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Assessment in Mathematics in Russian Schools
347
incomprehensible to them; thus, although solving these problems is by
no means necessary for passing the test or even receiving a good grade
on it, generally speaking this might produce a certain psychological
discomfort.
Let us give an example of a test whose authors may be characterized
as moderate adherents of the second point of view. This test is intended
for seventh graders, to be completed in one class period, and given after
students have studied equations and word problems that are reducible
to linear equations. Those problems on the test which belong to the
so-called “required level” are marked with the symbol
•. The authors
leave the criteria for grading up to the teacher’s discretion — pointing
out, however, that the teacher may decide to give a five both when all
problems have been solved and when one of the last two has not been
solved.
1.
• Solve the equation (a) 3x+2.7 = 0; (b) 2x+7 = 3x−2(3x−1);
(c)
2
x
5
=
x−3
2
.
2.
• Three seventh grade classes contain a total of 103 students.
Class 7b has four more students than class 7a and two fewer
students than class 7c. How many students are in each class?
3. Solve the equation
2
x−1
2
=
x+5
8
−
1
−x
2
.
4. In three days, a hiker walked 90 km. On the second day, he
walked 10 km less than on the first day, while on the third day,
he walked
4
5
of the distance that he covered on the first and
second days together. How many kilometers did the hiker walk
every day? (Zvavich, Kuznetsova, and Suvorova, 1991, p. 104)
As we have already remarked, checking and grading a student’s
work by no means consists in merely counting the correct answers. The
teacher usually first evaluates each problem individually, often making
use of the following system of classification and symbols:
+ The problem has been solved correctly; no comments.
± The gist of the solution is correct, but there is a small mistake.
∓ The problem has been solved incorrectly, but certain knowl-
edge and skills have been demonstrated.
− The problem has been solved incorrectly.
0 The student did not attempt to solve the problem.
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Russian Mathematics Education: Programs and Practices
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348
Russian Mathematics Education: Programs and Practices
Some teachers add a set of more subtle categories:
+. The problem has been solved correctly, but with an insignifi-
cant deficiency.
+! A difficult problem has been solved exceptionally well and
requires a special additional five (this test contains no such
problems).
−! The problem has been solved exceptionally poorly; the student
has demonstrated a complete lack of comprehension of this
problem.
The grade
± might be given, for example, if a student has correctly
solved a given equation but has made a computational error (in a
problem in arithmetic, of course, a mistake in arithmetic would cause a
grade to be lowered more substantially). The grade
∓ might be given,
for example, if in the solution to problem No. 4, the equation was
formulated in a fundamentally incorrect way, but then solved correctly.
Subsequently, the teacher usually analyzes the results obtained and
establishes criteria for grading (and different teachers’ opinions about
the lower and upper bounds of each grade do not necessarily coincide).
We would consider it reasonable to give a grade of three for the test
reproduced above if the results, for example, were as follows:
1a
1b
1c
2
3
4
+
+
∓
+
−
0
Another important factor that comes into play in grading tests, as
has already been noted, is the level of detail that the teacher demands
in a solution. Clearly, the more problems a test contains and the more
difficult these problems are, the lower will be the level of detail that
students can offer. If students are required, in solving a problem in
geometry, to indicate at every step what theorems they are using in the
course of their reasoning, then they will have time to solve no more
than two, or at most three, problems. While it is certainly necessary
to give such tests, it is not necessary that the problems on such tests
be difficult. One of the basic methodological aims of such tests is to
see how well students can express their ideas in written form. Other
kinds of tests, however, are possible and necessary as well. For example,
March 9, 2011
15:4
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch08
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