RUMEC/APOS Theory Glossary
as of February, 2001

Edited by David J. DeVries, authored jointly by RUMEC

APOS Theory and the RUMEC methodology continue to evolve. Indeed, it is central to our philosophy that, as we consider and analyze the evolving nature of an individual's knowledge, we consider and analyze the evolving nature of our own knowledge. Therefore the RUMEC methodology and APOS theory will never be set in stone. This document is an attempt to clarify what our understanding is right now. Reader be warned, it's been different in the past and will change in the future.

Taking that into account, RUMEC papers use certain terms in very specialized ways and these terms have very exact meanings. These are defined in our various papers, but we have collected them in one place for your (and our) convenience. Enjoy!

Defined Concepts

ACE Teaching Cycle:

Constructivism has implications for teaching and learning. One often begins instruction by presenting a situation that will create disequilibrium in the structures which the student has already constructed. In RUMEC designed instruction this is usually done through activities (A) exposing the student to new situations or information. For this purpose we often have students engage in computer programing as well as various cooperative learning activities. Part of the purpose of utilizing computer activities is to foster specific constructions. This is followed by class discussion (C) usually utilizing cooperative learning techniques to facilitate assimilation and accommodation. Classroom tasks often encourage construction through reflection. Finally the student is
assigned exercises (E) to reinforce and perhaps extend the ideas.
 

Example of ACE Teaching Cycle:

In a unit on Lagrange's Theorem in abstract algebra, students begin with an activity where they are required to write a computer program which accepts as inputs an element and a subgroup of a finite group and then returns the right coset. They then use this function to investigate the properties of these cosets and the set of all cosets for various groups. Part of the purpose of this activity is to help students construct a process conception of cosets. Class activities then involve reflection on what they have determined and note that all the cosets for a specific subgroup have the same number of elements and are either identical or disjoint. This reflection is important in fostering an object conception of the set of cosets. This is followed by a proof of Lagrange's theorem in a group discussion format. After class students are assigned exercises such as "prove that any group of prime power order as no proper subgroups."

Action/Action Conception:

An action is a transformation of an object which is perceived by the individual as being external. The transformation is carried out by reacting to external cues that give precise details on what steps to take. We say that an individual is at the level of an action conception of a given transformation, if her or his depth of understanding is limited to performing actions in carrying out that transformation. It should be noted that someone with a deeper understanding of a transformation may well perform actions when appropriate but is not limited to performing actions.

Examples of Actions /Action Conception:

1. One performs an action when given a formula for a function and a point, one calculates the value of the function at that point. A student who is unable to interpret a situation as a function unless he or she has a single formula for computing values is restricted to an action conception of a function. In such a case, the student is unable to do very much with this function except to evaluate it at specific points and to manipulate the formula.
2. Given a specific group with a binary operation algorithm one can apply, a specific subgroup and a specific element, an action is to calculate the elements of the left coset containing that element. Understanding a coset in group theory only as a set of calculations that are actually performed to obtain a definite set is an action conception.
3. An action is to solve a given equation by following the steps in an example for a similar equation. If one only understands solving an equation as searching for an example which can be imitated then one is at the level of an action conception of solving an equation. The example located may be a previously memorized example.
4. Given the general rule for finding the derivative of a polynomial function, and given a specific polynomial function an action would be to find the derivative by plugging the numbers into the general formula. One is at the level of an action conception of differentiation if one is only able to find the derivative of a function when at each step one is externally provided (by memory, for example, or looking it up in a list of rules) the rule which is to be applied and one just enters the specific numbers into the rule.
5. An action is to calculate the standard deviation of a specific set of data given the formula. If one's understanding of standard deviation is limited to the ability to calculate the standard deviation of a particular set given the formula then we say one is limited to an action conception of standard deviation.

Constructivism:

As is true of all the definitions in this glossary, we are attempting to clarify how RUMEC researchers use the term and we do not intend to give a general definition of constructivism. RUMEC conducts its investigations from the point of view that an individual's understanding of mathematical concepts results from the construction or reconstruction of mathematical actions, processes, and objects and organizing these in schemas so as to use them in problem solving situations. Progress in understanding is usually made by making a reconstruction in a problem situation where the problem is similar to but different in important ways from a previous problem situation. The reconstruction is not exactly the same as what existed previously, and may in fact contain one or more advances to a more sophisticated level. A further aspect of our approach is the belief that the construction or reconstruction of existing structures is enhanced by it taking place in a social context. Our point of view is related to the Piagetian ideas of disequilibrium, accommodation,
assimilation, reflection, the triad and schema development. Indeed our theoretical perspective is itself the result of a reconstruction of our understanding of Piaget's theory so as to extend this theory to learning post secondary mathematics.

Examples of Constructivism:

1. Our constructivist theory as to how students learn undergraduate mathematics leads to instructional strategies that are designed to foster specific constructions. The ACE teaching cycle, described in this glossary, is an example of such a strategy. The use of cooperative learning and computer programing are examples of strategies designed to encourage reflection which we see as the primary tool for building new understanding.
2. In order to understand the proof of Lagrange's Theorem in abstract algebra, one must have an object conception of the set of equivalence classes. This is required so one can compare the elements of this set and also count them. We believe that one reaches such an object conception through a building procedure involving actions, processes and reflection.

Genetic Decomposition:

The first step in the RUMEC methodology is to engage in a theoretical analysis of a mathematical concept in terms of the mental constructions a learner might make in order to develop understanding of the concept. The result of this theoretical analysis is called a genetic decomposition for the concept. It is a detailed description of the mental constructions an individual may make to deal successfully with a particular mathematical concept. It should be noted that a genetic decomposition for a topic cannot be assumed to be unique. (Vidakovic, 1993) For a particular concept under study, we present a genetic decomposition which is based on a general theoretical framework of learning, the totality of our observations and experiences, and our own understanding of the mathematics involved. This serves us as a possible 'roadmap' of one's learning of this concept. Much of the special terminology used in our genetic decompositions is discussed in this glossary.

Example of Genetic Decompostion:

In an abstract algebra context, the mental constructions a student might make in constructing an understanding of cosets might include the following items. Given an element x and a subgroup H of a group G, if an individual thinks generally of the (left)coset of H determined by x as a process of operating with x on each element of H, then this process can be encapsulated to an object xH. With such an object conception, cosets can be named and operations can be performed on them. Various actions on cosets of H, such as counting their number, comparing their cardinality, and performing operations can make sense to the individual who has an object conception of coset. The relationship between the process and object conceptions of cosets with regard to, say, checking the intersection of two cosets is as follows. To be able to think about the problem of investigating the intersection of any two cosets involves the interpretation of cosets as objects. The actual determination of the intersection requires that these objects be de-encapsulated in the individual's mind so as to make use of the properties of the processes from which these objects came(a certain kind of set formation in this case).

Object / Object Conception:

Individuals can construct cognitive objects in two ways. When an individual reflects on actions applied to a particular process, becomes aware of the process as a totality, realizes that transformations(whether they be actions or processes) can act on it, and is able to actually construct such transformations, then we say the individual has reconstructed this process as a cognitive object. In this case we say that the process has been encapsulated into an object. A second way to construct a cognitive object occurs when an individual reflects on a schema, becomes aware of the schema as a totality, and is able to perform actions on it, then we say the individual has thematized the schema into an object. We say that an individual has an understanding at the level of an object conception of a mathematical concept when that individual's depth of understanding of an idea or concept includes treating that idea or concept as an object. The individual is able to perform actions on the object. Such an individual is able to also de-encapsulate an object back into the process from which it came when necessary, or in the case of a thematized schema unpack it into its various components.

Examples of Objects / Object Conceptions

1. An individual who is able to think about a function as the sum of two functions without reference to specific examples is thinking about a function as an object. One has an object conception of a function if one is able to think about decomposing a function into the sum of two functions.
2. An individual who is able to think about counting the cosets in a finite group is thinking about cosets as objects. An individual who is able to define a binary operation on a set of cosets would be operating at the level of an object conception of cosets.
3. An indication that a student's chain rule schema has been thematized into an object is that the student is able to analyze a new situation and recognize how and why the chain rule is involved. Such an individual would have an object conception of the chain rule as a result of thematizing her or his chain rule schema and could act on it by combining it with other rules.
4. Euclidean Geometry is an example of a thematized schema, which is an object for someone who has several geometries he or she knows, moves among them, compares and contrasts them and selects an appropriate geometry to solve a problem.
5. A student who has thematized her or his schema for solving algebraic equations can select appropriate methods and understands the relationship between procedures for finding possible solutions and finding solution sets.
6. In differentiating functions one treats the rules for differentiation as objects by recognizing which general rules are needed and selecting them and using them correctly. The student is able to combine rules and explain how they have been combined.
7. When a student is able to think in terms of a generic binary operation which can be instantiated in two or more distinct ways on the same set, then they may be interpreting their binary operation schema as an object.

Process / Process Conception

When an action is repeated, and the individual reflects upon it, it may be interiorized into a process. That is, an internal construction is made that performs the same action, but now not necessarily directed by external stimuli. An individual who has constructed a process can describe, or even reverse the steps of the process without actually performing the steps. In contrast to an action, a process is perceived by the individual as being internal, and under one's control, rather than as something one does in response to external cues. We say that an individual is at the level of a process conception of a given transformation, if the individual's depth of understanding is limited to thinking about the transformation as a process.

Examples of Process /Process Conception:

1. One performs a process when one thinks of a function as receiving inputs and returning outputs or imagines the calculation of functional values without actually doing the calculation. An individual is at the level of a process conception of a function if she or he can differentiate functions specified by a formula but has difficulty breaking a function up into an algebraic combination of functions in order to find the derivative.
2. An individual performs a process when he or she can imagine constructing right cosets given a subgroup, and does not have to go through the actual construction. One is at the level of a process conception of cosets if one is able to imagine the formation of right cosets, but has difficulty defining a binary operation on those cosets.
3. A individual performs a process when he or she solves an equation guided by the form which would give a solution. In this case, an individual is able to describe the steps needed to solve an equation without actually doing them. Such a student may also be able to reverse the steps to show a possible solution is in fact a solution. One is at the level of a process conception of solving an equation if one has a process for finding the solution, but one is not able to perform an action on the solution set without actually finding the solutions.
4. One performs a process when one finds the derivative function of a given function using the standard rules. One has at most a process conception of differentiation if one can find the derivative of standard functions but can not utilize the idea of the second derivative of a function unless the first derivative has been calculated for a specific function.

Reflective Abstraction:

Reflective abstraction is a concept introduced by Piaget to describe the construction of logico-mathematical structures by an individual during the course of cognitive development. Reflective abstraction by an individual proceeds from two mechanisms which are necessarily associated. They are projection unto a higher level of that which was derived from a lower level, and secondly reflection which reconstructs and reorganizes within a larger system what is transferred by projection. (Piaget and Garcia in Psychogenesis and the History of Science)

Examples of Reflective Abstraction:

1. Interiorization of an Action: The mental construction of an internal process(a coherent totality) relative to a series of actions on cognitive objects that can be performed or imagined to be performed in the mind without necessarily running through all the specific steps. In this case we say the action has been interiorized to a process.
2. Encapsulation: Encapsulation is the mental transformation of a process (which is the interiorization of some action) into a cognitive object. This object can be seen as a total entity (or coherent totality) and can be acted upon (mentally) by actions or processes. In this case we say that a process has been encapsulated into an object. De-encapsulation is the mental process of going back from an object to the process from which the object was encapsulated.
3. Thematization of a Schema: When one reflects upon one's understanding of a schema, views it as "a whole", and is able to perform actions on the schema, then we say that the schema has been thematized into an object. (Piaget and Garcia say that a change from usage, or implicit application to consequent use, and to conceptualization constitutes what has come to be known under the term "thematization". [page 105, in Piaget and Garcia].)

Schema

A schema for a certain piece of mathematics is an individual's collection of actions, processes, objects and other schema which are linked consciously or unconsciously in a coherent framework in the individual's mind and may be brought to bear upon a problem situation involving that area of mathematics. An important function and defining characteristic of the coherence is its use in deciding what is in the scope of the schema and what is not.

Examples of Schemas:

1. A student may have a schema for solving equations which includes various methods for transforming equations and a conception of what it means to solve an equation.
2. An individual's schema for differentiation may include various rules for finding the derivative of a function.
3. An individual's group schema could be a coordination of various other schemas which might include a schema for sets as well as a schema for binary operations.
4. An individual may have a schema for limits, which enables one to coordinate in some fashion cognitive representations of closeness in the domain, understanding of closeness in the range, and a conception of function.
5. Mathematical rules, such as the chain rule for differentiation, which require coordinating two or more actions, processes, or objects may also be understood via a schema. The issue with understanding such rules appears to be more complex than simply encapsulating one process into an object.

Triad of Schema Development

Schemas develop through stages or levels in a specific order. Initially, in the development of a schema, it is at the intra stage or level. This is followed by the inter stage or level and finally by the trans stage or level. This terminology was introduced by Piaget and Garcia in their discussion of the development of knowledge and we have applied their ideas specifically to schema development. Sometimes these terms are expanded in order to reflect the nature of the schema. For example in a geometry schema the stages might be referred to as: intrafigural, interfigural and transfigural. In an analytic schema they might be referred to as: intraoperational, interoperational and transoperational. In the examples which follow we could use these expanded terms when that is appropriate but generally RUMEC simply uses the terms intra, inter and trans.

Intra Stage of Schema Development:

In the development of schema, the intra stage is characterized by a focus on individual items in isolation from other actions, processes and objects of a similar nature. The individual has not constructed any relationships between them.

Examples of the Intra Stage:

1. In the development of a chain rule schema, a student may use distinct, special rules such as the general power rule and not recognize them as related in any way or as special cases of the chain rule.
2. A student is at the intra stage of a binary operation schema when he or she is able to find inverses in some specific groups but does not recognize any relationship between the processes for finding inverses or see them as special cases of the general idea of finding inverses.

Inter Stage of Schema Development:

(A schema at this stage or level is referred to as a pre-schema)

In the development of schema the inter stage is characterized by the construction of relationships between actions, processes and objects. For example, one may begin to see several examples as special cases of a more general concept. At this level one begins to group together items of a similar nature and perhaps even call them by the same name.

Examples of the Inter Stage:

1. In finding the derivative of a function the student sees the general power rule and other special situations such as those involving the trigonometric functions for which rules have been learned as special cases of the "chain rule" and links them together under that description. The student does not yet understand why they are special cases but perhaps thinks in terms of similar procedures involving inside and outside functions. Such a student would be at the inter stage of development of a chain rule schema.
2. In group theory one begins to see the property of inverses instantiated in many examples and begins to collect these examples. One begins to see inverses in various situations as having a relationship, however one does not yet understand the general definition of an inverse in such a way as to be able to apply it in a new situation. In this case one's binary operation schema would be at the inter level.

Trans Stage of Schema Development:

The trans stage is characterized by the individual having constructed an underlying structure through which the relationships discovered in the inter stage are understood and which gives the schema coherence. An important function and characteristic of coherence is its use in deciding what is in the scope of the schema and what is not. At the trans stage understanding moves from a list to a rule. At the trans stage of development the collection can now be referred to as a schema as it now has the required coherence and may be thematized.

Examples of the Trans Stage:

1. One is at the trans stage of development of the chain rule schema if one understands various special cases as applications of the chain rule by identifying certain functions which are composed. The student is also able to apply the chain rule in some new situations by looking for a composition of functions. In this case we say that the student has a chain rule schema.
2. In abstract algebra a student has a trans level of development for a group schema when the student is guided by the definition or relevant theorems in determining whether a certain subset of an abstract group is also a group under the same binary operation.

Note 1: We do not speak of thematizing a schema before it reaches the trans stage, that is before it becomes a schema. We prefer to think in terms of very weak versions of a trans stage. We do not restrict the trans stage to something very sophisticated, but just look for whatever way the individual decides what is in and what is out of the schema. It is true that this leads to thematized schemas that don't do a very good job for the individual, but this is exactly what happens and it is important to note that in order to correct it, the schema has to be unpacked and reconstructed.

Note 2: Also some confusion can result when one gives the same label to schemas with essentially the same collection of elements but with different underlying structures which gives them coherence. For example, one can be at the trans level with respect to the chain rule schema for real valued functions of a real variable but at the intra level with respect to the chain rule schema for functions of several variables.

Note 3: When we use the term schema without a qualification of stage or level, we are referring to a schema which has reached the trans level of development.

Maturity of a Schema:

Discussions involving the strength of the schema are referred to by the label "maturity of the schema". This is similar to what happens with the concept of function. If the process is reduced to a syntactic process of organizing symbols in a formula according to certain syntactic rules, and if it is this process which is encapsulated to get an object, then the student ends up with a weak function conception. Again, the cause is not a failure to encapsulate, but the encapsulation of a process that was not very rich. In a similar fashion one may thematize a schema which has a weak coherent structure and thus one may not be able to apply the schema in many new situations.

Example of Maturity of a Schema:

One may have a chain rule schema which allows one to differentiate functions defined by standard algebraic formulas but is not able to recognize composition of functions involving functions defined by definite integrals and is thus unable to apply the chain rule. A weak function composition schema would lead to a weak chain rule schema.

Other Words which eventually may be discussed and added:

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Last Updated: February 2001