Alexandre V. Borovik
Mathematics
under
the Microscope
Corrigenda and Addenda
The book itself is available from
http://www.ams.org/
bookstore-getitem/item=mbk-71
November 26, 2012
American Mathematical Society
2
My ldi bednye i po bednosti svoe
i
melkoskopa ne imeem,
a u nas tak glaz pristrelmxi.
Levxa
Acknowledgements
A number of addenda in the text were stimulated by comments
from Mikhail Katz and Michael Livshits.
2.1 The starting point: mirrors and reflections
Fig. 0.1. The ability to recognize oneself in the mirror is equated, in popular culture, to self-awareness of a sentient being. Photo by roseoftimothywoods. Source: Wikipedia Commons and Flickr. Licensed under Creative
Commons Attribution 2.0 License.
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3.2 Number sense and grammar
Varley et al. [1 ] describe cases od adult patients with brain damage who lost ability to process correctly reversed and embedded
sentences (such as “The man killed the lion”, “The lion killed the
man”, and “The man who killed the lion was angry”), but retained
ability to manipulate with brackets in arithmetic expression (such
as 5×(6+2). Brannon [2 ] used this work to claim “the independence
of language and mathematical reasoning”.
We have here a case of a terminological confusion: for, me,
“mathematical reasoning” is not re-use of parsing rules for arithmetic expression learned in childhood; I argue that as far as parsing is concerned, “mathematical reasoning” means generation of
new parsing patterns.
3.5 Parsing, continued: do brackets matter?
Limitations of software
My comments on software are echoed in Christof van Nimwegens
paper The paradox of the guided user: assistance can be countereffective [3 ]
A recurring issue in usability guidelines is the importance of
minimizing “user memory load”, also referred to as computational
offloading (Scaife & Rogers, 1996). This means that the working
memory (WM) of a user is relieved so that a maximum of cognitive resources can be devoted to the task. To achieve this, certain information can be externalized to the problem solver and
thus be of assistance. Already two decades ago, Norman (1988)
proposed the idea that knowledge might be as much in the world
as it is in the head. He pointed out that the information embedded in technological artifacts (such as computer interfaces) was as
important to task achievement as the knowledge residing in the
mind of the user. [. . . ] Externalization is to make parts of the interface context-sensitive, e.g. by hiding or disabling functions that
are not applicable at the moment. By doing so, the user is “taken
by the hand” by limiting choices and providing feedback (Van Oostendorp & De Mul, 1999). Examples are wizards, help-options and
1
2
3
R. A. Varley, N. J. C. Klessinger, C. A. J. Romanowski, and M. Siegal,
Agrammatic but numerate. Proc. Nat. Acad. Sci. USA 102 (2005), 35193524.
E. M. Brannon, The independence of language and mathematical reasoning. Proc. Nat. Acad. Sci. USA 102 no. 9 (2005), 3177-3178.
Christof van Nimwegen, The paradox of the guided user: assistance
can be counter-effective, http://igitur-archive.library.uu.nl/
dissertations/2008-0401-200600/nimwegen.pdf. Accessed 27
September 2010.
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grayed-out menu-items that do not permit using them, thus offering a context-sensitive interface where only possible actions are
displayed. For example, in MS-Word, one cannot select “paste”
from the menu when nothing is copied first. “Paste” shown in gray
color indicates both the presence and the unavailability of the command. “Graying out” externalizes the applicability of an operator
at a certain moment, and makes memorizing this information unnecessary.
In the opposite situation, when no such features are provided,
a user has to internalize the information by himself, and store this
information in his memory. The above definition of usability implies that systems are tools we use to accomplish a certain goal
(text processing, spreadsheets, editing video, looking for information on the web). Rubin (1994) was probably right, though it can
be interesting to look at what these presumed friendly systems
actually do to humans and their behavior on a deeper level, and
whether the highest achievable usability possible in the classical
sense of the word is in fact always desirable.
Using experimental data, van Nimwegen concludes that “externalisation” harms learning. In his words,
In modern GUI’s, users often click around without control,
clicking and clicking on and on has become our second nature,
perhaps partly due to the World Wide Web. The undo facilities
that became almost mandatory in systems are mostly good and
handy features, but perhaps they changed our ideas about the consequences of our actions in computer environments.
[. . . ] one has to delve into human nature, and be aware of
the mechanisms that are natural to us, which include, unfortunately, laziness and shallow behavior. There is nothing we like
better than the feeling that things are being done for us, and in
many cases, this can be exactly what is desired. However, in some
situations one has to be careful to neglect these characteristics.
This research concerns exploring the conditions under which motivation and deep contemplation for the task can be provoked to
achieve better task performance, fewer errors, and having users
that constantly are on top of the task.
Verbalization and vocalization
Verbalization and its stronger form—vocalization— is an important but increasingly neglected channel of interiorization of mathematics. School culture in various countries treats it differently;
vocalization of arithmetic routines was a norm in Russian schools
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of my time, but, apparently, not in England. This is a testimony
from Alison Price4 :
I do not have memories of a particular content of mathematics
but do have vivid memories of myself as an 8 year old being taught
in a very formal teaching context. The teacher would show us how
to do a calculation on the board and we were then expected to work
through examples in our maths books in silence.
My problem was that I could not learn new mathematics without being able to talk about what I was doing.
As a result I frequently was hit with a plastic ruler (this was
allowed in the 1950s) for talking in mathematics class. As you can
imagine this was not conducive to learning.
And this is from current Russian mathematics curriculum for
ages 6–10:
The process of learning mathematics involves familiarisation
with mathematical language and formation of speech skills: children learn to express propositions using mathematical terms and
concepts, identify and emphasise words (phrases, etc.) that help to
understand their meaning, raise questions in the course of completion of an assignment, choose evidence of correctness or incorrectness of the action performed, justify the steps of the solution,
etc.
4.1 Parables and fables
These continues the theme of the subitizing/counting threshold.
I discovered an article [5 ] about the special role of number 6 in
the works of Daniil Kharms, the famous Russian absurdist writer.
Actually, there is a strong feeling that Kharms was writing about
the classical subitising/counting threshold. Two samples: one in
English translation, another is in original Russian (I cannot find a
decent translation of the most mathematical of all Kharms works).
FALLING -O UT O LD W OMEN
A certain old woman fell out of a window because she was too
curious. She fell and broke into pieces.
Another old woman leaned her head out the window and
looked at the one that had broken into pieces, but because she
was too curious, she too fell out of the window fell and broke into
pieces.
4
The Reverend Dr Alison J Price is English, female, Principal Lecturer
in Mathematics and Religious Education in a British university.
5
A. Gerasimova, N. Karrik. K voprosu o znaqenii qisel u
Harmsa: Xestь kak estestvenny
i predel. (In Russian.) http:
//xarms.lipetsk.ru/texts/geras1.html.
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Then a third old woman fell out of the window, then a fourth,
then a fifth.
When the sixth old woman fell out, I became fed up with watching them and went to Maltsevsky Market, where, they say, a certain blind man was presented with a knit shawl.
And another one, from the text “ vam hoqu rasskazatь . . . ”:
zametim, qto:
1. Dve ruki, dve nogi, poseredke sapogi.
2. Uxi obladat tem жe qem i glaza.
3. Begatь – glagol iz-pod nog.
4. Wupatь – glagol iz-pod ruk.
5. Usy mogut bytь tolьko u syna.
6. Zatylkom nenlьz rassmotretь qto visit na stene.
17. Obratite vnimanie qto posle xesterki idet semnadcatь. [6 ]
A translation of the last line: 17. Notice the six is followed by seventeen.
And, of course, I have to append Kharms’ glorious A sonnet7 :
An amazing thing happened to me today, I suddenly forgot what
comes first7 or 8.
I went to my neighbors and asked them about their opinion
on this matter. Great was their and my amazement, when they
suddenly discovered, that they couldnt recall the counting order.
They remembered 1, 2, 3, 4, 5 and 6, but forgot what comes next.
We all went to a commercial grocery store, the one thats on
the corner of Znamenskaya and Basseinaya streets to consult a
cashier on our predicament. The cashier gave us a sad smile, took
a small hammer out of her mouth, and moving her nose slightly
back and forth, she said: - In my opinion, a seven comes after an
eight, only if an eight comes after a seven.
We thanked the cashier and ran cheerfully out of the store. But
there, thinking carefully about cashiers words, we got sad again
because her words were void of any meaning.
What were we supposed to do? We went to the Summer Garden
and started counting trees. But reaching a six in count, we stopped
and started arguing: In the opinion of some, a 7 went next; but in
opinion of others an 8 did.
We were arguing for a long time, when by some sheer luck, a
child fell off a bench and broke both of his jaws. That distracted us
from our argument.
And then we all went home.
6
7
Does anyone know a decent English translation of “ vam hoqu
rasskazatь . . . ”?
With thanks to Donald A. Preece who brought my attention to it.
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And now I wish to mention an even more direct link between
Kharms’ “A sonnet” and modern cognitive science.
Indeed notice that Kharms writes, in a poetic form, about what
now is known as “mathematics anxiety”, and his observations find
a direct confirmation in the modern cognitive science [8 ]:
Individuals with mathematics anxiety have been found to differ
from their non-anxious peers on measures of higher-level mathematical processes, but not simple arithmetic. The current paper examines differences between mathematics anxious and nonmathematics anxious individuals in more basic numerical processing using a visual enumeration task. This task allows for the assessment of two systems of basic number processing: subitizing
and counting. Mathematics anxious individuals, relative to nonmathematics anxious individuals, showed a deficit in the counting but not in the subitizing range. Furthermore, working memory
was found to mediate this group difference. These findings demonstrate that the problems associated with mathematics anxiety exist at a level more basic than would be predicted from the extant
literature.
More papers on subitizing: [9 ], [10 ].
4.5 Convexity and sensorimotor intuition
Olson [11 ] quotes Abranavel [12 ] to say that
young children made accurate diameter estimates well before they
made accurate length estimates, both of which involve Euclidean
concepts. This observation was traced back to the fact that young
children automatically grasped the diameter by picking up the object, while length could not be apprehended without active search.
8
9
10
11
12
E. A. Maloney, E. F. Risko, D. Ansari, and J. Fugelsang, Mathematics anxiety affects counting but not subitizing during visual enumeration, Cognition 114, no. 2 (February 2010), 293–297.
DOI:10.1016/j.cognition.2009.09.013.
M. LeCorre and S. Carey, One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles.
Cognition 105 (2007), 395–438.
S. K. Revkin, M. I. Piazza, L. Cohen, and S. Dehaene, Does subitizing
reflect numerical estimation? Psychological Science 19 (2008), 607–614.
D. R. Olson, Cognitive Development: The Child’s Acquisition
of Diagonality. Lawrence Erlbaum Associates Inc., 1996. ISBN-10:
0805823026. ISBN-13: 978-0805823028.
E. Abravanel, The development of intersensory pattering with regard to
selected spatial dimensions. Monographs of the Society for Research in
Child Development 33, 2 whole no. 118 (1968).
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A classical study of tactile perception in young children is Piaget
and Inhelder [13 ].
5.5 The vanishing point
Here is an example of a wonderfully consistent theological perspective on time:
In his Systematic Theology, Vol. III, Wolfhart Pannenberg argues that God as eternal comprehends the different moments of
time simultaneously and orders them to constitute a whole or totality. The author contends that this approach to time and eternity
might solve the logical tension between the classical notion of divine sovereignty and the common sense belief in creaturely spontaneity/ human freedom. For, if the existence of the events constituting a temporal sequence is primarily due to the spontaneous
decisions of creatures, and if their being ordered into a totality or
meaningful whole is primarily due to the superordinate activity of
God, then both God and creatures play indispensable but nevertheless distinct roles in the cosmic process. [14 ]
6.2.1 Balls, bins and the Axiom of Extensionality
Thinking for themselves about potential and actual infinity, children can pick very subtle points. This is a one of childhood stories,
from Elizabeth Frenkel [15 ]
The first that comes to my mind is the Infinity Hotel. I think I
was 11 or 12 years old at that time. It was not that simple even for
a finite number of guests. I remember talking about the problem
with one of my parents—they were mathematicians by education.
It appeared very strange to me that when we want to accommodate just two guests we cannot send them “to infinity”. I prosed
to settle them “at infinity”. Indeed there is always one more room!
But everyone thought that this answer was not efficient. My parents (and a book) asked to name a concrete room number. What
should be room numbers for keys that a porter has to handle to
the guests? Such a “non-symmetricity” of infinity appeared to me
to be very unrealistic.
13
14
15
J. Piaget and B. Inhelder, The Child’s Perception of Space. Routledge and Kegan Paul, 1956.
J. A. Bracken, A new look at time and eternity, Theology and Science, 2
no. 1 (2004), 77-88. DOI: 10.1080/1474670042000196621.
EF is female, Russian, has a PhD in mathematics. This episode happened when she was 11 or 12 years old.
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It is dangerous to invoke irrelevant arguments when talking
to a bright child. Look how Lizzy turned the table on her parents’
argument:
The case when an infinite number of guests suddenly arrived
was perceived by me as more natural, even with the trouble of
adding the infinity at the end. But! The issue of efficiency came to
the fore here. How could this happen? For accommodating either
finite or infinite number of guests we have to do about the same
number of operations!
Lizzy was also unhappy with the Principle of Excluded Middle
(at least when it is used in conjunction with the Axiom of Extensionality):
Russell’s Paradox (of course, in the form of Barber’s Paradox)
caused resentment in me. Rephrasing my memories in today’s language, my feelings can be described as follows. By definition, we
have excluded the barber from the set of men who are shaved by
him as well as from the set of men who are not shaved by him.
Why then we ask which set the barber belongs to? Of course, my
question did not prove that I had understanding of foundations
of set theory, or even of any understanding of the essence of the
problem, but still, perhaps it contained something . . .
6.2.5 Finite differences and asymptotic at zero
Vladimir Arnold’s thesis that it is wrong to think of finite differences as approximations of differential operators is supported, with
a brimstone-and-fire passion, by Doron Zeilberger; see his paper
with the telling title “Real” analysis is a degenerate case of discrete
analysis [16 ].
16
D. Zeilberger, “Real” analysis is a degenerate case of discrete
analysis, in New Progress in Difference Equations (Proc.
ICDEA 2001) (B. Aulbach, S. Elaydi, and G. Ladas, eds.). CRC
Press, 2004. ISBN-10: 0415316758. ISBN-13: 978-0415316750. Available from http://www.math.rutgers.edu/˜zeilberg/mamarim/
mamarimhtml/real.html.
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6.3 Proofs by handwaving
A childhood testimony from AN which works as a comment on
Lakoff and Núnez [17 ] and Núnez [18 ]
As for the limits, we were told to imagine an animal getting
closer to point on the plane but not really arriving at it. This is a
very poor description and gave me many years of unnecessary pain
while dealing with limits. The things is this animal might actually
decompose itself into many different animals who have nothing to
do with the first one and whoever acts more dramatically close to
a certain point will guide the limit close to that point.
7.9 Winning ways
I quoted G. H. Hardy saying that the solution of a chess problem is
just that ‘proof by enumeration of cases . . . which a real mathematician tends to despise.
This feeling of contempt spread much further than chess. I met
quite a number of mathematicians who, say, hated traditional logic
puzzles—and I happily share this attitude. As one of my colleagues
put it:
I think for many mathematicians, perhaps even a majority, the
appeal of the subject lies in being able to solve particular, special
problems, such as logic puzzles, and being able to use (and know)
special tricks for doing so. Those are the sort of mathematicians
who excel at Olympiads and enjoy mathematical games.
For others of us, however, all that emphasis on puzzles and
tricks was - and is - a complete waste of time. I always thought
those puzzles were about as interesting as crossword puzzles or
card games, activities which I always found “mind numbing and
soul destroying”, to use the words of Yevtushenko. How is the
world a better place after solving a cross-word puzzle? What have
you, the solver, gained from solving a logic puzzle? Absolutely
nothing, as far as I could tell.
17
18
G. Lakoff and R. Núnez, Where Mathematics Comes From: How
the Embodied Mind Brings Mathematics Into Being. Basic Books,
New York, 2000. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. Basic Books, New
York, 2000.
R. Núnez, Do real numbers really move? Language, thought and gesture: the embodied cognitive foundations of mathematics, in 18 Unconventional Essays on the Nature of Mathematics (R. Hersh, ed.).
Springer, 2005, pp. 160–181.
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My mathematical interest was always in the large-scale formal, abstract systems and structures of the subject, not in particular puzzles or tricks. I remember being thrilled at about age 14 to
learn that “y = f (x)”, that two different concepts which my teachers had tossed around loosely for some time, the vertical axis “y”
in Cartesian co-ordinates and a function of some variable “x” were
in fact the same thing. At last these two disparate subjects were
connected, and lots of different things made sense as part of the
same over-arching picture. The same experience happened a year
or so later when we were first taught the differential calculus, and
I could finally see a general theory behind what we had been doing
with all those discussions of tangents to particular functions.
Some mathematicians certainly do like examples and particular problems and clever tricks. Others of us, however, think topdown and not bottom-up, and so desire general frameworks and
abstract structures. Any effective education system should recognize the existence of at least these two styles of thinking, and not
assume that all students are bottom-up thinkers.
Perhaps the emphasis on puzzles and tricks is fine for some
mathematicians—e.g., Paul Erdos seems to have been motivated
by puzzles and eager to solve particular problems. However, it is
not fine for others—Alexander Grothendieck comes to mind as
someone interested in abstract frameworks rather than puzzlesolving. Perhaps the research discipline of pure mathematics needs
people of both types. If so, this even more reason not to eliminate
all the top-down thinkers by teaching only using puzzles at school.
7.10 A dozen problems
Problem 10. Alexander Bogomolny offered a very simple solution:
Think of counting stones in a pile. This is done step-by-step,
every time splitting a pile and announcing how many smaller piles
are still there. The process stop when all piles consist of exactly
one stone.
8.4 The triumph of the heuristic approach:
Kolmogorovv’s “5/3” law
One of the recent papers devoted to deduction of Kolmogorov’s cascade from the Navier-Stokes equations [19 ] contains a considerable
bibliography.
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Fig. 0.2. Woman teaching geometry. 1309–1316, France (Paris). The
British Library. Source: Wikimedia Commons. Public domain.
From Wikimedia: Detail of a scene in the bowl of the letter ‘P’ with
a woman with a set-square and dividers; using a compass to measure
distances on a diagram. In her left hand she holds a square, an implement
for testing or drawing right angles. She is watched by a group of students.
In the Middle Ages, it is unusual to see women represented as teachers,
in particular when the students appear to be monks. She may be the
personification of Geometry. Illustration at the beginning of Euclid’s
Elementa, in the translation attributed to Adelard of Bath.
8.6 Women in mathematics
A comment from TE:
19
C. Foias, M. S. Jolly, O. P. Manley, R. Rosa, and R. Temam, Kolmogorov
theory via finite-time averages, Physica D 212 (2005), 245–270.
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I have to comment on the “intrinsic competitiveness” of mathematics.
In mathematics olympiads and in professional mathematics,
competitiveness is ritualized and contained by ethical rules. In my
olympiad training session, you were expected to explain things to
newer participants even though they might well be better than you
at the competition. In olympiads and real life, I have found that
this kind of ethics almost always corresponded to mathematical
capability.
So here is my anecdote on competitiveness:
In primary school (age 6–10), I had about half the running
speed of any other child. On the other hand, I understood everything in maths immediately and was very quick with mental calculations. In sports, we had weekly running competitions of groups
by four, where my group never had a chance at all. In maths, we
had weekly speed calculating competitions where each pair of students were posed a question and the quicker one would remain
standing until one student was left and got a little prize. I won
every time and was only allowed to participate every other week.
The lesson that good performance in math is bad for your social
standing with your peers could not have been more evident. [20 ]
A recent massive survey of research in gender issues in mathematics has appeared in [21 ].
Competitiveness and aggression
This is an important theme in the discussion of the place and role
of women in mathematics: an extreme competitiveness and aggression in research mathematics. This is a testimony from my colleague:
Part of my anger and frustration at school was that so much
of this subject that I loved, mathematics, was wasted on what I
thought was frivolous or immoral applications: frivolous because
of all those unrealistic puzzles, and immoral because of the emphasis on competition (Olympiads, chess, card games, gambling, etc).
I had (and retain) a profound dislike of competition, and I don’t
see why one always had to demonstrate one’s abilities by beating
other people, rather than by collaborating with them. I believed
that “playing music together”, rather than “playing sport against
one another”, was a better metaphor for what I wanted to do in
life, and as a mathematician.
20
21
TE is female, learned mathematics in German. She has a PhD in Mathematics (and a Gold Medal of an International Mathematical Olympiad),
teaches mathematics at an university.
D. F. Halpern, C. P. Benbow, D. C. Geary, R. C. Gur, J. S. Hyde, and M. A.
Gernbacher, The science of sex differences in science and mathematics,
Psychological Science in the Public Interest 8 (2007), 1–51.
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Indeed, the macho competitiveness of much of pure mathematics struck me very strongly when I was an undergraduate student:
I switched then to mathematical statistics because the teachers
and students in that discipline were much less competitive towards one another. For a long time, I thought I was alone in this
view, but I have since heard the same story from other people, including some prominent mathematicians. I know one famous category theorist who switched from analysis as a graduate student
because the people there were too competitive, while the category
theory people were more co-operative.
It may be worth mentioning that I am male. In other words, a
dislike of competitiveness is not confined to women. The statistics
department I entered as an undergraduate, for example, had no
women in it, yet was much less competitive than the pure mathematics department (which had once been headed by a woman). I
think it is disciplinary tradition rather than gender that is the key
factor here.
I am now a Computer Scientist. I have also found differences
in the competitiveness of people in different sub-domains of CS.
To generalize greatly, I have found people in Artificial Intelligence
(AI) much less macho and competitive than those in (say) Algorithm and Complexity Theory. Within AI, people in (say) Argumentation are generally much less macho and competitive than
those in Game Theory and Mechanism Design. In each case, the
more formal and mathematical the domain, the more competitive
it tends to be. It could be that these domains have acquired their
cultures from mathematicians, while the other domains have been
less influenced by the culture of mathematics.
Competitiveness as a virtue
A completely different approach to competitiveness comes, interestingly, from the Russian tradition of education in mathematics
and physics.
I quote a few fragments from a article Physics and Mathematics
School in a third of century written in 1988 about my alma mater,
the preparatory boarding school of Novosibirsk State University in
Siberia, by my physics lecturer at the School, Evgenii Ivanovich
Bichenkov, and translated in English by my old friend Owl (Otus
Persapience), also a former student of the School.
The document that Owl and I are to offer to your attention is a
manifesto of meritocratic eliticism in education, a recipe for building a highly selective22 and academically intensive school. A word
of warning: Bichenkov’s paper was written in a different historic
22
The catchment area of the School covered huge tracts of Siberia, Russian Far East and former Soviet Central Asia and had population of at
least 20 million people.
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epoch, in different socio-economic, political and cultural environments. It is not a recipe for modern day Britain!
What follows are quotes from Bichenkov; he is obviously very
fond of competition.
[. . . ] What new was brought by the school into the practice of
school education? What are the principal results of its activities in
teaching basics of sciences at a school level? [. . . ]
So, what has been achieved by selection of students? I am
deeply convinced that the very fact of selection and creation of childrens collective on the basis of selection is beneficial for a child.
When they come from their schools, where all roles and places
have been already assigned and fixed, children start their internal competitions for distribution over the scale of their hierarchy
of values. They cannot not to do that—such is their nature and
their age. It is important that at that age they are offered decent
moral and human rules of competitions and shown some good examples. It appears that the Novosibirsk FMSh has succeeded in
that.
Next. To what degree selection was determined by true abilities? Did results match the declared aims?
Here I cannot give a definite answer. In many ways, the selection can still be affected by chance. The selection is obviously influenced by personal aspirations and interests of the child, by the
family, teachers, friends, acquaintances; the results of olympiads
are affected by competitiveness, persistence, level of maturity, after all. And of course, choice manifests the personality of the
teacher, examiner. [23 ]
Please notice that Bichenkov accepts as inevitable that different
examiners apply different criteria for selection; it does not matter
for him whether the criteria are uniform; what matters is that they
should be fare and allow the examiner to select the best candidates.
11.4 Mathematics and Origami
The mathematical history of Origami, in particular, the history of
solving cubic equations by means of Origami, found a wonderful
exposition in Thomas Hull’s paper [24 ], complete with a very detailed bibliography. Giorgio Ferrarese’s website [25 ] contains texts
of a number of historic sources.
23
24
25
A fuller text can be found in my blog post Manifesto of meritocratic eliticism, http://micromath.wordpress.com/2010/09/28/
manifesto-of-meritocratic-eliticism-2/.
T. C. Hull, Solving cubics with creases: The work of Beloch and Lill, Amer. Math. Monthly 118 (2011), 307–315.
DOI:10.4169/amer.math.monthly.118.04.307.
G.
Ferrarese,
http://www2.dm.unito.it/paginepersonali/
ferrarese/paperfolding/. Accessed 5 May 2011.
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