I've been rereading Kiselev's two volume work, designed
for students just starting on Euclidean geometry.
The translator, Alexander Givental, provides some interesting
analysis at the end of volume 2 (Stereometry) trying
to debunk some of the myths that swirl around this
Why do we teach it again?
Is it really true that "formal reasoning" is the royal road?
Don't mathematicians also think heuristically, and don't
axioms often come towards the end of the day, more as
definitional and unifying than as "self evident truths"?
What is it about "rigor" that we're trying to get across?
Speaking of rigor, one thing I notice about the Euclidean
mindset is the notion of congruence teaches us to
overlook chirality or handedness.
In the opening pages, the notion is developed that
geometric figures may be slid around (translated)
and rotated, and that if, by these means, two such
figures may be shown to superimpose, such that
all features align, then these two figures must be
congruent. So far so good.
However, immediately after the isosceles triangle is
introduced, things get messy, as we're introduced
to symmetrical figures that cannot be made to
superimpose by translation or rotation. Anyone
who has played Tetris will know what I'm talking
A left-handed L is just not going to do the work
of a right-handed L, no matter how ya turn it.
Getting the "wrong or right L" can be a game-
changing experience. Yet in Kiselev, we're told
that to pick a figure off the plane temporarily,
and to set it down reversed, is a congruence-
I can sense students starting to squirm at this
point, as the logic hardly seems secure.
Consider the analogous situation in space. Having
two left-handed gloves or right-footed shoes is a
show stopper, in terms of having a proper pair.
The way to turn a left handed glove into a
right handed one is to turn it inside out. This
is like folding a tetrahedron back to a flat
plane-net, then creasing the folds the other
way, such that what used to be a convexity
is now a concavity and vice versa.
Should such a radical operation be considered
"congruence preserving"? How can a left shoe
and a right shoe be considered "the same"
when clearly they ain't?
At least we should empathize with students who
see some sleight of hand going on. We can
explain that these are just the conventions,
evolved over a long period. It's not like we can't
imagine alternatives, such as non-Euclidean
geometries that really care more about handedness.
On another list, I'm back to discussing Karl Menger's
"geometry of lumps" again. He proposed that
one way to branch away from Euclidean geometry
is precisely in this matter of initial definitions.
Instead of saying a point has no parts, or that a
line has no thickness, go ahead and have all
your entities made from "clay" ("res extensa"
in Cartesian terminology). Take a topological
approach: planes are not points because
they're flat and thin, not round or pointy.
Lines are like sausages, or thinner, but are
not infinitely thin (nor infinitely long).
So these objects are no longer distinguished
by "dimension number" then. Could a consistent
geometry be built in this manner? Menger proposes
this challenge in:
'Modern Geometry and the Theory of Relativity',
in Albert Einstein: Philosopher-Scientist ,
The Library of Living Philosophers VII, edited
by P. A. Schilpp, Evanston, Illinois, pp. 459-474.
Here's a web site about Karl:
The goal, in running by this alternative set of definitions
(or axioms if you prefer) is to sensitize students to what
is meant by "axioms" in the first place. Jiggering with
Euclid's 5th postulate is not the only way to set a branch
point, although one may do that too.
Zooming back from the sandy beach, locally flat, the
place where Euclideans inscribe their proofs (using
lines with thickness, traced with sticks in the sand,
a string for a fixed length), we see that said planar
surface is actually curved (globally speaking), as
are the lines thereon.
Flatness is a local / parochial phenomenon, not
something "to infinity". Yet Euclidean constructions
are possible in such a setting (or call it "ancient
And really, what standard Euclidean constructions
require "infinity" in the first place? Finite Universe
models work just as well (cite Knuth), in terms
of setting the stage for those various proofs about
triangles and parallelograms etc. So again, we have
our non-Euclidean alternatives.
Givental takes issue with tautological / dogmatic
expositions of geometry as the worst way to
approach the subject. Taking Euclideanism with
a grain of salt, rather than with worshipful
obedience, is probably a good way to inspire
critical thinking and a better appreciation for
what real mathematics is like. It's not about
developing rigid / set beliefs about the "one way"
it has to be.
Math is not a "my way or the highway" kind
of discipline, much as some authoritarian
types would like it to be.
Challenging the primacy or inevitability of
Euclidean geometry is a useful exercise, not
just for advanced students, but for those
just starting out. Why? Because we
don't want them to feel intimidated by
some onerous "one truth".
This subject of handedness is pretty hot
by the way, features in this debate about
what counts as the most primitive space-filler
(a pretty basic question, accessible to
laypersons). Hexahedra and pentahedra
don't count because they're topologically
too complicated. All the candidates are
tetrahedra, but which one is the winner?
You get some Archimedeans pointing to
the ortho-scheme of the cube, otherwise
known as "the characteristic tetrahedron"
or 1/48th of a cube. This is their champion
in the ring.
The problem with this selection is obvious:
said characteristic tetrahedron is either
left or right handed and will only fill space
in complement with its mirror. Ergo, we're
really talking about a *pair* of tetrahedra,
not a single space-filler.
Joining two such characteristic tetrahedra, to
form another tetrahedron, is arguably to generate
the true minimum space-filler, a tetrahedron
that fills space with identical copies of itself,
no worries about handedness. This was
D.M.Y. Sommerville's approach and his
argument is used by Senechal to explain why
some Aristotle fans have tended to bristle
over the oft stated claim that Aristotle was
wrong about tetrahedra filling space (see
Math World under space filling): they
most certainly do (he never said "regular").
Which Tetrahedra Fill Space? by Marjorie Senechal
Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981),
If this debate resolves the way I think it should, then
we have all the more reason for advancing our "minimum
tetrahedron" or Mite in more of the literature.
Sommerville identified this shape in his research,
as well as some of the shapes it makes in turn,
additional space-filling tetrahedra (e.g. the
disphenoid tetrahedron and the mono-rectangular
tetrahedron). It even assembles the characteristic
tetrahedron. Indeed, every space-filling tetrahedron
on Guy Inchbald's useful chart may be assembled
from our Minimum Tetrahedron (mite).
(note handed characteristic tetrahedron at the top --
the "tap root" of Archimedean thinking -- with
our champion, the Mite on the 2nd row, the three
Sytes below that: Lite, Bite and Rite respectively.
Four mites will make the characteristic tetrahedron
so it needn't be top row if we wish to organize
our thinking differently (no reason we shouldn't)).
These amateur geometry teachers discover the
disphenoid tetrahedron (rite) for themselves in this
oft cited Math Forum paper, but don't seem to realize
it's composed of two Sommerville tetrahedra (or
we could call them Aristotle's?).
An Amazing, Space Filling, Non-regular Tetrahedron
by Joyce Frost and Peg Cagle
These threads (handedness and space-filling tessellations)
are linked, in that Euclidean geometry may de-sensitize
students from appreciating handedness. Its notion of
"congruence" obscures it. Having a poor grasp of
handedness might lead students to think the so-called
"characteristic tetrahedron" is capable of filling space
without pairing up with its mirror image.
An interesting hypothesis will test whether our intro-
duction of non-Euclidean definitions helps students
grasp the importance of handedness. We have other
reasons for introducing Menger's geometry of lumps.
For example, in a ray tracing context, it makes little
sense to define anything "dimensionless" as any
object needs to reflect light, or might as well not be
there at all. This goes for points, lines, planes and
For those of you not familiar with Kiselev's Geometry,
these two volumes were at the heart of the Russian
school system, starting in the Tsarist period and
on through the opening decades of the USSR.
For a time, they were considered THE geometry
textbooks of that nation. They have only recently
been translated into English (2008) by A. Givental,
based at UC Berkeley.
Kirby wrote (in part)...
"Instead of saying a point has no parts, or that a
line has no thickness, go ahead and have all
your entities made from "clay" ("res extensa"
in Cartesian terminology). Take a topological
approach: planes are not points because
they're flat and thin, not round or pointy.
Lines are like sausages, or thinner, but are
not infinitely thin (nor infinitely long)."
We teach geometry exactly because the intuition as you espouse is insufficient, too imprecise and woefully not up to the task of describing the concepts and "things" of geometry or any technical endeavor. I would use such analogies as a very early starting point in a discussion of the abstractions of point and line but quickly get to the gist as to why there are invalid examples.
This thread of yours, and it has appeared before, is a form of primitivism. It is a natural instinct in all of us when we become overwhelmed by the enormity and complexity of human achievement and when it is presented all at once it cause discomfort and for an instant we tend to reject it. You have taken that instant and stretched it into a lifetime.
Thinking of lines like thin sausages is deficient and ineffective because we could not speak of tangents and other such constructions. Oh, I am sure you will counter that we can have a thin sausage in the shape of a circle and another in the shape of a line and the latter could be placed as to touch the former. And before I ask you what does "in the shape of a line or circle" means I would ask what does "touch" mean. And you would say "you know, like touch, like their edges touch". And I would ask "Edge? Is that another thin sausage? Does it have an edge as well?".
Rather than try to describe why you think we should not teach euclidean geometry, why not first impress us with why we do teach it? And then, when we have sufficient belief that you know what you are talking about, we might better entertain your ideas on how we might teach it better. But in your case you are making the argument that we shouldn't teach it at all. But that is your conundrum isn't it? If you were to actually state WHY we teach it and then state that we shouldn't teach it, some child in the audience will ask "How did you jump from WHY to SHOULDN'T" and your magic act would be kaput.
Out of curiosity, what part of "handedness" do you not understand? Or do you object to the definition of "congruency"?
"Consider the analogous situation in space. Having
two left-handed gloves or right-footed shoes is a
show stopper, in terms of having a proper pair."
Gloves as you describe are 3 dimensional and have (in varying degrees) knuckles and palms. If you had stayed in 2 dimensions as your first paragraph clearly was, you would have went down to Walmart and bought a box of 50 latex gloves that are all made identical (congruent) and found that they can indeed be stacked one on top of the other.
Your entire post serves as an example to the point of WHY we teach euclidian geometry. So that we don't confuse ourselves with imprecise terms and definitions and have the discussion turn into something better done over 3 or 4 beers.
On Fri, Jul 9, 2010 at 1:57 AM, Robert Hansen <[hidden email]> wrote:
> Kirby wrote (in part)...
> "Instead of saying a point has no parts, or that a
> line has no thickness, go ahead and have all
> your entities made from "clay" ("res extensa"
> in Cartesian terminology). Take a topological
> approach: planes are not points because
> they're flat and thin, not round or pointy.
> Lines are like sausages, or thinner, but are
> not infinitely thin (nor infinitely long)."
> We teach geometry exactly because the intuition as you espouse is insufficient, too imprecise and woefully not up to the task of describing the concepts and "things" of geometry or any technical endeavor. I would use such analogies as a very early starting point in a discussion of the abstractions of point and line but quickly get to the gist as to why there are invalid examples.
I wouldn't use words like "invalid" at the axiomatic level as this is
woefully imprecise and non-mathematical thinking. If we want to have
a "geometry of lumps" as Karl Menger suggested we might , then
here's a place to start.
The Euclidean definitions are another place to start.
Your response is dismissive, but also seems instinctively reflexive
("knee jerk" as they say).
> This thread of yours, and it has appeared before, is a form of primitivism. It is a natural instinct in all of us when we become overwhelmed by the enormity and complexity of human achievement and when it is presented all at once it cause discomfort and for an instant we tend to reject it. You have taken that instant and stretched it into a lifetime.
If by "human achievement" you mean the Euclidean tradition in
particular, then why not say so?
You seem to be practicing exactly what you decry: I put something out
there that's unfamiliar, yet humanly achieved, and you're
uncomfortable, have to resort to words like "primitive" and "instinct"
to cast aspersions on alternative definitions. I must defend my
mathematical freedoms? Even on math-teach?
Actually, your reaction comes across as typical of those with only a
traditional training in schoolish math: they have only a cursory
ability to philosophize.
When placed in a room full of youthful questioning minds, where
countering intuitions do occur ("how can dimensionless points add up
to form a line?"), you get these kinds of unthinking dismissals from
impatient teachers, trapped by the tests.
Rather than have a real discussion of the philosophical issues being
raised, they just say (or imply) "this is how it must be, don't
question your elders". This is how mathematics becomes such a
turn-off for so many -- it's taught in a dogmatic manner, like a
Thank you for amply demonstrating this widespread weakness in our math
teaching subculture. We should have a longer discussion, as I think
we're pinpointing something at the core of math's unpopularity.
> Thinking of lines like thin sausages is deficient and ineffective because we could not speak of tangents and other such constructions. Oh, I am sure you will counter that we can have a thin sausage in the shape of a circle and another in the shape of a line and the latter could be placed as to touch the former. And before I ask you what does "in the shape of a line or circle" means I would ask what does "touch" mean. And you would say "you know, like touch, like their edges touch". And I would ask "Edge? Is that another thin sausage? Does it have an edge as well?".
Every line ever drawn in any geometry book has thickness. Zoom in
more closely, and you'll find the ink blots in a mix with cellulose or
papyrus or whatever substance.
The doctrine of "essentialism" suggests that we must abstract from all
special case experiences to find what is essential to all of them
("family resemblance" is not good enough), and the resulting
distillate may result in an invalidation of every phenomenon as "an
Once in the euclidean namespace, no point, line or plane in the
geographical vista, nor even in the imagination, is a "real" point
line or plane, as these latter have no Earthly existence, only a
ghostly metaphysical presence as the "ideal forms" of a Platonic
That's a dogma we want students to swallow whole, without question,
and while still at too young an age to defend themselves against
Whereas in philosophy class it's maybe OK to call this essentialist
doctrine "a superstition" (or even "BS"), in a geometry class you just
get stern glares from harried teachers who need to be teaching to
those tests, have only limited time for those "smart mouths" who ask
the most uncomfortable questions (these might be future philosophers,
but more likely they'll wind up as high school drop outs, given their
low tolerance for authoritarian posers).
Raising these questions about what "edge" or "tangency" might mean,
when confronted with so many special case experiences (geographic
phenomena), is worthwhile. We're doing a philosophical investigation
into some related concepts, and that's to be encouraged.
There's no getting away from having so many special case experiences
in reality (where geometry is often applied), and unwanted ambiguity
does sometimes creep in. So I'm with you on seeking precision, look
for it in the Gulf (where mere PR seems to have an upper hand these
> Rather than try to describe why you think we should not teach euclidean geometry, why not first impress us with why we do teach it? And then, when we have sufficient belief that you know what you are talking about, we might better entertain your ideas on how we might teach it better. But in your case you are making the argument that we shouldn't teach it at all. But that is your conundrum isn't it? If you were to actually state WHY we teach it and then state that we shouldn't teach it, some child in the audience will ask "How did you jump from WHY to SHOULDN'T" and your magic act would be kaput.
You apparently are reading too quickly. I have not said we should not
teach euclidean geometry. I think we'll teach it much more
effectively when we set it next to other geometries that define their
core terms somewhat differently. These could be partially
overlapping, in terms of content. Even a non-Euclidean might agree
that diagonals of a rhombus bisect one another at right angles, while
also bisecting opposite angles. You can show this with a stick in the
sand and a two column proof.
> Out of curiosity, what part of "handedness" do you not understand? Or do you object to the definition of "congruency"?
I'm saying we should be clear that, in some language games, you can't
just "flip" the letter R to make it a backwards R. There's no way to
superimpose these or make them "the same" in the context of the rules
(I gave the example of Tetris).
So the notion of "congruency" is more subtle and nuanced than perhaps
many texts allow -- especially when we get to spatial geometry, where
you will not be able to assemble your cube from 48 "characteristic
tetrahedra" if these latter are delivered to the construction site in
any combination other than 24 left-handed and 24 right-handed.
Saying they should all be "congruent" is to obscure this requirement
(for equal numbers of left and right). It's that same lack of
precision, leading to unwanted ambiguity, that I was discussing above.
Teaching about "congruency" is OK, but lets make sure we spend ample
time on the fine print. For SSS and SAS to work, we must accept that
we allow "flipping" at will (pancakes off the griddle, spinning in
3-space -- ergo 2-d geometry was really 3-d in disguise (we shoulda
told them that (hard core euclidean is 3-d, not just flatlander stuff,
contrary to popular textbook dumbings down))).
While on this topic of left and right, is there a minimum tetrahedron
that will fill space yet *not* raise concerns about left and right
handed? This was the question posed by Sommerville and he came up
with a tri-rectangular tetrahedron some call the "minimum tetrahedron"
for this very reason (or "mite" for short). It's composed of two
face-bonded characteristic tetrahedra. It also builds the
characteristic tetrahedra (left and right).
The tetrahedron is our most important polyhedron, given it's the
simplest. We tend to use it as our unit of volume in some geometries
(not the one you learned in school of course -- that was strictly
cube-based, by convention, which is why you say "cubed" and "squared"
for 3rd and 2nd powering respectively, an ethnic characteristic of
your tribe and/or "calculation nation" (a plug for Thinkfinity,
sponsored by Verizon -- was just in a webinar about that yesterday)).
> "Consider the analogous situation in space. Having
> two left-handed gloves or right-footed shoes is a
> show stopper, in terms of having a proper pair."
> Gloves as you describe are 3 dimensional and have (in varying degrees) knuckles and palms. If you had stayed in 2 dimensions as your first paragraph clearly was, you would have went down to Walmart and bought a box of 50 latex gloves that are all made identical (congruent) and found that they can indeed be stacked one on top of the other.
"Three dimensional" is one way to talk, sure. If we go with Menger's
suggestion (he was a dimension theorist) we might dispense with the
"zero, one, two, three" dimension talk and craft a different language
game instead. This is something that mathematics allows. Pay no
attention to those authoritarians who warn against being too
instinctive or primitive.
Latex gloves can be turned inside out pretty easily, if they're really
simple. There's no fur lining, no special ribbing on the outside. A
theoretical featureless left handed glove can indeed be turned into a
right handed glove and vice versa. It's harder to imagine doing this
> Your entire post serves as an example to the point of WHY we teach euclidian geometry. So that we don't confuse ourselves with imprecise terms and definitions and have the discussion turn into something better done over 3 or 4 beers.
I'm not against teaching euclidean geometry, to again make myself clear.
But there's a lot of time in the day and one way to take the edge off
these dogmatic approaches (warned against by Givental, our translator
of Kiselev) is to compare and contrast the euclidean definitions with
some others. This is all within the purview of mathematics (we're not
breaking any rules of logic, per the Bertrand Russell school).
Lets show 2nd and 3rd powering using triangles and tetrahedra for
example. There's a consistent model for doing that, which, when set
against the cubic model, reveals the latter to be arbitrary, cultural
(as is the former -- though lets remember which of these two shapes is
more topologically primitive and structurally stable to boot).
I'd consider that nuanced discussion (of what we mean by "powering") a
requirement for any world class high school geometry curriculum.
Leave out tetrahedral mensuration and you're back to primitive,
imprecise, Euro-centric thinking. Chinese, in the meantime, will be
sharing about the volume 6 rhombic dodecahedron, made of 48 mites of
volume 1/8 (all referenced to some paradigm XYZ-like grid), leaving
their peers in the dust when it comes to geometric sophistication
(important in this nano-tech age).
We should do this kind of "philosophy for children" early and often,
lest students grow up to become adults such as yourself, easily
confused by alternatives and prone to reflexively retaliate in a
bigoted way, when confronted by a different ethnicity (one that
doesn't admire cubes so much, for example).
Mathematics hasn't been philosophy for 1000's of years Kirby, and in preparation to my reponse to Richard, Physics hasn't been philosophy for 100's of years.
You say that you put something out there that is unfamiliar and thus I dismiss it. I would say that what you posted is all too familiar and thus my analysis. Well, except for this obssession you have with polyhedra, and I have said before that I can't hook up on that one at all. But your want to regress and philosophize math again is pretty darn apparent.
Why would we regress? Do you understand that math, and all the sciences for that matter, divorced themselves from philosophy over the centuries because they outgrew philosophy? Well, except for psychology, it is still too far out of reach.
> Mathematics hasn't been philosophy for 1000's of
> years Kirby, and in preparation to my reponse to
> Richard, Physics hasn't been philosophy for 100's of
That's not my reading of history. Mathematics and
philosophy remain inter-twined, very deeply, at the
foundations, right down to our own time. I think
you are confusing how these subjects are taught
(compartmentalized from one another, artificially
divided), from how they in fact co-evolve.
Here's a good read for high schoolers, one which well
makes this point:
> You say that you put something out there that is
> unfamiliar and thus I dismiss it. I would say that
> what you posted is all too familiar and thus my
> analysis. Well, except for this obssession you have
> with polyhedra, and I have said before that I can't
> hook up on that one at all. But your want to regress
> and philosophize math again is pretty darn apparent.
Polyhedra are basic to the western tradition, from
Leonardo da Vinci back to Plato and forward to
contemporary mathematics. Below I've appended a mapping
between Fuller's modules (what he names them) and the
very long, awkward names that keeps this whole branch
of geometry locked away behind college level degree
programs where only over-specialists would venture.
Spatial geometry is made artificially difficult by
all the cruft that has grown up around it. We need
some fresh air, and that's what my thread is aimed
> Why would we regress? Do you understand that math,
> and all the sciences for that matter, divorced
> themselves from philosophy over the centuries because
> they outgrew philosophy? Well, except for psychology,
> it is still too far out of reach.
I'm not advocating regressing. I'm advocating getting
on to the next chapter. Philosophy is a potential
catalyst for doing this, in any age. We have not
"grown beyond" philosophy. Rather, we have imprisoned
children within a dogmatically delivered geometry
curriculum that is depriving them of their heritage.
In my book, this is inexcusable and I'm sorry to see
national standards spreading far and wide which perpetuate
this ugly state of affairs. Freedom-loving Americans
should fight this trend. Focusing on tetrahedral
mensuration, pointing out its absence, is a great way
to counter this mindless bureaucracy or "hive mind" that
would entrench itself to the detriment of our future
quality of life. Or maybe the Russians will have better
luck, given how totalitarian and hegemonic the US has
become (what Walter Kaufmann used to warn us might happen
- -- some philosopher I studied under **).
Note that once you get some of these dots on the map,
it's a lot easier to connect to the humanities and
intellectual history more generally, given the
syllabus involved. Fuller was an American
Transcendentalsit like his great aunt Margaret Fuller
Osoli, who worked with Emerson and Thoreau on Dial
Magazine. He features in 'The Pound Era' by Hugh
Kenner, one of the leading James Joyce scholars and
author of 'Geodesic Math and How to Use It'.
The link to Coxeter takes us to Wittgenstein (one of
his teachers) and Bertrand Russell (per Logicomix above).
Fuller received a Medal of Freedom from Ronald Reagan
and worked closely with the Marines in the 1950s (on
aviation logistics) as well as on the radomes.
'The King of Infinite Space', the recent bio on Donald
Coxeter, contains a lot of this information and is
the kind of summer reading any thoughtful young high
schooler might be considering right now (but probably
won't be, as our school system has dumbed things down
so bad that they've likely never heard of either
Coxeter *or* Fuller -- blame ETS?).
Appended info. Could be taught in high school, but is
today locked away on high level math tracks only. Yet
spatial geometry is at the heart of chemistry, biology,
architecture etc. Why surrender to past reflex
conditioning? "Are we not men?" (Devo).
Space-fillers (used in tessellations):
Inherited name = concentric hierarchy name
Trirectangular Tetrahedron = mite
Disphenoid Tetrahedron = rite
Skewed Trigonal Dipyramid = lite
Monorectangular Tetrahedron = bite
Rhombic Pyramid = kat
Oblate Square Pyramid = kate
Two Kites make...
Oblate Octahedron = coupler
Made of three Lites:
Aysmmetric Trigonal Dipyramid = octet
(based on unit volume tetrahedron -- see "cubing"
("qyoobing") vs. our alternative model of
3rd powering (aka "verboten math" -- nowhere
included on "hive mind standards")):
[ subtypes ]
MITE: 1/8 (= A + A + B modules each of 1/24)
[ the mite is a space-filling tetrahedron that, unlike
the "characteristic tetrahedron", does not need a
complementary mirror image -- other space-filling
tetrahedra may be made from it ]
SYTES: 1/4 (= Mite + Mite)
KITES: 1/2 (= Syte + Syte)
Kat (Bite + Bite or Rite + Rite)
Kate (Bite + Bite)
Kit (Lite + Lite)
RHOMBIC DODECAHEDRON: 6
I wonder why Robert thinks the way he does about philosophy, math and science? I see philosophy as intertwined in disciplines and very much cutting-edge.Looking at the polyhedra in new ways, through you, has opened my eyes, stretched my curiosity and I am delighted.
I think it is healthy to compare and contrast, connect with prior learning and wonder why and how in any subject, or on any topic, or observations within subjects. It is what educators commonly call higher order thinking - a tool that can prepare learners for more rigorous study in any subject.
Thinking of volume in terms other than cube seems to me like the "art" aspect of math. Why not investigate?
"I wonder why Robert thinks the way he does about philosophy, math and science?"
I'll answer your question with a question. If you couldn't play the piano, would it make much sense teaching others how to play the piano?
Being both a mathematician and a scientest, I would think I would know what goes into achieving these things.
And along the lines of my last response, consider this...
You watch someone play an instrument and you think to yourself how effortless they make it appear. And to them it is effortless. They do not even think about the effort, they are fluent in the effort, it is second nature to them. They are artists, not beholders of art, creators of it. They play in it, they live in it and they sing in it. They may have started life just like you or I but something in them, when presented with the possibilities, took to music. They might as well be in a different dimension. We call that talent.
That is how math works. That is the art in math. That is how many skills work. We call that life.
My concern is not about how to make math entertaining or delightful for the masses or to obfuscate it with polyhedra and philosophy. My concern is that students have available to them classes like I did. So that I don't have to sit through more meetings where the topic is filling positions with offshore talent.
You might want to think about that in your STEM endeavor. Your heart might be in the right place but you might want to visit some companies where STEM is played out. I'm there every day so this is second nature to me.
Is mathematics artful? Absolutely, but there is a problem. Consider the following link...
I can do that in math, and even envision the catchy soundtrack. I can't do that on the iPad though I certainly recognize the skill, the process, the attention and the personal creativity as being very much the same. The problem however is that my canvas is abstract and except for other mathematically inclined people, math art cannot be sensed like that. We all have built in senses for music and art, and excepting for taste, we immediately feel something when we experience it, without even having to be an artist. Math does not work that way and I know that people have tried for decades to make that truth untrue and to say that they have been anything but unsuccessful would be a complete lie. Unfortunately our sense of abstract appears to not be as universal as our tangible senses of music and art (damn Piaget). I fully realize that you want to drag math out of that abstract world into your tangible world so that you can touch it and feel it. But it cannot exist there, and I d!
oubt 1000 mathematicians could ever get that point across to you. Until you can solve the riddle of why some of us have a taste for the abstract while the rest do not, you should try to do something more productive. I mentioned spreadsheet math because I am a big believer in experiment and observation. The experiment was started some 40 years ago when myself and everyone else in my immediate generation started this journey of school and life. The results were that the majority of those that went on to professional lives (i.e. did any of their school work) are now using spreadsheets and forgot 90% of any algebra they might have learned. They are productive and effective. I don't know what else to say other than what is.
RE:On Mon, July 12, 2010 at 12:28 AM, Robert Hansen <[hidden email]> wrote:
"But it cannot exist there, and I doubt 1000 mathematicians could ever get that point across to you. Until you can solve the riddle of why some of us have a taste for the abstract while the rest do not, you should try to do something more productive. "
Are you suggesting that unless one has a "taste" for the abstract, that they are unable to build skills in abstract thinking? Should we only teach mathematics to those who have the "taste'? And further, how would you go about testing whether someone has the "taste"? Poor grades in courses are not the only indication of a student having little "taste" for any content. There are many other reasons why students do not complete their work and earn satisfactory grades, such as social, emotional, economic, etc. issues. How would you assess the "taste"?
Across all disciplines, I assert, that if a learner gets engaged, listens, thinks, asks questions and does their work that they can "grow" their abstract thinking skills and indeed, be successful over time, in work requiring abstract thought - regardless of any inherent "taste."
Is mathematics only for the elite in a stand alone module? For example, consider, how then, do story problems that require linguistic skill fit it to a stand alone mathematics module? I see inter-connectivity, here and in other areas.
What exactly are you referring to as unproductive?
Having an open mind to others' views, responding respectful and diplomatic is healthy. Isn't this something we would suggest to our nation's youth?
First of all, thank you for sharing your perspective. Secondly, I think you and I do not see learning in the same light. What I understand from reading what you have written, is that you go with the idea that students learn because teachers teach the right courses like you had the opportunity to take.
Well, I see it from a very different perspective. In my view, stemming from my research and experience - each learner constructs their own learning, while we, as educators, may introduce skills, concepts and assign homework, it is the learner who is taking this input and constructing their own learning.
When I teach, I think of myself as much as a learner as teacher - I am one facilitating, guiding - not the "all knowing expert" full of skills to impart. Instead, I encourage my students to think, ask silly questions and become engaged in the content, so that they take a personal interest and may learn whatever they need to reach their goals. Without the engagement of learners, it does not matter if the opportunities that you speak of are are present or not - very little learning or skill building can take place without engagement.
In my mind, the place to start is with good standards and methods of assessing whether they have been met or not. The next step would be the design or selection of learning activities to engage and support very focused learning objectives (directly from standards) - ones that encourage learners to use our natural born human curiosity about our world.
I suggest that finding more ways in which to engage learners in content, using interdisciplinary "hooks" is much stronger motivation, than just practicing skills, so that one day you may be able to do the "real" stuff.
Please see my embedded responses below:
On Sat, Jul 10, 2010 at 10:04 PM, Robert Hansen <[hidden email]> wrote:
"And along the lines of my last response, consider this...
"You watch someone play an instrument and you think to yourself how effortless they make it appear. And to them it is effortless. They do not even think about the effort, they are fluent in the effort, it is second nature to them. They are artists, not beholders of art, creators of it. They play in it, they live in it and they sing in it. They may have started life just like you or I but something in them, when presented with the possibilities, took to music. They might as well be in a different dimension. We call that talent."
Keeping with the musician scenario -
I am a musician, so I have a perspective to share. I was initially classically trained to read and play by the musical score. However, I always had a fascination with tones and rhythms, which led me to explore and create improvisational work. I now favor improvisation, over reading and playing the scores of other composers. I know many classically trained musicians who are great, but cannot play if they do not have a score in front of them. What I suggest is that it may be one part of our brain that plays mechanically by the music score and another one that creates.
An unexplained phenomenon comes to mind, how is that some persons cannot read a single note, yet they can create awesome compositions, often that they first hear in their heads? This has nothing to do with the musical score. When I am improvising, I often "hear" the next tone, set of tones, or rhythms in my head and through previous hands on time on my instruments, my hands know where to go to get those tones . I would call "knowing" where and how to place my digits on an instrument a skill and not necessarily related to creativity.
"That is how math works. That is the art in math. That is how many skills work. We call that life."
Thank you again for sharing your perspective. Although my perspective differs from my understanding of what you have written, I mean no dis-respect. I am not afraid to ask questions, as I have found is one of best ways that I learn. I have no doubt that you are not the only one that thinks that you cannot be artistic without skills training. However, I would suggest that sometimes, a singular "skill building type focus" only stymies creativity.
I am thinking that skills are the not only way to provide the scaffolding required for artistic expression. What of the young ones expressing themselves in artistic manners, would you not call it "art" because they are too young to have a strong skill set?
While you write "that is how.." "that is the..." and "We call that life."
I ask, "Why do you think that?"
And, I wonder, if you ever question, "Is this how math works?" or
"What is the relationship between math and art?" or
"In what ways do skills work?"
"Who can really be the expert on what is called "life"?
"My concern is not about how to make math entertaining or delightful for the masses or to obfuscate it with polyhedra and philosophy."
You may have misunderstood my prior post if you think my goal is one of making math entertaining or delightful for the masses. The goal I am reaching for is how to get students more engaged in their studies, so I begin asking what engages learners? The entertainment and delight that I am concerned with are the times when a learner has enough curiosity sparked to investigate deeper, and the elation that brings the "aha" moment when something difficult and new is finally understood and can be built upon. For me, I place little value on memorizing type skill building, and do not see it as an absolute requirement for creativity.
"My concern is that students have available to them classes like I did."
What classes are referencing?
Do you think that all learners learn just like you did (do)?
"So that I don't have to sit through more meetings where the topic is filling positions with offshore talent."
I am with you here - we need more American scholars and the over-use of offshore talent is not good for our nation.
"You might want to think about that in your STEM endeavor. Your heart might be in the right place but you might want to visit some companies where STEM is played out. I'm there every day so this is second nature to me."
First of all, it is not "my" STEM endeavor. The project you are referencing is one of collaboration taking place withing a non profit entity, https://myalaska.state.ak.us/business/soskb/CSearch.asp
Search the link for Thunderbird Early College Charter School, Inc. to locate us. Our directors and other interested persons are working collaboratively to design a new school in our community that builds scholars and exposes them to multiple STEM fields of study. Our non profit directors, are teachers, professors and other professionals providing us with many perspectives from which to draw from, in order to design our programs. The project is not "my STEM endeavor", it is about collaboration for the benefit of students in our community - the down to earth application and subsequent assessment of education reform ideas - more than just the ranting that characterizes the various edu-wars. We value active learning, not only for our students, but for ourselves. We see ourselves creating a "community of practice" where we as intellectuals, study our school and continually refine it.
Where is my heart, again? Anyway, why would community service given from the heart be something of less value than being a mathematical expert?
What do you see in "companies where STEM is played out" that could be researched further?
In response to your comment, I'm there every day so this is second nature to me."
I wonder, what exactly do you think is second nature?
Responding to Anna Roys' post of Jul 13, 2010 2:00 AM:
> First of all, thank you for sharing your perspective.
> Secondly, I think you and I do not see learning in
> n the same light.
What an excellent and well reasoned response that is to the issues that Robert Hansen(RH) has raised!! I concur with 99% or more with your perspective regarding what constitutes 'true learning'.
> >(RH) My concern is not about how to make math
> >entertaining or delightful for the masses or to
> >obfuscate it with polyhedra and philosophy."
I had one experience that may help illustrate. My granddaughter, aged about 13, had complained to me last year "Oh, Math is SO boring!!!"
>(Anna Roys) You may have misunderstood my prior post if
>you think my goal is one of making math entertaining or
>delightful for the masses. The goal I am reaching for is
>how to get students more engaged in their studies, so I
>begin asking what engages learners? The entertainment
>and delight that I am concerned with are the times when
>a learner has enough curiosity sparked to investigate
>deeper, and the elation that brings the "aha" moment
>when something difficult and new is finally understood
>and can be built upon. For me, I place little value on
>memorizing type skill building, and do not see it as an
>absolute requirement for creativity.
I wanted to convince her that this was emphatically NOT the case. So I looked up several 'math-associated' topics (this is now easy, thanks to the Internet), and thought of trying out some ideas on her about polyhedra (which RH considers to be 'obfuscating' the issue of learning math) as a route to demonstrate my contrary thesis that math is NOT boring.
There are surely many other routes to get 'there' - but specifically I looked at the 'Yoshimoto Cube' (see http://boingboing.net/2009/01/19/fun-with-the-yoshimo.html) as a potential way to 'engage her interest to learn math'. It took some doing, but I managed to construct a pretty satisfactory cube from medium-thick cardpaper using two different colours. That really got her attention - she was astonished! She wanted to learn how to make one herself - so I gave her the basic outline of how to do it - and then she constructed several Yoshimoto cubes for herself, some which she took to school a few days later. She got a number of her classmates interested, and then one of her teachers sent a request to me to conduct a small 2-hour session on math for those students. That too was done, using various constructs, like the Mobius strip, and other such constructs - the students were most interested in those demonstrations (particularly when I got them to try their hands on the artefacts), as they had never seen math in that kind of light. [I did NO school math problems at all - just demonstration of various math artefacts, explaining (non-technically) their connections to math, and emphasizing why they had to learn the regular school math if they wanted to truly understand these fascinating objects].
My grand-daughter has since changed her ideas about math being boring - she even started a 'math club' along with several other schoolmates of hers - AND they are all now managing to actually learn a good bit of the school syllabus math without feeling that 'math is boring'. (I'm NOT claiming that my grand-daughter will ever become a mathematician - most likely not, as her interest is art and biology - but she certainly has developed some interest in math, and she does NOT regard it as so boring anymore).
I believe it's entirely unlikely that I'd have got anything like this kind of response if I had just drilled her and drilled her with problems from the school math syllabus (which is probably the way RH would recommend).
I attach herewith a copy of a 'guiding model' that helped me work out this means of getting her interested in math. (RH: Sorry, this model contains several 'boxes' to which I believe you strongly object).
I look forward most keenly to see RH's rejoinder (as well as any thoughts you may have about the topic in general or my experience in particular).
To help students students get interested in their school math syllabus.doc (39K) Download Attachment
Responding to Kirby Urner,Jul 8, 2010 2:55 PM:
Thanks very much for that post, and especially for that link providing information about Karl Menger (and also for those other links, which I've not visited yet). There is plenty in your post and in those links that will provide much food for thought to anyone who wishes to develop the 'art and science of math learning' (and teaching, for that matter). I observe that Robert Hansen regards all of this as 'obfuscating the issue of learning math'.
Responding to Robert Hansen's post, Jul 12, 2010 12:58 PM"
Thanks for that excellent link to 'finger painting'. [This link, by the way, I may be using to demonstrate my thesis on math learning (and teaching) - and that, as you probably know, goes almost (but not exactly) diametrically opposite to the one you espouse. Anyway, thanks again].
GS, most of that post was my description of what it feels like to be good at math and how it is very similar to being good at art, but how it is also different because unlike the arts you can see and hear, math is abstract and only sharable with those inclined to see it. If I get your post correctly, you seem to be saying...
"thanks for sharing with me how you do the art you do but your personal reflection on how you do it and what it feels like doing it is all wrong and now I must rush off and tell the world how you do the art you do"
> Responding to Robert Hansen's post, Jul 12, 2010
> 12:58 PM"
> Thanks for that excellent link to 'finger painting'.
> [This link, by the way, I may be using to demonstrate
> my thesis on math learning (and teaching) - and that,
> as you probably know, goes almost (but not exactly)
> diametrically opposite to the one you espouse.
> Anyway, thanks again].
In reply to this post by GS Chandy
Responding to post on Wed, Jul 14, 2010 at 3:15 AM, GS Chandy <[hidden email]> :
As I looked over your attachment, I wondered about the numbering of the boxes. You might want to include some text or modify the graphic representation. My question, why are boxes numbered the way they are?
I immediately started thinking about branching off of box 4. This is how I would approach it. I would take each example individually as a branch, such as,Yoshimoto Cube, George A. Hart’s ‘mathematical sculptures’; fractals, etc.. After selecting the first example, I would then go to my state standards (GLEs Grade Level Equivalency) for the grade level I was teaching and identify specific learning goal(s) and common misconceptions that might apply. Following this, I would decide how I would asses whether students had met the objective or not. In this type of unit, I would probably choose student presentations of their learning. After this, I would brainstorm and come up with learning activities to support the selected objective.
While I think "hooks" such as these are super, they are only hooks. I think if not carried a step further and tied to standards and assessed, then the amount of student learning is substantially less. The value I find in using "hooks" is that they elicit students' personal emotional ties - a powerful engagement tool.
Thank you for sharing. I would be interested in knowing what learning goals you think might be set for any of the examples you suggested.
Robert Hansen posted Jul 14, 2010 7:48 PM:
Nope, that is NOT what I am saying (or seem to be saying). I have on several occasions indicated where you have provided useful insights.
> GS, most of that post was my description of what it
> feels like to be good at math and how it is very
> similar to being good at art, but how it is also
> different because unlike the arts you can see and
> hear, math is abstract and only sharable with those
> inclined to see it. If I get your post correctly, you
> seem to be saying...
> "thanks for sharing with me how you do the art you do
> but your personal reflection on how you do it and
> what it feels like doing it is all wrong and now I
> must rush off and tell the world how you do the art
> you do"
> Speaking of rigor, one thing I notice about the Euclidean
> mindset is the notion of congruence teaches us to
> overlook chirality or handedness.
There might be a notion of "properly congruent" that's
longer to say, and usually we don't care, but can haul
this out in a pinch.
The context, again, was like a beauty contest for the most
primitive space-filler. We went with D.M.Y Sommerville's
rules (see Wikipedia for bio) in that we wanted to dismiss
left and right complements (like two-party figure skaters),
leaving only properly face-bonding tetrahedra that could be
superimposed without inside-outing.
Yes, this is a stricter meaning of "congruent" than
There's a debate on Polylist as to how many total
such space-filling tetrahedra have been found for ordinary
(not higher dimensional) space. Math World likewise
contains a request for more of a treatment ("A modern
survey would be welcome.") You would think those
mathematicians would get busy eh? Actually, I bet
lots of 'em are working on it as we speak.
D.M.Y. Sommerville's 1923 paper entitled Space-Filling
Tetrahedra gives Mite, Bite, Rite, and quarter-Rite
as the only four (though not by those names). What
others do people know about?
Mite: minimum tetrahedron (24 build a cube),
plane-nets first published under that name in the
1970s (by R.B. Fuller), though depicted earlier
in other sources such as 'Regular Polytopes'
(Coxeter) and of course in Sommerville's paper.
Syte: three ways to face-bond Mites, only two
of which are tetrahedra, the Bite and the Rite
Quarter Rite: vertex at the Rite center, edges
to each vertex = 4 properly congruent space-
We could play with these in Montessori school,
other pre-K. The Mite is commercially available in
magnetized plastic. Yoshimoto has done the
Bites, though Mites also make it.
They needn't memorize any names at this age,
though we could have a few wall charts, giving
more of the taxonomy (per Inchbald). These
short simple names will also help them learn
Here's the opening paragraph of said paper by
"In the answer to the book-work question, set
in a recent examination to investigate the volume
of a pyramid, one candidate stated that the
three tetrahedra into which a triangular prism
can be divided are congruent, instead of only
equal in volume. It was an interesting question
to determine the conditions in order that the
three tetrahedra should be congruent, and this
led to the wider problem — to determine what
tetrahedra can fill up space by repetitions.
An exhaustive examination * of this required
one to keep an open mind as regards whether
space is euclidean, elliptic, or hyperbolic,
and then to pick out the forms which exist
in euclidean space."
But is this even mathematics?
Spatial geometry has become so alien that
it's considered mostly philosophy these days.
We're not supposed to confuse it with math.
The icosahedron went its separate way
thousands of years ago, we're given to believe.
> You get some Archimedeans pointing to
> the ortho-scheme of the cube, otherwise
> known as "the characteristic tetrahedron"
> or 1/48th of a cube. This is their champion
> in the ring.
The characteristic tetrahedron was never a contender in
this contest for most primitive space-filler, nor any
Why? Because the most primitive space-filler (an
elusive title) must be topologically a simplex, which
for our purposes means four-faceted, four-vertexed
And because we disallow left and right pairs from
competing as "one player" if they really can't be
superimposed (some can be, if you just rotate 'em).
All four of Sommerville's have this property, of
being tetrahedra and not needing left and right
> Which Tetrahedra Fill Space? by Marjorie Senechal
> Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981),
> pp. 227-243.
This is a much later paper that summarizes some of
Sommerville's results while providing a lot more of the
Until reading this paper, I had been unaware that Aristotle
had his defenders, against those who claim he misspoke
when he supposedly said "tetrahedra fill space" in
'On The Heavens'.
Some translators say he said "pyramid" whereas Greek
does have "tetrahedron" as its own word (like a cube
is a "hexahedron").
Be that as it may, the other pro-Aristotle argument is
he never said "regular", in which case the four
Sommerville tetrahedra all provide fine examples
of Aristotle's space-filling pyramids.
These four have longer names than Mite, Bite, Rite
and quarter-Rite, such as disphenoid tetrahedron (Rite),
but we'd like to include a pre-college set in this
conversation, as an adjunct to their building spatial
geometry skills -- CAD, VRML, GIS, GPS... whatever
welding) -- both Euclidean and non-Euclidean.
> If this debate resolves the way I think it should, then
> we have all the more reason for advancing our "minimum
> tetrahedron" or Mite in more of the literature.
I'm moving ahead, but more on other lists, more
related to literature in some cases.
Based on the feedback here, there's not much
receptivity to a more philosophical approach to
math content, nor much room for spatial geometry.
These themes could be removed to the visual arts
and architecture, or to chemistry perhaps.
The issue of chirality (handedness) is sometimes
extremely important when it come to the left and
right versions of a given molecule. Take the case
of thalidomide for example.
This chemical and biological aspect of handedness
is also a hot topic on PolyList these days (frequented
by polyhedron aficionados).
Anna Roys posted Jul 14, 2010 11:49 PM:
The numbering merely indicates the order in which the elements were recorded on paper - the position of the element in the list.
> Responding to post on Wed, Jul 14, 2010 at 3:15 AM,
> GS Chandy <
> [hidden email]> :
> As I looked over your attachment, I wondered about
> the numbering of the
> boxes. You might want to include some text or modify
> the graphic
> representation. My question, why are boxes numbered
> the way they are?
Indeed you are correct in all of your above observations. Properly tying the learning to standards and assessment is most important for effective learning. Your model would include all elements that appear important or significant to you. (If you'd like, I could use your thoughts as noted above to list a few ideas and then create a model that may reflect some of your thinking - if that exercise would lead to your creating your own model on the issues of interest, that would be excellent indeed!)
> I immediately started thinking about branching off of
> box 4. This is how I
> would approach it. I would take each example
> individually as a branch, such
> as,Yoshimoto Cube, George A. Hart’s ‘mathematical
> sculptures’; fractals,
> etc.. After selecting the first example, I would
> then go to my state
> standards (GLEs Grade Level Equivalency) for the
> grade level I was teaching
> and identify specific learning goal(s) and common
> misconceptions that might
> apply. Following this, I would decide how I would
> asses whether students had
> met the objective or not. In this type of unit, I
> would probably choose
> student presentations of their learning. After this,
> I would brainstorm and
> come up with learning activities to support the
> selected objective.
> While I think "hooks" such as these are super, they
> are only hooks. I think
> if not carried a step further and tied to standards
> and assessed, then the
> amount of student learning is substantially less.
> The value I find in using
> "hooks" is that they elicit students' personal
> emotional ties - a
> powerful engagement tool.
I should note that each of these models that I may display is only a representation of (part of) the mental model held by a specific person at that point of time (snapshots, in fact). These models develop and grow, and I've always found that my models keep developing and changing, usually reflecting some enhanced understanding of the issues treated. Several people using OPMS have found this change in models to be quite disconcerting, and it is a sizable part of my job as a facilitator to try and convince them that these changes in their models actually reflect the 'learning process' about their problems that is taking place in their minds. It's not a bad thing at all that your model today may be quite different from you model constructed a week ago!!
Also, your models would reflect your reality - which would generally be quite different in significant respects from (say) my models reflecting my reality. This too, people often find disconcerting till they understand the underlying reasons for differences in models. Such understanding, as it develops, can often lead to true consensus between different people on complex issues (consensus at a much deeper level than the usual compromises that we see on group issues. At one stage, Warfield used to call the process he recommended - from which OPMS developed - by the name "Consensus Methodologies").
> Thank you for sharing. I would be interested in
> knowing what learning goals
> you think might be set for any of the examples you
The model I showed was only one snapshot of the recreation of models made by a teacher to help get students interested. I attach herewith two models - one made by a teacher to help guide him in teaching his students; the other made by a student to help him develop his own learning program. (I observe that these too are recreations of models that had been made quite some time ago - and those models were only 'snapshots' reflecting mental models held at that time, as explained. What is important is the understanding of the 'development of the flow of contributions' through the model: as the users create these models, their understanding of their systems grow significantly).
[Aside: Robert Hansen might usefully note that these models "include" elements relating to the 'math problem-solving' which he brings up every now and then. (I note that "includes" is also a transitive relationship - and this relationship too is very useful indeed to help us arrive at an understanding of any complex system)].
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