Preservice Teachers' Problem-Solving Processes - merga

Mathematics Education Research Journal

1998, Vol. 10, No. 3,59-76

Preservice Teachers' Problem-Solving Processes Margaret Taplin The Open University of Hong Kong The purpose of the study reported in this paper is to explore some of the common difficulties with mathematical word problems experienced by preservice primary teachers. It examines weaknesses in students' content and procedural knowledge, with a particular focus on how they apply these aspects of knowledge to solving closed word problems" The SOLO Taxonomy (Biggs & Collis, 1982, 1991) is used to classify the processes used by students who attempted to solve a group of word problems of varying difficulty. Other characteristics of the students' processes that are analysed include the way they used the cues provided in the problem, the way they brought in additional concepts or processes, and the types of errors they made.

It has long been acknowledged that one of the responsibilities of a p~eservice mathematics education program is to ensure that student teachers develop adequate knowledge about mathematics (Cockcroft, 1982; Department of Employment, Education and Training [DEETt 1989; National Council of Teachers of Mathematics [NCTMt 1991). Such documents have recommended that action should be taken within teacher education institutions to ensure that graduating primary teachers are competent in mathematics .to at least the lower secondary school level and have developed positive attitudes towards the subject. Furthermore, it has been recommended that preservice programs should provide compulsory remediation courses to ensure that these competencies are developed in students identified as being "at risk" These recommendations force us to investigate more closely the concepts, skills, and procedures that students need to develop during their preservice education. The inadequacy of mathematical skills of many students entering primary education courses has been a long-term concern of mathematics educators (DEET, 1989). Tirosh and Graeber (1989) and Tirosh, Tirosh, Graeber, and Wilson (1991) suggested that student teachers are often able to perform mathematical operations but unable to successfully relate their solutions to real life, problem-solving contexts. Simon (1993) and Jones (1995) have also expressed concern about the number of teachers with weak conceptual backgrounds in the mathematics they are likely to teach. A study by Taplin (1995) identified several topic areas in which preservice primary school teachers performed poorly. These included multiplication and division of decimals; fraction concepts, including equivalents and operations; applying measurement formulae; the relationship between fractions, decimals, and percentages; and the application of geometric principles. Measurement and spatial tasks were particular weaknesses. The results also suggested that students performed well on items which called for simple recall and routine application of rules, and progressively worse as the items required an increasing application of knowledge. The items on which the students performed the most poorly were predominantly word problems, suggesting that one of the areas most in need of remediation is the transfer of procedural knowledge to unfamiliar situations. The Taplin study supports evidence cited above that many student teachers have inadequate problem-solving skills.

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Importance of Developing Competence in Problem Solving The concern expressed above, that preservice teachers lack investigatory or problem-solving skills, is a serious one since it is important for teachers to be competent problem solvers if they are to be able to teach mathematics effectively (Battista, Wheatley, Grayson, & Talsma, 1989; Bitter & Cameron, 1988; NCTM, 1991). Failure to address the students' deficiencies could have long-term consequences. Preservice teachers with low skills and high mathematics anxiety can become reluctant, ineffective teachers of mathematics. A probable consequence is that these teachers will produce yet another generation of pupils who are weak and who have· unfavourable attitudes (Bitter, 1987; Sullivan, Clarke, Spandel, & Wallbridge, 1992). The problem becomes particularly serious if the teachers have misconceptions about mathematics which they pass on to their students (Tirosh & Graeber, 1989). Since it is the responsibility of teacher training programs to help prospective teachers to overcome their anxieties and misconceptions (Tirosh & Graeber, 1989), it is dearly important to understand more about them (Eisenhart et al., 1993; Leinhardt, 1988; Simon, 1993; Sullivan et aL, 1992).

Knowledge and Skills Essential for a Sound Understanding of Mathematics To teach mathematics effectively, it is necessary for teachers to be competent in a complex web of knowledge domains: knowledge of and about mathematics, about the pedagogy of mathematics, and about the students (Borko et al., 1992; Cooney, 1994). The focus of the present study is on the first of these domains, knowledge of and about mathematics, which is a critical factor in teaching effectiveness (Ball, 1991). The most fundamental requirement for teachers is an adequate grasp of content knowledge (NCTM, 1991; Sullivan et al., 1992). This, however, is only one part of mathematical understanding (Borko et al., 1992; Simon, 1993). Ability to relate to the context in which the knowledge is embedded is as important a cOlnponent of mathematical competence as is the knowledge itself (Cobb, 1986; Reeves, 1993; Simon, 1993; Sullivan et al., 1992). Eisenhart et·al. (1993), Leinhardt (1988), and Simon (1993) have all described two complementary components of mathematical knowledge. The first of these is procedural, which is knowledge of the rules and algorithms and the symbol representation system. The second is conceptual knowledge, which refers to "the relationships and interconnections of ideas that explain· and give meaning to mathematical procedures" (Eisenhart et al., 1993, p. 9). The latter definition can be expanded to include the ability to make generalisations, describe relationships, and demonstrate higher-order reasoning skills (NCTM, 1989; Sullivan et al., 1992). These descriptions of procedural and conceptual knowledge are similar to Skemp's (1976) concepts of instrumental understanding (being in possession of a rule and able to use it) and relational understanding (knowing what to do and why). Many writers have suggested that teachers tend to place great emphasis on procedural

Problem-Solving Processes

61

knowledge and little on conceptual knowledge. Tall (1995) suggested· that excessive cognitive strain, brought about by an inability to make appropriate links, can cause children to revert to rote learned, procedural knowledge. To avoid this, teachers need to be able to help children to make links, to adapt their knowledge to a variety of contexts, and to construct their own conceptual knowledge. But teachers can only help their students if they themselves have learnt to make such links with ease--and teachers are also subject to cognitive strain. It is therefore necessary for teacher educators to understand mor~ about what types of tasks cause student teachers cognitive strain. As well as looking at the nature of the tasks themselves, we need to explore student teachers' responses to tasks of varying cognitive demand.

The SOLO Taxonomy The level of cognitive demand required to solve problem-solving tasks, as well as the quality of student responses, may be described using the SOLO Taxonomy (Biggs & Collis, 1982, 1991). The SOLO Taxonomy is based on the philosophy that thinking is multimoda1. At school level, there are two primary modes of intellectual functioning: ikonic and concrete symbolic. Ikonic functioning draws on . imagery and intuition. People operating at this level employ hunches, beliefs, or "workplace maths." The concrete symbolic mode is dependent on school teaching. It uses higher-order symbolisation which is the result of deliberate instruction and the application of processes according to symbolic rules. Modes are not exclusive, however. Collis et a1. (1991) describe an apparent interaction between modes "which enables the typical ikonic mode strategies to be applied to symbolic ways of representing reality" (p. 3). Watson, Campbell and Collis (1993) have described ways in which ikonic supportcan be given to concrete symbolic responses. The SOLO Taxonomy also describes a series of response levels which operate within ec:l.ch mode (Collis & Watson, 1991, p.67). The first of these is the prestructurallevel, at which cue and response are confused and the student closes without even seeing the problem. The second level is unistructural, where the. student uses only one relevant datum and provides a narrow answer based just on that datum. The third level is multistructural, in which the student uses a few limited and independent aspects of the data. The fourth level is relational, where there is a synthesis of relevant data. The fifth level is extended abstract, requiring the inclusion of ideas from outside the knowledge domain implied in the problem.

The Present Study The outcomes of an earlier investigation (Taplin, 1995) indicated that it would be valuable to investigate students' conceptual knowledge by focussing on their responses to a series of word problem tasks. For the present study it was decided to include just one type of task, namely closed word problems. Other types, including problems with multiple answers and open-ended investigations (NCTM, 1989), are equally important but could not be included due to time constraints. Consequently, the purpos'e of this study was to develop further the earlier exploration of students' content and procedural knowledge (Taplin, 1995) and to

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investigate the extent of their conceptual knowledge as revealed through their responses to closed word problems. The following research questions were posed: 1.

What characteristics differentiate tasks that students are able to solve successfully from those they find difficult? What levels of thinking do students use when solving word problems?

2.

Method

Design . The study was conducted among first and second year primary student teachers enrolled in a Bachelor of Education program. An average of 58 students were given the twelve word problems shown in Figure 1. The number of students varied slightly from problem to problem due to the fact that the tasks were administered on several different days and some absenteeism occurred. The twelve problems were judged to be typical of the kind the students are likely to use in their future teaching. They were selected from the items used by Taplin (1995) to provide a range from very easy to very difficult. The students were asked to show all the steps in their solutions to the problems.

Response Analysis To explore ways in which the students used information, concepts, and processes in their solution attempts, the structure of their responses was analysed using a mapping procedure based on that developed by Chick, Watson, and Collis (1988) and modified by Collis and Watson (1991) to assess the SOLO level of particular responses. This procedure, described luore fully in Taplin (1994), gives a visual representation of the task and provides a clear overview of three essential components: 1.

2. 3.

The way in which the problem solver used the initial information provided in the problem (represented on the maps by the symbol"'), The concepts, processes, and strategies used in attempting to solve the problem (represented by.), The results which are obtained at various stages in the solution of the problem (represented by. for a correct result, 0 for an incorrect result).

To illustrate how a response map can be constructed and interpreted, consider the map in Figure 2. The map shows a students' correct solution to Problem 6. The student selected data to which a process (i) was applied. This led to an interim result (ii). More data were needed for the next piece of processing to occur (iii). This lead to a further interim result (iv). The student then applied this result to one of the initial cues (v) and obtained the correct answer (vi). The students' response maps were classified by two evaluators who both had previous experience with the use of the SOLO Taxonomy to classify students' responses to various types of tasks. They first evaluated each response map independently and then discussed their classifications to reach a consensus


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1.

If nine teams, each of seven people, were rearranged so that there were nine people in each team, how many new teams would there be?

2.

There are 227 children in a school. Every child in the school belongs to either the music club or the sports club, and some belong to both clubs. The music club has 120 members, and 36 of these ~re also in the sports club. What is the total membership of the sports club?

3.

In the figure below, the small squares are all the same size and the area of the whole rectangle in 1 square unit. What is the area of the shaded part?

4. . There are 13 boys and 15 girls in a group. What fraction of the group is boys? 5.

The. distance between two towns is 150 kilometres. The distance is represented on a certain map by a length of 30 centimetres. What is the scale of the map?

6.

A concrete mix is made by volume: 3 parts gravet 2 parts. sand, and 1 part cement. In 12 cubic meters, what is the volume of (a) cement, and (b) sand?

7: . Twelve friends had enough money to buy five Mars bars between them. If the Mars bars were divided evenly, how many would each person get? 8.

An article is marked for $68. If a customer is given a discount of 12%, what is the amount paid?

9.

The area of a rectangle is 96 square centimetres and the perimeter is 560 millimetres. Find the length and width of this shape.

10. A box has a volume of 100 cm 3. Another box is twice as long, twice as wide and twice as high. What is the volume of this box in cubic centimetres? 11. At my shack I need to build a rectangular tank which will hold 1000 litres of water. What proportions should the tank be so that it will require the smallest quantity of metal? (Hint: a 10 cm x 10 cm x 10 cm cube holds 1 litre of water.) 12. Suppose a wire is stretched tightly around the Earth. (The radius of the. Earth is approximately 6 400 km.) If the wire is cut, its length is increased by 20m, and it is placed back around the Earth so that it is the same distance from the Earth at every point, could you walk under the wire? How far would the new wire be above the Earth?

Figure 1. Word problems used in the study.

Taplin

64 PRO C E SSE S Ie 0 N C E'P T S

CU ES

FIND

DATA (i) find total volume

RESULTS INTERIM FINAL (ii) 6

v 0 lu me

of cement

(v) double cern ent part

(vi) 2

Figure 2. Sample response map for Problem 6.

jointly. The response categories were those which reflected use of the ikonic mode, ·those which demonstrated concrete symbolic thinking, and those which combined the two modes. They were further classified into one of the three response levels defined by the SOLO Taxonomy (unistructural, multistructural, or relational). For example, the response shown in Figure 2 can be classified as multistructural because it requires the integration of three processes: finding the total volume of the cement, identifying the relationship between this and a volume of 12, and using the principle of ratio to find the equivalent volumes of the components. The response is classified as combining ikonic and concrete symbolic modes because the student used an intuitive approach but also drew on some knowledge of ratio and proportion. . Student responses were also analysed to ascertain whether they used all the cues given in the problem. Finally, incorrect responses were analysed for the types of errors which students made in pro~essing the given information and their interim results. Four categories of error were identified: 1.

2.

3. 4.

Translation errors involved the use of processes which were irrelevant or incorrect. For example, a typical processing error on Problem 7 was to divide the smaller number into the larger; another was to multiply them. An example of irrelevant processing occurred on Problem 6, where several students applied the volume formula to the three numbers given in the problem. Interpretation errors occurred when the student either ignored a piece of information or interpreted it incorrectly. For example, on Problem 11 many students gave only one example of possible dimensions-ignoring the request to find the smallest surface area. Knowledge errors constituted the use of an incorrect formula or inability to use the correct formula. Arithmetic errors were those which led to a wrong answer although the correct processes had been invoked.


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Results·

Three Sample Responses Three problems-an easy one, a moderately difficult one and a very difficult one-have been selected to illustrate the method of analysis.

. An easy problem. Figure 2 shows a typical response map of a correct solution to Problem 1. This was considered an easy problem because 95% of the students obtained a correct solution. CUES FIND

PROCESSES/CONCEPTS DATA

RESULTS INTERIM FINAL

multiply

9 teams

Ar---~--"'-~---1. 63

How many teams? 7 people per team ~--

9 people

7

teams

per team.

Figure 3. Response map of a typical correct solution to Problem 1. This response to Problem 1 was classified as ikonic thinking because the student operated in an intuitive manner on numbers that were already familiar enough to her to enable her to do so. The response map indicates that the student used multistructural thinking because she integrated hvo processes, multiplication and division. Approximately one half (53%) of the students who were correct used the same approach as this student, calculating an intermediate response of 63 by multiplying 9 x 7 and then dividing the product by 9. The other correct students applied the commutotive law directly, thus avoiding any interim response. This method requires knovvledge of an arithmetical law that is not provided in the problem, and was therefore classified as relational thinking. There were three students who had gave incorrect solutions. Two students Inade prestructural responses, writing down the question but making no attempt to solve it. The third student multiplied 9 x 7 and stopped, thus omitting the third cue about 9 people per team. This was classified as an interpretation error.

A mediutlz difficulty problem. Approximately one half of the students (45%) answered Problem 9 correctly. Like Problem 1, this problem also has three cuesbut they must be combined in a much more complex manner. The response map in Figure 4 shows a typical correct response. This solution was classified as relational level in the concrete symbolic mode because it integrates a sequence of separate pieces of information about area and perimeter. It requires three interim results in

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CUES FIND


•

rectangle

A.

area= 96cm 2

(i)

RESULTS INTERIM FINAL

(ii)

3 32 4 24 6 ·16

length

width perimeter =560mrn

2 48

--....:::::.._~~.,£---. . ~o..-_~---1__-r-",--+--~.

56cm

28cm

4 24

Key to' processes I concepts (i) area =length x width (ii) find factors of 96 (iii) make units consistent": convert to em (iv) perimeter = 2(length + width) (v) halve perimeter to get (length + width) (vi) find pair of factors of 96 that add to 28

Figure 4. Response map of a typical correct solution to Problem 9.

order to obtain the solution, and three processes need to be applied before the solution can be obtained. Furthermore, there are two instances where the student needs to combine two processes to produce an interim response. In one of these, the student recognised that he needed to find all the possible pairs of factors of 96 in order to apply the area formula (processes (i) and (ii) in Figure 4). In the second instance, the student brought in knowledge of the perimeter formula and then recognised that the given perimeter had to be halved to obtain the sum of the length and width (processes (iv) and (v». Of the 32 students who did not solve this question correctly, 21 gave prestructurallevel responses. One wrote down the first two cues and then stopped (ikonic, unistructural), while another 2 drew a picture of the rectangle and marked in the area and perimeter (ikonic, multistructural). Six students wrote down the area formula and then stopped (concrete symbolic, unistructuraD. The remaining 2 used their knowledge of the area formula and recognised the need to find factors of 96 (concrete symbolic, multistructural), but stopped before they actually carried this out. Three of the incorrect students ignored the first cue given in the problem, and 3 others failed to use at least one of the other cues. The most common type of processing error was in translating the information, with students recognising the need to apply the area and perimeter formulae but not having any idea of how to go about doing this. In addition, three students made interpretation errors, assuming that the rectangle was a square and only finding dimensions for this particular case. Three students made knowledge errors in writing down the area and/or perimeter formulae.


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A difficult problem. Figure 5 shows a solution to Problem 12. The students found this to be very difficult, only 13% obtaining a correct solution. CUES FIND


wire

...

(i)

(ii)

--!~..,._----:....,.~

RESULTS IN1ERIM FINAL

__.... circumference of Earth

distance

between new wire and Earth

radius (6400km) 20m

~~ _ _--jl_~. . . .

new circumference

__--r:~T-__::::4 new radiu~ (vi)

.a)3lSkm

increase (vii)

(3.18 m)

replace equidistant

Key to processes/concepts (i) apply circumference formula (ii) substitute radius into formula (iii) convert 20 m to 0.02 km (iv) add 0.02 km (v) rearrange circumference formula (vi) solve for r (vii) new r - old r

Figure 5. Response map of a typical correct solution to Problem 12. Problem 12 is more difficult than Problem 9 (see Figure 4) because it involves operations on decimals and Te, whereas the latter problem is concerned with whole numbers. The map in Figure 5 indicates a relational level solution in the concrete symbolic mode involving "an integration of the intermediate responses" (Collis & Watson, 1991, p. 76). There are five cues stated in the problem, and its solution produced three interim results. On three occasions, the student needed to combine two processes to produce an interim response. These were: 1. . knowledge of the circumference formula, and ability to substitute for the radius to be able to calculate the circumference; 2. recognition of the inconsistent units, a11d conversion of 20 metres to kilometres; and 3. ability to rearrange the circumference formula and to find the radius. All of these processes required the application of concepts from her previous expenence. Ahnost half (47%) of the students who were incorrect gave responses that were classified as multistructural in the concrete symbolic mode. They were able to identify that they needed to apply the circumference formula to the Earth's radius in order to find the length of the first wire, but were unable to complete the

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necessary application of this formula. The other students' incorrect responses were evenly divided between unistructural and multistructural in the ikonic mode, where they tried to find the length of the wire without using a formula, and unistructural in the concrete symbolic mode, where they wrote down the circumference formula and went no further. Twenty.,.nine of the" 48 students who could not solve this problem correctly used the first 3 cues given in the problem, but gave up before they had the chance to draw on any further cues. The most common processing errors, made by 25 of the students; were translation errors. For example, a common translation error was to add the 20 metre increase to the 40 228 kilometre circumference without converting to a common unit. Another translation error on this problem was an inability to reverse the circumference formula to find the radius. Ten students made errors resulting from incorrect knowledge about formulae and 13 made arithmetic errors. Table 2 Percentage distribution of SOLO/eve/sin incorrect responses, by problem Problem No.a

Level

PS

IK:U

IKM

CS:U

CS:M

Easy problems 1 (C5-IK:M)

2

2

0

0

0

2 (IK:M)

3

6

5

0

0

3 (C5-IK:M)

3

16

0

0

0

4 (CS-IK:M)

6

15

0

0

0

5 (CS-IK:M)

17

6

0

0

0

12 0 0 3

0 22 10 10

5 2 16b 3

Medium difficulty 6 (CS-IK:M) 7 (CS:M) 8 (CS:M) 9 (CS:R)

14 14 15 36

0 0 6 2

Difficult problems 10 (CS-IK:R)

25

35

0

8

3

11 (CS:R)

8

0

0

13

65 c

12 (CS:R)

0

11

185

11

47

Note. IK: ikonic, CS: concrete symbolic, CS-IK: combinations of concrete symbolic and ikonic; PS: prestructural, U: unistructural, M:" multistructural.

aThe SOLO level used in the majority of correct solutions is shown in brackets. blncIuding 2% CS-IK. CAll CS-IK.

I f

I f

l,


69

Summary of Response Analyses for the Twelve Problems Students' responses to all twelve problems were analysed in the same way as the examples described above. In the following summary, the problems have been grouped into three arbitrarily defined categories: those for which more than twothirds of the students obtained correct solutions (easy), those for which between one-third and two-thirds succeeded (medium), and those for which fewer than one-third of the students were able to obtain correct solutions (difficult).

SOLO level of students' solutions.. Table 2 shows the classifications of the incorrect responses, together with the level of the majority of correct solutions. It is clear that the easiest problems were those which students could solve in the ikonic mode, or a combination of concrete symbolic and ikonic thinking, using simple multistructural responses. Overall, the majority of the correct solutions used multistructural thinking in the concrete symbolic mode; but few of the incorrect .solutions showed this level of thinking. There were four problems on which correct solutions involved relational thinking in the concrete symbolic mode, but incorrect responses were all at lower levels. It can be seen from Table 2 that incorrect solutions were usually at a lower level than successful solutions. On the easy problems, which mostly required multistructural thinking in the concrete symbolic mode supplemented by some ikonic reasoning, most of the incorrect responses were at a prestructural or unistructurallevel in the ikonic mode. The medium problems all required concrete symbolic thinking at the multistructural level or above, but only a few of the incorrect solutions indicated the required level. The more difficult problems required relational level responses in the concrete symbolic mode, but the incorrect responses were multistructural or even unistructural. Use of cues. There does not seem to be any consistent differences in the· number of cues given in the easy, medium, and difficult problems. However, observation of the response analysis maps gives some clear indications of the way in which the students used the cues provided by each problem. Table 3 shows how the cues were used in the incorrect responses. . Relatively few incorrect solutions omitted cues. This only occurred on eight problems, most of them being among the more difficult questions. The most frequently occurring omissions were either when the students gave up before they had needed to use the last cues, or where they overlooked a linking cue which was vital for a correct solution. For example, a common error on Problem 6 was to ignore one of the three ingredients of the concrete mix and operate on only two. Three of the incorrect solutions to Problem 9 ignored either the area or the perimeter, and the most common omission on Problem 11 was to ignore the cue about surface area. On the easy Problem t the few students who obtained the wrong answer ignored the link between the seven-person and the nine-person teams. Use of implicit concepts. In the group of easy problems, only Problem 1discussed more fully in the previous section-draws on implicit concepts from the student's previous experience (the commutative law). Three of the four medium

difficulty problems require the use of at least one implicit concept. The difficult problems each require three or lTIOre implicit concepts to be used.

Processing errors. Table 4 summarises the types of processing errors made in incorrect solutions. The data suggest that the students were reasonably accurate with arithmetic calculations and that their knowledge (except perhaps of percentages) was adequate. Of more serious concern is that, on some questions, quite high proportions of students made translation and interpretation errors. There were no consistent patterns in the types of error made on the easy problems. On the medium difficulty problems, most of the errors were translation errors. The exception was Problem 8, where 13 of the 27 students who obtained incorrect solutions did not know that 12% of 68 meant J~~) of 68-most of them subtracting $12 from $68 instead. On the difficult problems, translation errors again occurred frequently. Interpretation seemed to cause problems for half the students who were incorrect on Problem 11. Interestingly, about one quarter of the students who were incorrect on Problem 12 made arithmetical errors; this may have been due to the fact that it required calculations with decimals.


71

Table 4

Percentage Distribution of Processing Errors in Incorrect Solutions Problem No.

Processing errors Translation

Interpretation

Knowledge

Arithmetic

Other errorsa

Easy problems "1

o

2

0

2

7

7

3

o

4

9

5

o

o o o

3

0

o o

2

7

10

7

o

5

0

5

17

0 2

o

14

2

14

o

Medium difficulty 6

12

5

7

21

22

22

2

5

o

33

26

o

8

o

o o

9

12

5

Difficult problems .

3

9

11

31 17

40

14

2 5

12

43

o

17

22

10

5

aErrors occurring in responses classified as prestructural.

Number of interim results required to find a solution. In most of the easy questions, no interim results are needed. The medium difficulty problems impose a slightly greater demand, with three of them requiring one interim result. The difficult problems can all be regarded as multistep, necessitating three or more interim results. There are also differences in the way the interim results may be obtained. None of the easy problems call for two processes to be combined to obtain an interim result. By contrast, three of the medium difficulty problems require this level of processing for one interim result and two of the difficult problelTIs require it on more than one occasion.

Discussion and Implications The mapping procedure used in this analysis to analyse student teachers' problem solving has provided some valuable insights into the problem-solving processes of preservice teachers on a selection of closed word problems of varying difficulty. The findings suggest that there were certain patterns in the students' solutions. They performed best on questions where they could use unistructural,

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ikonic thinking; where all necessary information 'was given in the problem; and where only one step was required to reach the solution. They performed least well on questions that required relational, concrete symbolic thinking; where it was necessary to employ several concepts from their previous experiences; where several interim results were required; or where more than one process was needed to calculate an interim result. There is support here for the argument presented by Tarmizi and Sweller (1988) that students are more successful in solving problems which do not impose a heavy cognitive load by requiring them to integrate multiple sources of information. While these results are not surprising, there are some clear implications for instructors of methods and content courses for preservice primary teachers. First of all, the outcomes suggest that a large number of students need help to be able to apply higher levels of thinking to difficult problems. In order to improve these skills, we should expose student teachers to and equip them with strategies for solving certain types of problems. In particular, they need to experience problems which do not provide all of the necessary information so they can learn to identify and include concepts from their previous experiences. We need to raise their metacognitive awareness of the strategies and processes they use to interpret and integrate the information stated in the problem as well as the concepts and processes they need to bring to the solution. In fact, it could be useful to use the mapping technique described in this article to show students how they are processing information. If they can be encouraged to map their own responses, this might help them to be more aware of what they are doing. Furthermore, we need to create opportunities for student teachers to discuss, and even brainstorm about, the rules and/or existing knowledge appropriate to use in certain situations, so that they can build up a store of appropriate experiences and procedures to draw upon. They also need to discuss different ways that the information can be processed-for example, whether one strategy is more effective than another in reaching a solution. They need to be given multistep word problems, so they learn how to sort out the data and not give up after the first step. If teachers are not able to solve this type of problem, the learning experiences they offer their children will remain restricted .to the lower-level problems with which the teachers feel more comfortable. If they are presented with tasks which are designed to develop their competence, then it is likely that they will als.o develop the confidence to use these types of tasks in their teaching. For the easy and medium difficulty tasks, the majority of the incorrect responses demonstrated prestructural or unistructural responses; those who .were correct gave multistructural solutions. For the most difficult problems, the students who were correct were able to raise their processing to the relational level; those who continued to operate at multistructurallevel or below could not obtain correct solutions. There was an alarmingly large number of students who were not equipped to give more than a prestructural response, even to the less difficult problems. This may add some support to Tall's (1995) contention that students tend to revert to lower levels of processing when the level required by the task imposes cognitive strain. Clearly there is a need to explore ways in which students can be encouraged to raise their levels of processing to appropriate levels. Biggs and Collis (1982, p. 172)


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suggested using the "plus one" strategy, w~ere the teaching is aimed at one level . above that at which the ~tudents can respond competently. This suggestion is based on the premise that if the teaching is pitched at the same level or lower, the students will not .learn to handle a more complex level; whereas if it is aimed at more than one level above, they will not comprehend. Relevant here is the Vygotskian notion of the zone of proximal development (Jaramillo, 1996)-the distance between actual problem solving ability when the student works alone and the level of potential development when solving problems under the guidance of an expert or a more capable peer. If Tall's suggestion is taken into account, however, there is a potential tension between the need to raise students' processing using the "plus one" strategy and the need .to avoid cognitive strain that might force them to revert to lower levels. This tension may be avoided through the use of scaffolding (Wood, Bruner, & Ross, 1976). Scaffolding occurs when an expert or competent peer helps someone less competent-·not by simplifying the task, but by holding the task constant and simplifying the learner's role through the graduated intervention of the teacher. Scaffolding is activated by teaching just above the current skill and knowledge level of the students. The results of the present study suggest that, when we set routine word problems for our students, we should include those which will challenge them to . respond at relational or more difficult multistructurallevels. However, since many students seem to be working at prestructural and unistructural levels when working alone, the intervention needs to be gradual. It would therefore be appropriate to have a spiral teaching program (Biggs & Collis, 1982), in which the same initial data and cues are used repeatedly in a sequence of problems that builds from less to more complex. Most students seemed to be adept at using all of the information provided by the problem, with relatively few omitting cues. The patterns obtained suggest, however, that many students were not able to use this information effectively. Translation and interpretation errors seemed to contribute to incorrect answers at least as much as lack of knowledge or arithmetic errors. The implication from this finding is that, while it is important to provide students with an appropriate hierarchy of problem structures, this in itself is not sufficient They also need to be taught strategies for interpreting the information given to them in word problems, to recognise the appropriate procedure to use, and to adapt (when necessary) known rules to fit new situations. Encouraging students to use the mapping technique metacognitively should stimulate them to consider the cues given in the problem, to explore the links between these clues, and to monitor the wClyS in which they process the given information.

Conclusion The analysis presented in this paper has contributed towards understanding the types of routine word prob~em-solving tasks which should be included in a teacher education program and the processing skills that student teachers need to be guided to develop. This project did not investigate whether individual students used consistent levels of thinking across problems or whether they raised or even lowered their levels with more difficult problems, and this is one area that requires

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further investigation. Also, there is still a need- to investigate. other types Of problems, including those that are non-routine and open-ended. Further research should explore whether it is possible to reduce the impact of cognitive strain when problem solving if students are exposed to a teaching program which helps them to locate the source of their processing errors and to overcome them.

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Author Margaret Taplin, The Open University of Hong Kong, 30 Good Shepherd Street, Homantin, Kowloon, Hong Kong. E-mail: .