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Tag: teaching

44 32’28.29″N, 4 05’08.61″E

Twenty
five years ago I was amazed that writing merely “Le
Travers,Sablieres,France” on an envelop did the job. Today I’m even more
surprised that typing just “Le Travers,Sablieres” into Google Maps or Google earth brings you there in seconds with an
offset of about 100 meters!

Actually, the Google mark may be more accurate as it depicts the spot on
an old mule-path entering ‘le hameau de travers’ which consists of two
main buildings : ‘le by’ just below us and what we call ‘the travers’
but locals prefer to call ‘le jarlier’ or ‘garlelier’ or whathever (no
consistent spelling for the house-name yet). If you are French and know
the correct spelling, please leave a comment (it may have to do
something with making baskets and/or pottery).

I’ve always
thought the building dated from the late 18th century, but now they tell
me part of it may actually be a lot older. How they decide this is
pretty funny : around the buildings is a regular grid of old chestnut
trees and as most of them are around 400 years old, so must be the
core-building, which was extended over time to accomodate the growing
number of people and animals, until some 100 yrs ago when the place was
deserted and became ruins…

The first
few days biking conditions were excellent. If you ever come to visit or
will be in the neighborhood and are in for an easy (resp. demanding,
resp. tough) one and a half hour ride here, are some suggestions.

Start/end
point is always the end of the loose green path in the middle (le
travers). An easy but quite nice route to get a feel for the
surroundings is the yellowish loop (gooing back over blue/green) from
Sablieres to Orcieres and gooing back along camping La Drobie. Slighly
more demanding is the blue climb to over 900 meters to Peyre (and back).
By far the nicest (but also hardest) small tour is the green one
(Dompnac-Pourcharesse-St.Melany). If you want to study
these routes in more detail using GoogleEarth here is the kmz-file. Btw.
this file was obtained from my GPS gpx-file using
GPS-visualizer. Two and a half years
ago I managed to connect the
place via a slow dial-up line and conjectured that broadband-internet
would never come this far. I may have to reconsider that now as the
village got an offer from Numeo.fr to set-up a
wireless (??!!) broadband-network with a pretty low subscription… But,
as no cell-phone provider has yet managed to cover this area, I’m a bit
doubtful about Numeo’s bizness-plan. Still, it would be great. Now, all
I have to do is to convince the university-administration that my online
teaching is a lot better than my in-class-act and Ill be taking up
residence here pretty soon…

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way too ambitious

Student-evaluation sneak preview : I am friendly and
extremely helpful but have a somewhat chaotic teaching style and am way
too ambitious as regards content… I was about to deny vehemently
all assertions (except for the chaotic bit) but may have to change my
mind after reading this report on
Mark Rowan’s book ‘Symmetry and the monster’ (see also
my post
)

Oxford University Press considers this book
“a must-read for all fans of popular science”. In his blog,
Lieven le Bruyn, professor of algebra and geometry at the University of
Antwerp, suggests that “Mark Ronan has written a beautiful book
intended for the general public”. However, he goes on to say:
“this year I’ve tried to explain to an exceptionally
good second year of undergraduates, but failed miserably Perhaps
I’ll give it another (downkeyed) try using Symmetry and the
Monster as reading material”.

As an erstwhile
mathematician, I found the book more suited to exceptional maths
undergraduates than to the general public and would strongly encourage
authors and/or publishers to pass such works before a few fans of
popular science before going to press.

Peggie Rimmer,
Satigny.

Well, this ‘exceptionally good
year’ has moved on and I had to teach a course ‘Elementary
Algebraic Geometry’ to them last semester. I had the crazy idea to
approach this in a historical perspective : first I did the
Hilbert-Noether period (translating geometry to ideal theory of
polynomial rings), then the Krull-Weil-Zariski period (defining
everything in terms of coordinate rings) to finish off with the
Serre-Grothendieck period (introducing scheme theory)… Not
surprisingly, I lost everyone after 1920. Once again there were
complaints that I was expecting way too much from them etc. etc. and I
was about to apologize and promise I’ll stick to a doable course
next year (something along the lines of Miles Reid’s
‘Undergraduate Algebraic Geometry’) when one of the students
(admittedly, probably the best of this ‘exceptional year’)
decided to do all exercises of the first two chapters of Fulton’s
‘Algebraic Curves’ to become more accustomed to the subject.
Afterwards he told me “You know, I wouldn’t change the
course too much, now that I did all these exercises I realize that your
course notes are not that bad after all…”. Yeah, thanks!

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attention-span : one chat line

Never
spend so much time on teaching than this semester and never felt so
depressed afterwards. The final test for the first year course on
grouptheory (60 hrs. going from nothing to Jordan-Holder and the Sylow
theorems) included the following question :

Question :
For a subgroup $H \subset G $ define the normalizer to be the
subgroup $N_G(H) = \{ g \in G~:~gHg^{-1} = H \} $. Complete the
statement of the result for which the proof is given
below.

theorem : Let P be a Sylow subgroup of
a finite group G and suppose that H is a subgroup of G which
contains the normalizer $N_G(P) $. Then …

proof :
Let $u \in N_G(H) $. Now, $P \subset N_G(P) \subset H $
whence $uPu^{-1} \subset uHu^{-1} = H $. Thus, $uPu^{-1} $, being of the
same order as P is also a Sylow subgroup op H. Applying the Sylow
theorems to H we infer that there exists an element $h \in H $ such
that
$h(uPu^{-1})h^{-1} = P $. This means that $hu \in N_G(P) $.
Since, by hypotheses, $N_G(P) \subset H $, it follows that $hu \in H $.
As $h \in H $ it follows that $u \in H $, finishing the proof.

A
majority of the students was unable to do this… Sure, the result was
not contained in their course-notes (if it were I\’m certain all of them
would be able to give the correct statement as well as the full proof
by heart. It makes me wonder how much they understood
of the proof of the Sylow-theorems.) They (and others) blame it on the
fact that not every triviality is spelled out in my notes or on my
\’chaotic\’ teaching-style. I fear the real reason is contained in the
post-title…

But, I\’m still lucky to be working with students
who are interested in mathematics. I assume it can get a lot worse (but
also a lot funnier)

and what about this one :

If you are (like me) in urgent need for a smile, try out
this newsvine article for more
bloopers.

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mathematics & unhappiness

Sociologists are a constant source of enlightenment as CNN keeps reminding

Kids who are turned
off by math often say they don’t enjoy it, they aren’t good
at it and they see little point in it. Who knew that could be a formula
for success?
The nations with the best scores have the
least happy, least confident math students, says a study by the
Brookings Institution’s Brown Center on Education Policy.
Countries reporting higher levels of enjoyment and confidence
among math students don’t do as well in the subject, the study
suggests.
The eighth-grade results reflected a common
pattern: The 10 nations whose students enjoyed math the most all scored
below average. The bottom 10 nations on the enjoyment scale all
excelled.

As this study is based on the 2003 Trends in International
Mathematics and Science Studies
and as “we” scored best
of all western countries
this
probably explains all the unhappy faces in my first-year class on group
theory. However, they seemed quite happy the first few weeks.
Fortunately, this is proof, at least according to the mountain of wisdom, that I’m on the right track

If too many students are too happy in the math
classes, be sure that it is simply because not much is expected from
them. It can’t be otherwise. If teaching of mathematics is
efficient, it is almost guaranteed that a large group or a majority must
dislike the math classes. Mathematics is hard and if it is not hard, it
is not mathematics.

Right on! But then, why is
it that people willing to study maths enter university in a happy mood?
Oh, I get it, yes, it must be because in secondary school not much was
expected of them! Ouf! my entire world is consistent once again.
But then, hey wait, the next big thing that’s inevitably going to
happen is that in 2007 “we” will be tumbling down this world
ranking! And, believe it or not, that is precisely what
all my colleagues are eagerly awaiting to happen. Most of us are willing
to bet our annual income on it. Belgium was among the first countries to
embrace in the sixties-early seventies what was then called
“modern mathematics’ (you know: Venn-diagrams, sets,
topology, categories (mind you, just categories not the n-stuff ) etc.) Whole
generations of promising Belgian math students were able in the late
70ties, 80ties and early 90ties to do what they did mainly because of
this (in spite of graduating from ‘just’ a Belgian
university, only some of which make it barely in the times top 100 ). But
then, in the ’90ties politicians decided that mathematics had to
be sexed-up, only the kind of mathematics that one might recognize in
everyday life was allowed to be taught. For once, I have to
agree with motl.

Also, the attempts to connect mathematics with
the daily life are nothing else than a form of lowering of the
standards. They are a method to make mathematics more attractive for
those who like to talk even if they don’t know what they’re
talking about. They are a method to include mathematics between the
social and subjective sciences. They give a wiggle room to transform
happiness, confidence, common sense, and a charming personality into
good grades.

Indeed, the major problem we are
facing today in first year classes is that most students have no formal
training at all! An example : last week I did a test after three weeks
of working with groups. One of the more silly questions was to ask them
for precise definitions of very basic concepts (groups, subgroups,
cyclic groups, cosets, order of an element) : just 5 out of 44 were able
to do this! Most of them haven’t heard of sets at all. It seems
that some time ago it was decided that sets no longer had a place in
secondary school, so just some of them had at least a few lessons on
sets in primary school (you know the kind (probably you won’t but
anyway) : put all the green large triangles in the correct place in the
Venn diagram and that sort of things). Now, it seems that politicians
have decided that there is no longer a place for sets in primary schools
either! (And if we complain about this drastic lowering of
math-standards in schools, we are thrown back at us this excellent 2003
international result, so the only hope left for us is that we will fall
down dramatically in the 2007 test.) Mind you, they still give
you an excellent math-education in Belgian primary and secondary schools
provided you want to end up as an applied mathematician or (even worse)
a statistician. But I think that we, pure mathematicians, should
seriously consider recruiting students straight from Kindergarten!

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Krull & Paris

The
Category-Cafe ran an interesting post The history of n-categories
claiming that “mathematicians’ histories are largely
‘Royal-road-to-me’ accounts”

To my mind a key
difference is the historians’ emphasis in their histories that things
could have turned out very differently, while the mathematicians tend to
tell a story where we learn how the present has emerged out of the past,
giving the impression that things were always going to turn out not very
dissimilarly to the way they have, even if in retrospect the course was
quite tortuous.

Over the last weeks I’ve been writing up
the notes of a course on ‘Elementary Algebraic Geometry’ that I’ll
be teaching this year in Bach3. These notes are split into three
historical periods more or less corresponding to major conceptual leaps
in the subject : (1890-1920) ideals in polynomial rings (1920-1950)
intrinsic definitions using the coordinate ring (1950-1970) scheme
theory. Whereas it is clear to take Hilbert&Noether as the leading
figures of the first period and Serre&Grothendieck as those of the
last, the situation for the middle period is less clear to me. At
first I went for the widely accepted story, as for example phrased by Miles Reid in the
Final Comments to his Undergraduate Algebraic Geometry course.


rigorous foundations for algebraic geometry were laid in the 1920s and
1930s by van der Waerden, Zariski and Weil (van der Waerden’s
contribution is often suppressed, apparently because a number of
mathematicians of the immediate post-war period, including some of the
leading algebraic geometers, considered him a Nazi collaborator).

But then I read The Rising Sea: Grothendieck
on simplicity and generality I
by Colin McLarty and stumbled upon
the following paragraph

From Emmy Noether’s viewpoint,
then, it was natural to look at prime ideals instead of classical and
generic points—or, as we would more likely say today, to identify
points with prime ideals. Her associate Wolfgang Krull did this. He gave
a lecture in Paris before the Second World War on algebraic geometry
taking all prime ideals as points, and using a Zariski topology (for
which see any current textbook on algebraic geometry). He did this over
any ring, not only polynomial rings like C[x, y]. The generality was
obvious from the Noether viewpoint, since all the properties needed for
the definition are common to all rings. The expert audience laughed at
him and he abandoned the idea.

The story seems to be
due to Jurgen Neukirch’s ‘Erinnerungen an Wolfgang Krull’
published in ‘Wolfgang Krull : Gesammelte Abhandlungen’ (P.
Ribenboim, editor) but as our library does not have this book I would
welcome any additional information such as : when did Krull give this
talk in Paris? what was its precise content? did he introduce the prime
spectrum in it? and related to this : when and where did Zariski
introduce ‘his’ topology? Answers anyone?

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