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Home , Concept learning, Jean Piaget, Jerome Bruner

Iman C. Chahine, Georgia State University, Atlanta, GA

Delineating the Epistemological Trajectory of Learning Theories: Implications for Mathematics Teaching and Learning

The purpose of this paper is to delineate the trajectory of fundamental learning theories and the way these theories have impacted the teaching and learning of mathematics over more than half a century. We argue that a critical examination of the depiction of learning theories and their inherent implications for the teaching of mathematics afford an understanding of the hierarchical of the field of mathematics . Needless to say, examining various learning theories in the context unpacks epistemological and ontological core issues underlying the teaching and learning of vital topics that are assumed to account for the changes in global economics and business in a STEM world.

Introduction fundamental problems facing education The quest to explore how people locally and globally. In a highly dynamic learn mathematics has been a perennial and versatile world, unfolding the best concern for decades. For some time now, practices particularly for teaching and learning theorists have mathematics and instigates a been incessantly searching for optimal pressing need for consideration by educators conditions under which learning occurs. A and policy makers. For mathematics tremendous bulk of contesting theories has education, the push toward standards-based evolved, thriving to explain the roles of the curricula governed by accountability and learner and the teacher when engaged in acts teaching effectiveness dogmas juxtaposed of learning and teaching. Notwithstanding with policies to enact cutting edge the fact that many learning theories differ in educational interventions in an ever- their empirical manifestations, nonetheless changing economy command inevitable the underlying are challenges that cannot be overlooked. The unequivocally analogous. highly political dynamics that depict the way At the turn of the 21st century, the standards are drafted and sanctioned in necessity to prepare children for a rapidly terms of content and are changing world has been consistently determining the way educational policies are reiterated in the literature (Csikszentmihalyi institutionalized and eventually endorsed at & Schneider, 2000; Cornell & Hartman, the grass roots level. 2007; Drucker, 2010). With the tumultuous Nevertheless, recent studies on political and economic climate prevailing student , attitudes and self- worldwide exacerbated by challenges efficacy have shown that unless students are emerging in the rapidly changing context of cognitively and emotionally invested in new technologies and globalization, there is what they are learning, little will an unprecedented need to capitalize on how be acquired (Friedman, 2006; Furner & best to educate future generations. A critical Gonzalez-DeHass, 2011). Naturally, examination of the trajectories that various designing learning environments that can theories of learning have taken since the last trigger students’ natural and century provides a clear explanation on the stimulate their interest will significantly 2013 Mathitudes 1

impact their learning. Thompson and thus provide sufficient support to students in Thornton (2002) found that when students learning the content. are intrinsically motivated they are eager to This leads us to the following learn. Yair (2000) argues that, “students’ questions: How did our understanding of interest in what they learn, and their of how students learn develop in light of enjoyment while learning, are highly widened exposure to subsequent research correlated with the outcomes of learning” (p. efforts in educational and cognitive 193). Furner and Berman (2003) found that ? And what are the major teachers need to do more in the way they learning theories that have impacted teach math to address attitudes toward math frequent paradigm shifts in the field of and mathematics anxiety in the classroom. mathematics education? However, research has suggested that The purpose of this paper is to motivation of adolescents decline as they delineate the trajectory of fundamental progress through junior and senior high learning theories and the way these theories school (Eccles, Midgley, et al., 1993; have impacted the teaching and learning of Gonzalez, 2002; Hidi & Harackiewicz, mathematics over more than half a century. 2000; Williams & Stockdale, 2004). We argue that a critical examination of the Additionally, the value children place on depiction of learning theories and their many academic activities, particularly inherent implications for the teaching of mathematics and their beliefs about the mathematics affords an understanding of the usefulness of school decline as they get hierarchical evolution of the field of older (Wigfield, Eccles, & Rodriquez, mathematics education. Needless to say, 1998). Today, students are more often examining various learning theories in the described as “physically present but mathematics education context unpacks psychologically absent” and thus are less epistemological and ontological core issues likely to actively and enthusiastically engage underlying the teaching and learning of vital in learning. Recently, student lack of topics that are assumed to account for the motivation in the mathematics classroom changes in global economies. has been a critical national concern in light Drawing on extensive literature of efforts to improve STEM (, related to learning theories that have Technology, Engineering, and Mathematics) emerged in the past two centuries, and education in the . One concern examining critically possible formulations in particular is that the pipeline of students on how mathematical knowledge and entering STEM does not meet the current problem solving activity can be constructed, demand for future scientists and engineers. we highlight important implications that, we One of the reasons attributed to the attrition argue, provide the most compelling rate of students embarking on a degree in explanation of how students learn STEM is students’ underperformance in mathematics. By virtually appealing to high school mathematics and their research on the cognitive as well as the inadequate preparation in rigorous social construction of mathematical mathematics content. Hence, addressing this knowledge, we explore various trends in national need requires research and teaching and learning practices proposed by development of the best pathways to each perspective and investigate their remediate the teaching of mathematics and practical significance and implications in the mathematics classroom.

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We begin with a brief overview of completely characterized in terms of stimuli major “grand” theories of learning, namely and responses . Almost all those of , , behaviorists describe "knowledge" as simply Constructivism, and . a succession of -response chain More concisely, we examine contributions acquired through conditioning. Behaviorists of cognitive and socio-cognitive theorists believe that learning is observable and is who advanced learning epistemologies that directly evidenced by a change in behavior. transformed mainstream perspectives on This theory has been criticized as being a learning mechanisms. We expand the theory of animal and human learning that argument further to discuss the implications only focuses on objectively observable of these theories with regards to addressing behaviors that discounts mental activities students’ attitudes and motivation in the (Tuckman, 1992). Attacked and refuted by mathematics classroom. Finally, we provide radical behaviorists, in particular by Skinner an account of possible contributions of (1938), behaviorism was abandoned in favor complexity science to the teaching and of operant conditioning where events in the learning of mathematics. environment determine and shape desired behavior (Post, 1988).Though this theory Theories of Learning was prevalent from the 1950’s through A handful of research studies 1970’s, it is still alive in the and classified learning theories according to practices of many educators in the 21st where these theories stand relative to four century. main categories: the of knowledge; In search for a more “humanistic” existence of mental representations; causal theory of learning, Neobehaviorism emerged relationship between mental relationships calling for some mental mechanism that and behavior; and the origin of knowledge mediates between situations that elicit (Illeris, 2004; Pritchard, 2005). Based on the behaviors i.e. stimuli and specific behaviors above-mentioned categories, Byrnes (2007) i.e. responses (Tuckman, 1992). In his book established three major groups that the conditions of learning, Gagné, a leading encompassed Meta-Theoretical Belief Neobehaviorist, explains “ the occurrence of Systems (MTBS): 1) Behaviorism, learning is inferred from a difference in Neobehaviorism, and Cognitivism; 2) human being’s performance as exhibited and Functionalism; and 3) before and after being placed in a ‘learning Nativism, Empiricism, and Constructivism. situation’” (1965, p.20). Each of these groups comprises a spectrum When identifying the conditions of , perspectives and belief necessary for learning to occur, Gagné systems that, we believe, can potentially (1965) cited five “categories of capabilities” explain the systemic evolution and which he defined as “conditions internal to acquisition of knowledge. the learner” (p.21). These capabilities include: intellectual, cognitive, verbal, Behaviorism, Neobehaviorism, and motor and attitudes. While for Thorndike Cognitivism learning is one and only, Gagné (1965) Behaviorists, including Edward L. spoke of eight varieties of learning, or Thorndyke (1898), argue that learning is the external conditions each necessitating acquisition of new behavior that can be different capability and internal conditions, manipulated by the environment and may be required from the learner. These types are

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seemingly cumulative and hierarchical such part in the learning process, uses internal that each type is built on its prerequisite, the mental structures to organize, transform and highest in the hierarchy being problem- retrieve information when acting on a solving or type 8. The hierarchy includes: learning environment (Chahine, under Problem-solving; Principle learning; review). As such, learning is intrinsic, learning; Multiple discrimination holistic and personal to the individual and learning; Verbal associations; Stimulus- can be enhanced through interactions with response learning, and Signal learning. others and with proper physical materials in Gagné also argued that for any learning to the environment. A closer look at the works take place at any level special attention of these theorists merits our discussion of should be given to its prerequisites. learning theories. Advocating the claims that the whole (goal or outcome) is the sum of its parts and that Piaget's Developmental Theory learning is rather specific and goal-directed, A prominent Cognitivist in the 20th Gagné (1965) established task analysis as a century is the Swiss biologist and technique that pinpoints conditions or (1896-1980). behavioral objectives under which learning Piaget is renowned for constructing a highly of a certain goal occurs. With a highlighted influential model of intellectual emphasis on “outcome content” (what to development and learning. Piaget's learn) rather than the process (how to learn), developmental theory is based on the idea instruction was perceived to proceed from that the child builds cognitive mental prerequisite skills to desired goals. Other structures in an attempt to make sense of his important personal international factors such or her environment (Piaget, 1952). These as motivation and establishment of attitudes stages, at least for Swiss school children, and beliefs are ignored in favor of achieving increase in sophistication with development behavioral objectives. Many drawbacks of and include: Sensorimotor stage (birth - 2 Gagné’s guided approach to learning have years old) where the child, through physical been frequently cited in the literature, such interaction with his or her environment, as limited transfer and neglecting other builds a set of about reality and important forms of learning such as informal how it works. Preoperational stage (ages 2- and (Post, 1988; 7) in which the child is not yet capable of Tuckman, 1992). abstract conceptualization and therefore On the other hand, theorists who needs exposure to concrete physical have given more weight to cognitive situations. The primary deficiency at this explanations and mental structures are stage is what Piaget (1952) calls the retained near the cognitive end of the reversibility principle where the child continuum and are referred to as Cognitivists cannot grasp the idea of conservation of (Byrnes, 1992). Cognitivists such as Piaget, quantity. Concrete operations (ages 7-11) Bruner, and Dienes see knowledge as where physical experience accumulates, the actively constructed by the learner in child starts to conceptualize, creating logical response to his/her interaction with the structures that explain his or her physical environment. In contrast to Neobehaviorists, experiences. He further claims that abstract Cognitivists emphasize the “how” of problem-solving is also possible at this learning rather than the “what”. In stage. Although a child’s actions are Cognitivists’ views, the learner, an active internalized and reversible, Piaget (1952)

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explains that the child cannot deal with Thus, Piaget does not view in possibilities that are outside the realm of terms of content or amount of knowledge, his/her direct experience, and finally formal but rather as an arrangement or structure and operations (beginning at ages 11-15) where a way in which information is organized. the child is capable of constructing formal Other developmental factors that contribute operational definitions and thus, abstract to include concepts maturation, active experience social One of the significant implications of interaction, and equilibration or self- Piaget’s developmental theory is related to regulation. teaching basic concepts. A direct A basic tenet of learning by assumption is that a child should be helped discovery or exploration requires a vigorous to progressively proceed from the concrete exposure to and involvement with the to the more abstract modes of . Much environment. A direct implication of of the mathematics taught following Piaget’s theory is the focus on development traditional curricula contradicts this of schemata that potentially facilitates assumption, through teaching by telling problem-solving. The developmental stages formal explanations of concepts are delineated by Piaget are highly relevant to presented to the child in a different mode of the teaching and learning of mathematical thought than his own. concepts. For example, at the primary level, In adapting to his/her environment students can be taught conservation thus avoiding and minimizing mathematics problems in late elementary and early anxiety, Piaget (1977) contends that middle level through tasks involving individuals use two mechanisms: seriation and classification. Moreover, Assimilation and accommodation. structuring the physical environment by Assimilation, which involves incorporating making multiple learning centers where new ideas into existing schemata in students can be actively and purposefully Piagetian theory, is somehow similar to the involved in the learning process is necessary Behaviorists’ concept of stimulus where to enhance students’ attitudes toward math having learned to respond to one stimulus and motivation. In addition, utilizing hands- makes it easy to respond to another similar on activities with varieties of manipulatives stimulus. Accommodation, on the other and multiple physical embodiments helps hand, involves modifying existing schemata children learn operations appropriate to their to fit the newly assimilated information. level of development. Piaget’s theory calls Striking a balance or equilibrium between for more emphasis on integrative themes, both assimilation and accommodation is the like probability and statistics in early grades basis of intellectual development. as well as of basic algebra Additionally, Piaget (1952) views concepts in primary grades, the case we see intelligence as “ to new now in Common Core State Standards circumstances” ( p.151) and explains that in (National Governors Association Center for any intelligent act “ the need which serves as Best Practices (NGA Center) and the motive power not only consists in repeating , Council of Chief State School Officers but in adapting , that is to say, in (CCSSO) (2010). Engaging in hand-on assimilating a new situation to old schemata explorations focused on these topics and in accommodating these schemata to reinforces students’ beliefs in the relevance new circumstances” (Piaget, 1952, p. 182). of mathematics to real life situations thus

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fostering positive math attitudes and Variability Principle, Dienes (1960) motivation for further learning. maintains that conceptual learning is enhanced by exposing the child to multiple, varied physical representations on the same Zoltan Dienes’ Theory concept. This allows the child to abstract One of the significant pioneers who similar elements underlying the different established a theory specifically directed embodiments. However, in the towards understanding the learning of Mathematical Variability Principle, Dienes mathematics is Zoltan Dienes. Dienes’ suggests that a concept dissected into theory of learning encompasses four major constituent sub-concepts can be generalized principles: The Dynamic Principle; the when “all possible variables [are] made to Perceptual Variability Principle; the vary while keeping the concept intact” Mathematical Variability Principle; and the (p.42). Finally, in the Constructivity Constructivity Principle (Dienes, 1960). In Principle, Dienes explains the mode of delineating the “skeleton” of his theory, thought involved in concept construction. Dienes (1960) acknowledges the works of When discussing the structure of a task Piaget, Bruner and Sir Frederick Bartlett appropriate for the child’s thinking, Dienes whose ideas resonate within each (1960) identifies two levels of “logical component of this theory. The Dynamic complexity”: constructive and analytical. He Principle outlines three basic stages for further argues that children, when young, are concept formation, each requiring a different involved in constructive types of thinking kind of learning: First, free stage which where they are actively engaged in episodes requires free unstructured, but rather of free play. Having had the proper purposive activities that allows open and opportunities to think constructively, Dienes informal experimentation with the task at asserts that the child at, around 12 years of hand; second, concept realization stage age, is then capable of more analytical mode where the child is exposed to varying of thinking and thus well prepared to decode experiences which are “structurally similar and analyze tasks using his initial (isomorphic) to the concepts to be learned” constructions. (Post, 1988, p.7). Finally, the third stage Other important implications of represents the development of the math Dienes’ work to increase motivation and concept and sufficiently applying it to varied invoke positive attitudes in the mathematics contexts. Dienes (1960) calls these stages as classroom involve re-structuring the “stages of growth necessary before a environment to include a variety of mathematical predicate or concept becomes manipulative and learning tools, fully operational” (p.42). He also argues encouraging group work, and reinforcing the that learning a mathematics concept role of the teacher as a coach and facilitator. necessitates a clear understanding of a set of variables underlying this concept as well as Structuralism and Functionalism other factors that are extrinsic to it and Founded by the British psychologist which are seemingly embedded in the Edward Tichener the theory of Structuralism experiences provided. This calls upon his capitalizes on the role of consciousness and second and third principles, the Perceptual in describing numerous Variability Principle and the Mathematical cognitive processes and mental structures Variability Principle. In the Perceptual including sensations, images and affections.

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Structuralists contend that the sum total of representation played in changing mental structures and their interactions performance either positively or negatively, comprise the conscious experience. A major depending on the circumstances (Kohl & implication of this theory is the role that Finkelstein, 2005). immediate experiences and reflection play in While proponents of structuralism stimulating and bolster capitalize on the nature and organization of students’ positive attitudes toward math thus concepts, functionalists focus on how the invoking learning (Carlson, 2010). operates in the course of problem- Structuralism as a movement lost its solving and (Byrnes, popularity in the 1960s with the emergence 2007). Millroy (1992) defined the Piagetian of post-structuralism, a French movement approach as “the structural developmental” that critiqued the basic tenets of approach for it specifically concentrates on structuralism and called for social individual cognitive development, with little constructionism as a means to expose emphasis on social influences. “subjugated ” (Foucault, 2003). In principle, Structuralism and Problem Solving Theory Functionalism symbolize two end-poles of a Most of the approaches to problem continuum vis à vis the degree of emphasis solving established during the past thirty each paradigm merits either conceptual or years practically fall under one of the two procedural knowledge. Rittle-Johnson, contradictory treatments: The information Siegler, and Alibali (2001) defined processing model of human thinking procedural knowledge as “…the ability to (Newell & Simon, 1972) and the social execute action sequences to solve problems” practice theory (Turner, 1982). The (p. 346). They also argued that procedural information processing model aims at knowledge involved mainly the use of describing the general processes of problem- previously learned step-by-step techniques solving, minimizing the role of individuals and algorithms to solve specific types of in the problem-solving process (Putnam et problems. Furthermore, the authors al., 1989). Schoenfeld (1983) extended by explained that conceptual knowledge entails far the scope of traditional conception of “… implicit or explicit understanding of the mathematical problem- solving highlighting principles that govern a domain and of the four fundamental dimensions of good interrelations between units of knowledge in practice: resources of mathematical a domain” (p.346). knowledge, heuristic strategies, control over A handful of research studies support the process of working on problems, and a the hypothesis that forming correct problem deep understanding of the nature of representations is one mechanism linking mathematical argumentation. improved conceptual knowledge to Current interest in problem-solving improved procedural knowledge (Jitendra, as a “practice” reflects a trend in which 2002; Hoffman & Spatariu, 2008). Rittle- learners are characterized as more active and Johnson, Siegler, and Alibali (2001) defined where problem-solving is viewed as a series problem representation as “the internal of activities (Lave et al., 1990). As a matter depiction or re-creation of a problem in of fact, it has been widely argued that working during problem-solving” emphasizing the value of teaching problem (p. 348). Furthermore, a number of studies solving in schools as a mathematical investigated the role that problem practice in contrast to rote procedural

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approaches enables the learner to gain more forms of work organization) contrary to in-depth understanding of mathematical cognitive psychologist focus on mental principles underlying such practice (Nunes representations. Jonassen and Murphy et al., 1993). The demand to integrate the (1999) argue that Activity theory presents a learner with the surrounding environment, new perspective for analyzing learning practice and culture is an approach that is processes and outcomes for designing embraced by the social practice theorists. In instruction. He adds: “rather than focusing principle, the social practice theory seeks to on knowledge states, it focuses on the interpret the of social activity in a activities in which people are engaged, the number of environments and to discern their nature of the tools they use in those causes. activities, the social and contextual A close akin of the social practice relationships among the collaborators in theory is the Activity theory. Activity theory those activities, the goals and intentions of has been widely employed in socio-cultural those activities, and the objects or outcomes and socio-historical research that aimed to of those activities” (p. 68). scrutinize human activity systems (Jonassen & Murphy, 1999). It has been extensively Nativism, Empiricism, and used as a framework for developing Constructivism Constructivist Learning Environments Theories of cognitive development (CLE) and cognitive tools for learning. The are situated under three fundamental views assumptions of activity theory resonate with related to the origin of knowledge: The those of constructivism, situated learning, empiricist view, the nativist view, and the distributed , case-based reasoning, constructivist view (Saxe, 1991). Byrnes social , and everyday cognition. (2007) argues that all nativists share a belief Activity theory has its roots in the that knowledge is innate. While the Soviet cultural-historical psychology of empiricist view favors the position that the Vygotsky, Leont'ev (1978) and it represents environment is the source of knowledge, the an alternative perspective to the claim that nativists advocate the need of knowledge learning must precede activity. The most structures to organize and categorize fundamental assumption of activity theory is experience (Saxe, 1991). The polar opposite that the “human mind emerges and exists as of nativism is Empiricism. In this view, it is a special component of interactions with the believed that individuals possess no a priori environment, so activity (sensory, mental, knowledge but rather that most knowledge is and physical) is a precursor to learning” perceived to be acquired through exposure (p.64). to the world. Although the learner has a central Constructivists, on the other hand, role in defining activity, very little, if any, adopt the premise that knowledge is not meaningful activity is accomplished inherent in the human mind nor in the individually. Activity theory contends that environment, but is rather actively learning and doing are inseparable, and that constructed by the individual as a result of they are initiated by intention (Jurdak & his/her interaction with the social and Shahin, 2001). physical environment (Millroy, 1999). As a Activity always involves an matter of fact, constructivism derives most ensemble of artifacts (instruments, signs, of its ideas from Piaget’s theory of cognitive procedures, machines, methods, laws, and development.

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Partly inspired by Piagetian ideas, interaction through which transmit to constructivism goes further to emphasize the the child the rich body of knowledge that role of others in the construction process. exists in the culture.This social context Through negotiation and communication shapes the range of potential each student with others, constructivists claim that people has for learning. As learning progresses, the receive continuous feedback as well as child's own language comes to serve as a agreement concerning their personal primary tool of intellectual adaptation constructions (Brooks & Brooks, 1993). In (Vygotsky, 1962). Therefore, Vygotsky this respect, constructivism agrees with (1962) viewed the process of learning as Vygotsky’s theory which asserts that mainly an internalization of a body of cognitive functioning occurs first on the knowledge and tools of thought that first social level, between people, and that the exist outside the child. child afterwards internalizes this in his As a result of his intense emphasis development (Vygotsky, 1962). on the social dimension of learning, Vygotsky’s view inherently diverges from and Socio-Cultural Theory that of Piaget. Kincheloe (2004) argues that Vygotsky was famous for Vygotsky was critical of the way Piaget introducing the term “the zone of proximal investigated children’s cognitive abilities development” which he defines as “the while working alone. He contends that a true distance between the actual developmental measure of individual ability is only level as determined by independent revealed through collective social problem-solving and the level of potential interactions. Kincheloe (2004) also explains development as determined through that, with his notion of learning as problem-solving under guidance, or in dependent on ZPD, Vygotsky discards collaboration with more capable peers” Piaget’s concept of development as a (Vygotsky, 1962, p.103). Through systematic shift from one discrete stage into scaffolding, an adult can facilitate and adjust another and highlights the role that artifacts the environment to enhance students’ i.e., sign systems, play in developing positive attitudes toward math and maximize cognition. learning. In a sense ZPD represents for Researchers influenced by Vygotsky Vygotsky the social context in which have basically emphasized the role of learning takes place. In this respect, cultural practices in analyzing the relation individual learning is seen as inherently between culture and cognition (Millroy, guided by the social world through the 1992). Their investigations have generally introduction of society’s tools and by focused on studying people’s use of math engaging with more experienced members outside the classroom (Gay & Cole, 1967; of society. As a socio-cultural constructivist, Jurdak & Shahin, 2001; Scribner, 1986; Vygotsky adopts the social cognition Lave, 1988; Lave, Smith, & Butler, 1990; learning mode, which asserts that individual Saxe, 1991, Millroy, 1992). The practice of cognition is socially and culturally mediated math has been explored in the contexts of (Vygotsky, 1962). Like most of the social everyday activities. Two main groups of cognition learning theorists, Vygotsky researchers have explored the use of math in (1962) believed that culture teaches children settings outside school: those interested in both what to think and how to think. To “everyday cognition” or “cognition in Vygotsky language is a primary form of practice”, where Lave is a prominent figure,

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and those interested in “ethnomathematics”, four major ideas: structure of the discipline, where D’Ambrosio (1985) is a key figure. readiness to learn which depends on rich Both groups of researchers call for a new learning experiences and enthusiastic conceptualization of mathematics that is teachers, intuition, and motivation. He also rooted in nonacademic practices. The work calls for curriculum to be organized in a of these groups focuses on three main spiral manner so that the student continually issues: Analysis of school practice, builds upon what he/she already learned. investigation of the transfer of school Perhaps one of the most important knowledge to out-of-school situations, and implications of Bruner’s work is the using the of practice to significant role that different modes of challenge conventional cognitive theory. representation play in learning mathematics. Cramer (2003) argues that experiencing the ’s Representation Theory benefits of using multiple representations as With the hypothesis that “any subject well as allowing possible translations among can be taught effectively in some these different modes of representations intellectually honest form to any child at any helps teachers become more aware of the stage of development” (Bruner, 1960, p.47), weaknesses inherent in any curriculum and Bruner envisioned learning as successively thus respond by incrementing it with outside proceeding through three hierarchical stages. resources. He held the view that experience is coded In addition to the significant role that and processed in such a way to ensure its external representations play in students’ retrieval when needed. Such a coding active construction of mathematical system Bruner calls representation. He concepts, many studies have investigated the converted his ideas about modes of impact of students’ beliefs, feelings and representation into chronologically affective representation in enhancing or structured stages of development and hindering mathematical understanding in the claimed that understanding in any domain classroom. Goldin (2003) calls for “a good must involve three modes of representation: balance between the standard manipulation enactive through habits of acting, iconic in of formal notational systems… and the the form of pictorial images and symbolic development of other representational through written symbols like language. “By modes: imagistic thinking, involving enactive, I mean a mode of representing past visualization; visual imagery, pattern events through appropriate motor response” recognition, and analogical reasoning; (Bruner, 1964, p. 69). In this regard, he heuristic planning, involving diverse agrees with Vygotsky’s notion of ZPD in problem solving strategies; and affective helping children understand and master a representation” (p.283). By the same token, concept by proceeding from physical Monk (2003) highlights the importance of practical actions then using imagery and providing concrete, multiple embodiments pictorial representation after which written that are meaningful to students, and further symbols can be used. explains “the goal is not to select one or two Not surprisingly, in his more recent representational forms for students to learn work, Bruner expanded his theoretical and use in all situations but, rather, to teach framework to encompass the philosophical students to adapt representations to a and sociocultural aspects of learning. In his particular context and purpose and even to approach to instruction, Bruner emphasized use several representations at the same time.

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This goal represents a shift from capture and model complexity in the world. representation to representing” (p.260). Such technologically-supported environments are transforming education Beyond Theories: Epistemological and expanding the concept of schooling Frameworks beyond school settings. In an article As has been noted by several entitled, Understanding Learning Systems: theorists (Byrnes, 2007, Lesh & Doerr, Mathematics Education and Complexity 2003), perhaps there no longer exist "grand Science, Davies and Simmt (2003) describes theories" such as Piaget's which attempt to mathematics classes as “ adaptive and self- explain many aspects of cognition. Instead organizing complex systems” where we have many "micro-theories" or what “learning is understood in terms of ongoing, Lesh and Doerr (2003) calls “models” recursively elaborative through designed to account for a “specific purposes which systems maintain their coherences in specific situations” (p.526). within their dynamic circumstances” An increasing number of theorists (p.138). are developing numerous epistemological While complexity scientists agree frameworks by combining constructs from with the views advocated by situated several theories which may potentially learning theories and social constructivism contribute to our understanding of how where the emphasis is placed on analyzing mathematics is learned. The current trend in the dynamics of emerging experiences in mathematics education research is context, however they transcend individual converging towards adopting a models and and social constructivism in calling for a modeling perspective for the teaching and conceptual shift away from mathematics as learning of mathematics (Lesh & Doerr, content and toward “emergent terms” 2003). The main gist underlying this call is (Davies & Sumara, 2007). This perspective not only to progress an agenda of research is concurrent with the view that mathematics that builds on recommendations of is socially and culturally constructed. constructivism, but also to include “a wealth Within the realm of complexity of recent advances in fields of mathematics science and where the world is seen as ranging from complexity theory to game becoming increasingly complex, the discrete theory- where a variety of different types of distinction between teaching mathematics systems thinking tends to be highlighted.” and teaching children collapses in favor of a (p.555) rather nested, integrated whole whose entities are inseparable. In this context, Complexity Science Davies and Simmt (2003) explain: “To teach Recent research in complexity children well, we argue, we must conceive science has revolutionized the conduct and of our activity in terms of active method of science by revealing new participation in the body of mathematics perspectives and possibilities that challenge knowledge by creating the conditions for the beliefs and ideas of contemporary learning emergence of bottom-up, locally controlled, theories. The exponential advancement of collective learning systems” (p. 163). innovations in digital technologies is Furthermore, the authors emphasize pushing the boundaries of learning beyond five basic conditions that help facilitate the existing traditional classrooms by creating creation of a classroom community: a) learning environments that seamlessly internal diversity which, calls for respecting

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and supporting individual differences in the published documents that delineated goals classroom; b) redundancy, which shapes the and standards for mathematical content and criteria for negotiation and participation in processes for grades K-12. In 1989, NCTM classroom activities; c) decentralized control published Curriculum and Evaluation where less emphasis is given to the role of Standards for School Mathematics the teacher as expositor and calls for more expanding on the recommendations of the thought-provoking activities; d) organized Agenda for Action publication issued in randomness where purpose is given to each 1980 and provided a “road map” for states activity; and e) neighbor interactions where and school districts in developing their ideas are productively negotiated and curriculum guidelines. Principles and provoked. Standards for School Mathematics (NCTM, To this end, we argue that building a 2000) followed, building on the preceding mathematical community that supports and publication and adding “underlying nurtures individual students’ understanding principles” for school mathematics. In of mathematical concepts is of the utmost spring 2010, The National Governors importance. Such a classroom collective Association Center for Best Practices (NGA represents a medium where meanings are Center) and the Council of Chief State negotiated and shared and where instruction School Officers (CCSSO) released the is captured in episodes of “teachable National Common Core Math Standards that moments”. These contemporary views specifically concentrates on fundamental present a shift of emphasis from individual shifts in content and pedagogy(National forms of learning to a rather social, Governors Association Center for Best collective mathematical knowledge Practices (NGA Center) and the Council of developed as a result of a network of Chief State School Officers (CCSSO) interactions in the context of the classroom. (2010)). Teachers don't exactly feel prepared to manage the bridging required to Impact of Learning Theories: What access quality instructional resources and lessons can teachers learn? teaching materials that will aid in faithful Much of the current debate about implementation of the standards for standards in mathematics education arose students’ academic success (Schmidt, from opposing views about how people Houang, & Cogan, 2011). As Confrey and learn mathematics. The questions: Should Stohl (2004) explain: “A successful automaticity and quick recall of facts with curriculum is impossible if it does not pay emphasis on procedural skills precede attention to the abilities and needs of problem-solving? Or should reasoning and teachers” (p.92). What seems to be essential constructive thinking reign in the is how teachers translate what they know mathematics classroom even before skill about learning from these theories into development? These questions remain practical everyday applications in their incessantly unanswered. design and delivery of instruction. In a Arguments initiated by national chapter entitled Beyond Constructivism: An standards have been loud for some time. Improved Fitness Metaphor for the Since the release of A Nation at Risk report Acquisition of Mathematical Knowledge, by the National Center of Education Lamon (2003) explains: “classroom Evaluation (NCEE) in 1983, many interpretations of constructivism are not organizations, including NCTM, have necessarily headed in a useful direction and

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that before the pendulum of reform sweeps Summary too far to the right, it may be time to The 21st century is an exciting and consider alternative, but not necessarily challenging time for mathematics teachers competing, perspectives on the development as the opportunities to expand teaching and of mathematical knowledge” (p. 436). With learning are becoming more and more so many voices contributing pervasive in formal and informal education. recommendations and standards, teachers Associated with this growth are the are faced with overwhelming challenges in increasing number of demands and this STEM world we live in. Interestingly expectations behooved on teachers to serve enough, while so many considerations are as leaders enacting cutting-edge directed toward “what” mathematics instructional practices in their classrooms. students should learn, only a dim initiative Our world, through the use of complex has been advanced on “how” children satellite systems, is connected with an should learn. invisible digital network that makes todays’ The impact that theories of learning classrooms inevitably global. Students now can have for the teacher in the mathematics learn from a multitude of resources that classroom is far more complex than what it range from textbooks to live seems to be. Lamon (2003) argues that the videoconferences with people powerful recommendations set by Piaget’s geographically separated by thousands of cognitive constructivist view, as well as miles. The world is becoming more open to Vygotsky’s sociocultural perspective of students through live, streamed videos acculturation and negotiation can be less enabling them to see the world “as it effective if misinterpreted and happens” without any controls. In this overgeneralized in the mathematics climate, teachers are expected to be well- classroom. Regardless of their graduate versed in the newest learning technologies training or experience, we argue that and products in order to best prepare teachers bring to their practice what might students for the global digital workforce. be called “personal theories” of teaching and However, in the midst of increased learning. These theories may be only partly technology, access to resources and to conscious; however these epistemologies are professional development in today’s highly what guide teachers’ everyday decisions diverse schools remains the fundamental about planning, subject content, and quandary dodging opportunities for classroom behavior. “Personal theories” of improvement in teaching and learning teaching and learning grow out of our (Smaldino, Lowther, Russell, 2012). While experiences as students and teachers and teachers are called upon to expand their begin developing while we are children. We professional knowledge and growth by develop predispositions toward certain staying informed of new technologies that learning "styles", just as we gravitate toward have positive impact on students learning, certain teaching styles. If we are to trace the very little effort is advanced to increase literature on best practices for improvement teacher capability to use assistive of teaching a useful focal point emerge technologies to facilitate student success. which is an analysis of the degree to which With heightened tension from policy teachers’ practice is consistent with their makers, the public as well as educational introspection about teaching-their “personal media, teachers find themselves pressured to theory.” deliver quality math instruction to ensure

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that all students achieve academic for American Progress. Retrieved proficiency in a STEM world (Chahine & from King, 2012). http://www.americanprogress.org/wp On February 17, 2009, President -content/ uploads/ issues/ Obama signed the American Recovery and 2012/03/pdf/rtt_states.pdf Reinvestment Act (ARRA), spurring the Brooks, J.C., & Brooks, M. (1993). In Race to the Top (RTT) initiative, which was search of understanding: The case a way to invest in the nation's education for constructivist classrooms. system in an effort to reform schools. Alexandria, VA: Association for President Obama declared: “America will Supervision & Curriculum. not succeed in the 21st century unless we do Bruner, J. (1960). Readiness for learning. In a far better job of educating our sons and J. Bruner (Ed.) In search of a daughters…The race starts today” (Bosser, pedagogy: The selected works of 2012, p.1). RTT focuses on more rigorous Jerome S. Bruner, (2006) Vol. 1 standards and closer evaluation of teachers (pp.47-56). London & New York: whose performance is inextricably linked to Routledge. students’ achievement. However, to Bruner, J. (1964). The course of cognitive empower teachers toward better teaching growth. In J. Bruner (Ed.) In search performance and thereby increase student of a pedagogy: The selected works of academic progress, perhaps we need to Jerome S. Bruner, (2006) Vol. 1 provide more incentives for teachers to (pp.68-89). London & New York: undertake the challenge of educating future Routledge. generations. Byrnes, J. (2007). Cognitive development In this critical educational climate, and learning in instructional it’s reasonable to argue that what contexts. Boston, MA: Pearson. mathematics teachers need more than ever, Carlson, N.R. (2010). Psychology: the is sincere support and collaboration that is science of behavior. Toronto, free from any political obligation. Instead of Canada: Pearson Canada Inc. focusing on issues of accountability and Chahine, I.C. (in press). The impact of using fueling efforts towards more exam–driven multiple modalities on students' instruction, perhaps teachers need more acquisition of fractional knowledge: support to understand how students think, to An international study in embodied uncover cultural and social backgrounds that mathematics across semiotic help interpret their ways of thinking, and to cultures. The Journal of capitalize on those practices that maintain Mathematical Behavior. equity, diversity and high quality instruction Chahine, I.C., King, H. (2012). Investigating for all students so that more student are Lebanese teachers’ mathematical, seeing success with math and liking it too. pedagogical and self-efficacy profiles: A . Retrieved References from http://dx.doi.org/10.5339/nmejre.201 Bosser, U. (2012). Race to the Top: What 2.2 Have We Learned from the States So Confrey, J. & Stohl, V. (2004). On Far? A State-by-State Evaluation of evaluating curricular effectiveness: Race to the Top Performance. Center Judging the quality of k-12

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mathematics evaluations. adolescents’ experiences in schools Washington, D.C: The National and in families. American Academies Press. Psychologist, 48, 90–101. Cramer, K. (2003). Using a translation Foucault, M. (2003). Society must be model for curriculum development defended. (Trans. David Macey). and classroom instruction. In R. Lesh Bertani, Mauro & Fontana, & H. Doerr (Eds.), Beyond Alessandro (Eds.). New York, NY: Constructivism: Models and Picador. modeling perspectives on Friedman, T. L. (2006). The world is flat: A mathematics problem solving, brief history of the twenty-first learning and teaching (pp.449-463). century. New York, NY: Farrar, Mahwah, NJ: Lawrence Erlbaum Strauss and Giroux. (Original work Associates. published in 2005) Csikszentmihalyi, M, & Schneider, B. Furner, J. M., & Berman, B. T. (2003). (2000). Becoming adult: How Math anxiety: Overcoming a major teenagers prepare for the world of obstacle to the improvement of work. New York, NY: . student math performance. Journal D’Ambrosio, U. (1985). Ethnomathematics of Research on Childhood and its place in the history and Education, 79 (3), 170-174. pedagogy of mathematics. For the Furner, J. M., & Gonzalez-DeHass, A. learning of mathematics, 5, 44-48. (2011). How do students’ mastery Davies, B., & Simmt, E. (2003). and performance goals relate to math Understanding learning systems: anxiety? Eurasia Journal of Mathematics education and Mathematics, Science & Technology complexity science. Journal for Education, 7(4), 227-242. Research in Mathematics Education, Gagné, R. M. (1965). The conditions of 34(2), 137-167. learning. New York, NY: Holt, Davies, B., & Sumara, D. (2007). Reinhart & Winston. Complexity science and education: Gay, J., & Cole, M. (1967). The new Re-conceptualizing the teacher’s role mathematics and an old culture: A in learning. Interchange, 38(1), 53- study of learning among the Kpelle 67. of Liberia. New York, NY: Holt, Dienes, Z.P. (1960). Building up Rinehart & Winston. mathematics. London, UK: Goldin, G.A. (2003). Representation in Hutchinson Educational LTD. school mathematics: A unifying Drucker, P. (2010). The changing world of research perspective. In J. the executives. Boston, MA: Harvard Kilpatrick, W.G. Martin, & D. Business School Publishing Schifter (Eds.), A Research Corporation. Companion to Principles and Eccles, J., Midgley, C., Wigfield, A., Standards for School Mathematics, Buchanan, C., Reuman, D., (pp.275-285). Reston, VA: The Flanagan, C., et al. National Council of Teachers of (1993). Development during Mathematics Inc. : The impact of stage- Gonzalez, A. (2002). Parental involvement: environment fit on young Its contribution to high school

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students’ motivation. The Clearing learning and teaching (pp.435-447). House, 75, 132–135. Mahwah, NJ: Lawrence Erlbaum Cornell, S., Hartmann,D. ( 2007). Ethnicity Associates. and race: Making identities in a Lave, J. (1988). Cognition in practice. changing world. Thousand Oaks, Cambridge, U.K.: Cambridge CA: Pine Forge Press. University Press. Hidi, S., & Harackiewicz, J. M. (2000). Lave, J., Smith, S., & Butler, M. (1990). Motivating the academically Problem solving as an everyday unmotivated: A critical issue for the practice. In R. Charles, & E. Silver 21st century. Review of Educational (Eds.), The teaching and assessing of Research, 70, 151–179. mathematical problem solving (pp. Hoffman, B., & Spatariu, A. (2008). The 61 - 81). Hillsdale, NJ: Erlbaum. influence of self-efficacy and Leont’ev, A. N. (1978). Activity, metacognitive prompting on math consciousness, personality. problem solving efficiency. Englewood Cliffs, NJ: Prentice Hall. Contemporary Educational Lesh, R., & Doerr, H. (2003). In what ways Psychology, 33 (4), 875-893. does models and modeling Illeris, K. (2004). The three dimensions of perspective move beyond learning. Malabar, Fl.: Krieger Pub. constructivism? In R.Lesh & H. Co. Doerr (Eds), Beyond constructivism: Jonassen, D., & Murphy,L.(1999). Activity Models and modeling perspectives theory as a framework for designing on mathematics problem solving, constructivist learning environments. learning and teaching (pp.519-556). of Research Mahwah, NJ: Lawrence Erlbaum and Development, 47(1), 61-79. Associates. Jitendra, A. (2002). Teaching students math Millroy, W.L. (1992). An ethnographic problem-solving through graphic study of the mathematical ideas of a representations. Teaching group of carpenters. Journal for Exceptional Children, 34 (4), 34-38. Research in Math Education Jurdak, M. and Shahin, I. (2001). Problem Monographs, 5 (0883-9530). solving activity in the workplace and Monk, S. (2003). Representation in school the school: The case of constructing mathematics: Learning to graph and solids. Educational Studies in graphing to learn. In J. Kilpatrick, Mathematics Education, 47, 297- W.G. Martin, & D. Schifter (Eds.), A 315. research companion to Principles Kincheloe, J.L. (2004). Critical pedagogy. and Standards for school New York, NY: Peter Lang mathematics, (pp. 250-261). Reston, Publishing Inc. VA: The National Council of Lamon, S. (2003). Beyond Constructivism: Teachers of Mathematics Inc. An improved Fitness Metaphor for National Commission on Excellence in the acquisition of Mathematical Education. (1983). A nation at risk: Knowledge. In R. Lesh & H. Doerr The imperative of educational (Eds.), Beyond constructivism: reform. Washington, Dc: Author. Models and modeling perspectives National Council of Teachers of on mathematics problem solving, Mathematics (2000). Principles and

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Standards for School Mathematics. iterative process. Journal of Retrieved from , 93, 346- http://www.nctm.org/standards/conte 362. nt.asp Saxe, G.B. (1991). Culture and cognitive National Governors Association Center for development: Studies in Best Practices (NGA Center) and the mathematical understanding. Council of Chief State School Hillsdale, NY: Lawrence Erlbaum Association. Officers (CCSSO) (2010). Common Schoenfeld, A. (1983). Beyond the purely core state standards initiative. cognitive: Belief systems, social Washington, DC. Authors. The cognitions, and metacognitions as Common Core State Standards from driving forces in intellectual http://www.corestandards.org. performance. In T. Carpenter, J. Nunes, T., Schliemann, A.D., and Carraher, Dossey, & J. Koehler (Eds.), D.W. (1993). Street mathematics Classics in mathematics education and school mathematics. Boston, research (2004), (pp.110- MA: Cambridge University Press. 133).Reston, VA: National Council Newell, A., & Simon, H.A. (1972). Human of Teachers of Mathematics Inc. problem solving. Englewood Cliffs, Schmidt, W., Houang, R., Cogan, L. (2011). N.J: Prentice Hall Incorporation. Preparing future mathematics Piaget, J. (1952). The origins of intelligence teachers. Science, 332, 1266-1267. in children. New York, NY: Scribner, S. (1986). Thinking in action: International University Press. Some characteristics of practical Piaget, J. (1977). The development of thought. In R.J. Sternberg & thought: Equilibration of cognitive R.k.Wagner (Eds.), Practical structures. New York, NY: The intelligence nature and origins of Viking Press. competence in the everyday world Post, T. (1988). Some notes on the nature of (pp. 13- 30). New York, NY: mathematics. In T. Post (Ed.), Cambridge University Press. Teaching mathematics in Grades K- Thompson, B., & Thornton, H. (2002). The 8: Research-based methods, (pp. 1- transition from extrinsic to intrinsic 19). Boston, MA: Allyn & Bacon. motivation in the classroom: A first Pritchard, A. (2005). Ways of learning: year experience. Education, 122, Learning theories and learning styles 785–792. in the classroom. Abingdon, VA: Tuckman, B. W. (1992). Educational David Fulton Publishers. psychology: From theory to Putnam, R.T., Lampert, M., & Peterson, application. Orlando, Fl.: Harcourt P.L. (1989). Alternative perspectives Brace Jovanvich. on knowing mathematics in Turner, J. C. (1982). Toward a cognitive elementary schools. Review of redefinition of the social group. In Research in Education, 16, 57-149. H. Tajfel (Ed.), Social identity and Rittle-Johnson, B., Siegler, R.S. & Alibali, intergroup behavior (pp. 15-40). M.W. (2001). Developing Cambridge, UK: Cambridge conceptual understanding and University Press. procedural skill in mathematics: An 2013 Mathitudes 17

Vygotsky, L.S. (1962). Thought and Williams, R., & Stockdale, S. (2004). language. Cambridge, MA: MIT Classroom motivation strategies for Press. prospective teachers. The Teacher Wigfield, A., Eccles, J., & Rodriquez, D. Education, 39, 212–230. (1998). The development of Yair, G. (2000). Reforming motivation: children’s motivation in school How the structure of instruction contexts. In P. D. Pearson & A. Iran- affects students’ learning Nejad (Eds.), Review of research in experiences. British Educational education (pp. 73–118). Washington, Research Journal, 26, 191–210. DC: American Educational Research Association.

About the Author:

Iman C. Chahine is an Assistant Professor of Mathematics Education in the Department of Middle-Secondary Education in the College of Education at Georgia State University (GSU). She is the director of three study abroad programs on Ethnomathematics and Indigenous Mathematical Knowledge Systems in Morocco, South Africa, and Brazil. She received her Ph.D. from the University of Minnesota in 2008 and her M.A. from the American University of Beirut in 1997. Her research interests include ethnomathematics, indigenous mathematical knowledge systems, situated cognition, and multicultural mathematics. Dr. Chahine has authored many articles in peer- reviewed journals and published a number of book chapters related to mathematics teaching and learning. Dr. Chahine received the GSU Instructional Innovation Award in 2012.

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