Epistemology and Constructivism are interrelated concepts. Epistemology is "the study of, or theory of, the nature and grounds of knowledge, especially with relation to limits and validity" (Webster, 1999). Constructivism has it's root in the word construct which means "to make or form by combining or arranging parts or elements" (Webster, 1999). It is in this light that constructivism can be viewed, as how one assembles knowledge. Before presenting my views of mathematical instruction from a social constructivist perspective, it is necessary to discuss the work of two influential individuals in this area, Jean Piaget and L. S. Vygotsky. Their general theories concerning epistemology and developmental learning theory shall be contrasted by first presenting the crux of each theory and then examining the relationship between their major components. It is in the context of these two individuals that I shall present my idea of mathematical instruction, drawing on some of their idea's, as well as my own.

Piaget's general theoretical framework was termed genetic epistemology, because of his primary interest in how knowledge took form in the human organism. Before presenting Piaget's genetic epistemology, I must address the philosopher Immanuel Kant and his epistemological views, for it is from this ideology that Piagets' theories evolve. Kant's world-view (and hence philosophy) comes from the outcome of the use of human reasoning faculties, this undertakes investigations a priori, or independently of human experience. Kant believed that sensory experience was not a reliable source of knowledge because of the fallibility of the senses, wherein we may see a phenomenon which is not completely accurate (*1). When Kant read of Hume and his views that sensory experience did play an important role in knowing, Kant agreed, but thought that Hume made a mistake in his formulation of knowledge coming into the human mind in pure form (*2). "Kant said that sensory experience (sensations from the environment), in which the empiricists placed so much confidence, must be interpreted before it can become knowledge" (Popp, 1997). Kant thought that the "mind" brought something to the interpretive process.

Piaget later termed the Kantian idea of interpretive process as assimilation, wherein he rejects the "direct input" view of knowledge (of Hume), and accepts that what is perceived is interpreted and assimilated by previous structures. This means that however we perceive our respective environments is interpreted by whatever perceptual framework we have set up. Piaget's inquiry and explorations in this area sparked great interest, and more questions. Piaget questioned how these structures came into being. How did a newborn infant construct these mental frameworks by which to interact with reality, and more importantly, how did it add onto these structures? Piaget devoted his life to these questions. Piaget referred to the process of formulation as "accommodation", where there are various degrees of assimilation by an organism in response to it's environment (*3). As an organism survives in existence it interacts with it's environmental surroundings (people, weather,…, stimulus). This interaction produces information (effects). We construct meaning out of this information. When an organism encounters idea's and information which exists in a contrary relationship with the organisms existing schema (set of ideas), then this creates a dissonance, or (as Piaget put it) a disequilibriation. When this dissonance occurs, the organism is faced with the task of balancing the harmony between it's pre-existing schema, and the new contradictory schema. When the organism successfully manages the disequilibriation, then it has accommodated itself. The more an organism is said to be accommodated, the more it is at peace. This is a task of modern science in essence (not to mention religion and almost everything else humans do), to offer rational (hopefully) schema for the understanding of our environment. A prime example of this in a mathematical perspective, would be the subject of conservation of quantity. In this experiment (Piaget, 1953) a child was given two receptacles of different sizes (one tall and thin, the other short and wide) and an equal number of red and blue beads. He was then asked to place each of the beads inside of the receptacles. All the red beads in one container and all of the blue beads in the other. He did so, and was then asked which container contained more beads. He was sure that both contained the same number of beads, but the size of the containers gave the "illusion" that the tall thin container contained more beads. The child is said to be disequilibriated. This now leads us into Piaget's theories on developmental stages of learning.

In Piaget's view, humans are said to be at certain levels of developmental learning. These are certain levels, or ages, at which children are ready to assimilate a more accurate world view. Many children when first entering school have a intuitive, or pre-operational way of thinking, while others will be thinking in terms of concrete operations, and others still may be thinking at a formal operative level. It is necessary to understand the distinction between these if we are to fully understand our role as educators.

"Piaget discovered that children do not learn to conserve matter, weight, and volume at the same time," and the inability to conserve all of these three is characteristic of the intuitive, or pre-operational child" (Popp, 1997). Piaget also used mathematics to explain these developmental stages, more specifically a child's sense of geometry starts out topologically (*4), which is equated with pre-operational thinking. In this sense, "a child's order of development in geometry seems to reverse the order of historical discovery (*5)" (Piaget, 1953). By saying a child is thinking topologically, Piaget is saying that he is thinking only of properties which are invariant under transformations. For example, a circle and a triangle may be viewed as the same, because they are both closed figures.

Piaget's level of concrete operational thinking is equated to projective geometry. In this perspective, this level of thinking is seen as a collection of discrete mappings which when taken together yield some result, but the holistic perspective is lost. In one study involving the analysis of angles (Epstein, 1980), the students could not generalize the results of their data set. "They viewed the results of their experiment as a collection of point-to-point relationships" (Popp, 1997). The particulars of the study are irrelevant for this essay, but the core idea of the concrete operational thinking phase is that the child cannot make generalities from given information.

Piaget's final geometrical analogy is equated with Euclidean geometry, and is termed formal operational thinking. It is in this stage that the child learns to think conditionally and extrapolate outcomes, to deal with abstract relations, and offer hypotheses. Piaget also suggests that "the discovery of logical relationships is a prerequisite to the construction of geometrical concepts, as it is in the formation of the concept of number" (Piaget, 1953). Let me make note that all of Piaget's notions of accommodation result from the organisms interaction with its environment, in the above cases, social interaction is emphasized. When approaching students from a teachers perspective, it is essential to consider the students level of thinking. In general, as students progress in school, the percentage of formalized thinkers increases.

Vygotsky operated in the theoretical framework that social interaction plays an essential role in the development of cognition. Critical to this theory is that the potential for cognitive growth for an individual is limited to certain periods of time. These times he terms, "Zones of Proximal Development" (ZPD). While the philosophy of Hume infested Kantianism is the foundation of Piaget's experiments in constructivism, with Vygotsky we have a Hegelian dialectical foundation. Let me discuss Hegel's philosophy in brief so we have a better grasp of Vygotsky. This may at times seem a bit abstract, but this is a necessary philosophical foundation in order to fully appreciate Vygotsky's theories.

Hegel's philosophy concern's the Absolute, the One, the Totality, All (Hegel, 1830). From this One, all has arisen. This one is the All, from which all comes. In an effort to know of itself the one has split into the many, but by being many, an individual piece of this many is not the totality, and so there is an absence. This "absence" naturally gives rise to a third by the fact that there are 'two, in opposition'. The science of philosophy is a way of the One to know itself (*6). Let us now place this general Hegelian overview into the context of thought, and then by way of Vygotsky, into a social context. For Hegel, thought is a process, it is not a discrete thing that we can discuss, but occurs in action. Hegel's concept of thought is called the dialectic method (*7). The negation is the core concept here which is described in three parts: thesis, antithesis, synthesis (*8).

Thesis is a thought which on itself proves itself incomplete, this incompleteness gives rise to antithesis, which is the affirmation of the existence of the negation, all in reflection. Both of these taken together (*9) gives rise to synthesis, which is a transcendence of both the thesis and antithesis in a more enlightened perspective.

One can already see differences in the core philosophies between Kant and Hegel, which in turn give rise to differences in Piaget's and Vygotsky's theories. In Kant's view the use of analytic logic is suited to make sense of the phenomenal world based upon ones sense experiences. Hegel's logic paradigm emphasized that analytical understanding was acceptable only for practicality and the natural sciences (not for philosophy). Kant viewed knowledge as transcendent, which was assimilated by an individuals interaction with it. Hegel's dialectic is not concerned with the transcendent (*10), but with the Totality which is reality. For Hegel (as stated previously) knowing is doing, with respect to the totality (for what else is there in his perspective). With this core philosophy in mind, I now turn to Vygotsky's socio-cultural theory.

Vygotsky's genesis of concepts from early childhood manifests in a theory based upon thought-language interaction. There are three phases to this interaction. The first phase, called syncretism, deals with the unification of objects only by relations formed by the organism, not by the intrinsic qualities possessed by the objects. Examples of this phase may constitute things in the child's visual field, or random trial and error experiments. The second phase, called complexes, deals with the unification of objects by personal relations (subjective) as well as any incidental facts associated with this connection. Examples of this phase may include the identification of a member of a baseball team (i.e. Fred is a member of the Red Socks ergo, Fred plays baseball), or connection by transitivity (i.e. aý b & bý c implies aý c). The third phase, called concepts, deals with the abstraction of an attribute to form a basis for a collective. An example of this would be generalizations concerning a mathematical property (all circles have area=2(pi)radius). These phases are all successive of their predecessors and yet co-existent and co-functional. The interaction of these three levels gives rise to increases in conscious awareness. Blunden describes the relation of the individual (a piece) to society (the totality) perfectly in the following quote:

Vygotsky's socio-cultural theory focuses on the idea that human intelligence originates from society (culture), and individual cognition occurs first through interpersonal environmental interaction, rather than just the assimilation of knowledge. One study based upon Vygotsky's socio-cultural theory had teachers promoting scaffolding (*11), thinking critically, and meta-cognitive thought processes among an ethnographic sampling (Miller, 1995). This study showed "how social environment can influence students' learning and thinking" (Hsiao, www). A critical idea to Vygotsky's theory is the Zone of Proximal Development (ZPD). The ZPD is a range of time in which an individual can have more of a potential for cognitive growth. This growth occurs with the interaction among the social denizens in the individuals environment. The contrast between Piaget's and Vygotsky's theories should now be apparent and kept in mind (*12). I would now like to offer my own view of mathematics instruction relating to both Piaget and Vygotsky.

I enjoy both the ideas and the pitfalls of adhering to a particular philosophy and ethos, but my approach to mathematical education & instruction stems not from a concept of truth (*13) in standard sense of the word, but a concept of use (*14). The functions of Kant's interaction with transcendental knowledge interpreted by mind and the dialectic of Hegel are only important to me as they are of use to me, in a case by case basis (*15) . This is of course in view of Piaget and Vygotsky. From these two individuals I "borrow" certain ideas which I like and discard the rest. If any of the ideas and findings of Piaget or Vygotsky do not seem to be helping in a particular case of teaching, then I would search till I found something that would. So, it is in this sense that I must make it clear that my view of mathematical instruction is one of functionality (*16) (C1).

In the context of a social constructivist paradigm I would include the following in my "null structure" approach to teaching, after which I shall give an example of how I may go about this. Piaget has offered much, but the idea which seems to be of greatest import (to me) is the concept of dis-equilibrium, which is what happens when an individuals cognitive equilibrium is upset. It is at these times that an individual is ripe for re-adjusting their existing schema, or adopting new schema for whatever concept the teacher is trying to convey. The dialectical method of Vygotsky (Hegel) is a wonderful stimulation to the disequilibrium effect, so from him I take this. Glasersfeld stated the crux of my position crystal clear in the following quote:

Glasersfeld again agrees with the functionality of Null Structure in the following exert:

In short, my idea of mathematical instruction from a social constructivist epistemological view resembles the idea of Null Structure that I have just established.

I would now like to describe some of the views and theories of a mathematical educator who draws from a social constructivist perspective, Paul Ernest. Ernest is a mathematics educator from the University of Exeter. Epistemology is a great concern for teaching and Ernest is concerned with the philosophy of mathematics (a branch of epistemology) from a social constructivist perspective. Ernest "builds on the principles of radical constructivism together with the assumption of the existence of the physical and social worlds," and proposes a social constructivist philosophy of mathematics (Ernest, www). Ernest views epistemology as a subject that must be continually questioned and re-evaluated. Traditional methods of inquiry and research should not be just accepted and utilized, but subject to re-verification and even variations on how knowledge is viewed (*17). Ernest has a view on Epistemology that is similar to Piaget's disequilibriation. To bring up and question an epistemological stance causes a dissonance in the social patterns. Society does not like this and seeks calm and complacent individuals which do not threaten the established order. But it seems that when this order is upset, growth occurs (*18). All of this in the context of mathematics education, is traced by Ernest down to the foundations of mathematics. One view asserts that mathematics is absolute and can be known, the other asserts that it is fallible (as everything else) and a social effect. The absolutists are those that believe "mathematical truth" to be universal, and independent of human existence. Ernest dispenses with the absolutists claims and offers social constructivism as a philosophy of mathematics (*19). In this view, mathematics is a social construct (and therefore fallible). He makes this assertion on two claims: (1) that mathematics origins are socio-cultural in nature, and (2), that mathematical justification is "quasi-empirical". His proof rests on two assumptions: the existence of the physical world and the existence of the social world (*20). This existence does not require that we have any explicit knowledge of these worlds. Now, this is where Ernest brings in social constructivism and build upon ideas from radical constructivism. He outlines six underlying assumptions:

• Knowledge is not passively received, but actively interpreted (*21).
• Cognition is adaptive.
• Personal views of reality must match physical and social realities.
• These personal views of reality must "achieve this by a cycle of theory-prediction-test-failure-accommodation-new theory" (Ernest, www) (*21).
• (4) gives rise to social theories, patterns and language rules.
• "Mathematics is the theory of form and structure that arises within language" (Ernest, www).

Ernest claims, that on the basis of these assumptions, he can account for the effectiveness of mathematics in modeling the world to such a high degree, and also account for the high degree of certainty in mathematics. He does this (in brief) by tracing our perceptions, to the formulation of mathematical structures, which are invariably linked to language, as we must share these theories (C2). Language, in turn is changed into mathematical language and truths taken from one language follow to the other; "Mathematical certainty rests on socially accepted rules of discourse embedded in our 'forms of life' (Wittgenstein, 1956)". (*22) Ernest is an example of a modern social constructivist with which I share views (*23).

Now I wish to describe a lesson plan which reflects ideas in Algebra which would exemplify teaching mathematics from a social constructivist perspective. In order to accomplish this, the lesson shall first be presented, then I shall critique it. I make commentary notes in the lesson's explanation denoted, C#, which are used for reference.

Source: EMT 708 Summer Quarter 1998 L. P. Steffe University of Georgia

Subject: The fundamental theorem of counting, unknown, arrays and binomials.

Description: This lesson's activity set is exerted from a quarter long problem solving class in which we (as graduate students) addressed various alternative constructivist approaches to mathematical problem solving. This activity set took a week and a half to discuss and complete. During this time our class engaged in group discussion, formulation, questioning, argumentation with mathematical presentations and proofs. Dr. Steffe challenged our engrained, "taken for granted" concepts of number, multiplication and algebraic formulations. The lesson did not follow a traditional format of presentation/ exploration etc… but was dynamic, totally interactive, and non-linear. Since the lesson is very large, I provide only exerted problem sets.

Activity Set II (C0)

The fundamental theorem of counting, unknowns, arrays, and binomials.

"We need to seek ways of helping students base their construction of algebraic knowledge on their natural number knowledge. The assumption is that if students construct algebraic knowledge using their natural number knowledge, they have a good chance of experiencing algebra as generalizing their knowledge of arithmetic and as being useful to them in human endeavors (C1). In this, they have a good chance of experiencing algebra as a constitutive part of their intelligence in a way that they experience their natural language (C2). As mathematics teachers, this goal is well worth working toward, and to work toward it we need to continue investigating constructive possibilities in the mathematics that we know.

The strategy here is to explore the implications of the fundamental theorem of counting for constructive possibilities. Remember, we need to emphasize students' quantitative reasoning and base algebraic reasoning on quantitative reasoning (C3). Let's consider the natural number 6, as the measure of a particular kind of quantity such as the number of possibilities of the occurrence of an event E. As example, consider E to be the possible outcomes of the experiment of tossing a die. The quantity is the numerousness of the possible outcomes and this quantity can be measured by counting. The result, six, is the numerosity of the possible outcomes. Thinking of natural numbers as measures of quantities is a very important connection between arithmetic and algebra.

Before we counted, we could denote the possible results of counting by using a letter, say a. In this case, we call a an unknown. In general, an unknown is the possible results of measuring a specific quantity. A known is the actual result of measuring a specific quantity. This should help students to regard a as a numerical symbol. Initially, at least, the students' algebraic symbols, like a, can refer to "real world" experiments. I use quotations to indicate that these experiments need not necessarily be recurring events in their everyday lived experiences, although they should hold that possibility in so far as possible." (Steffe, 1998) (C4)

1. Let E be the possible outcomes of tossing a coin four times. First, find a systematic way to count how many elements in E.
2. Let E1 be the outcomes of tossing a die and E2 tossing a tetrahedron. What experiment does E1 x E2 represent?
3. In (2), make a representation of the outcomes of a specific experiment E1 x E2 using a rectangular array. (This is a critical connection that can be used to relate the fundamental theorem of counting and multiplication of binomial).
4. A bowl contains 14 cards with the numerals from "1" to "14" printed on them, one on each card. A card is drawn and replaced, and then another card is drawn and then replaced. If E1 represents the outcomes of drawing a card on the first trial, and E2 the outcomes of drawing a card on the second trial, make a rectangular array that represents the outcomes E1 x E2. Then, develop a thinking strategy for finding the number of elements E1 x E2. Be sure to use your array when developing your thinking strategy. (it is important that students know how many elements in the square patterns through a 20x20 square pattern or that they engage in binomial reasoning to find such squares.
5. Develop a pattern of reasoning for finding n², where 0 < n <100.
6. Develop a pattern of reasoning for finding n², where 100 < n < 1000.
7. The computational form of multiplication of two natural numbers usually needs to be reconstructed by students (C5). Consider the following computational form of product: 23 x 7 =

If I were to explain the value of a social constructivist perspective for teaching mathematics to a college (or high school) mathematics faculty, I would use the word awareness. The value of this perspective is most present in the effect upon a persons awareness. Social constructivism is an active view of epistemology. It places mathematics in a context which directly relates to human consciousness and thought. It allows for a continuously transforming epistemological stance, that does not have to be self-consistent (as long as it works). These are my idea's on mathematical instruction.

• 1: A typical example of such is immersing a stick in water with one half of it protruding from the waters surface. The stick appears to be bent, when in actuality this is just a result of refracted light waves.
• 2: In short, Hume's view concerning knowledge was that knowledge "came into" the mind from the outside. Much similar to input "as is" into a computer, or a photo-copy machine.
• 3: It is these stages of cognitive growth that Piaget gained in notoriety.
• 4: "Topology 2a(1): a branch of mathematics concerned with those particular properties of geometric configurations (as point sets) which are unaltered by elastic deformations (as a stretching or a twisting) that are homeomorphisms (Webster, 1999)."
• 5: The historical order of discovery of these geometrical fields progress from Euclidean geometry, to the development of Projective geometry in the 17'th century, and finally to Topology in the 19'th century.
• 6: For further active study in this arena I recommend study of Tao, or Cabala. Even though Hegel himself was a Christian, modern views of Christianity are exceedingly lacking of the more esoteric components essential to this philosophy.
• 7: For our purposes, the Dialectic can be thought of as a thought dialogue (although the ideas of the tripartite way are explained).
• 8: The terminology of thesis, antithesis, & synthesis is of modern usage.
• 9: This taken together is the most interesting part that cannot be shown, it must be experienced to be known. One may think of this as a fusion of opposites, but this does not do it complete justice.
• 10: Unless of course we are opting to use this upon itself with a transcendental/ existential polarization. This can give rise to modern chaoist philosophy. I shall elaborate more on this later.
• 11: Scaffolding is a Vygotskian technique where a teacher uses the aid of a mediator to facilitate learning. A mediator is something that the child has already mastered. This mediator acts as a bridge, or tool to be used by the child to learn a new or more complex task. Once the child is able to perform the new task, the "scaffolding" (mediator) is then taken away. A scaffold is "a supporting framework" (Webster, 1999).
• 12: The main difference being in that Vygotsky viewed human thought developing from the social to the individual, while Piaget viewed this order of development in reverse. Kant vs. Hegel philosophy
• 13: "Truth?", "Philosophical Truth?", these and a buck fifty will get you on the bus.
• 14: This can be an interpretation of Chaoist philosophy spoken of previously; I choose to call my style of teaching a "Null Structure".
• 15: Although I particularly enjoy the concepts which Hegel has "borrowed" and formalized from older systems of thought and practice, in their own light.
• 16: As it should be for good teaching.
• 17: By slightly altering ones epistemological stance, this can radically alter the outcome of research. For example, if I view Mathematics Education as something to learn (outside of self) then I may study for years in academic institutions learning from the ideas of others, but if I view Mathematics Education as something to explore and experiment with I may uncover an entirely new branch of something! These are extreme examples.
• 18: "Epistemology is controversial, and it cannot be denied that controversy leads to argument and conflict, which is uncomfortable. It is also contrary to the conventional social morality, which seeks consensus, calm, and above all, stability. Controversy represents a threat to this stability. However, the irony is that we probably all subscribe to a belief in the key role of dissonance and cognitive conflict in the accommodation of schemas, and hence in the growth of personal knowledge. Without psychological conflict of this type the growth of personal knowledge is not possible... When a new theory threatens to replace an old one, it is not greatly welcomed by those who have invested their professional lives in developing the old theory. Our belief in the necessity of conflict for the growth of knowledge need not eliminate the sensations of discomfort, whether one is involved in conflict at the level of subjective knowledge, objective knowledge, or of social discourse. " (Ernest, www).
• 19: It is not my purpose here to detail the absolutist position, but to focus on social constructivism. Ernest does offer two criticisms of this position: one a proof using deductive logic, the other, citing Godels Incompleteness theorems.
• 20: Both of which are axiomatic in nature, and "proven" by you reading (social) this paper (physical).
• 21: Ideas similar to Piaget.
• 22: Ernest has not formulated this theory explicitly as of yet, and the specifics are omitted.
• 23: Please keep in mind notes 13 & 14. 

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(http://www.merple.net.au/~andy/txt/vygotsk1.htm)

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