I M P R O V E M E N T O F T H I N K I N G S K I L L S 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
I N 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 compu- tations, students -will have more oi)portunities to build I heir thinking skills, the abilities of Mathematic recognitions and computational skills. This pa- per 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 do follow up activities,
1. 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 teach- ers 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 do follow up activities.
2. 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 t h a t is the availabilit}' of easy-to-use and powerful calculators
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 3 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 con- cept 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 do not immediately learn that. If so, the teacher can make some sug- gestions 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 1. Recognise that (3x'^ + 3)' = 6x. Let u = 3x2 ^ 3^
Step 2. Find —- = b.c.
d.v
Step 3. Substitute u for Sx'-^ + 3 and 6x = —-, we have d.u dx
/ = j 6x{S.r' + 3)'^dx =
Step 4- Simplify the integral:
1= I u'^d.u
Step 5. Antidifferentiate with respect to u: / = -t/4 -|- C
Step 6. Replace u with ?>x' + 3 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 do 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
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 s u b s t i t u t i o n
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 - 3 with n = 3 1.
We can instruct students in the following steps:
Step 1. Let u = X - 3 Step 2. Find ~ = 1
dx
Step 3. Substitute u for x - 3, u -^ 3 for x and — for 1. we have:
dx
1=1 v{x - •?,)'"dx--^ l'{a + 3)n' ^~dx Step 4- Expanding the integrand:
/= l{n'/'+3u'/')du
Step 5. -Anti-differentiate with respect to u: I ^ —u^^^'^ + 3,-a"/'* 4- C Step 6. 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
to scDlve the problem, again, teachers need to make suggestions to (hem and instruc-t them carefully. Then it is neeessai}' for them lo do 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 - if^^'dx;
2
X - 0
2.3. Antiderivative involving trigonometric identities
This t}'pe of integral recpiires students lo remcMuber derivatix'es of Irigono- metric- 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 2 Step 1. Use identil}' to change cos-^ dx:
1= / ^ ( H - c o s . r ) f 7 . r - ^ /(l+e-()s.r)r/.r
Slej) 2. Antidifferentiate by the rule: / = -{x + sin.r) + C
Also, this problem is simple with those students who knov^^ well the trigono- metric 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 1. Factorise si:7?'^x as sinx and sin'x: I = J sin .7-sin" .rc/./-
Step 2. Use identil}' sni'x = 1 — cos'x: I = J s i i i , r ( l - cos'.r)dx
Step 3. 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.
3. 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.
R E F E R E N C E S
| 1 | 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 .1. .Morrow Margaret .1. Keniic}'. 1998. 'I'ln teaching and learning of al- gorith.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.
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