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Problem solving in school mathematics

Problem resolution has an of import topographic point in the universe of mathematics. Branca ( 1980, p. 3 ) quoted Lester 1977, ‘problem resolution has been said to be at the bosom of all mathematics ‘ to exemplify the importance of job resolution. However, in the field of school mathematics the primary end of learning mathematics is to develop the ability to work out a assortment of mathematical jobs. Monaghan, Pool, Roper, & A ; Threlfall ( 2009, p. 21 ) provinces that harmonizing to Lester ( 1994, p. 661 ) ‘most mathematics pedagogues agree that the development of pupils ‘ problem-solving abilities is a primary aim of direction ‘ but harmonizing to Schoenfeld, job resolution has ‘multiple and contradictory significances ‘ . Monaghan, et Al. ( 2009 ) raise the inquiry ‘What is the usage of pupils larning mathematics if they can non utilize it to work out jobs? ‘ ( p.21 ) and go on to state that this gives a signal that more job work outing demands to be done in schools and that pupils struggle in work outing simple jobs even when they are straight related to the mathematics that they have learned. Therefore, research on job resolution in mathematics has been done to happen schemes on how to work out jobs.

My chief focal point in this essay is to associate Polya ‘s, Burton ‘s and Schoenfeld ‘s attacks on how to work out jobs with the purpose of happening the most suited attack. First I will specify what a mathematical job is. Then I will discourse types of mathematical jobs and legion researches done on mathematical job resolution over the past decennaries. Following I will briefly illustrate stairss on each of the procedures: Polya ‘s, Burton ‘s and Schoenfeld ‘s episodes. Finally I will associate the three attacks with the purpose of happening the most suited attack.

2. What is a job in mathematics?

In this subdivision the definitions of ‘mathematics jobs ‘ will be discussed with the purpose of supplying information as to what a mathematical job is.

Frobisher ( 1994, p. 154 ) describes a job as a ‘situation that has involvement and entreaty to a kid, who hence wishes to research the state of affairs more to the full in order to derive apprehension of it. Goals arise of course during the geographic expedition and are determined non by the compositor of the job but by the kid. The kid in bend surveys the job state of affairs before researching avenues of involvement, following waies which may or may non take to a satisfactory decision. As Ernest ( 1991 ) so compactly puts it, ‘the accent is on the geographic expedition of the unknown land instead than a journey to a specified end ‘ ‘ .

Kilpatrick ( 1985, p. 2 ) defines a job as ‘ a state of affairs in which a end is to be obtained and a direct path to the end is blocked ‘ .

R. Mayer ( 1985, p. 123 ) provinces that ‘A job occurs when you are confronted with a given situation- Lashkar-e-Taiba ‘s call that the given province – and you want another state of affairs – Lashkar-e-Taiba ‘s call that the end province – but there is no obvious manner of carry throughing your end ‘ . Furthermore, Mayer ( 1985 ) gives the illustration of happening the volume of a frustrum of a right pyramid where the value of the sides of the two bases and the tallness is given. ‘ If you did non cognize a expression for volumes of frustrums, this would be a job for you ( Polya, 1965 ) ‘ ( R. Mayer, 1985, p. 123 ) .

Orton & A ; Frobisher ( 2005, p. 25 ) says ‘ a mathematics job for one scholar may be an exercising for another ‘ because if a pupil has had a similar state of affairs before the pupil may see the job to be an exercising, where another who has ne’er come across a similar state of affairs may acknowledge this as a job. He describes ‘a mathematical job can be said to be a state of affairs in which an single pupil:

Recognizes or believes that there exists a mathematical end to be achieved, normally an reply of some sort ;

Accepts the challenges to execute some mathematical undertaking in order to make the end ;

Has no readily known or recallable mathematical process available to enable the end to be attained straight ‘ ( p25 )

In Frobisher ‘s ( 1994 ) definition he mentions that the end is non determined by the compositor ( instructor ) but by the pupil. This statement is non valid for all job types. In some jobs such as schoolroom word jobs the pupil does non find the ‘goal ‘ ( reply ) . In some inquiries different waies can be taken to obtain the coveted ‘goal ‘ ( reply ) which is determined by the instructor. Furthermore, in job types like probe jobs the kids could obtain different replies ( ends ) .

It can be concluded by the above given definitions that a mathematical job can be a state of affairs in which a pupil recognizes the being of a mathematical ‘goal ‘ , performs some mathematics to accomplish the end although there is no known or recalled mathematical process for acquiring to the concluding ‘goal ‘ which is normally the reply.

3.Types of mathematical jobs

Harmonizing to LeBlanc, Proudfit, & A ; Putt ( 1980, p. 104 ) criterion text edition jobs ( word jobs ) and procedure jobs are the two chief job types that are widely used in school mathematics. The features of the two methods will be explained loosely.

3.1 Word ( standard text book ) jobs

Word jobs are normally used in simple school mathematics text editions and mathematics books. Frobisher ( 1994, p. 152 ) provinces that in a word job a undertaking is presented in words or symbols, and a end is set by inquiring a inquiry. Harmonizing to LeBlanc, et Al. ( 1980, p. 105 ) the chief feature of a word job is that it can be solved ( accomplishing of the end ) by utilizing antecedently mastered algorithms or operations. He farther states that the intents of the word jobs are to better the ability of remembering factors, beef uping of the accomplishments with operations and algorithms and constructing the relationship between operations and their relationship between existent universe state of affairss. The chief factors that affect the trouble of these types of jobs are the degree of mathematics, the complexness of the algorithm and the figure of stairss involved in work outing. ( LeBlanc, et al. , 1980 )

Orton & A ; Frobisher ( 2005, p. 27 ) describes ‘Routine jobs ‘ which is a class of word jobs that ‘uses cognition and techniques already acquired by a pupil in a narrow and man-made context ‘ . The illustration given below is a everyday job.

‘How many more than 432 is 635? ‘ ( p. 27 )

In these type of inquiries a pupil is expected to understand the ‘linguistic complexness ‘ and alter it to a theoretical account with mathematics symbols or operations. Narrative jobs are besides a type of word jobs that are set in a ‘ existent context ‘ which frequently needs an apprehension of the existent universe state of affairs. An illustration of a narrative job is given below:

‘A mailman has ninety four letters in his bag.

Twenty five of them are first category. How many are 2nd category? ‘ ( Orton & A ; Frobisher, 2005, p. 27 )

In the above job the apprehension of the different categories of letters is indispensable for a pupil to work out it.

As word jobs use the facts and techniques taught late, it is ‘ non an appropriate method for development of new cognition and its part to mathematics cognition is minimum ‘ ( Orton & A ; Frobisher, 2005 ) . Furthermore, Orton & A ; Frobisher ( 2005 ) argues that word jobs are mathematical exercisings instead than inquiries since they do non fulfill the standard that there is ‘no readily known or recallable mathematical process available to enable the end to be attained straight ‘ . Nevertheless, I feel that the reply ( ‘goal ‘ ) is besides predetermined by the inquiry compositor.

3.2 Procedure jobs

Procedure jobs are besides type of jobs that appear in mathematics text editions, but non available to simple school pupils in Sri Lanka. Unlike in word jobs this type requires ‘strategies and non algorithmic attacks ‘ ( LeBlanc, et al. , 1980, p. 105 ) and frequently has more than one reply.

Harmonizing to LeBlanc ( 1980 ) more accent is given to the procedure of obtaining the solution instead than the concluding solution. Furthermore the success of work outing the job depends on the usage of one or more schemes and non on the ‘ application of specific mathematical constructs, expressions or algorithms ‘ ( LeBlanc, et al. , 1980, p. 105 ) . Orton & A ; Frobisher ( 2005 ) takes a similar position and says that kids who reflects on different procedures develop the ability to work out other jobs.

LeBlanc, et Al. ( 1980, p. 105 ) provinces that procedure jobs ‘encourage the development and the pattern of job work outing strategies.. , provides an chance for pupils to invent originative methods of solution, to portion their method with other pupils, to construct assurance in work outing jobs aˆ¦and to bask mathematical job work outing ‘ . Furthermore, he explains that the trouble of procedure jobs depends on the figure of conditions that must be satisfied, the complexness of the conditions and the type of scheme used by the convergent thinker.

Butts ( 1980 ) in Orton & A ; Frobisher ( 2005, p. 158 ) describes ‘open hunt type of jobs ‘ as ‘one that does non incorporate a scheme for work outing the job in its statement. Therefore, the openness refers to the method of solution, non to the solution ‘ . He gives the undermentioned illustration:

How many different trigons with whole number sides can be drawn holding a longest side ( or sides ) of length 5cm? 6 centimeter? n centimeter? In each instance, how many of the trigons are isosceles?

As it does non hold one specific way to the solution I feel ‘open hunt jobs ‘ are a type of procedure jobs.

At this point, I would wish to show my thoughts about these two job types. First, in the simple school degree where pupils prosecute more on the basic operations and algorithms it is appropriate to utilize word jobs as they help the kids to familiarise themselves with the mathematical constructs. With my former experience as a mathematics instructor I have observed that when pupils do non get the hang in their basic mathematical accomplishments in lower degree categories they are unable to work out complex jobs or believe innovatively when they come to higher categories. More seriously pupils come to a decision that ‘Only masterminds are capable of detecting or making mathematics ‘ ( Schoenfeld, 1997 ) . Whereas, the existent trouble is the deficiency of basic mathematical cognition. On the other manus, I believe that as the pupils come to higher classs it is indispensable to prosecute in procedure jobs as it broadens the pupils mathematical cognition and the logical thinking abilities.

4. Research on mathematical job work outing

Harmonizing to Suydam ( 1980, p. 35 ) early research on mathematics job resolution has focused chiefly on word ( text edition ) jobs. The chief accent was on how kids solved jobs. In making so it was that a manner could be found to learn job work outing. It was during this period that Polya produces How to Solve It ( 1954 ) ‘a capturing expounding of the problem-solving self-contemplation ‘ ( Schoenfeld, 1987, p. 17 ) . Every research worker since so has based their research on mathematical job work outing on Polya ‘s work.

Harmonizing to English, Lesh, & A ; Fennewald ( 2008 ) most research workers have tried to look into the inquiries: ‘ ( a ) Can Polya-style heuristics be taught? ( B ) Do learned heuristics/strategies have positive impacts on pupils ‘ competences? ‘ but it was non successful.

Begel ‘s ( 1979 ) and Silver ( 1985 ) in ( English, et al. , 2008 ) have concluded after the reappraisal of literature in mathematics instruction that there is small grounds that transportation of acquisition has been successful. Though it has been reported in some surveies that successful acquisition has occurred, Silver suggests that it is due to pupils get the hanging the ‘mathematics construct, instead than from job work outing schemes, heuristics or job work outing procedure ‘ .

Schoenfeld ( 1992, p. 53 ) states ‘Polya ‘s word pictures did non supply the sum of item that would enable people who were non already familiar with the schemes to be able to implement them ‘ . He besides says that the ground for the deficiency of success was due to Polya ‘s heuristics being ‘descriptive instead than normative ‘ . Harmonizing to Schoenfeld in English, et Al. ( 2008 ) job work outing research should assist pupils to develop a larger figure of ‘specific job schemes ‘ , learn meta-cognitive schemes and to ‘improve beliefs about nature of mathematics, job resolution and their personal competences ‘ .

Lester Koehle ( 2003 ) in ( English, et al. , 2008 ) states that even ten old ages after Schoenfeld ‘s proposal, research on job resolution has failed. Harmonizing to him ‘Schoenfeld ‘s proposal merely moved the basic heuristics to a higher degree ‘ non accomplishing ‘prescriptive power ‘ . However, as a response to this Schoenfeld at the 2007 NCTN Research Pre-session proposed that research workers should concentrate on ‘meta-meta-cognitive procedures ‘ ( English, et al. , 2008 ) . But as in the earlier instances this had the same defects and was remarked as a ‘short list of descriptive regulations that lack normative power and a longer list of normative regulations involve cognizing when and why to utilize them ‘ ( English, et al. , 2008 ) .

Finally, as the field of research has failed more than 50 old ages, Lesh and Zawojewski ( 2007 ) concludes that it is clip to ‘re-examine the cardinal degree of premise ‘ and suggests that an option is to utilize theoretical positions and utilize methodological analysiss such as ‘models & A ; patterning position ( MMP ) on mathematics job work outing ( English, et al. , 2008 ) .

5. Polya ‘s Four Phases

Polya ( 1973 ) grouped his observations into four stages ( understanding the job, planning, transporting out the program and looking back ) depicting the four phases the individual passes through during the problem-solving procedure. These four stages would be a guideline to work out a job. Let us analyze them individually by utilizing a 5th class mathematical job: inquiry 1 ( appendix 1 ) .

5.1 Understanding the job

The pupil should be able to indicate out the unknown, the given informations and the conditions. The instructor ‘s function at this point is to inquire inquiries and steer the pupil. ‘Some likely inquiries the instructor can inquire are: What is the unknown? What are the informations? What is the status? ‘ ( Polya, 1973 ) . Depending on the inquiry sometimes it is easier to understand if the pupil draws a figure and usage notations.

As an illustration: in question1 ( appendix1 ) lines 1- 6 shows that the pupil tries to understand the inquiry by garnering information about the job. ‘Can you tell in your ain words what the job is inquiring you to happen? ‘ ( line 6 ) show that the instructor checks whether the pupil understood the inquiry by ‘asking the pupil to reiterate the statement ‘ ( Polya, 1973, p. 6 ) in his ain words.

5.2 Inventing a program

Mayer ( 1983, p. 44 ) provinces that in this stage the convergent thinker tries to utilize anterior cognition and experiences to happen a method of work outing the job. Planning is needed to make up one’s mind which computation or building should be done to obtain the unknown reply. Harmonizing to Polya ( 1973 ) it is frequently easy to look at a officially solved job and seek to associate to it or believe of the acquired mathematical cognition or even show a job which has been solved before and inquire the pupils to utilize it. The pupil can either happen out whether he can ‘restate the end in a new manner based on ‘ his anterior experience ( ‘working backwards ‘ ) or whether he can ‘restate the presumptions in a manner that relates ‘ to his anterior experience ( ‘working frontward ‘ ) ( R. E. Mayer, 1983, p. 44 ) .

As an illustration: in question1 ( appendix1 ) lines 7-9 the pupils figure out different methods they think would be more suited to near the reply.

5.3 Transporting out the program

Harmonizing to Polya ( 1973 ) when it comes to this measure because the most critical measure in planning is over it is merely a affair of transporting out the program patiently. If the pupil is convinced of the program he may transport out the program swimmingly, but in an case where he has reached the program by an outside beginning he might bury the program. Therefore, it is of import that the pupil understands the stairss of the program. The instructor excessively has to take a firm stand that the pupil ‘check each measure ‘ ( Polya, 1973, p. 13 ) .

As an illustration: in question1 ( appendix1 ) line 10 show the pupils transporting out the officially planned stairss.

5.4 Looking back

Polya ( 1973 ) states that it is of import to re-examine and reconsider the manner they solved the inquiry and the reply. It allows them to rethink about the inquiry and the method of work outing and sometimes to come up with a different, easier method. By making so, the kids learn to believe in themselves. The instructor could widen the job so that the pupil could see how the method fits to another job or for the pupil to obtain deeper apprehension. Shumway ( 1982, p. 134 ) provinces ‘one could reason job work outing terminals and the construct acquisition begins when one begins looking back, placing similar jobs, and prosecuting in other post-solution activities ‘ .

In question1 ( appendix1 ) lines 11-26 clearly shows the treatment between the instructor and the pupils which enables the pupils to reflect back on their replies. ‘When we draw up a list like Stephens ‘ and put the Numberss in some order, we call it an organized list ‘ ( line 24 ) shows that through treatment they decide which is the easiest and the most appropriate method. Furthermore, in lines 25-26 the instructor continues the treatment so that the pupil forms a broader apprehension.

6. Richard burtons ‘ three stages

Harmonizing to Burton three stages of activity viz. Entry, Attack and Review ( L Burton, 1984 ) can be observed in the procedure of job work outing. Let us analyze each stage individually utilizing the same inquiry used above.

6.1 Entry

This is the first measure where the convergent thinker tries to understand the inquiries as to what it is approximately and what needs to be done. Mason, Burton, & A ; Stacey ( 1982, p. 29 ) suggests to reply the three inquiries: ‘What do I KNOW? ‘

‘What do I Desire? ‘

‘What can I INTRODUCE? ‘

Harmonizing to Mason, et Al ( 1982 ) in this phase the inquiry needs to be read carefully and the relevant facts need to be extracted from the inquiry. The convergent thinker besides can repeat the inquiry in his ain words so that he understands what needs to be discovered. Introducing diagrams, symbols, images and notations non merely makes it easier to obtain a clear image but besides makes it easier for the following stage.

As an illustration in inquiry 1 ( appendix1 ) lines 1-7 falls under this stage and can be rephrased utilizing the cardinal words.

‘I WANT ‘ to happen the different coins Susan could utilize to pay for her confect. I KNOW ‘ the cost of a confect saloon. ‘I KNOW ‘ which coins the machine takes. ‘I KNOW ‘ Susan can non pay with quarters because the machine does non accept quarters. ‘I KNOW ‘ there are more than one reply to this inquiry so ‘I WANT ‘ to happen different ways of paying. ‘Can I INTRODUCE ‘ a tabular array, draw the different coins or compose down the denominations?

6.2 Attack Phase

It is at this phase that the inquiry starts to decide. In the effort to work out the inquiry the convergent thinker might take several attacks or different programs may be executed. Mason, et Al. ( 1982 ) states that it is rather normal for the convergent thinker to acquire ‘stuck ‘ and hence re-think of a different program or even travel back to the entry stage. ‘Stuck ‘ is ever accompanied by ‘aha! ‘ which means a different attack or a manner out.

When the convergent thinker is ‘stuck ‘ ‘TRY’aˆ¦.. , ‘MAY BE’aˆ¦aˆ¦ , ‘BUT WHY’aˆ¦aˆ¦ . ( Mason, et al. , 1982, p. 77 ) may assist him to get the better of the job. However, this stage is complete when the job is resolved.

In question1 ( appendix1 ) in lines 8-10 the kids get entries for the tabular array by seeking different combinations: Let us ‘TRY ‘ 4 Nis and 5 pennies. ‘MAYBE ‘ we should ‘TRY ‘ all possible ways with 2 Dimesaˆ¦aˆ¦ ‘BUT WHY ‘ ca n’t we use 3 quarters?

6.3 Review Phase

As the name suggests, it is the measure where you look back at what you have done, seek to widen it to a higher degree or better it. The three words that help to construction this stage:

‘CHECK’aˆ¦ the computation, statements to verify that the calculations are appropriate, that the declaration is right

‘REFLECT’aˆ¦on the cardinal points, statements, declaration

‘EXTEND’aˆ¦the consequences, by seeking a new way, by changing some of the restraints. ( Mason, et al. , 1982, p. 47 )

In inquiry 1 ( appendix 1 ) in lines 11-26 the pupils will ‘CHECK ‘ the replies and the instructor compares Becky ‘s and Steplen ‘s replies.

‘REFLECT ‘ on the cardinal point like where Stephen says, “ but with dimes foremost, so Nis and so pennies. ” ‘REFLECT ‘ on seting the Numberss in some order. They can ‘REFLECT ‘ on different consequences of other groups.

To ‘EXTEND ‘ the job the instructor or the pupils could inquire: if a confect saloon cost 30 cents show the ways Susan could set coins into the same machine. ‘How does the figure of ways alter if the machine will besides take quarters? ‘

7. Schoenfeld ‘s Episodes

Schoenfeld ( 1997 ) has parsed protocols into episodes for the intent of understanding the procedure. These are periods of clip where the convergent thinker engages in similar actions. Episodes include reading, analysing, geographic expedition, planning/ execution and confirmation. We shall seek to research the episodes utilizing the old inquiry.

7.1 Reading

This episode starts when the convergent thinker starts to read the job. This includes the clip he spends to understand the job and the rereading of the job. ( Schoenfeld, 1997 )

In question1 ( appendix 1 ) the kids would read the job, may be read it once more and seek to understand the inquiry.

7.2 Analyzing

Harmonizing to Schoenfeld ( 1997 ) in this phase the convergent thinker tries to to the full understand the job by choosing an appropriate perceptual experience, rhenium forming the inquiry consequently and presenting the mechanisms that might be suited for the job. In instances where the convergent thinker knows the relevant position this episode may be bypassed.

In question1 ( appendix 1 ) lines 1- 6 can be listed under this subdivision.At this point the pupil tries to understand the inquiry with the aim of happening an reply.

7.3 Exploration

This is where the convergent thinker explores for relevant information that can be used in other stages. This is less structured than analysis and slackly related to the conditions and the ends of the job. ( Schoenfeld, 1997 )

7.4 Planning/ execution

In these phases issues such as whether the program is good structured, whether it can be orderly implemented, whether its advancement is being observed and reported to the convergent thinker is addressed. ( Schoenfeld, 1997 )

In question1 ( appendix 1 ) the lines 7-10 falls to this class where the pupils program and implement it.

7.5 Confirmation

This is where the convergent thinker revives the solution. Some inquiries that are used in this episode are: ‘Does the job convergent thinker review the solution? , Is the solution tested in anyhow? If so, how? , Is at that place any appraisal for the solution? ‘ ( Schoenfeld, 1997, p. 300 )

Lines 11-24 in question1 ( appendix 1 ) shows that the instructor and the pupils verify the replies they obtained.

8. Protocol analysis

This subdivision contains the parsing of the protocol 1 ( appendix 2 ) utilizing Polya ‘s and Burton ‘s and Schoenfeld ‘s attacks. The analysis utilizing Polya ‘s and Burton ‘s Phases were done by closely detecting the kid ‘s work which is exhibited in the protocol 1 and the parsing by Schoenfeld ‘s episodes were extracted from Schoenfeld ( 1997 ) .

8.1 Protocol Analysis utilizing Polya ‘s Phases


Item 1- 4

apprehension and planning

points 5 – 8

Planing and transporting out the


Item 9-19

Look back

Item 20-21

Planing and carryout the program

Item 54-55

Look back

Item 56

T1: ( Item 2 )



T4: ( Item 22 )

8.2 Protocol Analysis utilizing Burton ‘s Phases

Entry Phase

Item 1- 8

Attack Phase

Item 9 – 19


Item 20 -24

Entry & A ; onslaught

Item 25- 39

Review and onslaught

Item 40 -48


Item 54-55

T1: ( Item 2 )



T4: ( Item 22 )


Item 56

8.3 Protocol Analysis utilizing Schoenfeld ‘s Episodes ( Schoenfeld, 1997, p. 311 )

E1: Reading

Item 1

( 1 minute )

E2: Analysis

Items 3 – 8

( 2 proceedingss )

Local Appraisal: Item 3

Local Appraisal: Items 7, 8

E3: Planning – Execution

Items 9 – 19

( 4 proceedingss )

Local Appraisal: Items 15, 16

Local Appraisal: Item 18

E4: Confirmation

Items 20, 21

( 30 seconds )

E5: Analysis

Items 22 – 39

( 4 proceedingss )

Metacomments: Items 24, 25

( Meta ) Appraisal: Item 33

Local Appraisal: Item 39

E6: Analysis

Items 40 – 48

( 3 proceedingss )

Local Appraisal: Item 43

Local Appraisal: Item 48

E7: Exploration

Items 49 – 53

( 3 proceedingss )

Metacomments: Items 49, 50

E8: Analysis – Execution

Item 55

( 35 seconds )

E8: Confirmation

Item 56

( 1 minute )

T1: ( Item 2 )



T4: ( Item 22 )

T5: ( Item 39 )

T6: ( Item 49 )

T7: ( Item 54 )


9. Associating the three Approachs

Under this subdivision the protocol analysis of Scheonfeld ‘s episodes will be compared with Polya ‘s and Burton ‘s Phases. As all three attacks have analyzed the same protocol by making so I wish to compare the three procedures against each other and to place how good each attack fits a existent job work outing state of affairs.

9. 1 Associating Polya ‘s stages and Scheonfeld ‘s Episodes.

Harmonizing to the protocol analysis utilizing the two theoretical accounts Polya ‘s Phases ( 1973 ) and the Scheonfeld ‘s Episodes ( 1997 ) the undermentioned observations can be made.

Line 1- 4, shows that the pupil reads the job and attempts to understand the inquiry therefore it can be categorized as Polya ‘s understanding stage. In Scheonfeld ‘s episodes line 1 falls under reading and lines 3-4 falls under analysing. In Lines 3-4 the pupil tries to understand the inquiry by sum uping the given information. The basic information he obtains by reading is non sufficient to work out the job. Although lines 1-4 is listed under apprehension, no proper apprehension will be acquired until the pupil investigates the information.

It is hard to categorise Lines 5-8 under a individual class of Polya ‘s Phases. In these lines the pupil tries to ‘ stress different parts, analyze different inside informations, examine the same item repeatedly but in different ways, combine the inside informations otherwise, approach them from different sides ‘ Polya ( 1973, p. 34 ) . Therefore, this falls under be aftering. Furthermore, the line ‘those trigons are similar ( line 6 ) ‘ shows that the pupil tries to place inside informations and to ‘contact with officially acquired cognition ‘ ( Polya, 1973, p. 34 ) . Furthermore, the pupil invariably argues with himself ( ‘Is n’t that what I want? ‘ line 7 ) to understand the job as he is unable to be after the method of work outing. Therefore, these lines could be categorized under apprehension every bit good as planning.

In points 9-19 it is hard to divide the planning and execution as the kid does computations while inventing the program mentally ( believing loud ) . The statements ‘so if I construct thea?s2aˆ¦..then I should be able to pull this line ‘ ( line 9 ) , ‘ so I merely got to retrieve how to do this building ‘ ( line 12 ) and ‘the best manner to make that is to build A ‘ illustrates that the pupil is explicating a program. Whereas the statements ‘ 1/a?s2 – allow me see here – ummm. That ‘s A? plus A? is 1 ‘ ( line13 ) and the manus written workings of the pupil shows the execution of the program.

However, in lines 22-38 and 40-48 the convergent thinker ‘works for better apprehension ‘ indicates understanding and ‘examines the same item repeatedly but in different ways ‘ indicates be aftering ( Polya, 1973, p. 34 ) and acquires a better apprehension of the job finally. But this can non be listed under any of the above two stages as this makes the pupil engage in a much complicated analysis than in a direct word job ( where the pupil is able to understand the job by analyzing the given informations ) . Furthermore, in these lines the pupil is ‘playing around ‘ ( Frobisher, 1994, p. 164 ) with the thoughts. Hence it is hard to place into which class these lines would fall into.

Harmonizing to Schoenfeld Lines 3- 8, 22- 38 and 40-48, are categorized as the ‘analyzing episode ‘ as this is ‘an effort made to to the full understand the job, to choose an appropriate position and redevelop the job in those footings and to present for consideration whatever rules or mechanisms might be appropriate ‘ ( Schoenfeld, 1997, p. 298 ) . In these lines the pupil tries mechanisms to near the job.

Item 49-53, shows that the pupil explores better ways of building the trigon by look intoing the information. Frobisher ( 1994, p. 164 ) provinces ‘ apprehension of a job merely emerges easy as it is explored ‘ and that ‘it is hard for a student to develop an apprehension of the job without first trying to research whatever appears appropriate ‘ . In these lines the pupil ‘explore ‘ possibilities. But Polya does non specify this type of pupil behaviour. So it can non be categorized under Ploya ‘s Phases. However these lines are listed under Schoenfeld ‘s researching episode.

However points

Associating Burton ‘s stages and Scheonfeld ‘s Episodes.

One common observation that is seeable throughout this protocol analysis was that when utilizing Burton ‘s stages, ‘chunks ‘ of lines could non be categorized into one peculiar stage. Let us analyze the analysis in item.

Lines 1- 8, can be recognized as the entry stage. The pupil reads the job ( line1 ) and tries to understand the inquiry by ‘organizing the information ‘ ( L Burton, 1984, p. 26 ) ( line 3 ) . Furthermore, in lines 4, 7 and 8 the pupil uses the rubric ‘I want to ‘ ( Leone Burton, 1984 ) which clearly defines this is the entry stage. On the other manus, Schoenfeld categorizes line 1 as reading and lines 3-8 as analysing. The pupil ‘organizing information ‘ and ‘exploring the job ‘ which is listed under entry is similar to ‘analysis episode ‘ . Therefore, entry stage can be interpreted as a aggregation of reading and analyzing.

Lines 9-19 can be categorized as the ‘attack ‘ . In this subdivision it can be seen that the pupil gets ‘stuck ‘ while seeking to happen declarations to the job and overcomes the state of affairs by acquiring an reply ( Ah huh! Ah huh! – in line 13 ) . Furthermore, Schoenfeld categorise these lines under ‘planning and implementing ‘ .

Lines 20-24 is the ‘review stage ‘ harmonizing to Burton and ‘verification episode ‘ under Schoenfeld ‘s episodes.

In Line 25-39 and lines 40 -48, the pupil analyses the information and attempts to place what should be done to work out the inquiry. Burton ( 1984, p. 38 ) says in ‘attack ‘ , ‘ several attacks may be taken and several programs may be formulated and tried out ‘ . Therefore, one could believe that this is ‘attack ‘ stage. But a proper program has non yet been formulated as the pupil is still seeking to understand the information by analysing. The pupil uses the rubric ‘ I know ‘ and ‘now it seems more possible ‘ ( line 39 ) makes it seeable that at the terminal of researching he gets an apprehension of what should be done.After detecting closely, I feel these subdivisions fall in to a stage in between ‘entry ‘ and ‘attack ‘ which is non individually defined by Burton. Although most activities the pupil is engaged in are defined by Burton, I feel that there is an vague spread which is really of import as the pupil can non continue farther without analysing and understanding the information.

Lines 49-53, the pupil tries to research the possibilities of ‘constructing a?s15 ‘ by remembering his past experience ( “ seeking to retrieve my algebra ” ) . Person could believe that these lines are ‘entry stage ‘ as Burton ( 1984 ) defines ‘ explore the job ‘ as the entry stage or even attack stage as he is taking action to get the better of being ‘stuck ‘ . Nevertheless it is hard to name these subdivisions under one class. Therefore the necessity for a broader and good defined stage arises.

In the visible radiation of the protocol analysis I like to raise these thoughts.

Schoenfeld ( 1987, p. 17 ) provinces that ‘Polya ‘s word picture of job work outing schemes were in kernel accurate drumhead description ‘ and that they are larger subdivisions of procedures. He continues to state that Polya ‘s stages are more descriptive but non normative, hence the stages do non supply sufficient information for pupils to utilize it as a job work outing scheme. Similarly when analyzing Burtons stages I felt that the inside informations are sufficient to acknowledge the process but non precise plenty to utilize as a guideline for the deduction of the scheme.

Polya ‘s and Burton ‘s stages fits absolutely for the word job where the pupil can believe of a method of work outing ( program ) straight. But in some jobs the pupil needs to look into the given information before coming up with a program. Schoenfeld ( 1997, p. 298 ) says ‘analysis leads straight into program development ‘ which shows the importance of the analysis episode. When a pupil attempt to work out a job he sometimes ‘search for relevant information so that they can integrate it to the analysis-plan-implementation sequence ‘ ( Schoenfeld, 1997, p. 298 ) . This hunt of information which is a critical point in work outing is non recognized by Polya or Burton.

The planning and execution is combined in Schoendfeld ‘s theoretical account. Though they are combined execution does non ever follow the program.

The procedure ‘look dorsum ‘ , ‘review ‘ and ‘verification ‘ are similar procedures in all three theoretical accounts, except that the ‘reflection ‘ and ‘extending ‘ procedure do non hold matching procedures in the Schoenfeld ‘s episodes as these two procedures ‘intends to develop job work outing expertness instead than assist in work outing a peculiar job ‘ ( Goos & A ; Galbraith, 1996, p. 242 ) .

9. Decision

Problem resolution has been one of the most of import activities in the school mathematics curriculum as it helps pupils to link the mathematical constructs in work outing existent life jobs. Therefore, many research workers have focused on the most effectual method of learning pupils job work outing. In this essay I have used Polya ‘s, Burton ‘s and Schoenfeld ‘s methods of job resolution.

When detecting closely the protocol analysis of the given job utilizing all three theoretical accounts the undermentioned observations can be listed.

Polya ‘s understanding stage is similar to Burton ‘s entry stage. Some of Schoenfeld ‘s reading and analysing stairss are similar to the apprehension and entry stage. But the analyzing has been defined more loosely so that the pupil engages in a series of activities seeking to understand the given information. In Polya ‘s and Burton ‘s stages explicating the program that should be taken occurs by merely reading and understanding the given informations. This may be possible in word jobs. But pupils engage in much complicated procedure jobs which needs to be investigated and analyzed more loosely before explicating a program. However I do non see a necessity of dividing reading and analysing. Reading excessively can be listed under analysing stage.

Harmonizing to Schoenfeld researching episode helps the pupil to understand the method of nearing the job when the inquiry is an unfastened job ( has different methods of work outing and different solutions ) where probe is needed to acknowledge the being of an reply. Polya ‘s and Burton ‘s first measure requires the convergent thinker to acknowledge the being of the solution utilizing the anterior cognition or experience which is rather impossible with certain procedure jobs. Therefore, I feel the exploring episode, can be really utile for the pupil to explicate the program.

Planning/ implementing is combined in Schoenfeld ‘s episodes, whereas in the other two stages it is separated. However, I feel it is better to divide the two as planning is ever non followed by execution. Particularly if pupils are working in a group after be aftering another pupil might raise a inquiry which would once more direct the pupils to another episode for elucidation

As I discussed earlier confirmation does non include reflecting on cardinal thoughts and seeking to widen the solution loosely. It merely refers to look intoing computation. However, when work outing a job the pupil foremost needs to happen the solution. Subsequently the instructor may widen the inquiry and in such a state of affairs the episodes could organize a spiral so that it starts another procedure of work outing.

In decision I would wish to propose the undermentioned stairss as the indispensable stairss for job resolution:

Reading/analyzing, Exploring, Planning, Implementation, confirmation.