The lived space of mathematics learning

Journal of Mathematical Behavior21 (2002) 25–47

The lived space of mathematics learning

Ngai-Ying Wonga,∗ , Ference Martonb, Ka-Ming Wonga, Chi-Chung Lama

a Department of Curriculum and Instruction, The Chinese University of Hong Kong, Shatin, N.T., Hong Kongb Department of Education and Educational Research, University of Göteborg, Göteborg, Sweden

Abstract

Students’ conceptions of mathematics and mathematics learning are important research issues in mathematicseducation, as they bear significant relationship to student’s mathematics performance. In this paper, as an attemptto reconceptualise students’ conceptions of mathematics, the notion of “thelived spaceof mathematics learning”is introduced, referring to the space experienced by learners and spanned by all the dimensions in which problemsin mathematics vary from each other, and from non-mathematical problems. This “lived space” constrains whatthe learners will regard as mathematics and as mathematics learning, and constrains therefore their conceptions ofmathematics and mathematics learning. Based on this understanding, the present study is aimed at investigating therelationship between students’ conceptions of mathematics and how they actually tackle mathematical problems.By using a variety of open-ended non-routine mathematical problems, the extent of variations in students’ con-ceptions of mathematics and in their lived space of mathematics learning were explored. Non-routine mathematicsproblems as tasks could bring about a sharp contrast with the usual problems in their classroom learning. Studentswere interviewed after they were asked to attempt several mathematical problems, some being routine and someopen-ended. Through a detailed analysis of students’ interview protocols, characteristic features of their concep-tions of mathematics and mathematics learning have been revealed. These were discussed in terms of the existingclassroom teaching and examination culture. It is argued that the present notion of the lived space can also point toways to broaden students’ current narrow conceptions of mathematics.© 2002 Elsevier Science Inc. All rights reserved.

Keywords:Mathematics learning; Conception of mathematics; Mathematical beliefs; Problem solving; Open-ended questions;Approaches to problems; Problem-solving strategies; Approaches to learning; Asian learners; Mathematics education

1. Introduction

Numerous studies have revealed that beliefs about mathematics as a discipline, beliefs about mathe-matics learning, beliefs about mathematics teaching, and beliefs about the self-situated in a social contextin which mathematics is taught and learned are closely related to the students’ motivation to learn and

∗ Corresponding author. Tel.:+852-2609-6914; fax:+852-2603-6724.E-mail address:[email protected] (N.-Y. Wong).

0732-3123/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.PII: S0732-3123(02)00101-3

26 N.-Y. Wong et al. / Journal of Mathematical Behavior 21 (2002) 25–47

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 keyto understanding their actions (Wittrock, 1986), and students’ failures to solve mathematical problemsare directly attributable to their less powerful beliefs about the nature of mathematics and mathematicsproblem solving (Schoenfeld, 1983).

It is a common prejudice that mathematics is culture-free and value-free (Bishop, 1988). In fact, thebeliefs on the nature of mathematics are as diverse as the subject itself (McDonough & Wallbridge,1993). Traditionally, mathematics has been defined as what mathematicians do and have worked out. Yet,classroom mathematics, i.e., the way mathematics is done in the classroom, has its own forms and traditionsdistinctive from mathematicians’ mathematics (Cobb, Wood, Yackel, & McNeal, 1992; Lampert, 1990;Schoenfeld, 1989). Indeed, mathematics learners, taken as a cultural group, generate their own conceptionof mathematics in their own forms and ways (Bishop, 1988; McDonough & Wallbridge, 1993).

Research has repeatedly revealed that mathematics is regarded as a body of absolute truth (Fleener,1996), and also for some others, as a set of rules for playing around with symbols (Clay & Kolb, 1983;Kloosterman, 1991; McLeod, 1992). There were 83 and 50% of the seventh-grade students participatingin the Fourth US National Assessment of Educational Progress who expressed agreement or strongagreement with the two statements “There is always a rule to follow in mathematics” and “Learningmathematics is mostly [memorising],” respectively, (Dossey, Mullis, Lindquist, & Chambers, 1988).With such a conception, knowing mathematics, to these students, is done through memorisation andalgorithms, and learning is a transmission process. In brief, knowing is believing, learning is developingand altering beliefs, and teaching is helping others develop and alter beliefs (Underhill, 1988).

Based on an investigation of junior high school students,Frank (1988)came to the conclusion that,in students’ eyes, mathematics is computation, mathematics problems should be solved in less than fivesteps (or else something is wrong with either the problem or the student), the goal of doing a mathematicsproblem is to obtain the correct answer, the role of the student is to receive mathematical knowledgeand to demonstrate that it has been received, and in the teaching-learning process, the student is passivewhile the teacher is active. Since there is always one correct way and/or rule that could be followed tosolve any mathematical problem, the task of the problem solver is virtually the searching of such a wayto solve the problem (Carpenter, Lindquist, Silver, & Matthews, 1983). Lester and Garofalo (1982)toofound that third and fourth graders believed that mathematical problems could always be solved by usingbasic operations and could always be solved in only a few minutes.

As Lampert (1990)remarked, “Commonly, mathematics is associated with certainty: knowing it, withbeing able to get the right answer, quickly. These cultural assumptions are shaped by school experience,in which doingmathematics means following the rules laid down by the teacher;knowingmathematicsmeans remembering and applying the correct rule when the teacher asks a question; and mathematicaltruth is determinedwhen the answer is ratified by the teacher. Belief about how to do mathematics andwhat it means to know it in school are acquired through years of watching, listening, and practising”(p. 32, italics in the original).

Naturally, such a classroom mathematics culture has been generated on a number of factors, amongwhich the teacher and the curriculum are some prominent ones. Teachers’ conceptions of mathematicsare frequently reflected in their teaching acts (Pehkonen & Törner, 1998; Shirk, 1972; Thompson, 1992).Those who see mathematics as following sets of procedures invented by others will provide studentswith little opportunity for making sense out of mathematics (Battista, 1994). Some teachers think thatmathematics is understood when students can successfully follow procedural instructions. For similar

N.-Y. Wong et al. / Journal of Mathematical Behavior 21 (2002) 25–47 27

observations,Cobb et al. (1992)have contrasted school mathematics with inquiry mathematics. Theypointed out that the school, the society and the textbooks enculturate students into the folk belief that it isimpermissible to use any methods other than the standard procedures taught in school to solve school-likemathematical tasks, and the use of these procedures is the rational and objective way to solve mathematicaltasks in any situation whatsoever (Lave, 1988).

The students’ (and probably the teachers’) views of mathematics, and even more so of classroommathematics, are very much narrower as compared to mathematicians’ views of mathematics. Differentfeatures mentioned above (namely, that it is computational, the solution of a problem does not involvemore than five steps, there is always one correct answer, etc.) must have been perceived by students oncontrasting mathematics with things of other characteristics (say, with something that is not computational,with other problems involving much more than five steps and having more than one solution). This mustbe how the students have arrived at seeing these features as constraining and defining mathematicsand classroom mathematics. In order to discern any specific features, we must experience variation indimensions corresponding to those features. Otherwise, we would never be able to discover them. If allproblems had only one answer, we would not have ascribed this feature to mathematical problems; ifeverything were computational, being computational would not have been an experienced defining featureof mathematics.

Discernment is always a function of variation. But variation in this particular case takes the formof a contrast between what is seen as mathematics and what is not. The same features could howevercorrespond to what is experienced as variationwithin mathematics. Some problems can be solved withinseconds, others take a lifetime; some are computational, others are not; some problems do have onesingle answer, others have infinitely many solutions, etc. Would the students be exposed to such kindsof variations, they would discern the features as features of (rather than constraints on) mathematics.We could imagine all the dimensions in which problems in mathematics vary from each other, andfrom non-mathematical problems, forming the space that defines mathematics. Let us see this space as aspace experienced differently by people. In this sense, we can speak of “thelived spaceof mathematicslearning.” It can be narrower for some learners and much wider for others, undifferentiated for someand highly complex for others. For every one of us our “lived space of mathematics learning” constrainswhat we take to be mathematics and what we take to be mathematics learning, what our conception ofmathematics is and what our conception of mathematics learning is.

Since discernment is an essential element to learning and variation is crucial to bringing about dis-cernment, the lack of variation in the “lived space” of learning mathematics experienced by the studentswould inevitably lead to a relatively narrow conception of mathematics (Marton & Booth, 1997; Marton,Watkins, & Tang, 1997; Runesson, 1999). Furthermore, they would also hold a narrow conception ofmathematical learning and would possess limited strategies when they are confronted with mathematicalproblems.

The relation between students’ belief in mathematics and their problem solving behaviour was inves-tigated bySchoenfeld (1989)using a set of 70 closed and 11 open questions administered to 230 Grade10–12 mathematics students. “You must know certain rules, which are a part of all mathematics. Withoutknowing these rules, you cannot successfully solve a problem” was a popular response. Thus, practicesand memorisation are important to learning. Students expect, or are expected, to master the subject “inbite-size bits and pieces” (p. 344). Schoenfeld has called it a “rhetoric of mathematical understanding,”and such an experience, year after year, has shaped students’ belief. “Students come to expect typicalhomework 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 12 min of work will turn out to be impossible”(p. 348). In such a learning environment, students come to separate school mathematics from abstractmathematics.

As mentioned above, such conceptions of mathematics would obviously constrain students’ conceptionsof mathematics learning and their approaches to solving mathematical problems. For instance, the beliefthat mathematics is a set of algorithms may imply rote-memorisation of facts, rules and procedures ofstereotypical problems as an appropriate learning strategy, resulting in a lower level of understanding(Confrey, 1983; Peck, 1984; Underhill, 1988). It is also found that the failure to recognise a givenproblem-solving situation as mathematical may be the cause of students’ apparent incompetence, asrevealed in mathematical assessment (Brown et al., 1988a, 1988b; Kouba et al., 1988a, 1988b; Kouba &McDonald, 1991).

Relationships between beliefs about mathematics and learning outcomes have been established byquantitative methods such as structural equation models (Kloosterman, 1991; Kloosterman & Stage, 1992;Stage & Kloosterman, 1995), whereas qualitative methods, including phenomenography, were utilised insome other related research studies (Crawford et al., 1994, 1998a, 1998b; see alsoPehkonen & Törner,1998). Wong (1993, 1995)used open-ended questions like “Mathematics is. . . ” and requested studentsto write any of their personal episodes in which they understood, realised, grasped or comprehended somemathematics. It was found that students have related mathematics understanding strongly to “getting thecorrect answer” (Wong & Watkins, 2001).

To date, though much work has been done in exploring students’ conception of mathematics and itsrelationship with solving mathematical problems, most of these studies have relied on one-way responsesof the students. Without appropriate dialogue, researchers lack the chance of clarifying what is in thestudents’ minds. The present authors (seeLam, Wong, & Wong, 1999; Wong, 2000; Wong, Lam, &Wong, 1998) began to tackle the issue by confronting 29 students with 10 hypothetical situations inwhich they were asked to judge whether “doing mathematics” was involved in each case (see alsoKouba& McDonald, 1991). Results revealed that these students associated mathematics primarily with itsterminology and content, and that mathematics was often perceived as a set of rules. Wider aspects ofmathematics such as visual sense and decision-making were only seen as tangential to mathematics. Inparticular, these features were not perceived as “calculable,” though mathematics was also recognised assomething closely related to thinking.

Despite the voluminous literature on students’ conceptions of mathematics and their relations withperformance in mathematics, not much effort has been devoted to the investigation of the relationshipbetween students’ conceptions of mathematics and how they actually tackle mathematical problems. Thiscan best be done by letting the students explicate how they actually approach mathematical problems,especially the non-routine ones. In the present paper, we are going to report on how students’ conceptionsin mathematics were investigated with the help of open-ended mathematical problems.

Open-endedness was used to introduce variations in the tasks so as to tap the extent of variations inthe students’ conceptions of mathematics and in their lived space of mathematics learning. Non-routinemathematics problems as tasks could bring about a sharp contrast with the usual problems in theirclassroom learning, and their conception of classroom mathematics could therefore be made more visibleagainst this background. Students were interviewed shortly after they were asked to attempt a number ofmathematical problems, some being routine and some open-ended. They were asked to recall how theyhad tackled these problems, how they had approached them and their views on these problems which wererarely met in their classrooms. With the actual experience of the process of solving mathematical problems


as stimulation (rather than hypothetical situations in a previous study), more in-depth expositions of theirconceptions of mathematics and mathematics learning were thus obtained.

2. Methodology

2.1. Participants and administration of tasks

Nine classes from each of Grades 3, 6, 7 and 9 (260, 277, 336 and 343 students, respectively, making atotal of 1216 students) were asked to respond to given sets of mathematics problems. Two students fromeach class (2× 9 × 4 = 72) were then interviewed individually on their strategies used in solving theseproblems. Owing to the limitations of the present research setting, the analysis could only be made withall the participants taken as a group. Analysis at the individual level has not been made.

2.2. Mathematical problems used

The students were requested to attempt three sets of mathematical problems before the interview. Thesewere computational problems, word problems and open-ended problems. For open-ended problems, thestudents were asked to give explanations of their working procedures, and the follow-up interview focusedmainly on this part. The number of problems in each part for different grade levels are listed inTable 1.

2.3. The open-ended problems

Cai (1999)defines an open-ended mathematics problem as one which allows openness in any of thefollowing: (a) given information, (b) goal to be achieved, or (c) the process of solving it (Fig. 1).

Becker (1998)has further identified three types of open-ended problems, with ascending levels ofdifficulties, namely, that (a) the process is open, (b) the end-products are open, and (c) the ways of

Table 1Mathematical problems used in the study

Grade 3 Grade 6 Grade 7 Grade 9

Computational problems 2 2 2 2Word problems 4 3 2 2Open-ended problems 4 4 4 4

Fig. 1. Definition of open-ended problems.


formulating the problem are open. In the present research, open-ended problems fromCai (1995)andCalifornia State Department of Education (1989)were adapted for use. The final sets of problemsinclude:

(a) Problems with irrelevant information (Low & Over, 1989).(b) “Problematic word problems” (Verschaffel, De Corte, & Borghart, 1997; Verschaffel, De Corte, &

Lasure, 1994; Yoshida, Verschaffel, & De Corte, 1997).(c) Problems which allow more than one solution.(d) Problems which allow multiple methods.(e) Problems with different interpretations possible.(f) Problems which ask for communication.(g) Problems which need judgement.(h) Problems which involve decision making.

Examples of these problems are given in Appendix.

3. Results

Both the students’ workings of the mathematical problems and their verbal responses in the interviewswere analysed. The focus of analysis included their performance in these problems, their conceptions ofmathematics, and how they approached these mathematical problems.

3.1. Performance

As expected, students’ performance in routine problems was generally good.1 A variety of meth-ods were found in their solutions too. However, their performance was not as good for those problemsin which there could be different interpretations of what was asked or when there was a conflict withthe realistic situation. They were also weak in providing evidence or making judgement. For problemshaving several answers, usually they gave only one. Students were weak in problems involving compli-cated situations or tedious calculations. The percentages of correct solutions of the problems are listedin Table 2.

In general, major student weaknesses included the misinterpretation of the question and their inabilityto express their thinking. They failed to give justifications when they were asked to. Careless mistakesin computations could lead to wrong answers too but this was not serious. As it was common practiceto leave it blank when the students could not solve a problem, we did not know how much the studentunderstood the questions with such blank solutions.

3.2. Conception of mathematics

Wong et al. (1998)has revealed that students associated mathematics with its terminology and content,and that mathematics was often perceived as a set of rules. Similar findings in students’ conceptions ofmathematics were found in this study.

1 Editor’s note: Recall that these students live in Hong Kong.


Table 2Percentages of correct solutions of the problems

Types of problems Grade 3 Grade 6 Grade 7 Grade 9

Computational problems #1: 90 #1: 85 #1: 9 #1: 90#2: 70 #2: 75 #2: 15 #2: 75

Word problems #1: 20 #1: 80 #1: 60 #1: 70#2: 90 #2: 90 #2: 2 #2: 55#3: 85 #3: 5#4: 85

Open-ended problems #1: 15 #1: 90 #1: 80 #1: 18#2: NA #2: NA #2: NA #2: NA#3: 5 #3: 2 #3: 22 #3: 3#4: NA #4: NA #4: NA #4: NA

NA: these problems do not involve single exact solutions.

3.2.1. CalculableMathematics was perceived as something calculable. “[It is]+, −, ×, ÷” (B3-3, T3-5),2 no matter

whether the students liked it or not. For instance, while one Grade 3 student said, “I love mathematicsbecause it is calculable” (B3-1), another said “I don’t love mathematics because it is too clumsy, sometimes+, sometimes−, and sometimes× and÷” (K3-1). Thus, they did not take our non-routine problems asmathematics, since they were not calculable (H3-8, S3-3), did not involve numbers (Y3-6), and involveddecision-making (H3-7). In brief, they held a segregated view of the subject of mathematics, which canwell be illustrated by the following dialogue:

S. (Student): This question looks like logical reasoning more than mathematics.I. (Interviewer): What is the difference between them?

S.: Logical reasoning involves words more, and for math, all are numbers.(P9-6; C9-6)3

3.2.2. Scarce use of diagramsAs observed in students’ solutions, most did not use diagrams in solving problems. Some of them

thought it “a waste of time” (M3-5, B6-8, Y3-12) and some even thought that it was not mathematics.

Mathematics should involve calculation, drawing figures [is] more like cartoon drawing. (K6-8)

Though the following two students expressed quite different views on using diagrams, most studentsthought that drawing diagrams was fine art more than mathematics: “I like using diagrams since I lovefine art” (M3-5), and “I won’t use diagrams since I am not good at art” (L6-3).

However, some secondary school students reflected they did use diagrams to help understand thequestion (Q7-6). Here are responses of two Grade 9 students:

If [I can’t understand the question, I would] look at the diagram or draw a diagram myself to help methink. (Q9-4)

2 For these quotations from interviews,Xn-p refers to citationp of a Graden student at schoolX.3 (A, B, C) means the citation here is from A but B and C are similar utterances.


I would if necessary. Diagrams can help thinking. Sometimes some problems need some [given]conditions to solve them and drawing diagrams can let me know clearly what these conditions are.(E9-9)

As mentioned inWong et al. (1998), spatial concepts have been under-represented.

3.2.3. Thinking involvedConsistent with our previous research findings, students expressed the view that mathematics is a

subject involving thinking. Some students even loved mathematics because it could provoke thinking(T3-1, Z3-1, H6-1, E7-2), and train the mind (H3-2, T6-1, E9-1). Their opinions were quite unanimous.Interestingly, a few did not like mathematics exactly for the same reason: “I don’t love math since I don’tlike thinking” (Y6-2). Here are more reasons why the students loved mathematics:

Mathematics is not straightforward. I always find difficulty but the more I need to think the more I findit interesting. (Y6-1)

Mathematics “activates the brain.” We can do it in one way. We can also do it in another. (M6-1)

Mathematics is challenging, tricky, uses the brain. And there is no need for rote memorisation. (T6-2)

The last response shows clearly that they thought understanding a better way to learn mathematics thanmemorisation (Marton, Tse, & dall’Alba, 1996; Marton et al., 1997). In some of their eyes, “Mathematicsis a subject that needs no revision” (L6-9; A7-1, E7-3, Q9-1) and precisely for this reason, some of them“love mathematics as rote memorisation [is] not necessary. Just use your brain to think” (J7-1, N9-1).The following two students gave their reasons why mathematics needed no memorisation:

Mathematics does not need rote memorisation. After knowing a formula you can apply it [and get theanswer]. (D9-3)

I love mathematics since it does not require much revision. Just be attentive in class, do some practicesat your free time and there is no need for you to revise. (M6-2)

Yet another student gave an interesting reason why he loved mathematics:

My memory is weak [thus I love mathematics], math does not involve too much rote memorisation,[but] needs thinking. (E9-3)

Some students did not find it conflicting to regard mathematics as something calculable and somethinginvolving thinking:

Mathematics differs from other subjects as it is a science subject, involves thinking and calculations.(E9-2)

Most of them found a distinction between the two notions when they compared these non-routineproblems with the types of problems they met day-to-day:

Most [routine] problems in textbooks involve calculations and these [non-routine] problems involvethinking. (Y3-3)

I think to calculate is easier than to think. (Y6-4)


3.2.4. UsefulnessAnother characteristic of mathematics perceived by the students is that mathematics “is useful in daily

life” (T3-2; L3-1), for instance, “see how much one has to pay in buying things” (L3-2) and “it can helpmake money” (S3-2).

3.3. Conception of classroom mathematics

Besides their conception of mathematics, the students also expressed much of what they perceivedmathematics in the classroom was like. This includes their encounters with mathematics and how mathe-matics was learned in the classroom context. We can see in the next section that their views are consistentwith their conception of mathematics.

3.3.1. Quick answersMost of the students found a difference between these non-routine problems with those they met

in the classroom. Requiring students to explain has been rare in school mathematics on working withexamples, exercises and examinations, although logical reasoning should be one of the major aims oflearning mathematics. Thus, correct answers are of utmost importance: “formulas are unimportant, whatis essential is a right answer” (C7-6). Some even suggested that those that needed explanations were notmathematics, “as they do not involve numbers” (B3-4; T3-4, L3-4). While we will elaborate more in thesection on writing mathematics, we see that not only is finding the answer stressed in most cases, but alsoobtaining the answer in a few minutes. One of the students reflected that “I did not think for too long” andon being asked for the reason, he said that “since I don’t want to think” (Y6-9). It is worth investigatingwhether such a classroom culture of asking for quick answers has indeed induced a habit of having noperseverance among our students in tackling mathematics.

3.3.2. Only one solutionNot only is it that the answer has been greatly stressed, but in nearly all cases, “each mathematics

[problem] has only one solution, unlike Chinese language [composition] which can have many answers,”as one student suggested. He continued to say, “hence, if you make a single mistake in mathematics, youwill get the wrong answer. This is not so in Chinese language” (H6-10). Close-endedness has been thenature of most mathematical problems they came across. “There is definitely one answer, very definite.One won’t be asked of possibilities” (W7-9). So, on meeting these non-routine problems, which mighthave more than one answer, they “didn’t proceed after getting an answer [one of the answers]” (K6-7).Furthermore, these answers were usually “tidy ones,” like integers or simple fractions. Teachers seldomset problems, which came up with an answer that looked very complicated. As one of the students said,“usually we have only one answer which is an integer” (C7-9). To some of them, these open-endedproblems did not look like mathematics very much: “These problems involve real life situations [ratherthan mathematics], not asking for a definite answer” (W9-3).

3.3.3. Solvable by routinesAs mathematics has been perceived as something calculable, students generally believed that mathemat-

ical problems could be solved by some routines, i.e., worked out step by step via a series of expressions.Thus, the search for an appropriate formula or rule would be of utmost importance. As one student said,“First I would ‘list out the expression concerned,’ then calculate slowly, [step by step accordingly]”


(H6-7). In general, they thought that they “just need to apply the formulas taught directly” (C7-10). Somefelt that these non-routine problems were more challenging while others “don’t like these problems asthey do not follow a fixed rule [for solution]” (P7-3). Thus, they did find difficulty in these non-routineproblems:

Usually, we are only given a formula, and you do it, do it, do it. There is not much application. These[non-routine] problems need problem solving skill and analytic power. (E9-5)

For routine problems, you can find a method after reading through the question. I can’t [even] find away to tackle them for these problems. (W9-4)

So they thought that setting an unknown and subsequently an equation would solve every problem: “Ithought I could calculate by setting an ‘x’ [in fact he couldn’t in that particular problem]” (W9-3).

I.: In general, how would you approach a problem?S.: Think of a formula.I.: What kinds of formula?S.: An equation.I.: Why do you use equations?S.: Use an equation to find the unknown. Set it [the unknown] as ‘x’ and then calculate it out.

(Z6-6)

As such, they tended to stick to what they had gotten to work without considering reasonableness in thecontext. A typical example can be found in Problem S1-III(d). Though some students realised that thenegative number they came up with was unreasonable, they still put it as the answer.

I did think that the answer was unreasonable, but I really got−40 according to the formula [so I putdown−40 regardless of its being unreasonable]. (P7-5)

Since I got a negative number [and I know that this is unreasonable] so I put down ‘0.’ (N7-3)

3.3.4. Writing mathematicsAccording to students interviewed, for problems they usually encountered in the classroom, only the

answer was stressed, and no explanations were required (e.g., H6-2). While younger students (mainlyfrom Grade 3) loved word problems, students from upper grade levels found it “a waste of time” (L6-2) anddid not have enough time to finish all the problems in this study. They rather preferred “using expressions,which is faster than writing in words” (H6-6). A few Grade 3 students, however, did appreciate thesenon-routine problems:

[I] Love word problems because writing enhances thinking. (H3-4; H3-5, T3-3, L3-10)

[I] like writing down reasons as it is interesting, and we can argue with classmates. (Z3-2; Z3-9)

Students in higher grade levels did not object to the use of writing in solving mathematics problems,but they simply felt that this would be too difficult for them (Y6-5), especially for more complicatedproblems:

I fear that [I am not expressive enough so that] the teacher couldn’t understand, so I did not write.(A7-2)


Certainly, there were some who thought that this could only be marginally regarded as mathematics. Wesee that their conception of mathematics learning was closely related to their conception ofmathematics:

I feel these problems are like composition rather than mathematics. (M6-8)

These problems require writing, like “arts subjects.” (E9-4)

3.4. Approaching a mathematics problem

Considering what the students said about their own ways of tackling problems, we have found thatthe different features of their approaches were consistent with features of their conceptions of classroommathematics.

3.4.1. Searching for toolsWhen mathematics problems are perceived as something solvable by routines, rule searching, as found

in Lee (1980), will be one of the major steps in mathematical problem solving in the classroom. A rule orroutine is a path that leads from what is given to what is being asked: “Look at what conditions I have inhand, what the given information is, and what is being asked” (E9-7). Such an approach was repeatedlyexpressed in students’ responses:

Read through questions, pick out the numbers, try out with+, −, , and see what the problem isasking for. (Y3-11; B3-7)

Read through the question,. . . think carefully what I should do. If I find out it should be division, andthen followed by multiplication, I would [get the answer and] check if the answer is correct. (Z3-7)

Getting the right formula was therefore regarded as very important (see, S6-2): “Understand the ques-tion, find out what is asked, and list out an ‘expression”’ (T3-7). Some students also told us how theycame up with such formulas:

Start thinking from what is being asked, then spot any relation between the “front” and the “rear” [i.e.,what is given and what is being asked], then try to list an expression. (Y6-7)

Sometimes they found the “trick” by looking into the details of the specific context:

Since the problem says 26 lamps and 14 lamps and then 5 in a row, it must be addition followed bydivision. (S3-7)

In the following dialogue, the student expressed quite clearly how he usually did in approaching amathematics problem:

S.: Read through the question again. If still can’t understand, I would read through again word-by-word.Think of their [the words’] meanings. Make guesses until I make the right guess.

I.: How do you know that you have made the right guess?S.: If I have made the wrong guess, the teacher will tell me. And if I can understand the terms [words]

and the meaning [of the question], I can come up with an expression [formula].I.: And how do you know which of the expressions is correct?


S.: Think again, write it down and look at it. Use your brain. If I feel [intuitively] that the expressionseems to be incorrect, I will try another one. If I think it is correct, I will proceed with it.

(M3-5)

3.4.2. Key wordsThe information in a mathematics question is not just given so that one can get the solution. Each piece

of information is usually so designed that it has to be used in the problem solving procedure. In mostschool mathematics problems, each number in the question is used exactly once. As one of the studentsaid, “Jot down the numbers in the question. Then read through the question to see how I can tackle it,e.g.,+, −, , and thus I get the answer” (K6-4). So if the given data are not adequate, the students haveto find out some more for themselves.

I.: Why did you use a ruler to measure [the dimensions of the figure]?S.: I can’t think of other methods. The only information is 140 km, and nothing else. So I have to find

out other information myself.I.: Usually, would you do similar things?S.: No. Usually all the information is provided and so you know how to calculate.

(J7-2)

The identification of key words is very important in problem solving too. As a student said, “I wouldpick out the keywords. Then I would concentrate [myself] in analysis to see which formulas we can use”(W9-7). Another student talked about his line of thought:

This question mentioned about 50%. So it looks like percentage stuffs, but it does not seem tohave any [numerical] relationship with the question. So it must be sort of “reasoning problems.”(M6-4)4

As most problems appearing in school mathematics are artificially set (so that all the numbers are usedjust once), some “rules of the game” are often embedded. For instance, one problem of this study saidthat 50% of the students love basketball and 50% love football (we deliberately introduced the fuzzinesshere), and many took for granted that “5 love basketball” meant “5 love basketball only” (see Problem(e) in Appendix) (H3-10, B3-6, Z3-8, Y3-9).

In another problem having many answers (which is quite uncommon in school mathematics), studentstook it for granted that they were asked for the smallest answer. “Since it is not mentioned in the question,I suppose that we are asked the smallest number which satisfies the conditions” (H6-5).

3.4.3. Juggling with tools at handBesides handling with what is given and what is being asked, students will also handle the problem by

choosing from among the (usually limited) number of tools at hand. In order to achieve this, an importantstep is, of course, to identify the topic involved, as it is uncommon to have problems in the mathematicsclassroom involving more than one curriculum topic. This is clear in the following responses:

S.: I would think how I would get the answer.I.: How do you know which formula you should use?

4 Literal reasoning and numerical reasoning are the two types of problems that appear in the academic aptitude test for secondaryschool place allocation at Grade 6.


S.: You will know when you are familiar [with doing similar problems]. I don’t know how myself.(Y6-6)

Try out those methods concerned that we have learned. (Y3-17)

Analyse which type the problem belongs to. Put in the numbers. Then you should be able to come upwith the answer. Only for those difficult ones would you fail to come up with an answer. (M6-6)

We can see more clearly from the following three responses that more competent problem-solverscould handle more skilfully the three aspects of problem solving, viz., what is given, what is being asked,and what tools we have at hand. Nevertheless, we hesitate to jump to the conclusion that they solvedmathematics problems with such skills without some “genuine” understanding of mathematical concepts.We believe that conceptual understanding is not segregated from problem solving skills (Wu, 2000):

See what the question asks for, e.g., if it asks for volume, then use the usual formula [for volume]:length× width × height. (A7-5)

S1.: See what the question is asking for and then think of a [related] formula.I.: How?

S1.: For instance, for this particular problem, since it concerns the circumference, so I looked at thoseformulas that involve circumference.

I.: If you can think of more than one formula, how do you know which is correct?S1.: One should know that each formula has its particular use.S2.: See what is the concern of the problem [involving which topic].

(N7-4)

S1.: Analyse the question to see what is being asked. Then use simple methods, and see if I can get tothe answer step by step.

I.: How do you know what methods you can use?S1.: Look at the question, e.g., if it concerns the triangle, then I would recall those formulas [related to

triangles]; if it concerns statistics, I would think of those in probability.S2.: More or less like his, but if it is a mixed type, usually I need much time to find out how to calculate.

(N9-2)

After having identified the topic, to imitate what the teacher has done with related worked examples isanother common practice (see, e.g., B3-8), as said by a student, “Read the question several times and seeif the problem has been taught before” (T6-3). And there was yet another one:

S1.: The first thing I do is to recall if there are similar problems I have done in the textbook, and see ifthere is any formula useful.

I.: How would you know that the formula is the right one?S1.: See if it is similar [in a similar context]. In fact I can’t tell exactly, just a guess.S2.: Read the question several times and see if I have been taught.

(J7-3)

3.4.4. Plausible reasoningIn the last dialogue, the student mentioned about making guess. In fact, the word “guess” appeared

repeatedly in students’ responses. On finer analysis, we have found different levels of guess making


among these responses, namely, from merely wild guesses to plausible reasoning. For instance, one said,“Just seek for a chance” (S6-3). Others said, “rely on wild guess” (Y3-15; Q7-4) or “rely on wild guess ifall other guesses have failed” (L3-9). Transcripts cited below, however, do show some more “grounded”guesses, where students guessed by picking up clues given in the questions:

First thing is to recall if I have learned it in the primary years. Look at the answer or the last expression[i.e., what is being asked], then see what numbers are offered in the question. Then [work out to] matchwith the meaning of the question. (C7-7)

Read through the question, write down [choose] a formula at random and test if it is correct. Considerthe question and see if the formula is the right one. (A7-4)

S1.: See if there is any relationship among the numbers. Usually you can get the solution.S2.: See [pick out] the main points of the question. Then make some guess.

I.: How would you get the answer given that you have got the main points?S2.: Calculate this and that.

I.: How?S2.: See if the answer thus obtained is reasonable.

(D7-4)

They judged the reasonableness not only from the meaning of the question, but also from the “rule ofthe game” implicit in the classroom culture.

I.: How would you know that you got it wrong?S.: When I’ve got the wrong answer? Sometimes if you’ve got an answer too large or too small, you

would think that it is unreasonable.(Q7-3)

To tackle how to help students move from merely wild guesses to plausible reasoning can shed light onhow problem-solving abilities can be enhanced. Some students expressed more clearly how they tackleda mathematics problem in general.

First, see how many [given] numbers there are in the question. Take out these numbers. See what therelations among them are, simplify these relations step by step, then we could work it out. (B6-6)

S.: Read through the question. Analyse how many points it has [what the essential points of the questionare]. Then work out by synthesising [the points].

I.: How do you know how to work out?S.: Look at the question. Usually you can find out [how to proceed]. If no, use an equation.I.: Why do you choose using equation when you can’t work out?S.: Calculate with the unknown [to set an equation]. You can come up with the answer sooner or later.

(M6-5)S.: See what the question is asking. If I found one difficult, I will skip it first and come back to it later.I.: If you still can’t do it afterwards?S.: Make some guesses and see which answer is most probable.I.: How do you know that an answer is probable?


S.: Look at the numbers. Perform+, −, , and see which answer looks more likely.(M6-3)

As expressed by these students, clues are of utmost importance. These clues not only come from thewording of the questions, but also from the “norms” of the classroom culture. For instance, as mentionedabove, the answers are usually integers and simple fractions, not too large or too small. “If an answer is‘out of scale,’ then it must be wrong” (Q7-5). All these provide additional hints for getting the desiredsolution:

Since putting in groups of five is given in the question, so it is either multiplication or division. Butmultiplication would make the number very large, so it must be division [The problem was: There are26 red lamps and 14 yellow lamps. If we put them in rows of 5, how many rows are there?]. (B3-9)

Since it mentions about putting in groups, it must be division [In fact he was wrong. The question was:There are 10 people in a group among which 5 like playing basketball and 5 volleyball. Are there anyin the group who do not like both? This did not involve division!]. (S3-5)

Another typical example of making reasonable guesses is found in Problem (c) in the Appendix. Hereare a couple of cases of plausible reasoning. Though they did not work in the “standard” way, theymanaged to get the correct solution by picking out all these clues:

Since the answer is not divisible by 2, it must be an odd number. And so I tested [with the otherconditions given in the question] for odd numbers, one by one, and found that the answer was 13.(H6-4)

I don’t think it could be using the greatest common divisor since it involves 2, 3, 4. So it must be usingthe least common multiple. (K6-6)

In the second case, the student read through the question and found it was most likely related to eitherthe greatest common divisor or the least common multiple. However, he got a clue that the greatestcommon divisor was not quite probable since the greatest common divisor of 2, 3, and 4 was 1. So it mustbe using the least common multiple.

3.4.5. When difficulties were foundWhen students found difficulties in mathematics learning, it was reported that the teacher (H3-15,

T3-8), family members (Y3-19, S3-9) and classmates (L3-8, K6-10, Z6-8) were most commonly askedfor help. Strategically, some of them would leave the problem aside for a while (K6-5), or maybe workon the next first (B6-7, M3-4). Some might choose to do homework of other school subjects first (L3-7),and some would try hard by reading the question several times (S3-11, K3-5). Of course there were somewho would simply give up (D9-7).

3.5. The classroom and examination culture

In the foregoing sections, we see how students’ approaches to solving mathematics problems wereconstrained by their views of mathematics learning in the classroom, and these views, in turn, wereconstrained by their conceptions of mathematics, which were largely shaped by their classroom learningexperience. In particular, problems in classroom mathematics are, admittedly, designed so that they can


be solved by using routines, which can be found by either identifying the key words, the curriculum topicinvolved, or other relevant clues. As said by a student, for most problems, it suffices “only to read the lastsentence” [which contains what is being asked] (D9-8), since most problems are of the “find the. . . ” type.One should find no difficulty in solving them if one “would recite the formulas fluently before sitting forexaminations” (D9-9), and “in such cases, it is simple just to find the relevant formula” [to get the rightanswer] (D9-5). This clearly indicates that “recitation,” at least for this particular student, is a result ofthe continued, narrow exposure to stereotypical problems.

The “examination culture” has perhaps played a major role in shaping this scenario too. One of thestudents said he “doesn’t like such problems as they require much thinking,” but he said he would “lovethem if unlimited time is given and no scoring is involved” (D7-2). The following response preciselyshows that the emphasis on scoring has affected how much the answer is being stressed:

Usually marks will only be granted to answers. “Steps” [of the working] won’t get any marks. Evenif you’ve got the [algebraic] “expression” but have not arrived at the answer, you would still get nomarks. (P7-8)

There are at least two “philosophies” of marking mathematics test papers. In examinations like theHong Kong Certificate of Education Examination (a secondary school exit examination at Grade 11),marks are usually given for any correct responses written in the answer books. Thus, candidates tendto write down anything likely to be related to the solution, or have been instructed by their teachers todo so. Conversely, in primary mathematics teaching in Hong Kong, the custom is to deduct marks forwrong answers. This policy may thus stop students making guesses (a step seen above as quite importantin solving problems), or writing down sketchy, incomplete expressions in their solutions. Here is theresponse of a student on a problem with more than one answer:

I.: Why did you finally choose that answer?S.: I did not care to think any more. I finished the work by choosing an answer randomly.I.: So you were not serious about it.S.: I was serious. If I gave two answers and both were wrong, then I would be done for.S.: You could make more explanations in your solution?I.: There was not enough time to finish the solution. In that case, you wouldn’t get any marks. If you say

more, there is more chance to have marks deducted.(L6-5, 6)

4. Discussion

As obvious to everyone, students inevitably spend vast amounts of time at school.Rutter, Maughan,Mortimore, Ouston, and Smith (1979)estimated that students have spent 15,000 h in the classroomby the completion of secondary school (see alsoFraser, 1986). As being part of this extremely longduration, such an extended exposure to the mathematics classroom culture, naturally, has shaped theconception of mathematics among our students. In the present study, such non-routine, open-endedproblems, as tasks, have brought about a sharp contrast with the usual problems in their classroomlearning, and their conception of classroom mathematics could therefore be made more visible againstthis background. Consistent with what has been found in previous research studies, students repeatedlyrevealed in the present study a conception of mathematics as being a body of absolute truth in which


there is always a routine to solve mathematics problems. The task of mathematics problem-solving isthus, in essence, the search for such routines. AsHeddens (1997)has pointed out, with such a narrowconception of mathematics, classroom mathematics seems to be the simple game of “memorising facts→applying algorithms→ following memorised rules→ calculating a result.” Moreover, the prevailingexamination culture reinforces such a status quo (Lee, Zhang & Zheng, 1997). Along this line of thought,every problem in the mathematics classroom has a unique answer, has only one way of tackling and canbe solved within minutes. Furthermore, students held a segregated view of the subject too.

Writing has not been regarded as mathematics, which has been perceived only as calculations withnumbers and symbols. Many students thought that by letting an unknownx to be the answer required andthen setting an appropriate equation, virtually all mathematical problems could be solved by such andsimilar kinds of routines. They also thought that one should only write down formal solution which wassure to be correct, in order to avoid marks being deducted for wrong statements. Therefore, leaving thesolution blank has been commonly found in their scripts for this study, though this does not necessarilymean that the students had not tackled the problem or that they had no initial ideas to solve it.

Students’ conceptions of mathematics and of mathematics learning are intertwined, and these concep-tions have direct impact on their approaches to tackling mathematics problems. When they see mathe-matics as something absolute and static, they will see mathematics learning as a transmission of a setof rules from the teacher. When students having such a conception are confronted with a mathematicalproblem, the first thing in their minds is to search for an appropriate rule for that particular problem.

Though the analysis of the present study has only been performed at the group level, taking all theparticipants as a whole, we have presented a clear picture of how students tackle mathematics problemsby picking up clues from the question and trying to find a suitable rule. In order to search for such rules,they look for clues embedded in the questions, which include the given information, the required un-knowns, the relevant curriculum topic and the format of the question. With the processes of perceiving,sub-goal formulating, rule searching, rule applying and verifying as identified byLee (1980), the modelof students’ mathematics problem solving can be reconceptualised in terms of students’ conceptions ofmathematics found in the present research (Fig. 2). However, students’ conception of mathematics is pri-marily shaped by classroom experiences and influenced by the societal mathematical beliefs (Pehkonen,

Fig. 2. A re-analysis of the process of solving mathematics problems.


1998; or “Anschauung” inSiu, 1995). In brief, owing to a lack of variation in the students’ lived spacein mathematics learning, which includes the variation in the problems they face in their day-to-dayclassroom learning, students show a lack of variation in their conceptions of mathematics and mathe-matics learning. They also lack variation in the strategies used when they approach mathematical prob-lems. By introducing variations into their lived space, it may be possible that this vicious circle canbe broken.

As we have indicated earlier, students’ ways of tackling mathematics problems are constrained bywhat they see as mathematics problems and what they see as reasonable ways of going about solvingsuch problems. And these constraints are derived from the variation they have been exposed to in thetypes of mathematical problems and in ways of trying to solve them. The experienced variations betweenmathematical problems and the experienced differences between mathematical and non-mathematicalproblems define “the lived space of mathematics learning” for the students. We have found that thisspace is rather narrow and most seriously constrains how the students, in the future, will be able to makesense of and handle problems which fall outside the space thus defined. If teachers keep on presentingproblems in their lessons that can be solved by guessing from non-mathematical clues, students’ attentionmay be shifted from coping with mathematical constraints and requirements to such non-mathematicalones. By systematically introducing variation into the problems encountered in the mathematics class,students have the opportunity to develop the capability of coming up with their own ways of solving novelproblems, rather than just searching for the rules. By widening the range of variation in the problems theyencounter, the range of variation in their approaches to solving problems, as well as their views of whatmathematics in and outside school is will likely be widened. And thus the lived space of mathematicslearning will expand. In thus doing, their conception of mathematics can be broadened too with thisbroader lived space of mathematics learning.

Acknowledgments

We would like to express our heartiest gratitude to Prof. Jinfa Cai for allowing us to use hisopen-ended questions in this study. The research project in this paper was supported by the DirectGrant for Research 1997–1998 of the Social Science and Education Panel, The Chinese University ofHong Kong. The opinions expressed are those of the authors and do not necessarily reflect views ofthe funding agency. All correspondences concerning the paper should be directed to the first authorat [email protected].

Appendix

Examples of the open-ended problems used.

(a) Problems with irrelevant information [Problem P3-II(a)]Papa bought a piece of string 29 m long, he cut out 15 m and then 9 m. How long did he cut

altogether?(b) Problematic word problems [Problem P3-III(a)]

101 teachers and students at Oxford primary school went to picnic. If each coach can accommodate25 people, how many coaches do they need?


Fig. 3.

(c) Problems which allow more than one solution [Problem P6-III(d)]Catherine told her brother Tony the game she played in her mathematics class today. Catherine

said, “Tony, we made use of wooden blocks in my mathematics class today. When I put them in pairs,1 was left; when I put them in groups of 3, again 1 was left; and when I put them in groups of 4, 1 wasleft too.” Tony said, “How many were there altogether?” How would you think Catherine’s answershould be?

(d) Problems which allow multiple methods [Problem P3-II(b)]There were 120 pots of flower in the playground. Peter, Paul and Mary each moved away 8. How

many pots were left?(e) Problems with different interpretations possible [Problem P6-III(c)]

Ida knows that 50% of her classmates love basketball and 50% love ping-pong. She takes it thatall of her classmates love either basketball or ping-pong. Do you think she is correct?

(f) Problems which ask for communication [Problem P6-III(b)]If you want to describe to a classmate the figure (Fig. 3) over the phone, how would you do so?

(g) Problems which need judgement [Problem S1-III(d)]Fong Siu-Ying wants to make a box out of cardboard of dimensions 14 cm by 8 cm. As shown in

theFig. 4, a box can be made by cutting out equal squares at its corners.

Fig. 4.


Table A.1

x

1 2 3 4 5

Capacity

(i) If Fong Siu-Ying cuts down from all four corners squares of side 1 cm, calculate the capacity ofthe box formed in this way.

(ii) If x is an integer, please calculate the capacity of the box formed in this way and put your answerinto Table A.1.

(iii) If we just consider thosex which are integers, does there exist a box of greatest capacity? If thereis, what isx and what is the capacity of this box? Please explain.

(iv) Now Fong Siu-Ying cuts another box withx = 2.5. Please calculate the capacity of this box.(v) If x is not restricted to be an integer, does there exist a box of greatest capacity? Please explain.

(h) Problems which involve decision making [Problem S3-III(d)]The school tuckshop sells either hot chocolate or cold orange juice (but only one of them) in a

singing contest. The cost price of either drink is $5 per cup, and the selling price is $10 per cup. Inhot days, the tuckshop can sell 300 cups of hot chocolate or 400 cups of cold orange juice on theaverage. In cold days, it can sell 700 cups of hot chocolate or 600 cups of cold orange juice.

Certainly the tuckshop would like to have the greatest profit but it is not known whether the day ofthe singing contest will be hot or cold. Can you help the tuckshop to decide whether they should sellhot chocolate or cold orange juice?

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The lived space of mathematics learning