Proceedings of the 30

Anna Sierpinska

Concordia University, Montreal, Canada

Universities experience an influx of mature students who return to school to continue their studies or re-orient their professions. Candidates are often required to take pre-university level mathematics courses as a condition of their admission ("bridging courses"). These are courses in elementary algebra, functions, and one-variable calculus. Many students experience frustration in these courses (FitzSimons & Godden, 2000: 28). There is a need to know more about the sources of students' frustration.

Two years ago, I embarked on a study aimed at a better understanding of frustration in such prerequisite mathematics courses, using a questionnaire and interviews. In this paper, I will focus on an analysis of two possible sources of the frustration: students' perceived irrelevance of mathematics for their future studies or professions and students' dependence on teachers for the validity of their solutions.

The questionnaire for the study is available at

The study is situated in the intersection of two research areas: affect (e.g., Hannula et al., 2004) and adult mathematics education (e.g. Coben, 2000). The population I am most interested in are "mature" students (in my university, this means 21+ years old and having spent some time away from formal education).

In mathematics education research, "frustration" is often mentioned in the context of descriptions of affect in problem solving, but the concept is rarely an object of study in itself. An exception is the work of Handa (2003). In the psychological literature, frustration is defined in various ways, depending on the adopted theory of emotion. According to Mandler (1975: 164-4), frustration is a kind of negative emotion aroused upon encountering an obstacle to satisfying one's needs, goals or expectations, which interrupts the ongoing activity. Mandler refers to previous laboratory research on frustration in claiming that the more alternative strategies are available immediately following the interruption, the greater the chances for relaxation of stress.

This is an important aspect for me: my hypothesis is that technical knowledge of a mathematical method without its theoretical justification, is not sufficient to provide those "alternative strategies" at the time of getting stuck on a problem. Frustration thus remains unresolved and may lead to abandoning the task. Yet, the bridging courses tend to focus on the teaching of rigid techniques: one technique for a given type of problem. Using the terminology of Chevallard's theory of teacher's practice (2002) and in particular, his distinction of "moments of study", these courses merge into one the moments of "first encounter" with a task, the "technical moment" and "institutionalization", and reduce the mathematical practice to its practical-technical bloc with little or no attention to the technological-theoretical bloc.

The perceived irrelevance of mathematics, and dependence on teachers for the validity of solutions, often linked with lack of interest for the validity, are well known "affective variables" in the literature. Perceived relevance of mathematics is known to be an important factor in career choice. For students' dependence on teachers in learning mathematics, researchers blame the traditional classroom culture, which appears to teach students not to be responsible for the mathematical validity of their solutions (e.g., Schoenfeld, 1989; Lampert, 1990; Stodolsky et al., 1991). Lack of interest in validity has been reported by several authors (e.g. Evans, 2000: 179).

The choice of the interventions and learning strategies is constrained by the

A questionnaire was designed and sent to about 800 students enrolled in the bridging courses; 96 responses, 63 from mature students and 33 from non-mature students were obtained. Interviews with 6 students were also conducted.

The questionnaire items were inspired by existing instruments (e.g., Haladyna et al., 1983; Schoenfeld, 1989) and experience of teaching the bridging courses. The organizing principle for the choice of items was to cover the respondent's

Responses were analyzed using simple descriptive statistics and the psychological and institutional perspectives outlined above.

The following abbreviations will be used: "ms" for mature student(s), "nms" for non-mature student(s).

"Math is extremely discouraging when you are forced to take it as a prerequisite. If I were going for a major in math, then I would understand that the course is necessary. However, in the commerce program, there is nowhere near as much or as difficult math as I have just taken. I also have another year of math prerequisites to take in order to get into the program I want. If I fail math, I don't get into commerce. So I feel math is the only thing that's stopping me from getting into the program I want…. I am currently spending 20 hours of studying math outside of class and I got 30% on my midterm. I'm starting to think that I'm the problem, and that's very discouraging." (Respondent # 39, ms)

About 2/3 of respondents (65% ms and 73% nms) reported taking the course because the academic advisor told them to (item 65) – we can consider these students to have been "forced" to take the course. Over 59% of all (54% ms, 70% nms) also agreed with "I'd rather not take this course if I had a choice" (item 13). Counting students who felt both forced to take the course and unhappy about it, we got 45% all, 40% ms and 55% nms. Mature students thus appeared to be more accepting of the situation than non-mature students. Not liking mathematics did not appear to be the main reason for students' reluctance: only 35% of the 57 students who would rather not take the course expressed their dislike of mathematics (in item 66 "I don’t like math" or in completing the sentence, "Math is…", item 76) (38% ms, 30% nms). It was slightly more likely to be a reason for ms than for nms. Therefore, sources of students' frustration with the bridging courses must be sought elsewhere than in their dislike of mathematics.

While uselessness was invoked in frustration, usefulness of math was not the main reason for being pleased with math. Of the 41 ms and 13 nms who said they liked mathematics, only 6 ms and 2 nms attributed it to some usefulness. One popular reason for liking mathematics was liking to solve problems and experiencing the "awesome feeling" – as one student put it – of finding "the correct answer" (17 students in all, 13 ms and 4 nms).

Unfortunately, it appears that students knew they got "the correct answer" not by checking or testing it themselves, but relying on teachers or books to tell them if they were right or wrong.

The questionnaire also contained items (74 and 75) where two kinds of solutions (labeled "

We chose these particular problems because, in my experience of teaching the bridging courses, some students displayed a remarkable resistance to adopting the theoretical approach, loudly protesting and arguing for the procedural one, which they "have always used", so "why would they have to forget what they have already learned". Thus, we hoped this question would provoke some stronger feelings in respondents and make them open up their hearts in responding to open items.

There was a very clear preference for the procedural solutions: 69% all, 65% ms and 76% nms chose solution

Students' perception of the prerequisite mathematics courses as irrelevant for their future studies and professions, and their disregard for mathematical validity are understandable. The courses are focused on techniques: simplification of algebraic expressions, solving equations of various degrees and types, differentiation and integration, solving typical word problems. It may be hard for the students to understand the purpose of achieving mastery in solving problems such as: "Factor completely:

At the same time as agreeing with "I need the teacher to tell me if I am right or wrong", many respondents (70%) agreed that, at the university, one is expected to be an autonomous learner (item 38). I consider this discrepancy between, on the one hand, their vision of the ideal university student, as well as their probable sense of control over their lives as adults who just made an important decision (returning to study), and, on the other, their lack of personal agency as learners (Bandura, 1989) as a

The word "deep" is used to reflect the fact students did not explicitly mention lack of control as a source of frustration. They didn't, because, maybe, this could feel like admitting to the loss of self-esteem, and maintenance of self-esteem is known to be very important for mature students (FitzSimons & Godden, 2000: 19-20). Students could maintain a positive self-esteem by

Institutions are difficult to change. They are based not only on conventions and rational rules of economy, but also on values that are considered "natural", and therefore impossible to change without destroying the world in which they live (Douglas, 1986: 46). Any attempt at changing or developing an institution in a certain desired direction must therefore be based on a thorough understanding of what constitutes its stable "genetic code" (Ostrom, 2005) and what are the things that can be changed without jeopardizing its existence.

In the case of the bridging mathematics courses at my university, the problem is to find if there is a possibility of making room for just enough theory to allow students to develop some minimal autonomy relative to the validity of their solutions. Without this autonomy, these courses are, indeed, irrelevant for their future studies, because knowledge learned this way is not open for further development; it is good only – at most – for passing a final examination.

1. This study was financially supported by SSRHC grant # 410 2003 0799.

2. The questionnaire was designed, and interviews conducted in collaboration with Christine Knipping and Georgeana Bobos.

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