Towards a Cognitive Model for Computer Mediated Learning in Mathematics

1. Papert and Logo: An Invitation to Dance

"There are two basic ideas of education. One is instructionism; people who subscribe to that idea look for better ways to teach. The other is constructionism; we look for better things for children to do, and assume that they will learn by doing."[1]
The origins of the development of a functional cognitive model for computer-based learning in mathematics date to the late 1950's. Having collected doctorates in mathematics at the University of Witwatersrand in South Africa and at Cambridge, Seymour Papert, a young, South African-born mathematician allowed his keen interest in cognitive development to lead him to Geneva for a research position at the International Centre for Genetic Epistemology. Founded and directed by the famous developmental psychologist and genetic epistemologist, Jean Piaget, the Centre was at the fomentation of the doctrine of constructivism in cognitive psychology. Strongly influenced by the ideas of constructivism, Papert would later remark that it was here that he first thought of "turning technology over to children to see what would happen as they explored it"[2].

The underlying metaphor of constructivism is one of structures. Understanding and knowledge represent cognitive structures assembled through the dual processes of assimilation and accommodation[3]. You are assimilating when you use or transform your environment to make it fit one of your existing cognitive structures. Having learned very early how to simplify two plus three, our math students use assimilation when learning to simplify '2x' plus '3x'. You accommodate when you change your cognitive structures to accept new information. Having learned to solve two equations in two unknowns, our students generally have few difficulties using accommodation to modify the solution schema to three equations in three unknowns. Genetic epistemology provides the theoretical foundation for the ideas of assimilation and accommodation... [Genetic epistemology]

In 1963, Papert left Geneva for MIT where he began work with Marvin Minsky at the Artificial Intelligence Laboratory. Two years later, he conceived of the idea of developing a computer language that children could use as a tool for exploration. With Minsky, Hal Abelson, other colleagues at MIT, and in collaboration with programmers from Bolt, Beranek, and Newman, he developed the programming language Logo. Based on Lisp, at the time, the standard for artificial intelligence programming, Logo allowed even very young students to become programmers. With Logo, he believed that his team had created an environment in which students could "learn by doing" and at least to some degree, take charge of their own learning.[4] The initial phase of Logo's development culminated in 1980 with Papert's publication of Mindstorms: Children, Computers, and Powerful Ideas. In Mindstorms, Papert presented the first widely recognized cognitive model for computer based learning. Motivated by observation of children's behaviour while playing and experimenting with Logo, Papert added the importance of a social interaction to Piaget's cognitive structure building. In effect, the structure building metaphor had now become "building and bragging". His model is now commonly referred to as "social constructionism". Papert describes it as follows:

"Constructionism - the N word as opposed to the V word - shares constructivism's connotation of learning as `building knowledge structures' irrespective of the circumstances of the learning. It then adds that this happens especially felicitously in a context where the learner is consciously engaged in constructing a public entity, whether it's a sandcastle on the beach or a theory of the universe... If one eschews pipe line models of transmitting knowledge in talking among ourselves as well as in theorizing about classrooms, then one must expect that I will not be able to tell you about my idea of constructionism. Doing so is bound to trivialize it. Instead, I must confine myself to engage you in experiences (including verbal ones) liable to encourage your own personal construction of something in some sense like it. Only in this way will there be something rich enough in your mind to be worth talking about."[5]
It is important to note that social constructionism is a cognitive model that attempts to describe how people, and especially children, best learn. Papert argues in favour of this model and he argues that by creating environments which encourage cognitive structure building, computer based technology such as implementations of Logo, should play an important role in education; particularly in math and science. In Mindstorms, he engagingly compares these environments to a dance school:

"LOGO environments are not samba schools, but they are useful for imagining what it would be like to have a 'samba school for mathematics.' ... The computer brings it into the realm of the possible by providing mathematically rich activities which could, in principle, be truly engaging for the novice and the expert, young and old." [6,p. 182]
In 1984, Papert was approached by LEGO, the Danish manufacturer of children's building blocks. This began a collaboration that led eventually to the production of LEGO/Logo; a robotic environment in which children use Logo to program LEGO robots. In 1988, LEGO would endow for Papert the LEGO Professor of Learning Research Chair at the newly formed MIT Media Lab. He continues to hold this chair today.

2. Distributed Constructionism: Cooperative Learning for the Information Age

Papert joined the Media Lab in 1985. With healthy corporate sponsorship and a talented pool of researchers, the lab rapidly established itself at the forefront of institutions conducting research into emerging information technologies. One of Papert's new graduate students was Mitchell Resnick, a returning student who had spent the previous five years as a science and technology journalist for Business Week magazine. Completing his Ph.D. in 1992, Resnick embraced Papert's social constructionism and extended it, introducing the model of distributed constructionism. Distributed constructionism is social constructionism with greater emphasis on the social. It is cooperative learning with levels of cooperation that are most easily facilitated through information technology. Resnick writes:

Distributed constructionism extends constructionist theory, focusing specifically on situations in which more than one person is involved in the design and construction activities. It draws on recent research in "distributed cognition" (Salomon, 1994), recognizing that cognition and intelligence are not properties of an individual person but rather arise from interactions of a person with the surrounding environment (including other people and artifacts). Recent research projects have attempted to use computer networks can facilitate the development of "knowledge-building communities" (Scardamalia & Bereiter, 1991), in which groups of people collectively construct and extend knowledge. In many of these projects, students share ideas, theories, and experimental results with one another. Distributed constructionism asserts that a particularly effective way for knowledge-building communities to form and grow is through collaborative activities that involve not just the exchange of information but the design and construction of meaningful artifacts.[7]
The emphasis on communication of ideas suggests that computer networks could play an important role in learning environments built to support the distributed construction cognitive model. Resnick identifies three major categories of distributed construction activities that can be supported by computer networks. These are:

1. discussing constructions
2. sharing constructions and
3. collaborating on constructions[8]

3. Confronting Complexity: The Impact of Parallel Programming

In the late 1980's, while conducting research that involved observing children working with LEGO/Logo, Resnick identified a major weakness of the robotic environment which surfaced whenever the kids created a robot that was made up of smaller robots. The problem was that the smaller robots could not be programmed independently. The example that he describes is that of an amusement park: if the amusement park contained two or more rides, it would be impossible to activate the park with the rides running independently of each other [9]. In the context of math education, we can imagine a situation where we might want our students to compare the evolution of two functions parameterized with respect to time. Ideally, our graphing application would plot the two functions at the same time. Even modern graphing calculators need to plot one and then the other. To address this problem, Resnick modified Logo to support parallel programming, calling the improved language "MultiLogo". Parallel programming is a programming paradigm that in essence allows a single processor to handle more than one program independently. With Multilogo, students could program two or more Logo turtles independently and observe their behaviour or, using the LEGO/Logo interface, write programs that set two robots into operation independently.

In terms of it parallel programming capabilities, MultiLogo was limited to a relatively small number of parallel processes. The rapidly evolving fields of complexity and systems science had led Resnick to imagine a programming environment which would model the complex behaviour of decentralized systems. He writes:

One reason for my interest in these "massively parallel" situations is that people seem to have great difficulty thinking about and understanding such situations. When people see patterns in the world, they tend to assume some type of centralized control. For example, when people see a flock of birds, they typically assume that the bird in the front is leading and the others are following. But that's not the case. Most bird flocks don't have leaders at all. Rather, each bird follows a set of simple rules, reacting to the movements of the birds nearby. Orderly flock patterns arise from these simple, local interactions. The flock is organized without an organizer, coordinated without a coordinator (Heppner & Grenander, 1990). [10]
With the goal of creating a programming language that could model the type of "decentralized" behaviour exhibited by a flock of birds, Resnick led a team in the development of *Logo (pronounced StarLogo). He describes *Logo as contributing three major extensions to Logo. The first extension concerns the degree of parallel programming that *Logo supports. While MultiLogo and its commercial derivatives might allow a small number of turtles to execute and display independent processes, *Logo permits routines for thousands of turtles to be executed in parallel. The second extension concerns the endowment of the turtles with senses. In the *Logo environment, Turtles can be programmed to respond to each other and to their "world". Finally, *Logo enhances the Turtle's world. Rather than simply acting as a drawing palette, the screen is divided up into patches. Each patch can be programmed in much the same way that turtles can except that the patches are stationary[11].

The cognitive model supporting *Logo is still distributed constructionism. However, distributed construction learning environments are systems themselves and *Logo has the potential to encourage students to reflect upon how they learn and behave within the system. Resnick concludes that tools such as *Logo "provide an opportunity for students (and others) to move beyond the centralized mindset [and suggest] an expanded set of models and metaphors for making sense of the world [12]".

Shortly after Resnick joined the Media Lab, he was joined by Uri Wilensky. Like Resnick, Wilensky sought to engage the model of distributed constructionism with the study of complexity. Wilensky's formulation of the model, however is more explicitly founded in math education. His unifying concept is that of "Connected Mathematics". He contrasts it with traditional math education as follows:

Traditional mathematics education has proceeded from a view that mathematics is "given" rather than constructed and is to be transmitted to learners primarily through formalism. As a result, mathematics is usually taught in isolation from other domains and the role of technology in mathematics education has primarily been to better inculcate or animate the existing formalisms. In contrast, the theory of Connected Mathematics sees the fundamental activity of mathematics as that of making and designing new mathematical representations and connecting these representations to each other and to other domains.[13]
The role of mathematics in helping a learner to make representations and connect those representations to each other is germane to Wilensky's distributed consructivist perspective. Representations, and the connections we make between them, help to render the abstract concrete and in developing understanding, concreteness is everything. In his 1990 paper entitled "Abstract Meditations on the Concrete and Concrete Implications for Mathematics Education", he writes:

I now offer a new perspective from which to expand our understanding of the concrete. The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object.[14]
An unfair assessment of Resnick and Wilensky's distributed connectionism might suggest that from the point of view of articulating a workable cognitive model, their main achievement has been to, take Papert's social constructionism and roll into it a selection of intellectual fashions of the early and mid 1990's. Along with their incorporation of the ideas of systems science and complexity theory, Wilensky acknowledges in the introduction of his Ph.D. thesis that his ideas of "connected knowledge" have antecedents in the feminist critique of educational practice. In fact, the efforts of Papert, Resnick, and Wilensky to formulate a cognitive model for computer-based learning of mathematics should properly be interpreted in the dual contexts of development of thought in the philosophies of education and mathematics.

4. The Constructivist/Behaviourist Debate

To properly locate the work of the Media Lab group in the development of educational thought, it is necessary to step back to the Moscow of the late 1920's. Born in, 1896, the same year as Piaget, a young Jewish scholar, Lev Vygotsky, in 1934 published "Thought and Language", the book for which he is now most famous. Vygotsky was a polymath. A graduate in law whose first love was literature, his varied interests had led him towards the study of cognitive psychology. His work, even more prominently than Piaget's, laid the theoretical framework for the constructivist theories of education that have dominated educational debate since the early 1980's. Vygotsky's theory of cognition is founded on three principles[15]:

1. learning is a social activity and is mediated by the student's social environment

2. learning is mediated by the students physical environment and the tools that she has at her disposal, and

3. learning takes place within a "zone of proximal development".
The zone of proximal development is the realm of the "almost understood", as opposed to the realms of the well understood and the completely unimagined. This concept is cited by modern constructivists as the theoretical foundation of "scaffolding". Educational activities built around scaffolding attempt to encourage learners to build comprehension from concepts that are well understood to concepts that are almost understood.

While the differences between the cognitive models presented by Piaget and Vygotsky are open to discussion, it is commonly accepted that the key difference is that Vygotsky emphasized and described the social origin of knowledge, while in Piaget's work, the role of social interaction in developing knowledge is not as clearly defined. In this respect, Papert's social constructionism is seen as being more in line with the work of Vygotsky than of Piaget. It is hard to say how much more Vygotsky might have been able to contribute to cognitive psychology had he not died of tuberculosis shortly after the publication of Thought and Language. The government had already restricted his research and suppressed his ideas. Vygotsky's work was rediscovered in the Soviet Union in the 1950's and Thought and Language was first translated into English in 1962.

Papert's social constructionism and the distributed constructionism of Wilensky and Resnick fall under the rubric of constructivist teaching practices. In essence, constructivist teaching can be broadly interpreted as any form of instruction that is sympathetic to the following six principles[16]:

--knowledge is constructed from experience;

--learning results from a personal interpretation of knowledge;

--learning is an active process in which meaning is developed on the basis of experience;

--learning is collaborative with meaning negotiated from multiple perspectives;

--learning should occur (or be 'situated') in realistic settings; and

--testing should be integrated into the task, not a separate activity.
Challenging constructivism, behaviourism dominated educational thinking from the 1950's through to the mid 1980's and although this dominance is being eroded, it continues to be the cognitive model supporting most classroom practice. In its most radical form, behaviourism is associated with the American psychologist, B.F. Skinner. This model proposes that students' learning can only be measured by their behaviour and that a teacher can mold this behaviour through an appropriate regime of positive reinforcement. From a behaviourist point of view, we reinforce desired behaviour in our students when we congratulate them for doing well on their math test. Whether we place a gold star at the top of a younger student's page or write a short, congratulatory note for an older student, the intent is the same: to encourage a similar result on the next test. From the point of view of computer based learning, a classically behaviourist design would be a well structured drill and practice tutorial that rewards correct responses with pleasant and congratulatory audio-visuals. While we use positive reinforcement with our students to express our natural inclination to acknowledge good results, through the lens of cognitive behaviourism, our students success will be entirely determined by how well we structure this positive reinforcement. Absent is any serious consideration about what takes place in a student's mind when he learns; learning, after all, is only recognized in desired behaviour, molded by appropriate reinforcement. A behaviourist learning environment is one that is constructed and carefully controlled by the teacher. Papert refers to these teacher controlled environments as epitomizing "instructionism" as opposed to the more learner-centred constructionist environment.

4. Beyond Constructivism and Behaviourism: A (Brief) Look at the Fuller Spectrum of Cognitive Theory

While the constructivist/behaviourist debate has dominated educational discourse concerning cognition, these two models are not the only ones to have implications for teaching and instructional design. Situationists identify themselves with one of a family of theories that are more radical in their perspective than constructivism. In a research report entitled "Constructivist instructional theories and acquisition of expertise", Law (1995) identifies the common ground shared by these theories as follows:

...all of them formulate their theories on the one hand with the basic philosophical stance that there is indivisibility between cognition and culture, and on the other hand with the unit of analysis that is commonly grounded on an amalgamation of person, activity and setting. They also share the core epistemological assumption that knowledge is actively constructed rather than passively absorbed by an intelligent agent.[17]
These theories present a holistic interpretation in which the learner cannot be considered separately from the environment in which she learns. The most widely discussed of the situationist theories is known as "situated cognition". Instructional design motivated by situated cognition is called "situated learning" and is presently the subject of considerable attention in educational discussion. In mathematics education, the ideas of situated learning surface frequently when we discuss how to make math relevant to our students. These ideas are also manifest in renewed attention to the importance of applied mathematics in the curriculum. For computer mediated math education, instructional design based on situated learning is being proposed that takes the form communication systems that support role playing activities. Recognizing the attention being paid to the techniques of situated learning, in their paper "Diving into Complexity: Developing Probabilistic Decentralized Thinking through Role-Playing Activities", Resnick and Wilensky propose role playing over the internet in which:

each person could control the behavior of a proxy object in a shared virtual space, and then observe the group-wide patterns that arise from the interactions. This type of activity could build on existing MUD environments (or architectures), but would have a greater emphasis on interactions of collections of computational objects (including representations of self) in structured activities [18].
When taken to their logical extreme, situationist theories of cognition reject cartesian mind/physical world dualism. The mind is the sum of an intelligent agent and the stimulus that interacts with it; one does not exist without the other. In his paper "An Ecological Approach to Cognitive Science", John Sanders writes:

It is exciting to see that, more and more, discussions of cognitive science are focusing greater attention on the importance for day-to-day cognitive activity of interaction with the changing environment, both at the level of individual cognition and, ultimately, at the level of understanding cognition more generally. It remains to be seen, though, how long it will take finally to shake the bonds of the Cartesian Theater and fully to embrace the ecological approach.[19]
Even more positivist and deterministic in nature than behaviourism, the underlying metaphor of cognitivism is that the human mind is a computer. Under this model, learning is analogous to programming and the goal of the teacher is to transfer a "cognitive map" to her students. Cognitivism comes in a number of varieties, however a good example is found in the work of John Anderson. Born and raised in Vancouver, Dr. Anderson is presently the Walter VanDyke Bingham Professor of Cognitive Psychology and Computer Science at Carnegie Mellon. He is most widely known for the development of the ACT-R theory of cognition and its accompanying computer simulation. In describing his research he writes:

We have taken on modeling the cognitive competences that are taught in the domains of mathematics, computer programming, and cognitive psychology. Much of the motivation for this research is to be able to tap into real situations where people learn and solve problems and understand the implications of these domains for the cognitive architecture. We have built larger-grain ACT-R simulations that are capable of solving problems in these domains and have developed computer-based instruction around these cognitive models. Many of these computer-based instructional systems have the cognitive models as a component and attempt to understand student behavior by actually simulating what the student is doing in real time.[20]
An example of the type of computer based instruction that he refers to is the Pat (Practical Algebra Tutor) program. The Pat tutor is an expert system that is designed to help students learn algebra by guiding them through problems based on real-world situations. The system provides feedback at decision branches in the solution process. The main criticism of systems such as these is their potential to isolate learners. The cognitivist model does not recognize the necessity of a social dimension to learning and consequently instructional design that is based on this model does not require social interaction. Having the students share computers builds interaction into the learning environment, however it compromises the effectiveness of the system for the individual.

5. Conclusion

In October of 1995, in testimony before a House of Representatives Panel on Technology and Education, Seymour Papert stated:

The question at stake is no longer whether technology can change education or even whether this is desirable. The presence of technology in society is a major factor in changing the entire learning environment. School is lagging further and further behind the society it is intended to serve. Eventually it will transform itself deeply or breakdown and be replaced by new social structures.[21]
Papert makes two points in this statement: that the mere presence of technology in society impacts the learning environment and that schools are lagging further and further behind. While both of these points could be debated, the fact that Papert was invited to express his ideas before this committee of legislators is an indication that, in education, the momentum for change is strong and no matter what form this change takes, the potential impact of technology cannot be overlooked.

The fact that the importance of educational technology has attracted the attention of legislators may be only a manifestation of two unsettled currents affecting education in general. First, and it is likely to always be there, is the issue of accessibility. This is a complicated issue as it binds together considerations of cost, the pervasiveness of computer technology, and the state of system interface design. In each of these three areas, the trend is towards increasing accessibility. Despite chronic funding concerns in education, recent cost reductions in processing speed, memory, and bandwidth are making "high enough" performance technology affordable to schools. Computer technology is also increasingly common and students are more likely than ever to have systems at home. Although it could be argued that there is still a lot of work to be done, interface design is also improving, making systems easier to use. With increased accessibility, suddenly much more is possible and consequently more is being attempted and even more is being contemplated.

The second issue concerns the philosophical debate that places the cognitivists on one extreme, the situationists on the other, with the behaviorists and constructivists staking out various regions of the middle ground. With the Language Arts curriculum, the popular manifestation of the philosophical debate is witnessed in the conflict between supporters of the "whole language" method of reading instruction, a decidededly constructivist approach, and the supporters of phonics based instruction, a more behaviourist approach. In a recent Atlantic Monthly article entitled "The Reading Wars", Nicholas Lehman characterizes the re-emergence of this debate is California as taking on the overtones of a full-fledged political war[22]. With the math curriculum, the philosophical debate is manifest in the conflict between those who support the implementation of the mildly constructivist leaning NCTM Curriculum and Evaluation Standards for School Mathematics and those who advocate a more traditional, behaviourist approach. Again in California, a group calling itself HOLD, for "Honest Open Logical Debate on math reform", is leading the opposition to implementation of the NCTM standards in the largest educational jurisdiction in North America.

Given the unresolved nature of the issues affecting educational implementation of technology, the question arises of how we can responsibly proceed lacking an undisputed cognitive model of how and what students learn in a technological learning environment and the changing nature of that environment. Here, I offer three suggestions: the first is that in the initial stages of implementation we restrict ourselves to domain specific analyses and refrain from attempting to make too many generalizations for models across the full range of subject areas. What constitutes a good implementational model in math, may not be a good model for Social Studies. The second is that we pay close attention to research in cognitive psychology and that we attempt to frame our analyses in the vocabulary of cognitive psychology and state the cognitive bias of the implementation. The third is that as part of the analysis of a particular implementation we build in follow up with those students who failed. It is important to attempt to determine to what extent their failure is due to the cognitive bias of the implementation and to what extent it is due to other factors.

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