Script identification: a way to understand how students choose

European Journal of Special Needs Education Vol. 21, No. 3, August 2006, pp. 301–319

Script identification: a way to understand how students choose strategy during mathematization? G. M. van der Aalsvoort*, F. J. H. Harinck and G. Gossé Leiden University, The Netherlands European 10.1080/08856250600810765 REJS_A_181018.sgm 0885-6257 Original Taylor 302006 21 G. [email protected] 00000August M. Van and & Article Francis Journal derAalsvoort (print)/1469-591X Francis 2006 ofLtd Special Needs (online) Education

The process of choosing strategies within mathematization is described in a multi-method study on solving additions and subtractions. The sample consisted of 40 dyads of 3rd-graders and their teacher from ten mainstream primary schools. From each classroom the two students rated as highest and the two rated as lowest were selected to take part in a session in which teacher and students worked together. Thus two research conditions were formed: high rating versus low rating. The teacher was not informed about the selection criterion for the students, and the students were not informed why they were invited to sit with the teacher and do sums. The task involved solving four sums, and it was video taped and analysed in three steps. In the first step, the conditions were compared quantitatively with respect to teacher’s social support, instruction quality, and regulation, and students’ response to social support. The second step involved identification of discourse patterns. The third step implied identification of student scripts following open-ended questions by teachers. The implications of the results for teacher training with respect to mathematics instruction are discussed.

Keywords: Sociocultural theory; Mathematization; Discourse analysis; Student scripts Introduction This study addresses the question as to whether script identification could be a way to understand mathematization with respect to choosing strategy. The main focus of our study was on visualizing the relationship between teacher expectations and behaviour in the classroom. Good and Brophy (1986) state that their studies on teacher evaluation consistently show that teachers behave accordingly in the classroom: they tend to call upon those thought to be better students more frequently, and these *Corresponding author. Centre for the Study of Early Learning Problems, PO Box 9555, 2300 RB Leiden, The Netherlands. Email: [email protected] ISSN 0885–6257 (print)/ISSN 1469–591X (online)/06/030301–19 © 2006 Taylor & Francis DOI: 10.1080/08856250600810765

302 G. M. van der Aalsvoort et al. students also receive more praise and more effective instruction. Several other studies lead to the same general conclusion, namely that there is a strong relationship between teacher expectations and future academic achievement. In research focusing on children as young as 2 years (Van der Aalsvoort, 1997a), at-risk kindergarteners (Smits, 1993) and 7th-graders (Blöte, 1995), it was found that there is a link between adult expectation and actual teacher behaviour during tasks. Teaching is also a cultural activity. It involves beliefs about how and what students should learn. Teachers make decisions about how much structure they will provide students. These decisions relate to perceptions of present ability and the student’s ability to grow as learner (Buzelli, 1996; Baker, 1999; Good & Nichols, 2001). Teachers tend to call upon those thought to be better students more frequently, and these students also receive more praise and more effective instruction (Good & Brophy, 1986; Good & Nichols, 2001). Good and Brophy, however, believe that teacher expectations are not automatically self-fulfilling: ‘To become so, they must be translated into behavior that will communicate expectations to students and will shape their behavior toward expected patterns’ (Good & Brophy, 1986, p. 9). Therefore the study investigated the process of mathematization during a task that involved teacher–student interactions. First, sociocultural theory as the conceptual framework for the study is outlined. Then teaching mathematics is described as an activity that results in processes of mathematization, which include both cognitive and social processes. The framework allows us to answer the question addressed in this study, namely whether identification of student scripts after open-ended questions by their teacher is a way to understand mathematization during additions and subtractions. Sociocultural theory In the investigation presented here the focus is on the dynamics of teacher–student interaction. According to Vygotsky’s sociocultural theory (1978), development of the mind takes place in the course of social experiences. Social experiences involve a mastering of mental tools, such as language, concepts, reading and mathematics. The zone of proximal development is one of the major concepts in this theory. The zone is defined as the distance between a child’s developmental level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance or in collaboration with more capable peers. However, an ideal match between learning and teaching is needed (Kraker, 2000). Ideally, ‘the teacher explains the problem and involves the child in the problem-solving process, giving her or him a chance to give an answer to the questions that arise at every intermediate stage of the search for the solution’ (Grigorenko, 1998, p. 209). In the case of academic domains, such as reading and mathematics, teachers should provide a path from complete dependence when the tool is just introduced, to complete independence when the tool is mastered: ‘By converging in their understanding of the situation at hand the teacher and the child co-construct knowledge’ (Grigorenko, 1998, p. 217). Academic knowledge thus develops socially: partners in

Choosing mathematization strategies

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the interaction explore each other’s roles in the interaction and their participation in communication is reciprocal. Working together as teacher and student in a task is not sufficient, however, to elicit learning progress. There must be a cycle of communicative challenge and inference between teacher and student. This interplay may come about verbally and nonverbally, and is driven by intentional conscious processes of affective guidance and instruction on the part of the teacher. The child will only be actively involved in the task when his or her language, cognitive and social skills are sufficient to engage in and profit from the teacher’s input. Contextualizing According to Van Oers (1998), learning is always based on experiencing a situation (a task), discerning relevant phenomena and relating them to that task. How the task is experienced and interpreted by the student is critical for actions and thus development. This ongoing construction of tasks is called contextualizing, and it is defined as a semiotic process of negotiation of meanings in which both teacher and student articulate their interpretations and negotiations of the activity and the actions to be performed (Van Oers, 1998). The major goal of this study is to clarify how contextualizing proceeds during mathematics. Ongoing discourse during additions and subtractions will be analysed. Therefore we refer to mathematizing in underlining that we investigate contextualization in processes in which mathematical knowledge is co-constructed. Student strategies during mathematics lessons in classrooms have been widely investigated (Elbers & Streefland, 1999; Wood, 1999; Wood et al., 2001). However, these studies have not explored the influence of discourse upon students’ strategy selection. One could expect that the choice of strategy is the result of mathematization. Therefore our study sought to employ a multi-method approach to investigate mathematization on a micro-analytic level in order to identify how students choose strategies during additions and subtractions. Teachers may follow a specific track within social interaction and elicit specific responses from students. We refer to students’ response as scripts. Soodak (1990) defines scripts as: cognitive representations of a prototypic event in which elements such as roles, props, and actions are specified. As there is a relationship between teacher expectation and his/her approach towards students in the classroom, a student in turn may also fall into a routine pattern. The study undertaken thus addressed the question as to whether scripts could be a way to identify processes of mathematization in order to understand how students choose strategies in relationship to teacher expectation. As emergent arithmetic with additions and subtractions is the main content of the study, we will briefly describe the current views on this subject-matter before presenting our results. Emergent arithmetic with additions and subtraction During the past two decades, a large number of studies have appeared concerning the way children solve sums. Siegler and Shrager (1984) and Siegler (2000) reviewed

304 G. M. van der Aalsvoort et al. young children’s problem-solving strategies in counting to 20. They found four strategies: counting finger strategy (lifting fingers, counting them); fingers strategy (lifting fingers, but not counting them); counting strategy (counting aloud or moving lips); and retrieval strategy (mental counting, no visual or audible signs). Children typically use several strategies in combination (Torbeyn et al., 2004). That is, they choose their strategy depending on the kind of problem they encounter when asked to solve a sum. If the problem is simple, the retrieval strategy may suffice, but the child may approach more complicated problems verbally or by counting on fingers. Counting in the early curriculum includes number facts when adding and subtracting up to 100. Discovery of simple patterns and easy structures, like abbreviated counting, is one of the activities that reveal emergent mathematical knowledge. Two main strategies have been identified for solving number facts. In the first strategy (splitting/sequential), one of the numbers is kept intact, and the other number is split and added up (or subtracted) to or from the first one: 65 + 28 = 93; 65 + 20 = 85; 85 + 5 = 90; and 90 + 3 = 93. This strategy is called ‘N10’ for its acronym (Number + 10). In the second strategy (decomposition), numbers are split in tens and units that are handled separately: 65 + 28 = 93; 60 + 20 = 80; 5 + 8 = 13; and 80 + 13 = 93. This decomposition strategy is called by the acronym ‘1010’ (10 = 10) (Beishuizen & Anghileri, 1998). The latter strategy is preferred, as the place value of the digits is an essential mathematical principle on which most methods are based. On other mathematical problems, children also use a variety of strategies. They often select the one they think most appropriate, drop old strategies they think less efficient and improve their skills with other strategies by generalization, abbreviation and mastery. Design The design involved a once-only phase of data collection with 40 students from 3rd grade and their teachers from 10 mainstream primary schools. The teachers involved had evaluated all the students from their class on a questionnaire with 25 items pertaining to classroom behaviour (details of the questionnaire in Van der Aalsvoort, 1997b). Moreover, all students performed a maths test on additions and subtractions to 100. From each classroom the two students rated as highest and the two rated as lowest were selected to take part in a session in which teacher and students worked together. Thus two research conditions were formed: high rating versus low rating. The teacher was not informed about the selection criterion for the students, and the students were not informed why they were invited to sit with the teacher and do sums. The task was performed while teacher and student sat together as a dyad within the classroom, and the class carried out sums on worksheets as part of their daily mathematics assignment. The researcher was present in the classroom and offered support when required so that the teacher could sit with the four students without being interrupted. The research task consisted of four two-digit sums on addition and subtraction (74 + 3; 76 − 60; 53 + 18; 93 − 25) and two three-digit sums (236 + 453; 643 − 332). The order of the sums was the same for all subjects. The teacher was asked to offer assistance when he or she thought that the student would not be able to solve the


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sums independently. Moreover, the teacher was invited to use the instruction that he or she was familiar with including supporting materials. The session with each child was planned for a period of 10 minutes. The session was video taped in such a way that both teacher and student were visible on the tape. Instruments As video tapes are relatively ‘raw’, in that they are not readily categorized or quantified, we could analyse the tapes with different coding schemes (Miles & Huberman, 1995; Stigler & Hiebert, 1997). We designed a three-step procedure to compare the research conditions quantitatively and qualitatively. ●

1st step: rating and categorizing the ongoing task

Each tape was scored by means of four instruments to answer the question whether quality of interaction was higher with students rated as high than those rated as low. The first instrument allowed us to rate teachers’ support during the task. The rating scales on social support, designed by Erickson et al. (1985), were used. The scales included five subscales that could be rated from 1 (poor) to 7 (high): Social support, Autonomy, Structuring and Quality of instruction, and Hostility. Each session was rated and the score could range from 5 to 35. The second instrument measured students’ response to the support. The rating scales on response to social support, designed by Erickson et al. (1985), were applied. The scales included four subscales that could be rated from 1 (poor) to 7 (high): Avoidance, Reliance on help, Persistence and Compliance. Each session was rated and the score could range from 4 to 28. The third instrument aimed at identifying instruction quality. The instruction given for each sum was categorized according to Van Parreren and Carpay’s (1972) outlines for instruction of sums. Four categories allowed assessment of instruction quality by scoring observed (1) or not observed (0): 1. 2. 3. 4.

Orientation: the teacher offers information on the task at hand. Instruction: the teacher explains how the sum is to be solved. Feedback on results: the teacher informs the student whether the sum is correct. Feedback on processes: the teacher offers feedback on the problem-solving strategy used.

For each sum a maximum score of 4 could be obtained. The results of the first four sums were used for statistical analyses. The fourth instrument aimed at identifying the way teachers steered students through the task. We identified direct and indirect regulation during each sum based on Wertsch and Sammarco’s (1985) study. Four categories for regulation were scored: observed (1) or not observed (0): 1. Student reads and solves the sum independently of the teacher. 2. Teacher reads and solves the sum. 3. Student is regulated directly by the teacher (e.g. ‘Read the sum aloud and write down the numbers first’).

306 G. M. van der Aalsvoort et al. 4. Student is regulated indirectly by the teacher (e.g. ‘How would you proceed with this sum?’). For each sum a maximum score of 4 could be obtained. The results of the first four sums were used for statistical analyses. Inter-rater reliability. The researchers rated three sessions together followed by discussing the findings. The opinions were comparable. Then separate ratings were collected in four sessions to compare inter-rater reliability by calculating the Pearson r-correlation coefficient. A mean of 0.80 was found for instruments 1 and 2. Cohen’s kappa was calculated and a mean of 0.90 was found for instruments 3 and 4. ●

2nd step: discourse analysis of sessions

The video tapes were analysed to collect discourse fragments initiated by the teacher with respect to: materials used; questions raised during the sum; supportive activities, e.g. offering help when the student seemed to fail in solving sums independently; and task structure, e.g. learning to apply strategic knowledge by using spatial and numeral representation of 1s and 10s correctly. ●

3rd step: identifying student scripts

The video tapes were employed to identify what students said after a question was raised about use of strategy to clarify roles that the partners in the dyad took. Four video tapes were screened together to define scripts as answers of students after having been posed an open-ended question on being offered a new sum. Scripts could be detected only when students met problems with the sum. Two types were identified. The first type was related to procedure—e.g. where to write down the formula on the worksheet. The second type was related to strategy—e.g. problem-solving algorithms for combining the 10s first (N10 strategy), or start with completing the amount until a 10 number (1010 strategy). The feedback from the teacher after the student’s reply was also categorized to identify patterns. Three types of feedback emerged: neutral: ‘Yes, that is a way to proceed’; positive: the remark is appreciated, e.g. ‘That is a nice way to solve it’; and negative: the remark is neglected, incorrect or the gesture that accompanies the utterance is stopped.

Sample The sample consisted of 40 children from 10 mainstream primary schools, divided equally over the conditions ‘high rating’ and ‘low rating’. An independent samples test was conducted to evaluate whether the research conditions were comparable. The means and standard deviations with respect to age, expressed in number of months (high rating: M = 96.8; SD = 3.65; low rating: M = 101.9; SD = 6.87), were significantly different between the conditions (t (28,965) −2.961, p = 0.006). The means and standard deviations of the number of boys and girls (high rating: 10 boys and 10 girls; low rating: 16 boys and 4 girls) also differed significantly (t (36,253) 2.042, p = 0.048). The results on the maths test, expressed as the proportion of


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correctly solved sums (high rating: M = 0.87; SD = 0.16; low rating: M = 0.77; SD = 0.21), were comparable between the conditions. Procedure The data collection took place as previously planned. During a 10-minute session, an average of four sums was done. The data collected in the first step were used for comparison of research conditions using SPSS. For the second step the video tapes were observed and typical parts of the discourse were selected to describe the findings qualitatively. The final analyses involved students’ scripts: these were identified by observing the video tapes followed by comparison of research conditions using SPSS. Moreover, typical parts of the discourse were selected to describe the findings qualitatively. Results 1st step: rating and categorizing the ongoing task Table 1 contains the means and standard deviations of the ratings on Social Support, Response to support, Instruction quality and Regulation type per condition. Independent samples t-tests were conducted to determine whether significant differences existed between the high- and the low-rated groups. The analyses showed that the groups differed significantly with respect to Response to support (t (38) 4.006, p = 0.000) and Direct regulation (t (37) −2.705, p = 0.010). The students rated as high showed more Response to support from the teacher than the students rated as low. The first step of the analysis confirmed that some elements of teacher behaviour and student behaviour were related to teacher ratings. The relationships found, however, do not clarify the processes during the task. 2nd step: discourse analysis of sessions The second step aimed at revealing how interaction unfolds as students and teachers shape their roles in working together. The teacher intends to communicate the goals Table 1. Means and standard deviations with respect to social support, response to social support, instructional quality and regulation of the condition high rating and low rating High-rating

Social support Response to social support Instruction quality Direct regulation Indirect regulation

Low-rating

M

SD

M

SD

30.7 25.2 6.8 1.9 2.0

5.15 1.64 2.05 1.17 1.56

27.7 22.8 6.3 3.1 1.4

5.67 2.12 2.31 1.60 1.54

308 G. M. van der Aalsvoort et al. of the task and to understand the child’s thinking, and the student tries to grasp the goals of the task and the teacher’s actions (Grigorenko, 1998; Hiebert & Stigler, 2000; Meijer & Turner, 2002). We started with a contrasting case analysis (Miles & Huberman, 1995) to clarify the differences between the research conditions. We selected two typical fragments of transcripts from each condition and we analysed these with respect to the ongoing interaction and problem-solving behaviour. The fragments are presented one after the other, followed by a comparison of the transcripts. The themes presented served as means to identify teacher initiatives with respect to student rating. We expected that the transcripts would reveal how mediating tools, such as writing the sum, reading aloud the sum, and using materials, supported and facilitated learning. Figure 1 (Zoran) represents the high-rating condition. Figure 2 (Yan) represents the low-rating condition. Figure 1 reveals that teacher and student together are responsible for the discourse flow. Zoran seems not to be intimidated by the difficulty of the sum: he may be familiar with it or he may have recognized the underlying strategy needed for its solution. He seems to trust his ability to solve the number problem. This shows right from the start. The teacher permits Zoran to solve the number problem without his assistance. Moreover, the teacher’s questions aim at independent thinking by discussing each step that Zoran takes to foresee the consequences. The teacher takes on the role as a supporter who guides Zoran’s arithmetic progress. Unlike Yan, most students from his grade level can solve this sum easily. He is lacking mathematical ability, and often elicits support from the teacher. The teacher seems to struggle to find a way to present the task easily, but Yan does not appear to profit from the instruction. He starts to solve the sum independently of the teacher, but fails to accomplish the task successfully. The teacher takes over, and solves the sum. There were, however, several opportunities to give informative feedback that Yan could have profited from. The teacher did not take on a role as supporter who guides the student’s progress, and the question remains whether Yan will be able to solve the sum the next time. Summarizing the two examples, the following conclusions emerge with respect to mathematizing: Figure 2. 1. Part of the transcript from Yan Zoran(low-rating (high-rating condition) condition)

Materials used ● ●

Zoran (Figure 1) does not require the use of teaching aids. The material is meant to help Yan (Figure 2), and to explain the strategy employed. He is not allowed to manipulate the material himself. At the beginning of the task, the teacher offers help. This gesture seems to elicit feelings of helplessness in Yan. Then a moment of confusion occurs when he is supposed to understand that the rod must be called a 10-rod when only one rod is lying on the table. This is a typical moment of mathematizing that confuses many students at the start of learning to use specific language for arithmetic materials. Yan responds to the teacher’s pointing gestures incorrectly, but does not receive feedback about the correct answer.

Choosing mathematization strategies Comments

Zoran Teacher

Sum 5: 74 - 27 T allows time to start using a

What will you do first?

strategy F initiates his strategy without

Starts subtracting 7 - 4

waiting for suggestions T structures the task by alerting What sum are you solving? Z 7-4 T structures the task by alerting Is that the sum? Z No So, what sum do you have? 4 - 7 but you can’t do that! T structures by using an open-

Do you know a solution?

ended question. T elicits Z’s independence by inviting Z to explain his strategy Well, going under 0 T structures by using an open-

That will be hard! How would you

ended question. T elicits Z’s

write that up?

independence by inviting Z to explain his strategy Well, minus 3 T supports Z: he expects him to Oh, minus 3. That’s quite difficult come up with the correct

but that is smart thinking!

solution this time

Figure 1.

Part of the transcript from Zoran (high-rating condition)

309

310 G. M. van der Aalsvoort et al.

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311

8 T repeats correct answer and

8, write that down: 53 + 10 + 8,

structures sum into small steps

write down: 53 + 10 + 8

T offers material to visualize

[

the sum. He does not explain

] Look, Y. Firstly, we

the use of the material, and uses

take a 10-rod, and then we add

it himself

.. [

] How much is 53 adding... ? [

] Y, write down

53 + 10 + 8. You can do it. Just write it down! Here? [

! Yes!

T starts to write the numbers

Yes, 53 + 10. ["

down. He manipulates the

!

material. He describes the task

as one that is shared by using

] We begin

we, but Y is not involved

with 53

T manipulates material, and

Y, this must be added [

structures the situation

] Firstly, we add ... 10

[ ] So, how much is this? [ #

] 6

]


T does not offer feedback but

And 3 ones is 60 and 3 is... ?

says the correct answer himself 6 and ... uhm ... 63 T structures task

[ ]... is 63

Figure 2.

Part of the transcript from Yan (low-rating condition)

Questions raised ●

●

Zoran’s teacher asks many open questions that are short and simple. The questions asked are helpful to him. Moreover, Zoran is allowed time to think before answering a question. The fragment involving Yan reveals that the teacher’s verbalizations consist of many rapid, closed questions about the same subject that leave little room for selfdirected answers. The teacher answers the questions himself most of the time. Yan is silent most of the time.

Supportive activities ●

●

The teacher supports Zoran strongly by open questions; he stays close to Zoran’s answers, and acknowledges answers by questioning; and he compliments Zoran at the end of the task. The teacher stresses the feeling of a shared task by using the expression ‘we’ with Yan. In reality, the teacher solves the sum almost alone. The teacher bends towards Yan, taps his arm at the end of the task, and says that the sum is hard, and that it cannot be expected that he would succeed with this sum on his own. The teacher compliments him from time to time.

Task structure ●

●

Zoran can solve the problem, step by step, by asking to write down parts of the final solution. The teacher asks Yan to write down the formula, but Yan does not know how to do this. The teacher fails to notice Yan’s hesitation, and becomes impatient and forceful. This, in turn, makes Yan even more hesitant. The teacher structures the solving process but does so without including the student. Structuring the task directly leaves Yan no room to show his competence.


313

The two protocols show that during the discourse the two teacher–student dyads coconstruct completely different patterns of problem-solving. In the Zoran path, ‘student initiatives’ and ‘experiencing success with sums’ prevail, and in the Yan path, the teacher takes over, leaving the student without opportunities to experience success as problem solver. The second step clarifies remarkable differences in the way teachers approach a student’s problem-solving behaviour while solving sums by means of the way they structure a task, offer support, pose questions and make materials available during the sum. We know from the first step of the analysis undertaken that the differences between the conditions were related to teacher rating. The second step of the analysis has clarified how teacher rating is translated into teacher behaviour such as that displayed during the discourse. At this point, however, we are still unclear with respect to the student’s part in the discourse. ●

Third step: student scripts

The question raised is whether student response offers a way of stepping out of the teacher’s path. Can students initiate mathematizing by showing that they are independent problem-solvers and thus also of teacher expectation? Teacher and student are both participants in the task and by converging in their understanding of the assignment mathematical knowledge is co-constructed. The scripts identified were the answers given by the students after an open-ended question from the teacher about how they would solve the sum. As the sample included 40 students, scripts could be found for a maximum of 160; 55 scripts were produced by 29 of the 40 students, or 73% of the sample. Scripts were detected concerning those sums that required a strategy to solve number facts, such as 53 – 18 and 93 – 25. Table 2 shows the number of scripts, the proportion of strategy types for both research conditions and the feedback type following the students’ reply. Two-way contingency table analyses (Pearson chi-square) were conducted to evaluate whether students in the high-rated condition differed from those in the low-rated condition. The two variables that were significantly related were type of feedback and Table 2. Number of scripts, raw scores and proportions of scripts related to strategy or structuring, and to feedback (neutral; positive; negative) for the conditions high rating and low rating High rating

Low rating

N

%

N

%

Number of scripts Scripts related to strategy

21 8

38

34 20

88

Scripts related to structuring

13

62

14

12

Neutral feedback

5

24

12

35

Positive feedback

7

33

2

6

Negative feedback

9

43

20

59


Comment

Teacher

Sum = 74 - 57

What will you do first?

Dennis

4 T corrects D

No, first take 50 or 7? 50 Yes! 50

T corrects D

No, write the sum first!

Figure 3.

Part of transcript from Dennis (high-rating condition)

research condition (2, N = 55, = 4.43, p = 0.035). Students from the high-rated condition asked more questions about what strategy to use for sums, and they received less negative feedback. This finding is illustrated in Figure 3, presenting part of the discourse with Dennis (high rating) and his teacher, and Figure 4 presents part of the discourse with Jack (low rating) and his teacher that includes a script. The fragments were chosen to show patterns that emerged after questions related to procedure when the student moved away from the teacher’s set of rules. In both examples the teacher’s response is not related to the problem-solving behaviour of the student. In order to allow mathematizing both partners should take part in the discourse. However, in both examples the students were not allowed to enter the discourse and thus co-construction could not take place, resulting in ongoing dependent problem-solving behaviours. The student’s behaviour was not taken into account even when the student became confused (for example, Jack). Figure 4. 3. Part of transcript from Jack Dennis (low-rating (high-rating condition) condition)

Discussion and conclusion This study addresses the question as to whether script identification could be used as a way to understand mathematization with respect to choosing strategies. A threestep multi-method analysis was carried out to answer the question. The first step, a quantitative comparison of conditions, revealed that low-rated students were less responsive to social support during the task and that the teachers directly regulated them more often than their high-rated peers. The second step, a qualitative approach to the discourse during the task, revealed remarkable differences in the way teachers

Choosing mathematization strategies Comments

Teacher

315

Jack

Sum: 49 + 46 How shall we do it? Firstly 40, adding 40 = 80 Write it down, 80, there, so you won’t forget J intends to write the

Writes down 8

ones behind the 8 later on 80! [with emphasis] 80? 80, and then add that is ... I want to continue the sum! T corrects with

Add! Firstly, add the sign, so that you

emphasis

won’t forget!

Figure 4.

Part of transcript from Jack (low-rating condition)

approached students’ problem-solving behaviour by means of the way materials were used, questions raised, support offered and the task was structured. The third and final step of the analysis included both a quantitative and a qualitative analysis and showed that students’ replies after open-ended questions in the low-rating group were followed more often by negative feedback than those from the high-rating group. The findings thus reveal that discourse during sums is related to expectation of teachers on students’ achievement level: high-rated students receive and elicit a different teaching arrangement compared to their low-rated peers. This differentiation is apparent especially when the sums become more difficult. The analysis presented allowed us to look at the discourse through a situative approach (Greeno, 1998) as we focused on the behaviour of the dyad in which the teacher and the student were participants. The results reveal that students can be more effective partners in processes that elicit mathematization if they are allowed to

316 G. M. van der Aalsvoort et al. take part in the meaning-making of the strategies involved in doing sums. The findings, however, also show that taking part should be addressed positively, so that the student is encouraged to stay involved as a participant who adds to the discourse through negotiation, or (dis)agreement (Wells, 1999). We expected that teachers would appreciate students’ efforts to solve a sum independently, but we could not confirm this in the study presented here. Although the meaning of ‘personal mathematical experience’, as suggested by Beishuizen and Anghileri (1998), during sums is regarded highly, we saw little evidence of eliciting these experiences in the sample. We found that the teachers tended to become forceful in structuring the problem-solving as soon as a student started to make mistakes. This emergence of so-called ‘funnel patterns’ (see Bauersfeld, 1988, in Wood et al., 2001) refers to a series of teacher questions leading students to producing preferred but superficial procedures, rather than meaningful mathematical solutions. The findings presented here may explain why students who perform as mathematically weak develop different strategies compared to their more competent peers, but show delays in efficiency and selection (Torbeyn et al., 2004). As the study was undertaken in mainstream primary schools, it may be discussed whether the findings can be replicated with students attending special schools. However, as Geary (2004) has stated, students with mathematical disability do not differ from those experiencing delays in mathematical progress other than in the persistence of instruction. This would mean that the findings presented in the study might well be comparable to those identified concerning mathematical disability. A replication study with a sample of students within the context of special education is needed to confirm this statement, however. Addison Stone (1998) suggests that in order to foster declarative knowledge and procedural skills students need to explore the efficacy of the techniques for adding and subtracting with sums to 100. The fragments of discourse that we have presented concerning students perceived as low achievers show that asking an open-ended question became a closed question in the end. The student is supposed to answer with a specific strategy choice, and when that is not presented to the teacher, a negative spiral starts in which students receive negative feedback on their answer followed by the teacher gradually taking over responsibility. Eliciting a student’s choice of strategy thus often becomes a formality rather than a situation that allows co-construction to take place. The student then becomes a passive onlooker in the task instead of an individual moving forward in his or her zone of proximal development. In case interactive construction is required, however, open-ended questions are required to allow co-construction to take place (Allal et al., 2003). The protocols illustrate contextualization from the perspective of the teacher, who acts out his or her implicit expectations of the student’s performance. At the same time, the student is modelled to become a person who grows into the role of dependent or independent problem-solver as the task proceeds. Not every instruction at the beginning of a task requires independent problem-solving at the end of the assignment. However, for co-construction to take place, a discussion strategy is required in developing judgements and in selecting reasons/strategies in order for independent problem-solving to occur later on. This specific ‘loop’ may be addressed in teaching


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courses that include the relationship between teaching content and addressing students as active participants. It is possible that experiences like this intensify the student’s sense of (in)competence with respect to the task he or she has to complete, and over the years lower his/ her self-esteem regarding mathematical ability (Stipek & MacIver, 1989; Good & Nichols, 2001; Hamre & Pianta, 2001). Our findings can be compared with those of Roth et al. (1999), who studied whole-class conversations in comparison to smallgroup dialogues over a four-month period and showed that physical and social arrangements influenced the discursive form and content. We argue that we found the same sort of influences, such as the use of materials and the types of questions that were asked during the task. The physical arrangements structured the interactional and temporal space. The arrangements formed the ‘scenario’ for actions and unfolding conversations as they determined topics and conversations. The research task supported the participants’ verbal and nonverbal communication by gesturing and pointing, and facilitated (or impeded) conversations in a remarkably different way for high- and low-rated students, however.

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