Influences from education theory
Influences from evolved-systems theory
Development of complex evolved systems
Overflowing variety from recombination
Connectivity and phase transitions
Winnowing and Coherent Precipitation
Phylogenetic Learning: The uneven foundation
Cultural Learning: The heritage of communities
Individual History: Students are already (distinct) people
Street learning and school learning
Style and specialization: communities of practice
Style and specialization: individuals
Instructional tactics: Learning at the edge of confusion
Strategies for complex-topic instruction
Confusion is the price of understanding
Combining confusion and clarity
The areas of educational theory that have contributed most to the analysis from which I designed the MFM course are those based on the classic work of Lev Vygotsky, the related later work of Lave & Wenger and Terezinha Nunes and her collaborators, and the “constructionist” analysis of Seymour Papert. The classroom-practice engineering of Ann Brown and her collaborators also had a substantial impact on my thinking, but is not as yet reflected in the MFM course materials, due to the challenges involved in adapting it to adult learning.
The aspect of Vygtosky’s work that has most influenced my thinking in this context was the idea that mature understanding evolves from a dialectical interaction between concepts derived from heredity or experience and those derived from instruction (Vygotsky, 1978). This process can be seen separately in the development of mathematical and practical-craft competencies, each of which is built by using culturally-transmitted ideas to organize and refine direct experience and innate capabilities (and to in turn gain meaning from them), with the resulting syntheses constituting mature viewpoints in each sector. But the particular relevance for the MFM course is that under modern conditions, fully-mature practical understanding results from a further synthesis of the mathematical and craft ways of thinking. While ensuring such a full synthesis for practical-arts students is beyond the ambitions of the MFM course, it is intended to facilitate communication between them and the engineers in their field who have accomplished it.
The idea that a leading role is played in learning by interactions with more-capable and more-knowledgeable people was also important, both as an explanation of the origin of many ideas held by practical-arts students and as a guide to instructional tactics. Lave & Wenger’s 1991 work on “legitimate peripheral participation”, especially its analysis of the mechanisms of the “communities of practice” within which craft traditions are developed and propagated, was particularly helpful in helping me relate my own experience as a technical practitioner (and a manager of such practitioners) to the issues of MFM course design. Their analysis also raises some serious questions about the feasibility of scholastic activity for promoting learning that will be usable outside a scholastic context; these are taken as valid concerns that must be addressed, in large part by efforts to situate the MFM course as a component in a larger community of shared practices that connects mathematics, science, engineering, and practical-arts disciplines.
The prior work that I found to be most directly relevant to the goals of the MFM course was that of Nunes et al (1993) relating “street mathematics” and “school mathematics”. Several elements of this multifaceted report illuminate the lack of impact that school mathematics often has on practical workers, who include independent small-business owners who successfully handle essential quantitative tasks. A point that I see as particularly relevant to MFM is the finding that for practical workers reduction of mathematical problems to written form often entails a disabling loss of meaning, while spoken consideration of the problem results in fewer mistakes, even for novel applications. This highlights the possibility that abstraction may be inferior to analogy as a mode of generalization for many people and situations, especially in the robustness of its conclusions in actual workplace decision processes.
Papert’s ideas about using computer programming as a concrete path to abstract thinking, and the suggestive principles he offers to guide topic selection in curriculum development, also provided guidance in the task of connecting the abstract realm of mathematics with the concrete realm of the practical arts. Although most of Papert’s work focuses on the articulation of mathematics with the early natural learning that makes programming Logo turtles an accessible task for elementary-school students, his “principle of cultural resonance” also provides a useful guide to articulation with the world of adult practice.
There should at this date be nothing novel in the realization that any successful comprehensive theory of learning will have to make substantial use of the most important single scientific fact about humans: that we are descended from simpler animals by processes that preferentially selected among natural hereditable variations in individual traits those that most furthered the survival and reproduction of their holders. Further, the striking superiority of human learning skills to those of our nearest kindred make it clear that important brain functions supporting learning must be part of our species’ recent evolutionary accomplishments. But the details of the impact of genetically-programmed structures on learning are currently obscure, although a rough outline of the situation can be sketched.
However, several aspects of theoretical analyses of evolved and complex systems can be used to illuminate portions of existing educational theory, and thus to guide a strategic analysis of how the goals of the MFM course may be achieved. I have found the work on natural order of Stuart Kauffman (1993, 1995, 2000) particularly useful in organizing my thinking about learning so that it makes more use of evolutionary perspectives. While I have not found any previous application to education of Kauffman’s ideas (which are rooted in genetics, and include a persuasive explanation of the origin of organic life), his demonstration that order can emerge “for free” from a variety of natural situations (and is thus more accessible than might be thought for use both directly and as a starting point for the accumulation of adaptive change) seems to me to have several important implications for learning theory. These are particularly relevant in this context because in some ways the interplay of natural order and experience-driven adaptation that Kauffman sees as driving biological evolution parallels the interplay of mathematics and practical experience that the MFM course is trying to foster.
Before directly discussing the educational implications of Kauffman’s ideas, here is a summary from the evolved-systems perspective of its main concepts (these will become clearer when later applied to explain specific learning-related concepts). While there are some sequence dependencies and shifts in the balance of the different processes as a system develops, all of these processes are active simultaneously over most of the development period. My summary glosses over many fine points that Kauffman addresses in detail in his 700-page book and dozens of journal articles — this is just an introduction.
The source of the variety that supplies novel material from which complex development can proceed is the recombination of existing entities in the system. Since the number of potential combinations is roughly a mulplicative function of the number of distinct entities capable of combination, any capacity for combination in a system can easily engender great variety. (The process of breaking up an existing entity into its components is seen as a part of the combination process, since the contribution of such breakups to system dynamics is generally to supply entities for subsequent recombination.)
For a system to exhibit the most complex behavior, the available modes of combination need to support repeated (preferably recursive) combination, and to be themselves capable of extension and adaptation. A linguistic example of this pattern is the use of connective and modifying words to form phrases, clauses, and sentences from a vocabulary. Note that a very large number of concepts can be expressed (usually in many different ways) with a quite modest vocabulary and grammar.
In interesting systems, many of the entities interact. These interactions form connecting webs of influence, which can[1] form mutually-supportive cycles. The overall level and pattern of connectivity in a system strongly influences its potential for supporting complex action, with low-connectivity systems showing few systemic effects (it is the connections that make a collection of entities into a system) and high-connectivity systems being largely static (effectively just one entity, and thus having limited development potential).
The most complex behavior is exhibited by systems “on the edge of chaos”, with many of the system entities having enough connections (of just the right sort) to participate in coordinated action on an appropriate scale, but with most not being so connected that their action can be fully predicted from outside. “Critical” systems that are complex and persistent are usually somewhat more connected than this, with “islands of stability” to anchor the dynamic constituents. Note that the potential for continued development is greatest if a system has internal processes that keep its connectivity at the critical level — this implies some feedback mechanism.
Because of the highly nonlinear response of the behavior of such systems to their level of connectivity, it is a natural feature for gradual increases in system connectivity to entail one or more “phase transitions” when critical connectivity thresholds are passed. For example, if a system is composed of numerous identical elements, each of which has an equal random chance of being connected to each other element (some will thus have no connections, others multiple ones), systems with just under half as many connections as elements will consist almost entirely of elements that are isolated or connected in quite small groups, while systems with just over half as many connections as elements will have a single large connected group that includes almost all the elements (Erdos & Rényi, 1960). This phenomenon can explain many situations in which small quantitative changes have effects that are so large as to be seen as qualitative.
Systems that persist and develop (as contrasted to just changing unpredictably) must include mechanisms that allow future states of the system to be conditional on past states. Each such memory record may entail some cost to the system (i.e., use of a limited resource), in which case there are efficiency benefits from minimizing the number of remembered entities required to achieve a given pattern of action.
A key tactic for achieving such efficiency is matching the structure by which memory is accessed to the connection structure of how the information involved will be used (along the lines of a thesaurus); this also facilitates building new combinations among related entities, since they already share part of their access methods. In human thinking, concepts (including concepts that connect or generalize other concepts) reflect such a structure.
A system may contain several memory mechanisms with different capabilities, capacities, and costs.[2] For example, humans have some limited “short-term” or “working” memory systems directly assessable to consciousness and much more capacious long-term ones, with a different subsystem used for facts from that used for processes – a finding clearly relevant to learning.
Rather than being viewed as entities and connections, a dynamic system can be considered in terms of the results that it produces. This view is particularly helpful when components of the system act in patterns of mutual support that constitute work cycles that have some useful role in the success of the overall system.
Work cycles can operate at various levels and time scales. Each can be considered as accomplishing a task in the overall economy of the system. Systems of sufficient complexity to be interesting typically have many different cycles, which are integrated enough for productive coordination but sufficiently decoupled that each cycle has some scope for individual development. Biological cells are an excellent example of this organization.
One type of work cycle of special significance is reproduction, in which a system produces similar systems. This often entails an individual-development process (i.e., an ontogeny), with the new system starting in a simple form and changing its characteristics substantially over time. Reproducing systems are of particular relevance to evolved-system theory because many biological examples exist whose mechanisms, history, and commonalities can be examined to provide insight into the subject. They are also relevant to education because it is a form of cultural reproduction. However, some systems that evolve but do not reproduce are currently present because they either are long-lived (e.g., ecosystems, galaxies), arise repeatedly (e.g., thunderstorms, mountains), or are products of other current conditions such as human culture (cities, mathematics).
An important
idea related to work-cycle tasks, called closure by Kauffman, describes
the situation when a system includes some work cycle to perform, at least
approximately, almost all potential tasks of some class. That
this condition can arise naturally in certain types of systems is one of
Kauffman’s key insights.[3]
When an evolving
system reaches a state of closure, its subsequent development is driven much
more by the characteristics of the task set (especially the benefits associated
with various patterns of task activity) than by the details of the system’s
earlier history. Closure can be a
system’s turning point from random development to convergence on an advanced
state that is more efficient, complex, and stable (until some later round of
evolution). Certain learning
processes show behavior of this kind when the complete set of concepts needed
to analyze a class of problems in an area are mastered — subsequent learning in
the area proceeds from that basis for all students, even if they reached that
level of understanding by different means.
Recursive combinatory processes usually are prolific, generating more (and different) entities until some constraint is encountered. During the frontier phase prior to that constraint, systems can become “supracritical” (Kauffman, 1993) with virtually all the potential work cycles present in at least approximate[4] form. This abundance implies that when resource constraints are encountered, a supracritical system will then evolve by deletion rather than by the accumulation traditionally associated with evolutionary development.
The survivors of such a scarcity-induced simplification will be, for each essential system task, one of the most efficient of the many work cycles by which it could be accomplished. The process does not require conscious direction (although it may well benefit from it if available), since the surviving complex will necessarily include all methods essential for its continued operation.[5] Viewed as it occurs, this process is clearly subtractive, since the “answers” were already somewhere in the mixture to begin with, and are just revealed by the consumption of all the elements that were not part of essential and efficient work cycles. Viewed retrospectively, however, it seems that a miraculously-coherent set of cycles precipitated[6] out of the chaos.[7]
One of Kauffman’s many contributions to my thinking was this demonstration that in some cases well-articulated complex systems can arise much more easily from such thinning than from building each component individually.[8] While this idea is of particular importance in explaining such getting-started puzzles as the origin of life, the process can also act at higher levels, using the results of earlier winnowing/adaptation sequences as components from which a new supracritical system is formed and then winnowed. As discussed later, learning may sometimes be just such a high-level occurrence of this process, when a fog of new ideas precipitates into understanding.
Even in a system with efficient work cycles, there may be interference between cycles with different tasks, setting limits to system capability. Sometimes the interference may spring from incompatible tasks, but other times the system may just lack the capacity to support both cycles at once. Partitioning the tasks between different variants of the system, which share some but not all subsystems and have mechanisms to coordinate their inputs and outputs, can be a way to avoid limitations due to incompatibility or incapacity.
One potential source of the variety embodied in such specialization is the winnowing process, which may produce different results in similar systems due to chance or to differences in local environment when the thinning process occurred. Another potential source is the combination of systems with previously-separate developmental histories[9] , as in symbiosis.
But specialization can instead easily arise via a gradual process that includes mutual adaptive adjustments even as the initially-similar variants become more and more distinct, and this pattern in fact seems to be the most frequent one. Darwinian natural selection supplies several obvious examples from biology, such as the differentiation of species in ecosystems and of cell types in organisms. Separation of initially-identical lineages plays an important role in many cases.
If a system reproduces (other than by simple fission), the ontogenetic process (i.e., individual development to a mature state) by which the pattern of specialization of its internal parts is reproduced in its descendents is distinct from the much slower (and probably less certain) phylogenetic evolutionary history via which that pattern of specialization first arose. Although there are likely to be parallels[10] , the ontogenetic process will be adapted for efficient progress to a predetermined type of mature state, with no requirement that intermediate states be fully-capable systems.
Application of this idea to education implies that individual learning (including specialization within a community) is basically ontogenetic, while cultural development is more nearly phylogenetic.[11] Thus it is not surprising that instruction seldom uses the authentic history of a subject area. In some areas (such as mathematics) history is almost completely ignored; many other fields (including most sciences and practical arts) use a mythic form of their history that has been compressed into manageable form and revised to emphasize points with current relevance.
Human learning is an interplay of information from several disparate sources with profoundly different characteristics. These sources — heredity, culture, individual experience, and current instruction — must be coordinated for teaching to be successful in promoting learning.
Heredity provides some basic concepts and mental skills, but provides very little of the specific informational content of human thinking. It does supply the mechanisms through which individuals gradually develop that content by interaction with their immediate physical and cultural environment. Much of the physical interaction (e.g., learning to move and see) unfolds along paths that are similar to those of many other species, and the associated mental developments are not unique to humans.
Another dimension of individual mental development, of central importance to humans, is provided by interaction with cultural features. The language of the child’s community is a leading example of such a feature, and language use plays a crucial role in both learning from others and in development of individual thinking. Of particular importance for learning are interactions within each individual between “spontaneous” concepts, which are derived from innate ideas and individual experience, and “nonspontaneous” concepts, which are culturally collected and transmitted (Vygotsky, 1978, 1986).
In addition to learning from those with more knowledge (directly or through artifacts they have produced), people learn from interactions with their peers both in shared work tasks, which provide convincing feedback on the effectiveness of performance, and in play, which for humans consists of activities in which action is made to follow meaning, thus promoting both rule consciousness and collective action (Vygotsky, 1976, Chapter 7). This work/play distinction is significant for the MFM course because the culture of mathematics uses a psychology of play in this sense, which can easily seem alien or frivolous to people with a scientific or practical orientation.[12]
For the teacher or designer of a particular course, all the prior sources of information for its students have to be taken as given, with instructional methods adapted accordingly.[13] This is particularly true for the MFM course, whose students are adults. But while there is no logical necessity that the immediate learning situation coordinated by the MFM teacher replicate the patterns that produced learning from other kinds of sources, it turns out that several strong parallels exist between evolved-systems mechanisms and effective teaching techniques. Of course, the effectiveness of teaching techniques is established by direct classroom experience rather than by correspondence with this or any other theory, but such parallels can be useful in using theory to suggest new techniques and to explain why particular techniques do or don’t work.
Many other aspects of consciousness that I cannot afford to significantly discuss here can be illuminated by similar analyses that take into account the mechanism that I have called coherent precipitation. Creative design is an example, as are paradigm-shift conceptual reorganizations and many elements of personal style, as discussed later. Even the basic process of decision-making has many aspects of this pattern[14] – if a decision is unobvious enough to engage serious conscious attention, its resolution may well emerge from the subconscious by this mechanism. This would explain how it is possible that a decision that had seemed confusing in prospect can be solidly adhered to after it is made – the alternatives fade away, and the chosen plan is refined so that alternatives can’t dislodge it. Thus the mysterious area of free will may have the same explanation as the origin of life, showing the amazing breadth and power of Kauffman’s idea.
But we will now narrow the focus. In the sections below, more detailed consideration is given to the learning theory, both established and derived from evolved-systems theory, that is associated with each of the kinds of information sources listed above.
While hereditary perceptual and mental mechanisms common to almost all humans are of fundamental importance to learning, it is important to realize that they are based on relatively little information[15] . Humans have approximately 40,000 genes, about the same as other mammals and only twice as many as some very simple animals such as nematodes. Only a few thousand[16] human genes differ significantly from their corresponding forms in chimpanzees (with almost all of these differences believed to affect balance between systems that already exist in chimps, rather than to introduce novel structural features).[17] But note that the evolved-systems theory discussed earlier shows that it is possible for quantitative changes in capacity and connectivity to entail qualitative changes in capability and behavior.[18]
Their modest information content implies that genetically-determined aspects of human learning must be concentrated in mechanisms that make good use of non-genetic structures and interactions, and are often shared with other animals. This select group probably includes several capabilities of particular interest to science and mathematics, such as basic ideas of space, time, motion, discrete-object distinction[19] and categorization, counting, connectivity, shape, inference, and causation. The general coherence of these basic math-related concepts results from the fact that they are adaptive responses to various aspects of the same reality, not because they are learned as a culturally-constructed system deduced from axioms.
Philosophers from Plato to Kant have seen such “innate” mathematical ideas as being independent of experience, and thus unquestionably true. A more modern view would be that innate ideas reflect the experience of the species lineage (via natural selection) rather than of the individual, and that innate ideas are often oversimplistic because the mechanisms supporting them are of limited flexibility and power, or because they provide heuristics that worked well enough in simple environments to remove significant selection pressure for further change, but may fail when used in novel or extreme situations. This is why innate concepts, including mathematical ones, so often benefit by being adjusted, extended, and connected by culturally-transmitted ideas and personal experience.
The human cultural repositories of information (such as societal and craft traditions) and tools (including the general mental tools of language and mathematics) have evolved over long periods. Cultural development has many similarities to the evolution of biological species, but also has many profound differences because techniques of action and thought (the units of culture) arise, spread, and combine much more easily than genetically-determined traits.
This is not the place for an extended analysis of cultural development in terms of the evolved-systems concepts described earlier, although several cultural patterns parallel similar patterns in individual learning. Further, while the rapid rate of cultural change characteristic of modern times makes cultural development an active (and often contested) issue in current educational practice, this acts to enhance rather than discredit the status of mathematics education, especially when directed to dealing with engineering concepts. Finally, the nature and behavior of communities is of particular interest in relation to the MFM course because the course is attempting to connect the mathematics and practical-arts communities.
Human culture is characterized by complex interpersonal relations with differentiated roles, dependence on the experiences and assistance of others, use of physical tools and learned procedures to enhance effectiveness of action, and deliberate change of the environment to serve human needs. All these factors are integrated by the use of language and supported by a high degree of mental agility and flexibility compared to other animals. The balance of influence among these factors in enabling physical evolution and cultural development is not clear; it has probably shifted at different times, and the factors are deeply interdependent in any case. In recent centuries the transforming effects of technological development have been prominent, but phenomena like the Internet show how interrelated these factors can be. One assessment of the situation is that development was (and still is) required on all these fronts to take advantage of the vast set of possibilities generated by the combination of earlier techniques and concepts.
Lave & Wenger’s analysis (1991) of the educational interaction of community and learner, which grew out of apprenticeship studies and centers on “legitimate peripheral participation” by learners in a community of practice, unites several of the most useful ideas I have encountered in this area. All three terms in their key phrase have essential implications. Legitimate points out that the participation of the learners is an integral part of the community structure, and thus at least as much an act of the community as of the learner. Peripheral connotes the existence of structure in the community, which includes elements of hierarchy but also means that even full participants have a periphery at which they are dependent on other community members — no one knows it all.[20] Participation implies that action on the tasks of the community is the key to learning, rather than detached observation or disconnected instruction or study.
This practice-centered analysis is generally applicable to educational issues, but has special resonance for the MFM course because of its students’ practical orientations. A key point of the analysis is that participation in a community of practice engages the whole person, with ramifications throughout the personality, implying that instruction directed at such a person must be adapted accordingly. It is particularly important to address the issue of whether a mathematics instructor can be accepted as part of a practical-arts student’s community of practice, which is unlikely unless the instructor is “sponsored” by respected practitioners in the student’s major field, typically the practical-arts faculty. It is felt that the cooperation of these faculty, who are usually members of both scholastic and working communities of practice, will help the MFM course escape from some of the serious difficulties that Lave & Wenger’s analysis implies in the relations between such groups.
The need to coordinate student learning with the knowledge already existing both in practical vocational fields and in the more abstract field of mathematics makes the MFM course particularly sensitive to correctly handling student interaction with the instructional material. One way in which this can be approached is to emphasize the communicative role of mathematics, showing that it enables more effective descriptions and directives in practical situations. A quotation captures both the danger and promise of the interaction between schools and practical-arts students: “For newcomers then the purpose is not to learn from talk as a substitute for legitimate peripheral participation; it is to learn to talk as a key to legitimate peripheral participation.” (Lave & Wenger, 1991, p. 109)
By Vygotsky’s analysis, “learning awakens a variety of internal developmental processes that are able to operate only when the child is interacting with people in his environment and in cooperation with his peers. Once these processes are internalized, they become part of the child's independent developmental achievement.” (1978, p. 90)
The idea of connecting instruction with existing knowledge and non-school activities has received extensive attention in connection with children’s education, where articulation of instruction with the early universal natural learning of language, movement, and social interaction is seen as being