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1、數學解題(Math Problem solving)初探第一節 數學解題的意義壹、問題(problem)的意義許多學者對於問題有不同解釋，一、Wickelgren(1974)指出問題由三種型式的訊息(informations)組合而成，即給定條件(givens)、運算(operations)及目標(goals)，茲分述其意如下： （一）給定條件：乃是指由一些objects、things、pieces of materials等所表達的方式，以及包含一些假設、定義、公設、公理、性質及定理等。 （二）運算：主要是指將給定條件中一個或數個表達方式轉換成新的表達方式，另一種說法是指變換(transfo
2、rmations)及推測法則(rules of inference)。 （三）目標：是指我們期望去完成的最終表達方式。貳、數學解題(mathematical problem solving)一、 解題的意義：（一）From Wikipedia, the free encyclopediaProblem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Considered the most complex
3、 of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills.1 Problem solving occurs when an organism or an artificial intelligence system needs to move from a given state to a desir
4、ed goal state.問題的解決是一個心理過程，這是較大的問題的過程， 包括問題發現和問題，塑造。 Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills. 1 Problem solving occurs when an organism
5、 or an artificial intelligence system needs to move from a given state to a desired goal state. 1解決問題時會出現一個有機體或人工智能 系統需要移動認為是最複雜的智力功能，解決問題已被定義為高階認知的過程，需要更多的常規或基本技能的調製和控制。從一個給定的狀態，一個理想的目標狀態。二、Problem-SolvingProblem-Solving Strategies andObstaclesByKendra Cherry,A GuideFrom organizing your DVD collect
6、ion to deciding to buy a house, problem-solving makes up a large part of daily life. Problems can range from small (solving a single math equation on your homework assignment) to very large (planning your future career). In cognitive psychology, the term problem-solving refers to the mental process
7、that people go through to discover, analyze and solve problems. This involves all of the steps in the problem process, including the discovery of the problem, the decision to tackle the issue, understanding the problem, researching the available options and taking actions to achieve your goals. Befo
8、re problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue if faulty, your attempts to resolve it will also be incorrect or flawed. There are a number of different mental process at work during problem-solving. These i
9、nclude: Perceptually recognizing a problem Representing the problem in memory Considering relevant information that applies to the current problem Identify different aspects of the problem Labeling and describing the problemProblem-Solving Strategies Algorithms: An algorithm is a step-by-step proced
10、ure that will always produce a correct solution. A mathematical formula is a good example of a problem-solving algorithm. While an algorithm guarantees an accurate answer, it is not always the best approach to problem solving. This strategy is not practical for many situations because it can be so t
11、ime-consuming. For example, if you were trying to figure out all of the possible number combinations to a lock using an algorithm, it would take a very long time! Heuristics: A heuristic is a mental rule-of-thumb strategy that may or may not work in certain situations. Unlike algorithms, heuristics
12、do not always guarantee a correct solution. However, using this problem-solving strategy does allow people to simplify complex problems and reduce the total number of possible solutions to a more manageable set. Trial-and-Error: A trial-and-error approach to problem-solving involves trying a number
13、of different solutions and ruling out those that do not work. This approach can be a good option if you have a very limited number of options available. If there are many different choices, you are better off narrowing down the possible options using another problem-solving technique before attempti
14、ng trial-and-error. Insight: In some cases, the solution to a problem can appear as a sudden insight. According to researchers, insight can occur because you realize that the problem is actually similar to something that you have dealt with in the past, but in most cases the underlying mental proces
15、ses that lead to insight happen outside of awareness. Problems and Obstacles in Problem-SolvingOf course, problem-solving is not a flawless process. There are a number of different obstacles that can interfere with our ability to solve a problem quickly and efficiently. Researchers have described a
16、number of these mental obstacles, which include functional fixedness, irrelevant information and assumptions. Functional Fixedness: This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options th
17、at might be available to find a solution. Irrelevant or Misleading Information: When you are trying to solve a problem, it is important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. When a problem is very complex, the easier i
18、t becomes to focus on misleading or irrelevant information. Assumptions: When dealing with a problem, people often make assumptions about the constraints and obstacles that prevent certain solutions. Mental Set: Another common problem-solving obstacle is known as a mental set, which is the tendency
19、people have to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can often work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.三、S
20、choenfeldOverview: Alan Schoenfeld presents the view that understanding and teaching mathematics should be approached as a problem-solving domain. According to Schoenfeld (1985), four categories of knowledge/skills are needed to be successful in mathematics: (1) Resources - proposition and procedura
21、l knowledge of mathematics, (2) heuristics - strategies and techniques for problem solving such as working backwards, or drawing figures, (3) control - decisions about when and what resources and strategies to use, and (4) beliefs - a mathematical world view that determines how someone approaches a
22、problem. Schoenfeld(1985)研究提出解題能否成功，取決於有關知識及技能所涉及的四個範疇，即(1)資源(resources):有關數學的程序知識與性質等。(2)捷思 (heuristics)：解題的策略及技巧。(3)掌握(control):能決定什麼是及何時使用上述所提及的資源及策略。(4)信念(beliefs)：從數學觀點如何確定能解決問題。Schoenfelds theory is supported by extensive protocol analysis of students solving problems. The theoretical framewor
23、k is based upon much other work in cognitive psychology, particularly the work of Newell & Simon. Schoenfeld (1987) places more emphasis on the importance of metacognition and the cultural components of learning mathematics (i.e., belief systems) than in his original formulation. Scope/Application:
24、Schoenfelds research and theory applies primarily to college level mathematics. ExampleSchoenfeld (1985, Chapter 1) uses the following problem to illustrate his theory: Given two intersecting straight lines and a point P marked on one of them, show how to construct a circle that is tangent to both l
25、ines and has point P as its point of tangency to the lines. Examples of resource knowledge include the procedure to draw a perpendicular line from P to the center of the circle and the significance of this action. An important heuristic for solving this problem is to construct a diagram of the probl
26、em. A control strategy might involve the decision to construct an actual circle and line segments using a compass and protractor. A belief that might be relevant to this problem is that solutions should be empirical (i.e., constructed) rather than derived. Principles: 1. Successful solution of mathe
27、matics problems depends up on a combination of resource knowledge, heuristics, control processes and belief, all of which must be learned and taught. References: Schoenfeld, A. (1985). Mathematical Problem Solving. New York : Academic Press. Schoenfeld, A. (1987). Cognitive Science and Mathematics E
28、ducation. Hillsdale , NJ : Erlbaum Assoc. For more on Schoenfelds work, see his home page at: http:/gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.htmlMathematics:摘自http:/www.instructionaldesign.org/domains/math.htmlA number of learning theories have been applied to the domain of mathematics. AC
29、T* has been used to develop a computer tutoring program for geometry. Repair theory provides a detailed analysis of the cognitive proceses involved in subtraction. Conversation theory served as the basis for studies in learning probability. Schoenfeld has developed a comprehensive theory of mathemat
30、ical problem solving that suggests four kinds of skills are necessary to be successful in mathematics: resources, heuristics, control processes, and beliefs. The Gestalt theory outlined by Wertheimer suggests some general mechanisms of problem-solving that are relevant to mathematics. The structural
31、 learning theory of Scandura has been applied extensively to mathematics. According to this theory, the most fundamental aspect of learning is the acquistion of higher-order rules that describe mathematical procedures. Bruner applies his constructivist framework to mathematics. The algo-heuristic th
32、eory of Landa also emphasizes the importance of rules in mathematics learning. In addition, theories of intelligence such as Gardner and Guilford. Research on mathematics instruction is reported in Charles & Silver (1989), Cocking & Mestre (1988), and Grouws & Cooney (1988). References: Charles, R.
33、& Silver, E. (1989). The Teaching and Assessing of Mathematical Problem Solving. Hillsdale, NJ: Erlbaum. Cocking, R. & Mestre, J. (1988). Linguistic and Cultural Influences on Learning Mathematics. Hillsdale, NJ: Erlbaum. Grouws, D. & Cooney, T. (1988). Perspectives on Research on Effective Mathemat
34、ics Teaching. Hillsdale, NJ: Erlbaum. 【註】Procedural knowledge 1.【Definition】Procedural knowledge is knowing how to control the revant factors for examing some phenomenon (Reber & Reber, 2001), performing a certain task or completing an activity. Procedure knolwedge also means knowing the method of m
35、anipulating a specific condition or the technique for implementing a task. This may include the procedures we use to do a science experiment, write an essay or solve a mathematical equation. Procedural knowledge is often thought about as certain skills we possess, tasks we can complete or processes
36、we are able to follow.A skill typically refers to a specific set of steps performed in a faily strict order and, ideally, without much conscious thought. A process is a more general set of steps that is performed with more conscious thought and careful consideration of what needs to be done next.(Ma
37、rzano, et.al.,1997 p. 49)2.Use of Procedural Knowledge in the Math classroom Procedures and processes are a big part of learning math. Many of the concepts we learn in math involve memorizing and following steps to get the correct answer. Basic addition involves a step by step process. Problem solvi
38、ng is a process. Solving an Algebraic equation involves following steps. As the students learn to complete basic math computations, they should be immersed in the processes they need to follow. The processes of the basic skills serve as building blocks for the future, more complex, skills they will
39、learn. There are three phases of acquiring procedural knowledge, construct models, shape and internalize. (Marzano, et al, 1997. p. 93) In the construct models phase, a model of the process to be learned is displayed and the steps involved shown. An math example could be working out a multi-digit mu
40、ltiplication problem. As the problem is being worked out, the steps should be discussed that are needed to complete the problem in a manner that is understandable to the learner. Work out the example, discussing or listing each step, so the learner has a model as a point of reference. Students can c
41、onstruct their own models and list their own steps, in their own words, which would demonstrate their understanding of the process. In the shape phase, the process originally followed will be modified to make it better. Adjustments should be made to improve the process and make it more efficient to
42、use. Some aspects may be added or dropped depending on what will make the process understandable to the learner. The learner may come up with ways of making the problem easier to solve and want to add to the steps they follow. In this phase it is important for the students to gain an understanding o
43、f the procedure they are performing. In the internalize phase, the learner needs extensive practice in order to get to a level of automaticity or fluency.(Marzano, et al, 1997. p.101). Certain skills need to be automatic, without having to think about what we are doing. Recognizing which basic math
44、skill is being indicated by a mathematical sign should become automatic, not requiring much conscious thought. Other skills need to become fluent, requiring a thought process, but known well enough to perform them with ease. Most mathematical concepts require the above phases to be repeated with add
45、itions to the knowledge base of the student. For example, when students originally learn long division they should go through the phases above. Then when they learn to do long division with the addition of decimal numbers, they will again need to go through the phases to allow for integration of the
46、 new knowledge. The process is made more difficult and should be repeated when the students do not reach mastery of the procedures they are to learn. 3. Conceptual / Procedural Knowledge(1) Conceptual Knowledge Knowledge rich in relationships and understanding It is a connected web of knowledge, a n
47、etwork in which the linking relationships are as prominent as the discrete bits of information. Examples concepts square, square root, function, area, division, linear equation, derivative, polyhedron By definition, conceptual knowledge cannot be learned by rote. It must be learned by thoughtful, re
48、flective learning. Is it possible to have conceptual knowledge/understanding about something without procedural knowledge?(2) Procedural Knowledge Knowledge of formal language or symbolic representations Knowledge of rules, algorithms, and procedures Can procedures be learned by rote? Is it possible to have procedural knowledge about conceptual knowledge?叁、解題策略(problem solving strategy)Strategy, a word of military origin, refers to a plan of action designed to achieve a particular goal
