Critical Barriers to Science Learning
By David Hawkins
Further Examples of Critical Barrier Phenomena
To investigate the ideas people have about mirror vision I have sometimes asked subjects to imagine that one wall of the
room we are sitting in is a large mirror and then to draw, on a map of the room, the direction in which they would look
to see a given object "in the mirror." If the object requires an oblique viewing angle I have found that subjects draw a
wide range of directions which cluster bimodally near two extreme positions. There is a small peak clustered near the
direction which geometrical optics require and a larger peak near the place defined by a
line drawn perpendicularly from the object to the mirror. For this larger group of subjects the mirror
image of the object is apparently thought of as analogous to the picture of the object pasted onto the surface of the
mirror "where it would see itself," or else the depth-dimension in The Looking Glass Room is radically foreshortened in
subjects' conception of it. I have found approximately the same statistics with upper elementary school children, elementary
school teachers, and two graduate classes in the philosophy of science. About fifteen percent cluster at the small, correct
peak, fifty percent at the other extreme, and the rest scatter in between. In the two graduate classes the fifteen percent
were mostly students in physics or mathematics, while the rest were from sociology, philosophy, psychology, etc. The
prevailing adult conceptualization of mirror vision, and vision in general, is a gold mine of the kind of phenomena I want
to discuss here.
This difficulty in understanding mirror vision illustrates a class of what I shall call critical barrier
phenomena, or simply critical barriers, in present-day science teaching. These phenomena are easily observed in many
contexts and represent barriers to learning for at least a clear majority of precollege, college, and adult students.
Though diverse in content, the phenomena share certain characteristics which I believe are uniform enough to sustain some
They appear early in any standard science curriculum; they are associated with extremely "elementary"
science topics. I put the word in quotes because "elementary" is often taken to mean "easy" or "obvious," and thus
appropriate to begin with. In fact, as I shall try to show, some "elementary" ideas are exceedingly unobvious to those who
have not yet assimilated them and are themselves only lately-won in the history of science. Elementary ideas are often
deep. Students who fail to assimilate them must often come to regard them as barriers to entry into any further science
learning. Often they give up, becoming frustrated and typically either dropping out or dropping upthat is, continuing
the course and managing to pass it without any valued or valuable precipitate of understanding.
Some students manage to avoid this impasse. They may have had early, self-directed interests and
talents; they may have had early successful teaching. They have already assimilated and can readily use elementary ideas
which, for others, are formidably opaque. In the few cases where I have some recorded statistics, this group is small and
typically consists of those who already have a conscious bent toward science as a career or an avocation.
My concern is with the general level of science education, not with the advanced education of scientific
specialistsit is with the size of the base of a social pyramid, not the height of its peak, though I am mindful of the
relation between the two measures. I believe that by carefully examining the classes of critical barrier phenomena it is
possible to arrive at some conclusions about present levels of scientific culture and modes of science teaching at all
except the highest levels. These conclusions do not automatically define remedies, though they suggest some. My concern is,
rather, to use them to define goals for science education policies, goals which I believe are crisp and definite enough to
suggest useful criteria for decisions about ways of working toward them.
In the following section I shall further illustrate, define, and interpret the class of critical barrier
phenomena. In the final section I shall attempt to define policy goals I have in mind. Chief among these is the need for
much more basic research, analysis, and experimentation. The experiments with mirrors, and others I shall cite, were
casually done and should be repeated with more carefully stratified sampling of subjects. I believe I am describing only the
tip of an iceberg.
Further Examples Of Critical Barrier Phenomena
My other examples of critical barriers come from contexts as rich and illuminating as the mirror difficulties. They are size
and scale, air and water, heat, and elementary mechanics.
Size and Scale. In some fifteen years of teaching a general physical science course for
non-science college majors, and in an equal period of time devoted to in-service teaching of general science to elementary
school teachers, I have found in both groups a marked conceptual difficulty in grasping, or gaining fluency with, the
elementary relations between length, area, and volume. The frequency with which this difficulty appears (if one looks for
it) is high it affects something in the neighborhood of eighty or ninety percent of both groups. Reasonably patient
explanation is no cure. For this reason, a teacher concerned to "cover the subject"meaning, of course, to get through
a textbook or promised outlinewill become exasperated with students' disabilities or with his own inability to make
such elementary things clear. The fact that patient explanation is no immediate cure is a hallmark of the class of critical
barrier phenomena. One can break through but not easily or uniformly, and failure may lead a teacher to say that some people
are just dumb. Another hallmark of the class, however, is that when the breakthrough does come with students they often have
a high emotional release, a true joy in discovery; "Is that what it means?" There is often a marked change in later
performance, as though a hitherto hidden secret had been revealed.
Returning to length, area, and volume (L, L2, and L3) and their relations to each other, both the
resistance to explanation and subsequent joy of discovery suggest that the student is not lacking in knowledge so much as
he is habituated or addicted to some congenial alternative way of thinking. My work with children of middle- and
upper-elementary ages reveals that with materials, time, and supportive interest they can arrive at these relations through
honest empiricismnot yet firmly built-in, perhaps, but without confusion or conflict. Generalization may be difficult;
it is one thing to see the scale-relations with larger cubes built out of smaller cubes and quite another to recognize them
in the scaling up or down of spheres of different sizes, of irregular shapes of dough or plasticine, of models, of doll
houses. That takes time.
Let me inspect this example in terms of what we call common sense or common knowledge. Length,
considered in isolation, is no problem. But, though farming, carpet-laying, and painting involve area, area is indeed
problematic, especially in relation to length or as a characteristic of irregular shapes. "You can't find the area of a
footprint; it's not a rectangle." An eight-by-ten rug is an eight-by-ten rug, but "is it really eighty square feet?" Volume
(Pace Piaget) is well understood in typical adult volumetric contexts but not as L3.
With such shaky foundations the next stepsthe principal of similarity and the elementary scaling
relationsare quite inaccessible. And here I think I have support from history. The ancient Greeks had formulated these
ideas. Euclid establishes them formally, though we would say awkwardly. Galileo was the first to elucidate their relevance
to the properties of material things in his discussion of the strength of beams. Extended to include time and mass they are
implicit in Newton and more or less formulated in nineteenth century physics. D'Arcy Thompson in the early twentieth century
was, so far as I know, the first to see living things as phenomena of scale. At about the same time Lord Raleigh elaborated
dimensional analysis as a style of simplified physical analysis.
I mention this history because it shows how long a time was required for a simple and widely
illuminating idea to show its full implications, even among the learned. At a relatively elementary level the now-old
P.S.S.C. (Physical Science Study Committee) physics text for high schools was written in the spirit of the scaling laws.
Philip Morrison, one of its co-authors, gave the Christmas lectures at the Royal Institution of London on this subject for
British school children some years ago. Ironically, the P.S.S.C.'s opening chapter on the scale of nature has been
eliminated from commercial editions, apparently because these ultimately simple considerations which give rough
intelligibility to the whole face of nature are still not considered to be physics.
I have used this example of a critical barrier phenomenon of science as part of my introduction to the
discussion of our failure to achieve a wide dissemination of scientific ideas and attitudes; it suggests that we are up
against something rather deep in the relation between science and common sense; we are up against a barrier to teaching in
the didactic made which has hardly been recognized, or if recognized has been seen mainly as a challenge to ingenuity in
teaching rather than as a challenge to a deeper understanding of human learning. It is the sort of phenomenon we tend to
acknowledge only in a spirit of despairing humor or complaint; we tend not to focus on such matters as worthy of
intellectual curiosity and excitement. Why are these difficulties at once so elementary and so abundant? That question is
too seldom asked.
As a first step of analysis, I suggest that the verb "to learn" implies a time scale; some things can be
learned in five minutes while some come only on a long developmental time scale. It is the great merit of Jean Piaget to
have emphasized the importance of the latter kind of learning, and his great demerit to have popularized the belief that
what takes place is not really learning at all but an age-specific biological development independent of a society's
educative potential; if an intellectual skill or scheme cannot be taught, just wait a while and it will appear anyway.
This does not reflect Piaget's best thinking but he has never repudiated it. The whole class of barrier phenomena I am
concerned with here represents the apparent inability of most adults in our society to get beyond what would have to be
classified as limitations that belong to early stages in the Piagetian taxonomy.
With respect to length, area, and volume, most adults have something in mind that is quite different
from, and potentially conflicting with, the geometrical sense for invariance and variation in scale. They have a
perceptual-commonsense way of taking things as "big" and "little" without reliance on the analytically defined concepts of
length, area, and volume. From the commonsense-perceptual point of view this is entirely reasonable. Immediate commonsense
judgment is geared to a great variety of perceptual cues and its practical reliability is typically very high. Since over
the range of normal experience length, area, and volume are highly correlated, it is plausible that in the commonsense
scale of big and little there is for most practical judgments of size no focal consciousness of any one of them. When
challenged to measure the area of a footprint, most students, most adults, will suggest measuring around it, measuring its
perimeter. The notions of perimeter and area are not clearly distinguished from one another.
In order to compel attention to such distinctions there are many ways of using the principle of the
extreme case, such as artificial or naturally occurring shapes with large perimeters and small areas, large areas and small
volumes, etc. This is not only an exercise; it leads naturally to the many biological examples of adaptation to
scalethe roots and leaf area of green plants, the elaborately branching lung tissues and guts of large animals, etc.
In extending curiosity and experience to these ranges of phenomenamany of them everyday phenomena accepted incuriously
by common senseordinary incurious perceptual habits of thought can be gradually cross-linked to those which are more
analytic and more consonant with the newly extended range of experience for which the history of science is responsible. A
deepening grasp of the significance for scale of invariance and variation is one of the major gateways to the modern world
of science. It represents the acceptance of an intellectual discipline upon the extraordinary subtlety and
pattern-recognizing capacities of ordinary perceptual learning, capacities that are geared to the great variety and
complexity of the human world and are basic to many forms of understanding and of art. In such perceptual matters the
axiomatic simplicity of geometrical scale is by itself almost useless; yet in extending our knowledge and intuition of the
ampler world of sciencewith which our life as a society must be increasingly concernedthe failure to develop
these axiomatic thought-habits and to link them fluently to perceptual modes will inevitably rob the mind of a power it
increasingly needs. The failure to grasp the planetary impace of present-day activities and practicesthe failure to
understand what it means to scale an explosion by a thousand or a millioncan be fatal to a society. Beyond that,
however, it is a failure which robs most of us of the possibility of any esthetic and moral framework within which we can
understand and enjoy, and thus be full participants in, the great and problematic era our history has created. Without it
most of us will remain or increasingly become what Arnold Toynbee called a "cultural proletariat," in but not of the
society we unwittingly constitute.
Air and Water and Beyond. My third critical barrier phenomenon of present-day science education
has an equally interesting and varied history. It is the scientific conceptualization of the states of matter. Aristotle
sorts them out as a matter of course, with fire suggestive of our "energy." Nothing is more obvious, and common sense has
no immediate trouble with the traditional introduction to the elementary text which began with the sorting into solids,
liquids, and gasses. Yet here again a large majority of "non-scientific" college students and adults develop deep
difficulties. Let me begin with the atmosphere. We live in it like the fishes in water and its very constancy as the medium
of our life renders it mainly unnoticeable except for special circumstances which common sense recognizes in its usual
piecemeal perceptual fashion. From history, again, we know that scientifically "obvious" things about the air are recent in
any human consciousness. The Greek astronomers appear to have deduced the "ocean of air," a terrestrial mantle of limited
thickness. This, I believe, was to explain the remarkable fact that an object so distant as the moon (whose diameter and
distance they had fixed from the geometry of the eclipse data) was still clearly visible, while distant mountains, so close
by comparison, were almost lost in the atmospheric haze. At any rate, Plato weaves a myth around the ocean of air. Yet the
impact of the ideaotherwise long forgottencame back to scholars full force only after Torricelli's and Pascal's
investigations and the visible fact of the Torricelli vacuum.
The elementary school science text or demonstration can prove that air has weight, and usually does so
badly, with balloons, avoiding the consideration of the ocean in which the weighing is done, of buoyancy and density. High
schools can evacuate a flask weighed before and after, and that is a neater demonstration; but neither demonstration can
produce any resonance in a mind which is unprepared, as most are. The siphon is a familiar phenomenon on the edge of
everyday experience, but for most of the group I speak of it is another of those mysteries which is only deepened by patient
scientific explanation. Elevate the top of a water siphon to thirty-odd feet, a silly trick just beyond the edge of common
experience; now the sense of mystery at the result will become palpable.
We often discuss, pro and con, the educational impact of television. News programs are
characteristically climaxed by a discussion of the national and local weather, complete with those marvelous satellite
pictures, accounts of new "systems" moving in or out, of the jet stream, of highs and lows. Some, at least, of those weather
experts are indeed good meteorologists, but like many scientific experts they have long since forgotten what most of their
audience does not know it needs to learn, the early slow steps by which they themselves assimilated a conceptual structure
which meteorology already presupposes. I discussed this once with a TV weatherman, a good meteorologist indeed, and
suggested some televised byplay with water barometers, rotating dishpan models of the atmosphere, and the like. He thought
it would be fun but explained that time constraints required rapid speech and bare daily essentials. Yet today good
climatologists are raising questions about man's own impact on the climate. What sense will these concerns make to
intelligent citizens for whom the global circulation of air and water is unrealfor whom water evaporates and condenses
only up and down, locally, and for whom, half the time, air is literally nothing, half the time reaches on to the moon, and
all the time is mysteriously able to support the flight of an airplane?
Another aspect of this topic concerns the elements of biochemistry and their relation to the green cover
of the globe. For thousands of years farmers have farmed well in the belief that their crops are earth-earthy, pushing up
from the maternal soil and somehow composed of it. Water and the heat of the sun were necessary but the stuff of life came
from below. That view, like some Jungian ancestral memory, still dominates the thought processes of most of us. It is only
a few generations since there was a scientific realization that trees are essentially shaped from air and water, that
sunlight drives their circulatory systems, that they grow from the outside in. A large majority of our adult students will
tend to believe the opposite: that plantsgrasses or treespush up out of the ground, their blades or branches slowly
rising, their newest growth in the center, and all this despite a forgotten course in biology.
At a slightly more sophisticated level are the ways of conceptualizing the interphase characteristics
of things, the simplest and most accessible being the water-air or water-oil boundaries. The idea of a "skin on water,"
being of negligible significance on the human scale, is hardly credible to common sense, though intelligent discussion of
it often raises up the phrase "surface tension" from some otherwise forgotten science lesson. This leads nowhere. Soap films
are not credited with thickness and their colors are rarely provocative. Here again scale is of the essence and a sense for
it is lacking. Evaporation and condensationup and downare believed in separately, but are not understood as shifts
of equilibrium in an always two-way exchange.
The missing ingredient here is any insistent realization of atomicity. Atoms are known about in the
verbal store as something to be believed in but not as things to be imagined in conceptualizing everyday physical,
biological, or chemical processes. The simplest reasoning of John Dalton, or even of Lucretius, is again a critical impasse
for most; explanation only heightens the impasse, though such now-accepted terms as "carbon monoxide" and "carbon dioxide"
Here again there is ample historical evidence of the recency of such ideas and of the discrepancy or
unresolved conflict between the scientific and the commonsense-perceptual modes of thought and imagery. The everyday physics
of qualitative change is still predominantly in the mode of the early Aristotelians and alchemists, a metaphysics of
dispositions and qualities --thus drying, cooking, dyeing, melting, dissolving.
Heat. Heat is another critical area, with temperature as associate. Thermometers are historically
recent, but are widely assimilated into the commonsense world. For most, what they measure is perceived as a refinement upon
Aristotle and the medical investigators of Galileo's time. Temperature is a measure of "temperament" in human bodies and
outside, of the balance between two principles called the Hot and the Cold. This ancient conceptual predilection is indeed
a nice match to the animal temperature sense, which measures something which is not physical temperature, although
correlated with it. What it measures is approximated by the scientific notion of heat-flow, in or out, but at this level
common sense conflicts with any notion of heat as substance, whether in the early form of "caloric" or the modern one of
This congenial notion of the Hot and the Cold conflicts with physics so long as we fail to recognize
that here again the commonsense-perceptual categories are inherently a different sectioning of experience than that of
modern science, more discriminating for many of the purposes of common life but less significant as abstractly universal.
This commonsense notion of the Hot and the Cold can he mapped into the scientific framework only after we know a great deal
not only about physical heat and thermodynamics but also about the temperature sense, its linking role in the homeostatic
regulating mechanisms of the animal body and its purely psychological aspects. If the physical concept of heat appears to
common sense as inaccessibly recondite, the commonsense notion of heat can be represented scientifically only by a complex
and perhaps still incomplete model. The transformation from one conceptual domain to the other is not one-to-one, is not
simple; it is one-to-many and many-to-one. Here as elsewhere, of course, the scientific concept has its roots in common
experience and thought, but the steps by which it has evolved took two centuries or more of analysis and research, reaching
into the last decades of the nineteenth century.
Elementary Mechanics. Historically the earliest modern science mechanics is beset by many similar
pedagogical troubles. Even the idea of balance of forces, which goes back to the Greeks and is treated as a dull little
subject introductory to the older texts, is in fact a fascinating thicket of these troubles. Archimedes derived the law of
the balance from pure considerations of symmetry, by a style of argument which is powerful and deep, anticipating that of
Liebnitz and of modern theoretical physics; it is close to common sense but not as a formal intuition, not for predicting
the stability or instability of structures made of wooden blocks or Tinkertoys. Almost none of our subjects knew ways of
thinking about the stability or instability of balance. In this context the image of the center of mass lying at the bottom
of a hill is radically difficult to reach. This does not imply the technical vocabulary I use; the image can be that of a
marble in a bowl, but the linkage of analogy is unavailable. Similarly, the transition from Aristotle to Galileo in the
discussion of motion is equally unavailable. For perceptual common sense, motion is always and inevitably in a medium; air
may not be thought of as real but space is definitely full, not empty. Mechanics derives Stokes' law for falling bodies by
adding a resisting medium to Galileo's law. Common sense, like Aristotle, has to go the opposite route, but it abhors the
distinction between air and the vacuum.
Mechanics is full of examples of things which for most of my subjects are unteachable by standard means
and which, if so taught, hardly go below the level of verbal discourse and artificial problem-solving. They certainly do not
become what Piaget called schemes, penetrating to what Dewey calls "the subsoil of the mind." Common sense says that the
wall does not push back on me when I lean on it; the flight of the airplane moves nothing downward to keep the plane up.
Perhaps the textbook science is stored for a while in some basket of recall but much of this learning can be unlearned; it
is not irreversible.
An alert and apparently very lively college sophomore had what appeared to be incurable difficulties
with the idea of the relativity of motion. The context was that of an introduction to astronomy, but homely examples were
to no avail, nor was patient explanation after class. An imaginative tutor finally got the student to pirouette
counterclockwise while observing the walls and ceiling, inviting her to imagine that she was stationary and the room
rotating clockwise. After two or three trials it suddenly worked, with the characteristic high emotional release. In this
case the change was major and unusually dramatic; she moved from failing grades to a very adequate final paper on the
kinematic equivalence relation between the Ptolemaic and the Copernican models of the solar system. It is not always so
simple; students more often must relive such transitions repeatedly. A teacher for whom kinematic relativity is second
nature may fail entirely to grasp the intellectual nature of this difficulty or to understand that explanation with
diagrams, no matter how patient, inevitably presupposes the very conceptual transition it seeks to explain. Historically,
we are reminded that even Galileo did not describe inertial motion with full generality of context, but only on a horizontal
plane. The thought experiment which requires a body moving arbitrarily in empty space was apparently not available to him.
It seems evident that in considering these critical barriers we must avoid a confusion of levels in learning. In many cases
less obvious than those discussed above verbal structures are often received and in some ways assimilated by students. These
structures may be returned on examinations or even applied to the solution of simple problems but what has been so learned
does not prove retrievable or applicable in new situations, especially those arising outside of class or in later years. The
loss rate of isolated knowledge transmitted in science classes is often about equal to the rate at which the knowledge is
gained. The partial recognition of these problems is very old, probably as old as formal instruction, but somehow they have
not been brought into sharp focus.
It is not appropriate to discourse here about the psychology of educationally significant learning, for
which in fact we have no widely received and powerful theory. It is, however, appropriate to distinguish between learning
conceived of as the reception, retention, and recall of verbally coded and transmitted information, and learning understood
as the development of Intellectual habits for transforming sensory or verbal information to bring it into congruence or
conflict with prior general knowledge or belief. The critical barrier phenomena suggest that it is this latter kind of
learning which has failed to take place. If such matters have been taught in a superficial wayverbally transmitted,
momentarily understood, and retrievable as fact but not transformed into tools or disciplines for further learningthen
loss or burial is unavoidable. A teacher who had been taught about the conservation of mass, in high school or college,
could maintain without conflict that a terrarium sealed for seven years now weighed more than when she had planted and
sealed it "because the plants are bigger." She could be reminded of her earlier learning, but only very slowly did she
acknowledge, with final delight, the logical quandary involved.
I have deliberately emphasized the prevalence of learning failures of the most elementary kind, but such
failures also occur at higher levels, even among the scientifically learned. The very high energies of cosmic rays were for
a long time regarded as a prime mystery. Only two or three decades ago Fermi pointed out that dynamic equilibrium between
stars and free atoms in space would imply even larger cosmic ray energies than those observedthe principle of
equipartition. Suddenly, as a result of Fermi's observation, the question was reversed; why aren't the cosmic ray energies
larger? In a recent popular television program on man-powered flight, many fine technical details were mentioned, but no one
thought to dramatize the simple fact that even a bird geometrically scaled up to the mass of a human being couldn't fly.
The difference between a bird and the man-powered Gossamer Condor is of a piece with the anatomical contrast between mice
and elephants. These two examples are at very different levels of scientific knowledge and sophistication but the latter
was as unavailable to the learned of three centuries ago as was the former to those of three decades past.
I have emphasized elementary examples for several reasons. First, they are commonly overlooked
prerequisites for even the kind of basic scientific culture we deem necessary to life in our present world. Second, what is
elementary from a scientific point of view was often unavailable even to the learned of a relatively recent past.
"Elementary" should not be thought of as meaning easy or innately understandable. A sense for powerful elementary ideas is
not the beginning of scientific knowledge but is typically a late product of its evolution. Individual learning does not
have to recapitulate history, but history can tell us a lot, commonly overlooked, about the dimensions of the learning and
teaching tasks we face.
A third reason for my emphasis is to combat the commonly received notion that widespread scientific
education and culture is increasingly problematic because of the vast increase in scientific knowledge, which allegedly
requires a specialization beyond any layman's possible understanding. But the power of even simple scientific ideas, fully
mastered and enjoyed, can make the scientific world-picture intelligible overall and in first approximation, and that is
the level at which I believe we have mostly failed. How else can we understand the prevailing level of PR about something
called the neutron bomb?
Since the immediate purpose of this essay is to propose a definition of goals sharp enough to suggest
directions of search and research into means of achieving these goals, I think it is proper to emphasize still further the
distinction between the two levels of learning mentioned abovethe "verbal structure" level and the level of true
conceptual understanding, of easy insight. It has been the historical aim of science both to extend our experience and to
reduce it to order, these two aspects being always interconnected. The ancient astronomersearly Greek or
pre-Greekextended their experience by carefully mapping the sky and its motions over centuries. At some point this
suggested, or allowed, the strange notion that Earth was not an indefinitely extended cosmological boundary but a thing,
a body, perhaps a sphere, poised in space. This was proposed as a fact which fitted all the data, but it was much more;
it was a reduction to order of many otherwise unrelated astronomical phenomena. But this new order conflicted with
commonsense intuition, which required a universal cosmological up and down. The conflation of these two ways of thinking
created the uneasy question about why the Earth, now a body among bodies rather than a cosmographic division, didn't fall,
and a question about upside-down inhabitants of the antipodes. Even Dante put the entrance to Hell down there. A century
or two after the early Greek discoveries, Aristotle announced, with a lingering note of triumphant understanding, that
"down is toward the center." The round Earth-body was not simply a new fact to be stored along with other facts; it was a
fact which required a radical reorganization of the whole category structure of geographical and cosmological thinking.
If it were taught merely as a fact, without appreciation of the need to help it penetrate into the subsoil of understanding
and to rebuild the mind's category structures in the process, it would remain something merely bookish and abstract, to be
entertained nervously and then forgotten. Perhaps children of today can grow up without this particular conflict of
understanding, one which many of us can remember from our own childhoods. The educational time scale here, that of the
transition from opaque fact to intuitive widespread grasp, has been at least a couple of millenia. We ought to do better.
This article is an edited version of a paper written for the Directorate for Science Education,
Division of Science Education Development and Research, National Science Foundation. It was commissioned during the course
of a study attempting to define needed research into the area of scientific literacy. David Hawkins is at
the Mountain View Center for Environmental Education, University of Colorado, Boulder, CO 80309
(Copyright permission granted by the author.)