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Teaching and Teacher Education 28 (2012) 1185e1195

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Teaching and Teacher Education

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Teacherestudent interaction in joint word problem solving. The role of situationaland mathematical knowledge in mainstream classrooms

Javier Rosales a, Santiago Vicente a,*, Jose M. Chamoso b,1, David Muñez a, Josetxu Orrantia a

aDepartment of Developmental and Educational Psychology, Faculty of Education, University of Salamanca, Edificio Europa, Paseo de Canalejas 169, 37008 Salamanca, SpainbDepartment of Didactics of Mathematical and Experimental Sciences, Faculty of Education, University of Salamanca, Edificio Europa, Paseo de Canalejas 169, 37008 Salamanca,Spain

h i g h l i g h t s

< 11 mainstream primary teachers were analyzed during word problem solving.< Problems included additional relevant mathematical and situational information.< Teachers did not use additional information to solve the problems in a meaningful way.< Teachers could be preventing students from solving the problems in a meaningful way.

a r t i c l e i n f o

Article history:Received 7 December 2011Received in revised form13 July 2012Accepted 15 July 2012

Keywords:Word problem solvingClassroom interactionTeaching methodsClassroom discourseSituational knowledge

* Corresponding author. Tel.: þ34 923294400x3441E-mail address: sanvicente@usal.es (S. Vicente).

1 Tel.: þ34 923294400x3469; fax: þ34 923294703

0742-051X/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.tate.2012.07.007

a b s t r a c t

Word problem solving involves the construction of two different mental representations, namely,mathematical and situational. Although educational research in word problem solving has documenteddifferent kinds of instruction at these levels, less is known about how both representational levels areevoked during word problem solving in day-to-day learning environments. The aim of this descriptivework is to analyze how mainstream teachers promote mathematical and situational knowledge whilesolving mathematically and situationally reworded word problems. The results suggest that wordproblem solving is faced by teachers as a mechanical and non-reflexive task which involves limitedsituational knowledge.

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1. Introduction

Word problem solving is one of the mathematical school tasksmost practiced around the world (Hiebert et al., 2003; Stigler &Hiebert, 1997). One reason for such frequency has to do with thesignificant role the task plays in developing meaning, which can beused for application of mathematical concepts and for integratingthe real world in the mathematics classroom (Verschaffel, Greer, &De Corte, 2000). Another reason could be that word problemsolving is considered a key component in learning mathematics(NCTM, 2000; OECD, 1999).

Furthermore, word problem solving has proved to be notori-ously difficult for students, the so-called “black hole” for middleschool mathematics (Bruer, 1994). This difficulty has been reported

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by international assessment programs. For example, Trends inInternational Mathematics and Science Study (TIMSS), Program forInternational Student Assessment (PISA), or National Assessment ofEducational Progress (NAEP) have reported that students all over theworld have difficulties when solving word problems, especiallyproblems that are close to real-life situations (Hiebert et al., 2003;National Research Council, 1989; OECD, 2010; Stigler & Hiebert,1997).

Although these reports arewell known, significant changes havenot taken place in the way in which word problems are usuallyapproached in mainstream educational practice in schools (Cuban,1993; Good, Clark, & Clark, 1997). It is therefore necessary to shedsome light on what happens when students solve problems in themainstream classroom in order to identify gaps between theoret-ical proposals and mainstream classroom practice. Specifically, thepresent study aims at analyzing how two specific aspects ofproblem solving, namely, mathematical and contextual knowledge,were evoked by 11 mainstream primary teachers when they solved

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e11951186

two non-standard word problems jointly with their pupils in theirmathematics class.

2. Theoretical framework

Word problem solving is a difficult task because to solve a wordproblem in a genuine way students must create different levels ofmental representation (in the sense of Van Dijk & Kintsch, 1983).That is, solvers must create individual and subjective mentalrepresentations that relate to, firstly, events or situations (the so-called situational model); and secondly, the mathematical struc-ture underlying the situation described (or mathematical model).In these mental models only the most relevant features of theproblem, both situational (related to the qualitative situationdescribed in the problem) and mathematical (related to numericalvalues and quantitative relations between them, in terms of themathematical structure of the problem2) are represented(Verschaffel et al., 2000). During word problem solving studentsmust generate an adequate situational model of the problem. Theymust consider the situation described by the problem and decidewhat information is essential and what information is less impor-tant. After the situational model has been created, the mathemat-ical model of the problem must be generated by using the solver’sprevious mathematical knowledge to fit the situational model andthe appropriate mathematical structure for the problem (in termsof quantities and the mathematical relations between them). Oncethis mathematical model has been constructed, the appropriatemathematical algorithms must be performed to obtain the result.Once the result is obtained, it must be interpreted in relation to themathematical model and the real situation described in theproblem. Finally, the interpreted and validated result needs to becommunicated in a way consistent with the question of theproblem.

However, students can also solve problems in a superficial wayin which some steps of the genuine approach are by-passed. Whenusing this superficial approach, there is no situational model of theproblem, and the mathematical model is not based on mathemat-ical reasoning. This is rather done in an automatic way by taking thedata of the problem and selecting the algorithm to be employedusing some meaningless strategy based on some salient element inthe problem, like the key word strategy (e.g., Hegarty, Mayer, &Monk, 1995; Nesher & Teubal, 1975; Verschaffel, De Corte, &Pauwels, 1992). In this strategy, the algorithm to be employed isselected by using certain words of the problem as a hint (forexample, “more” to add, “lose” to take away). Once the operationhas been selected and the algorithm has been performed, the resultis immediately communicated as the answer. Thus, students do notrefer back to the original problem situation to verify that it isa meaningful response to the original question or to check itsreasonableness. For example, considering the “bus problem”

(Silver, Shapiro, & Deutsch, 1993), “The Clearview Little League isgoing to a Pirates game. There are 540 people including players,coaches and parents. They will travel by bus, and each bus holds 40

2 Following Riley and Greeno (1988), for addition and subtraction problems likedthose used in this study, three mathematical structures can be distinguished:change, compare and combine. Change problems are those in which an initialquantity is increased or decreased by a change quantity to result in a final quantity(i.e., “John had 5 marbles. He won 2 marbles in a game. How many marbles doesJohn have now?”). In compare problems a compare quantity is compared doa reference quantity, there being a quantitative difference between them (i.e.: “Johnhas 5 marbles. Peter has 3 marbles more than John. How many marbles does Peterhave?”). Finally, in combine problems there are two quantities or parts that arecombined into a whole quantity (i.e.: “John has 5 marbles. Peter has 3 marbles. Howmany marbles do John and Peter have altogether?”).

people. Howmany buses will they need to get the game?”, the result ofthe division (540:40 ¼ 13.5 buses) must be interpreted by consid-ering the situation. Hence, the final answer must be, at least, 14buses because a bus cannot be split into 0.5 buses. However, whenusing a superficial approach students give “13.5” as the answerwithout checking its reasonableness according to the situationdescribed by the problem.

Previous research has shown that students usually face wordproblems in a superficial instead of a genuine way (Van Dooren,Verschaffel, Greer, & De Bock, 2006). Nevertheless, it is also wellknown that students can move from a superficial approach towarddeeper approaches. For example, there is abundant evidence sug-gesting that instruction in the processes associated with wordproblem solving allows students to perform better. Thus we knowthat students improve their achievement when they operate withboth the situational structure of the problem (Verschaffel & DeCorte, 1997) and the mathematical structure of the problem, interms of its change, compare or combine structure for addition andsubtraction problems (see footnote2) (Fuson &Willis, 1989; Jitendraet al., 1998; Xin, Jitendra, & Deatline-Buchman, 2005). Furthermore,empirical studies that added extra information to the wording ofthe problem (the so-called “rewording”) showed that the extrainformation might have a positive impact on students’ problem-solving processes and skills, especially when problems are math-ematically difficult.3 Mathematically reworded problems aim atclarifying the mathematical relations between the sets implied inthe problem (Cummins, 1991; Davis-Dorsey, Ross, & Morrison,1991; De Corte & Verschaffel, 1985; Vicente, Orrantia, &Verschaffel, 2007, 2008a,b) and situationally reworded problemspresent the (real world) situation referred to by the text of theproblem in a more enriched and elaborate way (Coquin-Viennot &Moreau, 2007; Cummins, Kintsch, Reusser, & Weimer, 1988;Orrantia, Tarín, & Vicente, 2011; Staub & Reusser, 1992; Stern &Lehrndorfer, 1992).

Despite the fact that reworded problems might help students toperform better, few studies have analyzed how situational andmathematical models construction is promoted during wordproblem solving in day-by-day classroom practices. Chapman(2006) probably provides the best description of the role ofpromoting mathematical and situational models construction inword problem solving. She explored how social/cultural contextsare used in instruction in mathematics classrooms, and based onthe modes of knowing proposed by Bruner (1985, 1986), sheconsidered two different modes of going about the task: a para-digmatic mode and a narrative mode. When the paradigmaticmode is used, teachers focus on the mathematical aspects of theproblem, such as data selection or mathematical reasoning, whichare relatively context-free. That is, this approach focuses students’attention on strategies and ways of thinking that are independentof a particular real-life context. In contrast, when the narrativemode is used, teachers focus on the surrounding context of theproblem, that is, they address the cover story of the word problemin order to understand or relate the storyline, plot, characters,objects, situations, actions, relationships or intentions to attain

3 Following Riley and Greeno (1988), for addition and subtraction problems themost difficult ones are: a) change problems with the initial quantity unknown(“John had some marbles. He won 2 marbles in a game. Now he has 8 marbles. Howmany marbles did John have at the beginning”); b) compare problem with thereference quantity unknown (“John has 5 marbles. John has 3 marbles more thanPeter. How many marbles does Peter have?”); and c), combine problems with thewhole quantity unknown (“John and Peter have 5 marbles altogether. Peter has 3marbles. How many marbles does John have?” The reader can compare the diffi-culty of all these examples to those in the previous footnote, which were the easiestof each problem type.

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e1195 1187

different goals. These goals may relate to having fun, establishingrapport in the class, considering non-mathematical solutions to beacceptable, or using contextual information to make sense of theproblem situation with the understanding that such informationmay or may not be necessary to obtain a mathematical solution.

Based on this differentiation, Depaepe, De Corte, and Verschaffel(2010) analyzed the paradigmatic and narrative interventions in twoFlemish sixth-grade mathematics classrooms in a seven-monthstudy. The results revealed that the paradigmatic approach wasmore dominant than the narrative approach. Depaepe et al. (2010)argued that one of the reasons for this absence of a strongmodeling approach, linked to the prevalence of the paradigmaticmode, could be the standard nature of the word problems usuallysolved in the mainstream mathematics classrooms. Standard prob-lems are those that can be properly modeled and solved bystraightforward application of one or more arithmetic operationswith the given numbers (for example, “Steve has bought 4 ropes of2.5 m each. How many ropes of 0.5 m can he cut out of these 4ropes?”, Verschaffel, De Corte, & Borghart, 1997, p. 340). Theseproblems, which are themost of the problem students has to solve inmathematics classrooms, do not imply a real challenge for students(i.e., problems do not contain any additional information orconstraints based on real-world knowledge). In turn, non-standardproblems are those that requiresmore than the simple application ofmathematical algorithms, so the problem solver has to createa deeper comprehension of the problem (for example, “Steve hasbought 4 planks of 2.5 m each. How many planks of 1 m can he sawout of these 4 planks?” Verschaffel et al., 1997, p. 340).

In short, empirical studies based on theoretical models of wordproblem solving have suggested that students can learn to generatesituational and mathematical models to solve problems successfully(Fuson & Willis, 1989; Jitendra et al., 1998; Verschaffel et al., 1997;Xin et al., 2005). In addition, different studies have proposed thatsituationally andmathematically reworded problems allow studentsto perform better, especially when the problems are mathematicallydifficult (Coquin-Viennot & Moreau, 2007; Cummins, 1991; Davis-Dorsey et al., 1991; De Corte & Verschaffel, 1985; Staub & Reusser,1992; Stern & Lehrndorfer, 1992). Although these results are wellknown by researchers, it is unknown whether mainstream teachersapproach word problem solving in a paradigmatic way by focusingmainly on the mathematical reasoning and not on situationalreasoning. In addition, Depaepe et al.’s (2010) study raises thequestion about what would happen if teachers solve non-standardword problems. If the results obtained by Depaepe et al. (2010) donot depend on the standard nature of the problems, no differenceswould be expected between the teacher’s approach when solvingstandard and non-standard problems.

In order to explore this possibility and go beyond Depaepeet al.’s idea of teachers’ paradigmatic or narrative approach to wordproblem solving, we designed a descriptive study to address threequestions: a) whether the behavior of mainstream teachersdepends on the kind of problem solved; b) whether teachers usethe explicit hints included in the text (about the mathematicalstructure of the problem) for mathematical reasoning promotionregarding the mathematical structure of the problem; and c)whether teachers focus the situational processing on the mostrelevant aspects of the situational content.

In other words, when mainstream teachers solve non-standard,situationally and mathematically reworded problems jointly withtheir pupils in their mainstream classrooms, will they:

- Use a paradigmatic or narrative approach?- Promote a deep mathematical processing of the problem?- Promote a deep qualitative comprehension of the situationdescribed by the problem?

3. Method

3.1. Participants

Eleven mainstream teachers (4 male, 7 female) and their pupils(ages ranging from 8.4 to 11.3 years old) from four primaryeducation state schools (grades 3e5) of a large urban school districtin Spain participated in the present study. Teachers were selected atrandom from an initial pool of 48 teachers from 10 differentschools, who volunteered to be audio-taped (their students alsoaccepted being audio-taped). The teachers were Spanish nativespeakers, belonged to the same race, ethnicity, and class that themost of the students, and were educated in culturally relevantpedagogy. This way, teachers can be considered as representative ofthe students they taught and no significant influence was expectedin this respect from the data generation, data collection and anal-ysis procedure. Teachers’ initial training was focused on developingcapabilities related to specific competencies for the training ofprofessionals to teach mathematics at the primary level; however,they did not receive any specific training on word problem solving.Their professional experience averaged 15.6 years (ranging from12.7 to 17.1 years). The schools where the teachers worked weremedium-size (500 students). These schools were embedded ina middle class socio-economic background and served predomi-nantly Spanish students. However, 8.2% of the students werePortuguese, Romanian, Bulgarian and Moroccan immigrants, andthey were of lower-middle socio-economic class. This percentage ison average of immigrant students of the region and slightly belowthe mean of Spain, so the classrooms analyzed can be considered asrepresentative. In any case, these classrooms analyzed can beconsidered as representative of the students in Spain.

The number of students in each classroom varied from 20 to 25,and the time spent for solving each word problem varied fromapproximately 3 to 10 min. The mathematical instruction receivedby the students was based on mathematics textbooks, and wordproblem solving is taught according to a four-step procedure: dataselection, operation selection, execution of algorithm and report ofthe solution. The textbooks contained more routine exercises thanword problems (about 20% of the activities were word problems),and these word problems were mainly standard (see Orrantia,González, & Vicente, 2005; Vicente, Orrantia, & Manchado, 2011).Students did not receive any specific training in mathematic orword problem solving beyond their regular mathematical instruc-tion. Teachers reported that students’ achievement, on average, waswithout noteworthy difficulties and did not deserve specificanalysis.

3.2. Materials

Two non-standard, mathematically difficult problems wereused in our study. These problems were used by Orrantia et al.(2011) and characterized as difficult for the same academicgrades. Previous studies have found that some kinds of additionalmathematical and situational information are necessary to solvethis type of difficult problem (Orrantia et al., 2011; Stern &Lehrndorfer, 1992; Vicente et al., 2007). Furthermore, the prob-lems were non-standard because they were reworded; hence, theypresented two types of additional information: mathematical andsituational. In order to ensure that the additional information wasadequate, the rewording procedurewas based on previous researchon mathematical and situational rewording. For example, studieson mathematical rewording found positive results by makingexplicit themathematical relations between the sets, but studies onsituational rewording focused on different aspects and some ofthem did not find any positive effect (for a review, see Vicente et al.,

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e11951188

2007). More specifically, two of them obtained positive results byusing similar situational rewordings on different types of wordproblems: Coquin-Viennot and Moreau (2007) and Stern andLehrndorfer (1992) added qualitative comparisons between thecharacters in problems involving comparisons between quantities,and Orrantia et al. (2011) highlighted the causal links between thegoals of the characters and the increases and decreases of thequantities in problems involving changes in the amounts (see Fig. 1,words in italics are situational information).

The design of our rewordings was in line with rewordings thatproved to be effective in previous research: firstly, the mathemat-ical structure underlying the problem was made explicit; andsecondly, the intentions of the characters, the causal relationsbetween these intentions, and the actions performed on the setsinvolved in the problem were made explicit by adding situationalinformation (see Fig. 2).

In Fig. 2, the mathematical information highlights the mathe-matical structure underlying the problem; that is, if we join thesheep devoured by the wolves to those that survived the attack weobtain the same quantity as if we join the sheep bought by theshepherd to those that he already had at the beginning. Thus, as thesheep devoured by the wolves and the sheep that survived theattack are known data, students can use them to begin to solve theproblem.

The rationale behind the added situational information was toestablish causal links between characters’ intentions and theactions performed in the problem (i.e., since there was good fodder,the goal of the shepherd was to increase the size of the flock;furthermore, due to the hungry wolves, a new action is performedon the flock and its size decreases).

3.3. Data collection

Teachers were audio-taped while they solved two differentarithmetic word problems (with the same mathematical and situ-ational structure and type of additional information) in theirclassrooms and during the time normally devoted to joint problemsolving in mathematics sessions. Problems were solved in twodifferent sessions. The time gap between sessions was approxi-mately two weeks to avoid teachers being able to use the firstproblem as a reference to solve the second one. All problems weresolved jointly with the students by means of whole-class interac-tion. Audio recordings were transcribed and analyzed. One class-room observer took notes to supplement the audio-tapeddiscourse. The notes provided the context for the recordings soeach teacher statement or action could be discerned. The classroomobserver field notes and the transcriptions of the audio-tapedsessions were integrated into a document describing the problem-solving session.

A shepherd was taking care of a flock of shee

He wanted to increase the size of the flock be

order to do so the shepherd went to a mark

sheep. One evening the shepherd saw a pack

hungry and then they devoured 11 sheep and

sheep did the shepherd buy in the market?”

Fig. 1. Example of situationally reworde

3.4. Analysis

The data analysis involved three different levels. Firstly, theconversational turns that took place in the course of the interactionwere grouped in exchange cycles (Wells, 1999). An exchange cyclewas considered the basic segment of the interaction analysis.Basically, an exchange cycle starts with an initial question/orderand it finishes when that question/order is completed, that is, whena common agreement between teacher and students is reached. Forexample, whereas in Cycle 1 the agreement is about the amount ofsheep the shepherd had, in Cycle 3 the agreement is about theamount of sheep the shepherd bought at the market (see Table 1)

Secondly, considering that our analysis focused on the informa-tion shared by teachers and students, the analysis unit was thecontent made public in the teacherepupils interaction. Publiccontents were controlled by teachers, that is, teachers decided whatkind of content would be made public by selecting the questions tobe made and the information to be shared. Therefore, each interac-tion cycle was identified and defined as a single idea generated bythe teacher or the students, or produced jointly by the teachers andthe students. For example, as shown in Fig. 2, the contents madepublic in Cycles 1e3 relate to: (1) The shepherd had 57 sheep, (2) Hewanted to increase the size of the flock, so he went to a market, and (3)We do not know how many sheep he bought because that is thequestion. One interaction cycle can include one or more publiccontents (see Cycle 9 of the example). However, only one kind ofcontent (mathematical or situational, as will be explained below)can be made public in each cycle. Interaction fragments devoted toreading the problemwere not included in the analysis because therewas no content made public by merely reading the problem.

Finally, the third level required categorizing the contents madepublic. To address the first question of our study e whetherteachers use a paradigmatic or narrative approach e a priori codingcategories for a teacher’s discourse were used. The transcripts werecoded into two broad categories: mathematical and situational. Toaddress the second and third questions e whether teacherspromote a deep mathematical processing of the problem, anda deep situational comprehension of the situation e further codingsubcategories were established for mathematical and situationalcategories. Mathematical subcategories were based on Verschaffelet al.’s (2000) distinction between superficial and genuine pro-cessing of word problems. A superficial understanding involvesselecting the data and certain key words in the problem statement,followed by the automatic triggering of the mathematical model,and finally, the execution of the calculations. However, a genuinemathematical modeling not only requires selecting data and per-forming operations, but also mathematical reasoning about themathematical structure of the problem. Considering these steps,three subcategories for mathematical public contents were estab-lished (examples in Table 1):

p. The shepherd had a flock of 57 sheep.

cause this year there was good fodder. In

et, where he decided to buy some more

of wolves in the area. The wolves were

now there are 96 sheep left. How many

d problem by Orrantia et al. (2011).

A shepherd was taking care of a flock of sheep. The shepherd had a flock of 57 sheep.

He wanted to increase the size of the flock because this year there was good fodder. In

order to do so the shepherd went to a market, where he decided to buy some more sheep

and he joined them to those he already had. One evening the shepherd saw a pack of

wolves in the area. The wolves were hungry and then, from the whole flock of sheep

they devoured 11 sheep and now there are 96 sheep left. How many sheep did the

shepherd buy in the market?

Fig. 2. Example of reworded problem used in the study. Underlined text indicates mathematical information added to Orrantia et al. (2011) situationally reworded problem.

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e1195 1189

- Data selection: contents devoted to selecting the data from theproblem

- Mathematical reasoning: contents related to a deep mathe-matical understanding of the problem, in terms of mathe-matical relations among the data involved.

- Mathematical resolution: contents related to the selection andexecution of mathematical algorithms.

Situational subcategories were based on the studies by Coquin-Viennot & Moreau, 2007, Orrantia et al. (2011), and Stern andLehrndorfer (1992), in which only the situational informationlinked to the mathematical model of the problem turned out toimprove students’ performance. Two situational subcategorieswere used (examples in Table 1):

- Relevant situational knowledge: contents related to the inten-tions and goals of the characters, and the actions performed toreach the goals to link the situation to the mathematical modelof the problem.

- Irrelevant situational knowledge: contents not related to thecausal chain generated by the character’s goals (e.g., descrip-tions of characters, places, objects).

Two independent coders analyzed a representative sample ofthe transcriptions. Interjudge reliability was carried out for each ofthe analysis dimensions by using Cohen’s kappa statistic. Given thata Cohen’s kappa of 0.65 or higher was considered to be a goodmeasure of inter-rater reliability, and the average kappa across alldiscourse categories for the 11 sessions ranged from 0.78 to 0.86(see Table 1), the coder agreement was considered appropriate.

Once the problem-solving sessions were categorized, thenumber of interaction cycles was calculated and the public contentswere turned into percentages. The mean percentage for each kindof public content by teacher was calculated.

3.5. Measures

Three different measures were taken. First, to explore whetherteachers use a paradigmatic or narrative approach, we analyzed thedifferences between percentages of exchange cycles in whichmathematical and situational knowledge were made public.Second, to explore whether teachers promote a deep mathematicalprocessing of the problem, we compared the percentage of publiccontents related to mathematical subcategories (data selection,mathematical reasoning, and mathematical resolution). And third,to explore whether teachers promote a deep qualitative compre-hension of the situation described in the problem, we compared the

percentage of public contents related to both situational subcate-gories (relevant and irrelevant).

4. Results

Mean percentages were compared using paired sample t-testsfor the comparison of mathematical and situational knowledge. Aone-way repeated measures ANOVA was conducted for thecomparison of the mathematical subcategories.

Regarding the first measure of the study, the proportion ofcycles that evoked mathematical knowledge was quite a bit higherthan the percentage of cycles that evoked situational knowledge(0.86 vs. 0.14, respectively, Table 2).

Paired sample t-test confirmed that whereas mathematicalknowledge was highly evoked during the interaction (M ¼ 86,SD ¼ 11.15), situational knowledge was scarce (M ¼ 14, SD ¼ 11.15),t(10) ¼ 10.70, p < 0.001; d ¼ 6.80, so the teachers used mainlya paradigmatic approach. The example of Table 1 illustrates thispreference of teachers to focus the interaction on the mathematicalknowledge. In turn, the interaction in which more situationalknowledge was found was that of Teacher 10 (T10) who, as showedby Table 2, just reached 30% of situational contents of the wholeinteraction:

T10: The problem is about a nice story in springtime. It is aboutsheep and fodder, and there is a shepherd and mountains.

S: Is there a river too?

[.]

T10: What did the shepherd want? What?

S: To increase.

T10: To increase. Did he achieve it? Did he increase his flock?

S: Yes.

T10: Why? What did he do to increase his flock?

A: He bought sheep in a market.

[.]

S: And where was the shepherd?

T10: It seems that he was not here. Where was he?

S: He went to the market.

T10: He went to the market, to do what?

S: To buy more sheep.

Table 1Example of analysis taken from Teacher 5. Observer agreement for each category, measured by Cohen’s Kappa, is shown in brackets.

Cycle (0.86) Category (0.86) Transcription Content made public (0.78)

Teacher (T): Let’s start by reading the problem. Carlos?Student (S): “A shepherd was taking care of a flock of sheep. Theshepherd had a flock of 57 sheep. He wanted to increase the size ofthe flock because this year there was good fodder. In order to do sothe shepherd went to a market, he decided to buy some more sheepand he joined them to those he already had. One evening theshepherd saw a pack of wolves in the area. The wolves were hungryand then, from the whole flock of sheep they devoured 11 sheepand now there are 96 sheep left. Howmany sheep did the shepherdbuy in the market?”

1 Data selection T: Let’s see. Carlos, how many sheep did the shepherd have? The shepherd had 57 sheepS: 57T: Ok, He had 57 sheep,

2 Relevant Situational Knowledge T: And he wanted to increase the size of the flock, so he went toa market.

Hewanted to increase the size of the flock,so he went to a market.

3 Data selection T: Do we know how many sheep the shepherd bought in themarket?

We do not know how many sheep hebought because that is the question.

S: NoT: Certainly, that’s the question. We don’t know how many.

4 Mathematical Reasoning T: But he had more sheep, hadn’t he? He had more sheepS: YesT: Right

5 Data selection T: The problem states: “One evening the shepherd saw a pack ofwolves in the area The wolves were hungry and then, from thewhole flock of sheep they devoured 11 sheep”. Is that another pieceof information?

Another piece of information is that thewolves devoured 11 sheep.

S: YesT: 11 sheep. Then, what do we have?S: The wolves devoured 11 sheep.T: They devoured 11 sheep.

6 Data selection T: And finally, the whole flock of sheep was. Finally, the whole flock of sheep was 96.S: 96 sheepT: 96 sheep, well done.

7 Data selection T: Then, the question is: How many sheep did he buy? The question is: How many sheep did hebuy?

8 Data selection What do we have to do? In order to solve the problem, the firstthing we should know is that there were96 sheep left in the whole flock.

S: (He’s thinking but he doesn’t answer)T: Let’s see. The first thing we have to know is how many sheepremain in the whole flock.96, ok?

9 Mathematical Reasoning T: If he has 96, what do we have to do? From the sheep the shepherd had at thebeginning, we have to take away 11 sheepwhich were devoured by the wolves

S: .Plus 11T: Plus 11? Noway. Plus 11 no.We know the sheep he had at thebeginning.but we don’t know the number of sheep he bought,so what do we have to do? From the sheep he had.S: We take away.T: We take away, how many?S: 11T: We take away those devoured by the wolves.S: Yes

Data selection T: Then, just do it. In the end, he has 96S: He had.57T: And now, how many sheep had he got in the end?S: 96

10 Mathematical Reasoning T: Then, if he has 96 and before he had. We have to take away 96 � 46S: 96 minus 46T: He had 46. Hence, that’s the solution

Table 2Mean percentage of cycle per content made public by each teacher.

Teacher Mathematical Situational

T1 95.83 4.17T2 87.63 12.37T3 81.55 18.45T4 96.67 3.33T5 91.67 8.33T6 92.74 7.26T7 73.81 26.19T8 85.56 14.44T9 68.75 31.25T10 68.71 31.29T11 100.00 0.00Mean 85.72 14.28

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e11951190

T10: Ok then, let’s see where the shepherd was. Where was he?He was in the market. Let’s make a drawing about him. He wasbuying in a market, OK?(the student draw a shepherd in the market)

[.]

T10: Ok, what can I see in this drawing? I can see that after goingto the market the shepherd has managed to increase his flock.

Regarding the second measure, namely, the comparison withinmathematical subcategories, a one-way repeated measures ANOVAtest showed differences between categories [F(2, 20) ¼ 49.54,p < 0.001, h2 ¼ 0.92 (Table 3). Pairwise comparisons using theBonferroni adjustment with p < 0.05 indicated that the teacherswere more focused on data selection (M ¼ 50.45, SD ¼ 14.86) than

Table 3Mean percentage of public content sub-category.

Teacher Mathematical Situational

Data Conceptual Resolution Relevant (%) Irrelevant (%)

T1 68.63 12.99 18.38 100.00 0.00T2 57.14 35.71 7.14 100.00 0.00T3 69.44 15.00 15.56 100.00 0.00T4 57.78 25.00 17.22 100.00 0.00T5 47.73 0.00 52.27 100.00 0.00T6 50.35 3.85 45.80 37.50 62.50T7 52.96 2.63 44.41 79.17 20.83T8 24.55 0.00 75.45 75.00 25.00T9 31.19 6.90 61.90 58.33 41.67T10 60.40 12.27 27.33 48.21 51.79T11 34.74 9.01 56.25 0.00 0.00Mean 50.45 11.21 38.34 100.00 0.00

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on mathematical reasoning (M ¼ 11.27, SD ¼ 11.02), (p < 0.001,d ¼ 3.12). Similarly, the teachers evoked less mathematicalreasoning than mathematical resolution (M ¼ 38.18, SD ¼ 22.29),although this difference only approached statistical significance(p < 0.06, d ¼ 1.16). However, no significant differences were foundbetween data selection and mathematical resolution subcategories(p > 0.85).

This result indicates that most of the interaction was devoted toexplicit public contents related to data selection, and to selectingthe operation and executing it, behaving in a superficial way. One ofthe clearest examples of this was Teacher 4 (T4). What is note-worthy about this interaction is that almost all the public contentswere devoted to data selection for choosing the suitable operationand its execution. Thus, only one public content was related to themathematical structure of the problem “[107 sheep] included thosehe had plus those he bought”, and it was not used to delve deeperinto the justification for the subtraction 107 � 57, but to justify thepreviously executed algorithm 96 þ 11 and to select the secondoperation without any further mathematical explanation:

T4: What did you understand from the problem?

S: Well, let’s see, he had 57 sheep.

T4: Yes.

S: and he wanted to increase them, he went to a market.

T4: And he bought?

S: And he bought some sheep. And then the wolves came andthey devoured 11.

T4: They devoured 11, yes.

S: And there were 96 sheep left in the flock. Then, by adding 11plus 96 you have 107, then you take away.

T4: Let’s do that on the blackboard. Let’s see, if you add the 96sheep he had to the 11 devoured by the wolves. right? Andhow many are 96 plus 11?

S: 107.

T4: 107 sheep, right?

S: Yes.

T4: These would include. Jose, which would be included?

S: Those he had, plus those he bought.

T4: All of them, those he had plus those he bought, very well. Solet’s see, what do we have to do now? Maria?

S: Mmm. To add 11 to. This was..

T4: Well, but following the problem as Jose was doing it, whatwould you do?

S: You have to add 96 plus 11, which are those that. and takeaway. take away 57.

T4: Sure, we take away 57; those are the sheep he had before,from 107, right? And then I will know how many sheep he has.Then the question says “Howmany sheep did the shepherd buyin the market?” The answer is “50 sheep”.

Regarding the third measure, that is, the comparison betweensituational subcategories, results showed that the teachers devoteda significantly large percentage of the cycles to relevant rather thanto irrelevant information (72% and 18%, respectively, see Table 3).Paired sample t-test indicated that the amount of relevant situa-tional contents evoked (M¼ 72.55, SD¼ 33.29) was higher than theirrelevant situational contents (M ¼ 18.48, SD ¼ 23.93, t(10) ¼ 3.61,p < 0.006, d ¼ 1.96). This means that the processing of the smallamount of situational information made public by the teachers wascarried out to promote deep situational comprehension. In otherwords, interactions such as this one from Teacher 7 (T7) andTeacher 9 (T9):

T7: . the problem states that the shepherd wanted to increasethe size of his flock. What does that mean? It means that hewanted to havemore sheep because there was very good fodder.

T9: The shepherd had a flock, and what happened to this flock?

S: Thenhewent to amarket to buy. to buy. to buymore sheep.

T9: And why did he go to buy more sheep?

S: Because he wanted to increase.

T9: He wanted to increase his flock, he wanted his flock to bebigger. were more frequently evoked than interactions such asthis one from Teacher 8 (T8) and Teacher 9 (T9) :

T8: The shepherd was taking care of the sheep in the mountain.OK? He was in the mountain and he was taking care of a flock ofsheep

T9: OK, come on, Mary, please explain: what is the problemabout?

S: It is about a shepherd that had four sheep.

T9: And where did he take care about the flock?

S: In a flock.

5. Discussion

The present study explores how 11 mainstream primaryeducation teachers solve non-standard, situationally and mathe-matically reworded problems jointly with their pupils.

Given the added mathematical and situational information,teachers were expected to promote genuine processing by usingsuch information and go beyond a superficial processing. Accordingto Verschaffel et al. (2000), the genuine processing means that twodifferent mental models should be generated: mathematical andsituational. To generate the mathematical model attention can befocused on superficial aspects of the problems (data selection,search for a key word to select the algorithm) or on deeper onesrelated to mathematical reasoning. The situational model isdevoted to connecting word problems to situations in which somehints can be used to solve the problem more successfully.

Depending on the amount of attention that teachers pay to themathematical or situational dimension, Chapman (2006) labeled

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teacher’s approaches to problem solving as paradigmatic ornarrative, respectively. Depaepe et al.’s (2010) study showed howeven expert teachers were more paradigmatic than narrative whenthey solved word problems with their pupils. The authors sug-gested that one of the reasons could be the standard nature of theproblems that were used. In this sense, the present study revealswhether teachers’ paradigmatic behavior might be due to thestandard nature of the problem or not, and also checks how deepthe mathematical and situational processing of mainstreamteachers would be when they solved non-standard, rewordedproblems. We designed our study to answer three main questions:Will teachers use a paradigmatic or narrative approach? Will theypromote a deepmathematical processing of the problem?Will theypromote a deep qualitative comprehension of the situationdescribed by the problem?

Regarding our first question, we found that the interaction wasbasically related to mathematical knowledge; that is, followingChapman’s categorization, their approach to problem solving wasmainly paradigmatic, even though our experimental problems werenon-standard. In fact, some teachers not only considered it unneces-sary to recreate in depth the situation model evoked by the problem(related to the reasons associated with characters’ goals and inten-tions) but also they twisted the situationalmodel of the problem to fitit to its mathematical model. Teacher 3 (T3) is a clear example of this:

T3: Did anyone else solve this problem in a different way? Let’ssee, Mary, how did you solve it?

S: I took away 11 from 57

T3: So, you think the wolves ate 11 sheep from the 57 sheep theshepherd had, don’t you? 57, isn’t it? And he has 46 now, hasn’the?

S: Yes, and then I took away 46 from 96

T3: From 46.and you have.

S: 50

T3: 50. Well done Mary.

Although the mathematical procedure to solve the problem iscorrect [i.e., 96 þ 11 � 57 ¼ 96 � (57 � 11)], it makes no sense froma situational point of view because it erroneously assumes that thewolves appeared in the first moment of the story and devoured 11sheep from the initial flock of 57 (whereas the wolves appearedafter the shepherd went to the market, and the decrease in theinitial size of the flock is not in accordance with the intentionalstructure of the problem). An additional commentary by theteacher should be necessary to underscore this situational incon-sistency, even if the final decision is to neglect it in favor of themathematical structure of the problem. Moreover, even thoughsome of the teachers detected that something was wrong with thecorrect (but situationally incorrect) mathematical procedure96 � (57 � 11), they tried to justify this situational inconsistency ina mathematical way. Teacher 8 (T8) is an example of this:

S: I added 57 to.I took away 11 from 57 and then I have 46 and Isubtracted 46 from 96 and then I have 50.

T8: Well, that’s a good algorithm, but that’s not the way youshould do it. It’s well done, but we have to follow a specific path.The problem usually shows us the path; it shows us the fullpath: at the beginning he had 57 sheep; we have the first pieceof information. In the end we have 96 sheep. Then we have toadd 11 more sheep that the wolf ate to those 96 sheep.that’sanother piece of information. And once the shepherd came backfrom the market he had 107 sheep. The question is how many

sheep did he buy at the market? So we have to subtract the 57sheep that he had at the beginning from the 107 sheep. Andthen, as Juan said, we have 50 sheep. He bought 50 sheep at themarket.

In this regard, the teachers in our study seemed to act like theteachers in the study by Verschaffel et al. (1997) by acceptinga mathematically correct (but situationally incorrect) problem-solving procedure.

Furthermore, given that we used the same reworded problemsas Orrantia et al. (2011) with the same level of difficulty (and theywere easier for students to solve), teachers might be preventing thestudents from using the situational rewording as a support in theproblem-solving process, sometimes by confusing the studentswith the way they represent the situational information, some-times by neglecting the situational information during theproblem-solving process.

Regarding the second question, namely, how superficial or deepthe mathematical processing was, we found a higher percentage ofpublic contents devoted to data selection and mathematical reso-lution than to mathematical reasoning. This result fits perfectlywith the superficialmodeling described by Verschaffel et al. (2000).In our sample, the problem-solving process included little mathe-matical reasoning, even though the experimental problemsincluded explicit information on the mathematical structure of theproblems that can be used to go deeper into mathematicalreasoning. The clearest example from our study was found inTeacher 1 (T1)’s interaction:

T1: Let’s see, the shepherd had 57 sheep, write “57 sheep”. Hewanted to increase his flock, because he hadmore fodder and hebought a number of sheep that we do not know, right?What weknow is that the wolf ate 11 of the sheep that he had. Then wetake away from the 57 sheep he had the 11 sheep the wolf ate,right? And at the end therewere 96 sheep left. Howmany sheepdid he buy? Ok, then from 57 we take away 11 and there are 46sheep left, right?

S: Ok

T1: Since at the end he had 96, what do we have to do?

S: We have to take away 46 from 96

T1: Ok, then he had 96 sheep at the end, we take away thosesheep that were left after the wolf ate them and these are whathe bought, Ok?

S: Yes

Despite choosing the most difficult way to solve the problem,96 � (57 � 11), Teacher 6 did not make explicit any mathematicalreasoning to solve the problem (the easy procedure would be96 þ 11 � 57, see Vergnaud, 1981, for a detailed explanation). Theteacher selected the data and jumped directly from the data to thefirst operation. She simply allowed the students to participate inthe interaction (by giving them the chance to select the secondoperation) after she had explained most of the problem. Althoughthere is no doubt that recalling data is necessary for problemsolving, mathematical reasoning is also necessary since it promotesa more reflexive context. In fact, we must note that, as we alreadystated for situational rewording, the mathematical rewording ofour experimental problems was identical to the mathematicalrewording by Vicente et al. (2007). Considering that mathemati-cally reworded problems were easier to solve in Vicente et al.’sstudy, it might be possible to suggest that Teacher 6 implicitly (andunintentionally) prevented students from using the mathematicalrewording to solve the problems.

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e1195 1193

Finally, regarding the third question, namely, whether teacherswill promote a deep qualitative comprehension of the situationdescribed by the problem, the results of the analysis of the situa-tional content made public suggest that teachers evoked morerelevant than irrelevant contents. They devoted most of these(rather scarce) cycles and public contents to generating anadequate situational model. This result might indicate that theteachers who decided to devote some parts of the interaction to thesituational aspect of the problem understood (at least implicitly)that situational information added to problems is important insolving the problem. Hence, they devoted most of these exchangecycles to understanding relevant information about the situationinstead of focusing on irrelevant details. However, some of theteachers could not resist the temptation to process all the infor-mation contained in the problem and they also made public theirrelevant situational knowledge. In any case, as we stated above,this behavior was not the most frequent.

A major outcome of the present study is that even though ourmainstream teachers were solving non-standard problems thatincluded explicit information on relevant situational aspects of theproblem, theymainly used a paradigmatic and superficial approachduring the task. In addition, this approach probably did not guar-antee a deep mathematical understanding of the problem becausethe teachers spent many more exchange cycles on data selectionand algorithm execution than on mathematical reasoning.

Therefore, it can be said that our mainstream teacher could befostering a paradigmatic and superficial approach to problemsolving, and this trend cannot be attributed to the standard natureof the word problems but to the way in which teachers solveproblems in their classes. In this sense, we wonder why teachersbehaved in this way and did not take any advantage of the addi-tional information provided in the problems of the study. Consid-ering that previous studies have documented that students are ableto take advantage of such information, this question is even harderto solve. Perhaps the answer to the question is related to what Ball,Hill, and Bass (2005) called “mathematical knowledge for teaching”(based on Shulman’s (1986) previous “pedagogical contentknowledge”). This knowledge is “a kind of professional knowledgeof mathematics different from that demanded by other mathe-matically intensive occupations, such us engineering, physics,accounting or carpentry” (p. 17). In particular, this knowledgewould include two different kinds of knowledge: first, “commonknowledge” of mathematics about the topics and procedures to betaught (not only word problem solving, but also factoring, primes,equivalent functions and so on); and second, “specialized” knowl-edge about the work of teaching that includes mathematical,psychological and pedagogical knowledge, and that only teachersneed know. While “common knowledge” is shared by any well-educated adult, specialized” knowledge is quite specific and mustbe developed specifically in the training of student teachers or byteacher education programs. It should include knowledge about theprocess of word problem solving and the way in which studentsconfront the task. This “specialized” knowledge should allowteachers to detect and understand students’ errors as indications ofan incomplete comprehension because some mathematicalprocedures that are automatic for adults are not obvious to studentsat all. In the case of word problem solving, this “specialized”knowledge could be related to the understanding of the needs thatsome students have during the task (especially when solvingdifficult problems). In this context, teachers should be able tosupport situational or mathematical reasoning and many of thestudents’ partially wrong answers that are on track to the rightanswer, but that are not completely acceptable (see examplesabove). In this way, when students have the opportunity to use theadditional information of the problems on their own, they can take

advantage of it. But when teachers solve problems with theirstudents they are most likely constrained by their lack of“specialized” knowledge about students’ need to use the mathe-matical and situational additional information.

6. Conclusions

Our study is an attempt to contribute to our knowledge aboutwhat is currently being done in mathematics classrooms. It is aneffective way to document the variety of complex aspects of wordproblem solving in the classroom. Then, this study deal with theimportance of considering situational aspects when Primary schoolteachers are solving word problems with their students in theclassroom that can help teachers of all over the world to learn moreabout their teaching. In this regard, we think the current studycould possibly add four contributions to the extant knowledge-base.

The first contribution of our work is the extension of previousresearch (e.g., Chapman, 2006; Depaepe et al., 2010) involvingcharacterizations of teachers’ behavior when solving word prob-lems in their classroom. Our study suggests that teachers might bemainly paradigmatic and superficial not only when solving stan-dard problems, as in Depaepe’s study, but also when solving non-standard problems with relevant additional information. Takingthis into account, it seems clear that not only our teachers (but alsothe teachers in Depaepe et al.’s study) should place more emphasison situational aspects when working on solving word problemswith their pupils, and that probably this is true for teachers fromvery different countries around the world.

The second contribution of this study is the international rele-vance that our findings could have. Different international studieson teaching mathematics, for example, the TIMSS video studies(Hiebert et al., 2003; Stigler & Hiebert, 1997) recorded, analyzedand compared mathematics classrooms of several countries andfound that the countries whose students showed a lower mathe-matical ability (like the U.S. or Germany) shared interactionpatterns in mathematics classrooms that led students to focus onthe mechanical aspects of mathematics instead of reasoning.Considering that Spanish students showed similar levels of math-ematical competence as students not only in the United States (butalso in Hungary, Ireland, Portugal, Italy and Latvia, see OECD, 2010),it might be suggested that teachers in these countries behave asSpanish teachers do. Further research should be done to addresswhether this situation holds in other cultural contexts or in coun-tries where students showed a high level of mathematicalcompetence, for example, in Japan, Korea, Finland, Liechtensteinand Switzerland (see OECD, 2010).

According to the results reported in the present study, the thirdcontribution would be to suggest that international community ofteachereeducators, taking the similarities and differences betweenthe Spanish programme and that of the other countries intoaccount, should develop a better understanding of the relevanceof mathematical and situational processing in teaching how tosolve word problems in a genuine fashion. Therefore, suchunderstanding should be internationally considered an importantgoal to be reached not only in training programmes of pre-serviceteachers, but also throughout the lifelong professional educationof in-service teachers. As we pointed out in the previous section,teachers need to develop “specialized” knowledge about how toteach mathematics. This means that some specific psychologicalknowledge about how students understand the mathematical andsituational aspects of word problems and how problem texts can beenriched to facilitate such understanding should be included inteacher training programmes. Furthermore, the same can be saidabout mathematical areas other than word problem solving, such

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as number properties and operations, geometry or algebra.Teachers need to know how children learn these mathematicalcontents in order to help their students to learn more easily.However, two additional steps should be taken. The first one is tolearn more about teachers’ beliefs in relation to the need toconsider this “specialized” knowledge and how difficult it would beto change the deep-seated beliefs underlying behaviors such asthose found in this study into more meaning-oriented ones. Thesecond step is to analyze and, if necessary, to modify mathematicstextbooks to include genuine models of word problem solving thatteachers could use as a support in teachingword problem solving ina meaningful way. In addition, textbooks should include chal-lenging word problems to which this genuine problem-solvingprocess could be applied.

The fourth contribution is that themethodology used in our studymight alsobeuseful for researchers of teacherestudent interaction inmathematics classrooms. Our measures can be considered ascomplementary to the ones used by Chapman (2006) and Depaepeet al. (2010). Our work and Chapman’s and Depaepe et al.’s studiesattempt to clarify the direction toward which teachers directstudent’s attention during word problem solving. However, somedifferences between our work and Chapman’s and Depaepe et al.’sstudies can be outlined. Whereas previous studies have analyzedwhat the teacher did (i.e.: interventions), we have analyzed in depththe meanings they constructed (i.e.: public contents). And we did soby using a clear categorization of public contents that can be directlyapplied for analyzing the interaction of teachers with their pupils.This could be helpful for international teachereeducators and couldbe used in future research for analyzing educational practices.

In addition, our analysis procedure could be considered morereliable than others for certain purposes because the publiccontents promoted by teachers through their questions, or theinformation provided directly by them, are completely under thecontrol of the teachers. Other methodologies, such as those basedon measuring the time spent on each activity, are more affected byelements not completely under the control of teachers, such as thesuitability of students’ answers to the question made, which couldmake it necessary to ask complementary questions, or the timestudents take to give an answer. For example, one teacher maymake one content public in a short time (one question and onecorrect answer by the student) while another teacher may spenda bit more time to make the same content public as a result of, forexample, a sequence of questions and wrong or incompleteanswers, or even silences on the part of the students. In this case,spending more time would not result in a deeper understandingbecause only one content is made public.

Nevertheless, although this study can be considered anotherstep toward understanding more about what is happening whenPrimary teachers are working on word problem solving with theirpupils, this is only the first step in changing instructional practicesto adapt what is already being done in mathematics classrooms towhat should be done. We also must take into account that ourresults have to be interpreted carefully because of their general-ization. Although our sample of teachers was bigger than Depaepeet al.’s, the problems solved were limited, and therefore weconsider that these results need to be generalized to a broaderrange of problems and classroom interactions before strong theo-retical and educational implications can be drawn.

Finally, the findings of the present study raise certain otherquestions that could be addressed by further research. Firstly, itwould be interesting to explore whether the results obtained maybe affected by variables such as academic grades or the topics beingstudied.

Secondly, further research might evaluate to what extent in-service teachers may change their paradigmatic approach after

receiving explicit instruction in genuine mathematical and situa-tional modeling, and how difficult changing their behavior topromote a deeper mathematical and situational reasoning may be.

Thirdly, it would be necessary to compare how word problemsare solved in the classrooms of teachers who have undergoneprofessional development in reform-based mathematics instruc-tion and how these problems are solved in classrooms of teacherswho have not. In this sense it would be quite interesting to askreform-based teachers to think aloud as they are scaffoldingstudents’ word problem solving. And then, if the reform-basedteachers have more elaborated situational elements included intheir problem representations, it would be fruitful to constructembellished problems from these teachers’ utterances.

Fourthly, itwouldbenecessary to analyzewhetheradifferent kindof rewording, one that more powerfully drew teachers’ attention tosituational aspects and mathematical relationships in the problems,would be more effective in changing teachers’ approach to problemsolving (in fact, thismight bea limitationof the study). In this sense, inorder to know to what extent our results could be due to the kind ofword problemwe used, further research incorporating other kinds ofnon-standard problems would be necessary. For example, realisticmathematical problems could beused as experimental items becausedue to situational limitations a mathematically correct solution maynot be the correct answer to the problem (e.g., John’s best time forrunning 100 m is 17 s. How long will it take him to run 1 km?).Furthermore, we also wonder what approach (paradigmatic ornarrative) teachers would use in solving these realistic problems orproblems embedded in an authentic context (Palm, 2002).

Fifthly, further studies including measures related to whatteachers think of the role of relevant mathematical and situationaladded information in reworded problems would be necessary todisentangle whether in-service teachers’ beliefs about non-standard problems may underlie the paradigmatic and superficialmode they used in our study. A study in this vein could followVerschaffel et al’s. (1997) study on pre-service teachers’ beliefsabout realistic problems and how they should be solved by pupils.Their results were clear: the teachers showed a strong tendency toavoid real-world knowledge and realistic considerations whenrating student’s answers. We wonder what in-service teacher’sbeliefs would be if they were asked to explain how to solvereworded problems with their pupils.

And finally, considering that most teachers follow mathematicstextbooks when teaching students to solve word problems, wethink that further research into how textbooks teach how to solveword problems would be necessary, particularly studies to explorewhether the way in which textbooks explain how to solve wordproblems may influence the superficial and paradigmatic approachshown by the teachers in our study. Some studies have documentedhow standard the problems solved by teachers are (Depaepe et al.,2010), and there is some evidence that allows us to identifydifferences between Japanese and U.S. mathematical textbooks(regarding the proportion of space devoted to explain the solutionprocedure for worked-out examples, to unsolved exercises, etc.)and how meaningful the instructional methods are (regarding theuse of representations or the organization of material, Mayer, Sims,& Tajika, 1995). Nevertheless, research is still scarce in most coun-tries as regards attempts to find a link between how textbooks dealwith solving word problems and educational practice in mathe-matics (for example, to what extent and how strictly teachersfollow textbooks in teaching students how to solve word prob-lems). In short, this study could open future lines of research indifferent senses because a lot of children, in almost all countries inthe world, work in word problem solving every day. That is thereason we considered the findings relevant to the internationalcommunity in Mathematics Education.

J. Rosales et al. / Teaching and Teacher Education 28 (2012) 1185e1195 1195

Acknowledgments

This research was supported by Grant EDU2009-13610 to J.Rosales and by Grant PSI2011-27737 to J. Orrantia from the SpanishMinisterio de Educación y Ciencia (Ministry of Education andScience).

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