thinking interacted in relation to single‐solution task performance, while visual divergent and convergent thinking interacted in relation to multiple‐solution task performance.ConclusionsChildren’s mathematics performance mainly relies on convergent thinking. The role of divergent thinking is twofold: it complements convergent thinking on multiple‐solution tasks and compensates convergent thinking on single‐solution tasks.

Students’ Mathematical Creative Thinking Ability with Posing-Exploring-Doing-Evaluating (PEDE) Productive Failure Model in New Normal

Students’ creative thinking ability is essential to be developed in Mathematics because it can help students solve everyday life problems. This study aimed to determine the impact of the Posing-Exposing-Doing-Evaluating (PEDE) Productive Failure model on developing students’ creative thinking ability in mathematics in the new normal. The study was conducted at Diplahan National High School of the Division of Zamboanga Sibugay to Grade-11 students enrolled for the school year 2020-2021. The study employed a quasi-experimental pretest-posttest control group design. The experimental group was exposed to PEDE Productive Failure Model and the control group was exposed to the Conventional Method. Multiple solution tasks were used to measure students’ level of creative thinking ability. A teacher-made test composed of 4-item open-ended questions with a rubric scale that measures mathematical creative thinking ability was the main instrument used in the study. The mean, standard deviation, and the analysis of covariance (ANCOVA) were used to analyze the data collected. The analysis revealed that students exposed to PEDE Productive Failure Model performed significantly better in terms of creative thinking ability posttest score. Researchers recommend that Mathematics teachers may use PEDE Productive Failure as a modular learning approach in the new normal to develop creative thinking ability in Mathematics. Teachers might also incorporate activities into their Mathematics modules that demand students to present various clearly defined solutions to foster students' creative potential.

Developing Students' Creative Constructs in Mathematics with Problem-Based (PB) and Problem Posing (PP) Tasks

The purpose of this experimental study was to examine the creative constructs of students enrolled in Differential Calculus at the University of Science and Technology of Southern Philippines, Cagayan de Oro City, Philippines. This study was composed of 132 engineering and mathematics education students enrolled in two different semesters. The experimental group was chosen randomly, exposed to the problem-based HALP model teaching approach and problem posing while the control group was taught using Polya’s problem-solving heuristics. Multiple Solution Tasks (MST) test in Calculus was administered to determine their creative constructs which can be measured in terms of their level of mathematical fluency, flexibility and novelty of solutions before and after the experiment. Results indicated that the experimental group exhibited a remarkable improvement of their mathematical fluency and flexibility but still in the developing level when they are required to posit novel solutions to problems as influenced by the problem-based HALP and problem posing activities. Hence, it is recommended that mathematics teachers may utilize these methods to successfully develop students’ mathematical creativity and future research may also be explored on integrating technology and how it can influence developing student’s mathematical creativeness as well as the mediating role of their affective domains and IQ (intelligent quotient).

A Systematic Literature Review: Discipline-Specific and General Instructional Practices Fostering the Mathematical Creativity of Students

The purpose of this systematic review is to reveal the research findings that suggest instructional practices to foster the creativity of students in mathematics. Although several studies have investigated the effects of various instructional practices influencing the mathematical creativity of students, little is known about how the findings of this collective body of research contribute to the understanding of what instructional practices should be integrated into a mathematic classroom to further foster the mathematical creativity of students. In this systematic review, the knowledge of instructional practices that foster the mathematical creativity of students were categorized under two main factors including: 1) discipline-specific instructional practices and 2) general instructional practices. The discipline-specific instructional practices were problem-solving, problem-posing, open-ended questions, multiple solution tasks, tasks with more than one correct answer, modeling/model-eliciting activities, technology integration, extendable tasks, and emphasizing abstractness of mathematics. The general instructional practices were providing students with ample time to think creatively about real-world related mathematical problems in a judgment free and collaborative classroom environment so that they take risks to share their mathematical ideas and use informal words. Integrating all of these instructional practices into mathematics classrooms can provide opportunities for students to discover their potential creative abilities in mathematics.

The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks

We live in the challenging times of the 21st-century with the increased need for humans to possess specific skills that will help them to be successful in this era. This means that education should in learners, develop these skills effectively. Different global countries have begun to recognize the significance of multiple solutions tasks in the teaching and learning of mathematics in the 21st-century. However, this practice is not visible in South Africa. Hence, the current study explores and synthetize the sparsely available literature on MSTs to answer the question: What is the significance of multiple-solution tasks (MSTs) in mathematics education and why is it relevant for South African mathematics education to make the exercise of producing multiple solutions accessible to learners? The literature that is being synthetized here is viewed through the optic lens of the social constructivism theory as proposed by Vygotsky and explicated in Jean Lave and Etienne Wenger`s Situated learning: Legitimate peripheral participation. In the conclusion I engage in an argumentation that illuminates the significance of MSTs in mathematics education and provide reasons why it would be beneficial for the South African mathematics curriculum to incorporate MSTs.

How the Novelty of Students' Answers in Solving Mathematical Problems?

The impact of giving a thinking freedom opportunity is the diversity of students' answers in solving mathematical problems. The smaller the similarity of students' answers compared to all the answers of students in a class, it means that they have higher creative thinking abilities than other students. It is based on a unique answer compared to the others or in other words it has a higher novelty value than the other answers. The objective of this study was to describe the attainment of the novelty of students' answers in solving mathematical problems. The measurement of novelty of students' answers was carried out on 1,002 students from six junior high schools in Indonesia who had different levels of student's input. The scoring system on the response to the completion of multiple solution tasks was by using the four categories of polytomous in which it is based on the Partial Credit Model (PCM). The data were analyzed by using Quest and Parscale program. The result of this study showed that the students have good skills in solving mathematical problems with new ideas on several items. The items referred to some abilities, such as resolving algebra problems involving money operations, designing a flat shape that has specific characteristics, arranging numbers based on certain patterns, and arranging numbers based on certain patterns. The novelty of this research is that this study considers every answer from students or provides a thinking freedom opportunity for students so that mathematical problems are not absolutely solved with one answer.

Embedding Proof-Writing in Phenomenon-based Learning to Promote Students’ Mathematical Creativity

The study aimed to identify the significant impact of phenomenon-based learning and embedding proof writing to this new teaching strategy on students’ mathematical creativity in tertiary education. The study used a quasi-experimental pretest-posttest non-equivalent control group design. There were 2 experimental groups, one exposed to phenomenon-based learning alone and the other was exposed to phenomenon-based learning with classroom proof-writing activities, and one control group exposed to the conventional method. Multiple solution tasks were used to measure students’ level of creativity. The data were analyzed using mean, standard deviation, and ANCOVA. The analysis revealed that students exposed to phenomenon-based learning with proof writing had the highest posttest creativity score. Students in this group had a significantly higher level of creativity among the three groups. Further, students exposed to phenomenon-based learning alone may not outperform students exposed to phenomenon-based learning with proof writing, but they still had significantly higher posttest mean scores compared to students exposed to the conventional type of teaching. The researchers recommend to teachers in tertiary education to use phenomenon-based learning with proof writing in teaching mathematics and its related courses since this instructional approach engages students in learning that is more focused to real-life issues, apply skills and knowledge from different subjects, and enhance important skills like creativity, problem-solving, communication, and teamwork.

Challenging students to improve their mathematical creativity in solving multiple solution task on challenge based learning class

As one of the competencies that must be possessed in the era of Industry 4.0, creativity also plays an important role for development of mathematics education in the 21st era. This study aims to determine the improvement of students’ mathematical creativity in solving Multiple Solution Task (MST) on Challenge Based Learning (CBL) class. The quantitative quasi experimental pretest posttest conducted at SMP Negeri 1 Semarang used a mathematical creativity test adopted MST as a data collection technique and data were analyzed using the t-test hypothesis and gain. The results revealed the improvement of students’ mathematical creativity in solving MST after the implementation of CBL and differences in the improvement of students’ mathematical creativity where the student’s gain of mathematical creativity on CBL (experimental) class was more than the student’s gain of the mathematics of the Problem Based Learning (PBL) (control) class. Other findings indicate a difference in achievement of mathematical creativity indicators between the CBL class and PBL class which shows that CBL provides a positive influence to improve each indicator of mathematical creativity.

Analysis of Mathematic Creative Thinking Ability of 10th Grade High School Students About Solution Mathematics Problems

The ability to think creatively is an important component needed in the 21 st century. Creative thinking is a person's ability to produce various ideas or concept and or new alternative solutions in solving a problem. Three important aspects of creative thinking are fluency, flexibility, and novelty. Teachers have an important role in preparing creative students, but there are still many teachers who do not understand what abilities students already have and how they should be treated. This research is a qualitative research about the level of mathematical creative thinking ability of 10th grade high school students in solving mathematical problems. This study uses purposive sampling techniques and triangulation methods. The research subjects are 62 students of 10 th grade of high school. The data are collected using test and interview guidelines. The results showed that the level of mathematical creative thinking ability of 10th grade high school students was still low. A total of 33 students (53%) were not creative (level 0), 26 students (42%) were less creative (level 1), 1 student (2%) were quite creative (level 2), 2 students (3%) were creative (level 3), and no student (0%) was very creative (level 4). So, students' mathematical creative thinking abilities still need to be developed, such as by getting students to work on a Multiple Solution Task (MST) and or applying a 3CM (Cool-Critical-Creative-Meaningful) model in the learning process