In this section, we describe school mathematics circles and electives.
circles in specialized schools with an advanced course in mathematics).
March 9, 2011
15:4
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch09
384
Russian Mathematics Education: Programs and Practices
in such circles; nonetheless, these circles have never been, are not,
and cannot be especially selective: the problem that they set before
themselves is not to prepare future winners of All-Russia or even
municipal Olympiads, but rather to facilitate general mathematical
development.
To give a more complete picture, however, we should point out
that school Olympiads are by no means limited to such high-level
competitions as the just-mentioned municipal or All-Russia Olympiads.
There is also a broad-based, district-level round, success in which is
generally encouraged. The quite numerous forms of accountability that
have existed and continue to exist in schools have included providing
reports about work not only with the “bottom” of the student body —
about the so-called struggle against academic failure — but also
with the “top” of the student body, for example about students’
achievements in Olympiads. Predictably, this has led to contradictory
results: on the one hand, teachers have often found comparisons
between their activities in this respect unfair (not without reason) —
obviously, students at more selective schools show better results than
students at ordinary schools, and it is hardly possible to blame teachers
at ordinary schools for this; on the other hand, this kind of official
attention has nonetheless motivated teachers (even if not all of them)
to devote more thought to working with stronger students.
The district-level rounds of Olympiads include problems which,
even though they are not, generally speaking, especially difficult,
nonetheless differ substantially from the problems ordinarily solved
in the classroom. An as example, consider the following problem from
a district-level round for sixth graders:
Each of three players writes down 100 words, after which their lists
are compared. If the same word appears on at least two lists, then it
is crossed out from all the lists. Is it possible that, by the end, the first
player’s list will have 54 words left, the second player’s 75 words, and
the third player’s 80 words? (Berlov et al., 1998, p. 15)
The solution of the problem is based on a simple line of reasoning.
If the described outcome were possible, then the first player would
have 46 words crossed out, while the second player and third player
March 9, 2011
15:4
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch09
Extracurricular Work in Mathematics
385
would have 25 and 20 words crossed out, respectively. But 20
+ 25
is less than 46. Therefore, not all of the 46 words crossed out on the
first player’s list could have been on the other players’ lists.
Although no special prior knowledge is required to solve such a
problem, those who have had experience with solving problems that are
not “school-style problems” have found themselves in a better position
at Olympiads. Teachers are advised to conduct so-called school-level
rounds (Olympiads within a school), which are supposed to, on the
one hand, prepare students for the district-level Olympiad and, on the
other hand, select those who will be sent to the district-level round.
In practice, such school-level rounds are very often skipped, and the
problems suggested for school-level Olympiads are used in some other
capacity (for example, put on display, along with their solutions, to
allow students to become acquainted with them) or not used at all, and
teachers themselves decide whom to send to the district-level round
(the selection is usually not rigid, however, and students who wish
to take part in the district-level round can usually do so). We should
emphasize once more, however, that a systematic mathematics circle
can help students to prepare for an Olympiad.
Officially, the differences between mathematics circles and electives
have been (and remain) quite substantial. Generally speaking, students
have the right to choose which electives they wish to attend, but
once this choice is made they are required to attend the elective
which they have selected; by contrast, participation in a mathematics
circle remains voluntary at every stage (to be sure, a teacher can,
in certain situations, prohibit students who skip mathematics circle
meetings too often from attending at all). The wages received by
teachers for teaching mathematics circles and electives are somewhat
different as well (it should be noted that teachers have sometimes
taught mathematics circles with no compensation at all). Nonetheless,
it is not always possible to make a sharp distinction between the
programs of mathematics circles and electives. A topic that has officially
been included in the program of electives may become the basis for
a mathematics circle. The more “adult” word “elective” is heard
more frequently in the higher grades; in grades 5–6, only the term
“mathematics circle” is used.
