Дипломная работа: Методические основы подготовки будущих учителей математики в условиях полиязычного образования

Известно, что нагрузка у всех работников разная, как и зарплата. Сколько человек работает в НИИ?

Такие задания часто рассматриваются и в предмете критического мышления, а в совокупности с фактором полиязычного обучения требуют от учащегося постоянного роста уровня английского языка в техническом направлении.

Например,

Exercise 1. There are given two statements:

1. That all translators know English on the excellent level.

2. Some writers are translators.

Which of conclusions are right?

а) Some writers know English on the excellent level.

Да

Нет

б) All writers know English on the excellent level.

Да

Нет

Justification of the answer ______________________________________

____________________________________________________________

Keywords

Exercise 2. «A manufacturer wishes to make an open-topped box out of the piece of cardboard shown below by folding up its sides. What is the volume of this box in cubic centimetres?» [16].

A 1600 B 2400 C 8000 D 10125 E 12500

Exercise 3. «The government blames schools and teachers for boys underperforming. However, science tells a different story. Evolutionary biology shows that females have evolved to have better verbal and emotional skills than males because of the need in prehistoric times for women to take the lead in bringing up children. By contrast, the need for males in prehistoric times to hunt in packs for food has made males more prone to violence and also skilled at calculating and planning. Neurologists have added to this insight by showing that the male hormone testosterone has an adverse impact on language skills. So clearly differences in educational performance between boys and girls cannot be explained in terms of failing teachers. Which one of the following is the best statement of the flaw in the above argument?

A It assumes that scientific explanations apply to the average male or female, ignoring exceptions.

B It assumes that biological differences come in degrees and are not absolute.

C It assumes that skills in calculating and planning have a role in educational performance.

D It assumes that the differences in performance between the sexes are due solely to biological differences.

E It assumes that teachers are not trying to improve the performance of failing boys».

Также стоит обратить внимание на то, что выделение «ключевых слов» является важным навыком и основой при решении задач на критическое мышление, в дополнении к этому являясь чаще всего являясь необходимыми к дополнительному изучению на английском языке.

В данном случае весь методический и практический материал необходимо составлять по технологии Hard-CLIL, для того чтобы обучающийся был полностью погружен в углубленное изучение английского языка и математики, но, тем не менее, преподавателю необходимо устно работать с аудиторией по методике Soft-CLIL, для того чтобы объяснять непонятные моменты.

Для выявления методических требований к программе развития профессионально-языковой компетентности учителя математики в условиях полиязычного образования, необходимо опираться на предыдущие тезисы, которые, в совокупности, помогут определить необходимые критерии и факторы.

Программа курса «Развитие профессионально-языковой компетентности учителя математики в условиях полиязычного образования» должна отвечать требованиям международных экзаменационных стандартов систем оценки качества знаний обучающихся техническим предметам на иностранном языке.

По официальным данным «College board», большая часть методического материала SAT направлена на задания с множественным выбором, но часть из них ориентирована на нахождение конкретного ответа на поставленный вопрос. Также материал разделен на условно «Calculator» и «No Calculator» и некоторые из них являются однотипными.

«Quick Facts

· Most math questions will be multiple choice, but some--called grid-ins--ask you to come up with the answer rather than select the answer.

· The Math Test is divided into two portions: Math Test-Calculator and Math Test-No Calculator.

· Some parts of the test include several questions about a single scenario» [17]. Также следует заметить, что «The SAT Math Test» можно условно разделить на три основных направления: «Heart of Algebra», «Problem solving and Data Analysis» и «Passport to Advanced Math». «The Math Test will focus in depth on the three areas of math that play the biggest role in a wide range of college majors and careers:

· «Heart of Algebra», which focuses on the mastery of linear equations and systems.

· «Problem Solving and Data Analysis», which is about being quantitatively literate.

· «Passport to Advanced Math», which features questions that require the manipulation of complex equations» [18].

Подобная методика ориентирована на выявление таких критериев как:

· Способность выполнять процессы гибко, точно, эффективно и стратегически правильно.

· Быстро решать поставленные задачи, определяя и используя наиболее эффективные подходы к их решению, путем проверки, поиска наикратчайшего пути или реорганизации предоставленной информации.

По аналогии с «SAT», программа развития профессионально-языковой компетентности учителей математики в условиях полиязычного образования должна отвечать всем необходимым стандартам и требованиям, а также быть ориентированной на аудиторию, не являющуюся носителями иностранного языка.

Исходя из текущих уровней знания английского языка, пользователей можно условно разделить на несколько уровней:

«По шкале CEFR:

A -- Элементарное владение (Basic User):

A1 -- Уровень выживания (Survival Level -- Beginner и Elementary)

A2 -- Предпороговый уровень (Waystage -- Pre-Intermediate)

B -- Самостоятельное владение (Independent User):

B1 -- Пороговый уровень (Threshold -- Intermediate)

B2 -- Пороговый продвинутый уровень (Vantage -- Upper-Intermediate)

C -- Свободное владение (Proficient User):

C1 -- Уровень профессионального владения (Effective Operational Proficiency -- Advanced)

C2 -- Уровень владения в совершенстве (Mastery -- Proficiency)» [25]. Стоит отметить, что данная программа рассчитана на аудиторию, владеющую английским языком на уровне B1 и выше, позволяя улучшить не только свои знания в вербальном техническом направлении языка, но и в решении задач профессионального характера на иностранном языке.

Ниже представлены темы, которые будут представлены в данном учебном пособии, включающие в себя краткий теоретический экскурс и набор необходимых заданий для усвоения материала.

Mathematics

1. The Number Line

2. Trichotomy Law

3. Absolute Value

4. Addition, Subtraction, Multiplication, Division

5. Integers

6. Exponent and Roots

7. Laws of Exponents

8. Squares and Square Roots

9. Logarithms

10. Change of Base Formula

11. Law of Logarithms

12. Distributive Law

13. Fraction, Decimals and Percents

14. Operations with Fractions

15. Operations with Mixed Numbers

16. Complex Fractions

17. Percents

18. Percent Increase and Decrease

19. Ratios and Proportions

20. Proportions

Algebra

1. Polynomials

2. Algebraic Fractions

3. Equations and Inequalities

4. Absolute Value, Radical, Fractional Equations and Inequalities

5. Quadratic Equations

6. Quadratic Formula

7. Exponential Equations

8. System of Linear Equations

9. The Addition Method

10. The Substitution Method

11. The Graphing Method

12. Solving Linear-Quadratic Systems

13. Word Problems

14. Rate Problems

15. Age Problems

16. Percent Problems

Plane Geometry

1. Lines and angles

2. Angles

3. Perpendicular and Parallel Lines

4. Triangles

5. Sides and Angles of a Triangle

6. Right Triangles

7. Pythagorean theorems and corollaries

8. Special right triangles

9. Perimeter and Area

10. Triangle inequality

11. Similar triangles

12. Quadrilaterals and Other Polygons

13. The angles of a Polygon

14. Special Quadrilaterals

15. Perimeter and Area of Quadrilaterals

16. Areas

17. Circles

18. Circumference and Area

19. Tangents to a Circle

THE NUMBER LINE

The word numbers always mean real number, a number that can be represented by a point on the number line.

A positive number is a number that lies to the right of 0 on the number line. A negative number lies to the left of 0 on the number line.

TRICHOTOMY LAW

For any real number a, exactly one of the following statements is true.

· a is negative

· a = 0

· a is positive

ABSOLUTE VALUE

The absolute value of a number a, denoted |a|, is the distance between a and 0 on the number line. Since 4 is 4 units to the right of 0 on the number line and -4 is 4 units to the left of 0, both have an absolute value of 4:

· |4| = 4

· |-4| = 4

Since 4 and -4 are the only numbers that are 4 from 0, if |x| = 4, then x = 4 or x = - 4. If |x| < 4, then x is less than 4 units from, which means - 4 < x < 4. If |x| > 4, then x is more than 4 units from 0, which means either that x < - 4 or x > 4.

Addition, Subtraction, Multiplication, Division

For any real numbers a and b:

· If either a or b is 0, then ab = 0;

· If ab = 0, then a = 0 or b = 0;

· If a and b are both negative, ab and are positive;

· If a and b are both negative, ab and are positive;

· If either a or b is positive and the other is negative ab and are negative;

· If both a and b are positive, a + b is positive;

· If both a and b are negative, a + b is negative;

· If either a or b is positive and the other is negative, a + b has the same sign as the number whose absolute value is greater;

· To evaluate a - b, write is as a + (-b) and use the above rules for addition.

Key words

Integers

The numbers in the set {…, -4; -3; -2; -1; 0; 1; 2; 3; 4,…} are called integers.

Consecutive integers are two or more integers written in sequence in which each integer is 1 more than preceding one.

For example:

· 3,4

· 15, 16, 17, 18,

· -3, -2, -1, 0, 1, 2

· n, n + 1, n + 2, n + 3, …

The sum, difference, and product of two integers is always an integer. The quotient of two integers may or may not be an integer. The quotient 86ч10 can be expressed as , 8, 8.6. You can also say that the quotient is 8 and the remainder is 6.

If m and n are integers, the following four terms are synonymous:

m is a divisor of nm is a factor of n

n is divisible by mn is a multiple of m

They all mean that when n is divided by m, there is no remainder (or, more precisely, the remainder is 0). For example:

4 is a divisor of 124 is a factor of 12

12 is a divisible of 412 is a multiple of 4

If m and n are positive integers and if r is the remainder when n is divided by m, then n is r more than a multiple of m. That is, n = mq + r where q is an integer and 0 ? r ? m.

The factors of 12: -12, -6, -4, -3, -2, -1, 1, 1, 2, 3, 4, 6, 12

The multiples of 12: …, - 48, -36, -24, -12, 0, 12, 24, 36, 48, …

Every integer has a finite set of factors (or divisors) and an infinite set of multiples.

Example: To find the factorization of 180 as a product of any two factors; then write any of those factors that are not prime as a product of two factors. Continue this process until all the factors. Continue this process until all the factors listed are prime.

180 = 18 10 = (92) (52) = 33252

This factorization is usually written with the primes in increasing order:

180 =

The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of each of them.

In addition, every integer than 1 that is not a prime can be written as a product of primes.

The product of the GCF (greatest common factor) and LCM (least common multiple) of any two positive integers is equal to the product if the two integers.

For any integers m and n:

· If either m or n is even, then mn is even.

· If both m and n are odd, then mn is odd.

· If m and n are both even or both odd, then m+n and m-n are even.

· If either m or n is even and the other is odd, then m+n and m-n are odd.

Key words

Exponents and roots

In the expression , which is read “b to the nth”, or “b raised to the nth power”, b is called the base and n is called the exponent.

For any number b and positive integer n:

· If b ? 0, then

·

· If n > 1, then where b is a factor n times

·

For example,

·

·

·

·

·

LAWS OF EXPONENTS

For any numbers b and c and integers m and n:

SQUARES AND SQUARE ROOTS

Although can be read “a to the second,” it is usually read “a squared.” You will see 2 as an exponent in many formulas. For example:

· A= (the area of a square)

· A=р (the area of a circle)

· (the Pythagorean theorem)

· ' =(x--y)Ч(x+ y) (factoring the difference of two squares)

Numbers that are the squares of integers are called perfect squares. You should recognize at least the squares of the integers from 0 through 15.

Of course if you need to evaluate, you can use your calculator. However, it is often helpful to recognize these perfect squares. That way, if you see 169, you will immediately think “that is .”

Two numbers, 5 and --5, satisfy the equation = 25. The positive one, 5, is called the square root of 25 and is denoted by the symbol . Clearly, each perfect square has a square root: 0 = 0, (25 =5, 100 = 10, and 169 = 13. However, it is an important fact that every positive number has a square root.

For any positive number a, there is a positive number 6 that satisfies the equation = a.

That number, b, is called the square root of a and is written a. So,

· for any positive number a, a·a= (= a

· for any positive numbers a and b:

· .

For example, .

The expression is often read “a cubed”. Numbers that are the cubes of integers are called perfect cubes. You should memorize the perfect cubes in the following table.

The only other powers you should recognize immediately are the powers of 2 up to .

In the same way that we write b = to indicate that :

We write to indicate that and call b the cube root of a.

We write to indicate that and call b the fourth root of a.

For any integer n ? 2, we write to indicate that and call b the nth root of a.

For example:

· because

·

· because

For any real number a and integer n ? 2:

· If n is odd, then is the unique real number x that satisfies the equation .

· If n is even and a is a positive, then is the unique positive number x that satisfies the equation .

For any positive number b and positive integers n and m with n ? 2:

·

·

For example:

Logarithms

Recall the statement is usually read as “2 to the 4^th power equals 16”. Another way to read this stresses the role of the exponent 4: “4 is the exponent to which the base 2 must be raised to equal 16”. Mathematicians have a special word for this exponent - logarithm. The statement is equivalent to the statement , which is read, “the base 2 logarithm of 16 is 4”.

If b is a positive number not equal to 1 and x > 0,

if and only if

For example:

Change of base formula

Laws of logarithms

For any positive base b and any positive numbers x,y and n:

(In particular, and )

For example,

if

=

= 3

=

Distributive law