Introductory Map Theory

As an introductory book, this book contains the elementary materials in map theory, including embeddings of a graph, abstract maps, duality, orientable and.
Table of contents

First knowledge learned by rote tends to be quickly forgotten, unless much rehearsed. Second, the knowledge structure or cognitive structure of the learner is not enhanced or modified to clear up faulty ideas. Therefore, to structure large bodies of knowledge requires an orderly sequence of iterations between working memory and long-term memory as new knowledge is being received and processed Anderson, We believe one of the reasons concept mapping is so powerful for the facilitation of meaningful learning is that it serves as a kind of template or scaffold to help to organize knowledge and to structure it, even though the structure must be built up piece by piece with small units of interacting concept and propositional frameworks.

There is still relatively little known about memory processes and how knowledge finally gets incorporated into our brain, but it seems evident from diverse sources of research that our brain works to organize knowledge in hierarchical frameworks and that learning approaches that facilitate this process significantly enhance the learning capability of all learners Bransford et al. Obviously, our brains store more than concepts and propositions. While the latter are the principal elements that make up our knowledge structures and form our cognitive structure in the brain, we pause briefly to discuss other forms of learning.

Iconic learning involves the storage of images of scenes we encounter, people we meet, photos, and a host of other images. These are also referred to as iconic memories Sperling, ; While the alphanumeric images Sperling used in his studies were quickly forgotten, other kinds of images are retained much longer. Our brains have a remarkable capacity for acquiring and retaining visual images of people or photos.

For example, in one study Shepard, presented pictures of common scenes to subjects, and later asked which of two similar pictures shown was one of the seen earlier? This and many other studies have shown that humans have a remarkable ability to recall images, although they soon forget many of the details in the images. Considering how often we look at pennies, it is interesting that the subjects asked to draw a penny in a study by Nickerson and Adams omitted more than half of the features or located them in the wrong place.

We believe that integrating various kind of images into a conceptual framework using concept mapping software like CmapTools described below could enhance iconic memory, and we hope research on this will be done. The learning and recall of sounds is also referred to as archic memory.


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Consider the musician who can play hundreds of songs without reading any music. Again we are dealing with memories that are not coded as concepts or propositions. Studies by Penfield and Perot , among others, indicate that regions of our brain that are activated when we hear sounds are the same regions that are active when we recall sounds.

While we can locate regions of the brain that are active in learning or recall of information using positron emission tomography PET scans, the specific mechanisms by which neurons store this information is not known. A full discussion of memory mechanisms is beyond the scope of this document. He has proposed a Theory of Multiple Intelligences.

His work has received much attention in education and has served to draw attention to the broad range of differences in human abilities for various kinds of learning and performance. It is good that schools are recognizing that there are important human capabilities other than the recall of specific cognitive information so often the only form of learning represented in multiple-choice tests used commonly in schools and corporations.

One reason we encourage the integration of the broad range of activities represented in our New Model for Education is to provide opportunities for these other abilities to be represented and expressed. Nevertheless, we seen the organizing opportunities afforded by associating the various activities with an explicit knowledge structure as very beneficial.

Time will tell if future research studies will support this claim. While it is true that some students have difficulty building concept maps and using these, at least early in their experience, this appears to result primarily from years of rote-mode learning practice in school settings rather than as a result of brain structure differences per se. It is not easy to help students in the former condition move to patterns of learning of the latter type.

While concept maps can help, students also need to be taught something about brain mechanisms and knowledge organization, and this instruction should accompany the use of concept maps. The information in the above paragraphs should become part on the instructional program for skillful use of concept maps. The information provided in this document could be part of this instruction. Other ideas for improving instruction to achieve understanding of the subject is available elsewhere Mintzes et al. To illustrate how difficult it can be for individuals to modify their ideas, especially if they learn primarily by rote, we cite the example of interviews done by the Private Universe Project PUP at Harvard University Schneps, The PUP interviewers found that 21 of the 23 interviewed could not explain why we have seasons, a topic that is taught repeatedly in school.

Included in this group was a graduate who had recently taken a course in the Physics of Planetary Motion, who also believed erroneously that seasons were caused by the earth moving closer to the sun in summer and further away in the winter. In fact, the earth is slightly closer to the sun when it is winter in Massachusetts, rather than in summer. The primary reason we have seasons in latitudes away from the equator is due to the tilt of the earth on its axis toward the sun in summer resulting in longer days and more direct radiation, thus greater heating.

In winter, the axis of the earth points away from the sun, thus resulting in shorter days and less intense radiation. What is interfering with these 21 Harvard people is confusion with the common experience that when we are closer to a fire or lamp, the heat is more intense than when we are further away. Thus, these people have failed to recognize that this same phenomenon is not operating to give seasons on Earth.

They are transferring knowledge from one context to another, but incorrectly. The only solution to the problem of overcoming misconceptions is to help learners learn meaningfully, and using concept maps can be very helpful. For more information on misconceptions in science and mathematics see Novak , and: One representation of the knowledge structure required required for understanding why we have seasons.

As indicated earlier, we defined concept as a perceived regularity or pattern in events or objects, or records of events or objects, designated by label. It is coming to be generally recognized now that the meaningful learning processes described above are the same processes used by scientists and mathematicians, or experts in any discipline, to construct new knowledge.

In fact, Novak has argued that new knowledge creation is nothing more than a relatively high level of meaningful learning accomplished by individuals who have a well organized knowledge structure in the particular area of knowledge, and also a strong emotional commitment to persist in finding new meanings Novak, , , Epistemology is that branch of philosophy that deals with the nature of knowledge and new knowledge creation.

There is an important relationship between the psychology of learning, as we understand it today, and the growing consensus among philosophers and epistemologists that new knowledge creation is a constructive process involving both our knowledge and our emotions or the drive to create new meanings and new ways to represent these meanings. Learners struggling to create good concept maps are themselves engaged in a creative process, and this can be challenging, especially to learners who have spent most of their life learning by rote.

Rote learning contributes very little at best to our knowledge structures, and therefore cannot underlie creative thinking or novel problem solving. As defined above, concepts and propositions are the building blocks for knowledge in any domain. We can use the analogy that concepts are like the atoms of matter and propositions are like the molecules of matter.

There are only around different kinds of atoms, and these make up an infinite number of different kinds of molecules. There are now about , words in the English language most of which are concept labels , and these can be combined to form an infinite number of propositions. Although most combinations of words might be nonsense, there is still the possibility of creating an infinite number of valid and meaningful propositions.

Poets and novelists will never run out of new ideas to express in new ways. We shall never run out of opportunities to create new knowledge! As people create and observe new or existing objects or events, the creative people will continue to create new concents and new knowledge. Creating new methods of observing or recording events usually opens up new opportunities for new knowledge creation.

While there is value in studying more extensively the process of human learning and human knowledge creation, this is beyond the scope of this document. The reader is invited to peruse some of the references cited. Some important considerations for construction of better concept maps and facilitation of learning will be discussed further below.

In learning to construct a concept map, it is important to begin with a domain of knowledge that is very familiar to the person constructing the map. Since concept map structures are dependent on the context in which they will be used, it is best to identify a segment of a text, a laboratory or field activity, or a particular problem or question that one is trying to understand.

This creates a context that will help to determine the hierarchical structure of the concept map. It is also helpful to select a limited domain of knowledge for the first concept maps. A good way to define the context for a concept map is to construct a Focus Question , that is, a question that clearly specifies the problem or issue the concept map should help to resolve. Every concept map responds to a focus question, and a good focus question can lead to a much richer concept map.

When learning to construct concept maps, learners tend to deviate from the focus question and build a concept map that may be related to the domain, but which does not answer the question. It is often stated that the first step to learning about something is to ask the right questions. Given a selected domain and a defined question or problem in this domain, the next step is to identify the key concepts that apply to this domain. Usually 15 to 25 concepts will suffice.

These concepts could be listed, and then from this list a rank ordered list should be established from the most general, most inclusive concept, for this particular problem or situation at the top of the list, to the most specific, least general concept at the bottom of the list. Although this rank order may be only approximate, it helps to begin the process of map construction.

We refer to the list of concepts as a parking lot , since we will move these concepts into the concept map as we determine where they fit in. Some concepts may remain in the parking lot as the map is completed if the mapmaker sees no good connection for these with other concepts in the map. The next step is to construct a preliminary concept map.

Post-its allow a group to work on a whiteboard or butcher paper and to move concepts around easily. This is necessary as one begins to struggle with the process of building a good hierarchical organization. Computer software programs are even better in that they allow moving of concepts together with linking statements and the moving of groups of concepts and links to restructure the map.

Chaos theory

When CmapTools is used in conjunction with a computer projector, two or more individuals can easily collaborate in building a concept map and see changes as they progress in their work. It is important to recognize that a concept map is never finished. After a preliminary map is constructed, it is always necessary to revise this map. Other concepts can be added.

The Map Reader: Theories of Mapping Practice and Cartographic Representation

Good maps usually result from three to many revisions. This is one reason why using computer software is helpful. Once the preliminary map is built , cross-links should be sought.

These are links between concepts in different segments or domains of knowledge on the map that help to illustrate how these domains are related to one another. Cross-links are important in order to show that the learner understands the relationships between the sub-domains in the map. The class identified concepts in the parking lot on the left, but this student was not successful in using many of these and her map makes little sense.

After a preliminary map is constructed, cross-links should be sought. Cross-links are key to show that the learner understands the relationships between the sub-domains in the map. It is important to help students recognize that all concepts are in some way related to one another. Therefore, it is necessary to be selective in identifying cross-links, and to be as precise as possible in identifying linking words that connect concepts.

Figure 6 shows an example of a string map. This is because they poorly understand the relationship between the concepts, or the meanings of the concepts, and it is the linking words that specify this relationship. Once students begin to focus-in on good linking words, and on the identification of good cross-links, they can see that every concept could be related to every other concept.

This also produces some frustration, and they must choose to identify the most prominent and most useful cross-links. This process involves what Bloom identified as high levels of cognitive performance, namely evaluation and synthesis of knowledge.

Concept mapping is an easy way to encourage very high levels of cognitive performance, when the process is done well. This is one reason concept mapping can also be a very powerful evaluation tool Edmondson, Thus, we see that concept maps are not only a powerful tool for capturing, representing, and archiving knowledge of individuals, but also a powerful tool to create new knowledge.

The software not only makes it easy for users of all ages to construct and modify concept maps in a similar way that a word processor makes it easy to write text, it allows users to collaborate at a distance in the construction in their maps, publish their concept maps so anybody on the Internet can access them, link resources to their maps to further explain their contents, and search the WWW for information related to the map. The software allows the user to link resources photos, images, graphs, videos, charts, tables, texts, WWW pages or other concept maps located anywhere on the Internet or in personal files to concepts or linking words in a concept map through a simple drag-and-drop operation.

Links to these resources are displayed as icons underneath the concepts, as shown in Figure 7. Clicking on one of these icons will display a list of links from which the user can select to open the linked resource. Using CmapTools, it is possible to use concept maps to access any material that can be presented digitally, including materials prepared by the mapmaker. In this way, concept maps can serve as the indexing and navigational tools for complex domains of knowledge, as will be illustrated later with NASA materials on Mars Briggs et al. A concept map about birds constructed by a high-school student.

Icons under the concepts provide links to resources e. There is a growing body of research that shows that when students work in small groups and cooperate in striving to learn subject matter, positive cognitive and affective outcomes result Johnson et al. Vygotsky introduced the idea that language and social dialogue can support learning, especially when members of the social group are at about the same Zone of Proximal Development ZPD. When students work cooperatively in groups and use concept maps to guide their learning, significantly greater learning occurs Preszler, In our work with both teachers and students, small groups working cooperatively to construct concept maps have proven to be useful in many contexts.

In our own classes and workshops, and in classes taught by our students and colleagues, small groups of students working collectively to construct concept maps can produce some remarkably good maps. CmapTools provides extensive support for collaborative work during concept map construction. The concept maps built using CmapTools can be stored on servers CmapServers, see: Through CmapServers, users of all ages and working in many disciplines have published thousands of maps on all topics and domains.

While concept maps on these public servers are only a sample of concept maps submitted by persons using CmapTools, and some do not meet our criteria of good concept maps, they nevertheless serve to illustrate diverse applications. Through the storing of concept maps in CmapServers, CmapTools encourages collaboration among users constructing the maps.

The high degree of explicitness of concept maps makes them an ideal vehicle for exchange of ideas or for the collaborative construction of new knowledge. We have also found that the obstacles deriving from personal insecurities and fear of embarrassment are largely circumvented, since critical comments are directed at the concept map, not at the person s building the map. The extensive support that CmapTools provides for the collaborative construction of concept maps by groups, whether they are at the same location or in distant locations, has encouraged the increasing use of collaboration during map building.

In a variety of educational settings, concept mapping in small groups has served us well in tasks as diverse as understanding ideas in assimilation learning theory to clarifying job conflicts for conflict resolution in profit and non-profit corporations e. Concept maps are now beginning to be used in corporations to help teams clarify and articulate the knowledge needed to solve problems ranging from the design of new products to marketing to administrative problem resolution.

Chaos theory - Wikipedia

In addition to a network environment that fosters collaboration and the possibility of constructing knowledge models, the software allows users, among other features, to a search for information based on a concept map Carvalho et al. The concept map can thus become an artifact around which the various activities of the learning process can be centered, as shown in Figure 8.

A concept map-centered learning environment implies that concept maps are used throughout the development of a learning unit or module. Concept maps within this environment are likely to be used as the mechanism to determine the level of understanding students have about the topic being studied before the topic is introduced.

The maps are then developed, extended and refined as the students develop other activities on the topic and increase their understanding, possibly concluding with complex knowledge models that link resources, results, experiments, etc. The whole spectrum of learning activities can be integrated using CmapTools, incorporating various learning activities recorded via the software creating a digital portfolio as a product of the learning.

Each student can construct the initial concept map individually, giving the teacher feedback on the level of understanding of every student. The concept map can be constructed by students working in couples or small groups, where the teacher must pay attention to the level of participation of every student.

Description

CmapTools has a recorder feature tht allows recording and playback of steps in map construction, including identification of each contributor. The concept map can also be a class effort, using a projector, where all students give their opinion and participate in the construction of the map. Would you like to change to the United States site? The Map Reader brings together, for the first time, classic and hard-to-find articles on mapping. This book provides a wide-ranging and coherent edited compendium of key scholarly writing about the changing nature of cartography over the last half century.

The editorial selection of fifty-four theoretical and thought provoking texts demonstrates how cartography works as a powerful representational form and explores how different mapping practices have been conceptualised in particular scholarly contexts. Themes covered include paradigms, politics, people, aesthetics and technology.

Original interpretative essays set the literature into intellectual context within these themes. Excerpts are drawn from leading scholars and researchers in a range of cognate fields including: The Map Reader provides a new unique single source reference to the essential literature in the cartographic field:. Request permission to reuse content from this site.

Robinson and Barbara B. Advancing the Agenda Alan M. MacEachren and Menno-Jan Kraak. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits. One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour; there is no chaotic system involving only two first-order, ordinary differential equations the system resulting in an equation of second order only.

Here, A is an adjustable parameter. The output of op amp 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative. Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model , four conditions suffice to produce synchronization in a chaotic system.

Examples include the coupled oscillation of Christiaan Huygens ' pendulums, fireflies, neurons , the London Millennium Bridge resonance, and large arrays of Josephson junctions. In the s, while studying the three-body problem , he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. Chaos theory began in the field of ergodic theory. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory , the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map.

What had been attributed to measure imprecision and simple " noise " was considered by chaos theorists as a full component of the studied systems. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand.

Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi 's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, , what he called "randomly transitional phenomena".

Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation.

To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0. This difference is tiny, and the consensus at the time would have been that it should have no practical effect.

However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.


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  8. In , Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. In , he published " How long is the coast of Britain? Statistical self-similarity and fractional dimension ", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device. In , Mandelbrot published The Fractal Geometry of Nature , which became a classic of chaos theory. Yorke coiner of the term "chaos" as used in mathematics , Robert Shaw , and the meteorologist Edward Lorenz.

    The following year, independently Pierre Coullet and Charles Tresser with the article "Iterations d'endomorphismes et groupe de renormalisation" and Mitchell Feigenbaum with the article "Quantitative Universality for a Class of Nonlinear Transformations" described logistic maps. In , Albert J. Feigenbaum for their inspiring achievements.

    There, Bernardo Huberman presented a mathematical model of the eye tracking disorder among schizophrenics. In , Per Bak , Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters [59] describing for the first time self-organized criticality SOC , considered one of the mechanisms by which complexity arises in nature. Alongside largely lab-based approaches such as the Bak—Tang—Wiesenfeld sandpile , many other investigations have focused on large-scale natural or social systems that are known or suspected to display scale-invariant behavior. Although these approaches were not always welcomed at least initially by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including earthquakes , which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg—Richter law describing the statistical distribution of earthquake sizes, and the Omori law [60] describing the frequency of aftershocks , solar flares , fluctuations in economic systems such as financial markets references to SOC are common in econophysics , landscape formation , forest fires , landslides , epidemics , and biological evolution where SOC has been invoked, for example, as the dynamical mechanism behind the theory of " punctuated equilibria " put forward by Niles Eldredge and Stephen Jay Gould.

    Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. In the same year, James Gleick published Chaos: Making a New Science , which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, though his history under-emphasized important Soviet contributions.

    Alluding to Thomas Kuhn 's concept of a paradigm shift exposed in The Structure of Scientific Revolutions , many "chaologists" as some described themselves claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, [62] involving many different disciplines mathematics , topology , physics , [63] social systems , population modeling , biology , meteorology , astrophysics , information theory , computational neuroscience , etc.

    Chaos theory was born from observing weather patterns, but it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are geology , mathematics , microbiology , biology , computer science , economics , [65] [66] [67] engineering , [68] finance , [69] [70] algorithmic trading , [71] [72] [73] meteorology , philosophy , anthropology , [11] [12] physics , [74] [75] [76] politics , population dynamics , [77] psychology , [10] and robotics.

    A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing. Chaos theory has been used for many years in cryptography. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of cryptographic primitives. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers, watermarking and steganography. Robotics is another area that has recently benefited from chaos theory.

    Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model. For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous , but recently scientists have been able to implement chaotic models in certain populations. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory.

    Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of fetal hypoxia can be obtained through chaotic modeling. In chemistry, predicting gas solubility is essential to manufacturing polymers , but models using particle swarm optimization PSO tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck. In quantum physics and electrical engineering , the study of large arrays of Josephson junctions benefitted greatly from chaos theory.

    Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately. Glass [95] and Mandell and Selz [96] have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior. Researchers have continued to apply chaos theory to psychology.

    For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion 's theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: Redington and Reidbord attempted to demonstrate that the human heart could display chaotic traits.

    They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics spectral analysis, phase trajectory, and autocorrelation plots , but when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.

    In their paper, Metcalf and Allen [99] maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented.

    The control parameter r operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r. Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased.