Peter Zelchenko
Prof. Vandervoort
June 18, 2010
Teaching Chaos Theory to the Adult Student
(Part II of “The Steep Slope to Understanding Chaos”)
In the first paper in this series, I described difficulties in reading and learning about
chaos: “The deficiencies in [the texts by Ian] Stewart and [Edward] Lorenz,” I wrote, “lie at the
threshold between the mathematical and the visual-spatial” (Zelchenko 2010).1 I also emphasized
how this relates to what I consider a “crucial bridge to understanding the systematic nature
underlying chaos,” that of closely understanding the link between mathematical and visual in
phase space.
The age of visual learning
I was suggesting that bridging the gaps in understanding chaos requires close attention to
translating physical and mathematical concepts into visual-spatial information wherever possible.
The 1950’s and 1960’s saw an explosion in the acceptance of “programmed learning”
technologies to emphasize visual learning (see, e.g., Zelchenko 1999). Filmstrip technologies
were definitively audiovisual, and this was considered to be a partial improvement over the
static, flat, silent nature of books. Beginning in the late 1960’s, computer learning systems
became popular. Although integrating auditory information using computers has actually taken a
step backward since filmstrips, and motion video is still also not a regular or well-regulated
fixture in education, the main advantage to any of these technologies is their marginally
enhanced exchange with the student. What desktop computers are now both mature for and
1
The intent of this paper and the previous one is not to indict any of the authors or the instructor. Indeed, chaos
teaching must explain to a popular audience the broad and ineffable abstractions of an as yet incomplete science.
One must congratulate the authors, but especially the instructor, for their efforts in this. In the teaching arts, there
may be no higher Everest to climb.
Zelchenko p. 1
uniquely suited to is providing an interactive visual experience. I believe this can be a crucial
component in learning chaos.
This is never to suggest that every teaching opportunity will benefit from full-motion
interactive visuals. There has been a mountain of literature over the years emphasizing that
interactive visual aids are not for all things (see, for example, Alesandrini 1984; Bennett 1981;
Lentz, 1982). On the other hand, numerous tests have demonstrated its effectiveness in certain
situations, particularly when illustrating a text (e.g., Alesandrini 1984, p. 65; Lentz 1982, p.
225).2 Alesandrini (p. 65) cited a study that suggested certain abstract concepts in mathematics
presented to adults as visual information did not facilitate science learning as unequivocally as
more concrete concepts. Researchers add that any positive effect is specific to the
appropriateness of the information provided (Lentz 1982, p. 226). Alesandrini offers a summary
of Dwyer (1978) that is particularly relevant to our discussion:
“Dwyer concluded that when visuals are used to supplement verbal information that
the learner is already familiar with, no facilitation will occur. On the other hand, if the
material to be learned is too complex, presenting a realistic visual may not facilitate
learning either. For certain learning outcomes, visual presentation can be facilitative.
For example, presenting a realistic visual can aid learning if the learning outcome
involves requiring the learner to draw or otherwise identify location and
interrelations among parts, specific patterns or functions, or content relationships”
(pp. 64-65). (Emphasis added.)
Readers familiar with the texts under review (Gleick 1987, Stewart 1989, Lorenz 1993)
and considering the above conclusion will recognize that certain aspects of the chaos-learning
material on the one hand may be too complex for some adult learners, despite often extraordinary
efforts of the authors and instructors to provide quality descriptive text and illustrations; and yet
there may also be opportunities for greater success if learners find ways to interact more with the
“When illustrations provide text-redundant information, learning information in the text that is also shown in
pictures will be facilitated” (p. 225). “...The positive learning effects of illustrations are specific to the information
provided by them” (p. 226). “Illustrations can help learners understand what they read, can help learners remember
what they read, and can perform a variety of other instructional functions.” (ibid.).
2
Zelchenko p. 2
visuals. Alesandrini (pp. 65-66) emphasized that concrete models and manipulatives were
particularly successful for science learning by learners who had reached the Piagetian stage of
formal operations — in the case of that study, high-school students.
Survey
A path to understanding chaos theory, as logically presented by two of the authors,
attempts to run from a starting point of sensitive dependency on initial conditions within a
dynamical system and concluding at complex attractors like the Mandelbrot set. In Stewart and
Lorenz, there is a certain continuity of the topics that permits us to approximate one curricular
order of presentation as a single ramp from rudimentary to advanced concepts, at least for this
discussion, so as to see where high and low points of mastery occur. Gleick’s order is different,
not to say inferior; but the other two present a more obvious logical progression.
I predicted that students studying chaos theory would come to similar learning impasses
from various conceptual gaps, and specifically that gaps would be seen at or near the following
places (although not exclusively these places):
1. between mastering (a) the general concepts relating deterministic versus
stochastic behavior, dynamical systems, differential equations, and obedience to
physical laws associated with given dynamical systems, and (b) those concepts’
association with example functions that specifically demonstrate chaotic behavior
under iteration;
2. between understanding (a) the single-pendulum attractor and (b) the more
counterintuitive functions underlying a compacted dynamical system like
Lorenz’s sled model and how it relates to the shape of its attractor;
Zelchenko p. 3
3. between comprehending relationships among (a) Poincaré’s visualization of phase
portraits and (b) the numerous other representations of phase spaces drawn from
the several disciplines;
4. between understanding (a) the stretching-and-folding transforms that these phase
spaces experience as the systems progress through time and (b) the related
mathematical concepts of period doubling and bifurcation; and,
5. understanding how (a) dynamical systems, attractors, and chaos theory in general
relate to (b) fractality, complex fractal forms, and attractors based on fractals, or
whether they relate at all.
Near the end of the quarter, I distributed a survey whose results are in the Appendix to
this paper. Two disclosures that I must make before reviewing the data: first, I am informant [4]
in the survey; and, second, I found after reviewing the above five points and comparing them to
the question flow as presented to viewers, that some of the questions needed to be reordered for
the flow to conhere. (This latter fact may serendipitously have helped to make the order of the
question set more opaque to subjects as to its intent. Refining and explicitly shuffling questions
in the future may provide better data.)
Deficiencies in this method include an inadequate matching of the question set to all
objectives; some gaps in the question continuity; a missed opportunity to shuffle questions; and
too small a population, which precluded dropping high and low values and which called for the
inclusion of myself as a subject. There does appear to be some positive match between my
hypothesis and the results. As shown in the line graph, there are some visible dips that
correspond approximately to locations where gaps were predicted in the mean comprehension
score of each item.
Zelchenko p. 4
Relevant learning models
Aside from the fact that many adult learners do not have much more mathematical
mastery than 8th graders do, their experiential opportunities share something with high-school
students as well as younger learners. There has been much discussion on differentiating and
individualizing learning types since the turn of the last century, when E.L. Thorndike, John
Dewey, and others began picking up on Rousseau and propagated the notion of individualized
learning. Kolb (1984) — in a detailed study based on a survey of learning styles he developed —
represented learning in a cycle of four steps: concrete experience, reflective observation, abstract
conceptualization, and active experimentation.
While it would be irrelevant to elaborate further on the many interesting explanatory axes
that have come to overlay the Kolb model over the years (not to mention several competing and
complementary cyclic models), they all readily agree with Gardner (1983) and his theory of
multiple intelligences when we understand that all of these experiences exist in the cycle of
learning styles in a single individual as well as in relationships within the instructor-student-text
triangle. What these various explanations appear to have in common is that the learning cycle
consists fundamentally of observing a process, then reflecting and explaining the theories behind
it, and finally the cycle repeating itself.
Learning itself therefore is, in this regard, an instantiation of science. And, as with the
cycle of the scientific method, there is no de rigueur starting point, and so a conclusion one
might draw from the analogy is that it may be useful at some junctures in a chaos curriculum to
start by exploring various visual models or experiments that do a good job of encouraging later
discussion of explanations of given aspects of chaos, and offering these models before reading
texts that go into deeper detail on the models.
Zelchenko p. 5
Related to Kolb and other models are the notions of “top-down” (or “rationalist”) versus
“bottom-up” (or “empirical”) learning processes (and processors). These are discussed by Winn
(1982, p. 7) and relate to the Kolb model in that certain learners at certain times construct
knowledge from things they already know, while certain learners at certain times build from the
ground up and wait to incorporate what they have learned into pre-existing knowledge.
In addition, this relates to and underscores the distinction between different kinds of
learners in this regard — that certain types of learners may benefit more by starting from the first
part of the cycle (observation or the bottom-up model) rather than the theoretical plan (or the topdown model), while others may benefit from reading the texts first, as a more rational or topdown approach. Still other highly abstract mathematical thinkers, using a top-down metaphor,
might prefer starting with the differential equations and the laws of physics before looking at any
visual examples.
There is one other important aspect of
modern learning theory we should incorporate here.
The idea of Piaget and Inhelder’s (1955) internal
schemata is mentioned in Winn (1982, pp. 6-7) as it
relates to similar ideas from artificial intelligence
theory, specifically of “frames” (Marvin Minsky)
and “scripts” (Roger Schank). Winn explains that
all of these are ways to represent a structural model
of the learning process. Schemata, as with frames
Figure 1. A cognitive model. Composite of Piaget
and Inhelder 1955, Winn 1982, Kolb 1984.
and scripts, all describe related internalized forms of
the world as experienced by a learner, and
Zelchenko p. 6
internalized actions which are understood by a learner to apply to these forms. For example, a
fire hydrant is a form and, once (and not until) a learner has observed how one operates, he or
she will then apply various predicates which apply to that form (“water gushes out”) and
gradually will develop related co-schemata and subschemata to assimilate it further into known
experience (e.g., “firemen use them,” “special wrenches are needed,” “they are fed from a central
water supply”).
In broad strokes, then, schemata are the knowledge products and the tools that work on
them, while the cyclical and structural models described above are the mechanisms behind these
products (Fig. 1). They clarify how and when an interactive visual device may be employed in
certain parts of the chaos-theory learning process to foster a comprehensive mastery of the total
theory of arguably one of the more involved challenges in learning today.
Visual aids that may work
I am relying on a 35-year career in problem solving for visual communication, which has
included, among other things, developing instructional graphics, software, and video for several
years. While studying chaos myself I subconsciously kept an eye on the threshold between the
theoretical concepts and the effectiveness of their visual aids. Coincidentally, in the middle of
this process I helped organize a symposium of the PLATO computer education system. I had an
opportunity to reunite with two colleagues who developed some of the more celebrated dynamic
children’s computer games of the 1970’s, several of which are still in use today. I offer them in
passing as eye-openers to the simplicity of core themes and yet at the same time the inspiration
that these women brought to bear when designing their products.3
3
The 50th anniversary symposium of the PLATO system was held this month at the Computer History Museum in
Mountain View, Calif. Dugdale and Seiler both sat on the education panel and demonstrated these games, which
reminded the author of their archetypal qualities.
Zelchenko p. 7
Figure 2. Sharon Dugdale’s “Darts” on PLATO (Dugdale 1973). The object is to break all the ballons on the
number line using darts aimed by entering decimal numbers. A fraction mode is also available.
Sharon Dugdale and Bonnie Seiler developed a number of pioneering computer games
for children that emphasized the visual, but two of the earliest and best are Dugdale’s “Darts”
(Fig. 2) and Seiler’s “How the West Was One + Two x Four” (Fig. 3). Both are still in use and
this testifies to their simplicity and effectiveness. The first teaches the number line and the
second teaches math operator binding. Hardly cases of learning sugar-frosted in entertainment,
these two activities in their early monochrome forms had only a small measure more excitement
than textbooks and at best could be referred to as mathematics with a superficial coating of play.
Figure 3. Bonnie Seiler’s truly pioneering “How the West Was One + Three x Four” on the PLATO interactive
system (Seiler 1976). Left: Title screen; right: the play field. Three random spinners yield up three numbers, and
student’s object is to use them in forming strategically bound equations in order to advance a stagecoach to Red
Gulch.
Zelchenko p. 8
Dugdale’s and Seiler’s lessons, as with almost all successful interactive experiential
learning units, demonstrate several important concepts relevant to our discussion:

they use interaction with the learner as their absolute primary function;

they provide a (minimally diverting) visual experience appropriate to the learner’s
interest levels;

they are learner-driven as to cause and effect and adjusting level of complexity
(acting as a flexible tool for both top-down or bottom-up learning); and,

they invite repetition of an activity to personal satisfaction of mastery (fostering
construction of schemata).
The operative difference in learning materials we produced on PLATO and later systems
is that, regardless of whether they were simulating lifelike events (as with “Darts” and “West”)
or using dynamic visual markup to emphasize parts of text and graphic data (see, e.g., Davis et
al., 1980), our techniques emphasized the visual and experiential foci in learning a concept, and
so-called “branched instruction” (selective routing based on a decision tree of the student’s
mastery) provided a cyclic return for review and repeating experiments until mastery was
achieved.
Drawing from these two games, there are two main guidelines that I feel are important to
keep in mind if we are to bridge the gaps I identified earlier:
1. When developing a visual aid, the designer must think like the student and attempt
to get to the bottom of what minute conceptual steps the student may be lacking
for gaining mastery. If the simple children’s examples above aren’t easy to
pinpoint on these matters, conceptual steps to understanding chaos theory will be
more difficult to pin down. And yet any attention away from attempting to see
Zelchenko p. 9
through the students’ eyes to the source of their difficulties will be compounded
into a teaching and will only cause the experience to wander further off the path
to mastery.
2. Entertaining diversions are ideal for children because they hold attention while
fostering cause-and-effect trial and error and other observational opportunities
from which to draw conclusions; when adults are learning chaos, a visual aid need
not be explicitly entertaining, but it should retain the trial-and-error and other
observational opportunities and tie them to symbolic knowledge (math, physical
laws, etc.) where possible. (For example, a graphing calculator may “point” to
chaos in a graph and show the iteration steps nearby that position.) If dynamic
updating and animation will foster this, then computer-driven simulations may be
welcome, but they are by no means a must, as I will demonstrate.
Low-tech examples
I cannot emphasize enough that one should always rule out low-tech solutions before
conceding to a higher-tech product (Zelchenko, 1999). A cheap single pendulum, double
pendulum, and multi-body magnetic pendulum, in the hands of a capable instructor methodically
stepping through a phase portrait on the chalkboard, may be all one
needs to visualize much of basic and complex dynamical systems
and get a start on understanding their attractors.
It is possible that the shape of the Lorenz sled attractor can
Figure 4. “Tenfold wrapping”
mockup helps explain system
stretching. This was a weak
design; a long elastic string is
called for, and marks should be
more distinct.
be clarified by nothing more than a wood, plastic, or ceramic 3-D
model of the compacted unit square of a mogul (Lorenz, p. 37, and
Fig. 11 on his p. 44). I sketched out a model of such a teaching
Zelchenko p. 10
device.
I attempted (Fig. 4) to attack the “tenfold wrapping” mapping concept (Stewart, pp. 99-102 and
his Fig. 44; pp. 107-108 and his Fig. 48), which explains system stretching, using a paper tube
with a marked string wrapped neatly around it, and although my initial rough efforts were not
rewarding, it demonstrates how simple one can get and in theory still be able to explain parts of
something as complex as chaos theory. I also looked briefly at my micrometer, because these
wrapping and logarithmic scaling concepts seemed to be approaching the theme behind my
Vernier4 adjustment that measures thousandths of an inch thicknesses of materials.
From simple dynamics to strange attractors
There is a visually identifiable causal relationship between the double pendulum’s erratic
motion and that of three bodies in space, in that the bodies are similarly pulling at one another
and thereby influencing their motion, and so it might make sense to present them sequentially
and encourage discussion of the similarities. This also offers an opportunity for a bridge from
laboratory systems to natural systems.
What might be beneficial at this step is an interactive that would show both the time
series as it develops (see Fig. 5), as well as a phase portrait, with the ability to pause the action
and show the relationship among all three. A next step would be to show the same parallel
visualization using the Lorenz sleds. This order will take students more seamlessly from the most
rudimentary systems that are not chaotic to some more complex ones that show chaotic behavior.
I am grateful to Prof. Peter Vandervoort for correcting my foolishness in conflating the Dutch master as “Vernier”
and the French mathematician as “Vermeer.” Thankfully, I was not in Holland at the time or else it might have
precipitated a serious diplomatic incident.
4
Zelchenko p. 11
Figure 5. Double pendulum online does a fine job of demonstrating chaotic motion and its graphs show period
fractionation, but mastery may not come without exploration of the point-by-point relationships among these
visuals. In this visualization, though the animated double pendulum quite clearly takes us to the next level of
complexity after a single pendulum and is helpful in that it graphically traces its complex path, the two graphs are
static and do not update synchronously, and so it the correspondence to the pendulum’s actions is not evident. This
computer simulation is therefore not of much greater value than a real double pendulum.
(http://scienceworld.wolfram.com/physics/DoublePendulum.html)
Stewart offers calculator chaos as a logical next step from Hyperion’s chaos, and that
may be a good strategy, but his transition to chaos using a calculator (pp. 12-16) was not
altogether clear, despite the lengths he goes to in his attempt (see Gap 1 in the survey). “What
you’re seeing,” he writes, “is a sort of Hyperion-in-microcosm” (p. 15). But most students I
polled were not clear on what that meant. Stewart seems to have substituted Hyperion’s
multivariate complexity with a “compacting” recursion in a calculator model without a clear
enough explanation. One student went so far as momentarily to think that Stewart was translating
Hyperion’s dynamics to his 2x2–1, but that misconception was not the only thing that caused
confusion.
I believe the problems emerged from a disconnect in students’ minds between function
and plotting, particularly when recursing long decimals and switching functions without
explaining each one’s plot-shape dynamics. A more gradual explanation of the action of plotting
at this step might be in order. I developed a Microsoft Excel spreadsheet (Fig. 6) that
dynamically updates the graph on every change of the function and allows observation of a
record of each step in the recursion. Although this may not get at the total source of the problem,
Zelchenko p. 12
it does satisfy the two gap-bridging proposals I provided above and may be useful alongside
Stewart’s text.
Figure 6. My Excel spreadsheet (after Stewart’s “calculator chaos,” pp. 12-16). This allows greater interactivity and
instant results, including a record of the iterations at left, but it still requires careful reading of Stewart alongside, as
well as a capable instructor’s guidance on the meaning of the spikes and oscillations, not to mention an explanation
of recursion and how it serves to “pack” complexity in a way analogous to Hyperion’s many variables.
I mentioned earlier the need for a more methodical visualization and analysis of the
Lorenz sled attractor. This attractor is, to my knowledge, the first and best entry-point for a
student to see a chaotic system that offers a strange attractor whose folds can be seen in their
development. It is potentially an excellent bridge from pendulum attractors to strange attractors.
But the shape of a phase diagram is defined partly by the nature of the system and partly by the
nature of whatever compaction scheme has been applied to it. Which part is which is often
difficult to contemplate. Lorenz’s compaction scheme on the sleds is highly counterintuitive; as
just one example, y is plotted as the vertical. (See Gap 2 in the survey.) Students need to
understand what is going on here.
Initially, I attempted to foster observation and analysis by depicting a physical model of
Lorenz’s unitized mogul (see my Fig. 7, based on Lorenz pp. 39-44 and his Fig. 11). A model
Zelchenko p. 13
shaped like this can be fashioned out of any real material, such as clay, wood, or papier maché; it
can simply be used in print as shown; or it can be developed as an animated computer activity.
Figure 7. My sketch of Lorenz’s mogul unit rectangle (Lorenz, pp. 39-44) which allowed him to calculate his
attractor in his Fig. 11 (represented here at lower right). I also plotted samples of where on the attractor two boards
would land. A more dynamic simulation (or an instructor) might venture into highlighting a region on the attractor
and showing where in general the sleds would be appearing on the model and why. This kind of activity should after
some repetition facilitate a complete understanding of the attractor’s odd contours and reinforce the general principle
of “packing” or “compacting” of data previewed in Hyperion, calculator chaos, and elsewhere.
Robert Lurie of MIT has been enthusiastic in modifying his Wolfram Mathematica
Lorenz sled demonstration since we began corresponding earlier this year. Finding the relation in
the Lorenz book between the model and its visuals difficult for learners to follow, Dr. Lurie
developed an interactive demonstration. On my encouragement, he has been working on changes
that will cause the attractor to update dynamically as each board crosses the attractor’s threshold
(Fig. 8). He is also working on zooming in on a remapped single mogul, as Lorenz did to transfer
his system onto a two-dimensional phase portrait. He is updating all three phenomena — the full
Zelchenko p. 14
ramp, the unit mogul, and the phase portrait — simultaneously. Because of its display
intensivity, it would be useful to be able to turn off any of these displays at any given time.
Figure 8. Above: Robert Lurie’s Mathematica project (Lurie 2010), a first attempt at a dynamic sled attractor. Note
the several adjustable sliders at top, the overview at left, a more local view, and the attractor. Below: Three details of
the local view (above) and partial attractor (below) show that dynamic, interactive development of the attractor may,
like my illustration, help visualize where parts of the attractor contour develop based on sled positions and
velocities.
I would also like to be able to zoom in and select a point or an area of the phase portrait
and see the entry point (and perhaps even the path through the mogul) for all of those sleds along
the unitized mogul rectangle.
Zelchenko p. 15
Stretching and folding, period doubling and bifurcation
Although the world of stretching and folding images merits further discussion elsewhere,
there is a logical bridge from the above discussions immediately into Smale’s horseshoe,
Hénon’s transformations, solenoids, and like models (e.g., Stewart Ch. 6 pp. 99-114, and Ch. 8).
An interesting interactive would be to allow actually visualizing the stretching and folding of a
given phase portrait such as Lorenz’s (or even allowing the student to do so), permitting students
to grasp the nature of the extenuation and distortion universal to all chaotic systems as they
progress in time.
However, it seems to me that once phase portraits are well understood, all that may be
necessary for mastery is a bit more summary association among these many aspects. My sense is
that our instructor did this quite well, but as it is built upon adult students’ weak foundation in
the graphic math (see Gap 4 in the survey), some students may need assistance in understanding
the association between each more or less clear visible transformation metaphor — such as the
horseshoe, U-tube, or stretching solenoid — and its more obscure graphico-mathematical
significance, as well as reminders that this is happening as a dynamical system is evolving and
that this appears to be a universal in chaos theory.
This then leads naturally to discussions of period doubling and bifurcation. As unlikely as
it may seem, these concepts may be more straightforward than those involved in stretching and
folding, since with a Feigenbaum tree we are working from a very high perch and then zooming
into something that is not a moving target. Prior knowledge must thus only include a moderate
understanding of both periodicity and perhaps how time-series diagrams translate to attractors.
The instructor used spring-loaded systems on the board to describe how an oscillator can express
period doubling in an attractor, and so this may be sufficient.
Zelchenko p. 16
The significance of the Feigenbaum 4.669 ratio, however, was probably completely lost
on all but two or three of the most diligent students surveyed anecdotally. But this is hardly
surprising: we are “full fathom five” deep into the waters of chaos theory by the time we get
there. If a student is missing any one of the necessary master links, they may have difficulty
connecting everything together and reveling in Feigenbaum’s truly remarkable discovery.
Fractals
Students experienced some apparently consistent confusion with how chaos theory
connects to fractal theory (see Gap 5 in the survey). Fractals seem somewhat far afield from
dynamical systems per se, and the sudden introduction of fractal dimensions, logarithmic scales,
and complex numbers may bewilder even the more diligent students despite careful reading.
There appears to be a logical path from attractor and system transforms such as stretching and
folding, as well as period doubling and bifurcation, to Cantor sets, and thence to other
mathematically interesting spatial divisions, particularly fractals. Lorenz ties all of this together
as mechanisms behind systems that are subject to sensitive dependency and declares that it was
therefore “but a short step for ‘chaos’ to extend its domain to fractals of all kinds… .” (p. 177178). But several students admitted they did not get the connection, and some were also looking
for some practical value.
Explaining fractal dimensions takes some doing, not least the logarithmic explanation and
the Sierpinski subtractive model, but also how these shapes can be called “attractors” in the
terms already so hard won. One very technically gifted student likened his mastery to building a
house of cards: “Every time I thought I finally understood, some new definition would cause the
whole structure to come crashing down again.” We know that stretching and folding, and period
doubling and bifurcation, all yield up self-similarity at finer levels of an attractor, and this
Zelchenko p. 17
relationship can vaguely be seen between the two, but why does it work that way? And what of
Lorenz? How does sensitive dependency associate with fractals, as he suggests?
I have no answers to these questions. But I did explore Julia sets in greater detail in a
demonstration by McClure (n.d.) that the professor directed us to (Fig. 9).
Figure 9. McClure’s (n.d.) Java Julia Set Generator. The user clicks near or upon the Mandelbrot attractor at left,
and the corresponding Julia set displays at right. Sets can be superimposed for comparison: one can, for example,
see the advancement (starting at 1, the circular form from the Mandelbrot origin) to elaborate “prisoner” contours (at
5, along the dense form of the Mandelbrot set). We understand now that Julia sets are unique sets based on complex
numbers plottable along the Mandelbrot space, and that the elaborate sets are near the edge of the attractor. But
why? What are these sets, why is the origin set a perfect circle (a clue!), why do the shapes diverge from a circle,
what do they have to do with complex numbers, and why are they called prisoner and escapee sets? And why is the
Mandelbrot set shaped that way? How is the circular nature of the Mandelbrot set related to the circularity of the
origin Julia set? (http://facstaff.unca.edu/mcmcclur/java/Julia/)
This exercise gave some insight into both the physical and mathematical relationships
between Julia sets and the Mandelbrot set, but it also raised new questions. Students will
discover that forms tend to be simple contours starting from a perfect circle at the origin, until
somehow they become true “prisoners” near the perimeter, but it is in no way clear why that
elaboration is developing, nor what the practical difference is between a prisoner and an escapee,
nor whether the inference is even correct. The circle at the origin provides some clue. In fact,
Julia sets have intriguing properties as they move away from the circle’s simplicity that may
Zelchenko p. 18
shed light on both Julia sets and the Mandelbrot set. Contiguous shapes skew as they leave the
center but do not tend to “floriate” until near the perimeter. They then begin “breaking up”
outside the perimeter. Likewise, as we move leftward into the secondary bay, the structures
begin twisting. There are definite patterns. These facts could not easily be gleaned from visual
evidence without this specific program (what we get from text, e.g., Stewart Fig. 99, does not
reveal these dynamics) and one could indulge in it for hours. Yet it does nothing to benefit
understanding of the concept; an instructor must relate these mysterious topological observations
to what we have previously learned.
Another activity seems to add more insight. Or, does it? Below (Fig. 10) is from a
Portuguese programmer:
Figure 10. António Miguel de Campos’ (n.d.) online demonstration shows “the evolution (orbits) resulting from the
iteration of zn+1 = zn2 + c, 50 times.” You can also zoom and see more detail. Left is clicked outside the perimeter
and shoots off into oblivion; right shows clicking inside. Is there some connection between the containment or
divergence and the concepts “escapee” and “prisoner”? Do the relative levels of containment of these two iterations
relate to how elaborate the points’ Julia sets are? This is not consistent, because clicking in the center of the small
blue bay at top or bottom yields only a triangle in this activity but a very elaborate shape in the Julia Set Generator
explored further above. Does this lead to more questions than answers, or is this getting us somewhere?
(http://to-campos.planetaclix.pt/fractal/mandelgen.html)
This discussion is not meant to be an exhaustive investigation into how to explain
fractals; it is only to emphasize that there are plenty of visual aids, and once assembled and
arranged they may help to provide a more comprehensive explanation of the concept of fractal
Zelchenko p. 19
geometry. But they may also simply add to the confusion. The important thing to keep in mind is
to assume the student does not know what the instructor knows.
Conclusion
Chaos theory — coalescing as it does from an unlikely assortment of disciplines — is a
famously broad subject and my two papers come nowhere near to doing justice to all aspects of
the problems. They are, in fact, meant primarily to emphasize the subject’s breadth and to
encourage enthusiastic study into all of the possibilities in learning and teaching it. Even the best
instructor will require far more than a few weeks with these three texts, and the available
teaching visuals are only under development. What is needed as a consequence of simple surveys
like this is a structured curriculum with clearly defined unit obectives and expected outcomes for
mastery. This in turn will encourage evolution of superior teaching aids and techniques.
In the interests of time and space, I omitted many important discussions here, most
notably the natural systems that are so well developed in all three books, logistical mappings and
the Lorenz attractor, and self-similarity. This paper is also bound to contain some misnomers and
faulty associations among certain concepts, based on certain misapprehensions. For these, I alone
am responsible — however, one must keep in mind that I am a neophyte to this science and a
layperson.
References
Alesandrini, K.L. (1984). “Pictures and Adult Learning.” Instructional Science 13(1), pp.
63-77.
[de] Campos, A.M. (n.d.) “Mandelbrot Set and Julia Sets.” Accessed June 2010 at
http://to-campos.planetaclix.pt/fractal/mandelgen.html.
Zelchenko p. 20
Davis, R., Siegel, M.A., Zelchenko, P., et al. (1980). PCP SYS IV Core Curriculum in
English. Urbana, Ill.: Computer-based Education Research Laboratory. In this series of lessons,
we presented information and negative and positive feedback (e.g., text markup, highlighting
incorrect answers, corrective re-insertion of incorrect answers into the drill set) by carefully
designed formulae based on research in direct instruction techniques developed by S. Engelmann
and refined by students Davis and Siegel. See, e.g., Siegel, M.A., and Misselt, A.L. (1983). “An
adaptive feedback and review paradigm for computer-based drills.” University of Illinois at
Urbana-Champaign, ERIC Research Report #ED234754.
Dugdale, S. (1973). “Darts.” PLATO computer lesson. Urbana, Ill.: Computer-based
Education Research Laboratory.
Dwyer, F. M. (1978). Strategies for Improving Visual Learning. State College, Penn.:
Learning Services. Cited in Alesandrini 1984.
Gardner, H. (1983). Frames of Mind: The Theory of Multiple Intelligences. New York:
Basic Books.
Gleick, J. (1987). Chaos: The Making of a New Science. New York: Penguin Books.
Kolb, D.A. (1984). Experiential Learning: Experience As the Source of Learning and
Development. Englewood Cliffs, N.J.: Prentice-Hall.
Lentz, R. and Levie, W.H. (1982). “Effects of Text Illustrations: A Review of Research.”
Educational Communication and Technology Journal 30(4), pp. 195-232.
Lorenz, E. (1993). The Essence of Chaos. The Jesse and John Danz Lectures. Seattle:
University of Washington Press.
Lurie, R.M. (2010). “‘Chaos While Sledding on a Bumpy Slope’ with variable
parameters and showing all hill, repetitive cell, and strange attractor.” Unpublished modifications
Zelchenko p. 21
to original Wolfram Mathematica demonstration project “Sledding on a Bumpy Slope: Chaos
and Strange Attractor” (n.d.). Available at:
http://demonstrations.wolfram.com/SleddingOnABumpySlopeChaosAndStrangeAttractor/.
McClure, M. (n.d.). “Java Julia Set Generator.” University of North Carolina at
Asheville. Accessed June, 2010 at http://facstaff.unca.edu/mcmcclur/java/Julia/.
Piaget, J. and Inhelder, B. (1955). Growth of Logical Thinking. London: Routledge &
Kegan Paul.
Seiler, B.A. (1976). “How the West Was One + Three x Four.” PLATO computer lesson.
Urbana, Ill.: Computer-based Education Research Laboratory. See also later editions, including
that by Pleasantville,. NY: Sunburst Communications.
Stewart, I. (1989, 2002). Does God Play Dice? The New Mathematics of Chaos. Malden,
Mass.: Blackwell Publishing.
Winn, W.D. (1982). “Visualization in Learning and Instruction: A cognitive Approach.”
Educational Communication and Technology Journal 30(1), pp. 3-25.
Zelchenko, P. (1999). “Technology in Education: Exploring Alternatives to Hype.”
Educational Leadership 56(5), pp. 76-81. Alexandria, Va.: Association for Supervision and
Curriculum Development.
Zelchenko, P. (2010). “The Steep Slope to Understanding Chaos.” Paper 1 for University
of Chicago sciences core course “Order and Chaos in the Natural World,” May 11, 2010.
Zelchenko p. 22
Four Adult Students'
Comprehension of Chaos Theory
Comprehension
Gap
1
2
3
4
5
10
8
6
4
2
0
1
4
7 10 13 16 19 22 25 28 31 34
Stage
Zelchenko p. 23
[1]
[2]
[3]
[4]
avg
Survey Question
[1]
[2]
[3]
[4]
avg
a
1
Deterministic behavior (Lorenz: "only one thing can happen
next")
8
9
10
8
8.75
b
2
Stochastic behavior (e.g., coin tossing, shuffling cards, or
Hyperion's odd orbit)
7
7
5
7
6.5
g
3
Dynamical systems
8
9
8
9
8.5
h
4
Discrete dynamical systems
8
7
5
9
7.25
c
5
Chaos: i.e., "stochastic behavior occurring in a deterministic
system" (Royal Society of London, 1986) or "lawless behavior
governed entirely by law" (Stewart, 1987)
8
7
8
7
7.5
d
6
Sensitive dependency on initial conditions ("small initial
changes can lead to widely diverging results")
10
9
0
10
7.25
e
7
How iteration (e.g., with "calculator chaos," or with Lorenz's
sleds) compounds sensitive dependency
(GAP 1 STAGES 7-8)
9
7
0
8
6
f
8
Showing chaotic behavior using simple equations on a
calculator or nonlinear plot
8
5
2
6
5.25
i
9
Basic attractors (e.g., the single pendulum, oscillators such as
loaded springs, etc.)
(GAP 2 STAGES 9-12)
9
7
2
10
7
j
10
Complex attractors (e.g., the double pendulum, Lorenz's
"sleds") and the idea of "strange" attractor
9
7
2
6
6
"It's the 'so what' of identifying a strange attractor that gets me
confused" (2)
k
11
How a time series relates to a phase portrait or attractor (see
figure in Gleick, p. 50)
8
7
0
9
6
"It's a series of snapshots" (2)
Zelchenko p. 24
"Although I prefer a slightly different definition of chaos" (2)
"It's the 'how' to illustrate the behavior that gets me confused"
(2)
l
12
Lorenz's "sled" strange attractor
5
7
0
7
4.75
o
13
Phase portraits: Limit cycles, sources, and sinks
8
7
10
5
7.5
p
14
Poincaré: Homoclinic tangles, and sections ("return maps")
5
7
5
7
6
"Same comment as j" (2). "I understand the concept but it
would be helpful to explain the shape of such an abstract
compacted attractor" (4)
"Visual diagram of conditions" (2)
(GAP 3 STAGES 14-16)
m
15
The logistic map equation
5
7
0
7
4.75
n
16
The Lorenz attractor (the "owl" from his simplified convection
equations)
4
7
0
6
4.25
s
17
Stretching and folding (e.g., Smale's horseshoe, Gleick p. 51)
3
7
8
5
5.75
t
18
The math underlying stretching and folding
2
7
0
3
3
"Same comment as j" (2)
(GAP 4 STAGES 18-21)
q
19
The period-doubling cascade
5
9
0
8
5.5
r
20
The math underlying period doubling
2
9
0
4
3.75
"Math of iterations" (2)
u
21
Is period doubling essentially the same concept as stretching
and folding?
2
7
0
2
2.75
"To a degree, but could a period doubling possibly vary
depending on the dimension(s) of the stretching and folding?
Stretch less, fold more, and still make it fit" (2)
v
22
Bifurcation (see, e.g., diagram Gleick p. 206)
4
7
2
9
5.5
w
23
May's bifurcation diagram (e.g., Gleick pp. 74-75)
4
7
2
8
5.25
x
24
The Hénon attractor
6
7
0
6
4.75
Zelchenko p. 25
"Iterations" (2)
y
25
Natural chaotic behavior: gravity and orbits (e.g., Poincaré);
population dynamics; weather systems; convection and
turbulence; and other natural systems
0
7
8
8
5.75
z
26
Renormalization and Feigenbaum's universal value of 4.669
2
7
0
8
4.25
A
27
Fractals in general
8
7
5
8
7
B
28
The fractal qualities of the bifurcation diagram
9
7
1
9
6.5
C
29
Cantor sets (e.g., Stewart, Figs. 49 "lines" and 50 "cheese")
9
9
5
8
7.75
D
30
How Koch curves, Menger models, etc., relate in their nature
to Cantor sets
9
9
4
9
7.75
"Don't ask me why I think I get this; perhaps it is because
graphics are easy for me visualize" (2)
E
31
The fractal qualities of Koch curves, Menger carpets and
sponges, Cantor sets, etc.
9
9
0
9
6.75
"More elaborated Cantor sets" (2)
F
32
Julia sets
7
6
0
6
4.75
"7 as a concept; 5 as to how one gets there" (2)
"7 as a concept; 5 as to how one gets there" (2)
"But I don't comprehend how this number, and/or several
other values that we discussed are used in life...besides in the
one example illustrated in the book & mentioned in class" (2)
(GAP 5 STAGES 32-34)
G
33
The Mandelbrot set
8
6
0
6
5
H
34
How fractals relate to attractors
7
9
0
6
5.5
215
252
92
243
201
Score (perfect = 340; practical mastery or "C" average ~= 200)
Zelchenko p. 26
"I understand the inverse correlation" (2)