How Students Integrate Mathematical Problem Solving and Computational Thinking: A Case Study of Two Grade 8 Students’ Described Strategies

In this systematic literature review, we investigate the connections between computational thinking and problem solving in the context of primary and secondary mathematics education. We do this by exploring how and at which steps in the mathematics problem-solving process seven peer reviewed studies report on the inclusion of computational thinking concepts, practices and perspectives. Overall, the studies show that it is possible and at times beneficial to include computational thinking in mathematics problem solving. However, more research is needed to see whether simply including computational thinking and its programming tools enhances students’ problem-solving skills in mathematics.

In this paper, we explore the challenges experienced by a group of Primary 5 to 6 (age 12–14) students as they engaged in a series of problem-solving tasks through block-based programming. The challenges were analysed according to a taxonomy focusing on the presence of computational thinking (CT) elements in mathematics contexts: preparing problems, programming, create computational abstractions, as well as troubleshooting and debugging. Our results suggested that the challenges experienced by students were compounded by both having to learn the CT-based environment as well as to apply mathematical concepts and problem solving in that environment. Possible explanations for the observed challenges stemming from differences between CT and mathematical thinking are discussed in detail, along with suggestions towards improving the effectiveness of integrating CT into mathematics learning. This study provides evidence-based directions towards enriching mathematics education with computation.

This study examined 12 preservice teachers’ understanding of computational thinking while planning and implementing a computational thinking activity for fifth grade students. The preservice teachers were enrolled in an add-on computer education license that would certify them to teach computer courses in addition to their primary major area (11 elementary education majors, 1 secondary social studies education major). The preservice teachers were asked to develop a 2 h instructional project for fifth grade students to build on the computational thinking concepts learned during the “Hour of Code” activity. Data was collected from preservice teachers’ initial proposals, two blog posts, video recordings of in-class discussions, instructional materials, final papers, and a long-term blog post 3 months after the intervention. Results showcased that the process of developing and implementing computational thinking instruction influenced preservice teachers’ understanding of computational thinking. The preservice teachers were able to provide basic definitions of computational thinking as a problem-solving strategy and emphasized that learning computational thinking does not require a computer. On the other hand, some preservice teachers had misconceptions about computational thinking, such as defining computational thinking as equal to algorithm design and suggesting trial and error as an approach to computational problem solving. We provide recommendations for teacher educators to use more directed activities to counteract potential misconceptions about computational thinking.

Challenges in the 21st century are increasingly complex, technology is developing rapidly and competition is getting tougher. Therefore we need quality human resources that can keep up with and anticipate the times. The use of technology involves computational thinking (CT) skills which are closely related to the problem-solving process. The stages in computational thinking are part of mathematical thinking, meaning that learning mathematics can support students' CT skills. Through the development of digital pedagogical models in CT integrated mathematics learning, it can improve problem-solving skills. This research uses design based implementation research with 4 phases including; preliminary research, prototyping, results, and design principle. The participants were 28 grade 8 junior high school students who took part in two rounds of experiment in direct CT activities and digital CT activities. In this paper, we present an iterative mathematical problem-solving process in the digital pedagogy model. The computational task, environment, tool and practices were iteratively improved over two rounds to incorporate CT effectively in mathematics. The results from CT environment demonstrated that direct CT activities are more effective than digital CT activities in mathematical problem-solving. Based on empirical research, we summarize the characteristic of the digital pedagogy model from computational tasks, computational environment and tools, and computational practices in mathematical problem solving.

Two studies were conducted to examine the use of grounded embodied pedagogy, construction of Imaginary Worlds (Study 1), and context of instructional materials (Study 2) for developing learners' Computational Thinking (CT) Skills and Concept knowledge during the construction of digital artifacts using Scratch, a block-based programming language. Utilizing a conceptual framework for grounded embodied pedagogy called Instructional Embodiment, learners physically enacted (Direct Embodiment) and mentally simulated (Imagined Embodiment) the actions and events as presented within pre-defined Scripts. Instructional Embodiment utilizes action, perception, and environment to create a dynamic, interactive teaching & learning scenario that builds upon previous research in embodied teaching and learning. The two studies described herein examined the effects of Instructional Embodiment, Imaginary World Construction, and Context on the development of specific Computational Thinking Concepts and Skills. In particular, certain CT Concepts, such as Conditionals, Variables, Thread Synchronization, Collision Detection, & Events, and CT Skills, such as abstraction and pattern recognition, were identified and measured within the learners' individual digital artifacts. Presence and/or frequency of these Concepts and Skills were used to determine the extent of Computational Thinking development. In Study 1, fifty-six sixth- and seventh-grade students participated in a fifteen-session curricular program during the academic school day. This study examined the type of instruction and continuity of Imaginary World Construction on the development of certain CT Skills and Concepts used in a visual novel created in Scratch. Main effects were found for learners who physically embodied the pre-defined instructional materials: embodying the pre-defined Scripts led to the learners using significantly more ‘speech’ Blocks in their projects and more Absolute Positioning Blocks for ‘motion’ than those who did not physically embody the same Scripts. Significant main effects were also found for continuity of Imaginary World Construction: learners who were instructed to continue the premise of the first digital artifact (Instructional Artifact) implemented significantly more computational structures in their second digital artifact (Unique Artifact) than those who were instructed to create a Unique Artifact with a premise of their own design. In Study 2, seventy-eight sixth- and seventh-grade students participated in a seventeen-session curricular program during the academic school day. This study examined the type of instruction and context of instructional materials on the development of CT Skills and Concepts during the construction of a video game using Scratch. Similar to Study 1, findings suggest that physically embodying the actions presented within the pre-defined instructional materials leads to greater implementation of many of these same structures during individual artifact construction. The study also showed that as the pre-defined Scripts become more complex (e.g. single-threaded to multi-threaded), the effect of physical embodiment on the development of CT Skills and complex CT Concept structures becomes less pronounced. Findings from this study also suggest that Context has a significant effect on identifying & implementing the CT Skill pattern recognition: learning CT Concepts from an Unfamiliar Context had a significant positive effect on the implementation of both Broadcast/Receive couplings and Conditional Logic & Operator patterns. In sum, the findings suggest that the type of instruction, the continuity of the Imaginary World being constructed, and the context of the instructional materials all play a significant role in the learners' ability to develop certain Computational Thinking Skills and Concept knowledge. The findings also suggest that a physically embodied approach to teaching abstract concepts that is grounded in an unfamiliar context is the most effective way to integrate a grounded embodied approach to pedagogy within a formal instructional setting.

[ILLUSTRATION OMITTED] Computational thinking is a fundamental skill for everyone, not just for computer scientists. To reading, writing, and arithmetic, we should add computational thinking to every child's analytical ability (Wing 2006, p. 33). A Framework for K-12 Science Education identified eight practices as essential elements of the K-12 science and engineering curriculum (NRC 2012, p. 49). These practices are deeply embedded within the Next Generation Science Standards (NGSS Lead States 2013a), where they are wedded closely to core ideas in the science disciplines. Most of the practices, such as Developing and Using Models, Planning and Carrying Out Investigations, and Analyzing and Interpreting Data, are well known among science educators. In contrast, the practice of Using Mathematics and Computational Thinking raises questions in the minds of many educators. As mathematics has long been integral to science teaching, the questions tend to revolve around the meaning of computational thinking. The Framework envisions computational thinking as a powerful intellectual tool: Since the mid-20th century, computational theories, information and computer technologies, and algorithms have revolutionized virtually all scientific and engineering fields. These tools and strategies allow scientists and engineers to collect and analyze large data sets, search for distinctive patterns, and identify relationships and significant features in ways that were previously impossible. They also provide powerful new techniques for employing mathematics to model complex phenomena---for example, the circulation of carbon dioxide in the atmosphere and ocean (NRC 2012, p. 64). and computational thinking We (the authors) asked ourselves how the different forms of computational thinking suggested in this definition differed from mathematical thinking. Mathematics, it's important to note, is not just a skill but also a way of thinking about the world. The Framework distinguishes mathematics from data analysis in the use of symbols to express relationships. So, for example, a student might graph data from an experiment and notice a pattern. Mathematics comes in when the student expresses the pattern as an equation that can predict additional data points. Students develop mathematical thinking when they approach a new situation with a range of mathematical skills in mind. Similarly, they develop computational thinking when they approach a new situation with an awareness of the many ways that computers can help them visualize systems and solve problems. The Venn diagram in Figure 1 shows how we see the relationship between mathematical and computational thinking. The diagram (itself a mathematical tool) shows which capabilities can be considered part of mathematical thinking, which are part of computational thinking, and which are part of both. As the diagram illustrates, analyzing and interpreting data is common to both mathematical and computational thinking, as are problem solving, mathematical modeling, and statistics and probability. Figure 1 also lists capabilities unique to computational thinking. In the remainder of this article we will illustrate how simulation, data mining, and automated data collection are important in today's science classroom. As you read these examples, keep in mind that simply using computers is not enough; students must be encouraged to re-orient and deepen their understanding about science using computational thinking. Your goal should be to help students build learning skills by recognizing the ways they can use computers to carry out investigations and solve practical problems. [FIGURE 1 OMITTED] The NGSS describes the practice of mathematics and computational thinking for high school as follows: Mathematical and computational thinking in 9-12 builds on K-8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. …

This paper aims at exploring students’ engagement in mathematical problem-solving through computational thinking (CT) and the programming language MATLAB. The work is a single case study conducted in the context of a first-year undergraduate course on programming with applications in mathematics. It uses a three-step approach based on theoretically derived insights from the research literature to address mathematical problem-solving by means of mathematical thinking, CT, and programming. The main method used is a semi-structured interview with two undergraduate students trying to solve a mathematical task related to Pythagoras’ theorem while responding to questions about the problem-solving process. The interviews were analyzed according to an inductive strategy based on the interplay between the three-step approach and the interview data. The results describe two students’ opposite experiences. While one was able to handle and solve the task using CT and MATLAB very easily, the other struggled to get acquainted with the task, CT, and the programming activity with MATLAB. Conclusions are drawn from the results to promote mathematical problem-solving through CT and programming in mathematics courses at the undergraduate level. Future work will address some of the research gaps found in the literature, in particular the links between mathematical thinking, CT, and programming to highlight their communalities and potential differences and deepen the knowledge about their connections.KeywordsAlgorithmComputational thinking (CT)MATLABMathematical problem-solvingMathematical thinkingProgrammingUsability

The notion of teaching experts’ habits of mind (e.g., computational thinking and scientific thinking) to novices seems to have inspired many educators and researchers worldwide. In particular, a great deal of efforts has been invested in computational thinking (CT) and its manifestations in different fields. However, there remain some troubling spots in CT education as far as how to teach it at different levels of education. The same argument applies to teaching scientific thinking (ST) skills. A remedy has been suggested to narrow CT and ST skillsets down to core cognitive competencies so they can be introduced in early and middle grades and continue to be nurtured during secondary and post-secondary years. Neuroscientists suggest that the act of (computational) thinking is strongly linked to the acts of information storage/retrieval by our brain. Plus, years of research have shown that retrieval practices promote not only knowledge retention but also inductive reasoning and deductive reasoning. Not surprisingly, these reasoning skills are core elements of both CT and ST skillsets. This article will mesh the findings of a teacher professional development with the existing literature to lay a claim that retrieval practices enhance CT and ST skills. The study offered training to secondary school teachers (n = 275) who conducted classroom action research to measure the impact of retrieval practices on teaching and learning of STEM and CT concepts. We used a quasi-experimental research design with purposeful sampling and a sequential mixed-methods approach focusing on the impact of professional development on teacher outcomes and, in turn, on student outcomes. A survey of teacher participants showed that the majority (96%) of survey respondents (n = 232) reported a good understanding of retrieval strategies, and how relevant ideas can be implemented and tested in the classroom. A large number of action research (target-control) studies by teachers (n = 122) showed that students who learned STEM and CS concepts through retrieval practices consistently scored 5–30% higher than those using the usual blocked practice. In most cases, the difference was statistically significant (p < 0.05). While the study contributes to retrieval practices literature, those looking for best practices to teach core CT and ST skills should benefit from it the most. The study concludes with some recommendations for future research based on the limitations of its current findings.

In this work, we examine whether repeated participation in an after-school computing program influenced student learning of computational thinking concepts, practices, and perspectives. We also examine gender differences in learning outcomes. The program was developed through a school–university partnership. Data were collected from 138 students over a 2.5-year period. Data sources included pre–post content assessments of computational concepts related to programming in addition to computational artifacts and interviews with a purposeful sample of 12 participants. Quantitative data were analyzed using statistical methods to identify gains in pre- and post-learning of computational thinking concepts and examine potential gender differences. Interview data were analyzed qualitatively. Results indicated that students made significant gains in their learning of computational thinking concepts and that gains persisted over time. Results also revealed differences in learning of computational thinking concepts among boys and girls both at the beginning and end of the program. Finally, results from student interviews provided insights into the development of computational thinking practices and perspectives over time. Results have implications for the design of after-school computing programs that help broaden participation in computing.

Even though the computer science education community has not definitively established exactly what ?Computational entails, most will agree that it is using a computing machine to solve problems. Like all tools for solving problems, this knowledge should be made available to everyone. Jeannette Wing sounded this call in 2006, writing that computational methods and models give us the courage to solve problems and design systems that no one of us would be capable of tackling alone. These skills are not just for computer scientists, but for anyone and everyone who plans to solve problems. We already teach other forms of problem solving to all our students including: Mathematical Thinking, Critical Thinking, and Scientific Reasoning. Computational Thinking, as well as engineering and design's Design Thinking, should be equally exposed to all students. It is also important to illuminate that these tools are not used in isolation to solve problems. Perhaps the key to the acceptance of Computational Thinking (as well as Design Thinking) as essential educational tools is to highlight their overlap and interdependence with other problem solving skills. The next generation will need practice with every tool at their disposal to be prepared to solve tomorrow's problems. I propose embedding Computational Thinking in a problem solving framework that leverages teachers' (and students') existing knowledge of problem solving in mathematics, science, and language arts in order to encourage teachers in non CS disciplines to teach Computational Thinking in their classrooms.

This article examines the results of Computational Thinking (CT) training using Scratch for high school teachers who are members of the Barito Kuala district mathematics MGMP (Musyawarah Guru Mata Pelajaran), facilitated by Computer Education Study Program of Universitas Lambung Mangkurat. Employing a case study method, the training focused on the experiences of teachers in understanding and applying CT based on the case given. Key CT concepts, such as abstraction, decomposition, pattern recognition, and algorithmic thinking, were introduced, with participants creating visual programs to solve problems or cases in mathematics. Pre- and post-training questionnaires were used to evaluate changes in teachers' perceptions and knowledge across seven aspects. The results show an increase in understanding of CT concepts for 34 participants by 60%, while 63.33% of participants reported improved confidence in using Scratch compared to 53.33% before the training. Additionally, 66.67% of participants assessed the training as effective in enhancing their teaching skills. This activity shows that integrating CT into mathematics education can significantly enhance teachers' problem-solving abilities and contribute to a more technology-driven educational approach to education in high school classrooms.

The study focuses on the identification of relationships and/or rules between computational thinking (CT) concepts among the undergraduate students of Applied Informatics due to their attitudes towards mathematics. We analyze three CT concepts - decomposition, pattern recognition, and algorithmic thinking. We assume that students who have a closer relationship with mathematics, a positive attitude towards mathematics, have better developed CT. We conducted the experiment during the 2022/2023 academic year on the Information Coding and Displaying subject. The results indicate that those students who have no relationship to mathematics, negative attitude towards mathematics, have no problem with decomposition and pattern recognition, but without significant algorithmic thinking. On the other hand, students who have a close relationship with mathematics are also able to decompose or recognize patterns, but moreover they have shown algorithmic thinking. The contribution of the study comprises the identification of relationships and/or patterns of computational thinking concepts among students who have a relationship to mathematics, who cannot assess their relationship to mathematics, as well as among students who have no relationship to mathematics. Our results indicate a different occurrence of computational thinking concepts as well as links and/or relationships between them.

Computational Thinking is a skill that guides the 21th century individual in the problems experienced during daily life and it has an ever-increasing significance. Multifarious definitions were attempted to explain the concept of Computational Thinking. However, it was determined that there was no consensus on this matter in the literature and several different concepts were mentioned in the definitions found in the literature. It was considered that this fact made it difficult to understand the concept of Computational Thinking. To establish a more comprehensive approach, the present study aimed to identify the concepts that are included in the Computational Thinking definitions that were presented in previous studies. It also aimed to reveal trends in the identified concepts throughout the years. As a result of the search, a total of 59 definitions were identified and a content analysis was conducted on these definitions. Analysis results demonstrated that Computational Thinking was defined based on several concepts such as problem solving, technology, thinking, individual and social qualities. Furthermore, it was determined that statements on thinking were prominent before 2006, and today, emphasis on problem solving and technology became more significant. It was considered that the present study would contribute to a better understanding of the Computational Thinking concept. At the end of the study, certain suggestions were presented for further research.

The progress of the times requires students to be able to think quickly. Student activities in learning are always associated with technology and students’ thinking activities and are expected to think computationally. Therefore, this study aimed to determine how learning with the concept of computational thinking (CT) using the Scratch program can improve students’ mathematical problem-solving abilities. An exploratory research design was conducted by involving 132 grade VIII students in Kuningan, Indonesia. Data analysis began with organization, data description, and statistical testing. The results showed that students performed the concepts of abstraction thinking, algorithmic thinking, decomposition, and evaluation in solving mathematical problems. There were differences in students’ problem-solving abilities before and after the intervention. Students’ activeness in solving problems using the CT concept through a calculator significantly affected 52.3% of the ability to solve mathematical problems.

Objective: The benefits of computational thinking have become an increasingly acknowledged and popular subject of investigation among researchers. This research aims to gather detail and comprehensive information regarding the learning of computational thinking skills in mathematics at various educational levels through a systematic literature review approach. Methods: With a focus on education level, instructional media, mathematics content, and the components of computational thinking addressed by previous researchers, this paper applied the PRISMA Systematic Review Protocol to offer a comprehensive synthesis of sixteen empirical studies retrieved from the Scopus database on the implementation of computational thinking in mathematics education. Findings: Most research on fostering computational thinking in mathematics education is concentrated at the elementary and junior high school. To optimize the development of computational thinking in mathematics, teachers should be reminded of strategies to support students, particularly through activities involving simulations using various instructional media. Examples of such media include programming platforms, visualization tools, and interactive simulations and games. Number operations and geometry are the mathematical content most widely used for fostering computational thinking. Algorithmic thinking, a crucial component in fostering computational thinking among elementary school students, helps them develop a strong foundation for understanding higher mathematical concepts. Conclusion: A systematic review of computational thinking in learning mathematics at the various education level is conducted in this study. The chosen studies were systematically analyzed for the advancement of computational thinking in learning mathematics to provide new an insight information for educators and stakeholders. Keywords: computational thinking, mathematics, systematic literature review. DOI: http://dx.doi.org/10.23960/jpmipa/v25i1.pp34-52