PHƯƠNG PHÁP dạy TOÁN BẰNG TIẾNG ANH

The main goal in teachingmathematical problem solving is that students develop a generic ability to solve reallife problems and apply mathematics in real life situations.. Skills Needed

0.1 Preface 1

0.2 Notations and Symbol 4

Chapter 1 Introduction 4

1.1 Notations and Symbols in mathematics 5

1.1.1 Some differences in the math symbols in English (Eng.) and Viet-namese (Vie.) 5

1.1.2 Geometry 6

1.1.3 Abbreviations and Notations 6

1.1.4 Notations for Numbers, Sets and Logic Relations 7

1.1.5 The Greek alphabet is commonly used in mathematics 7

1.1.6 Mathematical Symbols 8

1.2 Pronunciation of mathematical expressions 9

1.2.1 Logic and Set 9

1.2.2 Real numbers and operations 10

1.2.3 Functions 11

1.2.4 Some notation shortcuts are used in written English 12

1.2.5 Some notation shortcuts are often used in mathematics 13

1.3 Some Common Mathematical Symbols and Abbreviations (with History) 14 1.3.1 Binary Relations 14

1.3.2 Some Symbols from Mathematical Logic 14

1.3.3 Some Notation from Set Theory 16

1.3.4 Some Important Numbers in Mathematics 17

1.3.5 Appendix: Common Latin Abbreviations and Phrases 18

1.4 Skills Needed for Mathematical Problem Solving 20

1.4.1 Introduction 20

1.4.2 Mathematical problem solving as a process 21

1.4.3 Factors and skills involved in problem solving 22

i

1.4.4 Conclusion 32

1.5 Mathematical Writing 32

1.5.1 Some notes when writing mathematical 32

1.5.2 A Guide to Writing Mathematics 39

1.5.3 Mathematical Ideas into Writing 43

Chapter 2 Basic of Mathematics 49

2.1 Sets and Relations 50

2.1.1 Notation and Set Theory 50

2.1.2 Relations and Functions 54

2.1.3 Equivalence Relations and Classes 56

2.1.4 Natural Numbers, Integers, and Rational Numbers 58

2.2 Infinity and Induction 61

2.2.1 Countable Infinity 61

2.2.2 Uncountable Infinity 64

2.2.3 The Principle of Induction 67

2.2.4 The Real Number System 71

2.3 Logic 74

2.3.1 Negation of a Statement 74

2.3.2 Conjunction 76

2.3.3 Disjunction 79

2.3.4 Conditional Statements 80

2.3.5 Compound Statements 83

2.3.6 Biconditional Statements 85

2.3.7 Tautologies 89

2.3.8 Equivalence 91

Chapter 3 Methods of mathematical proof 92

3.1 Mathematical Induction - Problems With Solutions 93

3.2 The Pigeonhole Principle 100

3.2.1 Introduction 100

3.2.2 Applications of the Pigeonhole Principle 101

3.2.3 Examples 111

3.2.4 More difficult examples and exercises 113

3.2.5 Conclusion 118

3.3 Direct Proof 118

Contents iii

3.3.1 Theorems 119

3.3.2 Definitions 120

3.3.3 Direct Proof 123

3.3.4 Exercises 131

3.4 Contrapositive Proof 132

3.4.1 Contrapositive Proof 133

3.4.2 Congruence of Integers 136

3.4.3 Exercises 138

3.5 Proof by Contradiction 140

3.5.1 Proving Statements with Contradiction 141

3.5.2 Exercises 148

Chapter 4 Number Theory 150

4.1 Elementary properties of integers 151

4.1.1 Fundamental Notions and Laws 151

4.1.2 Definition of Divisibility The Unit 152

4.1.3 Prime Numbers The Sieve of Eratosthenes 154

4.1.4 The Number of Primes is Infinite 155

4.1.5 The Fundamental Theorem of Euclid 157

4.1.6 Divisibility by a Prime Number 157

4.1.7 The Unique Factorization Theorem 158

4.1.8 The Divisors of an Integer 160

4.1.9 The Greatest Common Factor of Two or More Integers 161

4.1.10 The Least Common Multiple of Two or More Integers 164

4.1.11 Scales of Notation 165

4.1.12 Highest Power of a Prime p Contained in n! 168

4.1.13 Remarks Concerning Prime Numbers 172

4.2 On the indicator of an integers 173

4.2.1 Definition Indicator of a Prime Power 173

4.2.2 The Indicator of a Product 173

4.2.3 The Indicator of any Positive Integer 175

4.2.4 Sum of the Indicators of the Divisors of a Number 177

4.3 Elementary properties of congruences 179

4.3.1 Congruences Modulo m 179

4.3.2 Solutions of Congruences by Trial 181

4.3.3 Properties of Congruences Relative to Division 182

4.3.4 Congruences with a Prime Modulus 183

4.3.5 Linear Congruences 185

4.4 The theorems of Fermat and Wilson 187

4.4.1 Fermat’s General Theorem 187

4.4.2 Euler’s Proof of the Simple Fermat Theorem 188

4.4.3 Wilson’s Theorem 189

4.4.4 The Converse of Wilson’s Theorem 191

4.4.5 Impossibility of 1 ⋅ 2 ⋅ 3⋯n − 1 + 1 = n k for n > 5 191

4.4.6 Extension of Fermat’s Theorem 192

4.4.7 On the Converse of Fermat’s Simple Theorem 195

4.4.8 Application of Previous Results to Linear Congruences 196

4.4.9 Application of the Preceding Results to the Theory of Quadratic Residues 197

Chapter 5 Other Topics 200

5.1 Complex Numbers 201

5.1.1 Square Roots 201

5.1.2 Complex Numbers 201

5.1.3 Cube Roots of Unity 203

5.1.4 Geometrical Representation 204

5.1.5 Product 204

5.1.6 Quotient 205

5.1.7 De Moivre’s Theorem 205

5.1.8 Cube Roots 206

5.1.9 Roots of Complex Numbers 207

5.1.10 Roots of Unity 208

5.1.11 Primitive Roots of Unity 210

5.2 Polynomials 211

5.2.1 Definitions and basic operations 212

5.2.2 Some applications 218

5.2.3 Analytic behavior 222

5.2.4 More identities 225

5.2.5 Integers and rationals 230

5.2.6 Polynomial problems with detailed solutions 236

Contents v

5.2.7 Exercises 242

5.3 Functional Equations 245

5.3.1 Introduction 245

5.3.2 Examples 245

5.3.3 Basic Methods For Solving Functional Equations 248

5.3.4 Cauchy Equation and Equations of the Cauchy type 250

5.3.5 Problems with Solutions 250

5.3.6 More Examples 270

5.3.7 Problems for Independent Study 279

5.4 A Tour of Triangle Geometry 285

5.4.1 Introduction 285

5.4.2 Isogonal conjugates 286

5.4.3 Simson line and line of reflections 295

5.4.4 Rectangular circum-hyperbolas 298

5.4.5 Conics 302

5.4.6 Further examples of reflections 309

5.4.7 A metric relation and its applications 313

5.4.8 The Apollonian Circles and Isodynamic Points 316

Chapter 6 Exercises with Solutions and Answers 336

6.1 Exercises 337

6.1.1 Grade 10 algebra excercises 337

6.1.2 Grade 10 math word exercises 339

6.1.3 Grade 10 geometry excercises 340

6.1.4 Grade 10 trigonometry excercises 342

6.1.5 Grade 10 math algebra excercises (advanced) 344

6.1.6 Grade 10 math word excercises (advanced) 347

6.1.7 Grade 10 geometry excercises (advanced) 349

6.1.8 Grade 10 trigonometry excercises (advanced) 354

6.1.9 Grade 10 excercises on complex (advanced) 357

6.2 Solutions and Answers to the Above Excercises 359

6.2.1 Grade 10 algebra excercises 359

6.2.2 Grade 10 math word exercises 360

6.2.3 Grade 10 geometry excercises 362

6.2.4 Grade 10 trigonometry excercises 364

6.2.5 Grade 10 math algebra excercises (advanced) 367

6.2.6 Grade 10 math word excercises (advanced) 373

6.2.7 Grade 10 geometry excercises (advanced) 379

6.2.8 Grade 10 trigonometry excercises (advanced) 387

6.2.9 Grade 10 excercises on complex (advanced) 395

Chapter 7 Examples and Exercises on Mathematical Training 397

7.0.10 Introduction 398

7.1 Operations on Rational Numbers 399

7.1.1 Basic Rules on Addition, Subtraction, Multiplication, Division 399

7.1.2 Rule for Removing Brackets 399

7.1.3 Ingenious Ways for Calculating 399

7.1.4 Examples 400

7.1.5 Exercises 403

7.2 Monomials and Polynomials 405

7.2.1 Definitions 405

7.2.2 Operations on Polynomials 406

7.2.3 Examples 407

7.2.4 Exercises 409

7.3 Linear Equations of Single Variable 410

7.3.1 Usual Steps for Solving Equations 410

7.3.2 Examples 411

7.3.3 Exercises 414

7.4 System of Simultaneous Linear Equations 416

7.4.1 Examples 417

7.4.2 Exercises 421

7.5 Multiplication Formulae 424

7.5.1 Basic Multiplication Formulae 424

7.5.2 Generalization of Formulae 424

7.5.3 Derived Basic Formulae 424

7.5.4 Examples 425

7.6 Some Methods of Factorization 429

7.6.1 Basic Methods of Factorization 429

7.6.2 Examples 429

7.6.3 Exercises 432

Contents vii

7.7 Absolute Value and Its Applications 434

7.7.1 Basic Properties of Absolute Value 434

7.7.2 Examples 434

7.7.3 Exercises 438

7.8 Linear Equations with Absolute Values 439

7.8.1 Examples 439

7.8.2 Exercises 443

7.9 Sides and Angles of a Triangle 445

7.9.1 Basic Knowledge 445

7.9.2 Examples 446

7.9.3 Exercises 449

7.10 Pythagoras’ Theorem and Its Applications 450

7.10.1 Examples 451

7.10.2 Exercises 456

7.11 Congruence of Triangles 458

7.11.1 Basic Criteria for Congruence of Two Triangles 458

7.11.2 Examples 458

7.11.3 Exercises 464

7.12 Divisions of Polynomials 465

7.12.1 Examples 467

7.12.2 Exercises 471

7.13 Congruence of Integers 472

7.13.1 Basic Properties of Congruence 473

7.13.2 Examples 474

7.13.3 Exercises 477

7.14 Decimal Representation of Integers 478

7.14.1 Decimal Expansion of Whole Numbers with Same Digits or Period-ically Changing Digits 478

7.14.2 Examples 478

7.14.3 Exercises 482

7.15 Perfect Square Numbers 484

7.15.1 Basic Properties of Perfect Square Numbers 484

7.15.2 Examples 485

7.15.3 Exercises 488

7.16 [x] and {x} 489

7.16.1 Some Basic Properties of x and {x} 490

7.16.2 Examples 491

7.16.3 Exercises 496

7.17 Diophantine Equations (I) 497

7.17.1 Definitions 497

7.17.2 Examples 499

7.17.3 Exercises 503

7.18 Diophantine Equations (II) 505

7.18.1 Basic Methods for Solving Quadratic Equations on Z 505

7.18.2 Examples 505

7.18.3 Exercises 511

7.19 Pigeonhole Principle 512

7.19.1 Basic Forms of Pigeonhole Principle 512

7.19.2 Examples 513

7.19.3 Exercises 517

7.20 Geometric Inequalities 519

7.20.1 Examples 519

7.20.2 Exercises 526

7.21 Solutions to given exercises 528

Bibliography 534

Preface 1

In recent years, due to the requirements of international integration, the demandfor the knowledge of English in some professional field is becoming increasinglyurgent, especially for the class teachers, high school students and college students

In Viet Nam, The Ministry of Education and Training has planned to set up theprograms for class teaching in bilingual Vietnamese - English, first for students ofnatural science subjects and then for students of social science ones in SpecializingUpper Secondary Schools

However, this is an extremely hard work because the knowledge of English ofmost teachers in this professional field is not good enough to carry out the task.They need to be trained again to meet the demand Students, also, need to betaught in such a way that they can be able to understand the lessons in English.Another obstacle is that teachers and students’ ability of English listening,speaking and writing is rather poor which has been considered an inherent weakness

of foreign language learning and teaching in our country today

To make some contribution to the ambitious program, I have decided to havethe lectures written and designed in English in order to help students specializing

in math understand and know the technical terms of solving exercises in English,

so that their general knowledge of English will be improved as well Some of thebeginning chapters and sections have been made directly by the teachers who areteaching in class, but most of the content of this research is for students to readand practice under the help and guidance of the teachers

This book includes 6 chapters and is divided into two parts

Chapter I Introduction: Provide the knowledge needed to understand thebook: The system of notation, Greek alphabet and the rules of mathematical word

Chapter V Other topics: Present the individual subjects in the school gram as well as for students of math.

pro-Chapter VI Exercises with Solutions and Answers: Offer some simpleexercises as rehearsals for students to do and explain exercises in English

Chapter VII Examples and Exercises on Mathematical Training: Offersome questions for students to practise, and test students’ ability to apply theirknowledge in solving real competition questions These examples and exercises aretaken from a range of countries, e.g China, Russia, the USA and Singapore, e.t.c.The books shown in the Bibliography are mostly sent to the author fromstudents who are studying abroad and hard to find in Vietnam With any luck,readers can find a pdf- files online, but if possible, you can order at amazon.comHowever, because of our limited level of writing, the limitation of time and thelength of the research, there are still some dissatisfaction in the discussion as thefollowing:

1 Some parts missing, such as: inequality, the transcendental equations, equations, and systems of equations (exponential, logarithmic and trigonometric),survey plot functions, applications of derivatives, etc

in-2 Some topics for gifted students have not been put in, such as (Invariant theory,game theory, extreme theory, combinatorial mathematics, etc.) or only superficialpresented in this textbook (Graph Theory, Principle Direchlet, etc.)

3 Also lack the classical geometry (plane and space), Vector and applications,Transformation, Method coordinates in Space, etc

In addition, the desired book is applied to three characters: mathematics ers, students specializing in math, and students of Specializing Upper SecondarySchools in general Hence the book style is not consistent While sections for stu-dents of math and for teacher are written with style accurately and scientifically,while other parts of text are freely written

teach-With full of hope, the author’s colleagues will continue to improve he research onthe shortcomings and with the sincere comments of your readers, the next versionwill be better

The book is written by software Viettex 2.5 and PCTEXv5.2 The picture drawn

by WinTpic, WinFig, Sketchpad and Graph4.3

0.1 Preface 3

A part of the lecture was set up by the author and MA Tran Thi Ha Phuong,who have used it to teach students of math in Bac Giang Specializing Upper Sec-ondary school for the last few years and has obtained some good results

We would like to express our sincere thanks to:

Professor, Doctor of Science, People’s Teacher NGUYEN VAN MAU,who has read and given many valuable comments on the content and the format

of the manuscript, and Doctor, Associate Professor Nguyen Vu Luong forhis strong support

The teachers: MA Bach Dang Khoa, MA Nguyen Anh Tuan, MA TranThi Ha Phuong, BA Nguyen Van Thao and MA Tran Anh Duc for readingand editing this manuscript

In particular, thanks to MA Tran Thi Ha Phuong and a group of students ofmath in mathematics courses K17, K18, K19, K20, K21 of Bac Giang SpecializingUpper Secondary school for their contributions to the manuscript

And, the most sincere thanks to some teachers of English in Bac Giang izing Upper Secondary school, Ms Do Thi Minh Hong, Ms Mai Thu Giang,

Special-Ms Vu Thi Hue and especially Mr Nguyen Danh Hao, who have checkedthe text carefully

Without timely support and help from these characters, the research could not

be completed successfully

The author would like to receive feedback and contributions from readers Allcomments should be sent to Bac Giang Specializing Upper Secondary school, HoangVan Thu street, Bac Giang City, Vietnam or send email to mailboxes

nvtiencbg@gmail.com

Sincerely thanks.Bac Giang, on 28/10/2012

Nguyen Van Tien

0.2 Notations and Symbol

In this book the authors use the notation system, and abbreviated as

1 ”:=” is the symbol of the phrases:

” called ” , or ” is denoted by ” , or ” instead ”

2 WLOG := Without loss of generality.

3 g.e := given equation ; g.i := given inequation ; g.s := given system of equations.

4 QED or ⊠:= "quod erat demonstrandum" (in Latin) or "Question est démontrée" (in France).

5 LHS := left hand side ; RHS := right hand side ; s.t.:= such that

6 "e.g." := "for example" (from "exempli gratia" in Latin) or "including"

7 i.e := it means that (from phrase "id est" in Latin) or "that is", or "in other word" or "it is".

8 wrt := with respect to ; "etc" := (from phrase "et cetera" in Latin) (or Et les autres choses

12 With X ⊆ R we denoted X ∗ ∶= X ∖ {0} ; X + ∶= X ∩ (0; +∞) ;

X+∶= X ∩ [0; +∞) ; X − ∶= X ∩ (−∞; 0) ; X − ∶= X ∩ (−∞; 0].

13 F (A):= Set of all functions f ∶ A → R with A ⊆ R.

14 With f (x) ∈ F(R) we denoted:

ˆ Df := The Domain of function f (x) ;

ˆ Gf:= The Graph of function y = f (x) ;

ˆ Rf := Set the value of the function f (x) ; R f = f (D f ) ;

ˆ Tf := Set of zero of function f (x), T f = {x ∈ D f ∣ f(x) = 0}.

15 C (A) [D(A)] := Set of continuous functions on A [differentiable on A].

16 k l ∶= {k; k + 1; k + 2; ⋯; l − 1; l} with k, l ∈ Z ; k < l.

17. (i)= := "equality occurs under conditions (i)"; Ô⇒:= "inferred according to claim abcd".abcd

18 "=∶" instead of the phrase "equality occurs if and only if" ; "⇛” instead of the phrase

"become".

19 The other symbols will be indicated as first appeared.

Chapter 1

Introduction

There is a good chance that you have never written a paper in a math class before.

So you might be wondering why writing is required in your math class now.

The Greek word mathemas, from which we derive the word mathematics, embodies the notions of knowledge, cognition, understanding, and perception In the end, mathematics

is about ideas In math classes at the university level, the ideas and concepts encountered are more complex and sophisticated The mathematics learned in college will include concepts which cannot be expressed using just equations and formulas Putting mathemas

on paper will require writing sentences and paragraphs in addition to the equations and formulas.

Mathematicians actually spend a great deal of time writing If a mathematician wants

to contribute to the greater body of mathematical knowledge, she must be able nicate her ideas in a way which is comprehensible to others Thus, being able to write clearly is as important a mathematical skill as being able to solve equations Mastering the ability to write clear mathematical explanations is important for nonmathematicians

commu-as well As you continue taking math courses in college, you will come to know more mathematics than most other people When you use your mathematical knowledge in the future, you may be required to explain your thinking process to another person (like your boss, a co-worker, or an elected official), and it will be quite likely that this other per- son will know less math than you do Learning how to communicate mathematical ideas clearly can help you advance in your career.

and Vietnamese (Vie.)

1.1.2 Geometry

No written as read as

1 ∣x∣ the norm (or modulus) of x

2. Ð→OA, Ð →u vector OA vector u

3 OA OA / the length of the segment OA

4 ∼ is similar to /Indicates two objects are geometrically similar

5 ⊥ is perpendicular to

6 ∥ is parallel to

7 ∠ Angle

8. Ða ⋅Ðb , Ð →aÐb the scalar product of vectors a and b

9. Ða ×Ðb the vector product of vectors a and b

10. Ði , Ð

j , Ð

k unit vectors in the directions of the cartesian coordinate axes

11 ∆ABC the triangle ABC

12 ≅ is congruent to

13 ∣Ð→OA ∣ , ∣Ð→ u ∣ the magnitude of vector OA the magnitude of vector u

AHSME American High School Mathematics Examination

AIME American Invitational Mathematics Examination

APMO Asia Pacific Mathematics Olympiad

ASUMO Olympics Mathematical Competitions of All the Soviet Union

AUSTRALIA Australia Mathematical Competitions

BMO British Mathematical Olympiad

CHNMO China Mathematical Olympiad

CHNMOL China Mathematical Competition for Secondary Schools

CMO Canada Mathematical Olympiad

HUNGARY Hungary Mathematical Competition

IMO International Mathematical Olympiad

JAPAN Japan Mathematical Olympiad

KIEV Kiev Mathematical Olympiad

MOSCOW Moscow Mathematical Olympiad

NORTH EUROPE North Europe Mathematical Olympiad

RUSMO All-Russia Olympics Mathematical Competitions

SSSMO Singapore Secondary Schools Mathematical Olympiad

SMO Singapore Mathematical Olympiad

SSSMO(J) SSSMO for Junior Section

UKJMO United Kingdom Junior Mathematical Olympiad

USAMO United States of American Mathematical Olympiad

VMO Vietnames Mathematical Olympiad

HOMO Hanoi Open Mathematical Olympiad

SOMO Singapore Open Mathematical Olympiad

1.1 Notations and Symbols in mathematics 7

N the set of positive integers (natural numbers)

N 0 the set of non-negative integers

Z the set of integers

Z+ the set of positive integers k m the set {k, k + 1, , m − 1, m}, k, m ∈ Z, k < m

Q the set of rational numbers

Q+ the set of positive rational numbers

Q+0 ; Q+ the set of non-negative rational numbers

R the set of real numbers

R− the set of negative real numbers

R−0 ; R− the set of non-positive real numbers [a, b] the closed interval, i.e all x such that a ⩽ x ⩽ b (a, b) the open interval, i.e all x such that a < x < b

⇔ iff, if and only if

⇒ implies

A ⊆ B A is a subset of B

A ∖ B the set formed by all the elements in A but not in B

A ∪ B the union of the sets A and B

A ∩ B the intersection of the sets A and B

a ∈ A the element a belongs to the set A

Rt Right triangle

(g.r.:= greek letter, Uppercase and lowercase letters)

No g.r read No g.r read No g.r read

1 A, α alpha 2 B, β beta 3 Γ, γ gamma

4 ∆, δ delta 5 E, ǫ epsilon 6 H, η eta

7 Θ, θ theta 8 I, ι iota 9 K, κ kappa

10 Λ, λ lambda 11 M, µ mu 12 N, ν nu

13 Ξ, ξ xi 14 F, f digamma 15 Π, π pi

16 P, ρ rho 17 Σ, σ sigma 18 T, τ tau

19 Υ, υ upsilon 20 Φ, φ phi 21 X, χ chi

22 Z, ζ zeta 23 Ψ, ψ psi 24 Ω, ω omega

Symbol How to read it Notes on meaning and usage

a ≈ b a is approximately equal to b Do not write = when you mean ≈.

(a,b) the point a b A coordinate in R 2

(a,b) the open interval from a to b The values between a and b,

but not including the endpoints.

[a,b] the closed interval from a to b The values between a and b,

including the endpoints.

(a,b] half-open interval from a to b The values between a and b, excluding

a, and includingb Similar for [a,b).

R the real numbers It can also be used for the plane as R 2

C the complex numbers C = {a + bi ∶ a,b ∈ R}, where i 2 = −1.

Z the integers Z = { ,−2,−1,0,1,2,3, }.

N the natural numbers N = {1,2,3,4, }.

∀ x for all x Something is true for all (any) value

of x (usually with a side condition

like ∀ x > 0).

∃ x there exists x Used in proofs as a shorthand.

∃ !x there exists a unique x Used in proofs as a shorthand.

f ○ g f composed with g or f of g Denotes f (g(⋅)).

n! n factorial n! = n(n − 1)(n − 2)⋯ × 2 × 1.

[x] the floor of x The nearest integer ⩽ x.

⌈x⌉ the ceiling of x The nearest integer ⩾ x.

f x→a= O(g) f is big oh of g when x to a lim

x→a ∣fg(x) ∣ =(x) C = const ≠ 0.

f x→a= o(g) f is little oh of g when x to a lim

x→a ∣f(x)

g (x) ∣ =0.

x → a + x goes to a from the right x is approaching a, but x is always

greater than a Similar for x → a −

o degree(s) Angular measure /Temperature /

Degree symbol

∶ is to / such that / Colon,ratio sign

it is true that

∣ such that / it is true Symbol following logical

that quantifier or used in defining a set

1.2 Pronunciation of mathematical expressions 9

The pronunciations of the most common mathematical expressions are given in the list below In general, the shortest versions are preferred (unless greater precision is necessary).

3 p ⇒ q p implies q / if p, then q

4 p ⇔ q p if and only if q /p is equivalent to q

/ p and q are equivalent

5 p&q p and q

8 x ∈ A x belongs to A / x is an element (or a member) of A

9 x ∉ A x does not belong to A

/x is not an element (or a member) of A

10 A ⊂ B A is contained in B / A is a subset of B

11 A ⊃ B A contains B / B is a subset of A

12 A ∩ B A cap B / A meet B / A intersection B

13 A ∪ B A cup B / A join B / A union B

14 A ∖ B A minus B / the difference between A and B

complement of B in A

15 A × B A cross B / the cartesian product of A and B

16 ∅ The null set / The empty set

24 {x ∶ t(x)} set of x having property t

25 ∀ x, P (x) For all x, P (x) holds

26 ∃ x, P (x) There exists an x such that P (x) holds

27 a ≡ b a is equivalent to b / a is indentically equal to b

28 a ≡ b (mod n) a and b are congruent modulo n

1.2.2 Real numbers and operations

9 x ≠ 5 x (is) not equal to 5

10 x ≡ y x is equivalent to (or identical with) y

11 x /≡ y x is not equivalent to (or identical with) y

12 x ⩾ y x is greater than or equal to y

13 x ⩽ y x is less than or equal to y

14 a < x < b a is less than x is less than b

15 0 ⩽ x ⩽ 1 zero is less than or equal

to x is less than or equal to 1

19 x 4 x to the fourth / x to the power four

20 x n x to the nth / x to the power n

28 ̂x ; x ; ̃x x hat ; x bar ; x tilde

29 x i xi / x subscript i / x suffix i / x sub i

30 ∑n

i=1

a i or ∑ n

i=1 a i the sum from i equals one to n of a i

/ the sum as i runs from 1 to n of the a i

1.2 Pronunciation of mathematical expressions 11

/ the third derivative of f with respect to x

7 f (4) (x) f four x / the fourth derivative

x→0 − f (x) the limit of f (x) as x approaches zero from below

12 logay log y to the base a / log to the base a of y

13 ln y log y to the base e

/ log to the base e of y / natural log (of) y

14 A T A transpose / the transpose of A

15 A −1 A inverse / the inverse of A

16 x −n x to the (power) minus n

22 ∫ f(x)dx indefinite integral of f (x) with respect to x

23 e x , exp (x) exponential function of x

24 g ○ f, gf the composite function of f and g

24 ∆x, δx an increment of x

26 lub (S) ; sup(S) least upper bound of the set S ; supremum of S

27 glb (S) ; inf(S) greatest lower bound of the set S ; infimum of S

28 yRx y is related to x by the ralation R

29 Rez ; Imz the real part of z ; the imaginary part of z

Individual mathematicians often have their own way of pronouncing mathematical expressions and in many cases there is no generally accepted ”correct”pronunciation.

1.2.4 Some notation shortcuts are used in written English

: Colon Period; Dot, Full stop; Decimal point

‘ Apostrophe " Open double quote; Open quote;

Open inverted commas

’ Open single quote " Close double quote; Close quote;

Close inverted commas

’ Close single quote > Greater than sign; Close angle bracket / Forward slash < Less than sign; Open angle bracket

/ Backslash ! Exclamation mark; Exclamation point

? Question mark ) Close parenthesis; Close paren

$ Dollar sign { Open brace; Open curly bracket

% Percent sign } Close brace; Close curly bracket

_ Underscore ] Close brace; Close square brace

− Hyphen; Minus sign; Dash © CopyRight

4 Three quarter sign

∣ Vertical bar & Ampersand; And sign

1

2 One half sign

// Double slash << Open angle quote

‘ Back quote >> Close angle quote

× Multiplication sign ⋯ Ellipsis; Dot dot dot

1.2 Pronunciation of mathematical expressions 13

Distinctions made in writing are often not made explicit in speech; thus the sounds fx may be interpreted as any of: fx, f (x),f x , (FX),FX,FX,ÐÐ→F X ? The difference is usually made clear by the context; it is only when confusion may occur, or where he/she wishes

to emphasise the point, that the mathematician will use the longer forms: f multiplied by

x, the function f of x, f subscript x, line F X,segment F X or the length of the segment

F X , the algebraic length of the segment F X, vector F X.

Similarly, a mathematician is unlikely to make any distinction in speech (except times a difference in intonation or length of pauses) between pairs such as the following:

some-x + (y + z) and (x + y) + z ; √ax + b and √

ax + b ; an− 1 and a n−1

1 l : the first letter of the word length

2 S : the first letter of the word square

3 V : the first letter of the word volume

4 R : the first letter of the word radius of a circle

5 D : the first letter of the word diameter

6 C : the first letter of the word circle

7 M : the first letter of the word Midpoint

8 h : the first letter of the word height

9 N : the first letter of the word natural number

10 Q : the first letter of the word quotient number

11 R : the first letter of the word real number

12 C : the first letter of the word complex number

13 i : the first letter of the word imaginary unit

14 r : the first letter of the word remaider

15 p : the first letter of the word prime number

16 d : the first letter of the word distance

17 m : the first letter of the word median

18 P : the first letter of the word Perimeter

19 R : the first letter of the word Radius of circumscribed circle

20 r : the first letter of the word Radius of incircle

21 R: the first letter of the word Relation

1.3 Some Common Mathematical Symbols and

Ab-breviations (with History)

◇ The symbol: = (the equals sign) means ”is the same as” and was firstintroduced in the 1557 book The Whetstone of Witte by physician and mathemati-cian Robert Recorde (c 1510-1558) He wrote, ”I will sette as I doe often in woorkeuse, a paire of parralles, or Gemowe lines of one lengthe, thus: =, bicause noe 2thynges can be moare equalle.” (Recorde’s equals sign was significantly longer thanthe one in modern usage.)

◇ The symbol: < (the less than sign) means ”is strictly less than”, and >(the greater than sign) means ”is strictly greater than” These first appeared inArtis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (”The AnalyticalArts Applied to Solving Algebraic Equations”) by mathematician and astronomerThomas Harriot (1560-1621), which was published posthumously in 1631 PierreBouguer (1698-1758) later refined these to ⩽ (”is less than or equals”) and ⩾(”is greater than or equals”) in 1734 Bouger is sometimes called ”the father ofnaval architecture” due to his foundational work in the theory of naval navigation

◇ The symbol: ∶= (the equal by definition sign) means ”is equal by inition to” This is a common alternate form of the symbol ”=Def ”, which firstappeared in the 1894 book Logica Matematica by the logician Cesare Burali-Forti(1861-1931) Other common alternate forms of the symbol ” =Def” include ”def.= ”and ”≡”, the latter being especially common in applied mathematics

def-◇ The symbol: ≐ (the approximately equals sign) means ”is nearly equalto” and was first used in 1875 by mathematician Anton Steinhauser (1802-1890)

in his Lehrbuch der Mathematik (This symbol was also briefly used in 1832 by ometer Farkas Wolfgang Bolyai (1775-1856) to signify ”equal by definition”.) Othermodern symbols for ”approximately equals” include ”≈” (read as ”is approximatelyequal to”), ”≅” (read as ”is congruent to”), ”≃” (read as ”is similar to”), ” ≍ ” (read as

ge-”is asymptotically equal to”), and ”∝” (read as ge-”is proportional to”) Usage varies,and these are sometimes used to denote varying degrees of ”approximate equality”within some context

◇ The symbol: ∴ (three dots) means ”therefore” and first appeared in print

in the 1659 book Teusche Algebra (”Teach Yourself Algebra”) by mathematician

Jo-1.3 Some Common Mathematical Symbols and Abbreviations (with History) 15

hann Rahn (1622-1676) (Teusche Algebra also contains the first use of the obelus,

”÷”, to denote division.)

◇ The symbol: ∵ (upside-down dots) means ”because” and seems to havefirst appeared in the 1805 book The Gentleman’s Mathematical Companion How-ever, it is much more common (and less ambiguous) to just abbreviate ”because”

as ”b/c”

◇ The symbol: ∋ (the such that sign) means ”under the condition that”and first appeared in the 1906 edition of Formulaire de mathematiqu’es by thelogician Giuseppe Peano (1858-1932) However, it is much more common (and lessambiguous) to just abbreviate ”such that”as ”s.t.”

There are two good reasons to avoid using ”∋” in place of ”such that” First ofall, the abbreviation ”s.t.” is significantly more suggestive of its meaning than is

”∋” Perhaps more importantly, though, is that it has become increasingly commonfor the symbol ”∋” to mean ”contains as an element”, which is a logical extension

of the usage of the unquestionably standard symbol ”∈” to mean ”is contained as

an element in”

◇ The symbol: ⇒ (the implies sign) means ”logically implies that”, and

⇐ (the is implied by sign) means ”is logically implied by” Both have an unclearhistorical origin (E.g., ”if it’s raining, then it’s pouring” is equivalent to saying

”it’s raining ? it’s pouring.”)

◇ The symbol: ⇔ (the iff symbol) means ”if and only if ” and is used toconnect logically equivalent statements (E.g., ”it’s raining iff it’s really humid”means simultaneously that ”if it’s raining, then it’s really humid” and that ”if it’sreally humid, then it’s raining” In other words, the statement ”it’s raining ⇔ it’sreally humid ” means simultaneously that ”it’s raining ⇒ it’s really humid ” and

”it’s raining ⇒ it’s really humid”.) The abbreviation ”iff ” is attributed to themathematician Paul Halmos (1916-2006)

◇ The symbol: ∀ (the universal quantifier) means ”for all” and was firstused in the 1935 publication Untersuchungen ueber das logische Schliessen (”In-vestigations on Logical Reasoning”) by logician Gerhard Gentzen (1909-1945) Hecalled it the All-Zeichen (”all character”) by analogy to the symbol ”∃ ”, whichmeans ”there exists”

◇ The symbol: ∃ (the existential quantifier) means ”there exists” andwas first used in the 1897 edition of Formulaire de mathematiqu’es by logicianGiuseppe Peano (1858-1932)

◻ (the Halmos tombstone or Halmos symbol) means ”Q.E.D.”, which is

an abbreviation for the Latin phrase quod erat demonstrandum (”which was to be

proven”) ”Q.E.D.” has been the most common way to symbolize the end of a logicalargument for many centuries, but the modern convention of the ”tombstone’ ’ is nowgenerally preferred because it is easier to write and is also visually more compact.The symbol ” ∃” was first made popular by mathematician Paul Halmos (1916-2006).

◇ The symbol: ⊂ (the is included in sign) means ”is a subset of ” and ⊃(the includes sign) means ”has as a subset” Both symbols were introduced in the

1890 book Vorlesungen uăber die Algebra der Logik (”Lectures on the Algebra ofthe Logic”) by logician Ernst Schrăoder (1841-1902)

◇ The symbol: ∈ (the is in sign) means ”is an element of ” and first appeared

in the 1895 edition of Formulaire de mathematiqu’es by logician Giuseppe Peano(1858-1932) Peano origi- nally used the Greek letter ”e ” (viz the first letter ofthe Latin word est for ”is”), and it was the great logician and philosopher BetrandRussell (1872-1970) who introduced the modern stylized version of this symbol inhis 1903 book Principles of Mathematics It is also common to use the symbol

”∋” to mean ”contains as an element”, which is not to be confused with the morearchaic usage of ”∋” to mean ”such that”

◇ The symbol: ∪ (the union sign) means ”take the elements that are ineither set”, and ∩ (the intersection sign) means ”take the elements that thetwo sets have in common” They were introduced in the 1888 book Calcolo ge-ometrico secondo l’Ausdehnungslehre di H Grassmann preceduto dalle operazionidella logica deduttiva (”Geometric Calculus based upon the teachings of H Grass-man, preceded by the operations of deductive logic”) by logician Giuseppe Peano(1858-1932)

◇ The symbol: ∅ (the null set or empty set) means ”the set without anyelements in it” and was first used in the 1939 book E’l’ements de math’ematique

by Nicolas Bourbaki (Bourbaki is the collective pseudonym for a group of primarilyEuropean mathematicians who have written many mathematics books together.)

It was borrowed simultaneously from the Norwegian, Danish and Faroese alphabets

by group member Andr’e Weil (1906-1998)

◇ The symbol: ∞ (infinity) denotes ”a quantity or number of arbitrarilylarge magnitude” and first appeared in print in the 1655 De Sectionibus Conicus(”Tract on Conic Sections”) by mathematician John Wallis (1616-1703) Possibleexplanations for Wallis’ choice of ”∞” include its resemblance to the symbol ”oo”(used by ancient Romans to denote the number 1000), to the final letter of theGreek alphabet ω (used symbolically to mean the ”final” number), and to the easewith which this simple curve (called a ”lemniscate”) can be endlessly traversed

1.3 Some Common Mathematical Symbols and Abbreviations (with History) 17

◇ The symbol: π (the ratio of the circumference to the diameter of a circle)denotes the number3.141592653589 ,and was first used by mathematician WilliamJones (1675-1749) in his 1706 book Synopsis palmariorum mathesios (”A NewIntroduction to Mathematics”) It was then the great mathematician LeonhardEuler (1707-1783) who popularized the use of π to denote this number in his 1748book Introductio in Analysin Infinitorum (It is speculated that Jones chose theletter ”π” because ”π” is the first letter in the Greek word perimetron (πǫιµǫτ ρoν),which roughly means ”around”.)

◇ The symbol: e= lim

n→∞(1+ 1

n)n (the natural logarithm base, also times called Euler’s number) denotes the number 2.718281828459 , and wasfirst used by Leonhard Euler (1707-1783) in the manuscript Meditatio in Experi-menta explosione tormentorum nuper instituta (”Meditation on experiments maderecently on the firing of cannon”), which was written when Euler was only 21 yearsold (It is speculated that Euler chose ”e” because ”e” is the first letter in the word

some-”exponential”.) The mathematician Edmund Landau (1877-1938) once wrote that,

”The letter e may now no longer be used to denote anything other than this positiveuniversal constant.”

◇ The symbol: i=√−1 (the imaginary unit) was first used by LeonhardEuler (1707-1783) in his 1777 memoir Institutionum calculi integralis (”Foundations

of Integral Calculus”) The five most important numbers in mathematics are widelyconsidered to be (roughly in order) 0, 1, i, π, and e, which are remarkably linked bythe equation

an irrational number

◇ The symbol: φ= 1+√5

2 (the golden ratio) denotes the number

φ ≈ 1.618033988749 Its use was first attributed to the American Mathematician

Mr Mark Barr in The Curves of Life: Being an Account of Spiral Formationsand Their Application to Growth in Nature, to Science, and to Art: With SpecialReference to the Manuscripts of Leonardo da Vinci (1914) by Sir Theodore AndreaCook (1867-1928):

The symbolφwas given to this proportion partly because it has a familiar sound

to those who wrestle constantly with p and partly because it is the 1st letter ofthe name of Pheidias, in whose sculpture this number is seen to prevail when thedistance between salient points are measured

The number φ is also often called the ”divine proportion” or the ”golden portion”, and it has been recognized since antiquity as an especially aestheticallypleasing ratio for the side lengths of a rectangle Such a rectangle is called a ”goldenrectangle”

ˆ i.e (id est) means ”that is” or ”in other words” (It is used to paraphrase astatement that was just made, not to mean ”for example”, and is always followed

ˆ et al (et alii) means ”and others” (It is used in place of listing multipleauthors past the first and is never followed by a comma.) The abbreviation ”etal.” can also be used in place of et alibi, which means ”and elsewhere”

ˆ cf (conferre) means ”compare to” or ”see also’ ’ (It is used either to draw

a comparison or to refer the reader to somewhere else that they can find moreinformation, and it is never followed by a comma.)

ˆ q.v (quod vide) means ”which see” or ”go look it up if you’re interested ”.(It is used to cross-reference a different written work or a different part of the samewritten work, and it is never followed by a comma.) The plural form of ”q.v.” is

”q.q.”

1.3 Some Common Mathematical Symbols and Abbreviations (with History) 19

ˆ v.s (vide supra) means ”see above” (It is used to imply that more tion can be found before the current point in a written work and is never followed

informa-by a comma.)

ˆ N.B (Nota Bene) means ”note well ” or ”pay attention to the following” (It

is used to imply that the wise reader will pay especially careful attention to whatfollows and is never followed by a comma Cf the abbreviation ”verb sap.”)

ˆ vs (versus) means ”against” or ”in contrast to” (It is used to contrast twothings and is never followed by a comma.)

ˆ c (circa) means ”around ” or ”near ” (It is used when giving an tion, usually for a date, and is never followed by a comma.) The abbreviation ”c.”

approxima-is also commonly written as ”ca.”, ”cir.”, or ”circ.”

ˆ ex lib (ex libris) means ”from the library of ” (It is used to indicate ership of a book and is never followed by a comma.)

own-ˆ vice versa means ”the other way around ” and is used to indicate that animplication can logically be reversed (This is sometimes abbreviated as ”v.v.”)

ˆ a fortiori means ”from the stronger ” or ”more importantly”

ˆ a priori means ”from before the fact” and refers to reasoning that is donewhile an event still has yet to happen (such as the impending page break in thislist)

ˆ a posteriori means ”from after the fact” and refers to reasoning that isdone after an event has already happened (such as the above page break in thislist)

ˆ ad hoc means ”to this” and refers to reasoning that is specific to an event

as it is happening (Such reasoning is regarded as not being generalizable to othersituations.)

ˆ ad infinitum means ”to infinity’ or ”without limit”

ˆ ad nauseam means ”causing sea-sickness” or ”to excessive”

ˆ mutatis mutandis means ”changing what needs changing” or ”with thenecessary changes having been made”

ˆ non sequitur means ”it does not follow ” and refers to something that isout of place in a logical argument (This is sometimes abbreviated as ”non seq.”)

1.4 Skills Needed for Mathematical Problem Solving

Abstract. Mathematical problem solving is one of the most important topics

to learn and also one of the most complex to teach The main goal in teachingmathematical problem solving is that students develop a generic ability to solve reallife problems and apply mathematics in real life situations Problem solving can also

be used, as a teaching method, for a deeper understanding of concepts Successfulmathematical problem solving depends upon many factors and skills with differentcharacteristics In fact one of the main difficulties in learning problem solving isthe fact that many skills are needed for a learner to be an effective problem solver.Also, these factors and skills make the teaching of problem solving one of the mostcomplex topic to teach This paper will discuss the idea that problem solving is

a process that needs to be understood by instructors so that they can developbetter and more effective classroom activities and tasks The necessary skills forproblem solving as well as the methods and strategies to teach or facilitate themare discussed

Mathematical problem solving related to counting and daily transactions ofcommerce has been present from the earliest days of human experience Geometrywas also widely used in land measurement Mathematics is now used to quantifynumerically and spatially natural as well as man-made situations It is used tosolve problems and with the advent of the computer technology, it has helped inmaking social, economical and technological advances that few decades ago wouldhave been unthinkable

Learning mathematical facts and contents is important but is not enough dents should learn how to use these facts to develop their thinking skills and solveproblems Mathematics educators have accepted the idea that that the develop-ment of problem solving ability deserves special attention and that one of themost important components in any mathematics curriculum or program is genuinemathematical problem solving If well facilitated, mathematical problem solvingmay help students

Stu-1 develop and improve the generic ability to solve real life problems,

2 develop critical thinking skills and reasoning,

3 gain deep understanding of concepts,

4 work in groups, interact with and help each other

1.4 Skills Needed for Mathematical Problem Solving 21

Most of the mathematics textbooks suggest few types of problems as exampleswith detailed solutions and then suggest similar problems as exercises Students’learning is limited if only routine problems are solved The problems used to creategenuine learning opportunities should be of the challenging type and not only thosesimilar to ones already solved in the past

Mathematical problem solving is a process that involves a set of factors and tasks

to achieve a defined goal It depends on many skills and factors which thereforemakes it challenging both to learn and to teach If the instructor’s understanding

of the process is limited, difficulties in teaching mathematical problem solving, willarise Hence the great need to understand these factors and skills if we want tohelp our students acquire this important process

In this paper, I am addressing issues and concerns related to mathematicalproblem solving However, much of what is said in this paper may be applied toany problem solving of the quantitative type such as those encountered in math-ematics, informatics, physics, chemistry, biology etc Our students, at the CBG,have difficulties in mathematical problem solving and we need to be aware of it

I believe that we need to address this issue and discuss it if we want to improvethe teaching of quantitative problem solving which will greatly improve our stu-dents’ thinking skills and problem solving abilities in general My purpose here is

to present some of these issues in order to stimulate discussions Suggestions onhow to facilitate certain aspects of mathematical problem solving, as I apply them

in my classes, are also presented

Mathematical problem solving is a process depending on many phases Thereare four main phases in mathematical problem solving

1 Understand the problem ;

Figure 1.1: Four steps in mathematical problem solving (linear model)

the right answer or it does not even work We therefore need to look back critically

at the formulation of the problem as well as the information given in the problem

A more practical model of the process is shown in figure 1.2 The learner need tounderstand that in solving a problem he needs to look back and reexamine thewhole problem during the problem solving process and be ready to restart again ifnecessary

One of the difficulties in problem solving is the fact that several steps are needed

to solve a given problem Also at each step, students need to use several skills Thereare also many factors and skills that determine the outcome of the process

The fact that mathematical problem solving is a process that depends on manyskills and factors makes it complex to learn and to teach Figure 1.3 shows themost important skills and factors necessary for successful problem solving Routineproblems may not need all the skills listed below, however challenging problemsneed a strategy involving most if not all the skills and attitudes shown in figure 1.3

In fact the latest development in the education of mathematics is to teach some ofthese skills through problem solving

1.4 Skills Needed for Mathematical Problem Solving 23

Figure 1.2: Four steps and looking back at the problem (More practical model)

Based on my experience and the experience of others, I will discuss and suggestways to facilitate some of the most important factors that affect the performance

of students when solving a mathematical problem

Facilitating and guiding students is one of the most efficient ways to ”teach”problem solving Facilitating is suitable for creating opportunities where studentslearn by doing Mathematical problem solving is a constructivist process wherestudent learn to solve problems by making connections to what they already know.Instructors can guide students to make these connections

Concepts and Facts in Problem Solving

Learners need a deep understanding of algebraic, geometric, statistical conceptsand facts to solve challenging mathematical problems Here I will explain how I used

a set of problems assigned as homework with my level (2) advanced mathematicsclasses to reinforce the concept of rate of work that was necessary to solve theproblem

Step 1: The concept of rate of work is first presented then discussed Then thewhole class discussed the solution to the following problem

Figure 1.3: Factors and skills involved in problem solving.

Problem 1 It takes Thao 1 hour longer to mow the lawn than it takes Tien tomow the lawn If they can mow the lawn in 5 hours working together, then howlong would it take each girl by herself?

Step 2: Students were then asked to solve the following problem as a home work,where the same concept as problem 1 is needed

Problem 2 Phuong takes 3 hours longer than Khoa to peel 500 pounds (lb) ofapples If together they can peel 500 lb of apples in 8 hours, then how long would

it take each one working alone?

1.4 Skills Needed for Mathematical Problem Solving 25

Although the above problem was discussed in class before they attempted tosolve it, to make sure that students understood it, they had difficulties in solving

it I decided that the concept of rate of work has to be discussed again Few dayslater few students managed to solve the given problem and the solution to problem

2 was discussed with the whole class

Step 3: In order to assess students’ understanding of the concept of the rate ofwork and the process of problem solving, I assigned the following problem as ahome work

Problem 3 It takes pump A 2 hours less time than pump B to empty a certainswimming pool Pump A is started at 8:00 A.M., and pump B is started at 11:00A.M If the pool is still half full at 5:00 P.M., then how long would it take pump Aworking alone?

Problem 3 also needs a deep understanding of the concept of the rate of work.Except for a few students, most found the problem very challenging and could notsolve it I carefully examined the solutions generated by students and I understoodthat the concept of rate of work was their main difficulty I decided to give a fullhour lesson on the rate of work with many examples and I made sure that notonly they understood the concept but also how to use it to formulate problems Irequested that they look again at the problem Few days later, more than half theclass solved problem 3 correctly

Step 4: I then assigned the following problem in a quiz

Problem 4 It takes pump B 2 hours more time than pump A to fill a swimmingpool Both pumps are started at 7 am At 10 am pump A breaks down It took 1hour to repair it and then was restarted again At 3 pm 80% of the swimming poolwas filled with water How long would it take each pump working alone to fill theswimming pool?

About half the class solved the problem correctly and a quarter of the classhad solution with minor mistakes I carefully examined the solutions generated

by students and it was clear that the students had a better understanding of theconcept of the rate of work and more importantly they knew how to apply it tosolve problems

Students fail to solve problems involving concepts that are not thoroughly derstood Moreover, mathematical problems may be used as teaching methodolo-gies not only to introduce concepts but also to help students gain a deeper under-standing of these concepts In fact some concepts cannot be thoroughly understood

un-unless they are used in problem solving or any other activity where critical thinkingand reasoning are involved.

Mathematical problem solving may also be used to introduce a new concept Anexample of a mathematical problem that may be used to introduce a new concept

is now presented

Problem 5 The present population of the Bac Giang Province is 1.5 million If weassume that the population grows at an annual rate r = 3% for the next 15 years,what will be the population P of the Bac Giang in t years?(assume t is smaller than15)

Students can easily be guided to use percentages and come up with the followingresult

Skills in Problem Solving

Basic mathematical skills such as solving equations and inequalities are essary for mathematical problem solving Formulating a problem can be very de-manding but simplifying and solving the equation obtained, for example, is nec-essary to answer the question in the problem Students who cannot manipulatealgebraic expressions will definitely have difficulties in problem solving

nec-Our students at CBG, have difficulties in reading with understanding and tracting the information from the text of the problem This skill has to be taughtexplicitly to our students It is the first step in problem solving and students cannotmake any progress if the problem is not understood

ex-Thinking and Reasoning in Problem Solving

Critical thinking is needed in all steps of problem solving Students do not lookback critically at the solution of a problem once it is solved They tend to acceptwhatever answer they have obtained Critical thinking is needed when extracting

1.4 Skills Needed for Mathematical Problem Solving 27

information from the text of the problem, formulating and solving the problem andanalyzing the solution obtained I now give an example of a problem that may beused to give opportunities to students to think critically

Problem 6 Two boats on opposite banks of a river start moving towards eachother They first pass each other 1400 meters from one bank They each continue

to the opposite bank, immediately turn around and start back to the other bank.When they pass each other a second time, they are 600 meters from the other bank

We assume that each boat travels at a constant speed all along the journey Is itpossible to find the width of the river using the given information?

Students are guided to make a diagram and write the equations to formulatethe problem

Solution: We first rewrite the equations obtained

Let t = 0 be the time when the two boats start moving Let S 1 andS 2 be theconstant speeds of the two boats and t 1 the time when they first pass each otherand t 2 the time when they pass each other for the second time Let X the width

to understand the problem and to come up with a solution and they are necessary

if we want our students to develop their thinking skills to the highest level possibleand become genuine problem solvers

It is not difficult to design problems that can activate students thinking Onehas to avoid problems with one obvious solution Note that the question in problem

6 does not say ”find the width of the river” but ”is it possible to find the

1.4 Skills Needed for Mathematical Problem Solving 29

width of the river using the given information?” A question of the type

”find the width of the river” already assumes that it is possible to find thewidth X However questions of the second type are more demanding and thereforesuitable to develop thinking skills

Open ended problems, usually with many valid answers, can also be used astools to generate activities where students have to think in order to come withanswers Useful discussions can be generated in order to compare all valid answersgenerated by students I now give an example of an open ended problem

Problem 7 Create a set of data points that satisfies the following conditions:

ˆ The set includes 8 data values

ˆ The range of the data set is 20

ˆ The median is equal to the mean

Show that your data set satisfies the conditions

Logical reasoning is fundamental to success in mathematics in general and lem solving in particular Inductive reasoning uses easy to understand cases toidentify patterns and it can also be used to establish relationships between mathe-matical objects Deductive reasoning uses mathematical definitions, axioms, rulesand theorems to draw conclusions Deductive reasoning is fundamental to criticalthinking Geometry problems may be used as tools to develop deductive reasonin

prob-I believe that both must be used and explicitly discussed with students

Attitude and Problem Solving

Learners must persevere and take the risk of failing in solving a given problem.They must also understand that learning takes place even when they are not able

to solve a problem What count is the time and efforts spent looking for a solution.Students do not learn much from problems that they can easily solve, they learnmore from challenging problems where they have to persevere However, studentsleft alone with challenging problems to solve may get frustrated and develop anegative attitude towards mathematical problem solving The instructor’s task isnot easy He must not give too much information so that students will not haveanything to think about But also he should give enough help so that students willnot be frustrated

I found that historical facts about mathematics and real life mathematical lems motivate students I use both to highlight the power of mathematics and itsapplications Below is a real life problem related to trigonometry about measuringthe circumference of the earth

prob-Problem 8 More than two thousand years ago Eratosthenes of Cyrene (276

BC-194 BC), a Greek mathematician, used the rays from the sun and the shadow of

a stick to measure angle a = 72 degrees Assume that the ray from the sun fallsvertical at Syene and the stick used at Alexandria is also vertical Assume also thatthe earth is circular and find the circumference of the earth

Figure 1.5: Rays from the sun and the shadow to measure angle.

Students are more interested in real life problems because they make sense andalso give reasons to study mathematics (see Figure 1.5)

Metacognition and Problem Solving

The term metacognition refers to a learners knowledge about his or her process

of cognition and the ability to control and monitor those processes as a function ofthe feedback the learner receives via outcomes of learning Learners must controland reflect on their cognitive processes in order to solve challenging problems It isthought that the best way to help students become aware about their own thinking

is to create opportunities where they have to explain their thinking explicitly As

an example let us look at the following problem

1.4 Skills Needed for Mathematical Problem Solving 31

Problem 9 Ahmed walked at a constant speed of 6 km/hour along a straight linefrom A to B, then walked back along the same line from B to A at a constant speed

of 4 km/hour What is the average speed over the entire trip? (Definition: Averagespeed = total distance / total time)

When the above problem was given to students in level (2) advanced matics, all students answered the question as follows: average speed = (6+24) = 5

mathe-km/hour

When asked to explain their answer, they said that any average is found byadding all data values and divide the sum obtained by the number of data values.They simply ignored the definition given to them I next guided them to work ingroups, look back at the problem and use the definition to answer the question.They had to justify the steps of their calculations and their thinking within theirgroup When the correct answer based on the definition was obtained, I asked stu-dents to discuss within their groups what happened and why all students answeredthe question without taking into account the definition I explicitly ask them tothink about the way they solved the problem and why

Metacognitive skills help students to analyze both the question and the solutiondeveloped They also help students to look back at the problem and restart if neces-sary In designing activities that may help students be aware of their metacognitiveskills, I have taken into account the following:

1 Metacognitive skills have to be taught explicitly

2 Only genuine mathematical problems, that students have not solved before,help them develop metacognitive skills

3 Students need to explain to other students and the teacher their way ofthinking

4 I sometimes use examples to explain my own thinking in solving problems

Group Work and Problem Solving

Experience shows that group work is very useful in solving problems in general.When the problem given to students is challenging, students are happy to work ingroups In fact is was shown that cooperative learning and metacognitive activitieshave positive effects on the students’ abilities to solve problems Group work alsoprepares students for the future where they have to work together on large problemsand projects

1.4.4 Conclusion

Many skills and factors are involved when genuine mathematical problems arebeing solved Instructors have to understand and be familiar with these factors andskills They also need to design activities and guide students to develop and usethese skills It is also possible to design problems that focus on a limited number

of skills and factors

Students develop these skills only if genuine mathematical problems solving istaking place The solved examples in the textbooks give the idea that problemsolving is a linear process with no false start or illogical attempts Also the waythe solution is presented does not show how much time and efforts are needed

to come up with a useful solution Even the problems suggested at the end of achapter are usually of the same type as those already solved Students may start

to practice on less demanding problems However as they become more confident,problems should be varied and more demanding Students need to understand thateven when no solution to the problem is obtained, learning is taking place It is thetime and efforts spent on finding that contribute to the learning process All thethinking taking place and the organization of one’s thoughts during the problemsolving process contribute to the learning process

The style and language of mathematical writing may not seem especially ural at first sight But–with practice and experience–one soon finds that adheringcarefully to mathematical writing conventions actually simplifies and streamlinesthe process of writing proofs, and it helps assure that your ideas are intelligible toothers

nat-Now that you have begun writing proofs, it is the right time to address sues concerning writing Unlike logic and mathematics, where there is a clear-cutdistinction between what is right or wrong, the difference between good and badwriting is sometimes a matter of opinion But there are some standard guidelinesthat will make your writing clearer Some of these are listed below

1 Never begin a sentence with a mathematical symbol The reason isthat sentences begin with capital letters, but mathematical symbols are casesensitive Since x and X or x can have entirely different meanings, putting