About mathematical preparation of applicants

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ABOUT MATHEMATICAL PREPARATION OF APPLICANTS

On December 22, 1970, at the Moscow Institute of Steel and Alloys, the rector’s office, together with the department of higher mathematics, held a meeting of mathematics teachers from secondary schools in Moscow to discuss the mathematical training of technical school applicants.

The joint report “On the required level of mathematical training of applicants” was made by the head. Department of Higher Mathematics, Professor I. E. Bazilevich and Professor M. A. Akivis. The report noted that when studying a higher mathematics course at a college, the following sections of the school mathematics course are especially often used: transformations of algebraic and trigonometric expressions, properties of the basic elements of functions and their graphics, solutions of various types of equations, equalities and systems of equations or inequalities, the concept of the absolute value of a real number and an arithmetic root, basic theorems and formulas of geometry, etc.

However, it is precisely these examples of sections that many applicants and first-year students have poor command of. The speakers emphasized that many applicants do not know how to correctly and competently format their work, which is associated with the inability to think clearly and consistently.

At the end of the report, some provisions were formulated, the implementation of which will help improve the level of mathematical training of applicants:

a) throughout the entire teaching of mathematics in secondary school, attention should be paid to the logical validity and coherence of the course; students must be able to think correctly, reason and correctly formulate their thoughts;

b) when solving various kinds of equations and inequalities, the concept of a region of acceptable values ​​should be used and students should be taught to monitor the change in this region during certain transformations;

b) special attention should be paid to the technical requirements of nalgebraic and trigonometric expressions; you should be required to fill out the basic formulas of algebra, trigonometrics and geometry;

d) solid values ​​of properties and graphs of the main elements of functions should be added; these properties should be used when solving equations and inequalities;

d) when solving geometric problems, students must be able to make a correct drawing and carry out a clear justification for all constructions and calculations;

e) the current school curriculum in mathematics does not contain some topics, knowledge of which is desirable when studying a mathematics course at a university. These include connection theory, Newton’s biome, Bezout’s theorem and solutions of two-valued equations. These topics should be included in elective mathematics classes. In addition, in these classes you should study complex numbers and inverse trigonometric functions more deeply than provided by the program.

The report of Associate Professor V.V. Goldberg highlighted the most common mistakes made by applicants during the entrance exams.

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The greatest number of errors in the work of applicants occurs when solving exponential and logarithmic equations and especially inequalities; equations and inequalities containing an unknown under the absolute value sign; when writing the general form of angles corresponding to a given value of trigonometric functions, and especially when solving the simplest trigonometric inequalities; when constructing graphs of the simplest transcendental functions.

Often comes across errors of a logical nature, especially in a geometry course: often going from a theorem to a definite one, a direct theorem from a converse; transfers the properties of some particular classes of functions to more general classes (it is believed that the median always coincides with the bisector or the height of the triangle, etc.); do not know the method of mathematical induction; do not see the need to prove the properties of some functions (trigonometric, exponential, logarithmic).

Once again attention was drawn to the flat design of written work by many applicants. When writing down solutions to problems, there is no logical sequence; a number of necessary justifications (especially in geometry) are missed.

The speaker pointed out a number of specific mistakes made by applicants when solving written work in 1970.

When solving an irrational equation

$$\sqrt[3]{x+1} + \sqrt[3]{3x+1} = \sqrt[3]{x-1}$$

Many applicants, after being cubed, replaced $$\sqrt[3]{x+1} + \sqrt[3]{3x+1}$$ with $$\sqrt[3]{x-1}$$, thereby acquiring the constructed root ($x=0$).

When solving inequalities, transformations are often made that lead to the construction of a solution. These

extraneous decisions are difficult to discard using verification, since they can fill an entire gap.

So, solving the inequality

$$x > \sqrt{2x+24}$$

by squaring, many received the answer $x > 6$, $-12 < x < -4$, without replacing that all values ​​$x$ from the second interval do not satisfy this inequality (this would not be accidental if it had been noticed that the inequality $x > 0$ follows from the condition). When solving the inequality

$$\left(x^2+x+1\right)^{\frac{x+5}{x+2}} \geq \left(x^2+x+1\right)^3$$

many correctly saw that it was necessary to consider the case of $x^2+x+1 > 1$ and $0 < x^2+x+1 < 1$, but they forgot that since the inequality is not strict, its solutions will not be the values ​​$x$, for which $x^2+x+1=1$, that is, $x=0$ and $x=-1$.

The inability to correctly operate with absolute value and root argument led to numerous errors. So, for example, solving the equation

$$\sqrt{1+\sin 2x} = \sqrt{2} \cos 3x,$$

Where

$$\pi < x < \frac{3}{2} \pi,$$

wrote a lot

$$\sqrt{1+\sin 2x} = \sin x + \cos x$$

instead of correct

$$\sqrt{1+\sin 2x} = |\sin x + \cos x| = -\sin x - \cos x,$$

since $\pi < x < \frac{3}{2} \pi$.

When simplifying the expression

$$\left(a+x\right)^{\frac{1}{2}} - \frac{1}{2} + \left(a-x\right)^{\frac{1}{2}} - \frac{1}{2}$$

at $x=4$ ($a-1$), $1 < a < 2$ we have to transform the expression $\sqrt{(a-2)^2}$.

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Many people wrote
$$\sqrt{(a-2)^2} = a - 2$$
instead of correct
$$\sqrt{(a-2)^2} = |a-2| = 2-a,$$
since $1 < a < 2$.
The main disadvantage in solving trigonometric equations was the flat value of trigonometric formulas and the inability to solve the simplest trigonometric equations.
Solving the equation
$$\frac{1 + tg x + \dots + tg^n x + \dots}{1 - tg x + \dots + (-1)^n tg^n x + \dots} = 1 + \sin 2x,$$
some applicants did not even know what to do with the left side of this equation. Those who saw two infinitely decreasing geometric progressions there correctly wrote that the left side makes sense if $|tg x| < 1$, that is, $-\frac{\pi}{4} + k\pi < x < \frac{\pi}{4} + k\pi$, but, having solved the equation, they left both series of solutions:
$$x_1 = k\pi \text{ и } x_2 = -\frac{\pi}{4} + k\pi.$$
Equation
$$3tg2x - 4tg3x = tg^23xtg2x$$
many solved it irrationally by expressing $tg2x$ and $tg3x$ in terms of $tgx$, resulting in a complex algebraic equation for $tgx$. The fastest way to achieve the goal was also the transformation:
$$3(tg2x - tg3x) = tg3x(1 + tg2xtg3x),$$
and, since in OD3 $1 + tg2xtg3x \neq 0,$
$$3\frac{tg2x - tg3x}{1 + tg2xtg3x} = tg3x,$$
or $-3tgx = tg3x.$

When solving the equation
$$8\cos^4x = 3 + 5\cos 4x$$
Some of the applicants arrived at a fourth-degree equation for $cos x$, which they failed to solve. This would not be an accident if we reduced this equation to an algebraic equation with respect to $y = \cos 2x$.
This solution yields a quadratic equation
$$2y^2 - y - 1 = 0.$$
When solving geometric problems, in addition to the logical errors mentioned above, the speaker noted illiterate drawings (incorrect drawing of the main section of a cone, placing a ball - on this subject, see the article by N.M. Beskin in our magazine No. 12 for 1970), ignorance of the properties of parameters with equilateral edges or faces, inability to construct a linear angle with two angles of the angle between adjacent lateral faces of parameters. or find the centers of the inscribed and circumscribed parameters of the balls. Great difficulties arise for applicants in tasks where data are nonlinear elements (area, volume) or a correspondence is given between some elements (sum of legs, ratio of lateral and total surface, etc.).
At the end of the meeting, the scientific secretary of the secondary scale section of the Moscow Mathematical Society A. Ya. Margulis and the senior methodologist in mathematics at the Institute for Teacher Development S. M. Gul spoke on behalf of the teacher, who, in part, pointed out the unconditional usefulness of such meetings and wished that similar meetings would be held in the future.

M. L. Smolyansky,
G. N. Dyachenko,
V. V. Goldbere