The participants were 4,198 13-year-old students in five Asian outperforming countries and five English-speaking Western countries (Table 1 shows the student number of each country); they filled in the student questionnaires and took the Booklet 3 mathematics tests in the TIMSS study in 2003. As there were some missing data for the student variables, these cases were deleted and ended with 3,269 students for the analysis of influential variables (Table 2).
Indicators
Four kinds of indicators were taken from the TIMSS 2003 study. (1) Country mathematics achievement. The individual country’s mathematics achievements were indicated by their average scale scores of each country, as provided by on Page 34 in the TIMSS 2003 International Mathematics Report. (2) Student mathematics achievement. One set of plausible values of student mathematical achievement provided by the TIMSS database were used as the indicators of individual students’
overall mathematical achievements. These values were good estimates of parameters of student populations. (3) Student responses to the focused problem. The focused problem in the present study was ‘Write a fraction that is less than 4/9’. This was the only ‘creative’ mathematical problem,
in the TIMSS 2003 study, which had endless correct answers. In addition, these correct answers can be clearly classified as ‘routine correct solutions’(3/9, 1/3, 4/10, 2/5), ‘novel correct solutions’ (other correct solutions), and ‘incorrect solutions’. (4) Student self-report results. There were 16 other variables selected from the self-report TIMSS student questionnaire (Table 2). The variables based on the Likert-scale were derived from factor analysis on the sample of 4,198 students in the present study.
RESULTS
There were significant differences between the Asian and Western countries in the four kinds of mathematical achievements, as presented in Table 1. The Asian countries outperformed the Western counterparts in the aspects of the overall mathematical achievements (t (8)=14.67) and the percentages of routine responses to the focused problem (t (8)=6.49).
As a reasonable result, the Asian countries had less incorrect responses than the Western countries (t (8)=-4.66). However, the Asian countries had fewer novel ‘correct’ responses than the Western ones (t (8)=-5.18).
This result was a contrary to the above trend, as it is sensible to predict that different samples should have similar distribution patterns of ‘correct answers’ of many kinds. The result also implied that there are different meanings of ‘novel responses to mathematical problem-solving’ between the Asian and Western students.
Table 1: Achievements by countries, and test results
Asian outperforming countries Mean English-speaking Western countries Mean T Hong
Kong Japan Korea Singapore Taiwan (N=5) Austral ia Engl
1For each country shows the number of students who solved the focused problem in Booklet 3, TIMSS 2003.
2Math achievements are the average scale scores as indicated in Exhibit 1.1, TIMSS 2003 International Mathematics Report, p.
34.
3Routine responses to the focused problem include 3/9, 1/3, 4/10, 2/5.
4Novel responses are the correct solutions to the focused problem, except the above routine responses.
** Significant at the .01 level
Discriminant function analyses were performed, in order to determine the influential student variables in defining the three kinds of responses. The analysis was conducted for the three samples of all the students, Asian students and Western students respectively, given the possibility that there were differential meanings of novel responses between the Asian and Western students. As can be seen in Table 2, Function 1s for the three samples can best distinguish routine responses from incorrect ones.
This implies that Function 1s address issues of achievements. Function 2s, on the other hand, can distinguish novel responses from the other two types of responses, addressing issues of novelty.
Table 2. Discriminant analysis: Functions at Group means
Country
test) LSD test results All students Function 1 90 % .78** .386 -.279 -.773 399.04** R>N,R>I, N>I
* Significant at the .05 level
** Significant at the .01 level
Table 3 reveals that, for all the students, the most significant influential variables that distinguish routine responses from incorrect responses are mathematics achievements and aspiration to higher education. Novel solvers were distinguishable by their test-language use at home, computer-use for learning, computer availability, positive mathematical affect (self-efficacy and value), and low disposition toward schooling. In addition, most novel solvers were boys and had fewer extra mathematics lessons.
Both Asian and Western routine solvers were distinguishable by their high mathematical achievements, self-efficacy in mathematics, and aspiration to higher education (Tables 4-5). The two groups of students, however, were different in their perceptions of mathematical teaching.
Transmission- and constructivist-oriented teaching approaches were both positive variables in influencing Asian students’ achievements. Western high-achievers experienced fewer extra mathematical lessons or tutoring.
Compared with their Asian counterparts, most Asian novel solvers’
parents were from other countries. They has low disposition toward schooling and negative mathematical affects (including self-efficacy, deep approaches, and value). Although Asian novel solvers had more
learning resources (such as speaking test-use languages at home and having computers at home and school), they perceived few opportunities of computer-use for learning purposes. They also experienced ‘freer’
mathematical teaching approaches or strategies: few extra mathematics lessons or tutoring, few transmission-oriented teachings, and little mathematical homework.
Table 3. Affective/contextual variables by all solvers
Routine 5. Aspiration to higher education 2.08 .76 1.95 .80 1.86 .78 24.08** R>N,R>I,N>I .25a .01
Math teaching approaches 10.Extra math lessons/tutoring 1.96 1.08 1.61 .88 1.81 1.01 23.45** R>N,R>I,I>N .17 -.51 a 11.Computer-use for learning 2.29 .84 2.41 .90 2.36 .94 4.83** N>R -.09 .21 a 17. Math Achievement 582.08 78.67 532.08 72.02 494.28 78.71 381.29** R>N,R>I,N>I .97 a .01
* Significant at the .05 level
** Significant at the .01 level
a Absolute correlation between the student variable and the Function 1 (or Function 2) equal to or larger than .20
Table 4. Student variables by responses for Asian students
Routine
17. Math Achievement 604.87 72.58 574.85 67.56 531.00 72.89 148.68** R>N,R>I,
N>I .95a .04
* Significant at the .05 level,
** Significant at the .01 level
a Absolute correlation between the student variable and the Function 1 (or Function 2) equal to or larger than .20
Table 5: Student variables by responses for Western students 17. Math Achievement 538.97 72.38 511.10 64.54 464.63 70.36 148.91** R>N,R>I,N>I .97a -.05
* Significant at the .05 level,
** Significant at the .01 level
a Absolute correlation between the student variable and the Function 1 (or Function 2) equal to or larger than .20
DISCUSSION
Although some Asian countries outperformed other countries on several international competition tests in mathematics and science, such as TIMSS and PISA, the present study highlighted their significant lack of novel problem-solutions in mathematics, even by analyzing responses to one ‘creative’ mathematical problem in TIMSS 2003 study. While routine solutions were related to their general mathematics achievements and aspiration to higher education, novel solutions was more related to positive affects about mathematics and plentiful learning resources. Novel solvers also appeared to have negative disposition toward schooling and did not rely on extra teaching. This result implies routine and novel mathematical solutions are cultivated by different contexts of mathematics learning, with ‘routine solutions’ determined by mathematical achievement and performance/ability goals, and ‘novel solutions’ determined by
Western novel solvers were distinguishable from their Asian counterparts by their abundant learning resources and high regard for mathematics.
Most Western novel solvers were boys and their mothers were natives.
They, however, had slightly low disposition toward schooling and low aspiration to higher education. A comparison between the three samples in the influential variables is presented in Table 6.
Table 6: Student groups by influential variables of routine and novel solving
Student group Influential variables of routine solving Influential variables of novel solving All students (+)Math achievement
(+)Aspirational affect (+) Learning resources (language, computer use/availability) (+) Math affect (self-efficacy, value)
(-) Schooling affect (-) Teaching (extra math) (+) Boy
Asian students (+)Math achievement (+)Math affect (self-efficacy)
(-) Math affect (self-efficacy, deep approach, value)
(+) Learning resources (language, computer availability) (-) computer use
(-) Teaching (extra math, transmission, math homework)
Western students (+)Math achievement (+)Math affect (self-efficacy) (+)Aspirational affect (-)Teaching (extra math)
(+) Learning resources (computer availability, language, computer use)
strong learning-resource supports, positive mathematical affects, and detached schooling/teaching experiences. In other words, routine solutions were developed by an achievement-centered learning context or centrally cognition-oriented context of mathematics learning; novel solutions came from a self-centered learning context or a peripherally affect-oriented contexts of mathematics learning. Mathematics has long been regarded by students as full of routine problems that can be solved by routine procedures (Schoenfeld, 1989). In order to become the ‘formal member’ of mathematics learning, i.e. becoming a high-achiever, providing routine solutions are likely to be the most significant means.
Given the critical aim of high-achievement in mathematics learning, novel solvers are the ‘peripheral’ (Lave & Wenger, 1991, p. 29) members of mathematics ‘classroom communities’ (Hamm & Perry, 2002, p. 135).
The tension between central-cognitive and peripheral-affective participation tends to be stronger for the Asian students than for the Western students. Both Asian and Western routine solvers have high mathematics achievement, self-efficacy in mathematics and high aspiration to higher education. Asian routine solvers benefited from both transmission and constructivist approaches of mathematical teaching, while Western routine solvers experienced few extra mathematics lessons/tutoring in school. In other words, Asian routine solvers were at the very center of mathematical learning, in terms of both achievement orientation and participation in teaching activities; Western routine solvers were at the center of learning, in terms of achievement orientation, but less in terms of teaching.
Asian novel solvers experienced a far peripheral and negative affective participation in their mathematical learning communities. The only positive factor for Asian novel solutions was learning resources of test-language use and computer availability. Asian novel solvers’ parents were not natives; they had various kinds of negative affective responses to schooling and mathematics; they experienced loose mathematical teaching. This picture is a mirror image of their Asian routine solvers’, a rather positive one. On the other hand, except for low disposition toward schooling and low educational aspiration, Western novel solvers appeared to possess advantages of strong support for independent learning from abundant learning resources, high regard for mathematics, and native mothers. The narrowly achievement goals fail to be their focus, Western novel solvers had positive or no less social/economic and affective support. This might encourage them to create their own novel/unique solutions for the sake of their own and mathematics. The differential achievements between Western and Asian novel solutions on the international competition test, or later achievement of mathematics expertise, are likely to be explained by the differences between the positive learning contexts of Western novel solvers and the negative ones of the Asian solvers.
A final point to make is that although some studies have shown that the difference between mathematics achievements of boys and girls are minimal and only in the top bands of 10% to 20% (e.g. Askew & Wiliam, 1995), the present study revealed that boys tended to produce novel solutions more than girls, especially in Western countries. In comparison with Vermeer et al.’s (2000) study, which indicated that boys tended to be better at complex ‘application’ than girls, there appears a need to study in depth gender differences in mathematics achievements in relation to problem types and affective issues.
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A situated self-regulated learning system:
Evidence from Taiwanese children’s constructs of mathematical problems Mei-Shiu Chiu*
National Chengchi University, Taiwan
Abstract
This study examined 75 Taiwanese children’s personal constructs of diverse mathematical problems by the repertory grid technique. These constructs were
‘situated’ in the contexts of their solving the diverse mathematical problems in the classroom, and regulated by the children as ‘psychological tools’ in order to successfully participate in their mathematical learning world. By coding, categorizing and formulating these constructs, there emerged a situated self-regulated learning system that comprised three cognitive components (situated-perceptions, -strategies, and -aims), and one affective component (situated-affect). The relatively differential emphases on these components for diverse problems also suggested a ‘situated’
characteristic of children’s sensitively regulating their authentic learning.
Summary
There have been a number of researchers who proposed models of self-regulated learning or meta-cognition (e.g. Boekaerts, 1996; Zimmerman, 2000; Pintrich, 2000;
de Corte et al., 2000; Whitebread et al., 2005). Most of the models were built upon the researchers’ rationales, along with past theories, and evidenced or elaborated by further empirical studies (e.g. Zimmerman & Martinez-Pons, 1998; Fuchs et al., 2003). There are, however, a lack of agreement between the models on the components and the systems of self-regulated learning (SRL). Research based on a
* I would like to thank Dr. David Whitebread, University of Cambridge, for introducing me the repertory grid technique for educational research. Later data analysis was supported by National Science Council, Taiwan (NSC 94-2522-S-004–001). Address for correspondence: Dr. Mei-Shiu Chiu, Department of Education, National Chengchi University, Taiwan. 64, Zhi-nan Rd. Sec. 2, Taipei 11605, Taiwan. E-mail: [email protected]
top-down design is theory-driven or hypothesis-verification, while research based on a bottom-up design is theory-finding or phenomenon-illumination. Aiming to identify significant components and formulate a system of SRL in relation to situated phenomena, this study drew on a bottom-up research procedure, the repertory grid technique (Kelly, 1955), by which children’s personal constructs of mathematical problems in their learning contexts, were induced.
Method
There were 51 children individually interviewed for the four focused problems in the fractions topic (Problems 1-4) and 24 children for the two focused problems in the coordinates topic (Problems 5-6). Children were selected for interviews by balancing classes, gender and attainments. The six focused problems are as follows.
Problem 1: Please use calculation procedure, 7 ÷5 = 1 2/5 , to make a mathematical problem.
Problem 2: Mother made several pizzas and Betty got 3/4 pizza. By which ways could the pizzas be divided?
Problem 3: Thirty-six scenery postcards are packed in a box. Equally divide ten boxes of postcards between nine persons. How much of a box of scenery postcards will one person get?
Problem 4: Two ribbons (of equal length) are equally divided between six persons. How much ribbon will one person get?
Problem 5: Bombing headquarters--Game rules: (1) Two people in a group, each person decides his/her coordinates for the headquarters. (2) Two people in turn bomb the other's headquarters. The person who correctly hit the other’s coordinates first wins the game.
Problem 6: In the following graph a silkworm is going to eat mulberry leaves. (1) What are the coordinates of the mulberry leaf? Mark it on the picture, and read it. (2) What is on the coordinates (5,4)? (3) The silkworm wants to eat the mulberry leaf. How many grids should it walk west? How many grids should it walk north? (4) After finishing the mulberry leaf, the silkworm walks east for three grids and south for five grids. Which position will it walk to? What are the coordinates of the position? Mark it on the picture and read it. (5) Try to describe what the coordinates of the baby silkworm are?
The interview began by asking children to solve the focused mathematical problems. Next, using the repertory grid technique, children’s personal constructs in relation to the focused problems were established. For the fractions topic, children randomly chose three problems from the four focused problems and separated the three problems into ‘two similar problems’ and ‘one different problem’. They were asked for their constructs of ‘similarity’ and ‘difference’ between the problems. The procedure was repeated until no further constructs could be formed. A similar procedure was used for the coordinates topic, but children were asked for the
‘similarity’ and ‘difference’ between the two problems directly.
The constructs obtained by the technique were coded and categorized by two coders, with the aim of forming meaningful components in mind. The components were formulated to develop a situated SRL system.
Results and discussion
A situated SRL system is formulated based on the quality and relationships among the components. The system comprises three cognitive components (situated perceptions, situated strategies, and situated aims) and one affective component (situated affects).
Situated affects are embedded in between the three cognitive components, rather than as a significantly separate part, as revealed in most SRL models. The rationales for this formation is that affect was much less perceived by the children as a construct than cognition. However, once it was perceived, affect was normally found interwoven with either cognitive component. Affect also serves as a trigger of strong emotions, substantial concerns, and complex events.
The research result also suggests a ‘situated’ characteristic of SRL: relatively differential emphases on the four components for diverse mathematical problems. For example, affect was significantly induced for a game problem (Problem 5); strategies were emphasized for close-ended problem (Problems 3-4); aims were the major concerns for open-ended problems (Problems 1-2); children significantly invested on situated perceptions in order to clarify a wordy problems (Problem 6).