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Introduction

The language of mathematics in the classroom. Some typical weaknesses of the students in using this language and ways which promote them to master it. Dr. Rebeka Pali Department of Mathematics Polytechnic University of Tirana, Albania.

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Introduction

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  1. The language of mathematics in the classroom. Some typical weaknesses of the students in using this language and ways which promote them to master it.Dr. Rebeka Pali Department of Mathematics Polytechnic University of Tirana, Albania

  2. "The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” Galileo

  3. Introduction As other languages the language of mathematics has two main components: the semantic and thesyntax. • semantic of the language of mathematics - the symbols of this language, its terms and words, and • syntax - the structures and its inner construction.

  4. Introduction Mathematics uses different signs. The entirety of all these specific signs form the mathematical symbolic. The finite system of letters, clearly distinguished between them forms the alphabet of the mathematical language. The alphabet of the mathematical language is the basic "material", and by means of it, based on some defined mathematical rules are structured the words and sentences of this language. In this alphabet are included: all letters and punctuations signs of the everyday language, and the mathematical symbolic.

  5. Everyday language plays a very important role in mathematics • It is a flexible tool, and mastery of it helps students in simultaneous usage of specific languages. • It serves as an intermediate link between mental processes, specific symbolic expression and their logical framework in certain mathematical activities. • It serves as an intermediate link between student’s experience and specific needs of their mathematical thinking, especially the need for interpreting and using reasoning that are different from the everyday ones. • It has an important role in creating mathematical concepts through argumentation. • It makes possible the interpretation of mathematical results. • It interlaces the basic mathematical concepts.

  6. Specifics of the language of mathematics • One of the particularities of the language of mathematics which is not present in other languages is its symbolism: theorems expressed by variable “x”, can be applied when “x” is replaced by “b” or “2x+7”. • Everyday language starts to be internalized by children as a spoken language, through voice sounds, related between them, in certain ways. While the language of mathematics starts to be internalized by children alongside spoken language as a written language. • The language of mathematics using terms of everyday language gives them a rigorously determined meaning. In the everyday language synonyms are considered as a wealth of the language. From the other side many terms have double or even multiple meanings. Differently, the language of mathematics is distinguished for the precise meaning of its terms. Each term of it represent a fully determined meaning or object.

  7. Specifics of the language of mathematics • Using or interpreting of connectives might cause that the meaning of a compound sentence in mathematical language to be different from its meaning in everyday language. • The language of mathematics and the everyday language differ regarding to the intention, suitability or implications of a sentence. • Small changes in verbal formulation in the mathematical language might influence the interpretation of the solution from a way to another. • Another difference between the math and everyday languages comes from the fact that the everyday language use many expressions with indexes, which are understandable from the context but can not be written in the symbolic language.

  8. What does it mean to master the language of mathematics ? • As a first step, to master the language of mathematics is necessary to master the native language. The language of mathematics includes it and can not exists without the arsenal of the native language. • Alongside, mastery of the mathematical language presumes the clarifying of the meaning of terms of the everyday language, which in mathematical language take a rigorously determined meaning. • A special importance in mastering the vocabulary of mathematical language that is used in the school,have “the cue words“. • Also, it is important for students to recognize and use patterns. Pattern recognition is the key to much of mathematics. • Mastery of mathematical language require that student to be familiarized with statements or sentences that have widely usage in mathematics.

  9. What does it mean to master the language of mathematics ? • Of a great value is the effort to make students aware about the fact that the major part of mathematical sentences or statements have two truth values: true or false. • Mastery of mathematical language is strongly relied in the knowing and precise using of the symbolic. • Another important demand in mastery the language of mathematics is the ability of students to do the two-way translations. • One more aspect to be mentioned is finding of a balance between the spoken language and written language.

  10. Study • The data are based on the observations. • 8 schools - 612 students - 20 teachers • Methodology: • Observation and recording of the verbal work of students and teachers; • written tests; • interviews with teachers; • study of the math textbooks.

  11. What is observed? • The knowledge and usage of symbolic of proper level; • The knowledge and usage of the terminology of the proper level; • The scale of respect toward the conventions of the math language; • The way teachers use the math language; • The ability to pass from symbolic language in terminology or in everyday language and vice versa; • The ability to use the mathematical language in giving explanations; • The ability of students to use a precise language in the formulation and solving problems.

  12. What is observed? • The ability of students to make questions and to answer to questions; • The ability of students to argue and to solve problems; • Their ability to express their ideas and to participate in the discussions; • The possibilities that the students really have to work in groups; • The questioning techniques that teachers use; • The evaluation techniques in mathematics; • The interaction that is going on in the classrooms.

  13. Findings Analysis of: • Some typical weaknesses of the students regarding the usage of the mathematical language. • Classroom learning environment and teaching techniques

  14. Some of the students mix up the terms that stands for the operations with terms that stands for the results of the operations. A - students express precisely B – have difficulties with “production” and “quotient” C – manipulate easily only with “sum” and “difference” D- are not able to interpret results

  15. Some of student’s responses • the double of its difference with 3– is translated: 2x-3 instead of 2(x-3); • the half of a number is 7 less than the number itself – is translated: ½ e x = 7 < x, or ½ e x = 7 – x, or ½ x < x 7 < x instead of x/2<x-7. • triple of its addition with 7- is translated: 3x+7 instead of 3(x+7).

  16. A great number of students have difficulty to do double-sided translations. Write with mathematical symbols • The half of a number is 7 less than the number itself; • A number squared is 6 less than the number itself. • The quintuple of a number is equal to the double of its difference with 3. • The quotient of a number with 4 is greater or equal to triple of its addition with 7. A – do precise translations B – can partially do the translations C – are not able to translate

  17. Inversely, Write by words the following equalities or inequalities: • 2 x = x + 8 • x: 2 = 3( x-1) • 4 · x > 2 + x • x : 4 = x - 3 A – make precise translation B – make partially C – can’t express in words

  18. Some of student’s responses • “x: 2 = 3( x-1)” – is translated: “ a numberx divided by 2 is equal to its difference with 1, multiplied by 3” • “2x = x +8” – is translated: “ double ofx is equal to x enlarged 8 times” • “4x > 2 + x” – is translated: “ quadruple of a number is greater than that number added with 2”, ( don’t use the term ”sum” but use the words “added with”).

  19. Many students don’t know the properties of arithmetical operations and don’t name them correctly. • Say in words the equalities: a)∆· 0 = 0; b) ∆ · 1 = ∆ ; c) (∆·○) □ = ∆ ( ○·□ ) A – Know the properties and generalize the answer B – can apply the property but can’t generalize the answer C – can’t say by words the properties and don’t know them

  20. Some student’s responses • “ when multiply a number by zero the sum is zero”; ( the result of multiplication is named “sum”) • “when multiply a number by 1 the production is equal to addend”, ( mix the term addend with the factor) • “for all numbers that are divided by 1 the production doesn’t change”, (mix the quotient with the production” • “ a + 0 = a is translated :” the sum is added with that number itself” etc.

  21. Determine the order of doing operations: (20+13) · 4 + 76. 51 % of the students gave correct answer. Write a numerical expression which has this order of doing operations: multiplication, addition, addition. 40% of the students created correct expressions. A considerable percentage of the students don’t know and respect the conventions of the language of mathematics regarding the order of doing operations and using brackets.It is easier todecide the order of the operations of a given expression that to create an expression when the order of the operation is given.

  22. Unclarities in naming of the variable • Students mix up the concept about the variable with the number concept. They solve the equation x + 5 = 8 and give the answer: “the solution of the equation is x = 3”. In fact, x = 3 is a new equation equivalent to the first one and the answer should be: ”the value of x equal to three is the solution of the equation, number 3 is the solution”. • This is reflected in written work as well. They have to find values of “x” that satisfy the inequality x  12, in general they write: x=1,2,3,4,5,6,7,8,9,10,11 instead of x1,2,3,4,5,6,7,8,9,10,11.

  23. Unclarities in naming of the variable Students show difficulties to • transform different expressions; • to evaluate the values of an expression for given values of its letters; • to evaluate the values of the function for given values of the variable; to make reduction of similar terms.

  24. Unclarities in naming of the variable • In the textbook: “Choose the correct answer for “x ≥ 7”: • smaller than or equal to 7; • greater than or equal to 7; • not smaller than 7; • not greater than 7; • at least 7; Such alternatives in fact correspond to “  7 ”.

  25. Students presents difficulty and inaccuracy in using the terminology and symbolic. • The symbolic notation “f(x) = 4x + 3” sometimes is red by words as: “multiply x by 4 and add 3” instead of “multiply by 4 and add 3” . • Students are not clear about the fact that: • The function f is the rule, • f(x) is the value of the function for a given value of its argument, and • f(x) = x -8 is the symbolic notation that is used to present the function in a analytic form

  26. Other difficulties and inaccuracies in using the terminology and symbolic. • Some students are not able to use the algebraic symbolism as a mean to express and to think about numerical relations. • They have incorrect imagination and usage of the symbol of equality . ( they write:“2x = 30 = x = 15”) • Can not distinguish an equation from an equality with a letter which is true for all values of the letter. • The symbolic they use for sets is incorrect.

  27. There are some weaknesses of the students regarding solving equations. • they don’t recognize the pattern - a pattern of operation and order represented by symbols. • feel a kind of uncertainty regarding the number of the solutions of an equation. They speak about “the solution” of the equation and not about “the set of the solutions”

  28. A considerable number of the students hesitate or have difficulties to give explanations. • “Explain why each positive number is greater then each negative number.” A – give exact explanation B – try to give explanation but they are not clear C – can’t give any explanation

  29. Student’s responses • “ because numbers are getting bigger when we go from the left to the right of the numerical axis”, • “because the positive numbers are on the right of zero and the negative numbers are on the left of zero”, or • “ because the positive numbers are greater than zero while the negative numbers are smaller than zero”. • “ negative numbers are lined up under the positive numbers ”; • “ the positive number has no sign, while the negative number has the sign”, • “because are far from the zero” • “ the negative numbers are behind of zero, positive numbers are ahead of zero” • “because the number which has “+” sign shows addition, while the number which has “–“ sign shows subtraction”

  30. The capability of students to formulate a problem. Two cases: • to formulate a problem based on the math knowledge that students already have; and • to formulate a problem that respond to a given numerical expression.

  31. First case “Formulate a problem based on your mathematical knowledge ” A – can formulate problems which are conform to their knowledge B – formulate problems which are not conform to their knowledge C – can’t formulate a problem

  32. What students do? • give more or less data than is needed; • do not keep relation between the data. Example: “ Three workers must prepare 100 models. The first one prepared 30 pieces. The second one prepared the double of the first one, while the third on e prepared the triple of the second worker”; • try to remember the formulation of problems from the textbook; • do not formulate the main question of the problem, sometimes they presume it and do not write. • during the solution are not focused to the question that problem propose but try to find all results that can be generated with given data, • do not use in a proper way the “cue words”. • some of them formulate two problems at the same time,

  33. Second case “Formulate a problem which respond to the numerical expression 150+ 4·30 + 2 ·20 “. What students do? • conceptualize the given numerical expression: 150+ 4·30 + 2 ·20 as (( 150 + 4) · 30 + 2) · 20 • use as data the productions “4·30” and “2 ·20” but the formulation doesn’t respond to the given scheme A – can formulate a correct problem B – try to formulate but don’t respond to the given expression C – can not formulate the problem

  34. Inaccurate usage of logical connectives • Write the setof odd numbers between23 and 30. Find some numbers that don’t belong to this set. A great number of students are able to find the elements of a set when a common property of them is given. But, if we give two common properties they are concentrated to the last one. A- can find the elements correctly. B – can find some of the elements and don’t take in consideration both of characteristics. C- do not fulfill any demand

  35. Is given the chain 6, 12, 18, 24, 30, 36.... Is it true that: All numbers of this chain are multiples of 2; This chain include all multiples of 2; Not all multiples of 2 belong to this chain; This chain include multiple of 4; In general, students respond exactly to the first question but get confused to the second one because the word “all” has changed its place in the sentence. They understand the sentence " Not all multiples of 2 belong to this chain” as “none of the multiples of 2 is included in this chain”, They are not sure if in the sentence “This chain include multiple of 4” is talk about “all” or “some”. It seems that it is difficult for students to show and to write a empty set. Usually they write it: “A = 0”, and say: “empty set include only zero”(?). Misunderstanding in using of the words like each, any, all, at least, some, not all, etc.

  36. Classroom learning environment & Teaching techniques • Students have limited possibilities to speak and discuss with each other. • Has no efforts for an active inclusion of all students. • Students work individually and not in groups. • Generally, teachers don’t enter deeply in questioning of the students to find reasons of their weaknesses. • Teachers tend to make closed questions.

  37. Classroom learning environment & Teaching techniques • Poor evaluating techniques. • It happens that teachers interrupt the natural curiosity of students and their prospector spirit because they don’t hear to the students. • Teachers don’t take in consideration the age development characteristics of the students. • There are a few efforts to enable students to read mathematics.

  38. How can teachers do to promote students to master the language of mathematics. • Compile exercises and problems which develop the language of mathematics at students • Design learning activities which make students to speak “mathematics”.

  39. Samples which have to do with the way we name the mathematical objects. • Give the proper term for the following symbolic notations: a) 3x+5; b) x+5=12; c) 2 < 6; d) 2 + 4; e) x=2x-7; f) x<4; g) 2 + 3 = 5; h)x(a+b)=xa+xb. (ex. 3x+5 expression with letters) • Write an equation of the variable “x”, and an inequation of the variable “x”. • Write an numerical expression and a numerical equality. • What is the difference between “ 3∙2+4” and “3∙x+4”? • Give the proper term for the following numbers: 2; 13; 5/9; 4,5; 2/3; 4 1/3 ; 5/15; 0,6; 22; -4 ;

  40. Game problem: Write the expression for each step in the number game. Steps Expression Your Number • Pick any whole number. • Add 12. • Double that number. • Subtract 8. • Divide by 2. • Subtract your original number. • The result is always _________. • Word problem: Sixty-three more than four-fifths of a number equals 111. What is the number?

  41. The following are examples where is required to generalize concrete examples in symbolic representation of math facts: • As you know 1 · 5 = 5, 1 · 97 = 97 . Write in symbols this fact about multiplicand 1. • 56 + 49 = 49 + 56. Name the property that generalizes this math fact. • As you know 12/1=12,321/1=321. Write with math symbols this fact about division by 1. • As you know 88+0=88,112+0=112. Write with math symbols this fact about addition of zero. • As you know 2(3+5)=23+25. Write with math symbols this fact about addition and multiplication. • Write with symbols the sentence: “ If you associate in different ways three addends of a sum the result doesn’t change” .

  42. The concentrating of the attention to the method of solving helps students to feel secure in solving all equations of that type. • What would you do to solve the equation : x-32=358? • Write your answer as an imperative sentence. • Write an abstract fact which correspond to your method. • What would you do to solve the equation : 42x=252? • Write your answer as an imperative sentence. • Write an abstract fact which correspond to your method. • Which is the rule that allows to pass from (a) to (b): (a) x -7 = 12 (a) 5 x = 25 (b) x = 19 (b) x = 5

  43. Some examples which would promote students in giving explanations: • How do we add a negative number to a negative number? Explain your answer. • 22. How do we multiply a negative number to a negative number. Explain your answer. • 23. How can we write a decimal number as a fraction? Explain your answer. • How we do multiply fractions? Express your answer in everyday language; Express your answer with math symbols. • Why positive numbers are greater than negative numbers? • Using numeric axis explain the following inequalities: -2 < 0 ; 7 > 0 ; 5 > -2 ; -5 < -1 ;

  44. What is the sign of the number (-5) · x? • What does it mean to solve an equation? • What does it means to prove an equality? • What does it mean to simplify an expression with letters? • Are equivalent the following pairs of equations? 2x+4=20 and 2x=16; x²=9 and x=3; 2x=10 and 2y=10 • When a set is called subset of a given set? Draw a Venn’s diagram to show this relation. • How we can find the elements that belong to the intersection of two given sets? • How do you find the area of a square. Give your answer by words and with math symbols.

  45. The two-way translations can be stimulated by the following examples: • Write with math symbols the given sentence: “The quotient of 32 with x is not smaller than double of x”. • Say by words the following equality: 4(x+2)=5x. • Write in words, as you understand the following symbolic notations: x – 2 = 12; x /3 = 21; x² = 16; x + 7 = 19 ; 2(x + 4) > 3. ( ex.: the difference of number x with number 2 is equal to 12) • Write with math symbols the sentence: “One number is twenty-eight more than three times another number.” • Turn into an equation the given problem: “Twelve pencils and one rubber costs 14 leks. The rubber cost 4 leks. How much does it cost one pencil?” • Find and formulate the characteristic properties of the elements in the given sets: A = { 1,2,3,4,5,…} ; B = { 2,4,6,8, ....} ; C = { 5,10,15,20,...}

  46. Rules and conventions of the mathematical languages: • Write an expression without brackets in which the first operation should be done the multiplication and the second should be done the addition. • Write an expression which have this order of operations: addition, multiplication, division multiplication, addition, multiplication. • Decide the order of operation in the given expression: 5(3∙2+5)+7 • Are equivalent the expressions: “x + 5x” and “ (x +5)x “

  47. The correct use of the symbols: Which of the following symbolic notations is written wrong: 2 < x > 10; 4 > x < 5; -2< x < 5 • What represent the symbolic notations: ( 2 , 5 ) , { 2 , 5 } • Which of the following symbolic notations is written right: {1 , 6 } = { 6 , 1 } ( 1 , 6 ) = { 1 , 6 } 5  { 3, 4 ,5 } 5  { 3, 4 ,5 } x < 2  x > 5 { 0, 1 }  {0,1,2, 3, 4} • Which is the difference between 7 and { 7 } ? • Which is the difference between “ ”and “” ? • Write with symbols “ the set of numbers smaller than 10”. • Write the set that has the only one element, 2. • What represent the symbol [AB] ? • Which of the following symbolic notations in written right: a + b = B + a ; a + b = b + a ; A + B = a + b; • The sentence “1 < x < 5” is the short form of an sentence that has two parts. Could you rewrite the sentence in the long way?

  48. To improve the language of mathematics that they use for function: • Give the mathematical expression in terms of “x” which express the rule: Add 20 and then divide by 3. • Create the function : “divide by 4” in the set: A= { 4,8,12,16,20}. • Let be y = 3x. When x = 4, y = 12. When x = 5, y = 15. When x change, y change as well. Can you tell what remains the same? • What represent the following symbols: a) f ; b) x; c) f(2); • Let be f(x) = 2x. Is it any difference between f(x) and 2· f(x)? • What is the difference between “f” and “f(x)”? • From a given plot, find the relation between x and y.

  49. Students can be practiced in formulating problems: • Formulate a problem when is given a number sentence. Example: How would you create a problem situation represented by 1∙4=4? • Formulate a problem when a picture is given. • Create a problem situation by yourself and formulate the problem. • Formulate a problem including in the data the multiple of a number. • Formulate the inverse problem of a given problem.

  50. To enrich the math vocabulary : • I’m a number. I’m greater than the production of 8 with 4. I’m smaller than the difference of 550 with 510. I’m even number. Can you find me? • I’m a production. I’ve two factors. One of my factors is 7 less than 30. the other factor is 5 times smaller than 20. Can you find me? • Which is the great number of the sweets: seven portions of eight sweets each; eight portions of seven sweets each fifty portions of one sweet each one portion of twelve dozen sweets. • I’m a six-digit number. To write me you need five zero. You don’t need to use the decimal comma. One of my digit is 2. can you find me? • I’m a fraction. My denominator is 14. I’m not part of a whole but I’m the whole. Which fraction am I?

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