Role of Heuristics for Problem Solving — Regina Bruder
The term "heuristic" traces its origins to a story about Archimedes, who was tasked by the King of Syracuse to determine if a wreath was made of pure gold While struggling with this challenge, Archimedes found inspiration in a bathhouse when he noticed water displacement as he entered the tub This revelation led him to solve the problem regarding the wreath, prompting him to famously exclaim, “Eureka, eureka!” The words "Eureka" and "heuristic" share a common root in ancient Greek, giving rise to the academic field of heuristics, which focuses on effective problem-solving methods As noted by Pólya (1964), heuristics encompasses various strategies for tackling complex challenges.
Heuristics focuses on task-solving by identifying key moments within a problem that can lead to effective solutions Its primary objective is to clarify the rationale behind choosing specific aspects of a problem for examination.
This discipline has evolved by closely analyzing various problem-solving approaches and identifying common heuristics Pólya (1949) highlights the significance of comparing these methods to uncover underlying similarities.
Engel (1998), Kửnig (1984), and Sewerin (1979) developed heuristics for mathematical problem-solving tasks, primarily focusing on challenges commonly encountered in talent programs and mathematics competitions.
In 1983 Zimmermann provided an overview of heuristic approaches and tools in
American literature which also offered suggestions for mathematics classes In the
German-speaking countries, an approach has established itself, going back to
Sewerin (1979) and Kửnig (1984), which divides school-relevant heuristic proce- dures into heuristic tools, strategies and principles, see also Bruder and Collet
Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problem-solving processes.
1.1.1 Research Review on the Promotion of Problem Solving
In the 20th century, research on mathematical problem solving advanced significantly, particularly during the 1960s and 1970s, focusing on heuristic strategies based on Pólya's model (1949) Studies indicated that teaching heuristic principles could enhance students' problem-solving skills (Schoenfeld, 1979) This approach, initially successful within talent programs, gained traction in the 1980s, emphasizing the need for problem solving to be central in school mathematics (NCTM, 1980) The understanding of problem-solving in regular classes evolved to include cognitive and heuristic aspects, as well as student-specific factors like attitudes, emotions, and self-regulated behavior (Kretschmer, 1983; Schoenfeld, 1985, 1987, 1992) Kilpatrick (1985) identified five promotional methods from the literature, which can be combined for effective teaching.
• Osmosis: action-oriented and implicit imparting of problem-solving techniques in a beneficial learning environment
• Memorisation: formation of special techniques for particular types of problem and of the relevant questioning when problem solving
• Imitation: acquisition of problem-solving abilities through imitation of an expert
• Cooperation: cooperative learning of problem-solving abilities in small groups
• Reflection: problem-solving abilities are acquired in an action-oriented manner and through reflection on approaches to problem solving.
Kilpatrick (1985) defines success in education as effectively teaching heuristic approaches through clear explanations, relevant examples, and practical problem-solving exercises The importance of familiarizing students with these methods is widely recognized in educational discussions The focus now lies in determining which specific problem-solving strategies or heuristics to teach and the best methods for imparting this knowledge, rather than debating the necessity of teaching them at all.
1.1.2 Heurisms as an Expression of Mental Agility
The activity theory, as advanced by Lompscher (1975, 1985), provides a comprehensive model for understanding learning activities and the varying processes and outcomes in problem-solving among learners (Perels et al., 2005) It posits that mental activity begins with an individual's goals and motivations Lompscher categorizes mental activity into content and process, where content in mathematical problem-solving includes specific concepts, connections, and procedures, while the process encompasses the psychological mechanisms involved in problem-solving Key qualities of this process include systematic planning, independence, accuracy, activity, and agility Furthermore, differences in motivation and expertise suggest that intuitive problem solvers exhibit notably high mental agility, particularly in specific content areas.
According to Lompscher,“flexibility of thought”expresses itself
The ability to shift perspectives and integrate various elements into different contexts highlights the relativity of situations and statements This adaptability enables individuals to reverse relationships and quickly adjust to new mental conditions while simultaneously considering multiple aspects of an activity.
Mental agility is often demonstrated through problem-solving techniques that utilize mathematical methods, aligning with the heuristics identified in the analyses of approaches by Pólya and further explored by Bruder (2000).
Successful problem solvers instinctively distill complex issues to their core elements, employing visualization and structuring tools like informative figures, tables, and solution graphs These heuristic aids not only facilitate the problem-solving process but also effectively document the intuitive approaches taken, making them comprehensible to others.
Successful problem solvers possess the ability to reverse their thought processes or recreate them in reverse order, often doing so instinctively in suitable situations, such as when searching for a misplaced key This capability aligns with a broader heuristic strategy known as working in reverse.
Effective problem solvers consider multiple facets of an issue simultaneously, recognizing interdependencies and adjusting variables strategically This approach often involves eliminating obstacles to promote sustainable ideas.
“hanging on”to a certain train of thought even against resistance Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry
(Engel 1998), the breaking down or complementing of geometric figures to cal- culate surface areas, or certain terms used in binomial formulas.
Effective problem solvers adapt their assumptions and perspectives to uncover solutions, considering various aspects of a problem intuitively By viewing challenges from different angles, they avoid stagnation and gain fresh insights For example, many elementary geometric propositions can be elegantly proven using vector methods.
Successful problem solvers excel at transferring established procedures to new and diverse contexts, as they can easily identify the underlying framework or pattern of a task This ability involves constructing analogies and consistently connecting unfamiliar situations to familiar ones.
Intuitive problem solvers, despite their natural ability to find solutions, often struggle to consciously access their flexible thinking skills As a result, they frequently find it challenging to articulate the processes behind their problem-solving methods.
Creative Problem Solving — Peter Liljedahl
The story of Archimedes highlights a distinct contrast to the heuristics discussed earlier When Archimedes immersed himself in the tub and experienced a moment of clarity, he did not rely on traditional methods such as osmosis, memorization, or cooperation Instead, his breakthrough came from a sudden insight rather than systematic approaches like reduction or transfer Ultimately, it was this moment of illumination that enabled him to resolve his dilemma.
Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.
According to some, such a scenario is the definition of a problem For example,
Resnick and Glaser (1976) define a problem as being something that you do not have the experience to solve Mathematicians, in general, agree with this (Liljedahl
A routine problem is one that can be approached with deliberate effort, making it less likely to lead to significant discoveries To tackle such issues, it's essential to engage in a process of trial and error, relying on persistence Ultimately, breakthroughs often come from sudden inspiration, intuition, or what some may refer to as luck.
Kleitman, participant cited in Liljedahl 2008, p 19).
Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl2008; Mason et al 1982; Pólya1965).
1.2.1 A History of Creativity in Mathematics Education
In 1902, the first half of what eventually came to be a 30 question survey was published in the pages ofL’Enseignement Mathématique, the journal of the French
Édouard Claparède and Théodore Flournoy, two Swiss psychologists, conducted a survey to explore mathematical discovery, creativity, and invention, aiming to engage mathematicians and gather insights for theoretical formulation The initial phase focused on factors influencing the decision to pursue mathematics, such as family background, educational experiences, and social context, along with attitudes towards daily life and personal hobbies In 1904, they published the second part of the survey, which examined the mental imagery associated with creative processes The collected responses were categorized by nationality and released in 1908.
Henri Poincaré (1854–1912), a prominent mathematician, significantly advanced the study of mathematical creativity In 1908, he presented "L’Invention mathématique" to the French Psychological Society in Paris, acknowledging the work of Claparède and Flournoy while asserting that their findings supported his own Poincaré's presentation and the subsequent essay remain among the most insightful explorations of mathematical discovery, creativity, and invention.
I recently departed from Caen, where I had been residing, to participate in a geological excursion organized by the School of Mines The journey distracted me from my mathematical studies Upon arriving in Coutances, we boarded an omnibus to explore the area.
As I stepped onto the platform, a sudden realization struck me: the transformations I had used to define the Fuschian functions were remarkably similar to those found in non-Euclidean geometry Although I didn't have the opportunity to verify this idea immediately, I felt a strong conviction about its validity Later, upon returning to Caen, I took the time to confirm my findings, ensuring that my intuition was correct.
His impactful presentation and profound insights into invention and discovery not only described mathematical creativity but also defined it Since then, discussions on mathematical creativity and creativity in general have consistently referenced Poincaré's contributions.
Jacques Hadamard (1865–1963), a contemporary and friend of Poincaré, was inspired by a presentation to embark on his own empirical investigation into a fascinating phenomenon He criticized Claparède and Flournoy for not adequately addressing key aspects of the topic, believing their extensive survey overlooked crucial questions, particularly concerning the reasons behind failures in creation.
While creativity, discovery, and invention may have distinct definitions, this book will treat them as synonymous terms This simplification, though seemingly minor, has significant implications, as noted by Hadamard (1945), who identified it as his second and most crucial criticism.
In 1943, inspired by his esteemed friend Henri Poincaré, mathematician Jacques Hadamard revised a survey on the topic of failure, seeking input from renowned figures like Albert Einstein Ironically, the updated survey omitted any direct questions about failure, reflecting the reluctance of prominent men to openly discuss such topics.
Hadamard gave a series of lectures on mathematical invention at theÉcole Libre des HautesÉtudes in New York City These talks were subsequently published as
The Psychology of Mathematical Invention in the Mathematical Field(Hadameard
Hadamard's seminal work explores the intersection of mathematics and psychology, offering a captivating insight into the eccentricities and rituals of mathematicians It highlights the beliefs held by mid-twentieth-century mathematicians regarding the processes through which they generate new mathematical ideas The text serves as a comprehensive investigation and a robust argument for the existence of unconscious mental processes in creativity By building on Poincaré's concepts and incorporating a stage theory inspired by the Gestaltists, Hadamard presents a framework that remains a compelling and relevant description of mathematical creativity today.
Mathematical creativity unfolds through four distinct stages over time: initiation, incubation, illumination, and verification, with illumination being just one part of the process (Hadamard 1945) The initiation phase involves deliberate and conscious effort, where an individual actively engages with a problem, often feeling that their attempts are unproductive as they draw upon their past experiences in search of a solution.
The inventive process relies heavily on the tension created by unresolved efforts, which ultimately paves the way for an emotional release during moments of illumination.
After the initiation stage, the solver often experiences a pause in conscious problem-solving, transitioning to the incubation stage where the mind unconsciously processes the issue, a phase that can last from minutes to years This incubation period may culminate in a sudden realization or illumination of a solution, often accompanied by feelings of certainty and positivity Despite the unconscious nature of these processes, it is evident that significant cognitive work takes place during incubation, as noted by thinkers like Hadamard and Poincaré.
In 1945, the phenomenon of illumination serves as compelling evidence of unconscious work, where solutions manifest suddenly after moments of relaxation or distraction This experience, often occurring during activities like walking, showering, or waking up, suggests that our subconscious mind plays a crucial role in problem-solving, as highlighted by Poincaré.
1952) Also deducible is that unconscious work is inextricably linked to the con- scious and intentional effort that precedes it.
Digital Technologies and Mathematical Problem
Solving — Luz Manuel Santos-Trigo
Mathematical problem solving is a key area of research that examines how problem-solving activities enhance learners' comprehension and application of mathematical concepts Central to mathematical practice, these problems are essential for developing the discipline and promoting student learning (Pólya 1945; Halmos 1994) According to Mason and Johnston-Wilder (2006), the significance of these activities cannot be understated.
“The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out”
Tasks play a crucial role in encouraging learners to express their ideas and engage in mathematical thinking In a problem-solving framework, the focus lies on the learners' objectives and their interaction with the tasks Even routine tasks can serve as a foundation for students to expand initial conditions and evolve them into more challenging activities.
Analyzing how mathematical problems are formulated and the processes involved in solving them is crucial for creating effective learning environments that support the development of mathematical concepts and problem-solving skills Additionally, mathematicians have shared their experiences in the field, providing insights into mathematical practices and their connection to learning processes Pólya's work laid the foundation for numerous research programs in problem-solving and has become an essential resource for teachers in structuring their mathematics lessons.
A key aspect of a problem-solving approach in mathematics education is the collaborative inquiry between teachers and students to explore mathematical concepts This involves formulating mathematical problems and engaging in discussions about significant problem types that enhance reasoning skills Understanding the mathematical processes and reasoning necessary for problem-solving is crucial, as is identifying the characteristics of instructional environments that promote these activities Additionally, assessing and characterizing learners' problem-solving competencies is essential, alongside exploring how digital technologies can aid in understanding mathematics and developing these skills Investigating how learners utilize digital tools in problem-solving contexts is vital, as it informs curriculum development and learning scenarios This section aims to highlight important themes that have emerged from research on problem-solving approaches that integrate various digital technologies.
Over the past four decades, significant advancements in the field of problem solving have clarified the nature of mathematical thinking and highlighted effective strategies for learners to develop a strong understanding within problem-solving contexts.
Over the past 40 years, the field has significantly explored the transformations that traditional learning environments undergo when integrating digital technologies in mathematics classrooms This review highlights the key themes and advancements that have emerged, emphasizing the critical role of both teachers and students in this evolving educational landscape.
Throughout history, traces of mathematical problems and solutions highlight humanity's enduring interest in exploring mathematical relationships (Kline, 1972) Pólya (1945) outlines the problem-solving process in four key phases: understanding the problem, devising a plan, executing the plan, and reflecting on the solution He also emphasizes the significance of heuristic methods in each phase of problem solving.
In 1985, a research program was developed based on Pólya’s (1945) concepts to explore how problem-solving heuristics aid university students in tackling mathematical challenges and fostering a mathematical mindset The study revealed that students' success in problem-solving is influenced by their mathematical resources, cognitive and metacognitive strategies, and their beliefs about mathematics Additionally, Schoenfeld (1992) highlighted that while Pólya’s heuristics are broad and applicable, they lack specific guidance for learners on how to effectively integrate these methods into their problem-solving processes He emphasized the importance of students engaging in discussions about key heuristics, such as examining special cases to derive closed formulas for sequences or analyzing relationships in polynomial roots and geometric figures.
Learners must engage with diverse examples to understand how specific heuristics apply across different domains Lester and Kehle (2003) highlight key themes in problem-solving research, such as the challenges students face, the characteristics of successful problem solvers, and the differences between expert and novice approaches They also emphasize the importance of metacognitive beliefs, emotional factors, context, and social interactions in problem-solving environments Research methods have shifted from a focus on quantitative designs to case studies and ethnographic approaches (Krutetskii, 1976) Furthermore, teaching strategies have evolved from teacher-centered methods to active student engagement and collaboration (NCTM, 2000) Lesh and Zawojewski (2007) advocate for extending problem-solving approaches beyond the classroom setting.
Model eliciting activities engage learners by exploring their ideas and thought processes, fostering the development of problem-solving skills Through participation in a collaborative learning community, students continuously refine their competencies while valuing the importance of modeling construction activities.
Recently, English and Gainsburg (2016) have discussed the importance of mod- eling eliciting activities to prepare and develop students’ problem solving experi- ences for 21st Century challenges and demands.
Türrner et al (2007) called on mathematics educators globally to explore the impact and evolution of problem-solving strategies in their respective countries Their findings reveal a strong connection between national mathematical education traditions and the implementation of problem-solving methods For instance, Chinese classrooms employ three instructional strategies: one problem with multiple solutions, multiple problems with one solution, and one problem with multiple variations In contrast, the Netherlands emphasizes a realistic mathematical approach to foster students' problem-solving skills, while France organizes problem-solving activities based on two key frameworks: the theory of didactical situations and the anthropological theory of didactics.
Problem-solving frameworks and instructional methods have traditionally been derived from the analysis of students' experiences with paper-and-pencil tasks However, there is a pressing need to reassess these principles and frameworks to better understand how learners develop skills in environments that systematically integrate digital technologies.
In exploring the impact of multiple purpose and ad hoc technologies on students' learning environments, it is crucial to identify how these tools enhance the representation and exploration of mathematical tasks Specifically, these technologies facilitate the emergence of mathematical reasoning by integrating both mathematical actions and versatile applications, ultimately enriching the educational experience.
Digital technologies are ubiquitous, significantly influencing various social and academic events Mobile devices like tablets and smartphones are revolutionizing communication, interaction, and daily activities.
Mobile technologies play a vital role in enhancing learning environments by facilitating continuous interaction among learners, allowing them to construct knowledge and solve problems effectively (Churchill et al., 2016) These tools provide resources, efficient connectivity for collaboration, and analytical capabilities that support learning activities As Schmidt and Cohen (2013) noted, mobile devices have become indispensable in daily life, significantly influencing cultural and technical advancements In education, mobile tools enable learners to extend mathematical discussions beyond traditional settings, engaging with online materials and interacting with experts and peers Dynamic geometry systems like GeoGebra further enrich the learning experience by allowing students to explore mathematical concepts interactively (Leung & Bolite-Frant, 2015) However, the integration of technology in mathematics education also presents challenges, necessitating a reevaluation of curricula and learning scenarios to accommodate new ways of representing and exploring mathematical problems It is crucial to understand the reasoning skills that learners develop through the use of digital technologies in their mathematical learning processes.