CONCEPTIONS OF MATHEMATICS AND THE LIVED SPACE OF MATHEMATICS LEARNING

Ngai-Ying Wong

The Chinese University of Hong Kong, Hong Kong

1. Introduction

Understanding what affect students’ learning has been a perennial question in education, and has been revisited time and again from different paradigms and perspectives under different background assumptions and constraints. Numerous efforts had been made which came out with innumerable publications even if we confine to the subject of mathematics. More and more attention has been given in recent years to the inter-relationship among modes of mathematics learning, outcomes of mathematics learning, and beliefs about mathematics (Leder, Pehkonen, & Törner, 2003). In fact, previous studies have revealed that students’ beliefs about mathematics as a discipline, beliefs about mathematics learning, and beliefs about the self situated in a social context in which mathematics is taught and learned are closely related to the students’ motivation to learn and their performance in the subject (Cobb, 1985; Crawford, Gordon, Nicholas, & Prosser, 1994, 1998a, 1998b; McLeod, 1992; Pehkonen & Törner, 1998; Underhill, 1988). Indeed, students’ beliefs are the key to understanding their actions (Wittrock, 1986).

Mathematics is often seen by students as a body of absolute truth (Fleener, 1996) and a set of rules for playing around with symbols (Clay & Kolb, 1983; Kloosterman, 1991; McLeod, 1992). A study in the United States has revealed that 83% and 50% of the seventh-grade students agreed or strongly agreed respectively with the statements “There is always a rule to follow in mathematics” and “Learning mathematics is mostly [memorizing]” (Dossey, Mullis, Lindquist, & Chambers, 1988). Frank’s (1988) study of junior high students has also discovered that students hold the following views:

(a)  mathematics is computation,

(b)  mathematics problems should be solved in less than five steps (or else, something is wrong with either the problem or the student),

(c)  the goal of doing a mathematical problem is to obtain the correct answer, and

(d)  the role of the student is to receive mathematical knowledge and to demonstrate that it has been received.

In 1996, a research team (other team members: Chi-Chung Lam and Ka-Ming Wong) was set up to investigate the conceptions of mathematics among students and teachers in Hong Kong and Mainland China and their relationships with mathematics learning and teaching. A series of research studies were conducted and came up with a number of findings (Lam, Wong, & Wong, 1999; Wong, 2002, 2004; Wong, Marton, Wong, & Lam, 2002; Wong & Watkins, 2001). Similar to previous studies, it was found that students generally possess narrow conceptions of mathematics. In particular, Hong Kong students take mathematics as “a subject of calculables,” and they believe that mathematics involves thinking and is useful (in daily life and other disciplines) (Wong, 2002).

Such narrow conceptions of mathematics have been found to constrain mathematics learning and their approaches to solving mathematical problems. Students who perceive mathematics as a set of rules may apply rote-memorization of facts, rules, and procedures of stereotypical problems as an appropriate learning strategy, resulting in a low level of understanding (Confrey, 1983; Peck, 1984; Underhill, 1988).

The relationship between students’ beliefs in mathematics and their problem solving behaviours was investigated by Schoenfeld (1989) using a set of 70 closed and 11 open questions administered to 230 tenth-grade to twelfth-grade mathematics students. A popular response was “You must know certain rules, which are a part of all mathematics. Without knowing these rules, you cannot successfully solve a problem.” Thus practice and memorization are important to learning. Students expect, or are expected, to master the subject “in bite-size bits and pieces” (Schoenfeld, 1989, p. 344). Schoenfeld has called it a “rhetoric of mathematical understanding,” and such an experience, year after year, has shaped students’ beliefs. For instance, “[s]tudents come to expect typical homework and test problems to yield to their efforts in a minute or two, and most of them come to believe that any problem that fails to yield to their efforts in 1 to 2 minutes of work will turn out to be impossible” (Schoenfeld, 1989, p. 348). In such a learning environment, students come to separate school mathematics from abstract mathematics.

2. Our project

Here is a summary of what we have done in these years (please refer to Figure 1 for our research model).

l  Prologue (1992-93): use of open-ended questions to tap when did students regard themselves as having understood some mathematics (Wong & Watkins, 2001).

l  Students’ conceptions of mathematics (1996-97): use of hypothetical situation and asked student to judge whether it is “doing mathematics” in each case (Wong, Lam, & Wong, 1998).

l  Further validation of results (1997-98): testing of reliability of a questionnaire developed from the results obtained above (Wong, Lam, Leung, Mok, & Wong, 1999).

l  The relationships between students’ conceptions of mathematics and their problem solving performances (1997-2002): use of open-ended mathematics problems to tap students’ approaches to tackling these problems in relation with their conceptions of mathematics (Wong, Marton, Wong, & Lam, 2002). Further investigations by interviews (Wong & Sun, 2002; Wong, Wong, Lam, & Law, 2004)

l  Teachers’ conception of mathematics in various regions (1998-2002): by questionnaires and by interviews (Perry, Vistro-Yu, Howard, Wong, & Fong, 2002; Perry, Wong, & Howard, 2002; Wong, Lam, Wong, Han, & Wong, 2003; Wong, Lam, Leung, Mok, & Wong, 1999; Wong, Lam, Wong, Ma, & Han, 2002).

l  Relationship between teachers’ conceptions of mathematics and their teaching (1999-03): the current state of the lived space (Wong, Lam, & Chan, 2002; Wong, Han, Wong, in press)

l  Changing students’ conceptions of mathematics (2001-03): by systematic introduction of variations (Wong, Chiu, Wong, Lam, in press; Wong, Kong, Lam, & Wong, 2004)

Figure 1. Research model of the present study

3. Students’ conceptions of mathematics

3.1. When did students regard themselves as having understood some mathematics ?

Two hundred and forty one Grade Nine students in Hong Kong were invited to respond to the open-ended questions on approaches towards mathematics problems: (a) “What would you do first in facing a mathematics problem ? (b) “What methods do you usually use in solving mathematics problems ?” (c) “What is the essential element in successful mathematics problem solving ?” (d) “Which do you think is the most important step of a successfully solved mathematics problem ?” and (e) “If you were to score a completed mathematics problem, which do you think is the most important part ?”. It was found that “trying to understand”, “revise and work hard” and “asking others for help”.

Another set of five open-ended questions on understanding mathematics was asked of 356 Grade Nine students. They were: “(a) When will you consider yourself to have understood a certain mathematics problem ?”, (b) “When will you consider yourself to have understood a certain topic ?”, (c) “Before actually tackling a mathematics problem, how can you be sure that you can solve it ?”, “(d) When do you discover that you don't understand a certain topic thoroughly enough ?” and “(e) Which part of the topic do you think you must understand in order to solve problems successfully ?” It was found that over half of the respondents took understanding to be “getting the correct answer”.

These 356 students were later requested to recall any instance in which they understood, realised, grasped or comprehended a mathematics topic, formula, rule or problem, after they responded to the open-ended questions. Results revealed that, for Hong Kong secondary school students, understanding mathematics, may mean the ability to solve problems, the knowledge of underlying principle, the clarification of concepts, and the flexible use of formulas.

3.2. Use of hypothetical situations

Twenty-nine students were confronted with ten hypothetical situations 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, 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. Views of mathematics were also sought from sixteen mathematics teachers. Mismatch between students’ and teachers’ views was found. Some of the views among the teachers were self-conflicting.

3.3. Test of reliability of a questionnaire

A questionnaire comprising the three subscales of “mathematics as calculables”, “mathematics involves thinking” and “mathematics is useful” was developed from the above research (Section 4. above). They consists of 14, 6 and 6 items respectively, put in a 5-point Likert scale (strongly disagree, disagree, slightly agree, agree, strongly agree). It was administered to a total of 6759 (2630 from Grade Six, 1357 from Grade Nine, 1419 from Grade Ten and 1419 from Grade Twelve). Satisfactory reliability indices (Cronbach’ alpha) were obtained (Grade 6: .71, .59, .70; Grade 9: .73, .69, .75; Grade 10: .72, .71, .78; Grade 12: .73, .69, .76). In general, the students agreed that mathematics is something calculable (mean scores of 3.38, 3.27, 3.32 and 3.21 for Grades 6, 9, 10 and 12 respectively), involves thinking (mean scores of 3.90, 3.92, 3.94 and 4.04 for Grades 6, 9, 10 and 12 respectively) and is useful (mean scores of 3.72, 3.24, 2.99 and 3.22 for Grades 6, 9, 10 and 12 respectively). As we move up the grade levels, the extents to which they think mathematics involves thinking increased and decreased for usefulness.

3.4. 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).

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. In fact, by analysing 1557 problems given by the teachers to the students among 4 topics in 15 schools in Hong Kong, it was found that most problems given to students lack variations, possess a unique answer and allows only one way of tackling them (Wong, Lam, & Chan, 2002). 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. So it is natural for us to turn our attention to the conceptions of mathematics and of mathematics teaching among the teachers themselves.

4. Teachers’ conceptions of mathematics and mathematics teaching

4.1. Questionnaire

An instrument developed and validated by Perry, Tracey & Howard (1998) was translated into Chinese and administered to 369 Hong Kong primary mathematics teachers, 275 Hong Kong secondary mathematics teachers. The questionnaire was also administered to 105 primary mathematics teachers in China Mainland (Changchun) and 156 primary mathematics teachers in Taiwan for possible cultural comparison. The questionnaire consisted of 7 items on “mathematics instruction as transmission” and 11 items on “child-centredness of mathematics instruction” and were put in a 5-point Likert scale (strongly disagree, disagree, slightly agree, agree, strongly agree). Satisfactory reliability indices (Cronbach’s alpha) were obtained (Transmission: 0.64, .61, .63 and .65 for Hong Kong primary, Hong Kong secondary, China Mainland and Taiwan respectively; child-centredness: .73, .69, .73 and .73 for Hong Kong primary, Hong Kong secondary, China Mainland and Taiwan respectively).