Secondary School Curriculum Review

Secondary School Curriculum Review: Review of Cambodian Mathematics Text books of Grade 9 and 12 By Mr. SOK Lin (RUPP) and Mr. SEAM Ngonn (RUPP) Email:[email protected], [email protected] Abstract This report presents an analysis of the Cambodian Mathematics text books of Grade 9 and 12 and makes a comparison with the knowledge, skills and attitudes at similar levels in the Singapore curriculum. Introduction The current review of the Cambodian curriculum is aimed at suggesting reforms that will contribute to economic development. The conceptual framework for this review is based on a comparison between the policies and curricula of Cambodia and Singapore. The Singapore education system is recognized as world class and so can act as a model from Cambodia can learn. In this initial stage of the review, only Grade-9 and Grade-12 text books have been considered as indicators of the quality and relevance of each subject. This is part of the Written curriculum, which may in fact be different from the Intended, Taught, Learned and Examined curriculum. According to the economic researchers at Royal University of Phnom Penh, the most important educational skills necessary for economic development are:    

Lower-order cognitive skills-literacy and numeracy Higher-order cognitive skills-analysis, decision-making, problem-solving, communication, foreign languages. Socio-emotional skills—work attitudes, e.g. being straightforward, honest. Technical skills and experience –TVET.

The answers to the four questions focused on the analysis the text books are the followings: 1. Are the knowledge, skills and attitudes presented in the grade 9 and 12 Cambodian text books appropriate for their grade levels? Why or why not?

The knowledge presented at each level was considered appropriate due to their contents being similar to those of the good education systems in the world like Japan, France Singapore and so on while the level of cognitive skills required were still commonly low, mostly at levels 1 (nearly absent), 2, 3 and with a small amount at level 4 in sense of Bloom’s Taxonomy. See the survey on Grade 10, Grade 11 and Grade 12 provided at the end. The lack of clear definitions, general approach or algorithm (which have been found in many sections of Grade 9) and of logical reasoning did not construct a strong Mathematical base for critical thinking towards other higher levels for example, levels 5 and 6. Moreover, due to less connection to daily life and being forced to remember so many unnecessary things (rather to understand), the students are frustrated and scared to learn Mathematics. The wrong interpretations and unclear explanations both contained in some sections of Grade 9 and 12 also caused the students to have bad attitudes towards Mathematics rather than to show the power and beauty lying in it. 2. How do the knowledge, skills and attitudes expressed in the grade 9 and 12 Cambodian Mathematics text books compare with those in Singapore? Mathematics is an international language where common things, for example, definition and theorem, are internationally known and agreed. Although the contents in the Cambodian Mathematics text books of Grade 9 and 12 are similar to those in the Singaporean’s, some formal definitions, which are used to explore some Mathematical properties and to create theorems, are not given. Moreover the Cambodian students are difficult to remember the definitions mostly given in the Grade-9 text book and in some parts of Grade 12 and hence difficult to understand the concepts which makes them learn to apply only the given properties or theorems to solve the problem rather than to analyze the imposed problem, to justify the solution to the problem and finally to create new idea (theorem) to solve other similar problems whereas the kind of such a critical thinking problem can be often found not only in the Singaporean text books but also in almost everywhere in Mathematics text book. 3. Are the knowledge, skills and attitudes expressed in the grade 9 and 12 Cambodian text books relevant and sufficient to meet predicted market demands? Although some sections in both Grade-9 and Grade-12 Cambodian Mathematics text books present some life skills, none of the Cambodian text books reviewed was considered to be relevant or sufficient to meet predicted market needs since the text books do not provide a deep knowledge to students while the skills and attitudes are still severely lacking. In the text books, students are often asked to apply Mathematical properties or theorems to solve the imposed problem without necessarily understanding it. The text books do not suggest students to make a Mathematical model, which transfers a real world problem into a Mathematical problem.

Predicted market demands will require higher order skills. Both Grade-9 and Grade-12 text books do not integrate higher order skills such as posing questions, formulating hypothesis, defining problems, generating possibilities, predicting, observing, using apparatus and equipment, comparing, classifying, inferring, analyzing, evaluating, verifying, communicating. 4. Are the knowledge, skills and attitudes expressed in the grade 9 and 12 Cambodian text books a sufficient foundation for higher education? Although the content in the Grade 12 (and some sections in Grade 9) is likely to prepare the Cambodian students for the “Foundation Year” at university, none of the Cambodian text books reviewed was considered to be a sufficient foundation for higher education in terms of knowledge, skills or attitudes. In particular the skills and attitudes conveyed by these texts were considered to be severely lacking.

Text book review: The analysis can be summarized as follows, 0 indicating none and 5 indicating sufficiency. Grade 9

Grade 12

Mathematics Knowledge Skills Attitudes Knowledge Skills Attitudes

Appropriateness to the grade

3

1

1

3

1

1

Compared with Singapore

2

1

1

2

1

1

Relevance to market demand

1

1

1

2

1

1

Sufficient for further study to HE

2

1

1

3

1

1

Cambodian Grade-9 Textbook

Lesson 1: Irrational numbers 1. Square roots 2. Cubic roots 3. Operations on surds

Knowledge -Finding square root of a number -Finding cubic roof a number -Operations on irrational numbers -Exercises 1-30, all at levels [1,2,3] but exercises 27, 29, 30 at level [4]

Skills

Numerical calculations

Attitudes

Comparison with Singaporean secondary three/four (Grade 9)

NA

NA

Lesson 2: Proportion 1. Proportion 2. Percentage 3. Interest

Bridge for higher education/Market demand

NO/NO

Secondary one : -Ratio rate and proportion:

-Solving problems related to proportion -Solving problems related to percentage -Solving problems related interest -Exercises 1-10, all at levels [1,2,3] except exercise 9 at level [4]

-Numerical calculations -Change from one unit to another one - Calculation of compound interest

• ratios involving rational numbers • writing a ratio in its simplest form • average rate

Appreciation, usefulness and interest

• problems involving ratio and rate -Percentage: • expressing one quantity as a percentage of another • comparing two quantities by percentage • percentages greater than 100% • increasing/decreasing a quantity by a given percentage

YES/YES

• reverse percentages • problems involving percentages -Speed : • concepts of speed, uniform speed and average speed • conversion of units (e.g. km/h to m/s) • problems involving speed, uniform speed and average speed

Secondary two: Algebraic manipulation:

Lesson 3: Algebraic expression 1. Product of algebraic expression 2. Factorization of algebraic expression 3. Operations on algebraic fractions

-Giving some examples related to algebraic expression, to expansion and to factorization as well as some approach for algebraic manipulation -Exercises 1-17, all at levels [1,2,3] but exercises 10, 14, 17 at level [4]

Algebraic manipulations

NA

• expansion of the product of algebraic expressions • changing the subject of a formula • finding the value of an unknown quantity in a given formula • recognising and applying the special products • factorisation of algebraic expressions • multiplication and division of simple algebraic fractions, e.g. • addition and subtraction of algebraic fractions with linear or quadratic denominator

NO/NO

Secondary one : Solutions of equations and inequalities:

Lesson 4: Linear equations in one unknown 1. Linear equations in one unknown 2. Equations with numerical denominator 3. Equations with first hand side as product of factors 4. Problems with linear equations in one unknown

• solving linear equations in one unknown (including fractional coefficients) • solving simple inequality

-Giving some examples related to linear equation in one unknown, some approaches for unknown with fractional coefficient and for quadratic unknown -Exercises 1-13, all at levels [1,2,3] but exercises 11, 12 at level [4]


NA

• solving simple fractional equations that can be reduced to linear equations • formulating a linear equation in one unknown to solve problems Secondary three/four :

YES/NO

Solutions of equations and inequalities: • solving quadratic equations in one unknown by: ∗ use of formula ∗ completing the square for ∗ graphical methods

• solving fractional equations that can be reduced to quadratic equations

Lesson 5: Linear inequalities in one unknown 1. Properties of Inequalities 2. Inequalities of one

-Giving some properties related to inequalities, solution to inequality using graph representation and some problems whose solution use


NA

Secondary one: Already included in linear equations in one unknown

YES/NO

unknown 3. System of inequalities in one known 4. Problem

inequality -Exercises 1-15, all at levels [1,2,3] but exercise is too critical Secondary one: Data handling: • data collection methods such as: ∗ taking measurements ∗ conducting surveys ∗ classifying data ∗ reading results of

Lesson 6: Frequencies 1. Frequencies in data collection 2. Cumulative frequencies 3. Grouped data

-Constructing data using table and interpreting in terms of bar graph, histogram and frequency polygon - Exercises 1-7, all at levels [1,2,3]

-Knowing how to represent data -Interpreting and analyzing data using graphs


observations/outcomes of events • construction and interpretation of: ∗ tables

YES/YES

∗ bar graphs ∗ pictograms ∗ line graphs ∗ pie charts ∗ histograms

• purposes and use, advantages and disadvantages of the different forms of statistical representations • drawing simple inference from statistical diagrams

Lesson 7: Statistical mean 1. Mean 2. Median 3. Mode

-Giving some approach to find mean, median and mode. -Exercises 1-7, all at levels [1,2,3]

Secondary two:

-Numerical calculations -Interpreting data

Usefulness and interest

Data analysis: • interpretation and analysis of:

YES/YES

∗ dot diagrams

4. Finding mean from cumulative frequencies

∗ stem-and-leaf diagrams

• mean, mode and median as averages • purposes and use of mean, mode and median • calculation of the mean for grouped data Secondary two: Probability:

Lesson 8: Probability 1. Probability of an event 2. Probability of a complementary event 3. Probability of multiple experiments

-Giving some examples of probability of elementary and complementary events -Exercises 1-16, all at levels [1,2,3] but 11 at level [4]

-Numerical calculations -Estimating chance of success

NA

• probability as a measure of chance • probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability) Secondary three/four: Probability: • probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate) • addition and multiplication of probabilities • mutually exclusive events and independent events

YES/YES

Lesson 9: Distance between two points 1. Distance between two points 2. Coordinates of middle point

-Giving an approach to find the distance between two points in the plane using Pythagoras’theorem, to find the coordinate of the midpoint. -Exercises 1-23, all at levels [1,2,3]


NA

NA

NO/NO

Secondary three/four:

Lesson 10: Equation of a straight line 1. Construction of a line 2. Equation of a straight line 3. Applications

-Constructing a straight line using a numerical table, determining directional coefficient of a line through two points, determining a line through two points, determining a line parallel or perpendicular to another line -Exercises 1-17, all at levels [1,2,3]

-Numerical calculations -Interpreting data using graph

NA

Coordinate geometry: • finding the gradient of a straight line given the coordinates of two points on it • finding the length of a line segment given the coordinates of its end points • interpreting and finding the equation of a straight line graph • geometric problems

YES/NO

involving the use of coordinates

Lesson 11: System of linear equations in two unknowns 1. Solving a system of linear equations using graph 2. Solving a system of equations using substitution method 3. Solving a system of

Secondary two: -Solving a system of linear of equations by two methods -Exercises 1-19, all at levels [1,2,3] but exercise 5 at level [4,5,6]


NA

• solving simultaneous linear equations in two unknowns by ∗ substitution and elimination methods ∗ graphical method • solving quadratic equations in one unknown by factorisation • formulating a pair of linear

YES/NO

equations in two unknowns or a quadratic equation in one unknown to solve problems

linear equations using elimination method 4. Problems of system of linear equation Lesson 12: Pythagoras’ Theorem 1. Pythagoras’ theorem 2. Applications of Pythagoras’ theorem

-Applying the Pythagoras’theorem in a right triangle -Exercises 1-10, all at levels [1,2,3] but exercise 9 at level [4]

Secondary two :


NA

• use of Pythagoras’ theorem • determining whether a triangle is right-angled given the lengths of three sides

NO/NO

Secondary three/four: Properties of circles

Lesson 13: Circles and lines 1. Properties of arc length 2. Tangent lines 3. Properties of tangent line 4. Circumference polygons 5. Positions of circles 6. Common tangent lines

-Describing properties of chords and tangents -Calculating arc length -Exercises 1-13, all at levels [1,2,3]


NA

• symmetry properties of circles: ∗ equal chords are equidistant from the centre ∗ the perpendicular bisector of a chord passes through the centre ∗ tangents from an external point are equal in length ∗ the line joining an external point to the centre of the circle bisects the angle between the tangents • angle properties of circles: ∗ angle in a semicircle is a right angle ∗ angle between tangent and radius of a circle is a right angle

NO/NO

∗ angle at the centre is

twice the angle at the circumference ∗ angles in the same segment are equal ∗ angles in opposite segments are supplementary

Lesson 14: Circular angles 1. Angles at center 2. Angles in circumference 3. Angles in semicirle 4. Angle in opposite segments of circumference polygon 5. Angle of a circumference polygon 6. Angle without vertices on a circle 7. Angle between tangent lines Lesson 15: Thales’Theorem 1. Parallel lines and secant 2. Thales’ Theorem 3. Applications in a triangle and polygon

-Describing properties of angles at center, in semicircle, in circumference -Calculating angles -Exercises 1-14, all at levels [1,2,3]


NA

-Proving Thales’ Theorem -Exercises 1-13, all at levels [1,2,3]


NA

Secondary three/four: Already included in Properties of circle

NA

NO/NO

NO/NO

Secondary two: congruence and similarity

Lesson 16: Similar triangles 1. Similar triangles 2. Proportional relations in a right triangle

-Describing similarity of triangles in three different cases and proving some relations in a right triangle -Exercises 1-25, all at levels [1,2,3]


NA

• congruent figures as figures that are identical in shape and size • matching sides and angles of two congruent polygons • similar figures as figures that have the same shape but different sizes • properties of similar polygons: ∗ corresponding angles are equal ∗ corresponding sides are proportional • enlargement and reduction of a plane figure by a scale factor • scale drawings • solving simple problems involving similarity and congruence

Secondary three/four: Congruence and similarity: • determining whether two triangles are ∗ congruent ∗ similar

• ratio of areas of similar

NO/NO

plane figures • ratio of volumes of similar solids

Secondary one: Angles, triangles and polygons

Lesson 17: Polygons 1. Definitions 2. Sum of internal angles of a polygon 3. Sum of external angles of a polygon 4. Measurement of angles in a regular polygon

-Describing properties of polygons and regular polygons -Exercises 1-17, all at levels [1,2,3] but exercise 15 at level [4]


NA

• right, acute, obtuse and reflex angles, complementary and supplementary angles, vertically opposite angles, adjacent angles on a straight line, adjacent angles at a point, interior and exterior angles • angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles • properties of triangles and special quadrilaterals • classifying special quadrilaterals on the basis of their properties • angle sum of interior and exterior angles of any convex polygon • properties of regular pentagon, hexagon, octagon and decagon • properties of perpendicular bisectors of

NO/NO

line segments and angle bisectors • construction of simple geometrical figures from given data (including perpendicular bisectors and angle bisectors) using compasses, ruler, set squares and protractors, where appropriate

Secondary one: Mensuration

Lesson 18: Solid 1. Area and volume of a solid 2. Ratio of areas and volumes of solids

-Describing regular solids and their properties -Exercises 1-23, all at levels [1,2,3]


NA

• area of parallelogram and trapezium • problems involving perimeter and area of composite plane figures (including triangle and circle) • volume and surface area of cube, cuboid, prism and cylinder • conversion between cm^2 and m^2 , and between cm^3 and m^3 • problems involving volume and surface area of composite solids

Conclusion for Grade-9 review The Grade-9 Mathematics textbook is written like an exercise book which can bring the student to the first levels and which makes them difficult to reach

NO/NO

Other higher levels 4, 5, 6 in sense of Bloom’s taxonomy (although there are some critical thinking problems given in the series of exercises). The substance in each lesson is not enough for the students to get formal concepts of Mathematics and thus the students could not have a strong base to analyze the problems and justify the answers of the given exercises with logical reasoning as well as to create, from the fundamental concepts, new things which are believed to be the main goal of learning (challenging) Mathematics. Some lessons in the text book exists in the Singaporean secondary two (Grade-8), some lessons (probability) are higher and full of serious mistakes and the other do not overlap. In general, the lessons in the Cambodian Grade-9 mathematics textbook provide less concepts and less applications compared with those of the Singaporean’s.

Cambodian Grade-12 Textbook (Advanced)

Chapter 1: Limits 1 Lesson 1: Limits of sequences

Chapter 2: Derivatives 1 Lesson 1: Application of derivatives

Chapter 3: Graphing 1 Lesson 1: Graph of irrational functions 2 Lesson 2: Graph of trigonometric functions

Knowledge - Calculating limit of geometric sequences -Calculating geometric series -Applying convergence of series to prove rationality and to calculate definite integral -Exercises 1-18, all at levels [1,2,3] but exercises 14, 16, 17at level [4] -Finding approximation values -Solving inequality problems -Finding approximation of extra costs, incomes and profits. -Exercises 1-16, all at levels [1,2,3] but exercises 8, 9 at level [4] Lesson 1: -Graphing irrational functions -Exercises 1-11, all at levels [1,2,3] but exercises 4-11 (similar approach) at level [4] Lesson 2: -Graphing trigonometric functions -Exercises 1-9, all at levels

Skills

Numerical skills

Attitudes

NA

Comparison with Singaporean pre-university(H2)

Sequences and series: 1 Summation of series 2 Arithmetic and

Bridge for higher education/Market demand

YES/NO

geometric series

-Numerical skills -Computing profit cost in business

Numerical skills


Calculus: 1 Differentiation 2 Maclaurin’s series

YES/NO

NA

Functions and graphs: 1 Functions, Inverse functions, Composite functions 2 Graphing and techniques 3 Equations and inequalities

YES/NO

Chapter 4: Definite integrals 1 Lesson 1: Definite integrals 2 Lesson 2: Volume of solids and arc length

[1,2,3] but exercises 6 and at level [4] Irrational and trigonometric functions: -Exercises 1-6, all at levels [1,2,3] but exercises 3, 5, 6 at level [4] Lesson 1: -Defining definite integral using sum of rectangle area -Calculating an area defined by a curve. -Calculating an area defined by two curves. -Exercises 1-7, all at levels [1,2,3] but exercises 5 at level [4] Lesson 2: -Calculating volumes of solids -Calculating a mean value of a continuous function in a close interval -Calculating an arc length of a curve -Exercises 1-9, all at levels [1,2,3] but exercises 4, 8, 9 at level [4] Lesson 1+ Lesson 2: -Exercises 1-15, all at levels [1,2,3] but exercises 2, 13, 14,

Numerical skills

NA

Calculus: 1 Integration technique 2 Definite integrals

YES/NO

15 at level [4]

Chapter 5: Differential equations 1 Lesson 1: First order differential equations 2 Lesson 2: Second order linear differential equations

Lesson 1: -Defining first order differential equation -Finding a solution of first order differential equation -Finding a solution of first order differential equation with initial condition -Graphing a function derived the solution. -Interpreting natural phenomena. -Exercises 1-11, all at levels [1,2,3] but exercises 6, 7 at level [4] Lesson 2: -Defining second order linear differential equation -Defining the characteristic equation of second order linear differential equation -Solving second order homogeneous linear differential equation with constant coefficient -Solving second order non-

Numerical skills

NA

Calculus: 1 Differential equations

YES/NO

homogeneous linear differential equation with constant coefficient -Interpreting natural phenomena and science facts

Chapter 6: Probability distributions 1 Lesson 1: Probability distributions 2 Lesson 2: Binomial distribution 3 Lesson 3: Normal distribution

-Exercises 1-10, all at levels [1,2,3] but exercises 6, 7 at level [4] Lesson 1+ Lesson 2: -Exercises 1-9, all at levels [1,2,3] but exercises 4, 5 at level [4] Lesson 1: -Defining random variable -Creating table of probability distribution -Calculating expectation and variance of a random variable -Exercises 1-10, all at levels [1,2,3] but exercises 8 at level [4] Lesson 2: -Calculating probability, expectation and variance of a binomial random variable -Exercises 1-9, all at levels [1,2,3] Lesson 3: -Knowing probability density

Numerical skills


Statistics: 1 Permutations and combinations 2 Probability 3 Binomial distributions 4 Poison distributions 5 Normal distribution 6 Sampling 7 Hypothesis testing 8 Correlation coefficients and linear regression

YES/YES

Chapter 7: Vectors in three dimensional space 1 Lesson 1: cross product 2 Lesson 2: Application of cross product

functions of a normal random variable -Solving problems related to expectation and variance of a normal random variable -Exercises 1-9, all at levels [1,2,3] but exercises 8 at level [4] Lesson 1+ Lesson 2+ Lesson 3: -Exercises 1-9, all at levels [1,2,3] but exercises 8, 9 at level [4] Lesson 1: -Determining vector product of two vector in the three dimensional space -Determining combination of a cross product and a scalar product -Applying in Physics -Exercises 1-9, all at levels [1,2,3] Lesson 2: -Determining equation of a line and a plane -Determining equation of a sphere -Determining distance from a point to a line -Determining distance from a

Numerical skills

NA

Vectors: 1 Vectors in two and three dimensions 2 The scalar and cross products

YES/NO

Chapter 8: Divisibility and Euclidean division 1 Lesson 1: Divisibility and Euclidean division 2 Lesson 2: Prime numbers 3 Lesson 3: common divisors and multiples

point to a plane -Exercises 1-14, all at levels [1,2,3] but exercises 12,13, 14 at level [4] Lesson 1+ Lesson 2: -Exercises 1-12, all at levels [1,2,3] but exercises 9, 10, 12 at level [4] Lesson 1: -Defining divisibility in Z -Defining Euclidean division -Exercises 1-17, all at levels [1,2,3] but exercises 15, 16 at level [4] Lesson 2: -Defining prime number -Giving some properties on prime numbers -Exercises 1-25, all at levels [1,2,3] but exercises 18, 24 at level [4] Lesson 3: -Finding the greatest common divisor of two numbers -Finding the least common multiple of two numbers -Exercises 1-13, all at levels [1,2,3] but exercises 12, 13 at level [4] Lesson 1+ Lesson 2 + Lesson 3:

Numerical skills

NA

NA

YES/NO

-Exercises 1-23, all at levels [1,2,3] but exercises 16, 22, 23 at level [4]

Chapter 9: Parameter equations and polar coordinates 1 Lesson 1: Parameter equations and polar coordinates

Lesson 1: -Exercises 1-9, all at levels [1,2,3]

Numerical skills

NA

Vectors: Three-dimensional geometry

YES/NO

Conclusion for Grade-12 review The Advanced Grade-12 Mathematics textbook is written better than that of the Grade-9 but it is still difficult to be accessible by the students since some parts are missing but only appearing in the fundamental Grade-12 Mathematics book. The substance in each lesson is still not enough for the students to get formal concepts of Mathematics and thus the students could not have a strong base to analyze the problems and justify the answers of the given exercises with logical reasoning as well as to create, from the fundamental concepts, new things which are believed to be the main goal of learning (challenging) Mathematics. We find that, in the Cambodian Advanced Grade-12 mathematics textbook, some lessons are similar to those of the Singaporean’s, some do not overlap and the other are less than that of the Singaporean’s, for example:  Chapter 1 only introduces the notion of geometric series while, in the Singaporean’s, some properties of series have been introduced and especially the Maclaurin’s series are used.  Chapter 2 only gives some simple graphing while, in the Singaporean, graphing technique like, coordinate transform and symmetry with respect to x and y are presented.

 Chapter 6 excludes, some counting techniques, which is essential in calculating probability, one probability distribution (Poison distribution), which is an approximation of binomial distribution, sampling and hypothesis testing, which are studied at the university level (Year 4 of RUPP) while those things are presented in the Singaporean’s.