Parents and Teachers may enjoy reading some of these comments published in the English journal of math teaching philosophy Cheods . Behind the heavy philosophy is the question of whether math can and should be taught in the old-fashioned sense of established fact vs. whether the underlying politics of power should take first place.
The Nature of Mathematics and the State of Mathematics Education
Stuart Rowlands and Ted Graham
The pedagogical effects of the emphasis placed on subjective experience have been shown in practice. The influential theories of both radical and social constructivism have emphasised subjectivity and denied the objectivity of mathematics as a body of knowledge. Radical constructivism emphasises knowledge with experience and social constructivism confuses origin with validity. The paper argues that mathematics, although fallible, is a practice that produces objective problem solving situations that are independent of individual cognition or social acceptance. This is important in the consideration of the teaching of mathematics from a Vygotskian perspective - that it is the teaching of decontextualised concepts that enables the teacher to facilitate cognitive growth. If the teaching of decontextualised concepts is replaced by a subjective approach then learners will be denied their full intellectual potential.
It could be argued that the mere fact that the syllabus requires Pythagoras' theorem to be known by students suggest that the syllabus is absolutist in conception. This can be problematic for a teacher who wishes to work from a radical constructivist perspective.
What is radical constructivism? It is an unconventional approach to the problem of knowledge and knowing. It starts from the assumption that knowledge, no matter how it is defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience.
(von Glasersfeld 1995)
The recent hysterical reaction to Melanie Phillips' (mostly) intelligent book, ironically titled `All Must Have Prizes', indicates a widespread acceptance of what she correctly notes as the influence of `personal knowledge' on primary educational theory. This theory is now centred on children's spontaneous creativity and experience and rejects as suspicious `facts passed down generations' and `other people's experiences'. Writing in the Observer, Phillips anticipates the reaction to her argument, while showing how this rejection of objective facts has redefined learning `into a wholly subjective process'
While it would seem fairly obvious that students are not `blank-slates' and that some form of construction process takes place in the development of learning, the idealist content of what has now become (according to Lerman 1994) the dominant theory of learning has shifted the emphasis away from the objective content of the subject matter to be learnt. This shift of emphasis from the objective content to the subjective experience has serious implications for both the subject to be taught (e.g. mathematics) and the learning of that subject. These implications can already be seen in practice. For example:
* Algebra has traditionally been regarded as a difficult topic to learn, the tendency has therefore been to dilute it rather than devise strategies to overcome the difficulties. The National Curriculum and GCSE have replaced most of the algebra that was taught by a mathematics that seems more relevant to the pupils everyday experience:
I believe the `supermarket curriculum', the fashion which says: `if you can't use it in the supermarket, let's not teach it,' is the biggest influence in the decline in mathematics skills and knowledge in the past ten years. And the tendency is to say: `if it's too hard, let's postpone teaching it'
(John Berry, Do Not Shirk a Testing Subject, Daily Mail, Tues., Dec. 3, 1996)
* The Plowden Report, apparently based on Piaget perspectives, has placed the emphasis on individual learning. This has resulted in many pupils under-achieving in linguistic, mathematical and cognitive skills by the time they have entered secondary education.
* The contradiction between the `improvement' in public examination performance of GCSE's and A-levels over the past period, and the concern over the poor standard of mathematics attainment compared with most other countries.
Creating a teaching practice guided by constructivist principles requires a qualitative transformation of virtually every aspect of mathematics teaching.
But it goes much further than that - the teaching practice itself would alter the very nature of the subject matter to be taught. If all knowledge is regarded as subjective, as in von Glasersfeld's (1984, 1995, 1996) claim that knowledge is a personal construct that makes sense of his or her experiential world, or inter-subjective, as in Ernest's (1991) claim that objectivity is conferred on the mathematical thought through its social acceptance, following publication, then the subject itself - as a body of knowledge - will be denied to many. Mathematics will be either reduced to the experience of the individual and his or her personal knowledge (the radical constructivist position), or to problem solving that is related in it's entirety to the narrow aspirations and social situation of the class (for example, the social constructivist position of Ernest 1991). To quote Fox in the Living Marxism article:
But what exactly is personal knowledge? It is what we know already, and have accumulated from our personal experience. For my 16-year old students, personal knowledge is limited to that of any teenager in Watford - beyond bands, family, TV soaps and adolescent love angst, there is little else. But does knowing who shot Ian in EastEnders really equate with understanding how electricity works?
The emphasis on personal or collective knowledge has become so subjective that it has become fashionable to elevate student intuitive and idiosyncratic misconceptions of scientific concepts above scientific knowledge itself. Consider the following statement by Trumper:
One implication of a constructivist view of learning is that it is neither possible or desirable to remove these intuitive ideas and to replace them with the accepted scientific concept.
It will be argued that in the very attempt to empower students with mathematics as a subject that is directly relevant to their lives will, in fact, dis-empower them from a very important intellectual resource.
On the objectivity of mathematics
While I was a secondary school teacher I attended an A-level mathematics INSET course. In one such session we had to participate in a circus of mathematics problem solving. One of the problems was to find the volume of a rugby ball. I was in a group of three who pondered over what we saw: one rugby ball, a note asking us to find its volume and a centrally placed table containing all the mathematical instruments that the participants could need. Using a measuring cylinder and a nearby sink we found the volume of the ball by measuring the volume of water displaced by the ball. On completing the task we felt uneasy as to whether our solution had any bearing whatsoever on the teaching and learning of A-level maths. Our unease led us to find the volume by: 1) attempting to find the equation of the curve that approximates to the shape of the ball and then using y2dx, and 2) approximating the volume of the ball to a set of cylindrical discs and finding the volume of each disc. We were told by the person giving the session that 1) and 2) were unnecessary given that we had already found the volume of the ball. When we complained: "But what about the A-level maths?", the supervisor simply similed and said "But what about it? You have found a solution to the problem - and it is that what really counts".
It was as if knowing the volume of a rugby ball was the answer to our prayers, and having the nous to find it was all that was necessary in fulfilling the object of the exercise. It was rather as if not bothering to acquire the skill of curve sketching (e.g. to find turning points, asymptotes, intercepts etc.) if you have the initiative to buy a graphics calculator. In fact, the whole circus seemed more to do with problem solving skills and the use of one's initiative rather than the development of conceptual understanding in A-level mathematics.
Finding the volume of a rugby ball might be a useful project, either in GCSE physics (water displacement method) or A-level mathematics (trapezium rule in 3-D). In either case it may be employed as a utility of existing knowledge (go find its volume!), or as part of a strategy to arrive at a target concept (e.g. Archimedes principle or trapezium method). A project or task so designed that the simplest solution is the best (`wouldn't it be just typical of A-level teachers to find the volume of a rugby ball the hard way!') may have value in itself (e.g. promotes divergent thinking) but has little value in the understanding of the concepts of mathematics!
Echoes of Platonism? The problem with mathematics and mathematics education is that if you mix the two then you will be placed in one of two camps: absolutism or fallibilism. Either you regard mathematics as a fixed and immutable body of objective knowledge that has certainty, and an existence that is independent of individual cognition or social acceptance, or you regard mathematics as a human creation, based on assumptions and conventions. In the former, mathematics is discovered; in the latter, mathematics is invented. The problem of mixing maths with mathematics education is that if reference is made to the teaching of mathematics as a body of knowledge then absolutism is seen to be implied. If I were to argue the case for the teaching of the theorem of Pythagoras in year 9, for example, then I apparently would have an `absolutist conception' (which might be a problem for me if I wanted to teach from a constructivist perspective!). The problem is in fact the confusion that is made between the origins of mathematics and its validity. The confusion is the objectivity of mathematics confused with the acceptance by the academic community. It is easy to see how the error is made: if you were to regard mathematics as fallible and yet to have validity independent of individual cognition or social acceptance, then it would be very tempting to ask where does that validity come from and where does it exist? The error is made in the move from mathematics is fallible to it is the social that confers objectivity to mathematics. Consider the following statements by Ernest (1991):
* Objectivity itself will be understood to be social (page 42).
* Publication is necessary (but not sufficient) for subjective knowledge to become objective mathematical knowledge (page 43).
* To identify the immutable and enduring objectivity of the objects and truths of mathematics with something as mutable and arbitrary as socially accepted knowledge does, initially, seem problematic. However we have already established that all mathematical knowledge is fallible and mutable. Thus many of the traditional attributes of objectivity, such as it's enduring and immutable nature, are already dismissed. With them go many of the traditional arguments for objectivity as a super-human ideal. Following [David] Bloor we shall adopt a necessary condition for objectivity, social acceptance, to be its sufficient condition as well (page 45).
* It has been established that objectivity is understood to reside in public, intersubjective agreement, that is, it is social (page 49).
* According to the social constructivist view, mathematical knowledge is fallible, in that it is open to revision, and it is objective in that it is socially accepted and publicly available for scrutiny (page 53).
One dare not even speak of the various ways of arriving at a solution to a problem, yet alone speak of arriving at the target concept, without the spectre of absolutism:
* The strength of the metaphor (a geographical metaphor of trail-blazing to a desired location) is that stages in the process (of problem solving) can be represented, and that alternative `routes' are integral to the representation. However a weakness of the metaphor is the implicit mathematical realism. For the set of all moves toward a solution, including those as yet uncreated, and those that never will be created, are regarded as pre-existing, awaiting discovery. Thus the metaphor implies an absolutist, even Platonist view of mathematical knowledge (page 285).
In his account of the `genetic fallacy' of the sociological reduction of all epistemological issues (with specific reference to David Bloor), Rom Harre states:
It does not follow that because one has given a correct account of how some belief came to be held that we are not entitled to ask about its truth as well. Someone may come to believe something that is true because he is frightened of a teacher, influenced by the attractive personality of a friend, or because he is a creature of his class position. The revelation of how that belief was caused has no bearing on its value as knowledge
(Harre 1986, Page 193, my emphasis).
Of course, I could always argue that social and radical constructivism has gained in popularity, and gets away with what has been written on its behalf, because of the post-modernist climate in which we live - academics cannot understand the nature of the international capitalist crisis (which may, incidentally, explain the popularity of chaos theory) and of the society that supports them - consequently everything is up for grabs. If knowledge is regarded as subjective, then any viable theory will do and damn any theory that attempts to be objective and explain what may actually be the case.
Knowledge does not constitute a `picture' of the world. It does not represent the world at all. It follows that what we ordinarily call `facts' are not elements of an observer independent world but elements of an observer's experience.
(von Glasersfeld 1995)
I could argue that constructivism is a petit-bourgeoise epistemology that, independent of any honest intentions and sincerity of its proponents, objectively serves the interests of capitalist society by mystifying! (Interestingly enough, it has been suggested that radical constructivism - architypical individualisation - had gained popularisation and support in a culture of right-wing Thatcher/Reagan-ite societies. See Woodrow 1996).
Of course, I shall not be arguing the sociological account of constructivism simply because the social origins and the nature of constructivism would not shed any light as to its validity (independently of whether or not such an argument is published or accepted by the maths-ed. community) - the validity of constructivism is independent of its social origins and class nature!
Mathematics is fallible, it is a human construct that has its origins in social and physical interaction. It is we who determine its axioms and its hypothetical/deductive logic. However, once its axioms and hypothetical/deductive system have been established then there exists an objective problem-situation. Like the game of chess, once the rules are made then consequences (a metaphor for theorems) follow. Mathematics is an autonomous objective practice that creates problems the solutions to which are independent of whether anyone has found a solution and independent of whether the academic community accepts the solution. Following Chalmers (1978) argument that science is a process without a subject, mathematics is a process without a subject because its development creates objective problem situations whether or not mathematicians realise the problem situation. A particular community may or may not practise a legitimate mathematics, and different social set-ups have created different contributions to a single (albeit multifarious) mathematics (e.g. Pascal's France and the China that created Pascal's triangle centuries before).
This is an objectivist position because it implies the existence of mathematics as an autonomous practice, independent of individual or consensus opinion, that constitutes the activity involved in its development. Although its development is dependent on the participation of the individual mathematician and of the community, nevertheless, mathematical concepts bear a relationship to each other independently of whether or not the individual or community realises it. Mathematical theory does not reside in Popper's third world of ideas (the world of the objective content of thought), nor does it reside in Plato's world of forms; it is, instead, constantly produced and developed as a result of mathematical practice The relation between consciousness and the objective, real, outside world is eloquently stated by Torrance:
If consciousness is an element of practice, what it reflects, or corresponds to, in reality, is always a specific problem-situation which practice confronts. As part of this reflection, it contains beliefs about the circumstances relevant to the problem. Such beliefs may reflect the circumstances more or less effectively, depending on their relevance, completeness, accuracy and so on. And these features depend, again, not only on the circumstances themselves, but also on the `position' or `angle' from which they are reflected, the condition of the reflecting instrument, and the purposes of the agent whose instrument it is.
It is mathematical practice that determines the objectivity of mathematics, yet its objectivity consists precisely in having validity independently of individual cognition or social acceptance. The objectivity of mathematics may be illustrated regarding the 4-colour problem. The proof of the 4-colour problem has been done by computer, but the difficulty with the proof is that no one has checked the proof to see if any error, say an error of design, has been committed. We do not know, with certainty, if the 4-colour problem has been proved. But supposing an able mathematician was granted a few thousand years and all the coffee he or she could drink to go through each line of working of the proof, and supposing the mathematician found an error in the proof (and published many papers as a result!) - the error would have been committed prior to its discovery by the mathematician or acceptance by the community. To say that mathematics is fallible and so any proof is fallible, or in this case to say that it is we who created the 4-colour problem and we who thought of the method of proof, lies outside the running of the computer programme. If the issue as to the fallibility of the proof by computer was settled, then it would be settled independent of the steps taken in the proof by the computer. If the error committed by the computer was a result of a malfunction in its circuits, then this would be akin to an error committed by a mathematician and would not reflect on the method of proof. If the error is an error by design then the error lies within the method of proof prior to the running of the programme. Either way, the error would have been committed before discovery. If no error has been committed, then the 4-colour problem has been proven before any realisation.
If the proof of the theorem of Pythagoras was demonstrated to a class, or, better still, constructed by the class, then it would seem very odd if the teacher concluded: QED, and oh, by the way, the proof is fallible so what we have isn't really a proof! The truth of the theorem of Pythagoras lies in its proof and not by consensus, and the issue as to the fallibility of the theorem lies outside the proof. History has shown, for example, that Euclidean geometry is fallible because if the truth-values of the axioms are reversed then the consequences lead to a set of consistent results. This does not, however, reflect on the truth of, say, the mid-point theorem (which, incidentally, can also be proved vectorially).
To say that complex numbers exist within an autonomous objective practice does not imply that complex numbers have a Platonic existence. Once the existence of complex numbers is assumed, then it makes sense to inquire as to their properties (e.g. do they satisfy the same laws of algebra as the real numbers - i.e. do they form a field?). Platonism may at times be the result of a confusion between ascribing qualities to an object with asserting it's existence. When the existence of complex numbers was assumed, the consequences were unforeseen and unintended by the original proponents of the assumption. These consequences existed as properties of complex numbers that were there waiting to be discovered by further mathematical practice (e.g. deMoive's theorem). Just as art can be aesthetically judged independently of the considerations of the historic and social conditions that led to the art, so in much the same way the validity of a mathematical theory is independent of the social and historical conditions that gave rise to the theory. Mathematics transcends the context from which it originates, although it is not difficult to see how origin can be confused with validity. To quote Lukacs:
The relations between origin and validity are much more complex here than in the case of the forms of the objective spirit. Marx saw the problem clearly: `But the difficulty does not consist in realising that Greek art and epic are bound to certain social forms of development. The difficulty is that they still give us artistic pleasure and that, in a sense, they stand out as norms and models that cannot be equalled.' Just as it is clear that Copernican astronomy was true before Copernicus but had not been recognised as such.
The pedagogical implications of mathematics as an autonomous objective practice
The Vygotskian perspective
Consider the following passage taken from J. Threlfall: Absolutism or not absolutism - what difference does it make? on the internet:
The use of the term `concept' to refer to the object of teaching and learning makes immediate sense only from an absolutist perspective in which the concept word refers both to the child's idea and to the notion in mathematics. The rejection of mathematical objects in the sense of `right' ideas strips the term `concept' of its reference. It is then not meaningful to refer to `the concept of perimeter' when trying to characterise a child's understanding or one's intentions for it. At the very least we have to adjust it to `a conception of `perimeter', and mean `some notion (yet to be described) concerning the area of experience and language the understanding of which I refer to as `perimeter' and which I take to be characteristic of the area of experience and language which other people also refer to as `perimeter'. The non-absolutist will accept that we cannot ever fully understand a child's thinking in mathematics, any more than anywhere else, but to try to get as much insight as possible about it, by using descriptive terms and metaphor
(J Threlfall, 11/14/96, Absolutism or not Absolutism - What Difference Does it Make? Philosophy of Mathematics Education Newsletter 9, http://www.ex.ac.uk/PErnest/pome/pompart6.htm.)
Not only is `concept' denied as a possibility held by the child, it also has no reference regarding mathematical objects. I wonder what a social constructivist would make of the following passage:
Absolute correctness is achieved only beyond natural language, in mathematics. Our daily speech continually fluctuates between the ideals of mathematical and of imaginative harmony.
From a social constructivist perspective it would seem that the Vygotskian perspective is absolutist. There is a disparity between constructivism on the one hand and the Vygotskian perspective on the other, yet Vygotsky is well referenced by radical and social constructivists alike as a means to credibility (e.g. Cobb 1996; Ernest 1991, 1994; Fosnot 1996; Greene 1996; Jaworski 1994; Steffe and Tzur 1994; von Glasersfeld 1995; to mention but a few).
For Vygotsky (1962, 1978), learning precedes development, and it is the learning of decontextualised concepts such as scientific concepts that cognitive development can take place in the classroom. It is very surprising that Vygotsky has not been criticised for his `absolutism' and the implied `anti-democratic' imposition of decontextualised concepts that are unrelated to the aims, aspirations and everyday experiences of the class. However, it would be worthwhile examining Vygotsky's perspective a little closer to see how mathematics - as a body of knowledge - can be taught as a means to develop cognitive and metacognitive skills along with the learning of the subject. Hopefully, what will become apparent in the following account is that the practical consequences of the radical and social constructivist perspective will leave the working class dis-empowered - not only from knowledge but also intellectual development.
Vygotsky's reference to scientific concepts also includes concepts in social science. Except for arithmetical systems, Vygotsky has made little reference to mathematical concepts. Nevertheless, what has to be remembered is the importance of decontexualised concepts that may be defined as either concepts that are explicitly well-defined by other concepts or concepts that are implicitly well-defined in relation other concepts via a set of axioms. Hopefully, what will become apparent in the following account is the need to teach decontextualised concepts (which would include mathematical concepts) the meanings of which are distinct from, and without reference to, spontaneous everyday concepts.
Although learning takes place in social contexts, according to Vygotsky (1978) there are higher psychological functions characterised by reflective control and deliberate awareness similar to metacognitive skills. What distinguishes higher mental functions is the shift of control from the environment to the individual, that is, the emergence of voluntary regulation and the emergence of conscious realisation of mental processes. Vygotsky believed that it was through contact with decontextualised concepts, concepts that are defined in relation to other concepts (e.g. scientific concepts), that the child can develop deliberate control over everyday concepts. It is through learning concepts separate from the immediate and the concrete that structures are provided:
Scientific concepts in turn supply structures for the upward development of the child's spontaneous concepts toward consciousness and deliberate use.
For Vygotsky, the school environment provides the context for learning decontextualised concepts within a discipline distinct from the everyday learning of everyday concepts. School learning is advanced by the teacher supporting the learning of decontextualised concepts, and it is by learning decontextualised concepts that the pupil develops metacognitive skills:
The child becomes conscious of his spontaneous concepts relatively late; the ability to define them in words, to operate with them at will, appears long after he has acquired the concepts. He has the concept (i.e., knows the object to which the concept refers), but is not conscious of his own act of thought. The development of a scientific concept, on the other hand, usually begins with its verbal definition and its use in non-spontaneous operations - with working on the concept itself. It starts its life in the child's mind at the level that his spontaneous concepts reach only later. One might say that the development of the child's spontaneous concepts proceed upward, and the development of his scientific concepts downwards, to a more elementary and concrete level. This is a consequence of the different ways in which the two kinds of concepts emerge. The inception of a spontaneous concept can usually be traced to a face-to-face meeting with a concrete situation, while a scientific concept involves from the first a `mediated' attitude toward its object.
(Vygotsky 1962, page 108; author's emphasis)
Children's' everyday concepts are unsystematised and characterised by a lack of conscious awareness. Children may be able to talk spontaneously and correctly about everyday concepts, but they have a difficulty in focusing on the concepts: the child is not conscious of the concept because his or her attention is always centred on the object to which the concept refers (Vygotsky, 1962). Children find it difficult to answer correctly abstract questions about concepts that have been placed in contexts separate from their immediate concrete experiences. However, Vygotsky (1962) argued that deliberate control over everyday concepts is developed through contact with scientific concepts. It is by learning scientific concepts divorced from immediate concrete experiences that structures are supplied (by the scientific concepts)
for the upward development of the child's spontaneous concepts toward consciousness and deliberate use.
(Vygotsky 1962, page 109)
This is only possible because:
In the scientific concepts that the child acquires in school, the relationship to an object is mediated from the start by some other concept. Thus the very notion of scientific concept implies a certain position in relation to other concepts, i.e., a place within a system of concepts. It is our contention that the rudiments of systematization first enter the child's mind by way of his contact with scientific concepts and are then transferred to everyday concepts, changing their psychological structure from the top down
(Vygotsky 1962, page 93)
Vygotsky provides empirical evidence to show that a sample of students was more successful using reasoning skills involving decontextualised social studies concepts than reasoning skills involving familiar everyday concepts. This goes against expectation as one would have thought that students would have performed better on reasoning skills involving everyday experiences. Vygotsky states:
How are we to explain the fact that problems involving scientific concepts are solved correctly more often than similar problems involving everyday concepts? We can at once dismiss the notion that the child is helped by factual information acquired at school and lacks experience in everyday matters. The child must find it hard to solve problems involving life situations because he lacks awareness of his concepts and therefore cannot operate with them at will as the task demands
(Vygotsky 1962, page 106)
Although the learning of scientific concepts is transferred to everyday concepts, the pupils everyday concepts must nevertheless already be at a certain level so that scientific concepts can be learnt (Vygotsky 1962). The understanding of causal (because) relationships in everyday speech is necessary if the notion of causation is to be understood within a scientific context. However, this does not mean to say that the pupil has deliberate control over the concept of causation within the context of everyday speech. Although the teacher can facilitate the notion of causation, Zeuli (1986) warns that Vygotsky makes no suggestion that teachers working within the zone of proximal development should make connections to what the learner already knows. According to Zeuli:
Bruner's examples of learning within the `zone of proximal development' support the view that the primary focus is on the adult's assistance as the student tries to understand the relationships between concepts - not how connections are made to the student's everyday concepts. Bruner also points out the importance of schooling as `joint culture-creating', and later compares the zone to the way `Socrates guides the slaveboy through geometry in the Meno - a kind of negotiation in which the abler frames the questions, the less able replies and gains in insight'.
Zeuli has reported on the way some researchers have used Vygotsky's theory to emphasise how the teacher should make connections to the students' everyday familiar concepts, and argues that this is inconsistent with Vygotsky's characterisation of school learning. For example, Greenfield studied how adults provide `scaffolds' to children learning within the zone of proximal development - the teacher builds on what the learner can do, thus closing the gap between task requirement and the skill level of the learner. Zeuli notes that:
1) Vygotsky does not suggest that, within the zone of proximal development, the teacher should make immediate connections to what the learner already knows. On the contrary - for Vygotsky - the school environment is the creation of a special context for understanding decontextualised concepts distinct from everyday learning.
2) That, according to Luria (a colleague of Vygotsky), students will initially fail to establish any connection between academic concepts and events in their everyday life.
3) Students' everyday concepts may interfere with learning unfamiliar scientific concepts: Connections to students' existing knowledge may not foster their understanding but may instead reinforce their misconceptions.
Zeuli (1986) further warns that while it is important for instruction to be sensitive to the cultural setting within which it occurs, nevertheless it does not mean that school learning must be continual with these cultures. On the contrary, schools should provide students with educative breaks from their everyday experiences to further students' objective judgement and scientific understanding. According to Zeuli, there is much research that uses Vygotsky's theory to justify the notion that school learning should be compatible with native cultures; yet this `compatibility' does not square with Vygotsky's position. Such research does not take into account Vygotsky's analysis of the limitation of students' everyday concepts or his emphasis on the discontinuity between everyday concepts and higher psychological functions.
Conclusion: The role of teacher as facilitator of knowledge and understanding
It is ironic that Vygotsky is well referenced as an authority yet not criticised as an absolutist. He may be described as a constructivist because of his perspective of the individual constructing knowledge facilitated by an adult or in collaboration with more capable peers. How does the teacher facilitate the construction of knowledge? Consider the following example:
In geography, for example, a number of different factors could affect rice growing in a country, such as fresh water, a fault area, fertile soil, and warm temperature. As Collins and Stevens point out, the teacher can use various strategies to help students understand the relationships between concepts in a discipline: `If a student says they do not grow rice in Oregon because it lacks a flat terrain (which is unnecessary), one can pick Japan which is also mountainous, but produces rice. If a student thought rice could not be grown in Wyoming because it is too dry (which is insufficient because it is also too cold), the teacher could ask, ``Suppose that it rained a lot in Wyoming, do you think they could grow rice then?''
By Socratic tutoring, or the giving of props and hints, the student's cognitive framework comes to life. It is not so much knowing that something is the case (such as the areas that affect rice growing) but why it is the case (such as the necessary and sufficient conditions for rice growing).
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