解決問題中數(shù)量關(guān)系模型教學(xué)的有效策略

1、在小學(xué)六年級(jí)分?jǐn)?shù)乘法教學(xué)時(shí)，筆者整理了這樣一組題:題買5支水筆需要10元，照這樣計(jì)算，買 3支水筆要多少元？題一本書有45頁(yè)，已經(jīng)看了 3/5。已經(jīng)看了多少頁(yè)？題整個(gè)長(zhǎng)方形表示 15,陰影部分表示多少？題草莓的單價(jià)是 25元，是香蕉的5倍。檸檬的單價(jià)是香蕉的 3倍，檸檬單價(jià)多少元？仔細(xì)閱讀分析之后，我們不難得到這組題的解答分別是：10+5X3, 45+5X3, 15+5X3, 25+5X3。這里有整數(shù)的解決問題、分?jǐn)?shù)的解決問題甚至還有圖形題，為什么四個(gè)不 同的問題都是幾除以 5乘3?其實(shí)，這組題我們都可以用下圖來解釋，都是先求1份再求3份。其實(shí)這就是模型化的思想。解決問題教學(xué)的關(guān)鍵是訓(xùn)練學(xué)生解

2、題的策略，其特點(diǎn)是用學(xué)生豐富的生活經(jīng)驗(yàn)，幫助學(xué)生理解解決問題的方法，從而提高教學(xué)效益。課改前在小學(xué)數(shù)學(xué)應(yīng)用題教學(xué)時(shí)，我們就很重視分析題目中的數(shù)量關(guān)系。隨著課改的實(shí)施，不少一線教師逐步把數(shù)量關(guān)系教學(xué)弱化甚至邊緣化，而學(xué)生解決實(shí)際問題往往是在生活經(jīng)驗(yàn)或者直覺的支持下進(jìn)行的。因此，學(xué)生對(duì)解決問題的過程缺乏有意識(shí)的體驗(yàn)，不利于學(xué)生形成解題的策略。 隨著課改的不斷深入， 筆者認(rèn)為應(yīng)從學(xué)生已有的知識(shí)出發(fā)，引導(dǎo)學(xué)生利用生活經(jīng)驗(yàn)理解問題，將分析數(shù)量關(guān)系作為解決問題策略的關(guān)鍵。2011年版義務(wù)教育數(shù)學(xué)課程標(biāo)準(zhǔn) 明確指出：“了解分析和解決問題的一些基本方法”；“在具體情境中，了解常見的數(shù)量關(guān)系：總價(jià)=單價(jià)X數(shù)量、

3、路程=速度X時(shí)間，并能解決簡(jiǎn)單的實(shí)際問題”。成功吸取課改前傳統(tǒng)應(yīng)用題教學(xué)的成功做法。像“總價(jià)=單價(jià)X數(shù)量” “路程=速度X時(shí)間”這種針對(duì)數(shù)量相依關(guān)系，采用形式化的數(shù)學(xué)符號(hào)和語言，概括性地表述出 來的數(shù)學(xué)結(jié)構(gòu)化語言（公式），就是數(shù)學(xué)模型化。一、加強(qiáng)運(yùn)算意義教學(xué)，建立基本模型小學(xué)階段數(shù)學(xué)的解決問題涉及知識(shí)，大都可以歸結(jié)為四則運(yùn)算模型（含圖形與幾何方面的主要問題）。因此，筆者認(rèn)為在教學(xué)過程中，應(yīng)該加強(qiáng)運(yùn)算意義的教學(xué)，以理解運(yùn)算意義 為基礎(chǔ)，讓學(xué)生進(jìn)行初步體驗(yàn)和歸納，把運(yùn)算與數(shù)學(xué)問題進(jìn)行溝通，建立最基本的數(shù)量關(guān)系模型，提高學(xué)生分析數(shù)量關(guān)系的能力。比如，在一年級(jí)學(xué)生認(rèn)識(shí)加法運(yùn)算的意義時(shí)，教師可以幫助學(xué)生

4、結(jié)合具體的情境理解加法意義：根據(jù)已知兩個(gè)不同的部分?jǐn)?shù)，要求總數(shù)是多少， 就是把這兩個(gè)部分?jǐn)?shù)合在一起的運(yùn)算，這種運(yùn)算叫做加法運(yùn)算。如一年級(jí)下冊(cè)有這樣一道題目：一件上衣50元、一條裙子40元、一條褲子30元，（1）買一件上衣和一條褲子多少錢？ （2）付給售貨員100元，應(yīng)找回多少錢？ （ 3）你還能提出什么數(shù)學(xué)問題？教師可以在學(xué)生仔細(xì)讀題之后幫助學(xué)生這樣理解 數(shù)量關(guān)系：衣服的價(jià)格+褲子的價(jià)格=總價(jià)；在具體情境中多次體驗(yàn)、感悟“數(shù)學(xué)模型”典型 實(shí)例的基礎(chǔ)上，理解、建立它們之間的數(shù)量關(guān)系模型就是“部分?jǐn)?shù)+部分?jǐn)?shù)=總數(shù)”。在教學(xué)認(rèn)識(shí)減法運(yùn)算的意義時(shí)，教師可以幫助學(xué)生結(jié)合具體的情境理解減法意義，如有這樣一

5、道題目：有 35本故事書借出2本，有35本動(dòng)漫書借出20本。（1）還剩多少本故事 書？ （ 2）還剩多少本動(dòng)漫書？教師可以引導(dǎo)學(xué)生理解這樣的數(shù)量關(guān)系：“總共的故事書-借出的故事書=剩下的故事書”和“總共的動(dòng)漫書 -借出的動(dòng)漫書=剩下的動(dòng)漫書”。其實(shí)這個(gè)題 目的數(shù)量關(guān)系模型就是“總數(shù)-分?jǐn)?shù)=另一部分?jǐn)?shù)”。在對(duì)兩個(gè)數(shù)量進(jìn)行大小比較時(shí)，教師可以讓學(xué)生借助具體情境，可以用減法運(yùn)算比出它們的大小，也就是從較大數(shù)中去掉與較小數(shù)同樣多的部分，余下的部分既是較大數(shù)比較小數(shù)多的部分，又是較小數(shù)比較大數(shù)少的部分，也是較大數(shù)與較小數(shù)相差的部分， 數(shù)量關(guān)系模型就是“大數(shù)-小數(shù)= 相差數(shù)”。當(dāng)然在教學(xué)乘、除法時(shí)也一樣要

6、注重運(yùn)算意義的教學(xué)。 總之在解決實(shí)際問題時(shí)，要把解 決問題與數(shù)學(xué)意義緊密聯(lián)系在一起， 潛移默化地滲透數(shù)量關(guān)系， 建立基本的數(shù)學(xué)模型， 為提 高學(xué)生解決問題的能力奠定基礎(chǔ)。二、結(jié)合情境教學(xué)，建立常見模型加強(qiáng)數(shù)量關(guān)系分析的指導(dǎo)，在用數(shù)學(xué)方法解決問題的過程中，注重常見數(shù)量關(guān)系的抽象概括與應(yīng)用，以數(shù)量關(guān)系的有效構(gòu)建提升學(xué)生分析問題和解決問題的能力。比如，有這樣一道題目：有兩個(gè)人在相距72千米的兩個(gè)地點(diǎn)同時(shí)相向而行，第一個(gè)人的速度是4km/h ,另一個(gè)人的速度是 8km/h ,有一只狗原來與第一個(gè)人在一起，與兩個(gè)人同 時(shí)出發(fā)，向第二個(gè)人的方向跑去，當(dāng)他追上這個(gè)人時(shí)，立刻向相反方向跑，去追另一個(gè)人， 這樣

7、重復(fù)下去，不停地在兩人之間運(yùn)動(dòng)，直到兩人相遇為止。如果狗的速度是 6km/h ,問這 只狗跑的距離。本題出現(xiàn)在小學(xué)高年級(jí)，但事實(shí)上這題難倒了不少初高中的學(xué)生，一些大人也束手無策！究其原因，學(xué)生們?cè)诓粩嗉?xì)究狗每一次的動(dòng)作，想算出每一次跑的路程再相加覺得很難。這時(shí)，從整體上分析問題、思考問題就輕而易舉：根據(jù)路程、時(shí)間、速度三者的關(guān)系s=vt,要求狗跑的路程，其實(shí)只要知道狗的速度和時(shí)間。速度已經(jīng)有了，只要知道狗跑的時(shí)間就可以了。兩個(gè)人同時(shí)用的時(shí)間與狗跑的時(shí)間相等！人所用時(shí)間：72+ （8+4） =6小時(shí)，狗跑的距離即6X6=36千米！多么發(fā)人深省的思想?。∥覀兙蛻?yīng)該給學(xué)生更多的機(jī)會(huì)，讓他們從整體上思

8、考、分析問題，看的不同了，境界也就不一樣了！其實(shí)在教學(xué)中，我們?cè)诔橄蟪鰯?shù)量關(guān)系模型后，還應(yīng)讓學(xué)生學(xué)會(huì)變式運(yùn)用，做到舉一反三，如根據(jù)“速度X時(shí)間=路程”，變化出“路程+時(shí)間=速度，路程+速度=時(shí)間”；根據(jù)“單 價(jià)X數(shù)量=總價(jià)”演繹出“總價(jià)+數(shù)量 =單價(jià)、總價(jià)+單價(jià)=數(shù)量”等。這些基本關(guān)系式具有 高度的概括性和廣泛的應(yīng)用性，我們可以用概括的語言和符號(hào)表示出來，建立數(shù)學(xué)模型，有助于培養(yǎng)學(xué)生抽象、 概括的思維能力，感受數(shù)學(xué)抽象的美。當(dāng)然數(shù)學(xué)模型建立后，教師應(yīng)引導(dǎo)學(xué)生將建立的數(shù)學(xué)模型遷移到他們不熟悉的情境中，作為實(shí)現(xiàn)解決問題的方法和措施。三、依據(jù)基本關(guān)系，以模型化繁為簡(jiǎn)小學(xué)階段的數(shù)量關(guān)系教學(xué)，既有簡(jiǎn)單的

9、基本數(shù)量關(guān)系教學(xué)，也有復(fù)雜的復(fù)合數(shù)量關(guān)系教 學(xué)。復(fù)合數(shù)量關(guān)系教學(xué)是小學(xué)中高年級(jí)的重要內(nèi)容，也是整個(gè)小學(xué)階段數(shù)量關(guān)系教學(xué)的重難點(diǎn)與核心。因此，學(xué)生在掌握基本數(shù)量關(guān)系模型的基礎(chǔ)上，必須了解和學(xué)會(huì)建構(gòu)復(fù)合的數(shù)量關(guān)系模型，以模型化繁為簡(jiǎn)。比如，新教材六年級(jí)上冊(cè)分?jǐn)?shù)除法單元新增這樣的題目：一套運(yùn)動(dòng)服共300元，褲子價(jià)錢是上衣的。上衣和褲子的錢分別是多少？數(shù)量關(guān)系是上衣的價(jià)格+褲子的價(jià)格=總價(jià)。但是仔細(xì)讀題我們還會(huì)發(fā)現(xiàn)：褲子的價(jià)格=上衣的價(jià)格X。讓學(xué)生對(duì)兩個(gè)數(shù)量關(guān)系進(jìn)行分析，發(fā)現(xiàn)上衣價(jià)格是3份，褲子價(jià)格是2份，那么整套衣服的價(jià)格就是5份。先求1份再求上衣和褲子的價(jià)格就可以了。這樣，這個(gè)題目的解決回到了本文

10、一開始的題組，先求1份再求幾份的問題。 只能讓學(xué)生在分析、解題與編題的過程中， 明白簡(jiǎn)單數(shù)量關(guān)系如何轉(zhuǎn)化為復(fù)雜的數(shù)量關(guān)系，從而提升學(xué)生思考與解決問題的能力。我們?cè)诮鉀Q問題教學(xué)時(shí)，應(yīng)該引領(lǐng)學(xué)生把書本知識(shí)與現(xiàn)實(shí)生活緊密聯(lián)系，把問題從具象到抽象概括，把解決問題從經(jīng)驗(yàn)式逐步提升到用數(shù)學(xué)方法解決問題。無論用等式、符號(hào)、語言還是圖形、模擬等各種形式表示數(shù)量關(guān)系，關(guān)鍵都在于讓學(xué)生經(jīng)歷建立數(shù)量關(guān)系模型的抽象過程，體驗(yàn)提煉、運(yùn)用策略的全過程，在經(jīng)歷建模、策略應(yīng)用的過程中，逐步提高學(xué)生數(shù) 學(xué)思維水平，幫助學(xué)生積累數(shù)學(xué)經(jīng)驗(yàn)，把其豐富的體驗(yàn)和認(rèn)知轉(zhuǎn)化為邏輯推理和數(shù)學(xué)抽象能力的發(fā)展，進(jìn)而有效地促進(jìn)學(xué)生思維品質(zhì)的發(fā)展，達(dá)

11、到教育教學(xué)的目的。In the sixth grade scores multiply teaching, the author compiled a set of questions like this: Problem (1) to buy five pen need 10 yuan, according to this calculation, buy 3 pen how much yuan?a book on page 45, has looked at 3/5. Have looked at how many pages?the whole rectangle said 15, th

12、e hatched section shows how much?the strawberry unit price is $25, 5 times that of bananas. Lemons price is 3 times of banana, lemon unit price how many yuan?Read carefully after analysis, it is easy to get this problem set solutions are: 10 present 5 x 3, present 5 * 3, 15 members present 5 * 3, 25

13、 members present 5 x 3. There is an integer problem solving, scores of problem solving and even graphics, why four different questions are divided by 5 by 3? In fact, this set of questions we all can be explained by the image below, is the first 1 to 3. In fact this is modeling.Problem solving teach

14、ing is the key to training students problem solving strategy, its characteristic is to use the students rich life experience, to help students understand the methods to solve the problem, so as to improve teaching efficiency.Curriculum reform in the elementary school mathematics word problems before

15、 teaching, we attach great importance to analyze relationship between number in the subject. With the implementation of the curriculum reform, many front-line teachers gradually weakening even marginalized, quantitative relationship between the teaching and the students to solve practical problems t

16、end to be carried out with the support of life experience or intuition. Students in the process to solve the problem, therefore, the lack of conscious experience, is not conducive to the students to form problem solving strategies. With the deepening of the curriculum reform, the author thinks that

17、we should departure from the students of the existing knowledge, guide the student to use life experience understand problems, analysis of quantitative relation as the key to the problem solving strategy.2011 edition of the compulsory education mathematics curriculum standards clearly pointed out: t

18、o understand some basic methods of analysis and solve problems . In a specific situation, understand the relationship between the number of common: total price = unit price * time, quantity, distance = speed and can solve simple practical problem. Successful absorbs the class changes before the succ

19、essful practices of traditional teaching of word problems. Like total price = unit price * number and distance x = speed time this number for dependency relationship, using the formal mathematical symbols, and language, a general expression of math structured language (formula), is the mathematical

20、modeling.A teaching, strengthening the operation significance, basic model is set upElementary school mathematics problem solving involves knowledge, mostly boils down to is arithmetic model (including graphics and geometric aspects of the main problems). Should, therefore, the author thinks that in

21、 the process of teaching, strengthen the operation significance of the teaching, on the basis of understanding operation significance, lets the student carries on the preliminary experience and induction, the problem of computing and mathematics communication, establish the number of the most basic

22、relational model, to improve students ability to analyze related quantities.For example, in first grade students to understand the meaning of an addition operation, teachers can help students understand addition: combined with the specific situation according to the known two different part Numbers,

23、 what is the total demand, is to put the two part number together operation, this operation is called additive operation. Grade as part ii has such a title: 50 yuan a jacket, a skirt for 40 yuan, 30 yuan, a pair of trousers how much money to buy a jacket and a pair of pants? (2) pay the salesman 100

24、 yuan, how much money should be recovered? (3) what you can put forward the math problem? Teachers can help students after the students read the questions carefully so that understand number relationships: clothes price + pants price = total price; Many times on the specific situation experience, fe

25、eling mathematical model, on the basis of typical examples, understanding, establishing the quantitative relation model between them is part number + = total.Know subtraction problem in teaching, teachers can help students understand subtraction significance, combined with the specific situation if

26、there is such a title: 35 this story book lend 2, 35 out 20 cartoon books. How much is left (1) this story book? (2) how much is left in this comic book? Teachers can guide students to understand the number of such relationships: the total story book - lend storybooks = the rest of the story book an

27、d a total of anime books - lend anime = the rest of the comic book. In fact the number of the subject relation model is - score = another part of the total number.When comparing the size of two quantities, teachers can make students with the specific situation, can use subtraction operation than the

28、ir size, which is out of large Numbers and the decimal part of the same, the rest of the is part of the much larger comparing Numbers, is relatively less decimal is large part, is the larger and smaller Numbers differ parts, the quantitative relation model is = a few larger - Numbers.When teaching m

29、ultiplication, division, of course, also should pay attention to the teaching of operation significance. Anyhow when solving practical problems, to solve the problem and the meaning of mathematics closely linked, subtly seepage quantity relationship, to establish a basic mathematical model, lay a fo

30、undation for improving the students ability to solve the problem.Second, the situational teaching, establish a common modelTo strengthen the guidance of the quantity relationship analysis, in the process of using mathematical method to solve the problem, pay attention to the common quantitative rela

31、tion of abstract, generalization and application are made valid by means of the quantitative relation of construction of improve students ability to analyze and solve problems.,for example, there is such a topic: there are two men in the 72 km away from the two sites at the same time each other, the

32、 first one is the speed of 4 km/h, the speed of another person is 8 km/h, HYPERLINK HYPERLINK there was a dog originally with the first one, with two people at the same time, ran to the direction of the second man, when he catch up with the person, we immediately to run in the opposite direction, go

33、 after another person, repeat, constantly in motion between two people, until two people meet. If the dog is the speed of 6 km/h, ask the dog run.Subject in the elementary school higher grades, but in fact this stumped a lot of high school students, some adults also helpless! The reason is that the

34、students constantly scrutinize the dog every movement, work out every time want to run away together again feel it is very difficult. At this time, as a whole to analyze and ponder over a problem is easy: according to the relationship of distance, time, speed, the three s = vt, request the dog ran a

35、way, in fact as long as know the dogs speed and time. Have the speed, just know the dog run time. Both at the same time, in time with the dog running time is equal! People use time: 72 present (8 + 4) = 6 hours, dogs run distance is 6 x6 = 36 km! What a thought-provoking ideas! We should give studen

36、ts more opportunities, let them from the overall thinking and problem analysis, look different, state is different!In fact in the teaching, we abstract the quantitative relation model, should also make students learn to apply, do the lines, such as according to speed x time = journey, change the dis

37、tance present time = speed, distance present speed = time; According to the unit price * number = total price to interpret a total number of members present = unit price, total price present price = number, etc. These basic formula has a high degree of generality and extensive application, we can us

38、e the general language and symbol, mathematical model is set up, helps to cultivate students abstract, general thinking ability, feel the beauty of mathematics abstraction. After the construction of the mathematical model, of course, the teacher should guide students to establish the mathematical mo

39、del of migration to they are not familiar with the situation, as the methods and measures to solve the problem.Three, based on the basic relations, in order to model change numerous for briefRelationship between the number of primary school teaching, both the simple quantitative relationship between

40、 the basic teaching, and also has a complex compound teaching quantity relations. Quantitative relationship between composite teaching is an important content in the elementary school higher grades, and the quantitative relationship between primary school teaching difficult point and the core. As a

41、result, students in mastering basic quantitative relation model, on the basis of must understand and learn to construct the complex relationship between the number of models, in order to model change numerous for brief.New teaching material, for example, grade six top volume fraction division unit n

42、ew topics: a total of 300 yuan, a set of clothes pants price is the coat. How much is the jacket and pants money respectively? Quantitative relation is the price of the coat + pants price = total price. But we also find carefully read the topic: the price of the pants = x coat price. Ask students to analyze the quantitative relationship between the two, find jacket price is 3, pants price is 2, the