Experiences of learning of equations from school

The teacher’s conceptions of equations have been described from three per-spectives: teachers’ experiences of the concept learning, their subject matter conceptions and their pedagogical content conceptions of the concept. The teachers have usually at least 16 years’ experiences as learners of mathemat-ics teaching and learning from compulsory school to university.

The teachers have apprehended their learning of equations, as follows:

as doing routine problems, as memorizing and reproduction of rules and models, as doing applications, and as interaction with other students.

The teachers’ conceptions about learning of equations in the two first categories have a focus on the act of learning and the consequences of the act, whereas the conceptions in two latter categories focus on the object of learn-ing itself. Similar approach to learnlearn-ing can be found in Marton’s & al. (1993) and Säljö’s (1982) study.

The research results indicate that the teachers feel that the concept learning was a mechanical drill especially at secondary and upper secondary school. The teachers have experienced that they have worked mostly with routine problems on all levels at school. It was essential for them to learn by heart rules and formulas, and if they did as the teacher said they could get the right answer. By doing as the teacher said they could get a good school re-port. Knowledge about equations was purely intended for consumption and its eventual use. One of the teachers points out:

It was important to remember the rules and to be vigilant. If you did ex-actly as the teacher said, it is fine and you got the right answer. I do not re-member if I understood what I did. I was careful to memorize rules and to follow them. In that way I got the mark 4 in math. (Maria)

Traditional classrooms usually consist of teachers presenting the ‘right’

way to solve problems or even the ‘right solution’. The teachers describe their experiences of learning of equations in quantitative terms—as doing routine problems, as learning certain typical examples, as memorizing rules,

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as reproducing models and patterns. The teachers’ main interest in this use is what they can reproduce in order to pass an exam, to get a right answer or to get a good certificate. It seems that learning of equations did not support students’ understanding. Knowledge, in a traditional classroom, is usually symbolic and isolated; learning does not typically motivate students or pro-vide them with problem-solving skills they can apply to other school levels or situations. As one of the teachers establishes:

When I began the upper secondary school the pattern didn’t work and I didn’t understand what I did (Rita).

Moreover, traditional views of interaction between teacher and students in traditional classrooms are characterized as distant, with the teacher as an authority figure (cf. Waller 1932; Burton 1989). One teacher tells:

The goal was that we must pass the exams…You could not ask a question, the teacher might get furious and lower your mark…(Mathias)

The teachers describe their learning experiences of equations on the way, which is typical in direct instruction: transmission of facts to students, who are seen as passive receptors (cf. Burton 1989, 17). In the classrooms where this type of teaching predominates, the teachers are active and they typically conduct lessons using a lecture format. They often instruct the entire class as a unit, write on the blackboard, and pass out worksheet for students to com-plete. It seems that mathematical knowledge about equations was presented as facts, and the students’ prior experiences were not seen as important.

The teachers’ experiences of concept learning indicate that school mathematics was apprehended as a collection of unintelligible rules. If the rules were memorized and applied correctly, they led to ‘the right answers’

(cf. Skemp 1971, 3).

Pedagogical processes in direct instruction deny the influence of indi-vidual or social context and present an imagined world of certainty, exacti-tude and objectivity (Burton 1989, 18). According to Burton (ibid) ‘by vali-dating a depersonalised model of mathematics, we ensure that the subject remains aloof from the concerns and interests of most members of society’.

For some of the teachers the understanding of equations came later, es-pecially in connection with applications in physics and in discussions with other students. Learning in this use was no more seen in a quantitative light.

It was not restricted to specific models and routine problems in mathematics.

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It had a focus on the object of learning itself and on understanding. The teachers connect the concept of equation to other subjects. The research re-sults indicate that when a mathematical concept will be connected with other school subjects and mathematical representations the teachers experienced that they acquired a deeper mathematical understanding of equations. By doing applications in physics, the teachers could understand the concept in a new light.

In discussions with other students understanding of equations is taken a step further. The teachers experience that learning and understanding of equa-tions is no more connected to only applicaequa-tions in physics. The new mathe-matical knowledge about equations was mainly gained in a communicative context, where language and reflection play an important role.

The teachers’ conceptions of learning of equations in the four categories of description are directly analogous to the difference between surface and deep approaches to learning: the former focusing on the tasks themselves and the latter going beyond the tasks to what the tasks signify (Marton & Booth 1997).

It looks like the experiences shaped during the mathematics lessons are still strong in the teachers’ memory (cf. Malinen 2000). As Silkelä (1998) says, significant and valuable experiences do not have only a momentary value. They will be a part of our personality. According to Malinen (2000, 134–140) a learner’s personal (experiential) knowledge originates from ac-quired life experiences. These experiences are always true for a learner. Ex-periences of this sort may be untested conceptions, incorrect theories and limited perspectives. These private and sometimes incomplete conceptions however constitute a holistic unity for a learner, because he seeks a unity among experiences. Thus, a learner’s personal (experiential) knowledge in-cludes such kinds of learning experiences that have an influence on the de-velopment of a person’s conceptions of mathematical concepts.

The earlier conceptions direct formation of new knowledge. Many mis-conceptions and learning difficulties of scientific concepts depend on the learners’ experiences and conceptions of concepts in everyday life (Duit 1995). Learners encounter, for example, the equals sign early in their life.

Outside mathematical classrooms the equals sign is often used to mean ‘is’ or

‘gives’ (cf. Kieran 1992). However, the limited view of the equals sign causes difficulties for the learners to understand mathematical entities, be-cause they may have, for example, a persistent idea that the equals sign is either a syntactic indicator, a symbol indicating where the answer should be

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written, or an operator sign, as a stimulus to action or ‘to do something’ (see e.g. Kieran 1981). The research results indicate that a restricted understand-ing of equals sign appears to persist through secondary school, continues into upper secondary and tertiary education and affects concept understanding at these levels. Research on misconceptions (see e.g. Confrey 1990a; Vinner 1983) has specially shown that direct instruction does not support students’

mathematical understanding (cf. Burton 1989). Misconceptions appear to be resistant to traditional forms of instruction. Efforts to develop forms of in-struction that overcome misconceptions have focused on the need to have students make their conceptual model explicitly (cf. Dunkels 1996; Mariotti

& Fischbein 1997).

The teachers in the investigation have experienced that they did not dis-cuss enough the innermost meaning of mathematical concepts either at sec-ondary or at upper secsec-ondary school. It seems that the teachers’ personal conceptions (knowing) were not important there, and they did not need to make their conceptions or concept images about equations explicitly. How-ever, learning cannot happen before the learners’ first-order experiences (personal knowing) ‘meet’ second-order experiences, i.e. experiences, for example, acquired in teaching and learning of mathematics (c. Malinen 2000, 135). All the second-order experiences are not yet educative. They instead prevent the learners’ capacity from learning and therefore, they cannot be integrated into personal knowing. The result may be non-learning (cf. Jarvis 1992, 1995), and the development of erroneous conceptions of mathematical concepts. The changing of the learner’s existing conceptions presupposes that there always must be a balance or continuity between a learner’s personal knowing and second-order experiences. Learning is a re-construction process, where more precise knowledge is created and already existing conceptions change (cf. Malinen 2000, 137).