Relationships between Mathematical Thinking and Computational ThinkingAbstract
This article presents an excerpt of the results from a doctoral research investigating, among other aspects, the relationships between Advanced Mathematical Thinking (AMT) and Computational Thinking (CT). The research, employing a qualitative and collaborative approach, analyzed a continuing education course for Mathematics teachers on Visual Programming (VP), offered in the small Massive Open Online Course (sMOOC) format. The VP language was chosen due to its recognized effectiveness in promoting Computational Thinking. The research aimed to address the question: How are the relationships between Computational Thinking and Advanced Mathematical Thinking established and manifested by Mathematics teachers during a training process on Visual Programming in a sMOOC format? The results showed that VP activities involving mathematical concepts are effective in developing both AMT and CT in teachers. Additionally, the study identified specific relationships between AMT processes and CT pillars, providing a detailed understanding of how these skills can be integrated and mutually enhanced. These findings underscore the importance of integrating these two types of thinking within the educational context, utilizing VP as an innovative pedagogical approach. The article highlights the relevance of such integrations for enhancing learning and promoting technological skills in Basic Education.
Keywords:
Computational Thinking; Advanced Mathematical Thinking; Visual Programming Language; Scratch; Teacher Training
Resumo
Este artigo apresenta um recorte dos resultados de uma pesquisa de doutoramento que investigou as relações entre o Pensamento Matemático Avançado (PMA) e o Pensamento Computacional (PC). A pesquisa, de abordagem qualitativa e colaborativa, analisou um curso de formação continuada de professores de Matemática sobre Programação Visual (PV), oferecido no formato sMOOC (small Massive Open Online Course), caracterizado por número reduzido de participantes e possibilidade de mediação pedagógica. A escolha pela linguagem de PV deve-se ao seu reconhecimento na literatura, considerado eficaz para a promoção do Pensamento Computacional. O foco deste artigo é identificar as relações entre os pilares do PC (abstração, reconhecimento de padrões, algoritmos e decomposição) e os processos do PMA (representação e abstração). A pesquisa responde ao problema: como são estabelecidas e manifestadas as relações entre o Pensamento Computacional e o Pensamento Matemático por professores de Matemática durante um processo formativo sobre Programação Visual no formato sMOOC? Os resultados indicam que atividades de PV que envolvem conceitos matemáticos são eficazes para desenvolver tanto o PMA quanto o PC dos professores. Além disso, o estudo revelou relações específicas entre os processos do PMA e os pilares do PC, oferecendo uma compreensão mais detalhada de como essas habilidades podem ser integradas e aprimoradas mutuamente. Essas descobertas destacam a importância de integrar o PC e o PMA no contexto educacional, utilizando a PV como uma abordagem pedagógica inovadora, e ressaltam a relevância de tais integrações para melhorar a aprendizagem e promover competências tecnológicas na Educação Básica.
Palavras-chave:
Pensamento Computacional; Pensamento Matemático Avançado; Linguagem de Programação Visual; Scratch; Formação de Professores
1 Introduction
Contemporary society, marked by technological transformation, has witnessed a profound impact of digital technologies (DT) on various aspects of everyday life and social practices. As highlighted by Lévy (2000, 2015), digital technologies have promoted significant transformations in social and productive processes. These changes have implications for the educational field, underscoring the need to adapt pedagogical practices to keep pace with the new realities and emerging demands of the digital society.
In this context, computational thinking (CT) emerges as a competence to be stimulated in basic education. It involves skills such as the systematic analysis of problems, the definition of algorithms and the resolution of complex issues, actions considered relevant for the formation of students in the 21st century (Brasil, 2018).
Despite the growing integration of CT into mathematics teaching, the literature reveals a significant gap in understanding the interactions between computational thinking (CT) and mathematical thinking (MT). The National Common Curriculum Base (Base Nacional Comum Curricular - BNCC) emphasizes the importance of CT, but does not provide a robust theory that integrates MT with CT (Brasil, 2018). The predominance of research on CT in mathematics education, as noted by Barcelos et al. (2015), underscores the need to explore the intersections between CT and MT in a more integrated, theoretically grounded manner. Existing investigations often treat the CT in isolation, without addressing how it can interact with the MT to promote a deeper understanding of mathematics.
This article is an excerpt from a doctoral research project that aimed to analyze these interactions in a formative context. The research investigated a continuing education course for mathematics teachers, focused on visual programming (VP) in the small Massive Open Online Course (sMOOC) modality, composed of a reduced number of participants and the possibility of pedagogical mediation. The initial hypothesis, that there are relationships between CT and MT, and that VP can be a promising approach to explore these relationships in basic education, has been proven. The analysis revealed that VP activities can effectively promote the simultaneous development of both thoughts.
The objective of this article is to identify the relationships between the pillars of CT (abstraction, pattern recognition, algorithms, and decomposition) and the processes of MT [AMT] (representation and abstraction), as part of the doctoral research that responds to the problem: How are the relationships between computational thinking and mathematical thinking established and manifested by mathematics teachers during a formative process on visual programming in the sMOOC format?
The article is organized as follows: first, we present the research methodology; then, the theoretical basis; and finally, a discussion of the results using the existing literature. The text concludes with considerations and indications for future research.
2 Methodological procedures
Given the nature of the study and the need for a deep understanding of the formative process, a qualitative approach was chosen, which allows the detailed exploration of the participants’ experiences and perceptions (Goldenberg, 2004; Minayo, 1994). The choice of a qualitative methodology aligns with the definition of qualitative research, which is more concerned with a deep understanding of social phenomena and practices than with statistical representativeness (Goldenberg, 2004). According to Minayo (1994), this approach seeks to understand the phenomenon from the actions and perspectives of the subjects involved.
To carry out the study, the research cycle described by Minayo (1994) was followed, which includes the exploratory phase, fieldwork, and data processing. In the exploratory phase, the methodology was defined and the research project developed, including a review of relevant literature and the establishment of the necessary theoretical foundation. The investigation of an online course for mathematics teachers led to the adoption of collaborative research, which integrates knowledge production and professional education (Desgagné, 2007). Although the course participants did not participate in the initial definition of the research directions, the investigation followed the principles of collaborative research, promoting the integration of knowledge production and professional education. The collaboration took place through active listening, the valorization of teaching knowledge and the joint analysis of the pedagogical practices emerging in the course. Thus, the research respected the fundamentals of the collaborative approach, albeit in a model of participation more concentrated in the phases of development and analysis.
The continuing education course entitled Visual Programming Course for Mathematics Teachers [Curso de Programação Visual para professores de Matemática] was developed based on Costa et al.’s (2015) and Nunes et al.’s (2017) guidelines for MOOCs, and made available on a platform created by members of the Research Group on Innovation and Technologies in Education (Grupo de Pesquisa em Inovação e Tecnologias na Educação - GPINTEDUC). It included both asynchronous and synchronous activities distributed across four units. The course had 80 hours of workload, was offered free of charge and certified by the platform, taking place from August 2021 to February 2022. The data were collected from the participants’ interaction with the platform, including questionnaires, tasks, discussion forums, and synchronous meetings. Participant observation was used to record and analyze the dynamics and interactions during the activities.
Although the application form reached 100% of the 100 vacancies, the MOOC platform registered 96 applications. Of these, 73 completed the initial questionnaire, and 22 completed all course activities. The certification criterion was to complete at least 75% of the activities, resulting in certification for 24 participants.
The collaborative research focused on certified participants because the complete formative process was essential for the analysis. The 24 certified teachers participating in the research ranged in age from 24 to 61 years. They represented all regions of Brazil, with the highest concentration in the southern region.
Most participants worked in public schools, with one teacher working simultaneously in two education systems. Teachers taught from the 6th grade of elementary school to the 3rd grade of high school, predominantly in the 9th grade of elementary school. Many performed in more than one grade throughout the school year. The participants’ professional experience varied, with the majority having about 15 years of teaching experience.
Data analysis followed the procedures recommended by Bardin (2016), including pre-analysis, material exploration, and result interpretation. The data were organized and analyzed to identify relationships between CT and MT. For this, it was important to understand in depth what these thoughts are about.
3 Computational Thinking
The growing speed of the internet has accelerated changes in the forms of human communication and information. The new generations experience the rapid pace of technological development, which aims to meet the needs of 21st-century society. The contemporary social scenario demands new professionals, new citizen action, and a new relationship between society and DTs (Lévy, 2000). These technologies can be integrated into various activities, including educational ones. In this way, the curriculum-guiding documents adapt to direct teaching actions, enabling the school to stay in tune with social transformations.
An example of this insertion is evident in the BNCC, which suggests the use of DTs so that students are not mere consumers of apparatuses, but can create, learn, and develop as citizens through the use of different technologies (Brasil, 2018). In line with the proposal to make students more active in using DTs, so that they can apply knowledge from computer science to everyday problem situations, CT has been researched in the educational context over recent years. According to Rocha and Pinto (2021, p. 101), CT corresponds to the “name given to human mental processes that lead individuals to solve problems, using elements of computer science, such as algorithms and abstraction.”
As Rocha and Pinto (2021) note, the CT can be developed from processes used to solve problems. Among them, abstraction, problem decomposition, algorithm development, and pattern recognition, known as the pillars of the CT, stand out (Brackmann, 2017).
Although the four pillars are widely accepted, there are variations in the literature. Brennan and Resnick (2012), for example, identify dimensions of CT that include concepts, practices, and perspectives, while Ribeiro, Foss, and Cavalheiro (2020) focus on abstraction, automation, and analysis. On the other hand, the International Society for Technology in Education (ISTE) and the American Computer Science Teachers Association (CSTA) list nine concepts related to PC (ISTE/CSTA, 2011), and Grover and Pea (2013) identify elements such as modularization and conditional logic. Later research, such as Code.Org (2016) and Liukas (2015), helped consolidate the four main pillars of CT, reflecting the evolution and refinement of the field.
Decomposition involves breaking down a complex problem into smaller, more manageable parts. The technique “consists of decomposing the problem into smaller problems, solving them and combining the solutions to obtain the solution of the original problem” (Ribeiro; Foss; Cavalheiro, 2020, p. 25).
Pattern recognition refers to the identification of similarities and regularities. In CT, patterns “are similarities or characteristics that some of the problems share and that can be explored so that they can be solved more efficiently” (Brackmann, 2017, p. 35). Recognizing patterns, therefore, corresponds to recognizing the similarities in the data or processes that involve problem solving. Ribeiro, Foss, and Cavalheiro (2020, p. 25) use the term generalization to address these characteristics, pointing out that it refers to “a technique that consists of building a more generic solution (algorithm) from another, allowing the new algorithm to be used in other contexts”. Thus, pattern recognition occurs when perceiving a certain constancy or similarities in some events.
The pillar of abstraction in computational thinking (CT) involves selecting and filtering data essential for problem solving, ignoring irrelevant information. Brackmann (2017) defines abstraction as the ability to identify and focus on important elements, creating a representation of the problem that facilitates its understanding and solution. For example, when using a map to plot a route, details such as altitude are omitted as they are not necessary for navigation. Brennan and Resnick (2012) and Ribeiro, Foss, and Cavalheiro (2020) highlight that abstraction simplifies reality and represents crucial aspects of a problem and its solution, using records, lists, and graphs to formalize ideas. Thus, abstraction allows the individual to select data and develop solving processes described by algorithms.
Finally, algorithms are sequences of steps to solve a problem or perform a task. According to Brackmann (2017), an algorithm is an ordered sequence of instructions to achieve a goal, which can be described in diagrams, pseudocode, or programming languages. For example, a cooking recipe or the programming of an application follows the same logic of instructions. Ribeiro, Foss, and Cavalheiro (2020) highlight that defining an algorithm requires understanding the basic instructions and operations. In mathematics, algorithms are fundamental, as in the teaching of division and algebra, which involve algorithmic procedures. The BNCC notes that algorithmic language has similarities with algebraic language, especially in the use of variables and pattern identification.
In the educational context, various CT approaches have been used over the years, including unplugged computing, digital games, programming languages, visual programming languages (VP), and pedagogical robotics. Each of these approaches offers possibilities to explore CT and can be adapted to meet students’ needs and skills. Programming, for example, can be performed from VP software, which provides an intuitive way to code and solve problems. VP is particularly useful in basic education, which does not aim to train professional programmers. At this stage of schooling, VP can provide students with a fundamental understanding of programming and problem-solving concepts.
VP refers to programming modes that do not require advanced textual descriptions of algorithms. Its commands are described in blocks or other graphical elements that connect to make the programming easier to understand. According to GPINTEDUC (online), VP “is one whose commands are described by blocks, mnemonics, or other graphic elements, not depending on advanced textual description of algorithms”.
For Resnick et al. (2009), programming in software that uses VP enables the learning of mathematical and computational concepts, fostering creativity, collaborative work, and a range of problem-solving and strategy development skills. We can mention, for example, the Scratch software, which has a scenario plotted on a Cartesian plane. The movement of characters that can be inserted into narratives or games is defined by pairs of ordered points in the plane and angles. In this sense, the user can use or improve the mathematical knowledge about these concepts. By applying concepts of decomposition, pattern recognition, abstraction, and algorithms across varied contexts, students enhance their ability to solve problems and think logically and structurally. This development is relevant for MT, which, like the CT, involves problem-solving and model-making processes.
4 Mathematical Thinking
The theory adopted for the research was developed by the mathematician Tommy Dreyfus, based on discussions of a group interested in the psychological issues of understanding in mathematics. This group aimed to advance students’ mathematical understanding, and the theories developed have implications for theories addressing advanced mathematical thinking (AMT). Since the creation of the working group on the MT, different perspectives have emerged on what constitutes an advanced understanding of mathematics. According to Selden and Selden (2005), the AMT can be analyzed in the final grades of secondary education and higher education, as an alternative to overcome epistemological obstacles or in advanced mathematics activities in the development of the MT.
Some scholars have sought to differentiate elementary mathematical thinking (EMT) from AMT. Dreyfus (1991) argues that there is no clear distinction between EMT and AMT processes, but notes that AMT involves more abstraction, definition, and deduction. For understanding in the student’s mind, a sequence of activities that allows interaction among mental processes and components is important, including the representation of concepts and properties, abstraction, discovery, intuition, verification, proof, and definition. The interaction between these components is what Dreyfus calls AMT. Thus, AMT is configured as the interaction between representation and abstraction in cognitive processes that involve mathematical interpretation (Dreyfus, 1991). MT can be understood as the cognitive process of solving non-trivial problems and describing the process of solving them, in addition to isolated knowledge or the memorization of algorithms.
Dreyfus (1991) discusses the importance of understanding mathematics, rather than merely knowing topics or isolated skills related to the discipline. The author mentions that students who complete degrees in calculus, algebra, or geometry often do not develop mathematical know-how. On the contrary, “they learn the products of the activity of many mathematicians in their final forms, but do not gain insights into the processes that led mathematicians to create those products (Dreyfus, 1991, p. 28, our translation).
Another relevant explanation by Dreyfus (1991) concerns the term ‘thinking,’ which is constituted by the student’s mental constructions. When a student builds a graph, it is not enough to use mathematical technical elements; it is necessary to create a mental representation that assists in interpreting and representing a problem. Dreyfus considers thinking the process of creating mental alternatives to represent, visualize, identify, select, and solve a problem. This thinking is mathematical when the process is based on mathematical representations and abstractions. It can be more elementary when these representations are limited, and the student cannot yet abstract. However, it can become more advanced as the repertoire of representations expands and the abstraction process matures.
According to Dreyfus’s theory (1991), MT involves processes that can be verified during students’ learning, the main ones being representation and abstraction.
To represent a mathematical object, the student can formalize it as a drawing, a symbol, an unknown or a digit. These representations are the forms of communication with the external world (Dreyfus, 1991). Thus, the ideas that we want to transmit mathematically will be understood as we make use of our own and the universal language. The use of several representations for the same mathematical concept can contribute to the understanding of the mathematical object and, consequently, to the development of AMT (Bussmann; Klaiber; Silva, 2017, p. 2).
Dreyfus (1991) explained that when faced with a mathematical problem, students seek cognitive structures that assist in understanding and resolving it. These structures can deepen and allow new possibilities of representation. The expansion of the repertoire of representations contributes to the advancement of MT from elementary to advanced, since “mental representations are created in the mind based on this concrete system of representations” (Dreyfus, 1991, p. 31, our translation). These representations allow students to use cognitive elements to interpret varied problems. The lack of a repertoire of representations can result in limited interpretive capacity, as when students know how to solve a type of exercise but face difficulties with small changes to the statement.
The representation process is explained in AMT theory by two sub-processes: (a) interpret and alternate; (b) model. According to Dreyfus (1991), the expansion of the range of representations involves the ability to interpret and alternate between elements that represent a mathematical situation. The student also needs to model the situation. As students acquire modeling and interpreting skills, they develop greater flexibility in problem solving. This is because they can transform problems into mathematical representations and manipulate these representations in different ways to find solutions (Dreyfus, 1991). Therefore, the representation process is essential to the advancement of AMT, as it deepens and sophisticates the ability to think and solve mathematical problems.
Another AMT process is abstraction. According to Dreyfus (1991), when a student consciously develops the ability to elaborate mathematical abstractions, they reach a higher level of MT. The genetic epistemology theory of Jean Piaget (1896–1980) presents that abstract thinking is related to the maturation process of the individual. According to Piaget (1995, p. 86), abstractions are located in the formal operative stage and refer to the “skills of engaging in propositional reasoning and logical deductions without the support of concrete objects”.
For the student to abstract mathematically, the sub-processes of generalization and synthesis are mobilized (Dreyfus, 1991). The generalization starts from the particularities of mathematical information, seeking validation through inductive methods. According to Dreyfus (1991, p. 35, our translation), “to generalize is to derive or induce from particularities, to identify similarities, to expand domains of validity”.
The synthesis, in turn, associates concepts and contents to solve the proposed problem situation. Synthesizing, therefore, “means combining or composing parts in such a way that they form a whole, an entity” (Dreyfus, 1991, p. 35, our translation). Dreyfus (1991) also explained that the processes of representation and abstraction are closely linked, and that when a student develops one, he inevitably advances in the other. The author also notes that MT goes beyond representation and abstraction, including discovery, intuition, verification, proof, and definition. Despite this, he considers the processes discussed to be the most relevant for advancing students’ mathematical understanding from EMT to AMT.
5 Starting the integration between CT and MT: organizing the analysis
This section presents the initial stage of the data analysis, following Bardin (2016). The author indicates three chronological poles for the organization of the analysis: (a) the pre-analysis; (b) the exploration of the material; and (c) the treatment of the results and the interpretation. According to the author, this organization must include selecting documents, preparing the material, and formulating grounds that will affect the final interpretation. The data must be processed and interpreted.
In reviewing the research data, we sought to verify evidence or direct mentions of possible relationships in the analyzed documents, considering the elements outlined in Dreyfus’s theory (1991) and the pillars of the CT. Thus, activities that could address these relationships, directly or indirectly, were selected.
The activities were previously analyzed in this investigation with the selection and organization of the data that would be used, from seven documents obtained in the formative course, which are: (a) Unit 1: Initial questionnaire; Activity on Unplugged Computing; Activity focused on the Programaê!; (b) Unit 3: Activity focused on the Scratch and (c) Unit 4: Final questionnaire; Word cloud; Mind map1.
Bardin (2016, p. 131) recommends that the researcher explore the material and interpret the results, “proposing inferences and advancing interpretations regarding the expected objectives”. In this process, it was observed that not all selected documents would be used in the next phase of content analysis. The Initial Questionnaire, for example, although it provided relevant information about the teacher’s initial knowledge of the CT and the MT, did not, of course, present concepts developed after the conclusion of the formative course. The word cloud and the mental map were important for supporting the categorization process, but, as they did not present concrete texts, they were explored only during the organization stage.
The Initial Questionnaire (IQ) aimed to understand the participants’ profiles and their initial understandings of CT and MT. Of the 24 teachers, 16 knew the term CT, with most associating it with problem solving and the use of DTs. Regarding the MT, 19 students had already heard of the term, but few of their answers reflected Dreyfus’s (1991) theory. Most linked the MT to problem solving using logical reasoning and mathematical knowledge.
In the Unplugged Computing activity, which did not require a description of the relationship with the MT, some teachers suggested that problem solving in mathematics could be emphasized. The students’ answers were recorded and classified as present or absent for MT subprocesses. Of the 24 activities analyzed, 16 showed no relationship with MT sub-processes. In eight answers, evidence of alternation and interpretation was identified, and one response indicated the presence of synthesis.
In the activity carried out on the Programaê! platform, teachers should identify the skills students developed and the mathematical content related to a specific activity. All 24 answers were recorded and classified according to the presence or absence of MT subprocesses. The students mentioned content in geometry, algebra, and arithmetic, as well as problem solving and logical reasoning. Quantitative verification revealed that 14 of the 24 responses indicated the presence of MT subprocesses according to Dreyfus’s (1991) theory. Despite being in the initial phase of the course, the students have already suggested possible relationships between CT and the Dreyfus subprocesses when considering the Programaê! resource.
The programming activity in Scratch was proposed in Unit 3, a unit that had already explored the concept of MT. The first activity in this unit featured a video of a basic education student programming in Scratch. The student should carry out a program that involves constructing a rectangle. The teacher was invited to watch the video and identify the MT processes, seeking to deteremine whether and how the student had developed interpretation, modeling, generalization, and synthesis. At this point in the course, the teacher had already studied the theory proposed by Dreyfus (1991) and already knew the characteristics of these processes.
All participants recognized interpretation and alternation; 21 identified modeling, 18 generalization, and 13 synthesis. Some students suggested that teacher mediation could improve the exploration of synthesis and generalization processes. Although the activity does not establish direct relationships between CT and MT, it does suggest that the latter can be developed through VP activities in Scratch. Students’ answers were fully used in the analysis categorization.
Students also participated in an activity in which they chose three representative words for both CT and MT, resulting in a word cloud. The word abstraction was the most mentioned as common to both types of thinking. Other terms highlighted included algorithm, decomposition, problem solving, pattern recognition, and generalization. These terms were used for the analysis categorization.
Examining the mental maps prepared by the 24 certified students, we realized the proximity between the MT and the CT, as identified and explained by each participating teacher. They highlight processes common to those types of thinking, such as algorithms, problem decomposition, pattern recognition, and representation. However, there was disagreement regarding the concept of abstraction. While some students consider abstraction similar in both types of thinking, most observe that, in mathematics, it involves more than the exclusion of irrelevant parts, encompassing the extraction of mathematical concepts, generalization, and synthesis, according to Dreyfus’s theory (1991).
The relationships presented by each of the 24 participants who produced mental maps revealed that the processes that appeared most often were abstraction, algorithm, decomposition, modeling, pattern recognition, representation, and problem solving, which align with the created word cloud.
The Final Questionnaire (FQ) was completed by 22 of the 24 certified students and revealed an expanded definition of CT, with all responses aligned with the pillars described by Liukas (2015). As for the MT, the answers were more detailed, reflecting the processes and subprocesses outlined in Dreyfus’s (1991) theory. All students confirmed the existence of relations between MT and CT. The FQ also evaluated the formative course, with positive feedback on the process and the use of VP activities in mathematics classes, although the number of participants who completed the activities was lower than the initial number of teachers.
6 Bringing the CT closer to the MT: coding
After organizing the analysis of the documents, the second phase proposed by Bardin (2016) began: coding. Considering the terms present in the word clouds and the relationships identified in the mental maps, some hypotheses for categorization emerged. In a previous listing, the frequent terms were: representation, abstraction, pattern, algorithm, and decomposition. The terms adhered to the theoretical foundation of the research, since they referred to the pillars of the CT. A previous coding was created, recognizing that all the material analyzed so far allowed us to verify the following relationships: (a) Representation with Abstraction; (b) Representation with Pattern; (c) Representation with Algorithm; (d) Representation with Decomposition; (e) Abstraction with Abstraction; (f) Abstraction with Pattern; (g) Abstraction with Algorithm; (h) Abstraction with Decomposition.
The units of meaning were defined inductively from the data and anchored in the theoretical foundations of Dreyfus (1991) and the pillars of computational thinking. The organization into subcategories allowed for greater refinement in qualitative analysis, highlighting how teachers understood and integrated mathematical and computational concepts. This structure was fundamental to interpreting the formative effects of the course on the participants’ teaching practice.
The above codes corresponded to the units of meaning that should be sought in the activities carried out by the course teachers. The texts for these activities were already available in electronic spreadsheets and, in view of the aforementioned units of meaning, all were read, and the coding process was carried out. The texts were cut and classified into one of the codes. It was not necessary to change the previous coding, as the codes created made it possible to encode the data.
7 Understanding the relationships between the CT and the MT: categorization and inferences
Coding led to the third phase of analysis, categorization. The coded texts were grouped into tabs in the spreadsheet and analyzed according to the theoretical framework of the research. This process allowed the definition of categories and subcategories. The texts of the first four codes showed the relationships between the CT and the process of MT representation, while the texts of the last four codes addressed the relationships between the CT and the process of MT abstraction. Thus, the feasibility of a new grouping from the eight text groups was perceived, as shown in Chart 1.
Each subcategory was analyzed to allow understandings to emerge from the participants’ texts in the fourth phase proposed by Bardin (2016): inference.
In the first category, the relationships between the CT and the representation process were investigated. The objective was to verify how CT could assist students in creating, selecting, and understanding mathematical models, as described in Dreyfus’s theory (1991). This category was subdivided into two subcategories: the first focused on texts that mentioned the alternation and interpretation subprocesses, while the second focused on elements related to the modeling subprocess.
In the first subcategory, which concerns the CT’s contributions to alternating and interpreting, participants indicated that abstraction plays a fundamental role. It allows the student to separate the non-essential parts of a problem by focusing on interpreting what is most relevant. According to the reports analyzed, this ability to abstract and simplify complex information facilitates the understanding and manipulation of mathematical representations. The data analyzed show that, by applying abstraction related to CT in the mathematical context, students can represent mathematical concepts in various ways, such as graphs, tables, equations, and algorithms.
In addition, the participants also pointed out the importance of decomposition in the process of alternating representations. The analysis revealed a comparison between decomposition and the ability to represent non-mathematical situations, using data division into different representative forms. The direct influence of two foundational pillars for alternating and interpreting was observed: abstraction and decomposition. The ability to abstract helps students simplify complex problems by focusing on what is relevant to interpretation. In turn, decomposition enables the problem to be divided into smaller parts, allowing them to be represented in different ways. Chart 2 presents examples of texts that show the contribution of the abstraction and decomposition pillars to the alternating and interpreting sub-process.
The elements present in the processes indicated by the students dialogue with Dreyfus (1991) theory. Interpretation, for the theorist, corresponds to the selection of different representations and the transit between them, which is present, for example, in the terms select, chose and alternated, and range of representations.
In the second subcategory of the relationships between the CT and the representation process, we explored how the CT contributes to the modeling sub-process. The data showed a connection between the modeling and the fundamental pillar of the CT, the algorithm. Teachers highlighted that, when using a programming language, students can model situations by creating detailed algorithms, which facilitate the accurate representation of real problems and allow them to better understand the situations, enabling simulations and tests before the final solution. In addition, the practice stimulates critical and analytical thinking, helping to identify the main elements of the problem, define the necessary steps, and evaluate the effectiveness of the proposed solution. Examples of analyzed excerpts that showed such a relationship are shown in Chart 3.
The excerpts dialogue with the theory of MT, since modeling, for Dreyfus (1991), is the transformation of a problem situation into a mathematical representation. This sub-process is evidenced, for example, in the terms graphical-geometric representation and represented mathematically.
In the second category, we sought to verify the relationships between the CT and the MT abstraction process, in two subcategories: relationships between the CT and the synthesis process; and relationships between the CT and the generalization process. The excerpts from the students’ answers indicate that decomposition helps divide complex problems into smaller parts, facilitating solution by grouping and combining them. According to Dreyfus (1991), solving complex mathematical problems involves identifying variables, solving equations separately and combining the final solutions, which simplifies solving the problem. The data from the continuing education course corroborate this idea, showing that decomposition in computational thinking (CT) is related to the synthesis of mathematical thinking (MT). In addition, selecting relevant data on the problem from the CT abstraction was mentioned as a useful factor for the synthesis. The data reveal that students understand synthesis as the ability to combine different elements and concepts to reach a solution through an integration process. They mention that, in the context of the CT abstraction process, synthesis involves combining parts or components to create a coherent and functional whole.
In this sense, it was observed that identifying and filtering the information necessary to solve the problem facilitates combining these selected elements. When selecting relevant data, the student or programmer focuses on specific aspects of the problem, discarding unnecessary or irrelevant information. Once the relevant data is identified, it can be combined and organized in different ways to solve the problem. In VP language, the student must select which resources they want to use to program, filter what is relevant to programming, and perform a structured mental organization of their goals. Some students mentioned that these processes require synthesis, as in AMT. Chart 4 shows excerpts of comments that illustrate these relationships.
It is noted that the terms association and combination used by teachers in the synthesis align with Dreyfus’s (1991) description. For him, synthesis refers to problem solving based on the combination of different mathematical concepts. In this sense, the contribution of the pillars of abstraction and decomposition to the synthesis sub-process in mathematics is evident.
The last subcategory showed that the CT can contribute to the generalization process through the pattern recognition pillar. This relationship was the most consensual among teachers and had more convergent texts. The data showed that pattern recognition is critical for generalization, as it involves identifying regularities or common features in data, situations, or problems. This skill enables the student to generalize concepts and apply them across different contexts. Chart 5 shows examples of excerpts present in this subcategory.
Regarding generalization, mentions such as constructing another, validating, and using the concept learned in other situations align with Dreyfus’s (1991) definition, which indicates the use of a concept in general contexts.
We observed relationships between the CT pillars and two abstraction subprocesses, which, according to Dreyfus (1991), are exclusive to AMT. Thus, we conclude that the CT can support the advancement of the MT by facilitating the development of the sub-processes of synthesis and generalization. The relationships between CT and AMT are summarized in Figure 1.
8 Final considerations
Throughout this study, the interactions between computational thinking (CT) and mathematical thinking (MT) were investigated in a formative process with mathematics teachers, using visual programming (VP) in the sMOOC format. The central question was: How are the relationships between computational thinking and mathematical thinking established and manifested by mathematics teachers during a formative process in the sMOOC format for visual programming? To answer it, we used Bardin’s (2016) content analysis, organized into spreadsheets, resulting in two main categories: (a) relationships between the CT and the representation process; and (b) relationships between the CT and the abstraction process, both with subdivisions that detail the connections between the CT pillars (abstraction, decomposition, algorithm, and pattern recognition) and the MT subprocesses (modeling, synthesis, generalization, and alternation of representations).
In the first category, we observed that abstraction and decomposition favored new forms of mathematical representation, deepening the conceptual understanding. Decomposition, in particular, was relevant to the alternation and interpretation between different representations. In the second category, mathematical modeling involved the development of algorithms, enabling participants to create step-by-step solutions to mathematical problems. Decomposition and abstraction also contributed to synthesis and generalization, articulating smaller parts into coherent and transferable solutions across different contexts.
The results indicate that CT relates not only to school mathematical thinking (SMT) but also to advanced mathematical thinking (AMT), thereby favouring the development of mathematical abstraction. For these relationships to materialize in the classroom, teachers must propose projects that integrate mathematical content with VP, using appropriate software and engaging in pedagogical mediation that stimulates students’ creativity and inventiveness.
Although the course was structured to explore these relationships, the empirical observation made by the participants is not trivial. The relevance of the conclusion lies in explaining how these relationships are manifested in teaching practice. Teachers not only recognized the connections between CT and MT but also described and applied them in concrete pedagogical contexts, evidencing a process of conceptual and methodological appropriation. The analysis of the manifestations revealed important nuances about how the pillars of the CT are articulated with the processes of the MT, offering valuable subsidies for teacher education and curriculum development.
Finally, we recommend that future research apply these practices to basic education students, investigate new VP resources, and analyze how students manifest CT and MT in their productions, thereby expanding the understanding of the impact of these approaches on mathematics teaching and learning.
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Data Availability:
The data generated or analyzed during this study are included in the published article.
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The complete description of the formative course, as well as the proposed activities, can be accessed in the original publication of the thesis: ROCHA, Flavia Sucheck Mateus da. Relações entre o pensamento matemático e o pensamento computacional: compreensões a partir de um curso de formação continuada de professores de matemática [Relationships between mathematical thinking and computational thinking: understandings from a continuing education course for mathematics teachers]. 2023. 237 f. Tese (Doctoral thesis in Education in Sciences and Mathematics) – Federal University of Paraná, Curitiba, 2023.
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Editor-in-Chief:
Prof. Dr. Marcus Vinicius Maltempi
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Associate Editor:
Profa. Dra. Ana Paula Jahn
The data generated or analyzed during this study are included in the published article.
Source: