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The way adults with orientation tomathematics teaching cope withthe solution of everyday real-worldproblemsAvikam Gazit a & Dorit Patkin aa Department of Mathematics Education, Kibbutzim College ofEducation, 149 Namir Road, Tel-Aviv, Israel

Available online: 29 Jun 2011

To cite this article: Avikam Gazit & Dorit Patkin (2012): The way adults with orientation tomathematics teaching cope with the solution of everyday real-world problems, InternationalJournal of Mathematical Education in Science and Technology, 43:2, 167-176

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International Journal of Mathematical Education inScience and Technology, Vol. 43, No. 2, 15 March 2012, 167–176

The way adults with orientation to mathematics teaching cope with

the solution of everyday real-world problems

Avikam Gazit and Dorit Patkin*

Department of Mathematics Education, Kibbutzim College of Education,149 Namir Road, Tel-Aviv, Israel

(Received 18 November 2010)

The article aims to check the way adults, some who are practicingmathematics teachers at elementary school, some who are academiciansmaking a career change to mathematics teachers at junior high school andthe rest who are pre-service mathematics teachers at elementary school,cope with the solution of everyday real-world problems of buying andselling. The findings show that even adults with mathematical backgroundtend to make mistakes in solving everyday real-world problems. Only about70% of the adults who have an orientation to mathematics solved thesample problem correctly. The lowest percentage of success was demon-strated by the academicians making a career change to junior high schoolmathematics teachers whereas the highest percentage of success wasmanifested by pre-service elementary school mathematics teachers.Moreover, the findings illustrate that life experience of the practicingmathematics teachers and, mainly, of the academicians making a careerchange, who were older than the pre-service teachers, did not facilitate thesolution of such a real-world problem. Perhaps the reason resides in theprocess of mathematics teaching at school, which does not put an emphasison the solution of everyday real-world problems.

Keywords: problem solution; real-world problems; mathematics; practicingteachers; academicians making a career change; pre-service teachers; adults

1. Theoretical background

Word problems are the ‘Achilles heel’ of mathematics teaching and learning of many

students in this discipline. There are but a few studies which examine the solution of

mathematical problems by adults.In a study conducted by Maier and Burke [1], the following problems were

presented to a population of male and female adults:

(a) A man bought a horse for $600 and sold it for $700. Then he bought the

horse back for $800 and sold it again for $900. How much did he profit or

lose in the entire transaction?

*Corresponding author. Email: patkin@netvision.net.il

ISSN 0020–739X print/ISSN 1464–5211 online

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/0020739X.2011.592614

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(b) A man bought a black horse for $600 and sold it for $700. Then he bought awhite horse for $800 and sold it for $900. How much did he profit or lose inthe entire transaction?

Whereas all the participants gave a correct answer to item (b), only about 40% ofthe respondents answered item (a) correctly. Both versions lead to the same solution:$200 regardless whether we refer to the same horse which has been bought and soldtwice or to two different horses which have been bought and sold one time each.Although this is an everyday real-world problem, people tend to represent the data ofthe two versions in a different way. In the version relating to the same horse, therespondents connected the actions in a sequence or related only to the first buy andto the last sale. They failed to see that this concerned two separate actions of buyingand selling. The incorrect solution was obtained by means of a technical algorithm,which the respondents perceived as suitable for solving the problem with thenumerical data. The incorrect solution in the study of Maier and Burke [1] was $300profit, resulting from the calculation of the difference between the final balance –selling for $900 – and the initial balance – buying for $600.

Another incorrect solution was $100, obtained from actions of two consequentprocesses in a sequence: the man bought for $600 and sold for $700, i.e. he madea $100 profit. Then he sold for $700 and bought for $800, namely a loss of $100.He bought for $800 and sold for $900 – a total profit of $100 [1].

The solution of word problems, even those dealing with everyday situations, hasbeen found to be one of the weakest links of mathematics teaching and theimplementation thereof.

Bruer [2], who relates to mathematics teaching, maintains that, ‘word problemsare the ‘black hole’ of mathematics at junior high school – extensive energy isinvested in it, but no light is coming out. . .’ (p. 32). One of the reasons for this ‘blackhole’ is that most of the problems pupils are required to solve at all age groups arenot experiential or authentic and are irrelevant to the pupils’ world. The problemsincluded in the textbooks engage in a limited range of topics in a dull and alienatedmanner [3].

In a study conducted by Gazit [4], 189 pupils from the 3rd to 6th grades wereasked what was the middle floor in a nine-floor building. About 2/3 of the 4thgraders gave the correct answer but only about 1/3 of the 6th graders gave a correctanswer. Approximately, 42% of them gave the answer of 4 1/2, which is a function ofthe dividing algorithm 9:2. Those pupils did not even attempt to reflect and considerthat their answer was unrealistic. The percentage of correct respondents to thisproblem in the 3rd, 4th and 5th grades were about 51%, 68% and 51%, respectively.This might indicate that as pupils go higher in their learning stages at school, thecorrect thinking in solving everyday real-world problems is being blocked [4].

Asman and Markovisz [5] conducted a study which explored to what extent dailyconsiderations are manifested while coping with problems in mathematics. The studycomprised 20 practicing elementary school mathematics teachers and 10 pre-servicemathematics teachers as well as 50 pupils at the 6th grade.

One problem presented to the participants dealt with transportation – dividing175 passengers among buses, each carrying 40 passengers. How many buses had tobe ordered? All the practicing and pre-service teachers solved the transportationproblem correctly, grounding their answer on their acquaintance with the reality oftransporting pupils and ordering buses. Some of them even answered that they would

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have tried to put more pupils on each bus so that four rather than five buses could

be ordered in order to save money. Unlike them, some of the pupils indicatedthe answer 4.375, without relating to the fact that it is impossible to order parts of a

bus. Among these pupils, some wrote ‘illogical result’ while others wrote 4(15),namely 4 buses with a remainder of 15 in order to cope with the issue of fractions.

The findings of that study show that pupils may solve problems without consideringthe reality.

Another problem engaged in averages: four families live in a building and

together they have 10 children. How many children on average does each familyhave? In this problem, 43% of the teachers answered that there are no half children

and, therefore, they rounded the answer to 2 or 3. They gave this answer although indaily reality, as well as in the communication media, they are exposed to averages

which are not integers in discrete values. Some of the practicing teachers argued thathad they known an average could be 2.5, then they would have taught it in class.

However, textbooks present data in which the average is always an integer.The conclusion of this study is that mathematics teachers too encounter difficulties in

solving problems which are connected to reality and that their typical answers aresimilar to those of the pupils.

In a study by Gazit [6], the findings illustrate that pre-service elementary school

mathematics teachers achieved a low percentage of success in solving real-worldword problems. Furthermore, the research participants included 10th graders,

learning advanced mathematics and others learning mathematics literacy (notadvanced) in a mainstream matriculation pathway (designed for pupils with learning

difficulties, learning disabilities, or other special needs). The highest percentage ofsuccess was achieved by the pupils in the mathematics literacy pathway, approxi-

mately 47%; the advanced mathematics pathway pupils’ percentage of success wasabout 43% and that of the pre-service teachers only 20%! The incorrect solutions

given by the pre-service teachers had a technical-algorithmic nature, without using a

proper representation of the problem conditions. These pre-service teachers are anoutcome of the education system, from which the 10th graders are also coming. The

admittance to the specialization in mathematics pathway requires an entry-level testas well as an examination checking the candidates’ knowledge in mathematics.

However, the requirements in the mathematics examination emphasize arithmeticand geometric knowledge, without any reference to problem solution.

Jozwiak [7] maintains that college professors are good in teaching technical terms

but their problem solution teaching is insufficient. In his paper, Jozwiak discusses theimportance of problem solution teaching and suggests several problem solution

methods.One of the reasons for the gaps between mathematics studied at school and

mathematics implemented outside the classroom is the uniform-stereotypical nature

of problems learnt at school [8].Russel [9] also criticizes the problems taught in mathematics lessons. According

to her, these problems do not represent reality and the use of real data is unsuitableand insufficient.

Research questions

(1) How do adults with orientation to mathematics teaching cope with the

solution of everyday real-world problems?

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(2) What are the incorrect strategies applied by adults with orientation tomathematics teaching in the solution of everyday real-world problems?

2. Methodology

2.1. Research method

This is a quantitative study in which respondents from three populations solved areal-world problem in mathematics.

2.2. Research objective

The goal of our study was to explore how adults with orientation to mathematicsteaching cope with the solution of an everyday real-world problem, similar to theproblem of buying and selling a horse [1]. However, the participants in the study ofMaier and Burke were adults without any specific characterization, whereas in thisstudy, the participants were practicing elementary school mathematics teachers,academicians making a career change to mathematics teachers at junior high school,and pre-service mathematics teachers for elementary school, all learning in a teachereducation college at the centre of the country.

2.3. Research population

The research population comprised 87 adults: 34 academicians making a careerchange to mathematics teachers at junior high school, half of them male and all ofthem between 30 and 50 years old; 29 practicing elementary school mathematicsteachers, 3 of them male and all of them between 30 and 50 years old and 24 pre-service mathematics teachers for elementary school who were at their third year ofstudies, all of them 20–35 years old.

2.4. Research tools

A questionnaire was used with an everyday real-word mathematical problem asfollows:

‘Danny bought a ball for NIS100.1 Some time later he sold it to Dana forNIS110. Then he bought it from Dana for NIS120 and finally sold it to David forNIS130.

How much has Danny profited on the entire transaction?

Please record the way by which you obtained the solution’.

One can solve the problem by applying three different strategies:

(a) Relating to each pair of buying and selling processes separately andcalculating the profit. Buying for NIS100, selling for NIS110 and making aprofit of NIS10. Buying for NIS120, selling for NIS130 and making a profitof NIS10. The total profit is NIS20.

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(b) Making a joint balance of payments and receipts. Buying for NIS100 andNIS120, a total of NIS220. Selling for NIS110 and NIS130, receiving a totalof NIS240. NIS240�NIS220¼NIS20 profit.

(c) Calculating the financial balance at every stage. Buying for NIS100 andtherefore the financial balance is �NIS100. Selling for NIS110, namely thefinancial balance is NIS110�NIS100¼þNIS10.

Buying for NIS120, bringing the balance to NIS10�NIS120¼�NIS110.

Selling for NIS130, the balance is NIS130�NIS110¼NIS20 profit.

2.5. Research procedure

In three different sessions, all the participants in each of the three groups received asheet with several word problems, including the investigated problem. The additionalproblems were introduced in order to create an atmosphere of thinking which issuitable to the solution, the investigated problem being part of the problems layoutwith which the participants had to cope. The solution of each of the problems did notrequire mastering of mathematical knowledge beyond that of elementary schoolpupils. However, people solving problems of this kind need to read the problemthoroughly in order to represent their data correctly, a fact which might encompassa strategy suitable to the solution [10].

3. Findings

3.1. The first research question

The first research question focused on the way adults with orientation tomathematics teaching (practicing elementary school mathematics teachers, acade-micians making a career change to junior high school mathematics teachers and pre-service elementary school mathematics teachers) cope with the solution of everydayreal-world problems.

3.1.1. The entire research population

The problem presents a situation of buying and selling with balances of financialprofit and loss.

Table 1 presents the distribution of the correct and incorrect answers accordingto the entire population.

The findings illustrate that only a little more than 2/3 of the entire researchpopulation (70.1%) responded correctly to the problem and used one of the abovestrategies for solving the problem.

3.1.2. Comparison of the three sub-populations

The common characteristic of the three groups is their orientation to mathematicsteaching. Table 2 presents the distribution of the correct and incorrect answersaccording to sub-populations.

Table 2 illustrates that the highest percentage of success was demonstrated by thepre-service teachers – 87.5% and only three of them applied one of the two incorrect

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solution strategies, resulting in the finding that Danny’s profit was either NIS10 orNIS30. The pre-service teachers have not yet accumulated any experience in teachingmathematics or in their daily lives since most of them are in their 20s.

Only about 3/4 of the practicing elementary school mathematics teachers, whohave a teaching experience of more than 3 years, gave a correct answer. Theseteachers have a more extensive life experience than the pre-service teachers since theyare 30–50 years old; seven of the teachers indicated a solution whereby they used oneof the incorrect strategies, the prominent incorrect solution being NISþ10, indicatedby five teachers (17.2%).

The lowest percentage of success was obtained by the group of academiciansmaking a career change to junior high school mathematics teachers. Only a smallnumber more than half of the participants – 18 (52.9%) – indicated a correct answer;7 (20.6%) and 6 (17.6%) of them used one of the incorrect strategies, writing theresult NISþ10 and NISþ30, respectively. Only in this group three participants(8.9%) indicated the result 0 – namely Danny neither profited nor lost. This is due tothe insertion of incorrect data to the strategy which usually leads to a correctsolution, i.e. making a balance of the total payments and receipts. This group ofacademicians making a career change consists of 30–50 years old adults with a BAdegree and life experience, but without any knowledge in mathematics teaching orlearning in a mathematics teachers’ education college.

The difference in success percentage between the pre-service mathematicsteachers – 87.5% – and the academicians making a career change – 52.9% – issignificant (�2=7.63, p5 0.006). The difference in success percentage betweenthe elementary school teachers – 75.9% – and the academicians making a career

Table 2. Distribution of the correct and incorrect answers according to sub-populations.

Answer population

Danny profitedNIS20

Correct answer

DannyprofitedNIS10

Incorrectanswer

DannyprofitedNIS30

Incorrectanswer

Danny did notprofit at allIncorrectanswer

Career change to junior highschool teachers (N¼ 34)

18 (52.9%) 7 (20.6%) 6 (17.6%) 3 (8.9%)

Elementary school teachers(N¼ 29)

22 (75.9%) 5 (17.2%) 2 (6.9%) 0

Pre-service teachers (N¼ 24) 21 (87.5%) 2 (8.3%) 1 (4.2%) 0

Table 1. Distribution of the correct and incorrect answers given by the entire researchpopulation.

Answerpopulation

Danny profitedNIS20

Correct answer

Danny profitedNIS10

Incorrect answer

Danny profitedNIS30

Incorrect answer

Danny did notprofit at all

Incorrect answer

3 (3.4%) 9 (10.4%) 14 (16.1%) 61 (70.1%) The entire researchpopulation (N¼ 87)

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change – 52.9% – is almost significant (�2¼ 3.55, p5 0.06). The difference insuccess percentage between the pre-service mathematics teachers – 87.5% – andthe elementary school mathematics teachers – 75.9% – is not significant. Thesignificance of the other bold values was not calculated according to the �2 rules,because the number of participants was less than five.

The distribution of answers among the three sub-populations can be summed upas follows: 7/8 of the pre-service teachers, who are currently at an advanced stage ofthe mathematics studies but do not have an extensive life experience, presented acorrect solution to the problem. About 3/4 of the practicing elementary schoolmathematics teachers, who are more experienced than the pre-service teachers,presented a correct solution to the problem. Only about half of the academiciansmaking a career change to junior high school mathematics teachers, whose lifeexperience is similar to that of the practicing teachers, indicated a correct solution tothe problem.

3.2. The second research question

The second research question related to the incorrect strategies applied by theparticipants in the solution of the problem. Little less than 1/3 of the entire researchpopulation (29.9%) gave an incorrect answer. This is due to three incorrectstrategies, showing they failed to understand the nature of the problem.

3.2.1. The answer þ10

The answer þ10, namely a profit of NIS10, was given by 14 participants from thethree groups (16.1%). The wrong strategy they used and its recording through thesolution was making an interim balance between every two consecutive transactionsof buying/selling or selling/buying and writing down the interim result. For example,one of the participants responded in the following way:

Danny bought for NIS100 and sold for NIS110, namely 110� 100¼þ10.

He sold for NIS110 and bought for NIS120, namely 110� 120¼�10.

He bought for NIS120 and sold for NIS130, namely 130� 120¼þ10

Hence, þ10þ (�10)þ (þ10)¼þ10. Danny profited NIS10.

The mistake or misconception of this strategy is due to the fact that the respondentsused twice the two middle transactions of selling and buying (NIS110 and NIS120).

3.2.2. The answer þ30

The answer þ30, namely a profit of NIS30, was given by nine out of the entireresearch population (10.4%). The incorrect strategy they used was calculating thedifference between the first buy and the second sale, 130� 100¼þ30, and ignoringthe two middle transactions of selling and buying (NIS110 and NIS120).

It is important to point out that in the first strategy, which resulted in anincorrect solution, the respondents used twice the same transaction of buying orselling. Conversely, in the second strategy, they ignored those two transactions.

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3.2.3. The answer neither profited nor lost

Three participants, all from the group of the academicians making a career change tojunior high school mathematics teachers, responded that Danny neither profited norlost. The strategy they used leads to a correct solution when there is a joint paymentsand receipts balance. However, although they used a strategy which should haveresulted in a correct solution, they made a mistake at the stage of inserting data tothe mathematical model, adding NIS100 to NIS130 in order to calculate theexpenses and NIS110 to NIS120 for calculating the income. Thus, their resultwas: 130� 130¼ 0. This mistake can be attributed to lack of attention, lack ofconcentration and/or to a problem of incorrect data reading and coding which isassociated with attention difficulty.

4. Discussion and conclusions

This article aimed to examine the ability of adults with orientation to mathematicsteaching to cope with an everyday real-world problem. The problem presented to theparticipants was similar in its mathematical structure to the question included inthe study of Maier and Burke [1]. The problem presented in both studies related tothe buying and selling of the same item (a horse there and a ball here) by the sameperson.

In the study conducted in the United States [1], about 60% of the respondents,both male and female without any additional characteristics specified, gave acorrect answer. Conversely, in this study, about 70% of the participants who have anorientation to mathematics teaching answered correctly. In the two studies, a highpercentage of the respondents, 40% and 30%, respectively, gave an incorrect answerto an apparently uncomplicated problem, in spite of the problem narrative whichpresented a chain of buying and selling transactions. Both studies were conducted infact on one question, approximately 44 years apart, in two different cultures (UnitedStates and Israel). They depict a rather similar picture of a relatively high percentageof incorrect answers to the question of buying and selling which is taken fromeveryday life. We believe that the higher percentage of correct answers given in thisstudy is due to the fact that the population was mathematics-oriented. However,these results cannot account for the high percentage of incorrect answers in spite ofthe mathematical background.

The incorrect answers in this study are identical to the incorrect answers in thestudy of Maier and Burke [1]. Moreover, the two strategies of incorrect solutions(profited NIS10 and NIS30) are identical in both studies. The exceptional answer,namely that Danny neither profited nor lost, given by three participants from thecareer changing group in this study, stems from wrong coding of the data or failureto comprehend the text [11]. If we combine the findings of this study to those of theAmerican one, it will support the conclusion that some of the adults – in our case30% of the adults with academic degree or those learning in academic frameworks –perform sterile calculations which are dissociated from the reality presented by theproblem.

Distribution of the answers given by the three populations show that theyoungest and relatively inexperienced group of the pre-service mathematics teachersat elementary school demonstrated the highest percentage of correct answers(87.5%). Perhaps the education process, stimulating thoughts combined with

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openness and lack of fixation, facilitated the solution of real-world problems

presented in such an unusual way. Another possible explanation for this resides in

the fact that more recent schooling has in fact supported the understanding of more

realistic math problems.The practicing mathematics teachers obtained a lower percentage of success than

that of the pre-service teachers. This stems from to the fact that, in spite of themathematical experience and background, teachers have a certain thinking blockwhen they are faced with a problem which is unlike problems commonly taught inelementary school. The group of academicians making a career change to junior highschool mathematics teachers attained the lowest percentage of success. Most of themhave academic knowledge in mathematics, sciences and computers. Yet they usedmore incorrect algorithms for solving the problem than the other groups. Thepercentage of success of this group (52.9%) is also lower than that of adults withoutspecific characteristics in the study of Maier and Burke [1] – 60%. About half of thecareer-changing academicians gave an incorrect answer. This finding might illustrate,within the research limitations, algorithmic-technical patterns of thinking without ause of tools for representing the problem. Both the practicing teachers and the careerchanging academicians constitute a distinct age group of adults between 30 and 50years old, compared to the young pre-service teachers, who are 20–25 years old. Thesuccess percentage of the younger participants (87.5%) was higher than that of thetwo older groups (75.9% and 52.9%). This finding, relating to the solution ofmathematical real-world problems, somewhat refutes the findings of another study[12], which found that adults at an advanced age solve everyday problems better thanyounger adults.

To sum up, even a daily real-world problem, associated with buying and selling,causes a cognitive block when the problem embodies an unusual component of thedata presentation. However, the data presentation does not include a ‘sting’ or a‘trick’ which should be decided or by-passed. By making a correct presentation of aproblem, it can be solved relatively easily without using equations or required specialalgorithms.

These findings cannot be generalized due to the limitations of this research whichexplored only a single question. It is recommended extending in future the study toadditional questions. Based on the research findings, it is recommended expandingthe issue of real-life problems, both in teacher education colleges and in mathematicslessons at school. An appropriate balance should be found between the learningof pure mathematics bodies of knowledge and topics associated witheveryday reality.

Note

1. NIS, New Israeli Shekel, equivalent to $0.30.

References

[1] N.R.F. Maier and R.J. Burke, Response availability as a factor in the problem-solving

performance of males and females, J. Pers. Social Psychol. 5 (1967), pp. 304–310.[2] J.T. Bruer, How children learn, Exec. Educ. 16(8) (1994), pp. 32–36.

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[3] A. Gazit, The teaching of mathematics, interest and beauty – Can they walk together andperhaps they were not meant to do it, in Teachers in a World of Change, S. Guri-Rozenblit,ed., Open University, Israel, 2004a (in Hebrew), pp. 356–389.

[4] A. Gazit, The strange case of the fourth-and-a-half floor: The way 3rd-6th-graders cope

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[7] J. Jozwiak, Teaching problems-solving skills to adults, J. Adult Educ. 33(1) (2004),pp. 19–34.

[8] K. Gravemeijer, Mediating between concrete and abstract, in Learning and TeachingMathematics: An International Perspective, T. Nunes and P. Bryant, eds., Psychology

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mathcurr.html[10] D. Patkin and A. Gazit, Effect of story problems, different word formulation and

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Int. J. Math. Educ. Sci. Technol. 42(1) (2011), pp. 75–87.[11] F. Arzarello, The role of natural language in pre-algebraic and algebraic thinking,

in Language and Communication in the Mathematics Classroom, H. Steinberg, M. Bussi,and A. Sierpinska, eds., NCTM, Reston, VA, 1998, pp. 249–261.

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