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February 25th, 2021
06:27 am
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August 6th, 2019
03:31 am
Мне сегодня исполнилось 33 года, и я никогда больше не буду вести лекции про
векторные пространства.
Мне так это надоело, что я записал на ютубе миникурс минилекций, к которому я
буду отсылать каждого, кто попросит меня рассказать что-то про векторные
пространства. Получилось неидеально, но по темам это именно то, что я хочу,
чтобы мои школьники знали. С запасом. Это мой подарок самому себе.
Курс в задачах. Назвал "Абстрактная теория векторных пространств". 11
минилекций. Куча задачек. Теоретический курс, совсем не вычислительный. Темы
лекций такие:
1) Поля.
2) Векторные пространства. Подпространства.
3) Линейные комбинации. Семейства векторов. Линейная оболочка.
4) Линейная независимость. Базисы. Теорема о существовании базисов.
5) Теорема о равномощности базисов. Размерность.
6) Линейные отображения. Ядро, образ.
7) Изоморфизм.
8) Матрицы.
9) Прямые суммы. Размерность суммы и пересечения подпространств. Расщепление
идемпотентов.
10) Фактор пространства.
11) Лемма Цорна. Доказательство теорем о существовании базиса и равномощности
базисов.
PS вижу, что можно было бы и лучше сделать, но сейчас уже нет сил переделывать.
Может, через какое-то время, когда выявится побольше недостатков, обновлю
некоторые лекции.
Школьники матшкол и первокурсники, пользуйтесь, вдруг кому пригодится. Ссылка
на слайды есть в описании каждой лекции.
https://www.youtube.com/playlist?list=PLoimxvgpOG5K6bfejCpgnpz8--xSDm3n_
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March 3rd, 2019
03:02 pm - slowpoke.jpg
The Moore method is a deductive manner of instruction used in advanced
mathematics courses. It is named after Robert Lee Moore, a famous topologist
who first used a stronger version of the method at the University of
Pennsylvania when he began teaching there in 1911.
F. Burton Jones, a student of Moore and a practitioner of his method, described
it as follows:
Moore would begin his graduate course in topology by carefully selecting
the members of the class. If a student had already studied topology
elsewhere or had read too much, he would exclude him (in some cases, he
would run a separate class for such students). The idea was to have a class
as homogeneously ignorant (topologically) as possible. He would usually
caution the group not to read topology but simply to use their own ability.
Plainly he wanted the competition to be as fair as possible, for
competition was one of the driving forces. […]
Having selected the class he would tell them briefly his view of the
axiomatic method: there were certain undefined terms (e.g., 'point' and
'region') which had meaning restricted (or controlled) by the axioms (e.g.,
a region is a point set). He would then state the axioms that the class
were to start with […]
After stating the axioms and giving motivating examples to illustrate their
meaning he would then state some definitions and theorems. He simply read
them from his book as the students copied them down. He would then instruct
the class to find proofs of their own and also to construct examples to
show that the hypotheses of the theorems could not be weakened, omitted, or
partially omitted.
When the class returned for the next meeting he would call on some student
to prove Theorem 1. After he became familiar with the abilities of the
class members, he would call on them in reverse order and in this way give
the more unsuccessful students first chance when they did get a proof. He
was flexible with this procedure but it was clear that this was the way he
preferred it.
When a student stated that he could prove Theorem x, he was asked to go to
the blackboard and present his proof. Then the other students, especially
those who had not been able to discover a proof, would make sure that the
proof presented was correct and convincing. Moore sternly prevented
heckling. This was seldom necessary because the whole atmosphere was one of
a serious community effort to understand the argument.
When a flaw appeared in a 'proof' everyone would patiently wait for the
student at the board to 'patch it up.' If he could not, he would sit down.
Moore would then ask the next student to try or if he thought the
difficulty encountered was sufficiently interesting, he would save that
theorem until next time and go on to the next unproved theorem (starting
again at the bottom of the class).
— (Jones 1977)
The students were forbidden to read any book or article about the subject. They
were even forbidden to talk about it outside of class. Hersh and John-Steiner
claim that, "this method is reminiscent of a well-known, old method of teaching
swimming called 'sink or swim' ".
https://en.wikipedia.org/wiki/Moore_method
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LJ.Rossia.org |