Mental computation and conceptual understanding

Learning and Instruction 10 (2000) 221–247 www.elsevier.com/locate/learninstruc

Mental computation and conceptual understanding Anke W. Blo¨te *, Anton S. Klein, Meindert Beishuizen Section of Developmental and Educational Psychology, Leiden University, Wassenaarseweg 52, 2333 AK Leiden, The Netherlands

Abstract The goal of this study was to assess the strategic flexibility of students in mental arithmetic up to the number 100. Sixty Dutch second-graders who took part in an experimental ‘realistic arithmetic’ program participated in the study. Results showed that students’ preference for certain mathematical procedures depended on the number characteristics of the problems. This indicates that the students had a good conceptual understanding of numbers and procedures. Their actual use of these procedures, however, was somewhat limited. Most problems were solved within a sequential structure. A completely different procedure was used for solving subtraction problems that had a very small difference between the two numbers. Furthermore, it was found that a substantial increase in the students’ use of a base-ten procedure occurred after the introduction of this procedure in the mathematics curriculum. Students’ preference for this procedure also increased, although to a lesser extent. Another finding of the study was that students exhibited more flexible strategic behaviour with context problems than with numerical-expression problems.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Arithmetic; Procedural flexibility; Conceptual understanding

1. Introduction In most countries the mathematics curriculum introduces written arithmetic procedures for addition and subtraction problems at quite an early stage. Many children, however, spontaneously use mental computation procedures when solving word problems (Verschaffel & DeCorte, 1990) or everyday contextual problems (Nunes, Schliemann & Carraher, 1993; Thompson, 1994). It is clear that students’ everyday * Corresponding author. E-mail address: [email protected] (A.W. Blo¨te). 0959-4752/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 9 - 4 7 5 2 ( 9 9 ) 0 0 0 2 8 - 6

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experience with numerical problems, and their individual cognitive (Siegler & Jenkins, 1989; Siegler & Shrager, 1984) and affective (see McLeod, 1992) characteristics influence the way they work with numbers, just as learning in the classroom does. The discrepancy between formal and informal computation procedures is currently seen as an impediment to the initial learning and understanding of mathematics (Anghileri, 1995; Hughes, 1986; Liebeck, 1990; Resnick, 1989; Treffers, 1991) as well as a hindrance in the development of number sense and use of flexible number operations at the end of primary school (Hart, 1981; McIntosh, Reys & Reys, 1992; Skemp, 1989; Treffers, 1991; Wijnstra, 1988). The focus of the present study was on flexibility in mental computation procedures with two-digit numbers. The setting was an experimental ‘realistic mathematics’ program for second graders (Klein, Beishuizen & Treffers, 1998) that was designed to evoke and stimulate strategic flexibility skills. We used a systematic approach that is based on recent research in the domain of addition and subtraction up to one hundred (see Beishuizen, 1993; Fuson, 1992; Klein & Beishuizen, 1994) as a guide in analyzing the students’ procedure use and procedural knowledge. The European tradition in mathematics education formed the background for the study. This tradition stresses the importance of mental arithmetic as the knowledge base for mathematical understanding, number sense, and flexible operations with numbers (Deboys & Pitt, 1995; Liebeck, 1990; Neumann, 1995). The emphasis on mental computation strategies in the lower grades, postponing the introduction of placevalue based written arithmetic, is a very important aspect of both the ‘realistic mathematics’ education movement in the Netherlands (Gravemeijer, 1994; Treffers, 1991) and the German mathematics curriculum (see Lorenz & Radatz, 1993; Selter, 1994). Beishuizen and co-workers (e.g. Beishuizen, 1993; Beishuizen, Van Putten & Van Mulken, 1997) obtained empirical evidence that two categories are prevalent among the variety of mental computation procedures that are produced in problems up to one hundred. Authors like Fuson (1992) and Cobb (1995) in the USA, Radatz and Schipper (1988) in Germany, and Thompson (1994) as well as Deboys and Pitt (1995) in the UK, all agree upon the distinction in two types of procedures. The two procedures are, theoretically and empirically, connected with two fundamentally different approaches to number operations. These approaches are referred to as ‘baseten’ and ‘sequential’ (Fuson, 1992; Lawler, 1990). In previous research we have coined the two procedures ‘1010’ and ‘N10’, respectively (Beishuizen, 1993). Table 1 presents a schematic overview of 1010, N10, and other important procedures. In the present study three main categories have been discerned: 1010, N10, and ‘short jump’ procedures. (1) In the 1010 computation procedure numbers are decomposed in tens and ones which are processed separately and then put back together. The 10s (1010 stepwise) is a 1010 procedure that conceptually can be located between the 1010 and the N10 procedure. (2) The N10 computation procedure (and also the variant of N10 ‘N10C’) starts with counting by tens up or down from the first, unsplit number. The A10 (Adding-on) procedure also starts from the first, unsplit number and goes from there to the next ten. (3) The ‘short jump’ procedure refers to bridging the difference in subtraction problems like ‘71⫺69’ in one or two steps instead of subtracting the second number from the first one (Treffers &

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Table 1 Computation procedures for addition and subtraction problems

N10 N10C 10s 1010 A10 Short jump

Addition: 45+39

Subtraction: 65⫺59

45+30=75; 75+9=84 45+40=85; 85⫺1=84 40+30=70; 70+5=75; 75+9=84 40+30=70; 5+9=14; 70+14=84 45+5=50; 50+34=84

65⫺50=15; 15⫺9=6 65⫺60=5; 5+1=6 60⫺50=10; 10+5=15; 15⫺9=6 60⫺50=10; 5⫺9=4(!); 10+4=14 65⫺5=60; 60⫺54=6 65傽59=6 59傽60傽65=1+5=6 65傽60傽59=5+1=6

Veltman, 1994). This procedure is therefore fundamentally different from 1010 and N10 procedures. In the US 1010-like procedures are favoured because they resemble place-value column arithmetic (that also decomposes numbers in tens and ones) which is taught from the early grades of primary school (Baroody, 1987; Fuson, 1992; Resnick, 1986). However, in the Netherlands and Germany maths curricula emphasize the use of N10 in mental arithmetic (Beishuizen, 1993; Radatz & Schipper, 1988). In addition, the majority of modern Dutch maths books are based on ‘realistic mathematics’ education (Freudenthal, 1983; Gravemeijer, 1994; Streefland, 1991). This approach stresses, among other things, the importance of flexibility in problem solving that is in conjunction with a realistic context for arithmetic. The student who has a flexible attitude will use 1010, N10, or any other procedure, depending on the characteristics of the problem. For example, a problem like ‘43+26’ could easily be solved with a 1010-procedure; whereas a problem like ‘54⫺18’ that requires decomposition would preferably be solved with an N10 procedure (cf. Table 3). Using 1010 for this latter type of problem (e.g. 50⫺10=40; take 10 to make 14; 14⫺8=6; 30+6=36) has two clear disadvantages (Wolters, Beishuizen, Broers & Knoppert, 1990): (a) the workload on the memory is relatively high and (b) the chance that errors occur, like 50⫺10=40; 8⫺4=4; 40+4=44, coined the ‘smallerfrom-larger bug’ (Baroody, 1987), is great. It is important to note that flexibility in the present study is defined in relation to the number characteristics of the problems, and not in relation to the type of problem. As far as actually choosing a computation strategy is concerned, we know from other studies that the problem type does make a difference in which strategy is chosen. For example, the change-type problem evokes an addition or subtraction strategy (Riley, Greeno & Heller, 1983; Verschaffel & DeCorte, 1990), while combine problems evoke addition, and compare problems are solved with a variety of strategies (see Klein & Beishuizen, 1994). Therefore, in order to stimulate flexible problem solving strategies it is considered important to use a realistic context (Mayer, 1987; Treffers, 1991).

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1.1. Realistic program design The students who participated in the study were involved in the Realistic Program Design (RPD; Beishuizen, Klein, Bergmans, Leliveld & Torn, 1995), a special arithmetic program for second graders. The instruction program is based on ‘realistic mathematics’ education (Freudenthal, 1983; Streefland, 1991; Gravemeijer, 1994). Teaching in school should not be isolated from the ‘real’ world but instead should relate to the world by using the knowledge children already have from their everyday experiences (cf. Resnick, Bill & Lesgold, 1992). One of the main points of this approach is that it builds on children’s informal strategies by presenting problems in a context. There is a chance that these problems become conventionalized; that the children will not respond anymore to the reality of the situations that are described. Nevertheless, compared to standard problems context problems have a much stronger appeal for ‘modeling’ or ‘mathematization’ (translating real-life situations into mathematical terms; see Greer, 1997) and for that reason are extremely important in teaching mathematics. An important principle of the ‘realistic’ method is that children construct (cf. Cobb, 1995) their own knowledge, rather than just applying the strategies they have been taught in their maths class. Context problems elicit informal knowledge and form a starting-point for ‘mathematization’ (cf. Resnick et al., 1992; Treffers, 1991). The ‘empty number line’ is used as a mental model of addition and subtraction problems up to 100 (Klein et al., 1998; Treffers, 1991). This number line is seen as a more challenging and open model than for instance the hundredsquare or arithmetic blocks. It promotes sequential procedures like jumping by tens from the first number (and as a consequence N10, N10C and A10; and not 1010) and is supposed to stimulate arithmetic processes on a higher mental level than on a procedural level (cf. Beishuizen, 1993; Gravemeijer, 1994): The relations between numbers and procedures are based on a visual representation which should facilitate students’ conceptual understanding. An additional advantage of students using the number line is that not only their answers to a problem are known but also their computation procedures.1 The RPD approach requires a lot of classroom discussion about the different procedures. About one-third of the time that is available for mathematics classes is spent on discussion. The discussion is facilitated by the fact that students’ computation steps are written down on the number line (or in some other way later on in the program). In this way students do not only keep track of their own strategies, they also have an instrument to show others how they solve the problems. Moreover, it facilitates the discussion about which procedure is the best one to solve a certain problem. When introducing a new procedure, teachers are building on examples of this new procedure as given by students (see below student Youri). In case the procedure has not yet been mentioned in the classroom teachers do not ‘teach’ it. They will use specific context problems (e.g. involving money with the 1010 procedure)

1 The term ‘procedure’ is specifically used here to indicate a numerical strategy used to solve arithmetic problems.

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to elicit it. Procedures like the earlier mentioned N10, N10C, A10 and the ‘short jump’ procedure are presented after students have been used to working with the ‘empty number line’. The 1010 procedure is introduced by the teacher later in the second semester, between March and June, because the number line is compatible with the N10(-like) and ‘short jump’ procedures, but not with the 1010 procedure (see Table 2). Furthermore, the students are told to use 1010 exclusively on addition problems, because it can lead to errors in some kinds of subtraction problems (see above). In the second semester rather abstract notation forms like ‘arrow schemes’ and ‘computation steps’ are taught. Both the number line and the latter two notation forms give a clear indication of students’ conceptual structures (see Fuson et al., 1997). Finally, it is important to note that right from the beginning of the program students are asked to solve problems in a variety of ways. For example, Fig. 1 shows how a student called Youri solved two problems during a clinical interview in October. He was asked to solve each problem in two different ways and after finishing a problem to draw a flag next to the procedure he thought

Table 2 Time schedule for the realistic program design Tests.......................................................................................................................September Number positioning on bead string as introduction to semi-structured number line up to 20

Sums ⬍20: 7+7, 8+7; 14⫺6, 11⫺9 Tests.............................................................................................................................October Number positioning on bead string as introduction to semi-structured number line up to 100

Introduction of the empty numberline practicing of “10-jumps”: +10, 20, 30 and ⫺10, 20, 30 Sums ⬍100: 74+8, 93⫺9; 45+32, 48+36, 51⫺49 Context problems as starting point to discuss different procedures like N10, N10C, short jump Tests..............................................................................................................................January Sums ⬍100: 85⫺32, 85⫺39; 81⫺79, 81⫺19 (subtraction of two digit numbers with carrying) Context problems as starting point to discuss procedures like N10, N10C, A10, shortjump Tests..............................................................................................................................March ‘Money’ context to discuss 1010 procedure for addition problems: 33+33, 38+35 Labels for different procedures Tests..................................................................................................................................June

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Fig. 1. Two problems each solved in two different ways. The flag indicates which procedure the student preferred.

was the best one of the two. In this way he chose the subtraction procedure as the best one to solve ‘12⫺3’; and the ‘short jump’ to solve ‘11⫺9’. When the interviewer asked him how he had come upon this latter procedure Youri answered that he had invented it by himself and that this happened long before his teacher introduced the short jump procedure in the classroom. Youri also said that his teacher used to call the short jump procedure ‘the Youri-way’ of solving a problem. He was very proud of this. Furthermore, he said he often explained ‘the Youri-way’ to his classmates, adding that ‘the Youri-way is not always handy’. At the end of the curriculum verbal labels are introduced to differentiate procedures. Fig. 2 shows a worksheet with pictures of children who solve certain problems, each in a different way. The students are asked how they would solve the other problems presented on the paper. They have to write down the name of the procedure they would use. After that, they are sometimes required to actually solve the problem and at other times they do not have to solve the problem. To their teacher’s surprise, most of the students learned to use these labels in an adequate way. 1.2. Conceptual understanding and strategy choice In the former paragraph we have sketched a cognitive approach to teaching arithmetic. In our view, like that of many others (e.g. Carpenter, Moser & Romberg, 1982 (pp. 9–24); Pressley & McCormick, 1995; Resnick, 1989; Resnick et al., 1992) it is important that students generate their own computation strategies instead of applying fixed strategies that they have learned in their maths class. At the time of this research project very little was known about how this new approach of teaching mathematics influences the way students learn. As DeCorte, Verschaffel and Greer (1996) stated ‘.. there is a strong need for additional theoretical and empirical work

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Fig. 2.

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Example of a worksheet. G: N10; S: 1010; SPV: N10C; 傽: short jump.

aiming at a better understanding and fine-grained analysis of the acquisition processes that this type of learning environment elicits in students’ (p. 537). Why do students choose certain computation strategies above others? How is that related to their teaching environment? What role does knowledge or the conceptual understanding of numbers and strategies play? As stated above, the focus of the present article was on procedural flexibility in relation to number characteristics. As far as the theoretical background of our research is concerned we refer to Hiebert and Wearne (1996). These authors have sketched a theoretical framework of mathematical understanding; in relation both to the creation of new strategies and the adoption of strategies that are used by others. Understanding was defined as the result of building connections or relations between representations of mathematical ideas; ideas concerning quantities, the ways to decompose quantities and how to regroup/recombine them. If an elaborate structure of connections already exists in the mental representations of students, they will then be capable of responding to problems with a greater degree of flexibility. The structure enables the student to relate one piece of information to

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many other pieces or combinations of pieces, offering the student the possibility to respond in a variety of ways. As far as selecting a strategy is concerned, the students understand that the efficiency of a particular computation procedure is related to the number characteristics of the problem to be solved. For example, some problems are easily solved using N10, whereas with other problems the N10C procedure seems much more efficient. Logically, the easiest way to solve a problem like ‘72⫺19’ is to first subtract 20 then add 1. In order to choose this strategy for this particular problem, the student needs a clear understanding of the characteristics of the two numbers in relation to other numbers and in relation to the type of problem (which kind of problem, numerical expression or context problem; addition or subtraction). In the case of the afore-mentioned problem, the knowledgeable student realizes that 19 is close to 20; that subtracting 20 is easier than subtracting 19; and, that after subtracting 20 1 should be added to the answer. Without this kind of knowledge of numbers and their relations to strategies it seems very unlikely that students will operate with numbers in a flexible way. The relevance of these theoretical aspects of the realistic approach to mathematics education is clear. It is expected that by using the number line and other models, by creating new strategies themselves, by discussing these new strategies in the classroom with their peers, and by trying the new strategies out on new problems, RPD students will be able to construct extensive mental structures. Consequently, they are then expected to show a good conceptual understanding of numbers and strategies. They should become flexible problem solvers who are able to choose their strategy in relation to the characteristics of the problem. In another study, the positive effect of the RPD on students’ strategic flexibility has indeed been established (Klein, 1998; Klein et al., 1998). However, the literature on strategic behaviour shows that students often do not use their strategic knowledge to the fullest (Garcia & Pintrich, 1994). Many students just guess when reading a difficult word instead of sounding out the letters, or, in a simple addition task keep counting their fingers, although the correct answer could be retrieved from memory (Siegler & Campbell, 1989). Why this discrepancy occurs between strategic knowledge and the implementation of that knowledge in a specific task situation is a very important question. Some authors link the ‘production deficiency’ (Flavell, 1970) to the development of children’s metacognitive abilities (Borkowski & Kurtz, 1987; Campione & Brown, 1977; Flavell & Wellman, 1977). Not only do children need to know when and how a certain strategy can be implemented (Borkowski, 1985), but they also need to recognize the value of the strategy (Kurtz & Weinert, 1989) and the relative usefulness compared to other strategies (Borkowski & Kurtz, 1987; Fabricius & Hagen, 1984). Students also have to manage how much effort they put into a strategy (Pintrich & De Groot, 1990). Effort management and motivational variables have been found to play a role, next to cognitive variables, in strategy use (Ames & Archer, 1988; Garcia & Pintrich, 1994; Meece, Blumenfeld & Hoyle, 1988; Pintrich & De Groot, 1990). Moreover, an alternative view in the study of cognition and learning places a strong emphasis on the kind of setting in which a child performs. Classroom context is considered to be a

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very important factor in influencing students’ strategic behaviour (Garner, 1990; Resnick et al., 1992). In summary, the RPD teaching environment stimulates students to form various conceptual structures for numbers, the relations between numbers, and the relations between numbers and procedures. At first, by using the number line, a sequential structure (relating to N10 and jump procedures) is constructed. Later on, the conceptual structure is extended to base-ten relations. However, it is important to note that the teaching environment and the student’s conceptual understanding together are necessary, but not sufficient conditions for students’ actual strategic flexibility in problem solving. 1.3. Questions asked in the study The aim of the present study was to investigate the development of RPD students’ procedural flexibility in relation to their conceptual understanding. We intended to perform an educational study of the variables that are important in learning mathematics in a realistic context. We looked at students’ spontaneous use of procedures; we ascertained whether they chose their computation procedures in relation to problem type and the specific number characteristics of the different problems. If the use of a procedure is determined by cognitive as well as metacognitive, affective, and teaching environment variables, a lack of flexibility does not necessarily indicate that a student has low conceptual understanding. Therefore, in order to get a clearer picture of students’ conceptual understanding, we also investigated students’ responses in a ‘procedure-valuing’ (or procedure-preference) task, in which students were asked which procedure from several alternatives presented to them was the best one, in their opinion, to solve a particular problem. Problems were chosen in such a way that each problem, depending on its type and number characteristics, could elicit a different computation procedure. In order to gain insight in students’ flexibility, we studied the effect of number characteristics of problems on the students’ use of and preference for certain procedures. This was done for numericalexpression (addition and subtraction) problems as well as context problems. In line with the programs rationale we expected students to be more flexible in solving context problems than in solving numerical problems. Furthermore, we expected that students would initially use the N10-like procedures and ‘short jump’ procedure. These procedures were compatible with the number line that was introduced at the start of the program. Later on, after the introduction of the 1010 procedure, we expected students to start preferring and using this procedure for addition. The following five questions have been asked: (1) Do students choose their computation procedures in relation to the number characteristics of the problems? (2) Do students think certain procedures are better than others to solve a particular problem, depending on the number characteristics of that problem? (3) Is the selection of a strategy (concerning use and preference) related to the problem type (i.e. addition or subtraction)? (4) Is the selection of a strategy related to the presentation of the problem (i.e. numerical-expression or context problem)? (5) Does the students’ selection of a procedure relate over time to what is practiced in the RPD program?

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2. Method 2.1. Subjects The subjects used in the study were second-grade students from two regular elementary schools that are situated in two middle-sized Dutch cities. The children in the first school were from lower-middle and lower class families. The children in the second school came from upper-middle and middle class families. The students’ mathematical skills were rated as average. The first school had one class in the study and the second school had two classes. Both schools were involved in the Realistic Program Design for teaching mathematics. Longitudinal data (four measurements over four months) were collected. Sixty students (out of the 80 students in the three classrooms) participated on all four testing occasions. These students were selected for use in the study. The sample consisted of 21 girls and 38 boys. The gender of one of the children is not known. Most of the children were 7 or 8 years of age. 2.2. The program In the research project both the instructional processes and students’ learning process were closely monitored. Teachers and students used the RPD materials instead of their regular arithmetic textbooks and teachers’ guides. In the RPD teacher’s guide, the instructions for the teacher were written clearly and in detail for each lesson. The lesson started with a group discussion, for example about a context problem; or with group exercises, for example making 10-jumps starting from 31. After that the students worked individually on the worksheets. The teachers used detailed instructions, which were given in the teacher’s guide, for all of the group discussions, the exercises, and the individual worksheets. Every fortnight one of the researchers had a meeting with the teachers to discuss their experiences with the program. 2.3. Instruments The use of the number line was validated several times during the school year by interviewing a representative sample of the students. The students were asked to solve problems by both thinking aloud and using a number line. Later on in the program, the students wrote down their computation steps on their sheet of paper next to the problem. They were free to write down either a number line, or an arrow scheme, or just the computation steps. It was found that the students’ oral and written computation steps were very similar. It was at that time decided to use the paper and pencil tests in the present study. The Arithmetic Scrap Paper Test was designed to measure flexibility in using strategies. In addition, clinical interviews were held. The Procedure Valuing List was used to measure the conceptual understanding of the students. The Arithmetic Scrap Paper Test (ASPT) (Klein & Beishuizen, 1995) is a paper and pencil test containing three types of problems, i.e. (a) four addition problems in

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numerical-expression format, (b) five subtraction problems in numerical-expression format, and (c) five context problems of the ‘change type’ that elicit a subtraction operation (see Table 3). As far as the number characteristics of the problems were concerned, Items 1–4 were similar for all three types of problems and Item 5 was similar for two types of problems (i.e. subtraction and context problems). The problems were chosen in such a way that, ‘logically’, Items 1–5 would elicit N10, N10C, A10, 1010, and ‘short jump’, respectively. The 14 problems of the test were placed in a random order. The pupils were asked to write down their computation steps next to the answer. This was not difficult for them, because they were used to writing them down during their daily maths classes. The format of the ASPT closely matched that of their daily worksheets. Responses were scored by trained graduate students according to the categories in Table 1; except for the 10s procedure that was placed in the 1010 category. Responses that could not be categorized into this scheme were placed in the category ‘else’. The Procedure Valuing List (PVL) was a revised version of the Strategy Valuing List (Blo¨te, 1993). The current version contained the same 14 problems as the ASPT. For each item a number of six or seven possible computation procedures were presented in a multiple-choice format. They related to the 1010-, N10-, N10C-, and A10-procedure; and for subtraction and context problems also to the ‘short jump’ procedure. N10 and 1010 were both represented in two alternative forms: the first one handles the tens first and the second one handles the units first (see Fig. 3). The items of the list were placed in a random order, as were the response categories of each item. The students were asked to choose one alternative: the one they thought was the best one to solve the problem. If the procedure they preferred was not among the alternatives they were given, they could write the preferred procedure down. If possible, these written responses were later categorized according to the alternatives on the list. Interviews were held with a sample of 40 children—students who were good at doing arithmetic as well as poor students—taken from the three classrooms. One part of the interview was that a child had to solve four pairs of problems orally. The problems were written down on a sheet of paper and the student was asked to state the computation steps needed to solve the problem. The two problems in each pair were the same type (subtraction) and had analogous number characteristics. After a Table 3 Items of the ASPT and PVL and expected procedures Item number/exp. procedure

Addition

Subtraction

Context

1. 2. 3. 4. 5.

57+36= 54+39= 69+23= 42+43=

75⫺36= 84⫺29= 61⫺23= 67⫺33= 71⫺69=

64⫺26= 73⫺29= 81⫺33= 87⫺44= 81⫺79=

N10 N10C A10 1010 Short jump

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Fig. 3.

Example of an item of the Procedure Valuing List.

student had solved the first problem, he/she then was required to solve the second problem in a different way (using a different procedure).

2.4. Procedure

The subjects were tested twice in March and twice in June, each time with a oneweek interval. The ASPT was administered on the first and on the third occasion; the PVL on all four occasions. The one-week interval retesting with the PVL was considered necessary because of the multiple-choice format of this instrument; possibly resulting in a great number of random responses. On Occasions 1 and 3 both tests were administered. The students took the ASPT on one day and the PVL on the following day. After the ASPT had been finished, the students were able to practice filling out a multiple-choice list on procedure valuing. The interviews took place on Occasions 1 and 3. However, the interview was not on the same day as the paper and pencil tests.

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3. Results and discussion 3.1. Testing for significance First, the stability of procedure valuing (PVL) over a one-week period was tested. Separate analyses were performed for each procedure, N10, N10C, and so on, using 0/1 scores: 1 if the procedure concerned had been chosen and 0 if a different procedure had been chosen. For each of the procedures two analyses were performed, one for Occasions 1 and 2 (March), and one for Occasions 3 and 4 (June). MANOVAs with a repeated-measures design were used (with occasion and item as within-subjects factors) to test the effect of occasion and the Occasion×Item interaction effect across all 14 items. None of the procedures had a significant main or interaction effect in either month. It was then decided to combine the scores over Occasions 1 and 2 and those over Occasions 3 and 4. In this way, two occasions (one in March and one in June) were created for both the ASPT and the PVL. The frequencies of the different procedures were calculated for each item of the ASPT and the PVL on both occasions (see Figs. 4–6). Visual inspection of these figures suggested that large differences existed between procedure use and procedure valuing. The variation in valuing seemed to be much larger than that in usage. Furthermore, the figures show possible occasion and item effects. We tested these differences using a MANOVA repeated-measures design with procedure as dependent variable and list (ASPT vs PVL), occasion (1–2) and item (1–4 for addition problems; or 1–5 for subtraction and context problems) as within-subjects factors. Separate analyses were performed for each procedure. In interpreting the results one has to keep in mind that the scores of the different procedures were not independent from each other and the respective MANOVAs on subsequent procedures were not independent tests. Addition problems. As far as the addition problems were concerned N10 was the dominant procedure used in March (see Fig. 4). In June, after the introduction of the 1010 procedure by the teacher, the students used both the 1010 and the N10 procedure. In March nearly 20% of the responses to Item 4 of the ASPT did not fall in any of the four categories of the study. (For example, some children used a vertical procedure in relation to 42+43, like 4+4=8, 2+3=5 →85.) As far as procedure valuing was concerned N10, 1010, A10, and N10C were chosen on both occasions; N10C especially in relation to Item 2. The procedure A10 was not frequently chosen. N10 was the only category with sufficiently large variance to test the difference between procedure use and procedure valuing across both occasions. The difference proved to be significant (F (1,48)=8.22, P⬍0.01). The other two main effects were also significant indicating that the frequency of N10 changed over time (F (1,48)=34.35, P⬍0.001) and was dependent on the number characteristics of the problems (F (3,46)=9.63, P⬍0.001). However, the interaction effect for List×Occasion was also significant (F (1,48)=17.65, P⬍0.001) indicating that N10 valuing and N10 use did not change from March to June to the same extent. No differences between use and valuing were found in their relation to number characteristics (List×Item interaction is not significant).

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Fig. 4. Procedure use and valuing in relation to addition problems; proportions of total number of procedures used/preferred.

We could not test this because of the near-zero variance in the 1010 usage data; we could however deduce that there was a significant increase in 1010 use from the results on N10 together with Fig. 4. Furthermore, we could test the effect of occasion on the 1010-valuing data. The differences in 1010 valuing between March and June proved significant (F (1,59)=4.54, P⬍0.05), while the Occasion×Item interaction was not significant. Thus, we concluded that the overall valuing of 1010 to solve addition problems increased. We also tested the difference between 1010 use and 1010 valuing for Occasion 2 only. The difference was significant (F (1,50)=7.16, P=0.01), and also depended on problem characteristics (List X Item interaction: F (3,48)=3.75, P⬍0.05). Summarizing the results it was concluded that although students did show flexibility in their valuing of computation procedures for addition problems, they did not act on it in their use of strategies. Students used N10 on Occasion 1, and 1010 or N10 on Occasion 2. However, on both occasions they valued N10, 1010, or N10C depending on the item characteristics of the problems (item effect for valuing N10, F (3,57)=6.92, P⬍0.001; item effect for valuing 1010, F (3,57)=6.08, P=0.001; item

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Fig. 5. Procedure use and valuing in relation to subtraction problems; proportions of total number of procedures used/preferred.

effect for valuing N10C, F (3,57)=5.12, P⬍0.01). In addition, the overall use as well as valuing of 1010 increased from March to June. However, they did not increase to the same extent. Possibly, the introduction of the 1010 procedure for addition problems between March and June had some influence on the use of certain strategies by students, but it did not change their conceptual structures very much. Subtraction Problems. On both occasions N10 was the dominant procedure for solving Items 1–4 of the subtraction problems (see Fig. 5). In solving Item 5, however, the ‘short jump’ was used more often. And, in relation to Item 2, some students used the N10C procedure. With respect to procedure valuing the responses were more diverse, although N10 was the largest category with respect to the first four items, and ‘short jump’ with respect to the fifth item (see Fig. 5). The difference in frequency between students’ valuing of the N10 procedure and their use of it was significant (F (1,49)=41.38, P⬍0.001). The effect of item characteristics on the use and valuing of this procedure was also significant (F (4,46)=33.32, P⬍0.001). However, the item effect on the use of N10 was different from that on

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Fig. 6. Procedure use and valuing in relation to context problems; proportions of total number of procedures used/preferred.

the valuing of N10 (interaction List×Item: F (4,46)=13.13, P⬍0.001). Finally, it was found that use and valuing of this strategy in June were not different from those in March. With respect to the other categories only the PVL had sufficiently high variance for further analyses. Valuing of 1010, N10C, A10, and ‘short jump’ all depended on item characteristics (F (4,56)=3.24, P⬍0.05, F (4,56)=3.22, P⬍0.05, F (4,56)=4.27, P⬍0.01, and F (4,56)=8.59, P⬍0.001, respectively). The effect of occasion and the Item×Occasion interaction effect were not significant. For Item 5 only we tested the difference between use and valuing of the ‘short jump’; the difference proved to be significant (F (1,53)=30.15, P⬍0.001). In short, students mainly used N10 to solve four out of five subtraction problems. A relatively very small number used N10C in relation to Item 2. To solve the fifth problem they mainly used the ‘short jump’ strategy. They also indicated that in their opinion this was a good strategy for the problem concerned. Next to N10 and ‘short jump’, students valued 1010, A10, or N10C as good strategies. And these choices

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were related to item characteristics, indicating that students’ variability in valuing was based on conceptual understanding. Context problems. Fig. 6 shows that procedure use in relation to context problems, like in subtraction problems, was more or less restricted to N10 and ‘short jump’. Procedure valuing, again, was much more diverse. The figure also shows that in relation to Item 2 the N10C procedure was chosen rather frequently. As far as N10 was concerned, the difference between the use and valuing of this procedure was significant (F (1,55)=77.58, P⬍0.001). N10 selection depended on item characteristics (F (4,52)=72.82, P⬍0.001), however, not in the same way for strategy valuing as for strategy use (List×Item (F (4,52)=16.78, P⬍0.001). There were no differences in N10 selection between March and June. With regard to the ‘short jump’ procedure in relation to Item 5 differences were found between use and valuing (F (1,55)=24.02, P⬍0.001), and between the measurements of March and June (F (1,55)=9.39, P⬍0.01). The ‘short jump’ was more often used in the ASPT than chosen as the best procedure in the PVL. The frequency of both use and valuing of this procedure decreased from March to June. With respect to procedure valuing (see Fig. 6) the procedures A10, 1010, N10C, and ‘short jump’ all had significant item effects (F (4,56)=3.64, P=0.01, F (4,56)=6.44, P⬍0.001, (F (4,56)=12.86, P⬍0.001, and F (4,56)=15.63, P⬍0.001, respectively), indicating that children understood the relationships between the number characteristics of the items and the different computation procedures. N10C had a significant Occasion×Item interaction effect (F (4,56)=3.09, P⬍0.05): the changes in N10C valuing from March to June mainly concerned Item 2 (see Fig. 6). In summary, the results for context problems were somewhat similar to those for subtraction problems. However, students were more flexible in their preferences with context problems. A difference with the findings on subtraction problems was that for context problems some occasion effects were found. For example, the valuing of N10C increased from March to June, and the use and valuing of the ‘short jump’ decreased. 3.2. Estimating variance components Next, we calculated the explained variance for the respective procedures in each the ASPT and the PVL. In this way we were able to directly compare the importance of all the different sources of variability in the data, also those on a within-person level. In the former section the significance of some effects was tested; in this section we describe the relative size of all effects. Using the G-theory method described in Shavelson and Webb (1991) the variance components of all sources of variability were estimated (see Tables 4 and 5). These sources were: person, occasion, item, Person×Occasion, Person×Item, Occasion×Item interactions, and the residual (three-way interactions and error). There were two occasions for procedure use and four occasions for procedure valuing. By using the PVL-data of all four occasions we were able to determine the explained variance components of single measurements. First, ANOVAs were performed for each category with non-zero variance. The mean squares of the different

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Table 4 Proportion of total variance; procedure usea

Addition

Subtraction

Context

a

N10 N10C 1010 N10 N10C Short jump N10 N10C Short jump

Person

Occasion Item

Pxo

Pxi

Oxi

Errora

0.00 0.00 0.00 0.00 0.05 0.00 0.00 0.03 0.00

0.28 0.00 0.42 0.00 0.00 0.00 0.00 0.00 0.00

0.34 0.00 0.38 0.04 0.03 0.00 0.10 0.11 0.00

0.00 0.57 0.00 0.16 0.45 0.15 0.05 0.27 0.06

0.00 0.02 0.00 0.00 0.01 0.00 0.01 0.01 0.01

0.36 0.38 0.21 0.20 0.39 0.10 0.25 0.54 0.14

0.02 0.02 0.00 0.59 0.06 0.75 0.58 0.04 0.79

The error term includes three-way interactions.

Table 5 Proportion of total variance; procedure valuinga

Addition

Subtraction

Context

a

N10 N10C 1010 A10 N10 N10C 1010 A10 Short jump N10 N10C 1010 A10 Short jump

Person

Occasion Item

Pxo

Pxi

Oxi

Errora

0.34 0.34 0.17 0.31 0.26 0.34 0.24 0.37 0.02 0.21 0.20 0.14 0.35 0.05

0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.10 0.07 0.20 0.02 0.09 0.04 0.19 0.03 0.06 0.09 0.03 0.18 0.05 0.02

0.07 0.10 0.06 0.00 0.15 0.13 0.02 0.05 0.24 0.18 0.19 0.08 0.09 0.22

0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.00 0.01 0.01

0.48 0.45 0.53 0.67 0.44 0.46 0.54 0.53 0.54 0.38 0.45 0.56 0.47 0.49

0.01 0.04 0.02 0.00 0.07 0.02 0.01 0.02 0.15 0.12 0.10 0.03 0.01 0.20

The error term includes three-way interactions.

sources in the analyses were used to calculate estimated variance components and percentages of total variance accounted for. Tables 4 and 5 show (as could be expected from the MANOVA results described in the former section) large differences between procedure use and procedure valuing in the relative importance of the different sources of variance. More specifically, the following conclusions were drawn: (a) Person was an important source of variance in procedure valuing, but not in procedure use. Some students valued, regardless of problem characteristics, the N10 procedure most, whereas others had a preference for either the N10C, 1010, or A10. It is remarkable, then, that students did not differ in their overall use of these procedures. (b) With regard to the ‘short jump’ procedure Item was an important source of variance, not only in procedure use but also in procedure valuing. In

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addition, the Person×Item interaction for procedure valuing was relatively high. This shows that some students valued this procedure in relation to Item 5 and others did not, whereas most of them used this procedure with the item concerned. (c) The Person×Item interaction was also relatively large with respect to N10C usage and with N10C-, and N10-valuing. That is, the relative standing of students with respect to these variables depended on item characteristics. (d) Item was not an important source of variance in addition problems (although the respective item effects for valuing N10, 1010, and N10C tested with a within-subjects design were significant; see the former section2). In subtraction and context problems, however, both use and valuing of N10 (and ‘short jump’; see point (b) depended on item characteristics. This was not the case for 1010-, N10C-, and A10-valuing, although the effect proved significant in the within-subjects analyses (see the former section). (e) Occasion played an important role with respect to procedure use in addition problems: the use of the N10 and 1010 procedures differed between March and June. As we have noted in the former section, a large number of students who used the N10 procedure in March, used a 1010-procedure in June. (f) The relative error component was considerably larger in procedure valuing than in procedure use. This is not surprising as valuing was measured with a multiple-choice list and use with open-end items. It seems probable that at least some children made their choices on the PVL more or less randomly and that others were not consistent over time in their differential responses to the items. 3.3. Interview data Table 6 shows which combinations of procedures were used when students solved four sets of two analogous problems each. In these oral interviews, like in the paper and pencil tests students mainly used the N10 and the ‘short jump’. As stated in the Introduction, N10 and ‘short jump’ are compatible with working with a number line. Therefore the children had practiced these procedures very frequently in class. In the interviews, a large number of students used N10 two times in a row, even after they had been asked to come up with a different procedure. In this instance, however, most students used two different forms of the N10; in one of them the tens were subtracted first and then the units. In the other instance it happened just the other way around. Table 6 also shows that N10 occurred in combination with all other procedures. The only combination in the table that did not involve N10 is N10C/A10. The frequency of this combination was very low. Visual inspection of the other frequencies showed that the non-N10 procedures each occurred with problems that, logically, would elicit them: N10C with problems in which the last digit of the second number was a nine; A10 with problems in which the last digit of the first number was a two or three; ‘short jump’ with problems in which the difference between the two numbers was very small. Only the 1010-pro-

2 It should be noted that the MANOVAs with repeated-measures design used averaged data over two occasions as input, whereas the variance components reported here are based on single measurements.

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Table 6 Procedure use by individually interviewed pupils for pairs of bare formula sums

N10/N10Ca N10/A10a N10/1010a N10/Short jumpa N10C/A10a N10/N10 Sh. jump/ Sh. jump N10C/ N10C Total a b

72⫺49/83⫺39

62⫺44/72⫺24

84⫺41/65⫺33

82⫺79/41⫺38

March

March

June

March

June

March

June

3 12 – – 1 19 (14) – – 35

3 2 2 1 – 28 (25) – – 36

1 4 1 – 2 27 (21) – – 35

2 1 2 2 – 29 (25) – – 36

2 – – 21 – 6 (3) 6 – 35

– – – 29 – 6 (4) 1 – 36

June

6 13 8 2 – 2 – 1 3 1 18 (13)b 16 (15) – – – 1 35 36

Also in reverse order. In brackets frequency of students who used two different forms of N10.

cedure did not seem related to a certain kind of problem. This can be explained by the fact that the procedure was introduced in the program with additions, whereas the test only covered subtractions. Furthermore, from March to June we saw an increase in N10C and a decrease in A10. It was concluded that most students (a) were able to solve subtractions in at least two different ways; (b) were able to switch between N10 and some other procedures; (c) used N10 as a widely applicable procedure in combination with more specific procedures (like N10C and ‘short jump’) tailored to the characteristics of a given problem; or, (d) used N10 in two different forms.

4. Conclusions With respect to the questions asked in this study we have drawn the following conclusions: 1. Students employed a variety of computation procedures that related to the number characteristics of the problems that were solved. Three different procedures were used: the N10, the ‘short jump’, and the N10C. The N10 was the most frequently used procedure, for items that elicit the using of N10 as well as for items that were expected to elicit the using of different procedures. It was also employed in the interviews along with the more specific procedures. Another procedure that was frequently used by the students was the ‘short jump’. This procedure was a very specific one: only the number characteristics of Item 5 (‘71⫺69’ and ‘81⫺79’ for numerical-expression and context problems, respectively) of the ASPT and Items 7 and 8 (‘82⫺79’ and ‘41⫺38’, respectively) of the interview elicited this procedure. Furthermore, a small group of students used the N10C procedure selectively, that is, when subtracting a number of which the second digit was a nine.

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2. When asked which procedure in their opinion was the best one to solve certain problems the students were rather diverse in their responses. Nevertheless, N10, again, was the most frequently chosen alternative for nearly all of the problems. Item 5 elicited again a rather high number of ‘short jump’ responses. Besides, in the context problems, the preference for N10C was related to item characteristics. That is, in March N10C was highly valued in relation to Item 2; and in June N10C was even the main choice for this item. Students were more flexible in their opinions about the value of procedures than in their actual use of procedures. Although some of the students must have just guessed when filling out the PVL, while others were very consistent in choosing the same procedure in relation to different problems, it is remarkable how well the remaining group chose their procedures in relation to the characteristics of the problems. An example from the PVL illustrates that it was no easy task to correctly choose which procedure related to the given characteristics of the problem (see Fig. 3). 3. Students were less flexible with addition than with subtraction problems. In the case of addition problems no item effect was found at all. Furthermore, in June many students used the 1010 procedure to solve addition problems, whereas, on that same occasion they used the N10 procedure to solve subtraction problems. The fact that they did not use the 1010 procedure to solve certain subtraction problems like ‘67⫺33’, indicates that this could have been due to an instruction effect (see also under point 5). After all, the 1010 is a completely adequate procedure to solve subtraction problems that do not require decomposition. 4. The results on context problems were somewhat different from those on numerical-expression problems. Students were more flexible in their preferences for various computation procedures when the problems presented to them were in a context format. 5. Use as well as valuing of 1010 increased after the introduction of this procedure in the RPD program. In accordance with the instruction of 1010, this increase only concerned addition problems. Surprisingly, the increase in use was much larger than that in valuing. This was explained in part by the fact that in March some students already considered 1010 a good procedure, before that it was introduced by their teacher in the classroom (and in spite of the fact that the number line did not favour the use of it). Another explanation is that some students used 1010 in June to solve the problems, and did so without choosing it as the best procedure in the PVL. It seems that these students only used 1010 because it had been introduced in their maths class, not because they thought it was the most adequate strategy. In general, the group of RPD students showed a high degree of flexibility in their responses to procedure valuing and in their computation procedures in the clinical interviews. Evidently, they developed a good conceptual understanding of the interrelations between numbers and procedures. At the same time, these students did not fully use their knowledge when solving problems in the pencil and paper test. Only a few aspects of the students’ extended conceptual structures came out in their actual strategic behaviour (cf. Fuson et al., 1997). Further research is needed to investigate

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the factors that caused students to use certain aspects of their total knowledge and not others. The results of the present study suggest that at least three factors play a role in the selection of strategies when solving a mathematics problem. The first one is the effect of classroom training (cf. Reys, Reys, Nohda & Emori, 1995). In the case that a training effect did occur, we must assume that the N10 and the ‘short jump’, next to the 1010 procedure later in the semester, got more attention in the classroom than the N10C and the A10 procedure. With the amount of training spent on the latter two procedures, students would have been able, when given the different procedures and problem characteristics, to recognize which procedure was best to solve a problem. They would not have used a procedure like N10C when they were solving a problem on their own. In that case a production deficiency would have occurred. A second possibility is that students used N10 by choice because they knew that with N10, if properly applied, no ‘bugs’ could occur (Beishuizen, 1993). It is an effective method for all kinds of problems and therefore could have become the ‘default’ procedure, even if students were aware of different procedures that were better fitted to the number characteristics of the problems. Increasing practice in solving number problems would then result in routine behaviour. Many authors (Carpenter, Moser & Bebout, 1988; Verschaffel & DeCorte, 1990; Nunes et al., 1993) see this phenomenon as a strong and negative force in traditional mathematics education. Third, from the viewpoint of ‘cognitive economy’, which is an important argument for the choice of a strategy according to authors like Baroody and Ginsburg (1986), a certain procedure would only be chosen above N10 if it was much easier to apply. This is probably what happened in the case of the ‘short jump’ procedure in relation to problems like 81⫺79. Most students will bridge the difference here in one or two short steps or retrieve the answer from memory, which uses less effort than subtraction in the N10-way. The N10C in contrast, is a much more demanding procedure in problems like 84⫺29. Many students will have to think hard whether, in a problem like 84⫺29, the jump ‘⫺30’ has to be compensated by +1 or by ⫺1. This complicated last step might be another explanation for the finding that N10C was more often valued than used. Because the PVL gave the choice of the right N10C procedure, (i.e. +1) it was easier for the students to recognize this procedure in the PVL than to generate it on their own. It is not clear which of these three explanations is the correct one. Probably, we have seen a combined effect of all three factors. With regard to the Realistic Program Design one has to conclude that the students in this curriculum attained a good conceptual understanding of addition and subtraction of numbers up to 100. At the end of the curriculum both base-ten and sequential relations were a part of the conceptual structures of the students. It seems more important that the students were able to obtain the new knowledge about the different procedures than that they used their new knowledge to the fullest when actually solving problems. The problems presented to them in this study were part of their everyday tasks and probably did not require them to be extremely flexible. They did not have to rely on specific procedures to find the correct answer. When confronted with new and more challenging tasks, however, their ability to be flexible will help them to invent new adequate

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procedures or to learn the ones that are taught to them (Hiebert & Wearne, 1996). Further research is needed to find conclusive evidence on this matter. Finally, earlier research (e.g. Riley, Greeno & Heller, 1983; Verschaffel & DeCorte, 1990) has shown that the semantic structure of word problems has an effect on the types of strategies students use. The present study presents empirical evidence that flexibility in strategy use is not restricted to semantic effects of word problems. The number characteristics of problems can also play an important role in which strategy is selected. 4.1. Limitations of the study and suggestions for further research Investigating the development of the mental computation procedures of students in relation to their conceptual understanding, is not a very straightforward task. Our method had its limitations. We will now discuss the most important aspects of these limitations. Measuring students’ flexibility in using procedures was mainly restricted to paper and pencil tasks. The validity of this method for students’ behaviour in their maths class was not a problem. As stated above the ASPT closely matched students’ daily worksheets, and students’ written computation procedures were closely related to their oral responses. However, further research is still needed to evaluate the relation between children’s cognitive processes on the one hand and their behaviour on oral and paper and pencil tasks on the other (e.g. Brown, 1987). All subjects in this study participated in the RPD curriculum. This had some consequences for the questions that were answered in the study. We could not reliably attribute changes in students’ use of and preference for certain procedures to specific characteristics of the RPD program because we lacked a control group. Moreover, the findings of this study cannot be generalized to students in different programs. Further research is still needed. In fact, we recently have already finished an extensive study that did use a control group. Most analyses were done on a group level and changes in children’s cognition were investigated by a comparison of their behaviour in March and in June. In order to gain a more detailed insight in the development of mental computation in relation to conceptual understanding it would be worthwhile to perform a fine-grained study of the process of change in individual children. To this purpose the microgenetic method could be useful (see Siegler & Crowley, 1991), with its large number of measurements in periods that changes occur. 4.2. Educational implications The distinction between conceptual understanding on the one hand and actual strategic behaviour on the other hand appears to be a very informative one (cf. Hiebert & Wearne, 1996). In the present study it was shown, among other things, that the conceptual understanding of students is sometimes far ahead of classroom instruction. For example, the students in our sample were initially taught to use the number line to solve two-digit addition and subtraction problems. The use of the

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number line favours the development of a sequential structure of number concepts. Nevertheless, some students thought a base-ten strategy was preferable in relation to certain problems. This occurred at a time that they had not yet received any instruction in this strategy and did not have any practice in using it in their maths class. In our view both teaching programs and research on arithmetic should take into account the distinction between strategic knowledge and strategic behaviour. It is not only important to know which procedures students actually use (something many teachers still may not know) but also which preferences and ‘tacit knowledge’ they have. Only then can the teacher adequately build on existing knowledge. Furthermore, the main objective should be that the conceptual structures of students are fine-grained and extensive. This will allow them to operate with numbers in a flexible way. Whether or not they actually use this knowledge when solving problems will depend on the situation and their own preferences. For many problems in their maths class, flexibility is not really needed. Students do need to be flexible, however, when they encounter difficult problems for which they have to create new procedures or modify old ones. Finally, it is considered important to use a realistic context in the constructivist approach to mathematics education if it is expected to enhance flexibility in proceduralization. The present study’s finding, that students’ valuing of procedures was more flexible in relation to context than to numerical expression problems, lends support to this viewpoint.

Acknowledgements We are grateful to Kees Van Putten for his critical comments on an earlier version of the manuscript.

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