Aho, A. V. (2012). Computation and Computational Thinking. The Computer Journal , 55 (7), 832-835.
In this article, Aho expounds on the concepts that form the foundational block in the linkage between mathematical thinking and computer science. The article emphasizes on the need for a clear definition of the terms and concepts in the two fields to aid knowledge acquisition and utilization in the fast-paced developments associated with them. The article insists on using computational thinking rather than mathematical because of the strong relationship between such thinking and computer science. The relationship is established in the definition of computational thinking as thought processes involved in the formulation of problems whose solutions can be represented as computational steps and algorithms. The relationship shows the use of mathematical thinking to address critical elements that form the basis of computer science. The article contends that the expansion in computer science applications increases the need for mathematical thinking to assist in understanding the world of interconnected machines. Therefore, Aho calls for close examination of the theory of computation to enhance understanding of its fundamental capabilities and limitations. The article advances that such a critical perspective is necessary to understand and implement the different mathematical thinking models in the world of computing.
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Bell, T., Rosamond, F., & Casey, N. (2012). Computer Science Unplugged and Related Projects in Math and Computer Science Popularization. In The multivariate algorithmic revolution and beyond (pp. 398-456). Springer, Berlin, Heidelberg.
The chapter recognizes the popularization of mathematical thinking as an integral element in advancing the aspect of creativity in all forms of education. Specific emphasis is given to projects that have been implemented to aid popularization of mathematics, which is synonymous to advocating for the crucial role of mathematical thinking in the education system. The article advances that the integration of mathematical thinking and computer science is an undertaking that evolved over the years since early activism for the same. The MEGA-Mathematics project used to illustrate the constructive co-existence between mathematics and computer science, is heralded for its effectiveness and efficiency in the development of mathematical thinking. The main point of the article is to focus on the need for a definitive framework that has been proven to work in the implementation of teaching and learning towards the development of mathematical thinking. The use of computer-based applications is cited to be an appropriate supporting strategy.
Calao, L. A., Moreno-León, J., Correa, H. E., & Robles, G. (2015). Developing Mathematical Thinking with Scratch. In Design for teaching and learning in a networked world (pp. 17-27). Springer, Cham.
The article plays a crucial role in highlighting the interdependence between mathematical thinking and computer science, a relationship that goes in both directions. While mathematical thinking forms the foundation for computer science, the article posits that computer programming can also be used to invoke mathematical thinking. The interchange between the two is central to the trends in modern educational landscape. Calao et al. (2015) recognized the existence of room for improvement in mathematical skills of students in the international context. As a result, the article suggests the integration of computer science in teaching mathematical skills. Drawing on the mantra of learn to code, code to learn, the article sought to examine the influence of computer-based applications and programming on students’ mathematical thinking ability. Calao et al. (2015) reviewed the application of Scratch to test the effectiveness of computer science in developing mathematical thinking by examining its capacity to address different elements in learning mathematics namely: modeling of processes and reality phenomena; reasoning; problem formulation and solving, and exercising through execution of procedures and algorithms. The findings of the study indicate a statistically significant gain in understanding mathematical knowledge in the test group, thus corroborating the vice versa relationship between mathematical thinking and computer science.
Euphony F. Y. Yang, Ben Chang, Hercy N. H. Cheng, & Tak-Wai Chan. (2016). Improving Pupils’ Mathematical Communication Abilities through Computer-Supported Reciprocal Peer Tutoring. Journal of Educational Technology & Society, 19 (3), 157-169.
The article recognizes the role of mathematical communication in enhancing students’ mathematical thinking. In addition, it argues that mathematical communication abilities can be improved through computer supporting learning strategies, such as the use of PCs tablets. The article establishes the basis for understanding the relationship between mathematical thinking and computer science. Most importantly, Euphony et al. (2016) offer useful insights into the concept of mathematical thinking to which mathematical communication is an important contributor. Mathematical thinking is recognized as an outcome of social and cognitive activities where mathematical communication plays a central role to increase learning interactions, sharing of mutual mathematical ideas, thoughts, development of mathematical concepts and strategies, and reflection on current mathematical understanding. The key highlight that can be drawn from the article is that computer science can be used to develop mathematical thinking. The resulting mathematical knowledge can then be applied to advance the field of computer science.
Kafai, Y. B., & Burke, Q. (2013). Computer Programming Goes Back to School. Phi Delta Kappan , 95 (1), 61-65.
The message in the article is straightforward: learning computer programming improves students’ problem solving and application designing capabilities. Mathematical thinking applies in such areas, hence justifies the teaching of programming in schools. The trend is driven by digitally based youth cultures that recognize the role computer science plays in the society. The article demonstrates that ability of the strategy to kill two birds with one stone: first through infusion of programming computer skills, and second by improving students' social and cognitive capabilities that are central to mathematical thinking. The advancement comes in the wake of increased need for computational thinking among students, a phenomenon that is instrumental in aspects of systems design, problem solving, and understanding of human behaviors. Therefore, Kafai and Burke (2013) present a range of strategies for teaching computational thinking to meet the demands of the changing cultures of computer science.
Kazimoglu, C., Kiernan, M., Bacon, L., & Mackinnon, L. (2012). A Serious Game for Developing Computational Thinking and Learning Introductory Computer Programming. Procedia-Social and Behavioral Sciences , 47 , 1991-1999.
The article highlights the diversity in application of computer science to enhance the development of mathematical thinking. Kazimoglu et al. (2012) examine a rather unlikely strategy, the use of games, to introduce students to learning about computational thinking. What can be deciphered from the article is that application mathematical thinking in computer science is diverse and teaching of the former varies with the context. For instance, the use of game-based learning (GBL) to impart theoretical and applied knowledge is attributed to the models ease of engagement and motivation for young age groups. Mathematical thinking is a skill that must be nurture through theoretical and practical strategies. For this reason, the article presents framework that uses GBL inculcate students with computational thinking. The article recognizes the importance of computation thinking in its objectives which address: creation and application of algorithms for problems; evaluation of algorithms; debugging algorithms and detecting errors; and simulating algorithms to observe consequences and complete abstractions. The procedures require competent application of mathematical thinking, which can be taught using models drawn from computer science.
Jankvist, U., & Misfeldt, M. (2015). CAS-Induced Difficulties in Learning Mathematics? For the Learning of Mathematics, 35 (1), 15-20.
The article contributes to the growing knowledge on the integration of mathematical thinking and computer science in educational settings. Nevertheless, it does so from a unique perspective by expressing contrary opinion to what is a consensus among researchers in the field – the problematic aspect of the relationship. Jankvist and Misfeldt (2015) use the case study of the use of computer-assisted algebra (CAS) in Danish upper secondary school where teachers have expressed dissatisfaction with its outcomes that are contrary to promised wonders of deep mathematical understanding. By illustrating examples of the difficulties in implementation of the CAS computer teaching and learning strategy, the article contends that the development of mathematical thinking skills remains elusive. The observation highlights the complexity in the relationship between mathematical thinking and computer science, particularly in the interchangeable skills development. The article contends further that success in integration of the two is dependent on blurring the differences using solve and dissolve approach.
Lee, I., Martin, F., & Apone, K. (2014). Integrating Computational Thinking across the K-8 Curriculum. ACM Inroads , 5 (4), 64-71.
The problem of integrating mathematical thinking and computer science in educational settings continues to present difficulties to stakeholders. The article examines the problem and proposes a solution for the same at the K-8 level in the US. The move is informed by the increasing role of computing in all spheres of life. Lee et al. (2014) advocate for clarity in defining and understanding the concept of computational or mathematical thinking including its application to computer science. In addition, the article calls for comprehensive strategies for understanding effective approaches to the development of mathematical thinking among the youth. In the same vein, strategies for integration of computational thinking must be developed. The article highlights the generalizability of evidence-based strategies for teaching and integration of mathematical thinking and computer science. However, the findings have an age bias.
Lye, S. Y., & Koh, J. H. L. (2014). Review on Teaching and Learning of Computational Thinking Through Programming: What Is Next for K-12? Computers in Human Behavior , 41 , 51-61.
The emphasis on the need for integration of mathematical thinking and computer science in educational settings is overwhelming. The article plays an important role in highlighting the effectiveness of existing computer-based teaching and learning models in improving students’ mathematical thinking abilities. Computational thinking using computer science concepts such as abstraction and decomposition is illustrated as an efficient approach to the development of skills in problem solving. Ley and Koh (2014) conducted an analysis of existing computer-based intervention in mathematical thinking in K-12 proposed the use of computational practices and perspectives in the classroom. The article strengthens the advocacy for computer-based learning models in the improvement of mathematical thinking abilities.
Voogt, J., Fisser, P., Good, J., Mishra, P., & Yadav, A. (2015). Computational Thinking in Compulsory Education: Towards an Agenda for Research and Practice. Education and Information Technologies , 20 (4), 715-728.
Computational or mathematical thinking is considered a universal competence. The article draws on the understanding to explore pertinent aspects of the phenomenon. First, it contends that mathematical thinking should be added to every child’s analytical ability to serve as a vital ingredient in learning. As a result, a trend has emerged where computational thinking is a prerequisite of computer science, hence a mandatory skill. The adoption and implementation of computer-based learning strategies imply learners need not only to interact with the systems, but also to understand their working principles and dynamics. The process calls for mathematical thinking, a critical skill in solving algorithmic problems associated with computer science. Therefore, the article recognizes the development and suggests deeper understanding of the concept of computational thinking, what needs to be taught, and how. Incorporation of the challenges and peripheral aspects of the phenomenon is imperative. Furthermore, the article proposes implementation of mathematical thinking aiding strategies in both formal and informal educational settings.
Integration Summary
Computer science is a dominant field in the modern educational settings and other aspects of life. Learners in formal and informal educational settings are increasingly being exposed to computer-based learning and teaching models. As a result, campaigns for adoption and implementation of frameworks shown to be effective and efficient have intensified. The bottom line is the integration of educational systems. Teaching and learning mathematics is one of the areas that continue to benefit from the technological developments. Aho (2012), Euphony et al. (2016), and Kafai and Burke (2013) observed that mathematical thinking is a critical component of computer science, which informed the integration of computer-based learning to advance it. The articles reviewed here in suggest that mathematical thinking is the foundation to understanding computer science. Such is the ideology advanced in Kafai and Burke (2013) who noted that computer programming is becoming a common concept in the classroom to aid the development of mathematical thinking.
Application of computer-based learning in the development of mathematical thinking assumes different strategies including the use of Scratch (Calao et al., 2015 ), Computer-Supported Reciprocal Peer Tutoring (Euphony et al. , 2016), use of computer games (Kazmoglu et al. , 2012), and programming models (Lye & Koh, 2014). The advancement towards education systems where computer-based learning would be mandatory (Voogt et al. , 2015) as effective and efficient frameworks for improving mathematical thinking, understanding the different applied strategies is pertinent. The articles highlight the mutually beneficial relationship between mathematical thinking and computer science, hence the justification for use of the latter to advance the former.
Research on the relationship between the two fields reveals the need for comprehensive understanding of the phenomenon of computational thinking and its application in computer science. Therefore, the articles emphasize on the presence of definition challenges that impede understanding and application. Aho (2012) defined the phenomenon as the “thought processes involved in formulating problems so their solutions can be represented as computational steps and algorithms” (832). The definition is applied across the literature because it recognizes the central of mathematical thinking, problem solving. Mathematical thinking imbues leaners with the capacity to understand computer language, enabling them to communicate with it through critical thought. The development has influenced the adoption of computer-based learning, which has been tested for effectiveness and efficiency at different levels. The review of the models used in teaching and learning across different educational settings has revealed promising outcomes, hence intensification of calls for implementation across the board. Proposals are offered for every model to be reviewed thoroughly prior to adoption.
The relationship between mathematical thinking and computer science is dominated by optimism, painting the image of a system that has no shortcomings. Jankvist and Misfeldt (2015) contend that such is not the case because some computer-based learning models have been established to present difficulties to learners. The contradiction highlights the needs to desist from a generalist approach when implementing models for teaching mathematical thinking and embraced evidence-based approaches. However, inadequate research on the potential of teaching models to yield desired outcomes has diminished progress in the field. Nevertheless, mathematical thinking remains to be a critical ingredient not just in the development of computer science, but also in its application. Therefore, a comprehensive review of the existing frameworks is overdue to ensure systematic and smooth integration of the concepts of mathematical thinking and computer science in diverse educational settings.
References
Aho, A. V. (2012). Computation and Computational Thinking. The Computer Journal , 55 (7), 832-835.
Bell, T., Rosamond, F., & Casey, N. (2012). Computer Science Unplugged and Related Projects in Math and Computer Science Popularization. In The multivariate algorithmic revolution and beyond (pp. 398-456). Springer, Berlin, Heidelberg.
Calao, L. A., Moreno-León, J., Correa, H. E., & Robles, G. (2015). Developing Mathematical Thinking with Scratch. In Design for teaching and learning in a networked world (pp. 17-27). Springer, Cham.
Euphony F. Y. Yang, Ben Chang, Hercy N. H. Cheng, & Tak-Wai Chan. (2016). Improving Pupils’ Mathematical Communication Abilities through Computer-Supported Reciprocal Peer Tutoring. Journal of Educational Technology & Society, 19 (3), 157-169.
Jankvist, U., & Misfeldt, M. (2015). CAS-Induced Difficulties in Learning Mathematics? For the Learning of Mathematics, 35 (1), 15-20.
Kafai, Y. B., & Burke, Q. (2013). Computer Programming Goes Back to School. Phi Delta Kappan , 95 (1), 61-65.
Kazimoglu, C., Kiernan, M., Bacon, L., & Mackinnon, L. (2012). A Serious Game for Developing Computational Thinking and Learning Introductory Computer Programming. Procedia-Social and Behavioral Sciences , 47 , 1991-1999.
Lee, I., Martin, F., & Apone, K. (2014). Integrating Computational Thinking Across the K-8 Curriculum. ACM Inroads , 5 (4), 64-71.
Lye, S. Y., & Koh, J. H. L. (2014). Review on Teaching and Learning of Computational Thinking Through Programming: What Is Next for K-12? Computers in Human Behavior , 41 , 51-61.
Voogt, J., Fisser, P., Good, J., Mishra, P., & Yadav, A. (2015). Computational Thinking in Compulsory Education: Towards an Agenda for Research and Practice. Education and Information Technologies , 20 (4), 715-728.
