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Two approaches to the study of mathematics in high school (Russia, Germany)

For centuries mathematics became to the integrated part of school education in all over the world. Mathematics as the science is promoting of intellectual development, formation of logic, abstract thinking and intuition.

By russian standarts [3] the main tasks of mathematical education were determined:

·                        obtaining a system of mathematical knowledge and skills which are necessary to apply in practice, study related subjects and continuing education;

·                        intellectual development, formation of personal skills necessary for a normal life in modern society: clarity and precision of thought, critical thinking, intuition, logical thinking, elements of algorithmic culture, spatial concepts, ability to overcome difficulties;

·                        formation of the concept concerning ideas and methods of mathematics as a universal language of science and technology and intrument for modeling of phenomena or processes;

·                        growing of the culture of personality, viewing to mathematics as part of human culture, the understanding of the importance of mathematics for scientific and technical progress.

German mathematical education seeks to ensure that the student appeared able to understand the importance of mathematics in the modern world and to be able to apply mathematical methods in their daily activities [4].

In Germany high school graduate should have the following mathematical skills:

·  to determine of the problems in reality which can be solved by mathematics;

·  to formulate these problems using the language of mathematics;

·  to solve these problems using mathematical knowledge and techniques;

·  to analyze of the used methods;

·  to interpret of the results in the perspective of the problem;

• to organize and record the final results of the task.

The success of mathematics-study strongly depends on the content of classbooks or textbooks. To the problem of writing of the textbooks in mathematics has always paid great attention in science, teaching and methodical literature.

In this work we have analyzed the main sections of mathematical textbooks for high school published in Russia and Germany [1], [2]. We took for analysis only the relevant sections of textbooks. The main purpose of our research is to determine the suitability of modern mathematical textbooks to the tasks of school mathematical education.

For the analysis we used the following criteria:

- Scientific and systematic exposition of the theoretical material;

- Accuracy of the statements and the establishment of concepts and definitions;

- The ratio of theoretical and practical material;

- Applied direction of exercises and tasks;

- Simplicity of presentation of the educational material;

- Clarity and brevity in presentation of the theoretical material;

Table 1 shows the main sections of the German and Russian textbooks in mathematics recommended by the ministries of education in these countries.

Sections to be studied in a German textbook

Sections to be studied in a Russian textbook

Function analysis using derivative

Function analysis using derivative

The use of differential calculus

The use of differential calculus

Fundamentals of integral calculus

Fundamentals of integral calculus

The use of integral calculus

The use of integral calculus

The rules for finding the derivative, derivative of a composite function

The rules for finding the derivative, derivative of a composite function

Trigonometric functions

Trigonometric functions

Elements of Probability theory

Elements of Probability theory

Random variables and probability distributions

Random variables and probability distributions

Exponential function

 

System of linear equations

 

Vector

 

Straight line in space

 

Scalar multiplication

 

 

Matrix

 

 

The real numbers.

Infinitely decreasing geometric progression.

nth root.

Exponentiation by a real index

 

Exponential function

 

Logarithmic function

 

1.     Power function. Analysis of the function with the use of derivatives.

 

 «Power function» and «Analysis of the function with the use of derivatives» are Separate topic in Russian textbook [1, C. 39-67, 257-279].

In the given textbook discusses in detail the different cases of the power function :

1. Index ð=2n is even natural number.

2. Index ð=2n-1 is odd natural number.

3. Index ð=-2n, where “n“ is natural number.

4. Index p= -(2n-1), where “n“ is natural number.

5. Index p – positive real noninteger.

6. Index ð – negative real noninteger.

For each case domain of a function, range, intervals of increase and decrease of the function are determined.

The graph of the function for a particular case is drawn.

Then the exercises for training are offered:

- to sketch the graph of function and to find domain of a function and it’s range;

- to identify the intervals of increase and decrease of the function;

- to compare numbers using the properties of power functions;

- to find the coordinates of the crossing points of  functions graphs.

All exercises are overly theoretical.

There are no tasks with practical content.

“Function analysis using derivative” is a final paragraph in section ”Derivative” in case of Russian textbook.

Before proceeding to the study of functions the students need to learn the concept of derivative, the geometrical and physical meaning of derivative, the continuity of function and the equation of tangent.

All "approaches" to all formulas concerning these concepts demonstrated using the limit.

There are many formal definitions.

During the study of some subject the proofs of the theorems taking place also.

In the 47-page textbook 34 pages are provided on theoretical material.

All the exercises are theoretical and there are no practical tasks.         

In German textbook the paragraph “Power function”[2, C.13-22] considered as a part of section ”Analysis of the function with the use of derivatives” while consideration of the theme begins with a practical example.

Traffic on the roads occurs regularly during the holiday season. This traffic occurs due to objective reasons sometimes. This may happens when the road is under construction for example. Then the traffic flow have to be restructured from several lines into two or just one. To avoid this it is necessary to determine the recommended speed which vehicles have to use per unit of time to pass this section of road. And it is aslo necessary to pay attention that if the vehicle is moving too slowly it will hamper the movement of other vehicles behind.

At the too high speed the limit of designated safe distance between vehicles is becoming too small which increases the accident rate on this part of road. The vehicle must keep some sufficient distance from other vehicles for safety. This distence is . Vehicle length is 5 meters. Task is to determine the optimal speed of the car in case of the traffic.

The formal model in the view of a power function is creating and after that it have to be analyzed.

This example is relevant to the modern world of course but authors have made a mistake.

This function was taken as the model: , where  distance measured in meters; v – speed [km/h].

In this case the question arises. From the physical point of view what is the value of 10?  According to the logic of the problem we see that:   . Such physical quantities we've never seen before.

There are four cases of the power function are under the attantion in the given textbook:

1. Index ð=0.

2. Index ð=1.

3. Index ð=2n – even natural number.

4. Index ð=2n-1 – odd natural number.

Clearly illustrated the dynamics of the function during of the index changing.

The main stages of the function analysis are clearly and concisely theoretically shown in the table: symmetry, zero of a function, local maximums and minimums, concavity, convexity, the ploting of the function. The meanings of tangent and normal are explained.

There is an approach: t(x)=mx+n

In our opinion, (2) is represented in not quite correct view.

Probably a better variant would be as following:

There is a lack of some geometric interpretations: concept of tangent as the limiting position of the secant, variable m as the angle of the tangent, variable n as the length of the segment which cuts off by tangent on the vertical axis (OY) from zero.

This topic contains four examples (one of them contains a practical sense - a game of mini-golf).

There are several self-paced exercises.

At the end of the topic is a brief theoretical overview of the studied material and and there is also a final test.

2. Using of the derivative.

Regarding the using of the derivative, in german textbook many of them are considered in relation to the practical point of view [2, C. 32-54].

9 examples are given for purpose to find the maximum area (m2) using the given conditions. For example the following.

Somebody has a cable length of 800 meters. It is necessary to enclose a rectangular area of land from three sides (one side is a sea coast). The textbook offers to calculate the optimal area of this space. It is should be maximal.

In addition there are four tasks concerning geometrical limitations (finding the maximum and minimum values).

Using the round timber having a diameter of 40 cm to saw out the square timber with maximum amount of it’s weight.

In our view this is also not very good example. It is fine that the authors of the textbook need for such tasks where there is only one variable. In this case it was possible to consider the task like that... Need to saw out the square timber. Cross-sectional area should be maximal.

The textbook also have some tasks solving the problem of finding the optimal size of the package for matches.

Then followed another 15 tasks to consolidate of knowledge. In the introduction the authors paid attention that this section provides a large number of tasks and it is not necessary to do all these exercises. However, we feel that it would be appropriate to put this part after the section "Derivative of a composite function". Then there was an opportunity to show the practical problems that are solved using of different functions and not just with a simple power law. Then in the example with a point on the screen would be better to just get the function  and do not to convert this to a strange function  and to calculate the derivative from it. The derivative from d(x) would have been enough.

In this paragraph the transition from the derivative to the optimization tasks using a system of linear inequalities is looks surprising. The relationship between the derivative and the system of linear equations is not quite clear.

The paragraph on "Recovery of function" is very interesting in our opinion. Some values ​​of the function: the value of the derivative at a given point, the zeros of fuction, etc. are presented. Using these conditions is necessary to restore the original view of equation. Such problems are, of course, contribute to the development of logical thinking.

The Russian textbook there are two sections showing the examples of the derivative application: "Using of derivatives to plot the graphs of functions" and "Using of derivative and integral to solve practical problems"[1, C. 305-314].

The paragraph “Practical application of the derivative” begins by introducing the concept of a differential equation. Then the two equations are solved in a general way. To illustrate the solution the examples are providing: relationship between speed of bacteria reproduction and mass of bacteria, equation of harmonic oscillation. In our opinion there are quite difficult formulated tasks. They are may be interesting for specialists in certain areas but not for school pupils. In this textbook there is a lack of simple tasks aimed to the practical content. Just such an excess in the German textbook.

3. Fundamentals of integral calculus.

The consideration of section “Integration” in German textbook [2, C. 70-75] begins with the following example: for given type of function , it is necessary to determine the derivative of which function is it.

Empirically, there are a variety of answers:  and a total solution for all is recorded in this view:  , where C is the constant of integration. After that gives a definition of antiderivative.

This task in our opinion is very interesting from the point of the pupil's thinking on the issue.

How to determine the original function using the given derivative?

Next step in texbook is examples of integral finding and description of the integral properties and showing of joint properties for the integration and differentiation.

Illustrates the geometric meaning of the constant of integration.

Then offers the tasks for independent work: to calculate the integral, to find the antiderivative for a given function.

The conception of integral is explained as the sum of an infinite number of the  rectangles areas.

It is shown that if the figure has a complex configuration, before finding an area it is desirable to mentally cut the figure for a few simple shapes.

Then shows specific examples. The properties of the definite integral are also considered in textbook. Offers examples for individual work.

As material for further study the numerical methods for finding the area of ​​a rectangle (the method of right and left rectangles) are presented. The final section summarizes the theoretical material in the form of test. 14 pages are discharged to this section. This section provides a brief, clearly, maintaining mathematical rigor.

The section "Integration" in the Russian textbook [1, C. 287-311] begins with the task: how to find a path traversed by a point based on point moving speed. (that is to say – how to determine the function s(t) knowing its derivative v(t)).

This task is not correct, because if we put the punctuation mark “full stop” at the end of the phrase " how to find a path traversed by a point based on point moving speed " we will find the absurd.

The example given in the German textbook, more successful, more clear and understandable.

Further, the Russian textbook gives a explanation of antiderivative and one example: how to prove that the given functions  are antiderivatives of function . Based on this example the concept of the integration constant “C” is introduced and explained; introduced so poorly, too theorized and too difficult.

 Then several examples for individual work are offers.

In the next section we can see the rules how to find the antiderivative and some tasks for independent work.

Paragraph "The area of the curvilinear trapezoid and integral" begins with the definition of a curvilinear trapezoid, then theoretically substantiated that the area of ​​the curvilinear trapezoid is the definite integral. Again it is necessary to focus the attention on the complexity of presentation and excessive formality. It is not clear why the pupil must to know the concept of "integral sum" at first acquaintance with the concept of the definite integral. In the textbook there are paragraphs: "Calculation of integrals" (only definite integrals are considered), "Calculating of the areas using the integrals". Why not combine them into one and call this a "Definite integral"? From a geometrical point of view the definite integral is the area of the curvilinear trapezoid. As a result 24 pages of textbook are dedicated to the section "Integration". Based on a comparison with the German textbook, the Russian textbook material concerning this topic is presented very poorly.

4. The application of integral calculus.

"The application of integral calculus" is a independent section in the German textbook [2, C. 86-109]. It begins with a consideration of the possible variants of the figure location (the area of this fugures has to be calculated).

Three cases are considerated:

1. figure is above the x-axis;

2.figura located below the x-axis;

3. part of the figure is below the x-axis, the other symmetric part lies above the x-axis.

In our view, the third case deserves more attention than it was realy paid by authors. The authors are considering an example:  It turns out an apparent contradiction. On the one hand a pupil sees schematically figure and understands that this figure has a specific area. On the other side of the definite integral is zero. Paradox! This example requires a logical conclusion. It was necessary to show that if part of the figure is below the x-axis and the other is above the x-axis, then the limits of integration must be divided into the separate sections. Then you need to calculate the module of the integrals at each site and sum up the obtained values. Indeed the other examples are discussed in this order. So why it is not done at once?

A variety of some practical tasks aslo deserve the attention. These exercises are interesting in content and not so difficult from a mathematical point of view. To the Russian authors of mathematical textbook I would like to recommend to study in detail the experience of German colleagues in the preparation of tasks with practical content.

I’d like to represent another exercise of the German textbook.

The tunnel is ventilated by two fans. The performance of each of them is about 80 cubic meters per minute. How much time is needed for completed replacement of the air inside the tunnel?

There are raw data: the length of the tunnel is 60 meters, width is 20 m, height is 8 meters and domed roof, represented by the equation of a parabola. The distance from the upper point of the roof to the rectangular base is 10 meters.

5. The rules for finding the derivative. Derivative of a composite function.

The topic in German textbook [2, C. 120-136] begins with a discussion of the question: How to find the derivative of multiplication. The authors draw an analogy with the derivative of the sum. They making the assumption that the derivative of the multiplication will be equal to the multiplication of the derivatives. Then the given example shows fallacy of this assumption. Probably it would be more successfully if the authors have shown a formula for finding the derivative of the multiplication at first and then researched or given to home-review that example which shows that the derivative of multiplication is NOT equal to the multiplication of the derivatives. Formula for the derivation is created not correctly. Written definition of the derivative: , and then after equality sign is written expression:  . It would be good to know how there was a transition?

Derivative of fraction are derived based on the derivative of multiplication. The derivative of a composite function is considered as an example of the power function. The derivative of a composite function is considered based on example of the power function.

For consolidation and completion of the material following tasks are offered: to find the derivative of a function at some point, to show the equation of the tangent, extremums, zeros of function, etc. Thus the previously learned material is repeated and fixed.

In the same paragraph the integral of the power function when the base is a combination of linear functions. At the conclusion of this paragraph is a brief summary of the theoretical material and provides a practical example.

The part “Differentiation rules. Derivative of a composite function” also considered in the Russian textbook [1, C. 236-251]. Formulas for the derivative of summation and  fro the derivative of multiplication (function × Constant) are creating thought the limits. Formulas for derivative of multiplication and derivative of fraction in their origin are not shown.

For the homework some tasks are proposed:

to find the derivative of a function at a point;

to find the intervals where the derivative is greater than zero or less than zero;

Practical tasks are required the knowledge of physics and chemistry. All of them are considered by the example of the power function. This tasks are little bit difficult in its formulation for non-specialist.

We saw two very different approaches to the study of mathematics: "Theoretical and formal" in Russia and "Practice-simplified" in Germany.

In our view the presentation of material in Russian textbooks too formalized, devoid of clarity. The theoretical material is too large in comparison to the exemplified. There is a lack of applied examples that demonstrating the importance of studying the topic. During the looking into a textbook the question are formed: Why should we learn it? Where it can be useful? Readers can only assume that it is necessary for the further study of mathematics.

The textbook published in Germany is obvious. It contains many life-examples, the simplicity and brevity of theoretical material.

In our opinion there is only one drawback in this textbook.

Trying to explain the theoretical material in an accessible form for a pupil, the authors lost accuracy of presentation and compliance of the presented material to mathematical sense.

References:

1. Alimov S. A., Kolyagin Y. M. Algebra and calculus: High Schools 10-11 class. M.: Prosveschenie, 2012. – 463 c.

2. Bigalke A., Kohler N. Mathematik. High Schools. Berlin: Cornelsen Verlag, 2011. – 560 c.

3. Standard of general education in mathematics. Russia. [Electronic resource] access mode: www.school.edu.ru.

4. Standard of general education in mathematics. Germany. [Electronic resource] access mode: www.standardsicherung.de.