TEACHERS’ BELIEFS AND MATHEMATICS CURRICULUM
REFORM: A STORY OF CHONGQING
Chen Qian [email protected]
Faculty of Education, The University of Hong Kong
Literature suggests that the consistency between teachers’ beliefs and the underlying
philosophy of reform-oriented curriculum can be an important indicator of the success
of curriculum reform. However, few studies have been conducted to directly deal with
this issue. This study used part of data collected for a larger ongoing research project
to investigate the consistency between teachers’ mathematics beliefs and the
underlying philosophy of the new curriculum at junior secondary level in Chongqing.
It was concluded that the underlying philosophy of the reform-oriented mathematics
curriculum in mainland China was to a great extent congruent to Constructivist ideas,
and the beliefs held by a large proportion of the subjects were mostly consistent with
the underlying philosophy, although some inconsistencies existed.
INTRODUCTION
At the turn of the 21st century, mathematics curricula of various countries and regions
around the world have experienced their reforms aimed at preparing younger
generations for an age in which the economy is globalized, and the society is
“knowledge-based” and information-rich (Wong, Han, & Lee, 2004, p. 27). Greatly
involved in the worldwide wave of reforms, China launched its new-round
mathematics curriculum reform in 1999 and has started implementing the
standards-based curriculum since September 2001.
It has been argued that most current curriculum reforms advocate the Constructivist
views of mathematics and its teaching and learning (Frykholm, 1995; Gregg, 1995;
Knapp & Peterson, 1995; Smith, 1996; Yang, 2003) which are substantially different
from those underpinning traditional curricula. Therefore, contemporary teachers are
called on to change their beliefs for the success of curriculum reforms. As a matter of
fact, the importance of teachers’ beliefs, particularly mathematics beliefs to
educational innovation has been increasingly emphasized by researchers (Battista,
1994; Handal & Herrington, 2003). Handal and Herrington (2003) observe that
teachers’ mathematics beliefs are critical in determining the pace of curriculum reform.
Mathematics teachers’ beliefs can play either a facilitating or an inhibiting role in
translating curriculum guidelines into the complex daily reality of classroom teaching.
If teachers hold beliefs compatible with the innovation, then acceptance is more likely
to occur. However, if teachers hold opposing beliefs or perceive barriers in enacting
the curriculum, then low-take up, dilution and corruption of the reform will likely
follow.
Unfortunately, an extensive body of research has indicated that very often, most
teachers hold ingrained beliefs which are incongruent to, even conflicting with the
underlying philosophy of innovation, and these beliefs engage them in the struggles
against the reform efforts, preventing them from implementing the reform
recommendations or ideas effectively (Battista, 1994; Cohen, 1990; Wilson, 1990;
Wilson & Goldenberg, 1998), thwarting the success of reform. The inconsistencies
between teachers’ beliefs and the assumptions of reforms have been concluded by
some researchers as one of the big obstacles to the success of reforms (Ross,
McDougall, & Hogaboam-Gray, 2002).
To sum up, literature suggests that the consistency between teachers’ mathematics
beliefs and the underlying philosophy of innovation can be an important indicator of
the success of curriculum reform. Despite the acknowledgement of the importance of
this consistency, few studies have been conducted to directly deal with this issue.
This study being reported here used part of data collected for a larger ongoing research
project to investigate the consistency between the junior secondary mathematics
teachers’ mathematics beliefs and the underlying philosophy of the new curricula in
Chongqing. The main three research questions were as follows:

What was the underlying philosophy of the new mathematics curricula of
Mainland China?

In Chongqing, what were the junior secondary mathematics teachers’
mathematics beliefs?

To what degree were the teachers’ mathematics beliefs consistent with the
underlying philosophy of new curriculum?
In view of the diversity of definitions of beliefs and mathematics beliefs in literature
(Leder, Pehkonen, & Torner, 2002; Pajares, 1992), this study borrowed Raymond’s
(1997) definition of mathematics beliefs as “personal judgments about mathematics
formulated from experiences in mathematics, including beliefs about the nature of
mathematics, mathematics teaching and learning”(p.552), considering it was
well-defined and appropriate.
METHODOLOGY
To address above-said research questions, data were collected through both qualitative
and quantitative methods, including document analysis and questionnaire survey.
Specifically, most important curriculum document issued by official organizations
concerned in mainland China, National Mathematics Curriculum Standards at the
Compulsory Educational Level (draft for consultation) (National Ministry of
Education, 2001) was used to analyse the underlying philosophy of new junior
secondary mathematics curriculum through ‘content analysis’ (Marshall & Rossman,
1999).
In Chongqing, a total of 114 junior secondary mathematics teachers from 11 schools,
including 55 male and 59 female, participated in the questionnaire survey conducted
from May to July 2007. Forty-one percent of these teachers taught grade 7, 42% taught
grade 8 and 17 % taught grade 9. In respect of educational level, 18% of the teachers
held junior college diploma, 78% held bachelor degree of mathematics, 4% held
master degree of mathematics or mathematics education. All participants had teacher
qualification certificate. The percentages of teachers having 0~6, 7~15 and 16+ years
of mathematics teaching experience were 25%, 45% and 30 respectively. Besides,
teachers varied in the number of times of attending in-service teacher training, 6%
never attended, 47% attended 1 to 3 times, 14% attended 4 to 6 times, 33% attended
more than 6 times.
The questionnaire (See Appendix) consisted of two parts: (a) Collier’s (1972) two
Scales measuring individual teacher’s mathematics beliefs, i.e. Beliefs About
Mathematics Scale (BAMS) and Beliefs About Mathematics Instruction Scale
(BAMIS); (b) personal particulars recording demographic characteristics of individual
teacher, including gender, educational level etc. Collier’s (1972) two Scales were
chosen because they are regarded as a reasonable measure of constructivist philosophy
and ideas about instruction that follow from that philosophy (Seaman, Szydlik, Szydlik,
& Beam, 2005). All items in BAMS and BAMIS had 6-point Likert scale response
options (from strongly disagree to strongly agree). Half of the items were phrased in a
positive manner (advocating informal/constructivist ideas) and half in a negative
manner (advocating formal approaches to mathematics). The possible range on each
scale was 20 to120, with 70 being a neutral score. A score higher than 70 was in the
“informal” direction and a score less than 70 was in the “formal” direction. In order to
ensure the research reliability and validity, the questionnaire was translated to a
Chinese version with the help of an expert and piloted before its use in this sample. The
reliability coefficients of the translated BAMS and BAMIS were both 0.7. One thing
worth mentioning is that the purpose of survey was neither to generalize the findings
from this sample to the whole population, nor to construct a valid questionnaire for
Chinese context, but to get a general idea about teachers’ beliefs by including teachers
with diverse characteristics in terms of educational level, teaching experience etc.
With regard to questionnaire data analysis, it has been suggested by many researchers
that because the rating scales (1-6) may be nonlinear, parametric tests such as
two-sample t-tests, ANOVA etc. used to compare means between categories are no
longer recognized as the most appropriate, and an acceptable alternative is to use a
chi-squired item level analysis (e.g. D'Ambrosio, Boone, & Harkness, 2004; Seaman et
al., 2005). In this study, a chi-squared test of significance was used to compare
responses between different groups of teachers at the item level (in comparing two
groups, when an expected cell count smaller than 5 was involved, Fisher’s exact test
was used). Furthermore, in the analysis, adjusted score response 1, 2, 3 were combined
(all indicating formal attitudes), and adjusted score response 4, 5, 6 were combined (all
indicating informal, or constructivist attitudes). This combining of scores made it
easier to identify the nature of the difference (Seaman et al., 2005).
FINDINGS
Underlying Philosophy of the Reform-oriented Mathematics Curriculum
The underlying philosophy of the reform-oriented mathematics curriculum referred to
the views of mathematics, mathematics learning and teaching expressed either
explicitly or implicitly in National Mathematics Curriculum Standards at the
Compulsory Educational Level (draft for consultation) through the following
statements:
Mathematics is an indispensable tool for people’s life, labour and study…mathematics is
basis for all advancements in technologies… mathematics play an unique role in
enhancing people’s ability to reason, abstract and imagine, create; mathematics is a kind of
human culture….One’s mathematics learning should be a lively, active and individual
process. Mathematics teaching should be built on students’ cognitive development levels
and existing knowledge and experiences. … Students are the masters of their learning,
teachers are organizers, guiders and co-operators of students’ mathematics learning
(National Ministry of Education, 2001, pp. 1-2).
Obviously, these views were to a great extent congruent with Constructivist ideas (e.g.
von Glasersfeld, 1987), because they acknowledged that mathematics is a cultural
product which does not lie outside of human action, mathematics is not learned by
transmission, but rather through an active process of construction wherein students’
existing knowledge and experience are crucial, and students take primary
responsibilities for their own learning, teachers act as facilitators.
Teachers’ Beliefs about Mathematics and its instruction
The means of teachers’ scores on BAMS and BAMIS were 92 and 84 respectively,
which indicated that on average, both teachers’ beliefs about mathematics and beliefs
about mathematics instruction were in the informal direction, but the latter was more
formal than the former. Besides, correlation tests indicated that teachers’ beliefs about
mathematics were significantly correlated with their beliefs about mathematics
instruction at the 0.01 level (r=.413), but not significantly correlated with their
mathematics teaching experience, educational level, and in-service training.
All subjects in the survey fell into five categories according to their scores on BAMS
and BAMIS, as shown in Table 1.
BAMIS
Formal
BAMS
Neutral Informal
Total
Formal
1
0
0
1
Neutral
1
0
0
1
Informal
8
1
103
112
10
1
103
114
Total
Table 1 Distribution of Teachers’ Response
For each item, the percentage of teachers selecting responses aligned with informal
belief was displayed in the following table:
Item
Percentage of
Teachers with
informal beliefs
Item
Percentage of
Teachers with
informal beliefs
1
18
21
42
2
97
22
81
3
76
23
40
4
82
24
97
5
98
25
97
6
96
26
95
7
93
27
75
8
86
28
63
9
31
29
56
10
97
30
97
11
78
31
49
12
83
32
46
13
87
33
79
14
51
34
87
15
90
35
72
16
64
36
81
17
74
37
88
18
97
38
60
19
100
39
64
20
62
40
59
Table 2 Percentages of Teachers Aligned with Informal Beliefs
It could be known from Table 2 that more than 50% of the teachers selected responses
aligned with informal belief on the majority of items (85%), except on item 1 ‘Solving
a mathematics problem usually involves finding a rule or formula that applies’, item 9
‘In mathematics, perhaps more than in other fields, one can find set routines and
procedures’, item 21 ‘The teacher should always work sample problems for students
before making an assignment’, item 23 ‘Students should be encouraged to invent their
own mathematical symbolism’, item 31 ‘Discovery methods of teaching have limited
value because students often get answers without knowing where they came from’,
item 32 ‘The teacher should provide models for problem solving and expect students to
imitate them’.
Chi-squared item level tests indicated that differences between female teachers and
male teachers on item 31 ‘Discovery methods of teaching have limited value because
students often get answers without knowing where they came from’ (p=.024) and item
36 ‘Teachers must frequently give students assignments which require creative or
investigative work’ (p=.028) were significant at the 0.05 level, and differences among
inexperienced, experienced and veteran teachers on item 16 ‘Mathematics is a rigid
discipline which functions strictly according to inescapable laws’ (p=.003) and item
29 ‘Teachers should spend most of each class period explaining how to work specific
problems’ (p=.045) were also significant at the 0.05 level.
The Consistency
Comparing the questionnaire data with the underlying philosophy revealed the
consistency between the two. On the whole, ninety percent of subjects (n=103) held
informal or Constructivist beliefs about mathematics and mathematics instruction,
which showed a high consistency between the two. Nevertheless, taking a closer look
at the items gave us more insights. High percentage of subjects agreed that

mathematics is a kind of human culture (e.g. 97% for item 2)

mathematics plays an unique role in enhancing people’s ability to think and
create (e.g. 97% for item 2, 98% for item 5, 83% for item 12)

mathematical problem-solving allows for multiple approaches (e.g. 93% for
item 7, 97% for 10, 100% for item 19)

mathematics learning requires very much independent thinking and inquiry
(e.g. 96% for item 6, 97% for item 18)

teachers should value and encourage students’ construction of mathematical
ideas (e.g. 97% for item 24, 95% for item 26)

students should find individual methods for solving problems and feel free to
use any method that suits him or her best (e.g. 97% for item 25, 97% for item
30)
On the other hand, less than half of the subjects disagreed that

mathematics is a collection of rules, formulas, and procedures (e.g. 18% for
item 1, 31% for item 9)

discovery methods of teaching have limited value (49% for item 31)

teachers should provide models for problem solving and expect students to
imitate them (46% for item 32),
which suggested that many teachers tended to have narrow views about mathematics
and its teaching and learning, and they may need more time to attempt discovery
methods of teaching and appreciate its value.
CONCLUSION
Based on the findings in this study, it was concluded that the underlying philosophy of
the reform-oriented mathematics curriculum in mainland China was congruent to
Constructivist ideas to a great extent, and more importantly, the beliefs about
mathematics and mathematics instruction held by a large proportion of the teachers
were mostly consistent with the underlying philosophy of the reform-oriented
curriculum, although some inconsistencies existed. Therefore, this study seemed to
give a quite optimistic picture of the mathematics curriculum reform in Chongqing.
However, as literature has indicated, teachers’ beliefs should be studied from a variety
of perspectives, including what they think (espoused beliefs) as well as what they do
(enacted beliefs) (Thompson, 1992). Thus further research aimed at examining
teaching practices of different categories of teachers emerged in this study is needed.
Studying the consistencies between teachers’ enacted beliefs, espoused beliefs and the
underlying philosophy of the curriculum may provide more implications for our
reform and research efforts.
References
Battista, M. T. (1994). Teacher beliefs and the reform movement in mathematics education.
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Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational
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Collier, C. P. (1972). Prospective elementary teachers' intensity and ambivalence of beliefs
about mathematics and mathematics instruction. Journal for Research in Mathematics
Education, 3(3), 155-163.
D'Ambrosio, B. S., Boone, W. J., & Harkness, S. S. (2004). Planning district-wide
professional development: Insights gained from teachers and students regarding
mathematics teaching in a large urban district. School Science and Mathematics, 104(1),
5-15.
Frykholm, J. A. (1995). The impact of the NCTM standards on pre-service teachers' beliefs
and practices. Paper presented at the AERA conference.
Gregg, J. (1995). The tensions and contradictions of the school mathematics tradition.
Journal for Research in Mathematics Education 26(5), 442-466.
Handal, B., & Herrington, A. (2003). Mathematics teachers' beliefs and curriculum reform.
Mathematics Education Research Journal, 15(1), 59-69.
Knapp, N. F., & Peterson, P. L. (1995). Teachers' interpretations of "CGI" after four years:
Meanings and practices. Journal for Research in Mathematics Education, 26(1), 40-65.
Leder, G. C., Pehkonen, E., & Torner, G. (2002). Beliefs: A hidden variable in mathematics
education. Dordrecht, Boston, London: Kluwer Academic Publishers.
Marshall, C., & Rossman, G. B. (1999). Designing qualitative research (3rd ed.). Thousand
Oaks, Calif.: Sage Publications.
National Ministry of Education. (2001). National mathematics curriculum standards at the
compulsory educational level (draft for consultation). Beijing: Beijing Normal University.
Pajares, M. F. (1992). Teachers' beliefs and educational research: Cleaning up a messy
construct. Review of Educational Research, 62(3), 307-332.
Raymond, A. M. (1997). Inconsistency between a beginning elementary school teacher's
mathematics beliefs and teaching practice. Journal for Research in Mathematics
Education, 28(5), 550-576.
Ross, J. A., McDougall, D., & Hogaboam-Gray, A. (2002). Research on reform in
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Seaman, C. E., Szydlik, J. E., Szydlik, S. D., & Beam, J. E. (2005). A comparison of
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and 1998. School Science and Mathematics, 105(4), 197-210.
Smith, J. P. (1996). Efficacy and teaching mathematics by telling: A challenge for reform.
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Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D.
A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.
127-146). New York: Macmillan.
von Glasersfeld, E. (1987). Learning as a constructivism in activity. In C. Javanier (Ed.),
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Hillsdale, NJ: Erlbaum.
Wilson, S. M. (1990). A conflict of interests: The case of Mark Black. Educational
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Wilson, S. M., & Goldenberg, M. P. (1998). Some conceptions are difficult to change: One
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Appendix
Questionnaire for Junior Secondary Mathematics Teachers in Chongqing
Please circle the number which best describes your agreement with each statement, and
students in the statements refer to junior secondary students.
Strongly
Disagree
Relatively
Disagree
A little
Disagree
A little Agree
Relatively
Agree
Strongly Agree
1. Solving a mathematics problem usually involves finding a rule or
formula that applies.
1
2
3
4
5
6
2. The field of mathematics contains many of the finest and most elegant
creations of the human mind.
1
2
3
4
5
6
3. The main benefit from studying mathematics is developing the ability
to follow directions.
1
2
3
4
5
6
4. The laws and rules of mathematics severely limit the manner in which
problems can be solved.
1
2
3
4
5
6
5. Studying mathematics helps to develop the ability to think more
creatively.
1
2
3
4
5
6
Strongly
Disagree
Relatively
Disagree
A little
Disagree
A little Agree
Relatively
Agree
Strongly Agree
6. The basic ingredient for success in mathematics is an inquiring nature.
1
2
3
4
5
6
7. There are several different but appropriate ways to organize the basic
ideas in mathematics.
1
2
3
4
5
6
8. In mathematics there is usually just one proper way to do something.
1
2
3
4
5
6
9. In mathematics, perhaps more than in other fields, one can find set
routines and procedures.
1
2
3
4
5
6
10. Mathematics has so many applications because its models can be
interpreted in so many ways.
1
2
3
4
5
6
11. Mathematicians are hired mainly to make precise measurements and
calculations for scientists.
1
2
3
4
5
6
12. In mathematics, perhaps more than in other areas, one can display
originality and ingenuity.
1
2
3
4
5
6
13. There are several different but logically acceptable ways to define
most terms in mathematics.
1
2
3
4
5
6
14. Mathematics is an organized body of knowledge which stresses the
use of formulas to solve problems.
1
2
3
4
5
6
15. Trial-and-error and other seemingly haphazard methods are often
necessary in mathematics.
1
2
3
4
5
6
16. Mathematics is a rigid discipline which functions strictly according
to inescapable laws.
1
2
3
4
5
6
17. Many of the important functions of the mathematician are being
taken over by the new computers.
1
2
3
4
5
6
18. Mathematics requires very much independent and original thinking.
1
2
3
4
5
6
19. There are often many different ways to solve a mathematics problem.
1
2
3
4
5
6
20. The language of mathematics is so exact that there is no room for
variety of expression.
1
2
3
4
5
6
21. The teacher should always work sample problems for students before
making an assignment.
1
2
3
4
5
6
22. Teachers should make assignments on just that which has been
thoroughly discussed in class.
1
2
3
4
5
6
23. Students should be encouraged to invent their own mathematical
symbolism.
1
2
3
4
5
6
24. Each student should be encouraged to build on his own mathematical
ideas, even if his attempts contain much trial and error.
1
2
3
4
5
6
Strong
ly
Agree
Relati
vely
Agree
A
little
Agree
A
little
Disagr
Relati
ee
vely
Disagr
Strong
ee
ly
Disagr
ee
25. Each student should feel free to use any method for solving a
problem that suits him or her best.
1
2
3
4
5
6
26. Teachers should provide class time for students to experiment with
their own mathematical ideas.
1
2
3
4
5
6
27. Discovery methods of teaching tend to frustrate many students who
make too many errors before making any hoped for discovery.
1
2
3
4
5
6
28. Most exercises assigned to students should be applications of a
particular rule or formula.
1
2
3
4
5
6
29. Teachers should spend most of each class period explaining how to
work specific problems.
1
2
3
4
5
6
30. Teachers should frequently insist that pupils find individual methods
for solving problems.
1
2
3
4
5
6
31. Discovery methods of teaching have limited value because students
often get answers without knowing where they came from.
1
2
3
4
5
6
32. The teacher should provide models for problem solving and expect
students to imitate them.
1
2
3
4
5
6
33．The average mathematics student, with a little guidance, should be
able to discover the basic ideas of mathematics for her or himself.
1
2
3
4
5
6
34．The teacher should consistently give assignments which require
research and original thinking.
1
2
3
4
5
6
35．Teachers must get students to wonder and explore even beyond usual
patterns of operation in mathematics.
1
2
3
4
5
6
36．Teachers must frequently give students assignments which require
creative or investigative work.
1
2
3
4
5
6
37．Students should be expected to use only those methods that their text
or teacher uses.
1
2
3
4
5
6
38．Discovery-type lessons have very limited value when you consider
the time they take up.
1
2
3
4
5
6
39．All students should be required to memorize the procedures that the
text uses to solve problems.
1
2
3
4
5
6
40．Students of all abilities should learn better when taught by guided
discovery methods.
1
2
3
4
5
6
Personal Information
Please put a √ in the appropriate box(es).
(1) Gender
Male
Female
(2) Contacts
Phone number:
Email box:
(3) Educational level (more than one choice)
□ Post-secondary certificate, diploma or associate degree Major:
Minor:
□ Bachelor or equivalent
Major:
Minor:
□ Master or above
Major:
Minor:
□ Other
(4) Teaching training or not
Yes
No
(5) Teaching certificate or not
Yes
No
(6) Teaching experience (including the current year)
General:
years
Mathematics:
years
New mathematics curriculum (started in 2001)
years
(7) Levels you are currently teaching mathematics
S1
S2
S3
(8) Number of times you have ever attended in-service training on the new curriculum
0
1～3
4～6
more than 6