MFM Theory Background

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.

Influences from evolved-systems theory

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.

Development of complex evolved systems

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.

Overflowing variety from recombination

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.

Connectivity and phase transitions

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 & R�nyi, 1960). This phenomenon can explain many situations in which small quantitative changes have effects that are so large as to be seen as qualitative.

Memory records

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.

Work cycles

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.

Winnowing and Coherent Precipitation

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.

Specialization

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.

Concept development

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.

Phylogenetic Learning: The uneven foundation

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.

Cultural Learning: The heritage of communities

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)

Individual History: Students are already (distinct) people

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 essential to fostering constructive engagement by students with the more abstract material that is dealt with in schools.

But to create an instructional context in which adult students can dependably succeed, allowance must also be made for the fact that adult activities and interests are strikingly diverse, in contrast to the broad similarity of early learning in all children. Respect for differences in origins and goals is particularly important in designing instruction for the diverse students in the MFM course. How this can be accomplished is examined in the instruction-focused section that follows, and in the actual trial instructional materials.

Implications for instruction

Street learning and school learning

Since the clearly-expressed goal of both students and society for practical-arts instruction is preparation of the students for effective action in the practical domains they are entering, information about what effects school instruction has on the ability of practical workers to deal with the situations they actually encounter �on the street� is of great interest. A series of well-constructed studies by Nunes and her collaborators (1993), in which the mathematics-related methods and capabilities of schooled and unschooled Brazilian children, self-employed artisans, and microbusiness owners were compared, provides several points of interest for the MFM course.

The first point is that computation in practical situations was often done by algorithms other than the standard models taught in school. For example, repeated and/or grouped addition was often used instead of multiplication in determining the total price for a purchase of several of the same item. It seems likely that this is done not simply because of ignorance of standard procedures (even schooled people generally used the typical practical method of their field, not the school-taught one), but because using an often-used, intuitively-checkable method to handle the relatively few cases of large purchases is in fact a more robust process. This implies that the MFM course should be careful not to scorn practical methods that seem inferior to mathematical ones � a better strategy is to explain and extend them (and to be alert for non-robustness in mathematical procedures).

A second point is that people showed much more competence in solving problems orally than in written form. This was true of schooled and unschooled people. Various possible explanations were investigated, and the one best supported by the evidence was that a disabling loss of meaning occurred for most people when the problem was reduced to written form. The subsequent computation thus lost corrective feedback from the person�s knowledge of the structure of the situation, leading to errors or dead ends. This seems to me to be of great relevance to the MFM course, since it touches on one of the main sources of alienation by practical-arts students, who often combine substantial informal competence with weak school-mathematics skills.

In fact, preservation of meaning was a central theme of the findings in several parts of the study. Practical workers needed to do so to solve the problems at all; they would give up if meaning was lost, but would seldom assert unreasonable answers. Students were more willing to continue, but were much more likely to produce erroneous results. The lesson for the MFM course is to support methods (such as consistently labeling numbers with the appropriate engineering units, and using written narrative to connect parts of a problem) that help preserve meaning, and to scrupulously ensure that problems describe realistic situations.

The study also showed that many practical workers could successfully transfer a computational skill (such as solution of problems involving quality-quantity-price relations) to an isomorphic problem domain. But an essential part of the transfer was for the workers to see that the pattern of meaning was isomorphic � if the problem was reduced to written form and handled just as an arithmetic problem, success was much lower. This suggests that for practical-arts workers (and no doubt many others), analogy may be a more effective mode of generalization than abstraction. This is why graph-based modeling (and graphs in general) will so often be more successful in this context than algebraic methods.

Overall, this study reinforces several of the design principles of the MFM course. Practical workers have existing, rational quantitative skills that are based on the meaning of the situations to which they are applied. Their need is to have these skills expanded, extended, and explained � not replaced.

Stages of learning

Vygotsky (1978, 1986) distinguished between �spontaneous� concepts (which arise without explicit instruction in response to direct experience, and thus start with familiar meanings but weak connections to other concepts) and �nonspontaneous� concepts provided by instruction (which start with good connections to other concepts, since that is how they are introduced, but have only vague meanings when first used). This �spontaneous/nonspontaneous� distinction can alternatively be expressed as �everyday/scientific�, or �individual/cultural�� the critical idea here is the contrast between inward and outward reliance for concept meaning.

This is not a judgmental division of concept classes into �authentic� and �imposed� (or, to take the other side of that poorly-framed argument, into �na�ve� and �professional�). Instead, Vygotsky perceived the productive interaction between the two kinds of concepts as learners use them in their communication with others and in thinking about and extending their experiences. Spontaneous concepts acquire more structural connections (via abstract intermediary concepts introduced by instruction) while such nonspontaneous concepts acquire more connotative connections to immediate perceptions (via observations of the concept's effect in use, especially when it touches familiar concrete objects). Thus the two concept-development processes reinforce each other rather than compete, transforming concepts from both sources into mature integrated concepts that are well connected both to other concepts and to experience and innate capabilities.

This concept transformation is an ongoing process, with each round of synthesis adding to an individual�s set of mastered concepts, which then can be used to interact^[21] with more advanced culturally-supplied concepts, leading to further synthesis and mastery. As the pool of mastered concepts expands, the number of connections between concepts grows even faster, since concepts of greater connective power become accessible and earlier connective concepts can be applied to already-mastered concepts other than those by which they were first introduced.

Linkage-driven processes of this kind can be expected to exhibit nonlinear shifts in power and behavior at a critical threshold in connectedness, a systems-theory finding described above that illuminates the long-observed existence of qualitative stages in the intellectual development of children. Hall (1904), Piaget (e.g., 1952), and Vygotsky (1978) give classic education-centered observations, but the idea is also implicit in the coming-of-age rituals (e.g., Christian first communion and confirmation) long common to all cultures. That such stages also obtain in adult learning is suggested by the tradition of apprentice/journeyman/master distinctions in skilled-craft work, as well as by the abrupt increases of capability people frequently exhibit during the process of learning particular technical skills (the classic example is learning to ride a bicycle).

Although stage transitions for some sectors may occur at predictable points in the accumulation of experience, learning is seldom simply cumulative. Expert thinking and behavior patterns usually replace (or at least substantially alter) those learned as novices, rather than just being added to them, largely because the expanded web of connective concepts reveals the contradictions that the na�ve ideas have with each other or with well-established related concepts. Much of the qualitative, �quantum jump� appearance of stage transitions can be explained by such feedback effects, which will often occur in cascades as concept transformation propagates until a condition without salient contradictions is reached.

The specific implication for the MFM course of these general principles is that the course�s topic choices should be designed to promote the establishment and extension of a well-connected conceptual framework, so as to facilitate the mutual transformation of related practical ideas and the use of existing practical knowledge to give meaning to the mathematical ideas used to examine it. This in turn implies that topics with a broad range of practical applications (e.g., right triangles, error sensitivity) are to be preferred to those, however elegant or powerful in their sphere, that provide only a few occasions for practical use. The value of the strategy of �increasingly sophisticated use of simple techniques� is thus explained, since it is the simple techniques that provide the wide-ranging connecting threads that lay the groundwork for the intellectual stage transition that results from development of a mathematical perspective.

Style and specialization: communities of practice

Mastery of an area of activity often entails a substantial reduction in the range of component actions (although not in the range of effects produced) compared with those known to be possible. Both particular communities of practice and master practitioners within them have distinctive styles that use only a fraction of the available methods and tools. One reason that this selectivity pays off is that along with its benefits, each physical or mental tool also has costs (in its associated maintenance and storage as well as in the initial construction and training).

But a more fundamental reason for a comparatively lean toolkit is that when the number of tasks becomes large, the strategy of developing additional tools (or words^[22] , or concepts) soon becomes much less fruitful than the strategy of developing more methods to use the tools already possessed in additional ways and combinations^[23] (such as techniques, sentences, or theories). This is especially true if a modest-sized set of connection/extension methods can be used repeatedly (and preferably recursively) on many different targets; such conceptual and procedural �meta-tools� underlie much of the power of human thought, and are used extensively throughout language^[24] , science, and mathematics.

This combinatorial strategy is often so prolific that it produces an embarrassment of riches, with an explosion of concept and technique formation that exceeds the retention capacity of any individual or coordinated group and, since many of the inventions are approximately functionally equivalent (or can be assembled into functionally-equivalent sets), the needs of any economy. Two types of solutions for this superfluity develop, both of which have practical implications for the MFM course.

One solution is specialization. If the tasks to be accomplished are partitioned among a variety of communities of practice in a society, each can concentrate on mastery of the subset of knowledge and skills needed for success in its sector. While there are some costs (mainly due to more difficult communication), the increasing dominance of highly-specialized urban and regional economies shows that current human conditions are very favorable to this approach. Both the newer scientific/professional specializations and the craft traditions of which modern practical work is the heir show that current society�s pervasive specialization is the culmination of a long-established trend, and is a natural pattern for social entities to take advantage of.^[25]

The other solution is winnowing, removing extraneous methods and concepts (but none of the essential ones!) from the overflowing collection produced by a combinatorial approach. As explained earlier, in some cases well-articulated complex systems can arise much more easily from such thinning than from building each component individually.

The winnowed complex will include particular solutions for the many tasks for which numerous different solutions were possible (the costs of maintaining distinct solutions will eliminate most duplicates from the winnowed set, yielding an efficient coverage of the task space). The exact form of the surviving solution may be determined by chance or by small relative advantages with respect to the set of tasks of that community of practice, but the ease and safety of copying compared to innovation^[26] will strongly promote the propagation of that particular form within the tradition once established (perhaps with suitable gradual adaptive modifications, with corresponding adjustments to related methods or concepts).

These two processes, specialization and winnowing, interact to produce the coherent and efficient^[27] technical language and methods characteristic of communities of practice. Of particular relevance to the MFM course is the fact that these processes mean that different communities of practice may well end up with different methods to solve identical problems, and even to use different concepts to characterize them. (Variation in technical terminology will also be frequent, but is much less problematic.) While such differences in thinking may exist even between communities of practice with the same or related tasks in different societies, they are particularly to be expected between practical-arts communities and those of mathematics practice, whose motivations and standards of value differ profoundly. This is why the topics and techniques used in the MFM course must be chosen with great care, since they need to be authentically mathematical but also positioned within the reach (i.e., inside the zone of proximal development [Vygotsky, 1978]) of all the communities of practice from which the course draws its students.

Style and specialization: individuals

The development of technical language and techniques, which occurs largely at the scale of communities of practice, provides most of the nonspontaneous concepts that are used in instruction in the related areas. But since the central issue of instruction is the development of individual mastery, the question arises of whether individual style and self-selected specialization are also advantageous for learners, or are instead unproductive diversions of the learner from the task of thoroughly assimilating the ideas and methods that have already been worked out by the culture.

That this question was one of the main battlegrounds of American educational policy during the 20^th century suggests that there is much to be said on both sides. Indeed, there are more than two sides to the question of support for student individuality, since it can be posed separately with respect to the content of the curriculum and to the learning methods that schools are organized to promote. I would characterize the general trend of American public education in the last half of the 20^th century as paying more respect to individuality in learning (partly because the higher goals set for universality of education cannot otherwise be met) but as narrowing curriculum diversity^[28] , with what amounts to a single pathway toward educational success laid out through the first year or two of college (�Onward to calculus!�).^[29] While some provision is made to provide courses for those who can�t or won�t follow the main track, these are clearly considered consolation prizes rather than well-respected alternative paths. A secondary-school student whose interests and learning style are oriented toward practical activities gets much less support now than in 1950.

There is movement to counter this trend. While the specialized �magnet� secondary schools (and the single-school honors-course equivalents) have proliferated mainly been along conventional arts/humanities and math/science lines (and reach only a small fraction of total students), the practical information-technology sector has enough prestige to be included in some cases. At the post-secondary level, the rapid rise of certificates as alternatives to degrees has permitted many practically-oriented people to finally get access to the training they desire. A further strong practical influence, also particularly prominent in community colleges, is the great increase in �life-long learning� reenrollments, which bring back into schools people with enough experience in employment-based communities of practice to recognize authentic learning and mastery (or its absence).

The MFM course was designed to encourage and participate in this counterflow. While enhancement of student ability to use some of the generalized tools that comprise mathematics is among the course objectives, the main purpose of the course is to promote effective student specialization by enabling them to more fully participate in the higher-level discourse of their respective chosen fields.

Although the issue of individuality in professional style is not directly addressed by the MFM course, the variety of professional orientations (and of depth of professional background) among its students makes it essential that substantial diversity of learning styles be supported. This was felt by the course authors to be good practice in any case, with the recent reform curricula^[30] in statistics and calculus being used as models. The unavoidable diversity of student styles is also taken advantage of as one explanation given to students as to why mathematical methods (which are generalized, although practical in orientation) must be used in the course rather than methods that are completely tailored to particular application areas.

Instructional tactics: Learning at the edge of confusion

Natural learning

Methods analogous to adaptive natural selection can be used to learn the kind of techniques that work better and better as each small change is made. Many physical adjustment processes are of this kind, with immediate feedback indicating the direction to modify the current situation in order to improve it. Using the same approach in less obvious areas usually just requires that attention be directed to the appropriate perception to use for feedback. People (and most other animals) have great natural skill at solving problems of this kind. Structured instruction is not needed in such cases.

In addition to providing a foundation of basic skills that are learned without being taught (except by experience), natural learning provides a style of learning from perception that recurs at higher levels. It is mainly because understanding extends the learner�s perceptive range that it is so valuable compared to simple knowledge � with understanding, learners are enabled to see for themselves in a new area, and can thus fruitfully draw again on many of the patterns they mastered as young children (and are able to look at and assess phenomena directly rather than depending on memory or authority).

Immediate-benefit instruction

However, natural learning does not work dependably (or at all, in some cases) if there is not informative feedback from small changes, and much of human culture consists of knowledge and techniques to which natural learning provides no path. But in many of these cases, an improvement is evident after a single step within the immediate capability of the learner if that step is taken in just the right direction. These are the cases where direct instruction is valuable, simple, and welcome. Questions that can be answered without introducing new concepts and proofs that require only a single, but non-obvious, logical step^[31] are examples of this category.

A substantial amount of culturally-transmitted information is routinely transmitted in this direct way, especially in the early stages of learning. This can easily create a situation in which there are expectations that all instruction will be as simple and as immediately satisfying. But in fact a great many very useful techniques cannot be learned in either the natural or the direct ways, because too many steps must be taken (or too many separate steps coordinated) before productive use can be made of the new technique. It is in these cases that instruction becomes both essential and problematic. And yet because things suitable for being taught by the other categories are so easily learned, it is precisely these unobvious techniques that are the target of most school-based instructional activity, especially after the elementary level.

Strategies for complex-topic instruction

There are two main instructional strategies used to promote learning of complex topics of this kind. Their proponents might characterize them respectively as �providing a clear path� and �teaching for understanding�. While each has advantages and disadvantages, and each depends on the (often unacknowledged) use of the other, they lead to significant differences in both goals and tactics. Because much of the policy-level debate about instruction is over which of these approaches to emphasize, any added insight into their nature is of potential use. My assertion below is that evolved-system theory provides a new perspective on what is actually happening when the teaching-for-understanding approach is used.

One step at a time

The first strategy is to create artificial goals spaced at achievable steps along the desired path (or as close to that path as can be managed) leading to the full technique that is the ultimate target. Single-step instruction is then directed at getting students to learn these intermediate goals. This you�ll-need-it-later, one-section-at-a-time approach dominates many areas of mathematics instruction, especially at the lower levels. Its advantage is that it provides a clear and predictable path, making it easy for both student and teacher to assess progress.

It also has important disadvantages: (1) it undermines student sensitivity to the authentic feedback for the final technique (since attention is directed to the artificial goals, which are not of real importance), and (2) many techniques require that several separate things be learned in coordination; it may then be impossible to construct the needed intermediate steps, and when separate paths are constructed their artificiality may fatally impede student retention during the time instruction is being provided on the other paths (thus necessitating constant review).

A further higher-level potential disadvantage is that this approach, especially if it is used exclusively, may cause the instructional discipline to choose its target topics based on how easy they are to teach this way, rather than on their utility; this is particularly a danger when schools are disconnected from the relevant communities of working practice. The one-section-at-a-time pattern also leads students to expect that all problems are to be solved in terms of a recently-discussed template � this is limiting at all levels, and becomes completely disabling at the college level.

A final potential disadvantage is that this pattern of teaching tends to ignore the fact that student learning about topics that are of real use (and thus recur in various forms in later lessons and actions) will continue over time, as the topic becomes better connected with others. In extreme cases, test-as-you-go assessment may be used as the reason for blocking student progress to working with the collateral uses of a topic even though such alternative modes of interaction are the very experiences that would lead the student to mastery of it.

Confusion is the price of understanding

The second strategy rejects the preference of continuous student clarity over authenticity. It accepts that students will be confused during much of the learning process. Indeed, it deliberately promotes a confusing variety of related experience and then uses the �aha!� felt when the confusion is resolved as one of its main modes of feedback to convince the student that something new and important has been mastered. This understanding-oriented approach is using the �coherent precipitation� evolved-systems mechanism described earlier, with the ideas that support each other and accord with experience remaining after the fading away of the ideas (perhaps including several erroneous preconceived ones) that are isolated or in disagreement.

The main advantages of the understanding-oriented approach are: (1) it strengthens rather than weakens student sensitivity to the essential aspects of the topic (and implicitly distinguishes the essentials from the incidentals), (2) the absence of a prescribed learning path permits students to learn according to their own style and skills, and (3) students get experience in the largely-undirected learning that must be mastered for mature competence and creative thought. Instruction is largely by way of well-chosen examples and anecdotes (including ones to guard against precipitation around an incorrect solution), not by immediate focus on explicit description of the target procedure or of the theory behind it.

The main potential disadvantages are (1) getting everyone past the confusion may take an inordinate amount of instructional time, (2) the lack of clear intermediate goals increases risk that students who are not productively engaged will not be noticed and helped, and (3) students who are not used to this approach will be intensely uncomfortable, and often even angry, with the confusion it entails (this may in turn impede their learning).

Although it is reasonable to take care that each teaching area makes good use of the limited amount of time available, the risk of inefficiency is not as great at it might seem. For one thing, quickly pretending to learn something is even less efficient than actually learning it slowly (provided that it really needs to be learned at all � the saving grace of many poorly-taught curricula is that people can easily do without the knowledge^[32] ). And because students are being trained to continue to function and explore even in the face of confusion, they are not disabled when some areas remain obscure for a while. They will eventually have both the standard answer to use as a focal point for their attention and subsequent related topics to help resolve the issue from other directions. For all students, the ideas related to important topics continue to develop � the first �aha� is not the end of the learning process.

The concern about students getting lost is more valid, and poses serious challenges (that have been vigorously addressed, if not to everyone�s satisfaction) in curriculum design and in teacher training. The performance of marginal students under the one-section-at-a-time approach is also unsatisfactory, however, so this area needs attention in any case. The typical hybrid strategy mentioned below, in which most student time is spent in clear activities (where disorientation or disengagement can be detected) but with these activities chosen and arranged to promote conceptual breakthroughs, will do much to ensure that everyone makes progress. Some modes of instruction may need special attention � distance learning courses, for example, lack the teacher-watching-student-expressions mechanism used to such good effect in classrooms.

The last objection cited, that students unfamiliar with the understanding-oriented process will resent the confusion and find it disabling, requires a steady effort to train students to use this kind of approach. The approach is not actually novel to them, since they use it extensively in non-school learning (it is the idea behind �Just do it!�, and is an essential element of dramatic storytelling^[33] ). It will be much easier to apply this approach to schoolwork if a systematic effort is made throughout the school, and broader success with it can be expected in that case. But something of the kind already has to be used in courses for which a solid foundation of understanding is essential for continued progress, which it is to be hoped includes most college courses. In the standard mathematics sequence, for example, the apply-the-right-template approach becomes clearly inadequate by about the college-algebra level. In the MFM course, the strategy is to relate mathematics to the understanding that students already have in their specialties � the �aha!� will be on the realization that math can be useful, not on fine points of the math itself.

Thus Kauffman�s �order at the edge of chaos� (1993) implies a parallel learning at the edge of confusion. As in the earlier case, continued development (although not maximal activity) will occur best just inside that boundary in the more orderly direction, where �islands of stability� form an anchor for the creative process. Supporting learners through the confusion stage is one of the reasons that instruction is needed, just as parental care is needed during the extended non-self-supporting period of human childhood. In both cases, the payoff for the period of relative helplessness is capabilities much greater than those that would otherwise be attainable.^[34]

Combining confusion and clarity

The selection between the one-section-at-a-time and understanding-oriented strategies is not really as clear-cut as is implied by the above presentation. This is well understood in educational practice, where the typical understanding-oriented curriculum provides many clear activities that promote exploration of the topic area even if they do not provide a fixed path. But a distinction is maintained between the incidental experiences and the conceptual framework, achieved by various paths, that reflects understanding.

Even when the one-section-at-a-time approach is the only one that is intentionally used, the actual learning of the more complex topics will generally occur by means of a coherent-precipitation process experienced as sudden understanding. This is one of the main modes by which people learn, whether they or their teachers know it or not.

Thus in many ways the evolved-systems analysis applied here to learning does not immediately show the way to new instructional practices. Appropriate combinations of teaching techniques may already have been worked out, at least approximately. In fact, it is likely that careful study of effective instructional techniques will be a fruitful source of information about the characteristics of the complex systems of human thinking and culture. This is a normal situation in the early stages of scientific explanation of an area of established practice � the artisans are more aware of the nuances than the savants are.

But the general experience of science and engineering indicates that knowing how what you are doing works can eventually lead to great improvements in practical techniques, and that scientific perspectives (when used non-dogmatically) can productively enrich professional discourse even while incomplete. Theories with broad applicability can also aid in rhetorical tasks � this evolved-system analysis may be useful in showing teachers (and students, and school boards) why confusion is an essential and appropriate component of understanding-based approaches, rather than an embarrassing flaw.

Theoretical background for the MFM course a component of the Mathematics For Measurement (MFM) thesis

Overflowing variety from recombination

Theoretical background for the MFM course
a component of the Mathematics For Measurement (MFM) thesis