What You Can Learn from Mathematical Problem Solving
A Mathematical Problem Solving Guide by the University of Dating
Mathematical Problem Solving (by Alan H. Schoenfeld) is a classic work that delves into the process of mathematical problem solving and highlights the elements that promote effective problem-solving. This book provides a framework for analyzing mathematical behavior and covers elements such as resources, heuristics (methods and strategies for problem-solving), control, and belief systems, all of which are necessary for enhancing knowledge about problem-solving.
1. Overview of Mathematical Problem Solving
Mathematical Problem Solving views mathematical problem solving not just as a method for obtaining answers but as an important process in itself. The author, Alan Schoenfeld, provides a detailed analysis of the thinking process involved in problem-solving and suggests educational approaches to nurture problem-solving skills.
Overview of Mathematical Problem Solving This book emphasizes the importance of deeply exploring the approach and mindset toward problems, rather than merely focusing on the process of obtaining answers. Schoenfeld explains how mathematical thinking develops and how mathematics can be effectively learned and taught.
2. Cognitive Processes in Problem Solving
The book discusses the cognitive processes involved in problem solving. Schoenfeld explains how mathematicians think and arrive at solutions during the problem-solving process. This allows learners to not just memorize theorems but to understand how they can solve problems on their own.
Cognitive Processes in Problem Solving The book provides a detailed explanation of how cognitive skills and strategies necessary for solving mathematical problems develop, and how these skills can be learned and enhanced. Schoenfeld portrays the problem-solving process as a cycle of "insight" and "trial and error," illustrating how mathematical thinking matures.
3. Teaching and Learning Problem Solving
Schoenfeld offers much advice to educators, especially on how to teach problem-solving skills to students and how to draw out learners' problem-solving abilities. The insights in this book are invaluable not only for teachers but also for researchers interested in educational theory.
Teaching and Learning Problem Solving This section explains specific strategies and examples for how educators can effectively teach problem-solving to students. It emphasizes the importance of fostering a mindset where students actively think and continuously challenge themselves as they build their problem-solving abilities.
4. Strategies for Problem Solving
Schoenfeld introduces specific strategies for mathematical problem solving. These strategies support the entire process of solving problems, from understanding the problem to constructing a solution, and even verifying the answer.
Strategies for Problem Solving Effective strategies for solving mathematical problems include breaking the problem into smaller parts, solving backwards, finding patterns, and reviewing the problem's conditions. Schoenfeld explains, through examples, how these strategies help in solving problems.
5. The Role of Metacognition
The importance of metacognition (the ability to monitor and adjust one's thinking) is also discussed. The ability to consciously reflect on one's thought process and adjust strategies as needed is essential for effective learning.
The Role of Metacognition This section explains how metacognition functions in the problem-solving process. It emphasizes that students' ability to reflect on their thinking and revise their approach to solving problems is a key factor in enhancing problem-solving skills.
6. Real-Life Applications
Mathematical problem-solving abilities can be applied not only in the classroom but also in real-life situations. This book explores how mathematical problem-solving can help address everyday challenges.
Real-Life Applications The ability to solve mathematical problems extends beyond theoretical study and is a valuable skill in real-life scenarios. For instance, it can be applied in budgeting, planning, and risk assessment, showcasing how mathematical problem-solving can be used in various everyday contexts.
Chapter Overview
Below, we explain the contents of each chapter.
1. A Framework for the Analysis of Mathematical Behavior
This chapter proposes a framework for analyzing mathematical problem-solving behavior. The author does not view problem-solving as just a sequence of operations but provides a detailed analysis of the cognitive and strategic actions involved in solving problems. The "framework" here serves as the theoretical foundation for understanding problem-solving and suggests how factors such as knowledge, strategy, and metacognition interact.
2. Resources
"Resources" refer to all the necessary assets for solving mathematical problems. This chapter discusses both cognitive resources (such as mathematical knowledge and skills) and external resources (such as notebooks and calculators). It explains how utilizing these resources effectively greatly impacts the success of problem-solving.
3. Heuristics
"Heuristics" refer to the rules of thumb or strategies used in problem-solving. This chapter explores how problem-solvers approach problems and derive solutions through trial and error. Heuristics are emphasized as effective methods under conditions where time and resources are limited, rather than aiming for perfect solutions.
4. Control
"Control" refers to metacognitive management during the problem-solving process. This chapter focuses on how problem-solvers adjust their actions, monitor progress, and modify strategies when necessary. It discusses how effective control is crucial in the problem-solving process and its importance in achieving success.
5. Belief Systems
This chapter centers on belief systems in mathematical problem-solving. It explains how a problem-solver's beliefs about mathematics and their own abilities influence the choice of solutions and approaches to problems. A positive belief system leads to a proactive approach to problem-solving, while a negative belief system affects resilience and the way challenges are tackled.
6. Explicit Heuristic Training as a Variable in Problem-Solving Performance
This chapter explores how explicit heuristic (problem-solving strategy) training impacts problem-solving performance. It demonstrates that consciously learning and applying heuristics can improve problem-solving abilities, and it offers specific methods for how such training should be conducted.
7. Measures of Problem-Solving Performance and Problem-Solving Instruction
This chapter discusses how to measure problem-solving performance and educational approaches to enhance it. It stresses the importance of establishing criteria for evaluating problem-solving and provides insights on how effective problem-solving instruction should be structured.
8. Problem Perception, Knowledge Structure, and Problem-Solving Performance
This chapter examines how "problem perception" and "knowledge structure" influence performance in problem-solving. It emphasizes the importance of how problems are framed and how knowledge is structured, as these factors directly impact problem-solving outcomes. It highlights that how one perceives a problem and organizes their thoughts significantly alters the results of problem-solving.
9. Verbal Data, Protocol Analysis, and the Issue of Control
This chapter explores how verbal data and protocol analysis can help uncover problem-solving patterns. It investigates how verbalized thoughts and the documentation of thinking processes reveal strategies and problem-solving patterns, as well as how "control" is related to this process.
10. The Roots of Belief
This chapter delves into the origins of belief systems. It examines where problem-solvers' beliefs about mathematics arise from and how these beliefs impact problem-solving. The chapter explores how education, past experiences, and social influences shape beliefs and emphasizes the psychological aspect of problem-solving.
Conclusion
Mathematical Problem Solving offers valuable insights for understanding the process of mathematical problem-solving and provides a crucial resource for developing students' thinking through that process. It approaches problem-solving from various angles, including strategies, cognitive processes, and the role of metacognition, making it an essential read for anyone involved in mathematics education. Furthermore, it provides useful content for considering how learners can enhance their problem-solving abilities.
This book offers meaningful insights for all those interested in learning and teaching mathematics. Through this work, it becomes clear that mathematical problem-solving is not just a set of computational skills but a complex cognitive and psychological process. To enhance problem-solving abilities, it is necessary to focus not only on strategies and knowledge but also on elements such as cognitive control and belief systems.
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