10.6.verschaffel

Laming

Pergamon

nnd Inrrrucrion. Vol. 7, No. 4, pp. 339-359. 1991 0 1991 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959.4752197 $17.00 + 0.00

PII:SO959-4752(97)OOOOS-X

PRE-SERVICE TEACHERS’ CONCEPTIONS AND BELIEFS ABOUT THE ROLE OF REAL-WORLD KNOWLEDGE IN MATHEMATICAL MODELLING OF SCHOOL WORD PROBLEMS L. VERSCHAFFELI, Center for Instructional

E. DE CORTE

and I. BORGHART

Psychology and Technology, B-3000 Leuven, Belgium

University

of Leuven,

Abstract Recent research has documented that many pupils show a strong tendency to exclude real-world knowledge from their solutions of school arithmetic word problems. In the present study, a test consisting of 14 word problems-half of which were problematic from a realistic point of viewwas administered to a large group of students from three different teacher training institutes in Flanders. For each word problem, the student-teachers were first asked to solve the problem themselves, and afterwards to evaluate four different answers given by pupils. The results revealed a strong tendency among student-teachers to exclude real-world knowledge from their own spontaneous solutions of school word problems as well as from their appreciations of the pupils’ answers. 0 1997 Elsevier Science Ltd

Theoretical

Background

For several years it has been argued that considerable experience with traditional school arithmetic word problems develops in pupils a strong tendency to exclude real-world knowledge and realistic considerations from the different stages of their solution processes, i.e., the initial understanding of the problem, the construction of a mathematical model, the actual computational activities, and the interpretation and eva1uatio.n of the outcome of these computations. Rather than functioning as realistic contexts that invite or even force pupils to use their commonsense knowledge and experience about the real world, school arithmetic word problems have become artificial, puzzle-like ‘tasks that are perceived as being separate from the real world.

Thus, pupils learn that relying on commonsense knowledge and making realistic considerations about the problem context-as one typically does in real-life problem situations encountered outside school-is harmful rather than helpful in arriving at the “correct” answer

‘L. Verschaffel

is a senior research

associate

of the National 339

Fund for Scientific

Research,

Belgium.

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of a typical school word problem (Davis, 1989; De Corte & Verschaffel. 1985; Freudenthal.

1991; Kilpatrick. 1987; Nesher, 1980; Reusser, 1988; Saljo, 1991; Schoenfeld, 1991; Silver.

Shapiro, & Deutsch. 1993; Treffers & de Moor, 1990; Verschaffel & De Corte, in press).

Recent studies by Greer (1993) Reusser (1995) and Verschaffel, De Corte, and Lasure (1994) have yielded strong empirical evidence for this argument. In these three studies, a paper-and-pencil test consisting of two kinds of word problems was collectively administered to a large group of 1l-l 3-year-old pupils: l

l

Standard problems (S-problem) which can be properly modelled and solved by the straightforward application of one or more arithmetic operations with the given numbers (e.g.. “Steve has bought 4 ropes of 2.5 metres each. How many ropes of 0.5 metre can he cut out of these 4 ropes?“).

Problems in which the mathematical modelling assumptions are problematic (P-problem), at least if one seriously takes into account the realities of the context called up by the problem statement (e.g., “Steve has bought 4 planks of 2.5 metres each. How many planks of 1 metre can he saw out of these 4 planks?“).

The analyses of the pupils’ reactions to the P-items yielded an alarmingly small number of realistic responses or comments based on realistic considerations (e.g., responding to the above-mentioned “problematic” problem with “8 planks” instead of “10 planks”, because in real life one can only saw 2 planks of 1 metre out of a plank of 2.5 metres). For example, in the study by Verschaffel et al. (1994). only 17% of all the reactions of a group of 75 fifthgrade pupils to the ten P-items involved in the test, could be categorized as realistic. Greer (1993) and Reusser (1995) reported similar alarming results.

While the findings of these investigations convincingly demonstrate the strong tendency among upper elementary and lower secondary school pupils to exclude real-world knowledge from their understanding and solution of school arithmetic word problems, they do not yield insight into the aspects of current instructional practice that are responsible for the development of this tendency. According to the above-mentioned researchers, this tendency is mainly caused and strengthened by the following two aspects of the instructional environment: (1) The impoverished and stereotyped diet of standard word problems, which can always (at least ostensibly) be modelled and solved through the straightforward use of one or more arithmetic operations with the given numbers.

(2) The way in which these problems are considered and used in current instructional practice and culture, and more specifically the lack of systematic attention to the modelling perspective by the teacher (see also: Davis, 1989; De Corte & Verschaffel, 1985; Freudenthal, 1991; Kilpatrick, 1987; Nesher, 1980; Reusser, 1988; Saljo, 1991; Schoenfeld, 199 1; Treffers & de Moor, 1990; Verschaffel & De Corte, in press).

However, so far there is hardly any empirical evidence supporting these claims.

In the present study we focused on a major component of the instructional environment, namely the teacher. More specifically, we analyzed (future) teachers’ own conceptions and beliefs about the role of real-world knowledge concerning the problem context in the interpretation and the solution of school arithmetic word problems as reflected by (1) their own spontaneous responses to a set of word problems with problematic modelling assumptions, and (2) their evaluations of pupils’ answers that do or do not take into account relevant real-world knowledge.

TEACHERS’

CONCEPTIONS

341

AND BELIEFS

Method Subjects,

Participants were 332 pre-service teachers from three different institutes for elementary school teacher training (~97, n=132 and ~~103) in Flanders. About two-thirds (~228) were preservice teachers who had just started their first year of training, while one third (~104) were third-year students who had almost completed their pre-service training. The age of the firstyear students was between 18 and 19, while the third-year students were 20 or 21 years old.

Materials

A paper-and-pencil test was constructed consisting of 14 word problems: seven “problematic” items (P-items) in which the underlying mathematical modelling assumptions are problematic from a realistic point of view, and seven non-problematic standard items (S-items) requiring the straightforward and unproblematical application of one or more arithmetic operations with the given numbers. The latter tasks functioned as buffer items.

Table 1 lists the seven P-items from this test.

Test I

The test was given twice to all pre-service teachers, but each time with a different task.

The first time (=Test 1), the student-teachers had to answer the 14 items themselves. However, Table I The Seven P-Items Involved

in the Study*

Name

Problem

Buses

450 soldiers must be bused to their training buses are needed? (Carpenter et al., 1983)

School

Bruce and Alice go to the same school. Bruce lives at a distance of I7 kilometres from the school and Alice at 8 kilometres. How far do Bruce and Alice live from each other? (Treffers & de Moor, 1990)

Runner

John’s best time to run 100 metres is 17 seconds. How long will it take to travel I kilometre?

1993)

Flask

This flask is being filled from a tap at a constant rate. If the depth of the water is 3.5 ems after IO seconds, how deep will it be after 30 seconds? (This problem was accompanied by a picture of a partly-filled cone-shaped flask) (Greer, 1993)

Rope

A man wants to have a rope long enough to stretch between two poles I2 metres apart. but he has only pieces of rope 1.5 metres long. How many of these pieces would he need to tie together to stretch between the poles? (Greer, 1993)

Planks

Steve has bought planks? (Kaelen,

Friends

Carl has 5 friends and Georges has 6 friends. Carl and Georges decide to give a party together They invite all their friends. All friends are present. How many friends are there at the party?

(Nelissen, 1987)

site. Each army bus can hold 36 soldiers. How many

4 planks of 2.5 metre each. How many planks 1992)

(Greer,

of I metre can he get out of these

*The problems were presented in Dutch. After each P problem, we have mentioned its source. However, in most of the cases the original problem was slightly reformulated and the original numbers were replaced by others.

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to prevent lack of motivation and/or suspicion about the real purpose of the study, we did not merely ask them to solve the word problems, but also to predict the most common error among a representative group of fifth-graders.

More specifically. the instruction printed on the top of the first page of Test 1 was as follows: This test consists of 14 arithmetic word problems. This test was recently administered to a representative group of fifth-graders from several elementary schools in Flanders. You are invited to predict the most frequent erroneous answer for each problem and to write it down in the box called “incorrect answer”. But before doing that, please write down the correct answer in the box called “correct answer”. Calculations and possible comments can be written down in the “comments box”.

This instruction was followed by an example of an S-item for which the correct and the incorrect answer box as well as the comments box were completed in an exemplary way.

Figure 1 shows how the word problems were presented in Test 1.

Test 2 In Test 2 the student-teachers were asked to score four different hand-written answers from pupils to the same 14 word problems as in Test 1 with either 1 point, l/2 point, or 0 points. More specifically, the following written instruction was given at the top of the first page of Test 2: This test contains the same I4 problems as Test I. But this time each problem is followed by four different answers that were observed during the above-mentioned study with the fifth-graders.

Please score each of these four answers with either I, II? or 0 points. Write your score in the small box in the bottom-right corner of the response alternative. When you think that an answer is (absolutely) correct, you give it a score of I point; with a score of 0, you indicate that this answer is (completely) incorrect; a score of l/2 point can be used in cases wherein you consider the pupil’s solution as partly correct and partly incorrect. For each

This flask is being filled from a tap at a constant rate.

If the depth of the water is 3.5 ems after 10 seconds, how deep will it be after 30 seconds?

Correct answer:

Expected incorrect answer:

Comments: Figure

I Presentation

of the problems

in Test

1.

TEACHERS’

CONCEPTIONS

343

AND BELIEFS

problem, all three possible scores can be used several times, and some scores can be left unused. At the bottom of each problem, there is a box for writing explanations or comments. If you want to comment on a particular response alternative, please specify this by mentioning the corresponding letter (A, B, C or D).

As for Test 1, this instruction was followed by an example of an S-item with four fictitious hand-written solutions from pupils, which were scored and commented upon in an exemplary way. Figure 2 shows how the problems were presented in Test 2.

The four response alternatives to the seven P-items in Test 2 belonged to four different categories: l

l

Non-realistic answer (NA), which results from the straightforward and uncritical application of the arithmetic operation elicited by the problem statement (e.g., for the abovementioned planks-problem, the NA was IO-the product of 4 times 2.5).

Realistic answer (RA), which follows from the effective and appropriate use of realworld knowledge about the context elicited by the problem statement (the RA for the planks-problem was S-the product of 4 times 2).

This flask is being filled from a tap at a constant rate.

If the depth of the water is 3.5 ems after 10 seconds, how deep will it be after 30 seconds?

3 x 3.5 cm. = 11.5 cm.

After 30 seconds the depth of the water will be 11.5 cm.

3 x 3.5 cm. = 10.5 cm.

After 30 seconds the depth of the water will be 10.5 cm.

A

B

3.5 cm. + 20 cm. = 23.5 cm.

After 30 seconds the depth of the water will be 23.5 cm.

It is impossible to give a precise answer.

C Figure 2. Presentation

D of the problems

in Test 2.

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Technical error (TE), which results from the straightforward and uncritical application of the arithmetic operation elicited by the problem statement, but which differs from the NA because of a purely technical mistake in the execution of the arithmetic operation(s) (e.g., responding to the planks-problem with 100 planks instead of 10 because of disregarding the decimal number in the multiplication 4x2.5).

Other answer (OA), involving an answer that could not be classified into one of the former categories; for instance, solving the planks-problem with the result of a wrong operation, such as addition instead of multiplication with the two given numbers.

l

l

Table 2 gives the non-realistic answer (NA) and the realistic answer (RA) for each of the seven P-items from Test 2.

By contrast with the seven P-items, the seven S-items could not, of course, have a realistic and a non-realistic answer. The four response alternatives to the S-items belonged to the following categories: (1) the correct answer, (2) a technical error, (3) a wrong-operation error, and (4) a response that stated that the problem was unsolvable.

Procedure

Test 1 and 2 were administered on the same day. All student-teachers received Test 1 first.

Immediately after they had finished and handed in this test, they were given the second one.

To prevent order effects between the different P-problems from each test, two different versions of Test 1 and of Test 2 with inverted sequences of presentation of the P-items were used in a counterbalanced design. The administration of the tests was always done by the teacher educator who was responsible for the mathematics education program. Oral instructions were kept to a minimum. Once the testing had started, the student-teachers were not

The Non-Realistic

Table 2 Answer (NA) and the Realistic Answer (RA) for the Seven P-Items in Test 2

Buses

NA: 450 divided by 36 is 12.5. So 12.5 buses are needed.

RA: 450 divided by 36 is 12.5. So I3 buses are needed.

School

NAI *: 17-8=9. Saskia and Bruno live at 9 kilometres from each other.

NA2: 17+8=25. Saskia and Bruno live at 25 kilometres from each other.

RA: You cannot know how far Saskia and Bruno live from each other.

Runner

NA: 17~10=170. John’s best time to run I kilometre is 170 seconds.

RA: It is impossible to answer precisely what John’s best time on 1 kilometre

will be.

Flask

NA: 3x3.5=10.5. .After 30 seconds, the level of the water will be 10.5 cm RA: It is impossible to give a precise answer.

Rope

NA: 12-I .5=8. Eight pieces of I.5 metres are needed.

RA: It is impossible to know how many pieces of rope you will need

Planks

NA: 4x2.5 metres =I0 metres. 10 metres -1 metre =lO. Steve can saw IO planks of I metre.

RA: Steve can saw 2 planks of 1 metre from one plank of 2.5 metres. 2x4=8. So, Steve can saw 8 planks.

Friends

NA: 6+5= I 1. There will be 1 I friends at the party.

RA: You cannot know how many friends there will be at the party.

*Contrary

to all other P-items, the school problem

was presented

with two non-realistic

response

alternatives.

TEACHERS’

CONCEPTIONS

AND

BELIEFS

345

allowed to ask questions aloud. If somebody raised his or her hand-for instance because (s)he was confused about one of the P-problems and did not know how to handle this confusion-the teacher educator went to that particular student and gave an individual response that was as non-directive as possible, such as: “Please, don’t explain your trouble to me now; if you have any trouble with this word problem, you can write it down in the comments box”.

Duta Handling and Analysis Test I Five different answer categories were used to categorize the student-teachers’ own responses to the seven P-items in Test 1. Four categories were similar to those of the response alternatives in Test 2 discussed earlier: (1) non-realistic answer (NA), which results from the straightforward and uncritical application of the arithmetic operation elicited by the problem statement, (2) realistic answer (RA), which follows from the effective use of relevant realworld knowledge, (3) technical error (TE), which differs slightly from the non-realistic answer because of a technical mistake or inaccuracy in the execution of the arithmetic operation(s), and (4) other answer (OA), involving all answers that could not be classified into one of the former categories, such as wrong-operation errors or given-number errors. A fifth category was added, namely “no answer” (NOA).

Besides the answers written down in the answer box, we carefully scrutinized the studentteachers’ writings in the comments box for any traces of hesitancy to perform the simple and straightforward mathematical operation due to the activation of real-world knowledge or realistic considerations. Examples of such traces are: criticizing the problem statement, supplementing the answer with a qualifying comment, etc. If such a trace was found, a “+” sign was added to the given answer code. If the comments box did not contain such a trace, the response code was followed by a “-” sign. In this respect, it is important to point out that a “+“code could coincide with any of the five answer categories mentioned above (RA, NA, TE, OA and NOA). For example, a student-teacher who solved the planks-problem with the non-realistic answer 10, but who noted in the comments box that to arrive at 10 planks of 1 metre “Steve would have to glue together the 4 remaining pieces of 0.5 metre two by two”, was scored as NA+, whereas someone who simply wrote down the formula “4x2.5=10” in the comments box was scored as NA-.

Starting from these data, a more global assessment of the student-teachers’ own spontaneous solutions to the P-items of Test 1 was made involving two major categories: realistic reaction (RR) versus non-realistic reaction (NR). The term “realistic reaction” (RR) refers to each case wherein a student-teacher gave either a realistic answer (RA+ or RA-) to the problem or an answer that was scored differently but that was accompanied by a realistic consideration in the comments box (NA+, TE+, OA+ or NOA+). A nonrealistic reaction (NR) refers to each case wherein evidence of activation of real-world knowledge could be found in neither the answer box nor the comments box (NA-, TE-, OA- or NOA-).

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Test 2

The analysis of the pre-service teachers’ reactions to the seven P-items in Test 2 focused on the score (1, l/2 or 0) given to the following two response alternatives: the realistic answer (RA) and the non-realistic answer (NA)2.

Hypotheses

and Questions

Test I

The first hypothesis with respect to Test 1 was that-due to their continuing experience with an impoverished diet of standard word problems and to the lack of systematic attention to the mathematical modelling perspective in their mathematics (education) lessonseven the student-teachers would demonstrate a strong tendency to exclude real-world knowledge when confronted with the problematic versions of the problems, and, consequently, would respond to these problems as if they are not at all problematic. Although we expected that the results of these 18 to 21 -year-old pre-service teachers would be better than those of the upper elementary school and lower secondary school pupils from the studies of Greer (1993) Reusser (1995) and Verschaffel et al. (1994) we still predicted an alarmingly low percentage of RRs on the seven P-items from Test 1.

Furthermore, it was hypothesized that the relative levels of difficulty of the different P-items observed in the studies with the upper elementary and lower secondary school pupils (Greer, 1993; Reusser, 1995; Verschaffel et al., 1994; Verschaffel, De Corte, & Lasure, in press), would be reflected in the student-teachers’ performances. More specifically, we predicted that-as in the above-mentioned studies with children-the buses-item would again elicit considerably more RRs than the other six P-items. In previous reports (Verschaffel et al., 1994, in press), the following two arguments have been proposed to explain this recurrent finding. First, in a number of current Flemish textbooks at least some minima1 attention is paid to the problem of making sense of the outcome of a division with a remainder in applications, while the realistic mathematical modelling difficulties involved in the other P-items are completely absent in these textbooks. Second, the likelihood of noticing the absurdity of the stereotyped, non-realistic answer is much greater for the buses-item than for the other P-items. Indeed, while the formulation of the typical non-realistic answer for the buses-item (namely “I 3.5 buses”) can still lead to the detection of the nonsensical nature of this response (e.g., “There are no half buses”), writing a response sentence for the other P-items (for instance, responding to the friends-item with “11 friends” or to the school-item with “25 kilometres”) does not yield an equally striking hint to detect the absurdity of the

‘Because one of the problems, namely the school-problem, involved two non-realistic answers instead of one (see Table 2) a specific rule was constructed for reducing the student-teachers’ scores given to both non-realistic response alternatives into a single I-, l/2- or O-score. According to this rule, a student-teacher’s evaluation of the NA to the school-problem was scored with a I if both non-realistic answers had received a l-score from the studentteacher; if the student-teacher had given at least one l/2-score, h&her evaluation of the NA became l/2; and a student-teacher’s evaluation of the NA was scored with a 0 if (s)he had scored both non-realistic responses with 0 points.

TEACHERS’

CONCEPTIONS

AND BELIEFS

341

answer from a realistic point of view. It was assumed that these arguments for the exceptionally high percentages of RRs for the buses-item among upper elementary school children would also hold for the student-teachers.

Finally, we wondered whether there would be a difference between the students who had almost completed their (third year of) training as elementary school teachers and the students who had just started their (first year of) training. On the one hand, one could argue that the third-year students would have a stronger disposition toward realistic mathematical modelling; for, during their theoretical and practical courses in mathematics education, these third-year student-teachers must inevitably have run into situations that invited or necessitated dealing with and reflecting upon the difficulties involved in realistic mathematical modelling. On the other hand, one could also conjecture that the quantity and/or the nature of these occasions were insufficient to have a significant positive impact on these student-teachers’ cognitions and beliefs about the issue of realistic mathematical modelling. One could even argue that, due to their three years of experience with a traditional approach toward elementary school mathematics in general and with school arithmetic word problems in particular, these third-year studentteachers might have developed a stronger disposition toward stereotyped, non-realistic modelling and interpreting of arithmetic word problems than the first-year students. Because of these conflicting hypotheses, no specific prediction was made about any difference in the percentage of RRs between the first-year and the third-year students.

Test 2’ First, we hypothesized that the student-teachers’ scores for the realistic (RA) and the nonrealistic answers (NA) to P-items from Test 2 would generally reflect their spontaneous solutions of the same problems during Test I. Consequently, we anticipated that the studentteachers would frequently consider the NA for the seven P-items from Test 2 as the correct answer and therefore score it with 1. while the RA would frequently be conceived as an inappropriate response and therefore be scored with a 0.

Second, we expected differences in the scores for the RA and the NA between the seven P-items, in the sense that the buses-item would elicit considerably better evaluations for the RA and considerably lower scores for the NA than the six other P-items.

Third, we wondered whether there would be a difference in the scores for the NAs and the RAs between the first-year and the third-year students. However, for reasons explained above, no specific prediction was made about the strength or the direction of this difference.

Relationship

Between

Test 1 and Test 2

A final set of hypotheses and questions concerned the relationship between studentteachers’ answers to a P-item in Test 1, on the one hand, and their scores for the NA and the RA on the same items in Test 2, on the other hand. Generally speaking, we expected a good match between the student-teachers’ reactions to each of the seven P-items in the two tests.

This implies the following two predictions. First, most subjects who answered a P-item from Test 1 with an NR, will give a l-score for the NA and a O-score for the RA for the same item in Test 2. Second, an RR on a P-item during Test 1 will typically be followed by a 0 for the NA and a 1 on the RA for the same item during Test 2.

But we also anticipated a number of learning and thinking processes in the student-teachers

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that could lead to a mismatch between a student-teacher’s own response to a P-item during Test 1 and the way in which (s)he scored the realistic and the non-realistic response alternatives for the same item in Test 2. A first possible source of such mismatches is the scaffolding effect of the confrontation with a realistic answer. We anticipated that subjects who had solved a P-item in a non-realistic manner themselves during Test 1 would sometimes notice the problematic nature of that item (and of their non-realistic response to it) after being confronted with the realistic response alternative during Test 2, and would therefore score the RA and the NA in a manner that does not match their own NR given during Test 1. This prediction was based on the findings from another recent investigation in which we analyzed upper elementary schoolchildren’s zone of proximal development with respect to realistic mathematical modelling (Verschaffel et al., in press). Indeed, in that study it was found that a small but significant number of pupils who spontaneously answered a P-item with an NR replaced it by an RR after being confronted with the realistic response of a fictitious classmate. A second possible source of mismatches between the student-teachers’ own solutions of the P-items during Test 1 and their evaluations of the realistic and the non-realistic response alternatives during Test 2 concerns their pedagogical knowledge and beliefs about the role of real-world knowledge in school arithmetic word problem solving. For example, some subjects who had themselves demonstrated awareness of the problematic nature of a P-item during Test 1, may nevertheless give scores for the RA and for the NA reflecting little or no appreciation for pupil answers based on realistic context-based considerations, because they believe that the activation of such considerations should not be stimulated but rather discouraged in elementary shool mathematics. We hoped that a detailed analysis of these mismatches-and especially of the student-teachers’ written explanations for them-would yield interesting information concerning their pedagogical knowledge and beliefs about the importance of real-world knowledge and realistic considerations in school arithmetic word problem solving.

Results Results for Test I

As predicted, the student-teachers demonstrated a strong overall tendency to exclude realworld knowledge and realistic considerations when confronted with the problematic word problems. Indeed, only 48% of all reactions to the seven P-items could be considered as realistic (RRs), either because the student wrote a realistic answer or made an additional realistic comment. This percentage is considerably higher than in previous studies with upper elementary and lower secondary school pupils (Greer, 1993; Reusser. 1995; Verschaffel et al., 1994), where overall percentages of RRs between 15”/0and 20% were observed. Nevertheless, it remains disappointingly low, as it implies that in more than half of the cases the student-teachers solved the P-items from Test 1 in a stereotyped, uncritical way, without any consideration for the realities of the context involved in the problem.

As in the previous studies with upper elementary and lower secondary school pupils (Greer, 1993; Reusser, 1995; Verschaffel et al., 1994), certain P-items elicited markedly more RRs than others. Table 3 lists the percentage of RRs for each of the seven P-items. As predicted, the buses-item again yielded the largest number of RRs, namely 90%. Just like school children,

TEACHERS’

Percentage

of Realistic

Reactions

CONCEPTIONS

(RRs)*

349

AND BELIEFS

Table 3 of the Student-Teachers

to the Seven P-Items From Test

Problem

Name

% RRs

1

Buses School Runner Flask Rope Planks Friends

90 48 31 39 31 64 29

2 3 4 5 6 I

I

48

Total

*RRs stands for “realistic reactions”, i.e., all categories reflecting activation of real-world knowledge either in the solution of the problem (RA + or RA-) or in the additional comments (NA+, TE+, OE+, or NOA+).

student-teachers seem more inclined to activate relevant real-world knowledge when dealing with the interpretation of the outcome of a division with a remainder, than when they are confronted with P-items involving other kinds of underlying realistic modelling difficulties such as the interpretation of additive situations involving sets with joint elements (as in the friends-item) or situations involving the illusion of linear proportionality (as in the flask-item).

There was a significant difference in the overall number of RRs between the first-year and the third-year student-teachers in favour of the latter group (r-test, two-tailed, t=3.40, p<.Ol). However, the overall percentage of RRs remained low in both groups, namely 45% and 54% for the first-year and the third-year students, respectively (see Table 4). Interestingly, the size of the difference between the two years was dissimilar for the three teachertraining institutes involved in the study. In two institutes, a significant difference between the first-year and the third-year students was found: in School A (r-test, two-tailed, ~2.40, pc.05) and in School C (t-test, two-tailed, ~3.38, pc.01); but in School B there was hardly any difference (t-test, two-tailed, t=.60, ns) (see Table 4). This finding suggests that studentteachers’ ability and disposition toward realistic modelling of arithmetic word problems are at least partially influenced by the courses on mathematics education received during their pre-service training. Of course, this tentative conclusion should be investigated more systematically in further research.

Results for Test 2 The student-teachers’ strong disposition toward non-realistic modelling was also revealed by their evaluations of the realistic answer (RA) and the non-realistic answer (NA) on the

Percentage School

of Realistic

Table 4 Reactions (RRs) to the P-Items From Test I for the First-Year Student-Teachers From the Three Teacher-Training Institutes First-year(n=228)

and the Third-Year

Third_year(n=l04)

A B C

45 48 41

60 51 59

Total

45

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seven P-items during Test 2. Only in 47% of cases did the RA receive a score of 1; 6% of the RAs received a l/2-score and in 47% of the cases the RA was scored with a 0 (see the total percentages given at the bottom of Table 5). On the other hand, the NA was scored with a 1 and a l/2 in 56% and 26% of the cases, respectively; only 18% of the NAs received a O-score.

A r-test revealed that the difference between the percentage of l-scores for the NA (56%) and for the RA (47%) was significant (t-test, two-tailed, ~6.25, pc.01); the same holds for the difference between the percentage of O-scores for the NA (18%) and for the RA (47%) (t-test, two-tailed, t=-21.56,p<.Ol).

Summarizing these findings. it means that the student-teachers’ overall evaluation of the stereotyped, non-realistic answer to the P-items was considerably more positive than for the realistic answer based on context-based conderations.

As in Test 1, certain P-items elicited remarkably better scores for the RA and remarkably lower scores for the NA than others. In Table 5. the percentages of I-, l/2- and O-scores for the RA and for the NA are given for each of the seven P-items from Test 2.

There was a significant difference between the first-year and the third-year studentteachers (see Table 6). The third-year students gave significantly more I-scores (t-test, twotailed, t=3.29, p<.Ol) and less O-scores (t-test, two-tailed, r2.63, pc.01) for the RAs than the first-year students. Correspondingly, the third-year students produced significantly less I-scores (t-test, two-tailed, t=2.30, pc.05) and more O-scores (r-test, ?=2.33, pc.05) for the NAs than the first-year students.

As for Test 1, the size of the difference between the first-year and the third-year students was not the same in the three teacher-training institutes involved in the study. Only in School C were all four above-mentioned differences between the two groups of students statistically significant (%I l-scores for RA, t-test, two-tailed, t=3.55,p<.Ol; ‘%O-scores for RA, t-test, two-tailed, ~3.49, pc.01; ‘%, l-scores for NA, t-test, two-tailed, tz2.11, pc.05; %I O-scores for NA: t-test, twotailed, t=2.77;p<.Ol).

In School A. only one of these differences was significant (‘%Il-scores for RA, t-test, two-tailed, t=2.14,~<.05; ‘%O-scores for RA. ?-test, two-tailed, ~1.87, ns; % l-scores ?or NA, t-test, two-tailed, t =1.55, ns; ‘%jO-scores for NA: r-test, two-tailed, t =0.60, ns), and for School B no significant differences were found (‘%,l-scores for RA, r-test, two-tailed, ~0.62, ns; ‘%jO-scores for RA, t-test, two-tailed. ~0.12, ns; ‘%,1-scores for NA, f-test, two-tailed, 2=.68, ns; ‘%IO-scores for NA: t-test, two-tailed, t= 1.09, ns).

Percentages Item

1 2 3 4 5 6 I Total

Table 5 of l-, l/2- and O-Scores for the Realistic Answer (RA) and the Non-Realistic Seven P-Items From Test 2 Name

Buses School Runner Flask Rope Planks Friends

RA

Answer

(NA) to the

NA

I

l/2

0

1

l/2

0

95 46 24 42 24 68 28

3 5 5 3 I 13 8

2 49 72 55 69 19 64

6 66 81 59 72 33 75

73 19 12 22 20 18 15

21 15 7 19 8 49 10

47

6

47

56

26

18

TEACHERS’

CONCEPTIONS

351

AND BELIEFS

Table 6 Percentage of I-, l/2-, and O-Scores for the Realistic Answer (RA) and the Non-Realistic Answer (NA) on the Seven P-Items From Test 2 for the First-Year and the Third-Year Students From the Three Teacher-Training Institutes (A, B. C) Year

RA

School

NA 0

I

II2

0

Total

47 45 38 44 58 49 55 53

41 46 56 49 39 46 39 42

55 58 62 58 48 54 52 52

23 26 25 25 28 26 26 26

22 I6 13 17 24 20 22 22

1+3

41

41

56

26

18

I A B C

1

Total 3

Results

A B C

Concerning

112

the Relationship Between the Student-Teachers P-Items From Test 1 and Test 2

Reactions

to the Seven

The previous sections focused on the results for Test 1 and Test 2 separately. Now we turn to the relationship between the student-teachers’ own reactions to a P-item during Test 1 (RR or NR), on the one hand, and their scores for the realistic answer (RA) and the nonrealistic answer (NA) on the corresponding item of Test 2, on the other. If student-teachers’ scoring on Test 2 was a mere reflection of their own (in)ability to solve a P-item in a realistic way, one could expect (1) that all subjects responding to a P-item with a RR during Test 1 would score the RA with a 1 and the NA with a 0. and (2) that all subjects responding to a P-item with an NR during Test 1 would score the RA with a 0 and the NA with a 1. In this section, we will investigate to what extent the student-teachers’ evaluations for the NAs and the RAs during Test 2 indeed matched their own performances during Test 1, by separately analyzing the scores for the RA and the NA following the 52’Y0non-realistic reactions (NRs) and the 48% realistic reactions (RRs) on Test 1.

Scores for the 52% non-realistic

reactions on Test I

Table 7 presents the distribution of the different combinations of RA scorings (1, l/2 or 0) and NA scorings (1, l/2 or 0) over the seven P-items of Test 2 for the 52% non-realistic reactions on Test I, as well as the overall distribution of the scorings for the RA and the NA over the three possible scores (1, l/2 and 0).

Table 7 Combinations of RA Scorings (I, l/2 or 0) and NA Scorings (1, l/2 or 0) Over the Seven P-Items of Test 2 for the Total Number of Non-Realistic Reactions on Test I, asWell as the Distribution of the Scorings for the RA and the NA Over the Three Scores (I, I12 and 0) RA-scores NA-scores I l/2 0 Total

I

112

0

3.4 3.8 2.8

6.6 0.1 0.2

79.3 2.4 1.2

Total 89.3 6.5 4.2

10.0

6.9

83.1

100.0

352

L. VERSCHAFFEL

et al

As expected. we found a strong and straightforward relationship between the nonrealistic reactions on a P-item during Test 1, and the evaluations of the RA and the NA on that item during Test 2. In 89.3% of all cases wherein an NR was given to a P-item during Test 1, the NA to that item was given a l-score in Test 2 (see Table 7). Correspondingly, 83.1% of the student-teachers’ NRs during Test 1 were followed by a O-score for the RA in Test 2. The match of a l-score for the NA and a O-score for the RA occurred in no less than 79.3% of all cases wherein a P-item from Test 1 was answered with an NR. This scoring combination is perfectly in line with the student-teachers’ own non-realistic interpretation and solution of the P-item, as evidenced by their NR to the corresponding item during Test 1. They scored the NA with 1 because this response was exactly the same as the stereotyped, non-realistic answer they had given themselves on this item during Test 1, and they scored the RA with 0 because they could not suspect, and, therefore, appreciate the context-based considerations underlying this answer. Support for this interpretation is also provided by the fact that critical remarks or reflections almost never accompanied the combination of a l-score for the NA and a O-score for the NA following an NR during Test 1. The only written comments that we found were a few rare statements which revealed the student-teacher’s complete lack of understanding of the (realistic) reasoning behind the realistic response alternative, as illustrated in the following examples: “This pupil did not understand anything about the problem” (comment accompanying a O-score for the RA to the friends-problem).

“I am afraid that the pupil who wrote these answers hadn’t learnt anything about mathematics yet” (general comment on the O-scores for all NAs that involve a statement about the unsolvability of the problem).

Most df the remaining 16.9% scoring combinations produced by the non-realistic reactors involved also at least one l-score, namely 6.6% for the combination NA-1, RA-l/2.

3.8% for NA-l/2, RA-1, 3.4% for NA-1, RA-1, and 2.8% for NA-0, RA-1 (see Table 7).

Taking into account the student-teachers’ notes and reflections written down in the comments box, the following possible explanations for these scoring combinations can be suggested.

For 10.00/o of the NRs to a P-item during Test 1, a l-score for the RA was given during Test 2. This suggests that in those cases the confrontation with the RA during Test 2 had functioned as a scaffold toward (more) realistic modelling. This interpretation is exemplarily supported by the following written comment on the l-score for the RA on the flask-item from a student-teacher who reacted in a non-realistic manner during Test 1: As the form of the flask is cone-shaped, it is indeed impossible to give a precise answer to this problem.

Therefore, I scored this answer (= RA) with I. But I have to confess that I was not aware of this when solved this task myself during the previous test.

The finding that this scaffolding effect of the confrontation with the realistic response occurred only in a small minority of cases (i.e., lO.O%), can be interpreted as additional evidence for the strength and the resistance of the tendency toward non-realistic mathematical modelling among student-teachers (see also Verschaffel et al., in press).

Interestingly, these 10.0% scorings showing insight into the appropriateness of the RA during Test 2-as evidenced by the l-score for the realistic response alternative-were accompanied by a diversity of appreciations of the NA. As shown in Table 7, in 2.8% of the cases the l-score for the RA was matched with a 0 for the NA, in 3.8% the NA was scored with l/2, and in 3.4% with 1, even. A detailed analysis of the written comments accompanying the latter two combinations (RA with 1 and NA either with a 112 or a 1) revealed that-

I

TEACHERSâ&#x20AC;&#x2122; CONCEPTIONS

AND BELIEFS

353

although in those cases the student-teachers had detected the problematic nature of the P-item-they nevertheless believed that it was unfair to blame elementary school children for responding to arithmetic word problems in a stereotyped and uncritical way by giving them a low score for their NA. Typical remarks explaining such l/2- or l-scores for the NA were: This word problem is (very) badly formulated; pupils should not be punished for it.

You cannot expect a fifth-grader to be able to handle such complex problems.

This is a tricky problem. I would never give such a problem to a fifth-grade child

How, finally, can we understand the scoring combination NA-1, RA-l/2, which followed 6.6% of the NRs on Test l? Because the RA was not scored with a 0 (but with a l/2), we assume that the non-realistic reactor became aware of the problematic nature of that particular item during Test 2. But if so, the question arises why there was nevertheless a higher appreciation for the NA than for the RA. We believe that this scoring combination is the result of the complex interplay of two pedagogical considerations which will be discussed at the end of the next paragraph.

Scores for the 48% realistic reactions on Test 1

Table 8 presents the distribution of the different combinations of RA scorings (1, l/2 or 0) and NA scorings (1, l/2 or 0) over the seven P-items of Test 2 for the 48â&#x20AC;?/0realistic reactions on Test 1, as well as the overall distribution of the scorings for the RA and the NA over the three possible scores (1, l/2 and 0).

As expected, the distribution of the combinations of RA-scores and NA-scores related to the realistic reactions during Test 1 (see Table 8), is completely different from the distribution of the scores associated with the non-realistic reactions (see Table 7). However, Table 8 also shows that the congruence between the realistic reactions on Test 1 and the scorings of the RA and the NA during Test 2, was less straightforward than for the non-realistic reactions. On the one hand, the evaluations of the RA were generally in line with our expectation; indeed, 85.4% of the realistic reactions on Test 1 were followed by a l-score for the RA on Test 2. However, their scorings for the NA were rather surprising: only 33.5% of the subjects who reacted in a realistic way to a P-item during Test 1 scored the NA with a 0 during Test 2 (almost always in combination with a 1 for the RA). Apparently, these 33.5% combinations of RA-1, NA-0 conveyed the judgment that an elementary school pupil who

Table 8 Combinations of RA Scorings (I, I12 or 0) and NA Scorings (I, l/2 or 0) Over the Seven P-Items of Test 2 for the Total Number of Realistic Reactions on Test I, as Well as the Distribution of the Scorings for the RA and the NA Over the Three Scores (I 1112and 0) RA-scores NA-scores

I

l/2

0

I l/2 0

10.2 42.3 32.9

3.2 2.0 0.4

1.4 1.4 0.2

20.8 45.1 33.5

Total

85.4

5.6

9.0

100.0

Total

354

L. VERSCHAFFEL

et al

demonstrates some awareness of the realistic mathematical modelling difficulty involved in a P-item should be fully complimented, while a non-realistic answer based on a stereotyped, uncritical solution process should be totally discouraged. But because of the lack of informative written explanations in the comments boxes accompanying these 33.5% RA-1, NA-0 combinations, this interpretation cannot be supported by qualitative data.

A closer look at Table 8 reveals that one scoring combination occurred even more frequently than the combination of a 1 for the RA with a 0 for the NA. In 42.3% of the cases wherein an RR was given to a P-item from Test 1. the 1 score for the RA on that item was accompanied by a l/2 for the NA. In addition, the combination RA-1, NA-1 also occurred in a substantial number of cases (10.2%). These results indicate that in many cases where student-teachers reacted themselves in a realistic manner to a P-item, they were nevertheless quite understanding and tolerant towards elementary school pupils who interpreted and solved these P-items without seriously taking into account the relevant realworld knowledge. According to their written explanations in the comments box, they thought that it is unfair to punish a fifth-grader for solving the P-item in a stereotyped, non-realistic manner. This is illustrated by two comments accompanying the scoring combination RA- 1, NA-1 with respect to the runner-item: I scored alternative

D (= RA) with I because this pupil knows that it is not realistic to assume that John will be able to run at his record speed for I kilometre. But I also gave I tbr alternative A (= NA) because from a purely computational point of view this is the correct answer.

Mathematically

the response

of pupil A (= NA) is a correct

answer, but in real life it is impossible

The two remaining cells from Table 8 which contain a considerable percentage of scoring combinations are the one involving a score of 1 for the NA combined with a score of l/2 for the RA (3,2X), and the combination of a 1 for the NA with a 0 for the RA (7.4%). At first sight, these scoring combinations are incomprehensible: although in these cases an RR was given to the P-item during Test 1, this was nevertheless followed by attributing a higher score to the NA than to the RA during Test 2! Although it is not impossible, we doubt that these bizarre scoring combinations are due to the fact that the student-teachers who reacted in this way had lost their awareness of the realistic modelling difficulty during the second confrontation with the P-item in Test 2. Based on a qualitative analysis of the written protocols, we rather assume that these strange scorings resulted from the interplay of the following two considerations.

First, the student-teachers scored the NA with 1 because they believed that one should not blame fifth-graders for not resisting the tendency to interprete and solve a P-item in a non-realistic way, Indeed, several l-scores for the NA (in combination with a 1 or a l/2 for the RA) following a realistic reaction during Test 1 were motivated by arguing that it is undesirable or even improper to confront fifth-graders with such complex, ill-formulated or tricky word problems. Second, the RA was not scored with 1 or even with l/2 because of the vagueness of this realistic response; to deserve a better score (a 1 instead of a l/2 or even 0), the RA should have been more precisely formulated and/or better motivated. This is illustrated in the following exemplary comments: (Explanation for a l/2-score for the RA on the rope-item): “In Pact this boy is right because you do not know how much you will lose for making the ties. but he should have explained this in his answer”.

(Explanation for a I/2-score for the runner-item): “If he would have explained that a runner cannot run constantly at his record speed for the 100 metres. I would have scored this alternative (= RA) with a 1”.

(Explanation for a O-score for the RA on the rope-problem): “In the case of such an answer, at least some kind of explanation is required. If this pupil would have done this, I would have given him 112 point”.

TEACHERS’

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355

Differences Betu’een Student Groups and Between Problems The percentages given in Tables 7 and 8 are the overall results for all student-teachers and for all seven P-items. We also analyzed here whether the relationship between the results on Test 1 and Test 2 was the same for the two groups of student-teachers (first and third year) and for the seven distinct P-items. Only very small and non-significant differences between the first-year and the third-year students were found and will, therefore, not be reported.

With respect to effects of problem type. we will focus on the P-item which again elicited scoring patterns that deviated most strongly from the general pattern presented in Tables 7 and 8. namely the buses-item.

The most striking finding with respect to the buses-item (see Table 9) was the extremely high percentage of non-realistic reactions on Test 1 accompanied by a l-score for the RA on Test 2. namely 86.2% as compared to only 10.0% for all P-items together (see Table 7).

Apparently, the scaffolding effect of the confrontation with the realistic response was much greater for the buses-item than for all other P-items. A first explanation for this finding is the intrinsic easiness of the realistic modelling difficulty underlying this problem as compared to the other P-items (see the section “Hypotheses and Questions”).

But a closer look at the RA to this item in Test 2 yields a plausible additional explanation.

Contrary to most other P-items, the RA for the buses-item is not merely a statement about the unsolvability of the task, but a precise, numerical response which differs distinctly and understandably from the NA. This characteristic of the RA for the buses-item may also have contributed to its great attraction and positive acceptance among those who gave already a realistic answer to the buses-item in Test 1. Indeed, the percentage of l-scores for the RA on Test 2 matching the realistic reactions on Test 1 was also very high (i.e., 95.6% as compared to 85.6% for all P-items together).

The buses-item elicited also a very high percentage of l/2-scores for the NA both among the non-realistic reactors (65.5% versus only 6.5% for all P-items altogether) as well as among the realistic reactors to this item (73.6% versus 45.7% for all P-items altogether). Almost all these l/2-scores for the NA came from student-teachers who demonstrated awareness of the realistic modelling difficulty underlying the buses-item during Test 2, as evidenced by their l-score for the RA. Why did the student-teachers give so little l- and O-scores for the NA to the buses-item, in comparison to the other P-items? On the one hand, the strong refusal among these student-teachers to score the NA with 1 revealed their pedagogical opinion that fifth-grade pupils should be aware of, and capable of, making proper sense of the Table 9 Combinations of RA Scorings (1, l/2 or 0) and NA Scorings (1, 112 or 0) for the Buses-Item of Test 2 for the Total Number of Non-Realistic Reactions (10%) and Realistic Reactions (90%) on This Item During Test 1, as Well as the Distribution of the Scorings for the RA and the NA Over the Three Scores (1, 112 and 0) Non-realistic

reactions

Realistic

RA-scores

1

l/2

0

I 112 0

13.8 65.5 6.9

0.0 0.0 3.5

3.5 0.0 6.8

17.3 65.5 17.2

Total

86.2

3.5

10.3

100.0

NA-scores

reactions

RA-scores Total

NA-scores

1

112

0

l/2 0.0

3.1 71.5 21.0

1.3 2.1 0.0

0.7 0.0 0.3

5.1 73.6 21.3

Total

95.6

3.4

1.0

100.0

I

Total

356

L. VERSCHAFFEL

et al

outcome of a division word problem with a remainder. With respect to the other kinds of realistic modelling difficulties underlying the other P-items, they made considerably lower demands for fifth-grade children. But on the other hand, the relatively small number of O-scores for the NA suggests at the same time that the student-teachers did not feel too strongly about answers lacking proper sense-making of the outcome of a division. It seems that these student-teachers considered that the problem-solving process that resulted in the NA was largely correct and that the mistake originated only in the very final stage of the solution process, namely the interpretation of the outcome of the proper calculation; therefore, it seemed unfair to them to punish this pupil too severely for this (small) mistake by scoring the answer with a 0.

Theoretical, Methodological,

and Educational Implications

Recent studies have provided ample evidence for a phenomenon whereby children solving word problems in school often produce answers without regard for realistic constraints (Greet-, 1993; Reusser, 1995; Verschaffel et al., 1994, in press). The present study provides some insight into one of the instructional factors that are considered responsible for the development of this tendency among children, namely the teacher’s conceptions and beliefs about the importance of real-world knowledge in arithmetic word problem solving. As in the abovementioned studies with elementary school children, we found a strong and resistant tendency among pre-service teachers to exclude real-world knowledge and realistic considerations when dealing with arithmetic word problems as instructional tasks. even among those studentteachers who were at the end of their pre-service training and ready to start working as elementary school mathematics teachers.

This tendency toward non-realistic modelling of school arithmetic word problems was first of all evidenced by the student-teachers’ own stereotyped, uncritical responses to a first test (Test 1) consisting of seven problematic word problems (P-items) that could not be solved in a straightforward and unproblematic way, at least if one seriously takes into account the realities of the problem context. Using the same criteria as in the previous studies with elementary school children (see Verschaffel et al., 1994, in press), only 48”/0of the pre-service teachers’ spontaneous reactions to the P-items from Test I could be categorized as realistic (RRs). Although the performance of the third-year student-teachers was better than that of the first-year students, the number of RRs in the former group was still disappointingly low. As in the previous studies with children, only the item dealing with the interpretation of the outcome of a division with a remainder-a variation of the well-known buses-item from the American NAEP mathematics assessment (Carpenter, Lindquist, Matthews, & Silver, 1983)--elicited a satisfactory number of responses based on realistic, context-related thinking.

The student-teachers’ tendency toward non-realistic modelling was also evidenced during a second test (Test 2) wherein they were asked to evaluate the appropriateness of a realistic answer (RA) and a non-realistic answer (NA) to the same seven P-items as in Test I, by giving a score of I, l/2 or 0 to each of these response alternatives. Generally speaking, the scores given by the student-teachers reflected their own disposition toward non-realistic mathematical modelling as already revealed by their spontaneous solution endeavours during Test I: in 47”/ of the cases the RA was scored with a 0, while this O-score was given to the NA in only 18% of the cases. In line with the findings for Test 1, the third-year students

TEACHERS’

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351

demonstrated more positive evaluations about the realistic responses and more negative evaluations about the non-realistic ones than the first-year students. Finally, the results for the buses-item were again exceptional, in the sense that only for this P-item was the RA considered the (most) appropriate one by almost all student-teachers.

A confrontation of the reactions to the same P-item during Test 1 and Test 2, revealed further insight into the processes and considerations underlying the student-teachers’ evaluations of the RAs and the NAs during Test 2. A first important outcome of this confrontation is that subjects who answered a problematic item in a non-realistic manner themselves only rarely discovered the problematic modelling assumptions underlying their own nonrealistic solution during the confrontation with the realistic answer of a fictitious pupil. In about 80% of the cases, these subjects scored the RA with 0 and the NA with 1, which suggests that they stuck to their own stereotyped, uncritical approach toward school arithmetic word problems and therefore could neither suspect nor appreciate the context-based reasoning behind the realistic response alternative. This result can, therefore, be considered as a clear indication of the strength and the resistance of student-teachers’ own tendency to model and interprete school arithmetic word problems in a non-realistic way. A second important result of the confrontation between the student-teachers’ reactions to the P-items in Test 1 and Test 2 is the mildness of their evaluation of the NAs during Test 2, even if they had themselves produced an RR to the same item during Test 1. Indeed, the NA received considerably more l/2- and l-scores than O-scores by student-teachers who gave a RR during Test 1.

The m,ost frequent argument was that it was undesirable or even totally inappropriate to confront fifth-grade children with such complex, ill-formulated or tricky problems. More or less explicitly, these comments conveyed their pedagogical belief that such non-standard, ill-structured problems are unimportant and even harmful for teaching arithmetic word problem solving in the elementary school, because they lead the children away from what these student-teachers consider as the crucial goal of teaching word problem solving, namely learning to find the correct numerical answer to such a problem by performig the formalarithmetic operation(s) “hidden” in the problem. A third important result was that the busesitem elicited considerably more positive evaluations of the RA and more negative evaluations of the NA than the other P-items, both among student-teachers who solved the problem in a non-realistic manner and among those who answered it in a realistic way. Several complementary explanations were proposed for this third finding. One of these explanations concerns the nature of the RA for the buses-item. Indeed, contrary to most other RAs in Test 2, the RA to the buses-item involved a numerical answer that was distinctly and understandably different from the NA, rather than a vague and unmotivated statement about the unsolvability of the problem (as was the case in most other P-items, see Table 2).

Therefore, one could argue that the student-teachers’ overall evaluations might have been more positive for the RA and more negative for the NA, if all RAs had consisted of a precise, numerical response (such as the following possible RA to the rope-item: “12+1..5=8; at least 9 pieces of rope will be needed”) and/or a clearly motivated non-response (such as the following possible RA to the friends-item: “You cannot give one precise answer to this problem because the two boys may have common friends”).

While the present study demonstrates that many future teachers have knowledge and beliefs about teaching and learning arithmetic word problems which are problematical from a realistic point of view, it does not, of course, yield direct evidence that these teacher conceptions and beliefs are responsible for children’s strong tendency to exclude real-world

358

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et al

knowledge and realistic considerations from their word problem solving endeavours. However, based on the recent literature on mathematics teaching (Fennema & Loef, 1992; Thompson.

1992) there is good reason to assume that these teacher cognitions and beliefs about the role of real-world knowledge in the interpretation and solution of school arithmetic word problems have, indeed, a strong impact on their actual teaching behavior and consequently on their students’ learning processes and outcomes. But this aspect remains to be systematically investigated in further research.

In many current reform documents relating to mathematics education. a strong plea is made for making problem solving in school mathematics more closely related to the experiental worlds of children by using more complex and more authentic problem situations in the mathematics lessons (see e.g., De Corte, Greer, & Verschaffel. 1996; National Council of Teachers of Mathematics, 1989; Treffers & de Moor, 1990). However, if we want to connect problem solving in school mathematics to the experiental worlds of children, we will also have to stimulate and help (student-) teachers to construct the proper concepts, skills and beliefs that are needed for realistic modelling of problem situations and for realistic interpreting of outcomes of arithmetic calculations, as part of a genuine mathematical disposition.

The problematic items used in the present study could function as good starting-points for valuable teacher-training activities and discussions on the issue of mathematical modelling and interpretation.

More specifically, they could be used to produce a cognitive conflict (or disequilibrium) in the student-teachers, which is necessary to produce the desired reflection and revision of their beliefs about the role of everyday knowledge in mathematical problem solving. However, further research is needed to assess whether such preservice or in-service instructional experiences can indeed be helpful in changing (student-) teachers’ inappropriate conceptions and beliefs about mathematical modelling and problem solving and about the role of real-world knowledge in it.

Acknowledgments-The authors participated in this study.

thank

the headmasters,

the teachers,

and the pupils

of the three schools

that

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