**I. INTRODUCTION**

Although it is difficult to explain the concept of understanding briefly, we will try to give a brief account of its meaning and consequences in the context of teaching from the viewpoint of constructivist learning.

Is it sensible to assume that ‘Everything which has been taught has also been understood’? Accumulating research on understanding reveals that it is not that simple. Not every teaching generates understanding. Von Glasersfeld (1990), commenting on the present situation, argues that something must be wrong as some children are leaving school unable to read and write or handle numbers properly, and knowing nothing about the scientific nature of the world. He relates this to teaching practices derived from behaviourist models; he is critical of the removal of the distinction between training on the one hand and, on the other, teaching which aims at generating understanding. Whether training manages to generate understanding is, he says, a matter of chance.

Understanding, in fact, is a very complex phenomenon to describe both philosophically and practically. In one attempt to do so (Skemp, 1989), two categories of understanding are defined: relational and instrumental. Relational understanding corresponds to intelligent learning. Instrumental understanding means habit-learning or learning ‘rules without reasons’. The advantages of instrumental understanding are: its being easier to grasp; providing more immediate and obvious rewards; and yielding the right answers more quickly. The advantages of relational understanding, on the other hand, are: it is more adaptable to new tasks; and what is learnt is easier to remember.

A shift towards relational understanding may provide the necessary motivation for students to learn more effectively. However, teaching for relational understanding suffers a major drawback. Selinger (1994) states: ‘Encouraging pupils to learn more relationally can be problematic for both teachers and their pupils; the investment required to make connections is greater than in instrumental learning and the contents of the curriculum need to be considered, so that connections can be easily and readily established.’

**II. CONSTRUCTIVISM IN THE CLASSROOM**

In constructivism, knowledge is always contextual. The so-called “objective” mathematical knowledge is the product of active construction which we make and share with others (Wood, 1995). Understanding, then, means constructing acts on the shared mathematical objects (ibid.).

The two major assumptions of constructivism can he summarised as follows (Caprio, 1995):

a) Knowledge is not objective, and does not have an absolute structure. What we know and understand is only our perceptions of reality.

b) Teachers help students to develop new insights. This is done by leading students to develop their own analogies, examples and non-examples, proof methods, new ways of solving a problem etc. The teacher, in short, attempts to teach thinking.

In the behavouristic model, the body of knowledge in. a subject-matter is the object of teaching. The teacher’s task is to transmit (or inject) that knowledge into the learner (who does not have it) through an adequate discourse. Constructivism questions this transmission-based approach. It assumes that students have their own paradigms before coming to class, some of which are invalid or incomplete. The duty of the teacher is to guide students on their way through reaching out to that in the knowledge they bring to class which is viable.

According to Pateman and Johnson (1990), constructivist ideas have influenced teaching practices. Teachers need to set up the temporal, discursive and cognitive space for students to construct concepts (Mousley, 1993). Mousley uses the term “moments” to describing the events in the classroom in which teachers and students create space for themselves; this term reflects the immediate nature of very small periods in which the potential of the lesson changes continuously. Her observations indicate that the decisions taken by the teacher in those moments help or prevent students to make sense. That is, the learners do not have command of the learning situation. Learning still depends on the teachers’ residual authority. Therefore, she suggests, teachers should leave more interpretive space in the learning process by presenting the students with loosely-defined tasks.

Jaworski (1994), commenting on the demands of constructivism on teaching methods, says that getting inside the head of the student helps teachers to build a model of the student’s conceptual level; language is a tool for that purpose, and the errors and apparent misconceptions of students can provide the feedback information the teacher needs to understand and adapt to the student’s level of understanding.

Teaching is heavily reliant on what the learner already knows. The duty of the teacher, in the constructivist model, is to help the student to construct new knowledge or discriminate alternative concepts, and to guide them to commonly shared concepts.

Constructivism, being a theory of learning, does not say much directly about teaching methods. Instead, it specifies the methods that are not appropriate (Orton, 1994). But since teaching and learning are complementary and reciprocal, it is inevitable that this theory has many implications for teaching. Von Glaserfeld (1987), for instance, listed some possible consequences of constructivism for teaching:

a) There will be a separation between teaching of understanding and training which aims (only) at repetition of behaviours.

b) Educators will be interested in what is going on in the minds of the children rather than in their overt responses.

c) The teacher will realise that language is a tool to help learners to construct understanding, rather than a mechanism for transferring knowledge.

d) There is a shift in the perception of errors from being something undesirable (which teaching should eliminate) to being possible indicators of the process whereby students make sense of their experiential worlds. The teacher’s duty is to detect those errors which yield information about how and why the child deviates from the teacher’s expectations.

e) There is a shift toward seeking knowledge about learners’ conceptual structures and finding ways to modify them.

A metaphor for teaching in the constructivist framework is “facilitate” (Lerman, 1993), as opposed to “inject” for the typical behaviourist approach.

**Making Students Make Sense**

Mason (1989) lists some aims of a teacher concerned to help his or her students make sense. He would like students to have seen connections, experienced a crystallisation of experiences subsumed under a general concept, gained a sense of coherence of a topic etc. Elaborating on the issue of the aims of teacher and students, Mason states that “subordination of teaching to learning” and “starting where the students are” do not mean that the teacher’s aims are the same as those of students. Rather, the aims should be confluent. The factors that determine student success in these aims depend on: teacher attitudes toward learning and teaching; the interest and commitment of the teacher; the extent to which teaching evokes students’ capabilities; and the extent to which students share the teacher’s goals.

Listening to the students while they are trying to make sense is a powerful way to get to know how they understand a topic. This talking can be among themselves or with the teacher. The ideal would be to talk to students about their understanding individually, but that is practically impossible. If they are talking with each other, the teacher can listen to different groups and gets an overall sense. Teachers’ dependence on that “overall sense” may not always be a reliable basis for making decisions about any individual student, but in a classroom situation it is inevitable.

**III. TEACHING FOR UNDERSTANDING**

Helping children to develop their perceptions is more productive than helping them to acquire particular ideas or bits of information. The necessary and sufficient condition for achieving this is to have a rough understanding of their current perceptions. It is essential for the teacher to have knowledge about pupils’ past experiences so that remedial actions may take place accordingly (Threlfall 1990).

Discovery and inquiry-based methods are acceptable ways to facilitate understanding. What is not good for the teacher is to sit back and await answers from the students without putting in any effort. Rather, the teacher should arrange the learning environment in order to enable useful experiences for the students. Although they can use some directive statements sometimes, much talking on the part of the teacher does not contribute to better understanding on the part of the learner (Orton & Frobisher, 1996).

**Rote Learning vs. Meaningful Learning**

Meaningful learning occurs if previously isolated bits of information are organised and restructured into new relationships. In other words, meaningful learning takes place as new knowledge is integrated into the learner’s existing knowledge. There is no one-and-only methods (e.g. discovery learning) for making this happen. Any method, such as expository teaching, is appropriate if it ensures that new learning is linked to existing knowledge (Orton, 1987).

In rote learning generally, the schema remain fixed. What is learned remains closely tied to where it has been learned, therefore it cannot be generalised to other contexts. It stays the memory as isolated bits of information. The learning of bare facts and procedures falls in this category.

Despite having little effect on learning, repetition and drill may be useful for the acquisition of declarative knowledge (Mayer, 1983). Repetition influences not only how much is learned but also the structure of what is learned. In fact, there is a misunderstanding, says Von Glasersfeld (1990), about the role of rote learning and memorization. He states that, they are not unnecessary. There are matters that simply have to be learned in a direct mechanical way. Orton (1994) argues that rote-learning has unfairly been associated with behaviourism and is a necessary method for learning. It takes place in a natural way.

**Discovery Learning**

Students in discovery learning situations are led, with limited teacher support, to discover the abstract from the concrete, and encouraged to make connections from seemingly disconnected events. In other words, discovery learning transfers the burden from the teacher to the student (Corno and Snow, 1986).

Discovery learning is advocated by constructivism because of the opportunities for discussion, negotiation and the exchange of ideas (Orton, 1994). The teacher’s role here is to initiate students to make their own interpretations which makes it easier for them to learn. But case studies conducted in classrooms reveal that what the teachers claim to be discovery learning is not discovery learning in its fullest sense (Wood, et al., 1990). The teachers rather impose on students what they are supposed to discover.

**IV. CONCLUSION**

No two students are same. Hence, it is impossible to find two identical teaching experiences. That is, “there is no royal road to teaching and learning.” The suggestions for good teaching do not comprise a method but rather promote a perspective. The teacher cannot do the learning for any pupil (Mason, 1989). At the end of the day, the so called “transmission process” (Jaworski, 1989) depends on students’ own construction of what the teacher says. It, indeed, is not surprising that there frequently is a discrepancy between the two.

Finally, understanding is not the comprehension of an absolute reality but establishing a fit between the old and the new experience (Von Glasersfeld 1984 in Jaworski, 1994). To make students attain understanding, teachers should keep in mind that the teacher’s role is as a facilitator, setting up constructivist learning environments which provide useful experiences for students; there is no ready-made teaching strategy for facilitating understanding: a constructivist approach can use a repertoire of methods, the only thing excluded is passivity on the part of teachers or students; finally, teachers’ knowledge of what their students already know is the only source enabling them to plan their actions.

**References**

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*rno, I. & Snow, R.E. (1986) ‘Adapting teaching to indiviual differences among learners’ in Wittrock, M.C. (ed.) Handbook of Research on Teaching, New York & London, Macmillan Publishing Co, pp.605-29.*

*Jaworski, B. (1989) ‘Mathematics teaching; belief and practice’ in Ernest, P. (ed), Teaching Mathematics; The State of the Art, New York & London, Falmer Press.*

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