Investigating Conceptions of Math by the Use of Open-ended Mat Problems, presented as ICME-9 TSG11
N.Y. Wong
Investigating Conceptions of Mathematics by the Use of
Open-ended Mathematical Problems (*)
Ngai-Ying Wong (nywong@cuhk.edu.hk)
Department of Curriculum and Instruction, The Chinese University of Hong Kong, Hong Kong
1. Introduction
The research team (#) started off the investigation as a curriculum issue. Students’ conception of
mathematics was taken as part of the “attained curriculum” (learning outcome), a result of the
“intended” and “implemented” curricula (Travers & Westbury, 1989). First of all, Asian students in
general and Hong Kong students in particular outperformed their Western counterparts in international
comparisons in mathematics, (Cai, 1995; Leung and Wong, 1996, 1997, Leung et al, 1999; Stevenson
& Lee, 1990), but we suspected that they may lack conceptual understanding. As one facet of it, we
suspected that students only possess a narrow conception of mathematics. Secondly, the mathematics
curriculum was under reform in which the author was actively involved. As a first step, we need to
make a full situational appraisal, to identify the strength and weaknesses of the current curriculum.
There, students’ conception of mathematics was again one of the foci of investigation (Wong et al,
1999). Thirdly, we made some tried out the use of history mathematics in teaching (Lit & Siu, 1998).
To evaluate the effectiveness of such a curriculum, we saw that conventional tests do not suffice. We
incorporated assessments in the affective domain, including attitude and beliefs. All these led us to a
more in depth investigation on the conceptions of mathematics. At the same time, we realised that a
well-established methodology for such a series of studies may be lacking.
Numerous studies have revealed that beliefs about mathematics influence students’ motivation to learn
and their performance in the subject (Cobb, 1985; McLeod, 1992; Underhill, 1988). Mathematics is
often regarded as an absolute truth (Fleener, 1996) and is seen as a set of rules (Carpernter, et al,1983;
Clay & Kolb, 1983; Frank, 1988; Confrey, 1983; Dossey et al, 1988; Kolb, 1983; Kloosterman, 1991;
Lester & Garofalo, 1982; McLeod, 1992; Peck, 1984; Wong, 1993, 1995). With such a conception, to
the students, knowing mathematics is done through memorisation and algorithms and learning is a
transmission process (Underhill, 1988).
2. Initial research
Relationships between belief in mathematics and learning outcome or problem solving behaviour have
been studied too (Kloosterman, 1991; Kloosterman & Stage, 1992, Stage & Kloosterman, 1995,
Schoenfeld, 1989). However, most of these studies rely on one-way responses of the students and lack
the chance of clarifying what is in the minds of the students by creating dialogues. The research team
began to tackle the issue by confronting 29 students with ten hypothetical situation in which they were
asked to judge whether “doing mathematics” was involved in each case. Most of the situations were
taken from Kouba and McDonald (1991). Some examples are “Siu Ming said that half a candy bar is
better than a third. Was he doing mathematics ?”, “An elder sister lifted her younger brother. She said
that he must weigh about 30 pounds less than she. Did she do mathematics ?” and “Dai Keung and
Siu Chun went to take a photo at the spiral staircase at the City Hall. When the photo was processed,
We would like to thank Prof JinFa Cai for allowing us to use his open-ended questions in the study. The
research in this paper was supported by the Direct Grant for Research 1997-98 of the Social Science of
Education Panel, The Chinese University of Hong Kong. The opinions expressed are those of the authors
and do not necessarily reflect views of the funding agency.
(#) The team comprises Chi-Chimg Lam, Ka-Ming Wong and the author.
(*)
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Investigating Conceptions of Math by the Use of Open-ended Mat Problems, presented as ICME-9 TSG11
N.Y. Wong
Dai Keung discovered that the staircase looked like a sine curve. Did he use mathematics when he
looked at the photos ?”. Results revealed that students associated mathematics with its terminology
and content, and that mathematics was often perceived as a set of rules. Wider aspects of mathematics
such as visual sense and decision making were only seen as tangential to mathematics. In particular,
they were not perceived as “calculable.” However, students did recognise mathematics as closely
related to thinking. The details of which can be found in Wong, Lam & Wong (1998).
3. Use of open-ended problems
3.1. Research setting
In 1998, we proceeded to make more in-depth investigations by the use of open-ended mathematics
problems. Nine classes (around 35 students each) of each of Grades Three, Six, Seven and Nine were
asked to tackle to a set of mathematical problems. Each set comprised 2 computational problems, 2-4
words problems and 4 open-ended questions. Two students from each class (2  9  4 = 72 students)
were then asked how they approached these problems. The original hypothesis was, a narrow
conception of mathematics (as an absolute truth, say) is associated with surface approaches to tackling
mathematical problems and a broad conception is associated with deep approaches (Marton & Säljö,
1976; Wong, Lam, & Wong, 1998). Crawford et al (1998a, 1998b) already found that fragmented
view is associated with surface approach and cohesive view is associated with deep approach, but they
were investigating general approaches to learning rather than on-task approaches (Biggs, 1993).
3.2. Mathematical problems used
The students were requested to attempt three sets of mathematical problems before the
interview. They were computational problems, word problems and open-ended questions.
In the third part, the students were asked to give explanations of their working procedures
and the follow-up interview focused mainly on this part. The number of questions in each of
the grade levels is listed in Table 1.
Table 1. Mathematical problems used in the study
Grade 3
Grade 6
Grade 7
Computational problems
2
2
2
Word problems
4
3
2
Open-ended questions
4
4
4
Grade 9
2
2
4
Cai (1999) defines an open-ended mathematics problem as one which allows openness in any
of (a) given information, (b) goal to be achieve or (c) the process of solving it (Figure 1).
process
Given
Open
Goal
Open
Open
Fig. 1 Definition of open-ended questions
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Investigating Conceptions of Math by the Use of Open-ended Mat Problems, presented as ICME-9 TSG11
N.Y. Wong
Becker (1998) further identified three types of open-ended questions, with ascending levels
of difficulties, with (a) the process is open, (b) the end-products are open, and (c) the way of
formulating problems are open. In the present research, open-ended questions from Cai
(1995) and California State Department of Education (1989) were adapted for use in the
present study. The final set of problems include
(a) Problems with irrelevant information (Low & Over, 1989)
(b) “Problematic word problems” (Verschaffel, et al, 1994, 1997, Yoshida et al, 1997)
(c) Problems which allows more than one solution
(d) Problems which allows multiple methods
(e) Problems with different interpretation of question possible
(f) Problems that ask for communication
(g) Problems that need judgement
(h) Problems that involve decision making
3.3. Results
Consistent with what was found in previous research, students repeatedly showed in this study a
conception of mathematics being an absolute truth where there is always a routine to solve problems in
mathematics. The task of mathematics problem solving is thus the search of such routines. In order to
search for these rules, they look for clues embedded in the questions including the given information,
what is being asked, the context (which topic does it lie in) and the format of the question. Students
held a segregated view of the subject too. Writing is not mathematics and mathematics is calculations
with numbers and symbols. Many of them thought that by letting the answer to be found as an
unknown x and by setting an appropriate solution, virtually all mathematical problems can be solved
by such kinds of routines. Furthermore, one should only write down those things that are formal and
that you are sure to be correct (otherwise, marks will be deducted for wrong statements). Therefore,
leaving the solution blank is common found in their scripts which does not mean that the student did
not tackled the problem nor that the student had no initial ideas of solving it. It is also clear from this
study that students’ conception of mathematics is shaped by classroom experience. Most problems
given to students lack variations, possess a unique answer and allows only one way of tackling them.
It is thus not surprising that students see mathematics as a set of rules, the task of solving
mathematical problems is to search for these rules and mathematics learning is to have these rules
transmitted from the teacher (Wong, Marton, Wong, & Lam, in preparation).
4. An outlook to possible future research
There are a number of areas that worth further research. First, we may need a more powerful and more
established instrument to tap the conceptions of mathematics and mathematics learning. The interrelationship among students’ conceptions of mathematics and mathematics learning, their antecedents
and consequences worth further exploration. By antecedents we mean the shaping of such conceptions
by their exposure to mathematics, the mathematics problems they encounters and the conceptions of
mathematics and mathematics learning of their own teachers. And by consequences, we refer to their
approaches (surface vs deep) to tackling mathematics problems, their performance in solving routine
and non-routine mathematics problems. Intervention programs to broaden students’ conception of
mathematics could be another exciting area of research too.
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Investigating Conceptions of Math by the Use of Open-ended Mat Problems, presented as ICME-9 TSG11
N.Y. Wong
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