vol. 1, no. 1, 2009, 27-43 modeling mathematical...

Modelling mathematical Actiotopes with CLARION 27Talent Development & Excellence

Vol. 1, No. 1, 2009, 27-43

Modeling mathematical Actiotopes: The potential role of CLARIONShane N. Phillipson1,* and Ron Sun2

Abstract: The challenges facing research into giftedness are considerable, including the lack of a clear meta-theoretical framework and agreement on terminology. If giftedness, in broad terms, refers to the interactions between complex patterns of actions, the psychological aspects of self and the social environment toward the attainment of achievement excellence, then it is not surprising that the study of giftedness presents a number of conceptual and practical difficulties. Using mathematical achievement as an example, this article explores the potential benefits and problems of modeling the Actiotope Model of giftedness using the cognitive architecture CLARION. In using simulation studies to explore giftedness, the meta-theoretical framework, includingterminologies, is clearly defined, helping to direct future field studies using the more traditional methods of research.

Keywords:

achievement excellence, Actiotope Model of Giftedness, CLARION, cognitive architecture, mathematics, modeling

Introduction

The current state of research into giftedness has been described by a number of researchers as fragmented and contradictory. The aim of this article is to argue that the Actiotope Model of Giftedness (AMG) (Ziegler, 2005) provides a much needed sense of coherence and unity to this research. The AMG focuses on the development of action repertoires necessary for the attainment of excellence. In broad terms, the development of action repertoires is dependent on physiological, psychological and environmental (or social) variables, and excellence is viewed within social contexts.

Although the AMG is difficult to test experimentally using traditional research methods because of its reliance on these different conceptual levels, we also argue that it is possible to simulate the AMG using techniques developed by researchers currently working in the area of cognitive modeling. Using excellence in the domain of mathematics as an example, we describe the CLARION cognitive architecture as one possible way to model the development of higher forms of mathematical thinking from its antecedents of innate number sense. In describing the general principles of both the AMG and CLARION, this preliminary discussion then shows how the two can be mapped onto each other and the immediate challenges in using this approach and thereby providing some suggestions for future research.

Conceptions of giftedness

In reviewing the status of research into giftedness, Eyre (2009) pointed out that, despite the many decades of work, conceptions of giftedness amongst psychologists are fragmented and often contradictory. Recent volumes that specifically address different conceptions of giftedness (Phillipson & McCann, 2007; Shavinina, in press; Sternberg, 2004b; Sternberg & Davidson, 2005), the related topics of intelligence (Sternberg, 2000), creativity (Kaufman & Baer, 2006; Kaufman & Sternberg, 2006) and terms such as “high ability”, “talent” and

1 Shane N. Phillipson, Department of Educational Psychology, Counselling and Learning Needs, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, NT Hong Kong

* Corresponding author. Email: [email protected] Ron Sun, Cognitive Science Department, Rensselaer Polytechnic Institute, Troy, NY, USA.

ISSN 1869-0459 (print)/ ISSN 1869-2885 (online) 2009 International Research Association for Talent Development and Excellencehttp://www.iratde.org

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S. N. Phillipson & R. Sun28

“precocity”, for example, would tend to confirm this impression. Given this status, it is not surprising that the broad field appears confusing to researchers working either within or outside the field. It is this confusion that provides detractors with ample opportunity to criticize and, in some cases, deride the research into giftedness altogether (Marsh, 1998; Seaton et al., 2008).

If conceptions of giftedness amongst psychologists are numerous and fragmented, it is not surprising that the practice of gifted education is fragmented. The reasons for the development of the three broad “educational paradigms” (Eyre, 2009) in gifted education lie partly in the different agendas of the various educational systems around the world, and partly in the different conceptions of giftedness that are adopted by the policy makers working within these systems. In Hong Kong, for example, policy documents supporting gifted education, either explicitly or implicitly, simultaneously refer to multiple intelligences, creativity, the Marland report, IQ and Gagné’s Differentiated Model of Giftedness and Talent as conceptions of giftedness. The documents also mention disabilities and underachievement as characteristics of some gifted individuals (Phillipson et al., in press).

Using the Hong Kong context again, the policy of including as many conceptions of giftedness is politically attractive. Being “inclusive” means that the cohort of gifted students can be as wide as possible, reducing the likelihood of alienating certain members of the community (Eyre, 2009). As Eyre pointed out, the cohort approach depends on the prior identification of students who are gifted in some way and depend, therefore, on psychological and/or personality commonalities amongst these students. Perhaps the best known examples of research involving the cohort paradigm are those that centre on mathematical precocity (Lubinski & Benbow, 2006). Significantly, this research has identified some of the important environmental and personal components that work together to enable the attainment of mathematical excellence.

Eyre (2009) has identified “definitions” as one of the key questions for future research in gifted education. It is timely that psychologists researching in the field of giftedness begin to think about how to bring coherence to the diversity of conceptions of giftedness. This coherence needs to take into account cultural diversity as well as personality variables such as differences in measured intelligence, creativity and motivation, for example, and differences in conceptions of achievement. As Eyre concluded, the field must reorganize itself if it is to move beyond the “educational margins” and take its rightful place in identifying the “conditions that enable and encourage exceptional ability to flourish” (pp. 17, 18).

Eyre’s (2009) concerns are not new. Ziegler (2005) has argued that most of the theoretical conceptions of giftedness do not have a sufficient empirical basis, that the relationship between gifts and achievement is often unclear, and that research methods and definitions are often characterized by tautologies. In discussing the relationship between intelligence and giftedness, for example, Callahan (2000) observed that the commonly cited conceptions of giftedness such as Renzulli’s three-ring definition, Tannenbaum’s psychosocial definition, Sternberg’s triarchic and WICS models, and Gardner’s multiple intelligences model rely on different explanatory levels, such as cognitive skills, expressions of appropriate sets of behaviors, production of knowledge, bio-physical potential, personality, luck and culture. Furthermore, combinations of these different explanatory levels sometimes exist in the one model. Of course, models that rely on different explanatory levels are not easily falsifiable.

Last, Ziegler and Heller (2000) pointed out that much of the reported research to date lacks the rigor normally expected of scientific research, such as appropriate control groups. Heller and Schofield (2000) also concluded from their content analysis of international trends on research on giftedness that the field is overwhelmingly dominated by applied rather than basic research, and that there was an urgent need for improvements in the “quality of research designs and measurement techniques …[and] … most importantly, basic research … [requiring] the intensification of cooperation with researchers” (pp. 136-137). Although there is an urgent need for a systematic analysis of research since 1999, Eyre (2009) concluded that the situation has probably not changed very much since that time.

Modelling mathematical Actiotopes with CLARION 29

Generally, research on giftedness centers on exceptional performance of some kind, and on the antecedents and processes that lead to exceptional performance. Despite the problems in research approaches and methods, there is broad consensus that giftedness involves interactions between aspects of self (such as intelligence, cognitive processes, self-efficacy and motivation, for example) and the environment (parents, education, and culture, for example) that help propel the individual or the group to exceptionality (Sternberg, 2004a). Last, exceptional performance is contextual, meaning that it refers to different things at different times and for different sociocultural groups (Phillipson & McCann, 2007).

The critical state view of giftedness

Each of the psychologists working within the field of giftedness brings to their research, of course, their individual prejudices and preferences. This should be encouraged since it is the source of debate and the engine of further research. In order to bring a much needed unity to this debate, Ziegler and Heller (2000) described a meta-theoretical framework for research into giftedness. Termed the critical state view of giftedness (CSG), the framework describes a probabilistic rather than deterministic relationship between the critical state (CS) and achievement excellence (AE). Boundary conditions operate directly on the CS, to either promote or decrease the probability of an individual reaching AE (Ziegler & Heller). The term talented refers to someone who has the necessary psychological components, but not yet fulfilled the conditions of the CS. Gifted refers to someone who has fulfilled the conditions of the CS but, because of unfavorable boundary conditions, not yet reached AE. Expert refers to persons who have reached AE.

The two stages of the CSG refer to the temporal relationship between the critical state (CS) and achievement excellence (AE) (Figure 1). “Gifts” are the psychological components within the CS but there are other aspects of personality such as motivational states with all being necessary and sufficient for AE. Individuals achieve AE after fulfilling the conditions within the CS. Within any domain there can be many types of AE and there may be many ways to achieve the any one AE.

Achievement excellence (AE) is at first glance the most contentious area of the CSG. Ziegler and Heller (2000) proposed that AE needs to be understood using a sequential series of problems beginning with a clarification of domain, focus, frame of reference and significancerespectively. Ultimately, however, solutions to these problems are more likely to be found by persons working within rather than outside the domain (Phillipson & Callingham, in press).

Figure 1. Achievement excellence in Domain X originates from one of Critical State (CS) a , CS b … CS n under optimal boundary conditions for each critical state. Each CS operates independently of the others and the gift, a psychological process, fulfills the inus condition of Postulate 2. (Adapted fromZiegler & Heller, 2000, p. 15).

Achievement eminence in Domain X

CS a

CS b

CS c

CS n

Critical State (CS) a

Personality characteristics inducive to gifts (unnecessary, sufficient)

Gift (insufficient, non-redundant)

Environmental conditions

Development, learning processes


Nature of gifts

Within each CS, the components of interest are the psychological processes. Ziegler and Heller (2000) referred to these processes as gifts, hence the study of giftedness should focus on the processes that lead to AE. The gift is a necessary but in itself insufficient for AE.

According to Ziegler and Heller (2000), gifts must satisfy four postulates:

Postulate 1 is that the gift temporally precedes the AE and there must be a causal and non-trivial relationship between the gift and the achievement. Postulate 2 is the fulfillment of the inus condition. In other words, the psychological process is in itself an “…insufficient but non-redundant part of an unnecessary but sufficient condition” (p. 13). Gifts are therefore necessary but in themselves do not guarantee the AE. Ziegler and Heller pointed out that the gifts are one component within the CS, and all components must be present if there is a chance of achieving AE. Postulate 3 is that gifts are personal (i.e. psychological processes) and not environmental variables. To use the example provided in Ziegler and Heller, intelligence does not qualify as a gift because a direct and causal relationship between it and AE is not easily determined. At best, the relationship between intelligence and AE can only be described as correlational. Postulate 4 is that the gift must hold some significance in the explanation of the AE, and that a probabilistic rather than deterministic relationship between the gift and AE is made explicit.

To conclude, the critical state view of giftedness (CSG) clearly defines the nature of giftedness and how it might be researched. Although Ziegler and Heller (2000) focused on the psychological processes within the CS, it is clear that any understanding of the development of AE must also be directed toward the interactions between the various components. The CSG suggests that the research focus should centre on the psychological processes, termed gifts, which contribute directly to the achievement of excellence. When applying the CSG, a number of guiding questions can be formulated (Ziegler & Heller), including:

• What is the domain of excellence and in how many ways can it be defined?• What are the components of the critical state that lead to AE?• What are the psychological processes which contribute directly and non-trivially to the

AE?• What interactions between the psychological, personality and environmental conditions

are likely to contribute to AE?• Under which conditions is the psychological relationship between CS and AE valid?

Of these questions, the first question is perhaps the easiest to answer. Excellence within a domain is defined by persons working within that domain and, hence, varies across cultures and time. The psychological processes that contribute to AE will also be different from domain to domain, as will the interactions between the processes, the personality and the environment. These interactions are not easy to describe.

The Actiotope Model of Giftedness

The Actiotope Model of Giftedness (AMG) is a systems approach to the development of achievement excellence (AE) (Ziegler, 2005; Ziegler & Stöger, 2004). It describes the complex interactions between a person’s action repertoire, their subjective action space and the environment. In the AMG, the focus is on the development of behaviors that lead to AE.

The development of action repertoires during each stage depends primarily on the individual setting and realising new goals. Individuals are free to choose goals and to determine the best possible pathway towards these goals. In determining the pathways, individuals make subjective assessments of the relevancy, value and usefulness of their current action repertoires. These subjective assessments, termed subjective action space, are psychological structures that represent action opportunities and they may either over or under-estimate the true nature of the action repertoires. These behaviors are both affected by


and, in turn, affect the environment and so undergo a series of progressive adaptations – but only as long as the person is pursuing excellence.

Not surprisingly, the terminology in the AMG is consistent with that of the CSG, including the use of the terms talent, gifted and expert. In the CSG, a gift is a psychological process that satisfies the four postulates. In the AMG, a gift is also a psychological process and its nature is dependent on the observer and context. Whether or not actions satisfy the four postulates can only be answered through research.

The components of the Actiotope, including the psychological components, actions, goals and personality variables (such as motivation and self-efficacy) are described as a dynamic system in accordance with the requirements of the critical state (CS) within the CSG. Since these components, by definition, directly interact with the environment in a purposeful way toward the AE, the Actiotope has a clear developmental trajectory.

The development of an Actiotope leading to mathematical excellence, for example, proceeds through a number of phases:

• Phase 1 – Pre-natal and early childhood development. An individual may show exceptional learning or precocious achievement and their actions are termed talented.

• Phase 2 – Attainment of a critical state. An individual has attained all of the necessary knowledge and behaviours (actions) necessary for excellence. The actions of an individual at this stage are termed gifted.

• Phase 3 – Attainment of excellence. An individual demonstrates outstanding performance in a domain. Their outstanding actions are termed excellence.

Of these three, Phases 1 and 2 are more likely to correspond with the individual’s development through their formal education, although it is clear that the domain(s) in which future AE occurs is likely to be unknown and that the domain may or may not correspond with the immediate aims of education. Phases 1 and 2 are, however, critical periods in the development of action repertoires necessary for AE.

The probabilistic relationship between an individual’s Actiotope and AE is not, however, clearly articulated in Ziegler (2005). In this regard, modeling the development of the Actiotope from talent through to expert stage using Structural Equation Modeling (SEM) may at first glance offer one way to more precisely describe this relationship (McCoach, 2003). There are two related problems, however, that limit the usefulness of SEM to model the development of AE. First, SEM requires a sample size of at least 100 if the modeling is to produce reliable results. This means that we need to identify at least 100 individuals with a trajectory toward the same AE. If these individuals are at the expert stage then this identification is made much easier. If the individuals are at the talent or gifted stage, then it is almost impossible to identify these individuals since they have not, by definition, reached AE.

One possible solution would be to identify the action repertoires of a number of experts within a domain, and then use these as markers for actions within individuals at the talent and gifted stage. There is the complication that some of these individuals will exercise their free will and choose different domains to be an expert. The situation is in fact much more complex than this because the AMG proposes that different Actiotopes may develop the same AE.

The second problem is that the AMG focuses primarily on the individual rather than a population (Ziegler, 2005; Ziegler & Stöger, 2004). Hence, the AMG is a model of individual differences, albeit at a systems level. The development of an action repertoire, together with the subjective action space, goals and environment do not guarantee AE. An individual’s Actiotope, therefore, is an unnecessary but sufficient condition, with the action itself being both insufficient and non-redundant. In other words, the action within a successful Actiotope does not in itself guarantee AE since the other components are also necessary, even though AE without the action is impossible. It is this variability that renders the SEM a conceptually inappropriate tool.


Gardner’s theory of multiple intelligences (Gardner, 1993) is often criticized for its lack of heuristic value (Anderson, 1992; Klein, 1997; Morgan, 1996). Anderson’s criticism is based on the observation that multiple intelligences theory is derived using evidence from a number of different conceptual levels, including culture, environment, personality, cognition and neurobiology. Similarly, the AMG is dependent on a very broad conceptual framework, including environment (for both defining actions and domains), meta-cognitive regulation, self-evaluation, psychological structures, skills and IQ, rendering it susceptible to the same criticism.

Despite the difficulties in the heuristic potential of the AMG, it is a unique approach to understanding the nature of giftedness. The theory is pleasing because of its focus on the development of an individual’s action repertoire rather than unitary (or pluralistic) views of intelligence. The AMG takes a holistic perspective of giftedness, focusing on the development of these action repertoires within a changing and interactive environment. Although it is possible to identify the action repertoires and subjective action space of an individual at any given time (Ziegler & Stöger, 2004), modeling the interactions of the many different components of the Actiotope present greater research challenges.

The Actiotope Model of Giftedness and achievement excellence in mathematics

In discussing mathematical giftedness for the International Handbook on Giftedness(Shavinina, in press), Phillipson and Callingham (in press) used the Actiotope Model of Giftedness (AMG) as their conceptual framework. Drawing on research that focused on the development of mathematical giftedness, Phillipson and Callingham drew on wide ranging research that reflected aspects of self (number sense1, intelligence, motivational states, beliefs and attitudinal states, gender, and self-efficacy and self-esteem, for example), the development of action repertoires (memory, computation and logical reasoning, for example) and the environment. Phillipson and Callingham also discussed the domain problem, concluding that there are at least three broad ways to describe mathematical excellence, including societal perspectives, educator’s perspectives, and the perspectives of professional mathematicians.

Phillipson and Callingham (in press) noted that the AMG provided a useful conceptual framework that helped bring unity to their discussion of mathematical giftedness. At a fundamental level, however, the AMG emphasized the developmental nature of AE in mathematics, showing how it may be possible to describe the development of complex mathematical thinking from the small number of innate numerical abilities such as numerosity and additive expectations2. Furthermore, Phillipson and Callingham proposed that future research could meaningfully focus on computer modeling of the complex interactions between an individual and their environment leading to AE using cognitive architectures such as SOAR and CLARION (Sun, 2006a, 2006b).

In agreement with the generalizations from almost all conceptions of giftedness to date, the attainment of mathematical AE in the AMG is dependent on both the social processes interacting with the individual and the characteristics of the individual. In other words, AE is a social phenomenon since its attainment depends on the recognition by significant others. Reaching AE also depends on the characteristics of the individual. In the parlance of cognitive modeling, it is the interaction of a single agent within a broader multi-agent system.

The next section describes how CLARION might be used as a tool to model the development of mathematical AE within the framework of the AMG. It begins by describing the essential features of CLARION, including the interactions between the various components of the architecture. Next, the various levels of the AMG are mapped onto the components of CLARION. Finally, some of the challenges in using CLARION to model the development of mathematical excellence are discussed.


The CLARION cognitive architecture and the Actiotope Model of Giftedness

Modern computational models of cognition recognise the important role of the social environment. In exploring the contributions that cognitive science can make to the social sciences, Sun (2006a) drew a link between the ability of modern cognitive architectures such as ACT-R, SOAR and CLARION to model complex cognitive processes, and the need of social scientists to explain complex sociocultural phenomenon. Describing “computer simulations of social phenomenon” (p. 6) as the third way of doing science3, Sun believed that these simulations can provide data to support both inductive and deductive approaches to understanding the processes involved in the phenomenon. According to Sun, however, the key to modeling complex sociocultural phenomenon is to

incorporate realistic constraints, capabilities, and tendencies of individual agents in terms of their cognitive processes (and also in terms of their physical embodiment) in their interaction with the environments (both physical and social) (p. 7).

To date, the simulation of complex cognitive processes using cognitive architectures include the dynamics of youth subcultures (Holme & Grönlund, 2005) and new religious movements (Upal, 2005). The general issues surrounding computer simulations of complex social phenomenon have also been discussed (Sawyer, 2003; Sun, 2006a, 2001; Sun & Naveh, 2007; Wilensky & Rand, 2007)

The AMG describes a number of broad groupings related to the development of achievement excellence, broadly grouped into aspects of self (physiological and psychological state) and the social environment. Possible sources of individual differences into aspects of self and the social environment are easy to find - we merely need to consult an introductory textbook on educational psychology. For example, individual differences in the physiological state include pre-natal environmental damage (i.e. alcohol, drug, nicotine, other), nutrition, gender, genetic dysfunctions and/or damage, sensory integrity and sensory-motor deficiencies. Individual differences in the psychological state include existing schemas. In the case of mathematical thinking, the earliest schemas are likely to be those related to number sense.

Other types of individual differences in the psychological state include goal orientations, motivations, meta-cognitive skills, self-esteem, self-efficacy, cognitive skills and behaviours, speed of information processing, learning style, and working memory. In the social environment, examples include significant other persons such as peers, teachers and parents, and domain specific mentors.

This list, of course, is not exhaustive. When modelling these individual differences using cognitive architectures such as CLARION, it is important to recognise the dimensions of variability and onset. Furthermore, the means to “translate” the physiological and psychological dimensions into syntax that is recognisable by the architecture is required. For CLARION, much of this syntax is already available.

In devising a computational model for the AMG, the cognitive architecture must be able to incorporate the interactions between components within and across different conceptual levels. Furthermore, the attainment of excellence by an individual in a domain reflects both the individual’s demonstration of an appropriate suite of actions (action repertoires) and the production of something of value, as well as recognition by a significant group of other persons.

The CLARION4 cognitive architecture (Sun, 2006b) was devised in the late 1990s to model the cognitive processes of individual cognitive agents within a social context. According to Sun, CLARION is a modular system “intended to capture the essential cognitive processes within an individual agent” required for learning (p. 82). To date, CLARION has been used to simulate a variety of cognitive tasks, including the collective processes of academic publication (Naveh & Sun, 2006), food distribution and enforcement of law in a tribal society (Sun & Naveh, 2007), and organizational decision making (Sun & Naveh, 2004).


The following description of the overall structure of architecture is derived from Sun (Sun, 2001, 2003, 2006b, 2007; Sun & Naveh, 2007; Sun & Zhang, 2004, 2006; Sun, Zhang, & Mathews, 2006). The essential features of CLARION in relation to the present work include:

1. The relationship between cognition and the environment;2. The social orientations of both needs and motivations;3. Two separate dichotomies, including the interactions of declarative processes and

procedural learning processes, and the interactions of explicit (rule based) and implicit (trial and error) processes;

4. Its focus on both top-down approaches to skill development, where, for example, procedualization of explicit knowledge leads to skilled performance (implicit knowledge), and bottom-up approaches, where explicit knowledge can arise from implicit skills or knowledge;

5. The development of implicit knowledge is gradual and incremental, and as a result, the development of explicit knowledge may often be graduate and incremental as well;

6. The development of domain specific (abstract) knowledge is possible;7. Explicit knowledge of needs/desires/motivations can be acquired through bottom-up

learning;8. The inclusion of innate biases and behavioural propensities (individual differences); 9. The continual interactions of the subsystems;10. The opportunities of social interactions, for example, through the reflective possibilities

of meta-cognition; and11. Agents can cooperate with others often because agents can understand the

motivational structures of other agents.

Top level

Bottom

level

Inpu

t

Output

Action centred explicit

representation

Action centred implicit

representation

Non-action centred

explicit representation

Non-action centred

implicit representation

Goals

Drives

Reinforcement

Goal setting

Filtering goal setting

regulation

Meta-cognitive subsystem (MCS)Motivational subsystem (MS)

Action-centred subsystem (ACS) Non-action-centred subsystem (NACS)

Figure 2. The CLARION architecture. Adapted from Sun (2006, p. 80) and reproduced with permission.


The four subsystems of CLARION are shown in Figure 2 with each having two levels of knowledge representation, namely explicit (top-level) knowledge and implicit (bottom-level) knowledge, where:

1. Implicit representational processes are meant to represent the inaccessible nature of implicit knowledge, including implicit procedural knowledge; and

2. Explicit representational processes represent accessible knowledge.In CLARION, learning occurs through both top-down (implicit knowledge from explicit knowledge) and bottom-up processes (explicit knowledge from implicit knowledge).

Implicit knowledge is represented by sub-symbolic, distributed representations, and captured by backpropagation neural networks. A multi-layered neural network is used to implement a mapping function, where the parameters of the mapping function are adjusted to change the input/output mappings through trial-and-error interactions.

Explicit knowledge can be learned in many ways. One-shot learning, based on hypothesistesting, is preferred when interacting with the world5. Explicit knowledge can be derived from implicit knowledge (bottom-up), and it can be assimilated into implicit knowledge (top-down).

Action-centred subsystems (ACS) represent knowledge that is important in action decision making, where actions change the world in some way. The role of the ACS is to control actions (physical movements and internal mental operations). The ACS receives information from the NACS. When comparing the changed state with the original state, the agent learns.

In the ACS, the bottom level is termed the Implicit Decision Networks (IDNs). The top level is referred to as the Action Rule Store (ARS). The overall algorithm for deciding on an action is:

1. Observe current state x.2. In the bottom level (the IDNs), compute the Q-values6 of x associated with all possible

actions ai’s: Q(x, a1), Q(x, a2), ... Q(x, an). (The set of all possible actions along with the process of deciding on an action in the bottom level constitutes the IDNs.)

3. At the top level, find all possible actions (b1, b2, … bm ) (the ARS) based on the rules in place.

4. Consider values of ai and bj, and choose an action b.5. Perform b, and observe next state y, and (possibly) reinforcement r.6. Update bottom level according to Q-learning-backpropagation.7. Update top level using Rule-Extraction-Revision (RER).8. Go to Step 1.

(Sun, 2006b, pp. 84-85)One example is Q-learning, where an evaluation of the “quality” of an action a is calculated for a given state x as described by a sensory input. One way to choose an action is to use maximum Q-values, i.e. choose a if Q(x, a) = maxi Q(x, i). To ensure adequate exploration of potential actions, a stochastic decision process7 based on the Boltzmann distribution8 can beused.

Bottom level learning allows for both input and output of information. Input to the bottom level consists of three sets of information:

1. Sensory input. The sensory input has a number of dimensions, each with a number of possible values.

2. Working memory items. The working memory items have a number of dimensions, each with a number of possible values.

3. The selected item of the goal structure. The goals have a number of dimensions, each with a number of possible values.

Thus, the input state x is represented by a set of dimension-value pairs: (d1, v1), (d2, v2) … (dn, vn).

Output of the bottom level consists of action choices, working memory actions (for temporary storage of information or removal of information), goal actions, and external actions.


Note that the combination of Q-learning and backpropagation enables the development of implicit knowledge based solely on the agent exploring the world and does not require an external teacher or a priori domain-specific knowledge.

Top level learning allows for the development of explicit knowledge through rules and chunks. The condition of rules consist of three sets of information, namely

1. Sensory input;2. Working memory; and 3. Current goal.

The output of a rule is an action choice (working memory actions, goal actions, and externalactions). The condition of a rule and the conclusion of a rule constitute a distinct entity known as a chunk.

Bottom-up learning is captured using Rule-Extraction-Refinement (RER) algorithm. Here, the rules at the top level are learned by using information from the bottom level.

• If an action of the bottom level is successful, then an explicit rule is extracted and later refined.

• If the outcome is not successful, then the condition of the rule is made more specific.Top-down learning may happen when the agent relies on explicit rules at the top level for making decisions. As more knowledge is acquired by the bottom level through observing actions directed by explicit rules, the agent becomes less reliant on the top level and more reliant on the bottom level.

In addition, a set of response time equations specifies the response times of different components of the ACS, as well as overall response times. Response time equations are based on “base-level activations” (Anderson & Lebiere, 1998).

Non-action centred subsystem (NACS) represents knowledge that is static, declarative and generic: That is, it stores general knowledge about the world. The role of the NACS is to maintain implicit and explicit general knowledge. Information is stored as associative rulesand associative memory. The NACS is under the control of the ACS.

At the top level knowledge is stored as associative rules. At this level, general knowledge stores (GKS) encode explicit non-action-centred knowledge. At the bottom level, knowledge is stored as associative memory. Associative memory networks (AMNs) encoded non-action-centred knowledge by mapping input with output, using, for example, backpropagation learning algorithms.

As well as associative memory and associative rules, similarity based reasoning may be employed, whereby a known chunk can be compared with another chunk. If there is sufficient similarity then the latter chunk can be inferred.

In order to allow for complex patterns of reasoning to emerge, thereby capturing the essential features of human thinking, similarity and rule based reasoning can be mixed. Furthermore, a set of response time equations specifies the response times of different components of the NACS, as well as overall response times. Response time equations are based on “base-level activations”.

A key point in the AMG is the development of action repertoires and subjective assessments of their effectiveness by the individual. According to Ziegler (2005), an action within this repertoire includes schemas that reflect knowledge and rules, both implicit and explicit. Schemas related to mathematical thinking change and adapt in response to learning. Clearly, there is a match between the requirements of the AMG and the capabilities of CLARION.

Motivational subsystems (MS) provide the underlying motivations for perception/action/cognition, including the impetus (reasons) for actions and feedback (are the outcomes un/satisfactory?). The MS “supervises” (Sun, 2006b, p. 90) processes in the ACS and NACS (Figure 3). The MS allows for:


• Proportional activation, where activation of drives should be proportional to corresponding offsets, or deficits, in related aspects such as food or water.

• Opportunism, where an agent’s preferences are taken into account (for example, the agent may prefer to satisfy x instead of y).

• Contiguity of actions, where current actions sequences are preferred over new actions.• Persistence, where actions persist beyond a minimal level of satisfaction.• Interruption when necessary, where current actions are interrupted when a new and more

urgent need arises.• Combination of preferences, where different drives can be combined to form a higher

overall preference.

goal stimulus

High-level drives

Secondary drives

Low-level drives

Sensory

input

Goal

actions

from ACS

and MCS

goal to

ACS and

MCS

Drive

strengths

to MCS

Figure 3. The structure of the motivational subsystems (MS). Adapted from Sun (2006, p. 91) and reproduced with permission.

A dual system of motivational representations is used, where the explicit goals (“find water”) of an agent are based on internal drives (“being thirsty”). CLARION refers to primary drivesas those that are essentially “hard-wired” to begin with. They include, for example:

• Get-water Strength is proportional to .95*(water-deficit, water-deficit*water-stimulus). The actual strength depends on the water-deficit felt by the agent and water-stimulusperceived by it.

• Get-food Strength is proportional to .95*(food-deficit, food-deficit*food-stimulus). Again, the actual strength depends on the food-deficit felt by the agent and food-stimulusperceived by it.

• Avoid-danger Strength is proportional to .98*(danger-stimulus*danger-certainty). It is proportional to danger signal (its distance, severity) and certainty. Danger signal is captured by danger-stimulus, and danger certainty captured by danger-certainty.

(Sun, 2006b, p. 92)These drives implemented as hard-wired instincts using backpropagation neural networks. In addition, higher level drives include belongingness, esteem, self-actualization, and more can also be hard-wired. Secondary drives can also be implemented. They are derived mostly through the satisfaction of primary drives as well as being gradually acquired or externally set (Sun, 2006b, p. 92).


Meta-cognitive subsystems (MCS) refer to an individual’s knowledge about their cognitive processes and outcomes. It also refers to the active monitoring and regulation of these processes, usually toward the fulfilment of a goal. The MCS in CLARION operationalizes these conceptions (Figure 4). The MCS is linked with the MS to monitor, control and regulatecognitive processes for the purpose of enhancing their performance. Control and regulation can be achieved through:

• The setting of goals in the ACS;• Interrupting and changing ongoing processes in the ACS;• Setting essential parameters in the ACS and NACS; as well as others.

evaluation

goal setting

level selection

learning selection

reasoning selection

input selection

output selection

parameter setting

goal change

Monitoring

buffer

goal

state

drives

reinforcement

goal action

filtering,

selection

and

regulation

Figure 4. The structure of the meta-cognitive subsystem (MCS). Adapted from Sun (2006, p. 94) and reproduced with permission.

There are many types of metacognitive processes in the MCS, including:

• Behavioural aiming (setting of reinforcement functions, setting of goals),• Information filtering (focussing on input dimensions in the ACS and NACS),• Information acquisition (selection of learning methods in the ACS and NACS),• Information utilisation (selection of reasoning methods in the ACS and NACS),• Outcome selection (selection of output dimensions in the ACS and NACS),• Cognitive mode selection (selection of explicit processing, implicit processing, or a

combination thereof in the ACS), and• Setting parameters of the ACS and NACS (including the IDNs, ARS, AMNs and GKS).

(Sun, 2006b, p. 93)


The MCS consists of a number of modules in two levels: The bottom level consists of goal setting network, reinforcement function network, input selection network, output selection network, parameter setting network and so on, whereas the top level consists of explicit rules (if they exist) and is similarly divided.

The AMG describes the importance of motivational and metacognitive states, and goal orientations in the development of action repertoires, including those in the domain of mathematical thinking. This motivational and metacognitive states change in response to the needs of the individual and the pressures of the environment. Success, for example, is highly motivating to the individual. For some social groups, however, motivation can be externally imposed and just as successful. Goal orientations are also “programmed” within CLARION. Once again, there is a clear match between the requirements of the AMG and the capabilities of CLARION.

CLARION parameters describe the constraints of the learning processes as being of two sets. The fundamental properties of Set 1 include:

• Learning rate of neural network;• Reliance on top vs. bottom level learning (expressed as probability of choosing each

level);• Temperature (degree of randomness); as well as others.

In Set 2, the parameters concerning rule-extraction learning include:

• RER positivity threshold (which must be exceeded to count a step as positive),• RER generalisation threshold (which must be exceeded for a rule to be generalised),• RER specialisation threshold, as well as other performance metrics.

The AMG describes aspects of the psychological state as being important in the development of Actiotopes. Manipulating these parameters offers the researcher opportunities to change learning parameters related to speed and ways of learning, matching individual differences in the aspects of the psychological self in the AMG with CLARION.

Antecedents of action repertoires

Motivational states, metacognitive processes and starting parameters are relatively simple to set in CLARION. A more important and perhaps more challenging consideration in using CLARION to model the development of mathematical AE is to show that CLARION can learn complex mathematical rules from the small number of innate mathematical abilities,collectively known as number sense. Several types of number sense are evident in human species, including numerosity, subitizing, additivity and subtractivity and magnitude representation. Hence, the first step is to represent each type of schema at the top-level (explicit level) in both the ACS and NACS as fixed rules (FRs). In CLARION, FRs take the following syntax:

current-state-condition à action

Chunk ID: (d1, v1)(d2,v2)… à action

(Sun, 2003, 2006b)

To take numerosity as an example, the current state condition for a one-to-one correspondence of objects would involve identifying them as “same” with a two-to-one correspondence being identified as “different”. For magnitude representation, a mental number line is thought to underpin many of these arithmetic skills, where magnitude is represented on a number line. Two types of number line are postulated – a number line representing approximate solutions to arithmetic problems, and a number line representing exact solutions to arithmetic problems. Approximate representations of numbers are thought to be represented as positions on a line with a logarithmic scale. The fixed rule for position of the number on the line is represented as a normal distribution with mean logx and a fixed


variance (SD2). Hence the numbers “1” and “2” are represented as:

• approx-1: (total, 1)(1, ND), with fixed SD2; and• approx-2: (total, 2)(2, ND), with fixed SD2 respectively,

where “total” refers to total number of items and ND is a normal distribution.

On the other hand, exact representations of numbers are represented as positions on a line with a logarithmic scale. Hence the fixed rules for “1” and “2” are represented as

• exact-1: (total, 1)(1, log1).• exact-2: (total, 2)(2, log2), respectively.

where “total” refers to total number of items.

Representations of the various types of number sense may present the most challenging aspects of modeling the development of higher forms of mathematical thinking using CLARION. However, its success may provide important clues as to the possible pathways and important milestones in Phase 1 of this development. More importantly, the model may provide clues regarding the influence of other psychological states such as speed of information processing and working memory, and other environmental factors such as nutritional state and schooling on this development.

The way forward

The value of the AMG as a model of giftedness is that it provides a unifying basis for explaining how aspects of the psychological self, the physical self and the social environment work together in the attainment of excellence. In the AMG, giftedness is a process by which an individual acquires a sufficient number of actions that enables them to attain excellence within a particular domain. There are three phases in this process, defined in terms of stage in the development of these action repertoires. In Phase 1 an individual may have developed an action repertoire that is relatively more sophisticated in comparison with their peers. These individuals are termed talented because of the nature of this repertoire. In Phase 2, an individual may have attained all the knowledge and behaviors that are necessary for excellence. The actions of these individuals are gifted. Once these individuals have passed through Phase 2 and demonstrated outstanding performance in their domain, their actions are termed excellent.

The process of achieving mathematical excellence begins with the development of talentedmathematical action repertoires from their antecedents in number sense. Modeling this process within Phase 1 using CLARION should be the first step in testing the AMG. Once the challenges of creating FRs that reflect the various types of number sense are overcome, CLARION could be used to find answers to the following questions, including:

• Which of the innate basic arithmetic abilities within number sense are important in the development of higher levels of mathematical thinking?

• How do higher levels of mathematical thinking develop from the interactions of the various types of number sense?

• What are the significant learning milestones in the development of higher levels of mathematical thinking?

• What are the significant aspects of the physiological and psychological self that promote or inhibit the development of higher levels of mathematical thinking?

• What are the significant variables within the social environment that promote or inhibit the development of higher levels of mathematical thinking?

• Which combination of self and the social environment lead to talent?The answers to these questions may be tested in the field using more traditional approaches to psychological and social science research.

If research into giftedness is to be reinvigorated, new avenues of research need to be proposed and implemented. If we accept that giftedness is as much a social phenomenon as it is a reflection of the physiological and psychological state of an individual, then tools such as


CLARION can be employed to simulate the interactions between individuals and their social environment. Simulations can provide the testable hypotheses that are required for the advancement of research, not only because they generate data, but because the simulations themselves depend on a clearly defined meta-theoretical framework.

Acknowledgments

The authors acknowledge a number of helpful discussions with Prof. Dr. Heidrun Stöger at the University of Regensburg in the development of these ideas. The comments made by the anonymous reviewers of this article are gratefully appreciated

Notes1 Number sense refers to a number of innate arithmetic abilities (or schemas) and include

numerosity (Wood & Spelke, 2005) (capacity to make representations of numbers of objects), use of number words (Lipton & Spelke, 2006), additive and subtractive expectations (Bisanz, Sherman, Rasmussen, & Ho, 2005) ordinal numerical knowledge (Brannon, 2005), magnitude (Dehaene, 2003), and arithmetic expectations (Kobayashi, Hiraki, Mugitani, & Hasegawa, 2004).

2 Recently, Halberda, Mazzocco and Feigenson (2008) established a relationship between numerosity and mathematical achievement amongst junior high school students.

3 The first (or deductive) approach to conducting social science is to construct mathematical models of social phenomenon in order to make deductions about changes in variables within the models. The second (or inductive) approach is to make generalizations from a large number of observations. According to Sun (2006), the insights from such observations are mostly qualitative in nature (p. 6).

4 CLARION stands for Connectionist Learning with Adaptive Rule Induction.5 In this model, an agent explores the world, acquires, representations of the world, and

modifies the representations as needed.6 Q-learning algorithm used to calculate Q-values. In the algorithm, Q(x, a) estimates the

maximum (discounted) cumulative reinforcement the agent receives from the current state xon. The updated Q(x, a) is based on:

(x, a) = (r + e(y) – Q(x, a)),

where is a discount factor, e(y) is maxaQ(y, a) and y is the new state resulting from action a.7 Stochastic decision process describes a framework for modeling optimization problems

involving uncertainty,

( )∑

=

i

axQ

axQ

e

exap

),(1

),(1

α

α

,

where controls the degree of randomness (or temperature) of the decision-making process.

8 The Boltzmann distribution predicts the distribution function for the fractional number of particles occupying a set of states, each state possessing energy.


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The authors

Dr Shane N. Phillipson is an Associate Professor in the Department of Educational Psychology Counselling and Learning Needs, in the Hong Kong Institute of Education. His research interests include cultural conceptions of giftedness and models of achievement.

He has been awarded a number of research grants, resulting in research publications in many international peer reviewed journals, including High Ability Studies and Educational Psychology. His books include Phillipson, S. N. (2007). Learning diversity in the Chinese classroom: Contexts and practice for students with special needs. Hong Kong: The Hong Kong University Press, and Phillipson, S. N., & McCann, M. (2007). Conceptions of giftedness: Socio-cultural perspectives. Marwah, NJ: Lawrence Erlbaum Associates.

Ron Sun is Professor of Cognitive Science at Rensselaer Polytechnic Institute, and formerly the James C. Dowell Professor of Engineering and Professor of Computer Science at University of Missouri-Columbia.

His research interest centers around the study of cognition, especially in the areas of cognitive architectures, human reasoning and learning, cognitive social simulation, and hybrid connectionist-symbolic models.

He is the founding co-editor-in-chief of the journal Cognitive Systems Research, and also serves on the editorial boards of many other journals. He is the general chair and the program chair of CogSci 2006, and the program chair of IJCNN 2007. He is a member of the Governing Boards of Cognitive Science Society and International Neural Networks Society.

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