Alexandrov was, as we have already noted, a committed supporter of the classic geometric method, which goes back to Euclid. Nonetheless, he formulated proofs that were fundamentally new in school geometry.
One of them is given below.
The classic school theorem “a line L that is not perpendicular or parallel to planeP(an inclined straight line) is perpendicular to a lineM in planePif and only if the projection ofLonto planePis perpendicular toM” has usually been proven using congruent triangles. In Kiselev’s textbook, for example, this was done as follows (Fig. 4):
LetABbe a perpendicular to planeP,AC an inclined straight line, and BC the projection of that straight line onto plane P. On the straight line, let us mark off equal segmentsCE andCDfrom point Cand connect pointsDandEwith pointsAandB. Now we can see
Fig. 4.
that if AC⊥DE, it follows thatADEis an isosceles triangle, from which in turn follows the congruence BD = BE, and because of the properties of an isosceles triangle this means that BC⊥DE. If it is given thatBC⊥DE, practically the same argument leads to the conclusion thatAC⊥DE.
In place of this proof, Alexandrov suggested the following argu- ment, which is based on the notion of distance and the following proposition: the minimum value of the distance from point A, lying outside a straight line, to the points of this straight line is found at a point that is the base of the perpendicular dropped from A to this straight line.
Let us take a variable pointXon the given straight line and consider the two valuesAX2andBX2. The triangleABXis a right triangle.
Therefore,AX2=AB2+BX2. Therefore, the valuesAX2andBX2 differ by a constant term. Therefore, these quantities have their least values simultaneously — for the same point, X. If X is the base of a perpendicular dropped from A, then it is also the base of the perpendicular dropped fromBand vice versa. (Fig. 5)
What is important is not so much that Alexandrov’s proof is shorter than Kiselev’s (for students, the former is unlikely to be easier than the latter), but that it makes it possible to understand in a new way the essence of a classic theorem — that the theorem is about shortest distances — and in this capacity may be applied and generalized.
Fig. 5.
In giving this new proof, Alexandrov proceeded as a modern geometer, who is not confined to thinking in terms of Euclid’s categories and methods. Attempting to generalize what has happened over the course history, one could say, with all the necessary qualifica- tions, that in school-level instruction there has been a strand oriented toward the geometry of figures and another strand oriented toward the geometry of functions. The former — which stems, for example, from Euclid — finds the basic content of the subject to consist of the examination and study of the various figures that surround us and their interrelations; the latter, which stems from Klein, and in a certain sense from Descartes, pays the greatest attention to the functions that are important in geometry — geometric transformations. It is likely not by accident that Kolmogorov, who contributed possibly to all branches of 20th century mathematics, inclined toward the latter approach, which connects geometry with other mathematical disciplines, while the geometers Alexandrov and Pogorelov probably found greater affinity with the former, purely geometric approach.
In saying this, however, we must stress that talking about the purity of an approach, so to speak, is completely out of place in this context.
The attempt to transform school geometry into a part of some general mathematical theory about functions, matrices, and so on — although it might gladden the research mathematician due to its generality — deprives the student of the experience of direct investigation and reasoning. On the other hand, it would be strange to deliberately conceal from the students the new understanding that has come from the development of science.
What is old, traditional, and Euclidean is supplemented in Russian textbooks with what is new and post-Euclidean. This is accomplished in various ways, and one can argue about the relationship and balance between these two sides of the curriculum. Transformations, vectors, and coordinates, in the opinion of the authors of this chapter, must have a definite place in the school course, although second-generation standards devote little attention to them. On the other hand, we also believe that studies should begin, as history did, not with these materials, but with Euclidean methods. But what is perhaps more important than adding comparatively or even genuinely new sections to the traditional material is to read the classic material in a new way.
The degree to which it will be possible to connect traditions accumulated over the centuries with new mathematical conceptions and new pedagogical and social demands will define the development of school geometry in Russia in the 21st century.
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4
On Algebra Education in Russian Schools
Liudmila Kuznetsova, Elena Sedova, Svetlana Suvorova, Saule Troitskaya
Institute on Educational Content and Methods, Moscow, Russia
1 Algebra as a School Subject
Algebra as a science has undergone a whole series of transformations, which have radically changed its content. For Newton, algebra was “the universal arithmetic,” which used letter notations to solve arithmetic problems. For Bertrand, “algebra is aimed at shortening, making more precise, and in particular simplifying the solutions of questions that can be posed concerning numbers” (Goncharov, 1958, p. 41).
For Lagrange, “algebra may be seen as the science of functions;
however, in algebra only those functions are investigated which derive from arithmetic operations, generalized and transposed into letters”
(Goncharov, 1958, p. 41).
By the end of the 19th century, the view of algebra as the study of integral rational functions became firmly established in science; it was from this perspective that the university course in “Advanced Algebra”
was taught. By the middle of the 20th century, the science of algebra had taken a new step: relying on set-theoretical premises and using the axiomatic approach, it declared its problem to be “the study of
‘arithmetic’ operations, performed on objects of an arbitrary nature (‘group,’ ‘ring,’ ‘field’)” (Goncharov, 1958, p. 41).
The content of “Algebra” as a school course studied in Russian schools includes the foundations of the science of algebra in each of the interpretations given to it by Newton, Lagrange, and Bertrand;
however, at the present time the meaning of the subject “Algebra” is not limited to these interpretations. Of course, the school course in algebra does not involve studying algebra at the level of operations on abstract objects. But its contents are expanded due to the inclusion of the foundations of related branches of mathematics. At present,
“Algebra” as an academic subject studied in grades 5–9, as well as its sequel “Algebra and Elementary Calculus,” which is studied in grades 10–11, constitute “conglomerate” subjects, addressing basic probabil- ity theory, calculus, and analytic geometry, no less than basic algebra.
These school subjects acquired such a form mainly as a result of the influence of the content of education in institutions of higher learning, which is increasingly becoming a form of mass education among young people, and in which calculus, analytic geometry, and probability theory invariably occupy places of paramount importance.
2 The Algebraic Component in the System of School Mathematics Education
Mandatory universal education in Russia consists of three stages:
elementary schools (grades 1–4) for children between the ages of six and 10; basic schools (grades 5–9) for children between the ages of 11 and 15; and high or senior schools (grades 10–11) for students who are 16–17 years old. The last stage of education introduces a furcation, which means in particular that students have the opportunity to study subjects of the mandatory sequence — of which, naturally, mathematics is a part — at two different levels: basic and advanced (“profile”).
Broadly speaking, the algebraic component is represented at all stages of education. In mathematics classes in elementary school, students gradually learn to use letters to denominate numbers, to put together elementary equations, and so on. In this way, when they reach basic school, they already have a certain minimal experience of
“interacting” with letters.
School subjects at the basic and high school stages of education —
“Algebra” and “Algebra and Elementary Calculus” — contain, as has
already been noted, several different strands of content. A special place among them is occupied by educational material that can in essence be regarded as strictly algebraic. Possessing a history of almost three cen- turies, it is today isolated into an independent component that interacts with the other components of the course, which have their own aims and goals. This is algebra in its classic definition — computations and equations involving letters.
It is precisely this purely algebraic component of the school course
“Algebra,” which aims to develop the ability to design mathematical models, abstract from inessential details, and form skills pertaining to the formal manipulation of numerical and literal data, in keeping with the essence of mathematical science that produces that mathematical apparatus without which it is impossible either to investigate problems internal to mathematics or to solve practical and applied problems. It is precisely this purely algebraic component, therefore, that is called upon to demonstrate to students the power of the mathematical method.
3 The Content of Algebra Education in Russian Schools
The content of mathematics education in contemporary Russian schools is prescribed by two main documents approved by the Ministry of Education and Science. These are the basic time allocation plan and the federal component of the educational standards for general education —Standard(Ministry, 2004a, 2004b).
The basic time allocation plan allocates no fewer than 875 hours for the study of mathematics in basic school, estimating five hours per week in grades 5–9, and in high school four class hours at the basic level and six class hours at the advanced level. Thus, there are 280 and 420 class hours in all, respectively (the time for the study of the advanced course may be increased up to 12 hours per week by using the school’s allotment of elective courses). Of these, about 350 hours are designated for the study of algebraic material in basic school, and in high school about 90 hours at the basic level and no fewer than 140 hours at the advanced level.
The Standard defines the objectives of studying each school subject, a mandatory content minimum, and requirements for graduation.
In basic school, the study of algebraic material is aimed at the formation of a mathematical apparatus for solving problems drawn from mathematics, related subjects, surrounding reality, the acquisition of practical skills necessary for everyday life, and the creation of a foundation for the further study of mathematics. It is intended to facilitate logical development and the formation of the ability to use algorithms. The language of algebra underscores the significance of mathematics as a language for constructing mathematical models of real-world processes and phenomena. One of the basic purposes of studying algebra is to develop students’ algorithmic thinking, which is indispensable, for example, for the assimilation of the course in computer science; the study of algebra also helps students acquire the skill of deductive reasoning. The manipulation of symbolic forms contributes in its own specific way to the development of imagination and the capacity for mathematical creativity.
In the process of assimilating algebraic material, students have the opportunity to acquire the symbolic language of algebra, to develop formal-operational algebraic skills, and to learn to apply them in solving mathematical and nonmathematical problems. In a broader context, algebra, along with the other components of school-level mathematics education, facilitates the development of logical thinking, speaking skills, and an understanding of the concepts and methods being studied as crucial means for the mathematical modeling of real-world processes and phenomena (Ministry, 2004c, p. 2).
In the study of algebra in high school at the basic level, educa- tors solve the problem of teaching students new types of formulas;
improving practical skills and computational literacy; expanding and improving the algebraic apparatus formed in basic school; and using it to solve mathematical and nonmathematical problems (Ministry, 2004d, p. 2).
In high school at the advanced level, the content of algebraic education represented in basic school is developed in the direction of constructing a new mathematical apparatus based on an expansion of number sets from real to complex numbers; and the development and improvement of techniques for algebraic transformation, solving equations, inequalities, and systems.
The study of algebraic material, along with the other components of school-level mathematics education, contributes to solving the general problem of developing students’ ability to construct and investigate ele- mentary mathematical models in solving applied problems, and solving problems from related disciplines, thereby increasing knowledge about the special characteristics of the application of mathematical methods in the study of processes and phenomena in nature and society (Ministry, 2004e, pp. 1–2).
As already stated, the content of school-level mathematics education is defined by the Standard. Following the Standard, we offer a description of the content of the algebraic component at the basic school and high school stages; material that must be studied but is not part of the graduation requirements is indicated in italics (Ministry, 2004a, 2004b).
Grades 5–9
Algebraic expressions. Literal expressions (expressions with vari- ables). Numerical values of literal expressions. Permissible values of variables in algebraic expressions. Substituting expressions in place of variables. The equality of literal expressions. Identities; proving identities. Transformations of expressions.
The properties of powers with integer exponents. Polynomials.
Adding, subtracting, multiplying polynomials. Short multiplication formulas: squares of sums and squares of differences;cubes of sums and cubes of differences. Factoring polynomials. The quadratic trinomial.
Completing the square of a quadratic trinomial. Viète’s theorem.
Linear factorization of the quadratic trinomial. Polynomials with one variable. Powers of polynomials. Roots of polynomials.
Algebraic fractions. Reducing fractions. Operating with algebraic fractions.
Rational expressions and their transformations. Properties of square roots and their use in computation.
Equations and inequalities. Equations with one variable. Roots of equations. Linear equations. Quadratic equations: the quadratic formula. Solving rational equations. Examples of solutions to higher- degree equations; methods of variable substitution, factorization.
Equations with two variables; solving equations with two variables.
Systems of equations; solving systems of equations. Systems of two