1、談問題解決-以數學科教學為例 左太政/高雄師範大學數學系數學解題的概念數學解題歷程及策略範例解說一、緒論(一)何謂數學問題?許多學者對於問題有不同解釋 Wickelgren(1974)指出：問題由三種型式的訊息(informations)組合而成，即1.給定條件(givens)、2.運算(operations)、3.目標(goals)。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。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(transformat
2、ions)及推測法則(rules of inference)。目標 是指我們期望去完成的最終表達方式，簡言之，就是題目要求或證明什麼。數學中的問題解決數學中的問題解決問題,是數學的核心,學習數學就是學習如何解決問題,包括那些可以轉換成數學題的各類問題(即外在連結)。由於解題的態度和學習方法的不同,將影響其學習成效。問題與習題的區別問題解決中的問題，並不包括常規數學問題，而是指非常規數學問題和數學的應用問題。常規數學問題，就是指課本中既已唯一確定的方法或可以遵循的一般規則、原理，而解法程序和每一步驟也都是完全確定的數學問題。解題的意義 知識的表現指解題者擁有特殊解決問題的學科知識，如幾何學、代數
3、學、數論、機率與統計等；解題的表現指解題者以已知一般的學科知識，以程序性的方式，如四則運算、作圖表等，靈活運用來解決問題。數學解題的目的 要訓練、培養學生、使他們有能力與自信面對並解決非例行性的數學問題。在建構理論的觀點下，教師是佈題者（problem poster），而不是解題者（problem solver）；是讓學生自行提出有效的解題活動，而非只是讓學生做一個模仿者。教師能依下列方式教導學生，將有助於數學的學習 能教導學生將解題視為是研究的觀點,將所解的每一道題目,仔細加以研究。在解題之前,必須探討其與已知的數學知識、方法及過去解題經驗之間的聯繫,從中找出一條或多條解題思路。教師能依下列
4、方式教導學生，將有助於數學的學習 解題之後,需再研究其多種不同的解法；嘗試將題目條件的變化或推廣,進而產生新的題目。如果能這樣做,達到貫通數學知識和數學方法的目的,以提高學生解題能力及學習的成效。簡言之，數學解題是指：解題係指當某人在解一個數學問題時，這個人為獲得答案所從事的一系列活動。數學解題係指在解決數學問題過程中需要用到一些數學概念、原理或方法等。二、解題策略及 解題歷程參考下列二位學者1.G.Polya2.A.SchoenfeldSchoenfeld 提及：解題成功的因素解題能否成功，取決於有關知識及技能所涉及的四個範疇 (1)資源(resources):有關數學的程序知識與性質等。(
5、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。(一)解題策略的意義策略是指完成任務的方法。解題策略是指解決數學問題所使用的方法。解題策略的種類解題策略的種類(Problem-Solving Strategies)Algorithms Heuristics Trial-and-Error Insight 1.Algorithms An algorithm is a step-by-step procedure that will always produc
6、e 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.2.Heuristics Heuristic refers to experience-based techniques for problem solving,learning,and discovery.
7、Examples of this method include using a“rule of thumb”經驗法則經驗法則,an educated guess,an intuitive judgment,or common sense.In more precise terms,heuristics are strategies using readily accessible,though loosely applicable,information to control problem solving in human beings and machines.Polya對Heuristi
8、c 的說明：If you are having difficulty understanding a problem,try drawing a picture.If you cant find a solution,try assuming that you have a solution and seeing what you can derive from that(working backward).If the problem is abstract,try examining a concrete example(如求長方體的對角線長如求長方體的對角線長).Try solving
9、a more general problem first(the inventors paradox:the more ambitious plan may have more chances of success).捷思策略(Heuristic Strategy)A general suggestion or technique which helps problem-solvers to understand or to solve a problem.表示一種一般的建議(general suggestion)或策略，可協助解題者瞭解題意與有效地利用他們的資源去解題。3.Trial-and
10、-Error A trial-and-error approach to problem-solving involves trying a number 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 t
11、he possible options using another problem-solving technique before attempting trial-and-error.4.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 hav
12、e dealt with in the past,but in most cases the underlying mental processes that lead to insight happen outside of awareness.George Plya,(Hungary,1887-1985)Plya worked in probability,analysis,number theory,geometry,combinatorics and mathematical physics.The aim of heuristic is to study the methods an
13、d rules of discovery and invention.Heuristic,as an adjective,means serving to discover.its purpose is to discover the solution of the present problem.What is good education?Systematically giving opportunity to the student to discover things by himself.Polya的至理名言：If you cant solve a problem,then ther
14、e is an easier problem you cant solve:find it.這是所謂類比這是所謂類比(屬於Heuristic解法的一種)G.Polya and“How to Solve It!”An outline of Polyas framework (即解題歷程)1.Understanding the Problem Identify the goal The first step is to read the problem and make sure that you understand it clearly.Ask yourself the following q
15、uestions:(1)What is the unknown?(2)What are the given quantities?What is the conditions?Are there any constraints?Polya taught teachers to ask students questions such as:Do you understand all the words used in stating the problem?What are you asked to find or show?Can you restate the problem in your
16、 own words?Can you think of a picture or a diagram that might help you understand the problem?Is there enough information to enable you to find a solution?2.Devising a Plan Find a connection between the given information and the unknown that will enable you to calculate the unknown.It often helps yo
17、u to ask yourself explicitly:”How can I relate the given to the unknown?”If you do not see a connection immediately,the following ideas may be helpful in devising a plan.(1)Establish subgoals(divide into subproblems)In a complex problem it is often useful to set subgoals.If we can first reach these
18、subgoals,then we may be able to build on them to reach our final goal.(2)Try to recognize something familiar Relate the given situation to previous knowledge.Look at the unknown and try to recall a more familiar problem that has a similar unknown or involves similar principles.(3)Try to recognize pa
19、tterns Some problems are solved by recognizing that some kind of pattern is occurring.The pattern could be geometric,or numerical,or algebraic.If you can see regularity or repetition in a problem,you might be able to guess what the continuing pattern is and then prove it.This is one reason you need
20、to do lots of problems,so that you develop a base of patterns!(4)Use analogy Try to think of an analogous problem,that is,a similar problem,a related problem,but one that is easier than the original problem.If you can solve the similar,simpler problem,then it might give you the clues you need to sol
21、ve the original,more difficult problem.For instance,if the problem is in three-dimensional geometry,you could look for a similar problem in two-dimensional geometry.Or if the problem you start with is a general one,you could first try a special case.One must do many problems to build a database of a
22、nalogies!常見類比原則 Fewer variables 三維空間幾何題化為二維平面題(5)Introduce something extra It may sometimes be necessary to introduce something new,an auxiliary aid,to help make the connection between the given and the unknown.For instance,in a problem where a diagram is useful the auxiliary aid could be a new line
23、 drawn in a diagram.作補助線作補助線 In a more algebraic problem it could be a new unknown that is related to the original unknown.引進新的變數引進新的變數(6)Take cases We may sometimes have to split a problem into several cases and give a different solution for each of the cases.For instance,we often have to use this
24、strategy in dealing with absolute value.(7)Work backward 倒推法(assume the answer)It is often useful to imagine that your problem is solved and work backward,step by step,until you arrive at the given data.Then you may be able to reverse your steps and thereby construct a solution to the original probl
25、em.This procedure is commonly used in solving equations.For instance,in solving the equation 3x5=7,we suppose that x is a number that satisfies 3x5=7 and work backward.We add 5 to each side of the equation and then divide each side by 3 to get x=4.Since each of these steps can be reversed,we have so
26、lved the problem.(8)Indirect reasoning Sometimes it is appropriate to attack a problem indirectly.In using proof by contradiction to prove that P implies Q.we assume that P is true and Q is false and try to see why this cannot happen.The skill at choosing an appropriate strategy is best learned by s
27、olving many problems.You will find choosing a strategy increasingly easy.A partial list of strategies is included:Guess and check Make and orderly list Eliminate possibilities Use symmetry Consider special cases Use direct reasoning Solve an equation Look for a pattern Draw a picture Solve a simpler
28、 problem Use a model Work backward Use a formula Be ingenious(心靈手巧的)3.Carrying out the Plan In step 2 a plan was devised.In carrying out that plan we have to check each stage of the plan and write the details that prove that each stage is correct.A string of equations is not enough!4.Looking Back Be
29、 critical of your result;look for flaws in your solutions(e.g.,inconsistencies or ambiguities or incorrect steps).Be your own toughest critic!Can you check the result?Checklist of checks:Is there an alternate method that can yield at least a partial answer?Try the same approach for some similar but
30、simpler problem.Check units(always,always,always!).If there is a numerical answer,is the order of magnitude correct or reasonable?Check special cases where the answer is easy or known.This might be a special angle(0 or 45 or 90 degrees)or the case when all masses are set equal to each other.Use symm
31、etry.Does your answer reflect any symmetries of the physical situation?If possible,do a simple experiment to see if your answer makes sense.數學解題策略 瞭解問題-審查題意,發掘概念內涵;若題意不了解,不妨再閱讀二至三次,直至了解題意。數學解題策略擬定計畫擬定計畫-分析問題及產生聯想,尋求解題途徑(1)儘可能畫出圖形或表格(2)檢查特例如令問題中整數取 1,2,3,4,5 等 代入,看看是否可歸納出規律來。(3)嘗試簡化問題如利用對稱性、採用不妨假設 而不
32、失問題的一般討論方式。(4)保留任何解題的紀錄,以便先做別題後再回頭解本題時參考使用。(5)將一個問題分成一系列子問題 數學解題策略實行計畫-選擇策略及綜合運用知識去進行推理計算解決問題回顧解答-驗證答案是否合理及思考結果或方法能否用於解其他問題,甚至於自己修改原問題或推廣其結論,形成另一個問題,亦可考慮作為專題研究之題目。3R解題策略解題活動先從題目待答或待證明的地解題活動先從題目待答或待證明的地方著手方著手(Request),(Request),適時引進題目的已知條件及潛在的性適時引進題目的已知條件及潛在的性質質(Response),(Response),最後導出結果最後導出結果(Resu
33、lt).(Result).這是所謂的這是所謂的3 R3 R策略。策略。Alan H.SchoenfeldProfessorCognition and DevelopmentUniversity of California,Berkeley Degrees Ph.D.in Mathematics,1973,Stanford University M.S.in Mathematics,1969,Stanford University B.A.in Mathematics,1968,Queens College,New York Alan Schoenfeld Schoenfeld從事：Resear
34、ch and development in human AI，於1980年提出研究報告：Teaching problem-Solving Skills該篇研究報告主要處理二個問題該篇研究報告主要處理二個問題是否能夠明確描述出解題熟手所使用的解題策略為何？我們是否能夠將這些解題策略教導學生如何解題？Schoenfeld的解題歷程 分析(Analysis)設計(Design)探究或探索(Exploration)履行或完成(Implementation)驗證(Verification)解題策略(一)分析與瞭解問題題意：(二)設計與計畫求解（Designing and planning a solut
35、ion）(三)對於困難問題進行其解的探索（Exploring solutions to difficult problems）(四)驗證其解（Verifying a solution）(一)分析與瞭解問題題意 1.如有可能的話可作圖或表。2.考慮特殊情形可由下列方式進行（Examine special cases to）：(a)視為問題的特例（exemplify the problem）；(b)探究可能取值的範圍（explore the range of possibilities through limiting cases）；(c)將整數參數從1,2,3,依序探討起，尋找其規律性（find
36、 inductive patterns by setting integer parameters equal to 1,2,3,in sequence.）。3.利用對稱性或為不失一般性嘗試簡化題目（二）設計與計畫求解 1.有系統地計畫求解（Pan solutions hierarchically）2.對於求解的任何觀點，都能夠去解釋你正在做什麼及為什麼這樣做；經過此過程後你將會做什麼(Be able to explain,at any point in a solution,what you are doing and why,what you will do with the result
37、 of this operation)。(三)對於困難問題進行其解的探索 1.考慮各種等價的問題（Consider a variety of equivalent problems）2.考慮將原問題作些許的修改 3.考慮將原問題作大幅度的修改（四）驗證其解 1.利用特殊情形作檢驗（Use these specific tests:Does it use all the data?Conform to reasonable estimates?Stand up to tests of symmetry,dimension analysis,scaling?）2.利用一般情形作檢驗 分析與瞭解問題
38、題意 1.如有可能的話可作圖或表。2.考慮特殊情形可由下列方式進行（Examine special cases to）：(a)視為問題的特例（exemplify the problem）；(b)探究可能取值的範圍（explore the range of possibilities through limiting cases）；(C)將整數參數從1,2,3,依序探討起，尋找其規律性（find inductive patterns by setting integer parameters equal to 1,2,3,in sequence.）。3.利用對稱性或為不失一般性嘗試簡化題目（Tr
39、y to simply it by using symmetry or“without loss of generality）設計與計畫求解（Designing and planning a solution）有系統地計畫求解（Pan solutions hierarchically）對於求解的任何觀點，都能夠去解釋你正在做什麼及為什麼這樣做；經過此過程後你將會做什麼(Be able to explain,at any point in a solution,what you are doing and why,what you will do with the result of this
40、operation)。對於困難問題進行其解的探索（Exploring solutions to difficult problems）1.考慮各種等價的問題（Consider a variety of equivalent problems）（1）（Replace conditions with equivalent ones）（2）（Recombine elements of the problem in different ways）（3）（Introduce auxiliary elements）（4）（Reformulating the problem by）a.（a change o
41、f perspective or notation）b.（arguing by contradiction or contrapositive）or c.（assuming a solution and determining the properties it must have）Consider slight modifications of the original problem （1）（Choose subgoals and try to attain them）（2）（Relax a condition and then try to re-impose it）（3）（Decomp
42、ose the problem and work on it case by case）Consider broad modifications of the original problems （Examine analogous problems with less complexity（fewer variables）（Explore the role of just one variable or condition,leaving the rest fixed）（Exploit any problem with similar form,“givens“or conclusions；
43、try to exploit both the result and the method）（四）驗證其解（Verifying a solution）1.（Use these specific tests:Does it use all the data?Conform to reasonable estimates?Stand up to tests of symmetry,dimension analysis,scaling?）2.（Use these general tests:Can it be obtained differently?Substantiated by special
44、 cases?reduced to known results?Can it generate something you know?）heuristic strategies:1.Draw a diagram if at all possible.2.If there is an integer parameter,look for an inductive argument.3.Consider a logical alternative:arguing by contradiction or contrapositive.4.Consider a similar problem with
45、 fewer variables.5.Try to establish subgoals.三、範例解說問題1.已知 為介於0和1之間的正實數，試證：,a b c d(1)(1)(1)(1)1abcdabcd 能否描述本題的解題策略？利用類比(Analogy)Fewer variables方法，先從二個變數著手，即先作 其次，不等式的兩邊再分別乘以 及 去說明三個變數及原式的結果。(1)(1)1abab(1)c(1)d問題2.試求：之值。1232!3!4!(1)!nn如何教導學生解題策略？通分？還是參考解法一 將原式引進sigma符號來化簡，即1111(1)!(1)!nnkkkkkk參考解法二
46、考慮 ，再尋找規律性。1,2,3,n 類題 計算111111111 32 43 5(2)nn 問題3.試證：當 為質數，則 必為質數。21nn目前已知最大的質數 News note:GIPMS the 47th known Mersenne!為12,978,189位數，可能是the 47th known Mersenne!Announced August 23,2008.參考網址參考網址 http:/primes.utm.edu/largest.html http:/www.mersenne.org/default.php4311260921特殊與一般的轉化:數學新知試問試問:的個位數字及末二
47、位數為何的個位數字及末二位數為何？4311260921問題試問43,112,609是否為質數?特殊與一般的轉化若 為質數，則 必為質數。(利用反證法利用反證法)p21p參考解法假設 為合數，令 為合數，與題意不合。p,1,1pmn mn122121(2)1(21)(2)(2)21)pmnmnmmnmnm 問題4.在單位球面上取二點，以一條圓弧連接此二點，試證：當此圓弧長小於2時，則必有一個半球與此弧長不相交。解題策略：利用類比策略 先將原問題化為二維的情形(Fewer variables)問題5.設 為正實數，試證：及 這三數中不可全都大於 。14,a b c d(1),(1)abbc(1)c
48、a問題6.試求四維空間球的體積。又如何求其表面積？Fewer variables Strategy Consider lower dimension cases問題7.試證：當 ，則 。2222abcdabbccddaabcdConsider two-variable case 試證：當 ，則 。註：不宜考慮下列情形：當 ，則 。22ababab22ababbaab問題 8.給定二條相交於一點的直線，在其中一條上取一點 ，如圖所示。試利用尺規作圖作一圓相切於此二直線，使得此圓與其中一條直線相切於點 。PPP問題 9.在 中，試證可利用尺規作圖作一直線，平行於底邊 ，且將此三角形平分成二個面積相
49、等的圖形。試問是否能夠利用類似的方法將此三角形分成五個面積相等的圖形？A B CABCBC問題 10.已知一圓的半徑為 ，試問此圓的內接三角形面積最大者為何？需說明理由。R正整數與其數字之間的關係應用問題設 a,b,c 均為異於零的三個不同的數字，共可組成六個相異的三位數,已知其中五個三位數的和是 3185,試問這六個三位數中最大的是多少?正整數與其數字的關係 設 a,b,c 為三個都不是零的數字,試問用 a,b,c 三個數字能組成若干個三位數？並說明這些三位數的和與 a,b,c 的關係。提示：abcacbbacbcacabcba)_222()abc參考解答 設此三位數為 由上題及題意知，故滿
50、足題意10010,abcabc3185 10010222(),3185 100222()3185999,15,16,17,18abcabcabcabc 16abc練習題設 a,b,c 均為異於零的三個不同的數字，共可組成六個相異的三位數,已知其中五個三位數的和是 3194,試問這六個三位數中最大的是多少?類題114 這個數有一特點:將114 的各位數字的數字和乘以 19 就得到 114.試問是否還有這樣的三位數嗎?如果有,請找出所有這樣的三位數。你能否觀察出本問題何以需乘以 19?是否可以改為乘以其他數而有類似結果呢?參考解答 設此三位數為 由上題及題意知，10010,abcabc100101
