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•lOURNAL OF SCIENCE OF HNUE Edm ational Sei 2011, ''''ol r,(i, No I, pp 38 42 I M P R O V E M E N T OF T H I N K I N G SKILLS A N D A B I L I T I E S OF R E C O G N I T I O N F O R S T U D E N T S B Y U[.]
•lOURNAL OF SCIENCE OF HNUE Edm-ational Sei 2011, \'ol r,(i, No I, pp 38-42 I M P R O V E M E N T O F T H I N K I N G SKILLS A N D A B I L I T I E S OF R E C O G N I T I O N F O R S T U D E N T S B Y U S I N G A L G O R I T H M IN T E A C H I N G M A T H E M A T I C S A T S C H O O L S Ngiiyeui T h i T i n h Hanoi Nalton.al Un.ivcr.sity of hy due a lion l>mail: (inhn( fi'hnue.edu.vn Abstract Algorithmic- thinking aic vr.vy important f(H students studying Mathematics It is believed that if teac:hers apply algorithms in teaching Mathematics and reciuii-e students to work more cjn j)encil-and-paper computations, students -will have more oi)portunities to build I heir thinking skills, the abilities of Mathematic recognitions and computational skills This paper presents some examples in teaching integral parts where teac-hers cau improve the abilities of mathematic reccjgnitions and algorithmic thinking skills to students by providing procedures or algorithms of each problem to students and require them to follow up activities, Introduction An algorithm of a mathematic j^roblem is a ste^p-ln'-stej) iMocedure de\signecl to obtain the results of the problem in a finite period of time Sometimes, an algorithm has some steps that can be repeated in a numbei of limes It is believed that if teachers apply algorithms in leaching Mathemalics and reciuire students to work more on pencil-and-paper computations, students will have more opi)orlunities to build their thinking skills, the abilities of Mathematic recognilions and computational skills This paper presents some exanij^les in teaching integral parts where teachers can improve the abilities of mathematic recognitions and algorithmic thinking skills to students by providing procedure's or algorithms of each problem to students and recpiire them to follow up activities Content In the past, experienced teachers when teaching Mathematics at schools put more focus on imi)roving the ability of recognition of students in finding out the way to solve the problem faced and applying algorithmic thinking to work out the steps leading the solution of the problem, Howe\'er, these da}^s the role of algorithms in teaching Mathematics at schools have been changing [2], Perhaps, one of the main reasons for that is the availabilit}' of easy-to-use and powerful calculators 38 Improvement of thinking skills and abilities of recognition for students and computers As a result, many students are facing difficulties in carrying out simple algorithms on pencil-and-paper computation such as addition, subtraction, multiplication, division,, , let alone to solvc^ more complicated mathematic problems There are often complaints that many students seem not to have the abilities of algorithmic recognitions in Mathematics [3], The follcjwing are examples, where teachers can use algorithms to show students the way to solve> the; problems, 2.1 Using substitution algorithm * Problem: Find the antiderivati^'e / = /'()./-(3;/'" 4- 3)-V/./Students with good knowledge on deri\'ali\'es and understand clearl}' the concept of anti-derivati\'e can easil}' to recognize that the deri\ative of (3x- -t 3) is 6x and ciuickly find the result for the problem, Howe\'er, in realily, there are a lot of students who not immediately learn that If so, the teacher can make some suggestions for them and to instruct them to work out the best wa}' to find the result The instructions or the algorithm for this problem can have the following steps: Step Recognise that (3x'^ + 3)' = 6x Let u = 3x2 ^ 3^ Step Find —- = b.c d.v d.u Step Substitute u for Sx'-^ + and 6x = —-, we have dx / = j 6x{S.r' + 3)'^dx = Step 4- Simplify the integral: 1= I u'^d.u Step Antidifferentiate with respect to u: / = -t/4 -|- C Step Replace u with ?>x' + we have: / = -(Sa:^ + 3)"^ -f C This example is simple, however, teachers then need to give students more examples and emphasize that by using this algorithm, we can make some difficult integrals simpler and easier Then the teacher can remark that: If we recognize that some part of the expression is the derivative of the other part, then we can use substitution algorithms In the classroom, after teacher's instructions by the above example, we can let students the exercises by themselves, such as finding anti-derivative of the following functions: y = (6x5 ^ i)/y(a;6 + x): y = (x'^ - 1) cos (3x- x'^); v = sin^xcos^x,,, At first, teachers can require students to work individually Each student has to write all the steps of the algorithm leading to the solution Then teachers can ask 39 Nguyen Thi Tinh students to work in groups of two or three to discuss and c-ompare> the results and the steps in the algorithms for eae4i problem Next, teachers can encourage> students as volimteeis (o go (o the board to |)resent the algorithm By these activities, all sliidcMils can uiideistand more on his or her algorithm and the ones provided by other students .At the same time, they can improve (heii- algorithmic thinking, Malhemalic rec-ognilion when solving those problems Inirthermore, they can learn from each other and also tliev can build up communication and lu'esentation skills Following are some more examples on the topic- 2.2 Using linear substitution If ant idei-i\'al i\'e having the Corm /•(.r)[.ey(.r)]"e/.r,„^() where g(x) is a linear funetion that is (jiie of the t}'pe g(x) ax b and f(x) is not the deri\'ative of the g(x) the substitution u - g(x) is often siic-(-essrul in finding the integral * Problem: Find the antiderivative /= /".r(.r-3)^/'r/,r In this example f(x) = x, g(x) - x - with n = We can instruct students in the following steps: Step Let u = X - Step Find ~ = dx Step Substitute u for x - 3, u -^ for x and — for we have: dx 1=1 v{x - •?,)'"dx ^ l'{a + 3)n' ^~dx Step 4- Expanding the integrand: /= l{n'/'+3u'/')du Step -Anti-differentiate with respect to u: I ^ —u^^^'^ + 3,-a"/'* 4- C Step Replace u with x - 3: / = — (:r - ^Y^'^ + — (,r - 3)'/^ + C For those students who are very gopd at Mathematics, perhaps, this algprithm comes immediately and naturally However, for many students, who find difficulties 40 Improvement of thinking skills and abilities of recognition for students to scDlve the problem, again, teachers need to make suggestions to (hem and instruc-t them carefully Then it is neeessai}' for them lo more exereises of this l}'pc' until they master the> algorithm, such as ~^~d.r: I s/4x~i le/./-: I v{x + 3)'d.x; I 2,r(l - 2.r)dx; j {x - 2)(2.r X - 2.3 if^^'dx; Antiderivative involving trigonometric identities This t}'pe of integral recpiires students lo remcMuber derivatix'es of Irigonometric- functions in order to reeogni/e and find the wa}' lo substitute or transform the integrand to the easicM' form DiffeM'ent trigonometric identities can be used to antidifferentiate sin"x or cos"x with n has the natural number * Problem 1: Find the anliderivati\'e / - /'e-os^ dx Step Use identil}' to change cos-^ dx: 1= / ^ ( H - c o s r ) f r - ^ /(l+e-()s.r)r/.r Slej) Antidifferentiate by the rule: / = -{x + sin.r) + C Also, this problem is simple with those students who knov^^ well the trigonometric identities, but we can't guarantee that there are no students in am' classes are confused when solving this for the first time, * Problem 2: Find the antiderivative I = [ sin'^ xdx Step Factorise si:7?'^x as sinx and sin'x: I = J sin 7-sin" rc/./Step Use identil}' sni'x = — cos'x: I = J s i i i , r ( l - cos'.r)dx Step Let a = cc^.r so du = —sinxdx and the antiderivative rule can be applied, / = / {u' - \)du / = ~y/'^- n + C Step 4- Substitute u for cosx: / ~ -rc;.s'^.r - cosx + C Each step in the algorithm helps students de\'elop thinking skills, deploying and linking the knowledge learned before (factorizing, trigonometric identities and substitution) to and applying them to solve the problem Skills are only obtained with enough practice [1] Students should be given enough exercises to work in the classroom and at home As a result, they can improve the abilities of recognition and quickly building up and designing the algorithm for each problem 41 Nguyen 44ii Tinh Conchision Algorithms and thinking skills i)la}' ver\' importanl roles in ])roblem solving al)ililie>s of each student, 44iose skills can be built up in many different wa}'s at school leve4s I( is said thai mai hematic (eaeliers can lie4i) s(iiden(s highly fh'velop those skills (hroiigh rec|uiriiig sliidents to work out llie algorithms for each problem and i)reseul it in the Corm of step b\' step 44ien s( ueleii( should be given enough exerc-ises lo work in the elassioom individually At (he same time students should be reeiuired to work logelher (o e-om])are algoiilhms e4' the same problem with nihev students B\ these aetivilies, sludents can leaiii from eaeh olhcM- and deeply imdersland the algorithms the}- learned and improve (heir Mathematic ree-ognition abilities and thinking skills REFERENCES | | Caroll, \\',M,, 19f)7 Mental imd wrillen compulalion: Abdilies of siudents in reformed-bused cu.rnculum I'lie \lal heuial ics Ivdue-ateu-, 2(1): pp 18-32 (2| Lorna .Morrow Margaret Keniic}' 1998 'I'ln teaching and learning of algorith.m.s in School Malh.emulies Heston, \'.A : Naliejual Coune41 of Teachers of Mathematics cl998 |3| Edmonds Jeff 20U8 Hoin lo llun.k ubonl ulgorilhin.-^ Cambridge: New Aork: Cambridge Uni\'ersit}' Press TOM TAT Nang cao ki nang tii va kha nang nhan thxic cho hoc sinh qua su" dung thuat toan giang da\ mon Toan d trifdng thong Thuat loan va nhrmg ki nang sii}' luan denig vai trd rat quan troiig kha nang giai c|U}'et van de eiia iue')i hpc sinh .XliUng ki nang co the dupc thiet lap vk tich Ifi}' bKug nhieu each tfr l)ac ]A\C) tlie'ing CVic giao \'i("'n Toan co the giiip hpc sinh phat trie'-n nhung ki nang na\- rat te')t tluHig ciua \'ic~>c yen can hpc sinh tim thuat loan cho mdi bai to;in \'a trinh ba}- no dudi dang cac budc giai Bai bao gidi ihie'u mot se') thuat loan giao \-ie"'n e-o the sir dung de day cho hpc sinh each giai cac bai loan nham nang cao ki nang tU \-a kha nang nhan thue- cho hpc sinh 42 ... TOM TAT Nang cao ki nang tii va kha nang nhan thxic cho hoc sinh qua su" dung thuat toan giang da\ mon Toan d trifdng thong Thuat loan va nhrmg ki nang sii}'' luan denig vai trd rat quan troiig... ''ic~>c yen can hpc sinh tim thuat loan cho mdi bai to;in ''a trinh ba}- no dudi dang cac budc giai Bai bao gidi ihie''u mot se'') thuat loan giao \-ie"''n e-o the sir dung de day cho hpc sinh each giai... \-ie"''n e-o the sir dung de day cho hpc sinh each giai cac bai loan nham nang cao ki nang tU \-a kha nang nhan thue- cho hpc sinh 42