4.2 High School (grades 10–11; students aged 16–17)
4.2.2 The study of algebraic expressions in grades 10–11
The content of the material pertaining to the study of algebraic expressions in high school, like the other sections of the course in mathematics, is prescribed by the Standard. It should be noted that certain topics are listed in the Standard in italics; these topics must be included in the curriculum but are not part of the final attestation.
Also, the Standard does not require that the high school curriculum include a section specifically devoted to numbers. Issues connected with expanding the concept of number thus belong to the algebraic part of the curriculum.
As can be seen from the passages from the Standard cited above, the content that pertains to the study of algebraic expressions differs substantially in the basic and advanced courses. Their common part is connected with the study of roots of degreen, powers with rational exponents, and the logarithm of a number. In these sections, students receive virtually the same set of theoretical facts, so the main difference is in the depth of their assimilation of this material.
As an illustration of this difference, consider how the students learn the topic “Roots of thenth degree and their properties.”
In the textbook by Kolmogorov et al. (2007), the emphasis is on learning definitions and algorithms. Thus, in studying this topic, students must assimilate certain techniques for transforming algebraic expressions. At the mandatory level, they must learn how to solve the following types of problems:
• Move a factor outside the radical sign (a >0,b >0):
(a)√6
64a8b11; (b)√5
−128a7.
• Move a factor inside the radical sign (a >0,b >0):
(a)−b√4
3; (b)ab8 5b3
a7 . (Kolmogorovet al., 2007, p. 205)
A somewhat higher level is illustrated by the following problem:
• Put the following expression in the form of a fraction whose denominator does not contain a radical: (a) √3 1
2−√3
3; (b) 2
a−√3 b. (Kolmogorovet al., 2007, p. 206)
The corresponding technique, as we will see below, is used in solving irrational equations.
In the advanced textbooks of Dorofeev, Kuznetsova, and Sedova deliberately learning to carry out elementary algorithms is not an end in itself. Transformations involving radicals as a rule play a secondary role and have the character of technical work, which must be carried out in the process of solving more substantive problems.
For comparison, consider several problems on the topic examined above from the problem book of Dorofeev, Sedova, and Troitskaya (2010). Of course, as in Kolmogorov et al.’s (2007) textbook, this problem book includes problems that involve elementary simplifica- tions of expressions with radicals. But this problem book also examines the opposite problem: under what conditions (constraints on variables) is an already-transformed expression equal to the one given? (Dorofeev, Sedova, and Troitskaya, 2010, p. 16).
For whatxandyis the expression(y−5) x−y−155 equal to:
(a)√
(x−15)(y−5); (b)−√
(x−15)(y−5); (c)√
(15−x)(y−5)?
Considerable emphasis is placed on the understanding of the connection between this new concept and other areas of mathematics.
Thus, for example, for practical purposes, a student has no need to think about the fact that the root of a positive integer cannot be anything other than a positive integer or irrational number, but the future mathematician must understand this.
Students may recall how, in basic school, they proved by contradic- tion that certain roots are irrational: for example, that the number√
2 is irrational. Now they possess an instrument that makes it possible to prove at once that all numbers of this form are irrational (the textbook discusses how the rational roots of the polynomialf(x) =xn−kcan only be divisors of the numberk, i.e. integers). Therefore, a problem
that requires students to prove this fact is included among the problems in this section.
Problems involving transformations of expressions with radicals by means of multiplying them by a “conjugate factor” are traditionally widespread (Kolmogorovet al., 2007, p. 206). The advanced problem book offers a more substantive problem:
Is the following function monotonic? y = √
x+1 − √ x−2.
(Dorofeev, Sedova, and Troitskaya, 2010, p. 15)
This problem is solved precisely by multiplying by “conjugate factors”: after the corresponding transformation, we obtain another expression for the same function,y = √x−2+3√x+1, from which it can be seen that this function is a decreasing function (the numerator of the fraction is a positive constant, while the denominator is increasing).
Of course, by including one problem within another in this way, we always obtain a problem that allows for several solutions; and indeed the student has the right to dispense with multiplying by a “conjugate factor,” which, however, does not seem worrisome. The problem book also contains problems involving proofs of formulas with double radicals:
(a) a+√
b= a+√2a2−b+ a−√2a2−b; (b)
a−√
b= a+√2a2−b− a−√2a2−b.
This problem constitutes both an exercise in transforming expres- sions with radicals and a certain addition to students’ algebraic arsenal:
of course, they do not need to memorize this formula, but it is important for understanding the fact that sometimes (if the number a2 − b > 0 is a perfect square and a > 0) one can eliminate a complicated radical by turning it into the sum of two simple radicals;
this technique is sometimes used to solve biquadratic equations.
As an example of an even more difficult assignment that involves a proof, consider the following problem:
Prove that 53 < √126+√127 + ã ã ã + √135 <2. (Dorofeev, Sedova, and Troitskaya, 2010, p. 19)
Here, too, the solution cannot be reduced to an operation involving radicals. The student must note that
√10
35 < 1
√26 + 1
√27 + ã ã ã + 1
√35 < 10
√26,
since each of the given ratios, beginning with the second one, is less than that preceding it — after which it is not difficult to see that
5
3 < √1035 and √10
26 <2.
This section also covers the topic “Divisibility.” At this point, we must explain in greater detail our understanding of what an advanced course in mathematics in high school must achieve, and the fundamental difference between such a course and a course that results simply from the addition of certain topics to the basic course in mathematics (unfortunately, there is a common but — in our view — erroneous opinion that this latter type of course is just what constitutes an advanced course in mathematics).
As an example, let us consider precisely the topic “Divisibility.”
Why was this topic included in the content of the curriculum? We can point to many reasons for this, but the main, most “conceptual” one apparently had to do with the fact that issues connected with divisibility are far more important for mathematics than, say, solving irrational equations; in other words, this topicbrings the content of the school course closer to real mathematics. In particular, knowledge of this material makes it possible, in studying the topic “Polynomials with one variable”
(another topic that distinguishes the advanced course from the basic, examined in greater detail below), to study questions connected with the rational roots of polynomials with integer coefficients, i.e. to solve a broader range of higher-degree equations, and in turn to make use of this knowledge in studying rational and irrational numbers, and so on.
We might also mention that in solving problems pertaining to these topics, students learn to use not so much the algorithms for solving some narrow class of problems as the methods and techniques of mathematical activity in general.
At present, problems connected with divisibility are generally thought of as belonging to the category of so-called Olympiad prob- lems; but this is so only because in the existing course in mathematics,
this content is covered effectively only in grades 5–6 and essentially has a narrowly directed aim — to develop certain well-defined arithmetic abilities and skills.
The stylistic aspect of this topic’s presentation is determined first and foremost by the objectives associated with studying mathematics in basic school, where it is by no means assumed that most students will take the advanced course in mathematics in high school. It is also limited objectively by the age-dependent characteristics of the students — the highly concrete nature of their thinking, which makes it difficult for them to interact with abstract objects, and with letters in particular, because of their insufficiently developed capacity for making theoretical generalizations, and for understanding the essence of proofs and their role in mathematics; because of their lack of any felt need to prove propositions “in the general form” when confronted with conclusive concrete examples; and so on.
However, these traits are no longer characteristic, by and large, of 16–17-year-old teenagers, especially those who have gone through three more years of schooling in a different style that is more in harmony with the essence of mathematics and, above all, those who have chosen an advanced course of study, designed essentially for the formation of the country’s “technical–scientific elite.”
This position became more or less central in the general approach of the textbooks and problem books of Dorofeev, Kuznetsova, Sedova, and Troitskaya. “Divisibility” is the first topic presented in these textbooks, mainly with a view of providing continuity with the content of the basic school curriculum, but also in consideration of the objective simplicity of its content and its proximity to experiences that students already have. The difficulties with its assimilation (both on the level of theory and, to an even greater degree, on the level of exercises) are connected with a purely psychological barrier: the unfamiliarity of the mathematical activity that corresponds to the content of the material.
In particular, in treating this topic, the authors of this textbook use material that is quite simple to form and develop the students’ ability to formulate proofs; this ability, as is well known, is one of the most significant weak spots in the mathematical preparation of students. In doing so, the authors have not deemed it necessary to fill in all the
logical gaps that have been left by the study of divisibility in basic school; for example, they do not consider it necessary even to prove the criteria for divisibility by 3 and 9 in the general case. Thus, for example, they present the proof of criteria for divisibility by 11 in basic-school fashion, based on presenting an example, the generality of which is obvious to any mathematician and must be equally obvious to any student. A formal proof of this fact requires only a complicated mathematical “ornament” and, apart from logical rigor (which in this instance seems superfluous), adds nothing to the mathematical content of the argument or, most importantly, to the basic problem of developing the students’ mathematical thinking. Moreover, the very fact that students have understood the generality of an example that conclusively demonstrates the mechanism of a potential formal proof constitutes an important contribution to their mathematical thinking, promoting those peculiar features of thought which are characteristic of mathematicians and necessary for assimilating mathematics.
Let us note that the concept of logical thinking, the thinking that is used in mathematics and to an even greater degree by representatives of other sciences, is substantially broader than that of deductive thinking — a fact that many representatives of the methodological disci- plines and practicing teachers sometimes forget, losing or substantially weakening theproductivecomponent of thinking by doing so.
Everything that has been said above pertains, of course, not just to the topic “Divisibility,” but illuminates the way in which an advanced course in mathematics must differ from the basic course, what the general principles governing the design of the advanced course must be, and what approach must be used, in our view, to solve the corresponding methodological problems.
Let us examine concrete problems for students that reflect the authors’ approach to the topic “Divisibility” in the aforementioned textbooks by Dorofeevet al.
1. Prove or refute the following statements:
(a) All even numbers are composite; (b) if an even number is divisible by 15, then it is divisible by 6; (c) if an even number is divisible by 15, then it is divisible by 20.
2. Prove that:
(a)32003+32004+32006is divisible by 31;
(b)20186+18253is divisible by 19.
3. Find the remainder after dividing:
(a)6n+5(n— integer) by 3;
(b)6n+5(n— positive integer, greater than 1) byn; (c)22005by 7.
4. Which of the progressions
5, 8, 11,…; 4, 7, 10,…; 6, 9, 12,…
contains the number11ã3820−4ã2510? (Dorofeev, Kuznetsova, Sedova, and Okhtemenko, 2004, p. 38)
As we can see, there is no general rule, no algorithm, and no general ability for solving these problems except one: the ability to reason. Not for nothing was the topic “Divisibility” traditionally a favorite topic for problems on college entrance exams, at a time when there was no Uniform State Exam.
Clearly, despite the simplicity of the formulations of these problems, the basic level of preparation is not enough to solve them — and this has to do not with new, additional criteria for divisibility (for example, criteria for divisibility by 11), which may or may not be present in the textbook of the advanced course; or with new concepts and theorems that the Standard prescribes for the advanced level (such as “Congruences”). It has to do with the depth with which those concepts are assimilated, which are already known to all graduates of basic schools. The Standard does not stipulate the study of any ready- made algorithms for solving such problems; rather, what is required of graduates here is the ability to engage in mathematical reasoning in nonstandard situations.
With regard to significant differences between the content of the basic and advanced courses, we should also look at the topic
“Polynomials,” which is studied in advanced classes. The main purpose of this topic, according to the Standard for advanced schools, is to improve the general mathematical preparation of the students, and to
help them learn simple and effective techniques for solving problems, especially algebraic equations.
Without presenting any fundamental difficulties, the study of polynomials gives students the possibility of solving many problems that belong to all other parts of the course. In particular, this theoretical content can be effectively used in solving problems connected with prime and composite numbers, while the ability to find the rational roots of polynomials with integer coefficients allows the students not to be too afraid of cubic equations and higher-degree equations — in many cases, to stop relying on the art of grouping (i.e. heuristic techniques) and to make use instead of the algorithmic methods of the theory of polynomials; to simplify standard proofs; and so on.
It should also be noted that the study of polynomials provides a fitting conclusion to the generalization of the concept of number, while the parallelism between the theory of polynomial factorization and the outwardly very different theory of integer factorization, unexpected for the students, is important from a general educational and general cultural point of view.
Let us consider some examples pertaining to this topic. The following problem provides a useful illustration:
Is the expression 1
x2+1a polynomial?
This problem calls for a well-founded answer. Naturally, the main point here is for students to grasp the concept of a polynomial in a substantive sense, and therefore excessive attention to formalities in defining this concept is unlikely to be fruitful. Attempts to give a logically impeccable definition of a polynomial will merely lead to formulations with which probably not even all professional mathe- maticians are familiar. On the other hand, in trying to identify a polynomial among other expressions, a logically developed student must understand that, strictly speaking, this cannot be judged merely by the external form of an expression. Thus, for example, the expression
1
x2+1 is not a polynomial not because there do not appear to be any algebraic transformations that can be used to put it into the appropriate form, but because there actually are no such transformations. Indeed, supposing that the given expression is a polynomial, then from the
equality 1
x2+1 =f(x), wheref is a polynomial of degreen, there would follow the equality1 = (x2 +1)f(x); but this equality is impossible, since its left-hand and right-hand sides have different degrees.
The algorithm for searching for rational roots must be worked on until it becomes a familiar skill. Students must not experience difficulties when they encounter problems of the following type:
• Find all the roots of the following polynomial:
3x6−14x5+28x3−32x2−16x+16=0.
• Factor the polynomial f(x) = 3x4−2x3−9x2+4 into linear factors.
When studying divisibility and division with a remainder, there is no need, for most polynomials, to list completely, much less to memorize, the criteria of divisibility. On the contrary, it is far more useful to emphasize to the students that many properties of the divisibility of integers that are known to them are present in the divisibility of polynomials as well. But the students must also be asked to prove these properties (or some part of them) on their own, and in the process of formulating these proofs they will conclude for themselves that the arguments differ only because of their terminology and symbolism.
Another theme that distinguishes the advanced course from the basic one is connected with the concept of a symmetric polynomial. In our view, students who have chosen the advanced course could have already learned at the basic-school level how to solve various problems that require only identity transformations aimed, to put it in a lofty way, at expressing any symmetric polynomials in terms of elementary symmetric ones. Note that various ordinary identity transformations effectively constitute the central content of algebra in basic school, but are often lacking in ideas and aimed mainly at simplifying expressions.
Such a situation even compromises mathematics to some degree in the eyes of the students: it almost seems as if someone had deliberately complicated simple expressions to create difficulties for students.
Meanwhile, the concept of the symmetric polynomial makes it possible to introduce substantive problems of another type. For example, expressing the sum x3 +y3 +z3 through the elementary symmetric
polynomials
x3+y3+z3 =(x+y+z)3−3(x+y+z)(xy +xz+yz)+3xyz makes it possible to solve the most varied problems. Thus, using the identity just given, it is not difficult to deduce that the last three digits of the number4233+2553+3223−423ã255ã272are zeroes, since x3+y3+z3−3xyz is divisible by x+y+z. From the same identity follows the inequalityx3+y3+z3−3xyz ≥0, i.e. in essence an inequality between the arithmetic mean and the geometric mean.