## Curriculum: Mathematics

The below is an *indication* of the topics covered; our program focuses heavily on the solving of word/non-standard problems, which be believe best develops analytical ability, and students are grouped based on competence level.

**PRE-PROGRAM YEARS – SUGGESTED CURRICULUM**

The Year 1 – Year 4 foundation years should develop competence in the areas of arithmetic, geometry and logic.

**Year 1 (age 5 or 6)**

**Year 1 (age 5 or 6)**

**Arithmetic**

– Numbers from 0 to 20: names, symbols, addition and subtraction without regrouping.

– Nomenclature and signs: Addition, subtraction, addend, sum, minuend, subtrahend, difference, equality, inequality, expression, value.

– Addition: commutativity.

– Compute value of expressions with addition and/or subtraction

– Addition and subtraction within 0 to 10

– Tens numbers to 100: names, addition, subtraction

– Arithmetic sequences

**Geometry**

– Points, lines, line segments: construction, length measurement, detection, counting.

– Square, circle, rectangle, rhombus. Detection and counting within composite shapes; changes by moving sides; assembly and disassembly.

– Units of length

**Logic/combinatorics/set theory
**

– Permutations and combinations without repetitions: simple cases with numbers, colors and shapes.

– Simple logic problems.

– True and false statements (first examples).

– Sets, attributes

**Year 2**

**Year 2**

**Year 3**

**Year 3**

**Year 4**

**Year 4**

**PROGRAM CURRICULUM**

**Year 5**

**Methods/logic**

– Learning mathematics – approaches.

– History of math – curiosities.

– Computer tools.

– Comparison expressions – equal; less than; greater than; more; less; at least; at most;

not; and; or; for all; there exists

– The concepts certain, possible and impossible

– The language of mathematics – interpreting and creating various texts

– Problem solving and results verification

– Arranging and ordering according to given criteria

– Choosing from a list and ordering elements

**Arithmetic, algebra**

– Natural numbers – concept, location on number line, comparison up to one million;

integers; fractions, decimals

– Larger numbers in the place value system

– Negative numbers – definition

– Fractions – two definitions

– Negative and absolute value of a number

– Digit value, place value

– Decimal system

– Binary system

– Conversion from decimal to binary system and vice versa

– Operations – written and oral; illustration on the number line

– Natural numbers – divisors and multiples

– Addition and subtraction using integers and positive fractions

– Multiplication and division by 10, 100, and 1000

– Positive fractions – multiplication/division by natural numbers

– Decimal numbers – multiplication and division by natural numbers

– The role of zero in multiplication and division

– Results of operations – verification

– Properties of operations – commutativity, associativity and distributivity

– Operation priority – the role of brackets

– Rounding and estimating

– Word problems that can be solved using linear equations and inequalities, graphical

solutions. Checking the solutions by text substitution

– Proportions, measurements and conversion between units

Relations, functions and sequences

– Number line and intervals – description

– Less than, greater than, at least, at most – diagram interpretation

– Determining locations in practical situations

– The Cartesian coordinate system

– Variable quantities – the relations between them

– Simple linear relations – graphs and tables, filling in missing elements on the basis of

known or recognized rules

– Sums, differences, products, quotients and how they change

– Sequences – forming sequences using rules and some given elements

**Geometry and measurements**

– Building solids – properties. Edges, faces, vertices

– Solids – grouping by given properties

– Cubes, cuboids – properties and nets

– Segments – parallel and perpendicular

– Convexity

– Solids – mutual loci

– Plane figures and polygons – the concepts and properties

– Triangles and quadrilaterals – basic properties

– Concept of distance. Finding points with specified properties

– Circle and sphere – concepts and natural occurrences

– Finding points that are equidistant from two given points

– Perpendicular bisector of a segment – sketching

– Perpendicular line passing through a given point of a line

– Triangle – construction from three sides

– Angle – concept, measurement and types

– Rectangle and square – perimeter, area

– Cuboids and cubes – surface area and volume in chosen units

– Standard measurements and conversions – length, area, volume

**Probability, statistics, measuring**

– Probability – games and experiments; data collection and arranging

– Measuring – length, volume, time, temperature

– Measuring data – systematization in tables and illustrations on graphs

– Prefixes

– Bar charts – construction

– Simple graphs – definition and interpretation

– Mean – calculation

**Year 6**

**Methods/logic**

– Logical value of statements – true and false statements

**Arithmetic, algebra**

– Rational numbers – concept and location on number line. Reciprocals

– Rational numbers – operations – multiplication and division by fractions and decimals

– Negative numbers – operations

– Operations – priority

– Fractions – rounding and estimating

– Fractions and decimals – conversion

– Divisibility – simple divisibility rules using the last digits or sum of digits

– Greatest common divisor, least common multiple

– The notion of powers with positive integer exponents

– Prime factorization

– Fractions – simplification

– Number of divisors

– Proportions – direct and inverse; their graphs

– The notion of percentage – working with percentages

– Word problems that can be solved using linear equations and inequalities, graphical

solutions. Preparation for algebraic solution methods

**Relations, functions and sequences**

– Variable quantities – relations between them

– Linear functions – graphs

– Sequences – examples and graphing

**Geometry and measurements**

– Figures and solids in 2D and 3D

– Simple transformations – examples. Paper folding

– The reflection about an axis. Reflection symmetry

– Triangles and their classification according to the sides and angles

– Properties of quadrilaterals and their types

– Concepts in the circle – radius, diameter, chord, secant, sector, segment of circle

– Angle bisection

– Rectangles – construction

– Line perpendicular to a given one and passing through a given point – construction

– Polygons – perimeter

– Solids and nets – surface area and volume of a cuboid

**Probability, statistics**

– Probability – games and experiments

– Collecting data – systematization and illustration. Pie charts

– Frequency of events

– Various properties of data – the most frequent element, outliers

**Year 7**

**Methods/logic**

– Rewording, verifying and negating simple statements containing for all and there

exists using real-world examples. Logical relations between notions and statements

– Examples of concrete sets – subset, complement, union, intersection

– Solving word problems

– Solving various combinatorial problems – permutations and combinations (with only

a few objects)

**Algebra**

– Rational numbers – operations

– Notion of powers with positive integer exponents. Identities of exponentiation –

particular examples. Scientific notation.

– Ratio, proportion, proportional division – relations of proportionality in practical

cases and in scientific tasks

– Percentages and interests – calculation.

– Prime factorization. The greatest common divisor and the least common multiple of

two numbers. Simple divisibility rules – divisibility by 3, 9, 8, 125 or 6

– Using letters as variables. Notion of algebraic expressions – monomials, polynomials

in one or more variables with integer coefficients. Homogeneous expressions

– Simple algebraic expressions – transformation and evaluation. Algebraic expressions –

elementary operations – multiplication of monomials and binomials

– Equations and inequalities – solving by using various re-arrangements

– Word problems – solutions

**Relations, functions, sequences**

– Two sets – correspondences – concrete examples. Unique correspondences and their

graphs in the Cartesian coordinate system

– Linear functions and their special cases – representations with or without tables.

Simple properties – intercept, slope, increase, decrease. Examples for non-linear

functions (e.g. the 1/x function)

– Linear equations in one unknown – graphical solution

– Sequences – examples. The arithmetic sequence

**Geometry**

– Units conversion – concrete practical problems using rational numbers

– Special lines in a triangle – perpendicular bisectors, angle bisectors, altitude, median.

The medial (or auxiliary) triangle

– Triangle – area

– Perimeter and area – parallelogram, trapezoid/trapezium, rhombus, kite, circle

– Angle pairs – corresponding angles, alternate angles, complementary and

supplementary angles

– Point reflection. Figures symmetrical about a point in the plane

– Regular polygons, number of diagonals

– Constructions of special angles. Construction of triangles – the basic cases

– Triangles – congruence

– Triangle angles – sum of interior and exterior

– Quadrilateral angles – sum of interior and exterior

– Polygon angles – sum of interior and exterior

– Nets of right prisms and cylinders – properties

**Probability, statistics**

– Probability experiments in the universal sample space – solving simple tasks

– Frequency and relative frequency – notions and properties

– Collecting and systematizing data. Representation of data – drawing graphs

**Year 8**

**Methods/logic**

– Oral and written expression of thoughts – problems, conjectures, relations, etc.

– Mathematical proof – preparations – making a conjecture, doing experimentation;

systematic trials, counterexamples, disproving a conjecture

– Famous unsolved problems. Curiosities from the history of mathematics

– Understanding word problems, problem solving, solution verification

– Arranging elements into sets, listing elements of a set – examples. Using known set

operations in exercises

– Simple combinatorial problems – solving using various methods – tree diagram, path

diagram, tables

– Combinatorial methods – systematization through examples

**Number theory and algebra**

– Rational numbers – finite and infinite decimals. Examples of non-rational numbers –

infinite non-repeating decimals

– Square root

– Natural numbers, integers and rationals. Real numbers

– Operations in the set of rational numbers. Estimation of results

– Systematization of identities of operations. Transformations of algebraic expressions,

formulae and physics formulae. Factorization in simple cases. Multiplication and

division of algebraic expressions – simple cases. Determination of the domain

– Linear equations and inequalities – domain, solution set

– Systematization of problem solving methods – solving word problems

**Relations, functions and sequences**

– Functions and their graphs in the coordinate system – square and square root

– Basic properties of functions – intercepts, increasing and decreasing functions,

symmetries, function value

– Linear transformations of functions.

– Loci in the coordinate system

– Graphical solution of equations and inequalities in one variable

– Sequences – the geometric sequence

**Geometry**

– Revision – triangles, quadrilaterals and regular polygons

– Revision of known solids. Introduction of the cone, pyramid and sphere

– Translation in the plane. The vector as a directed line segment. Sum and difference of

two vectors

– Central enlargements and contractions by given scale factors

– Revision of geometric transformations. Constructions

– The Pythagorean Theorem – applications in 2D and 3D

– Perimeter, area, surface area, volume

– Geometric computations

**Statistics and probability theory**

– Probability experiments in the sample space. A priori estimation of probability and

the suggestive concept of probability

– Analysis and interpretation of data – mode, median. Illustration of data – creation and

analysis of diagrams

**Year 9**

**Review**

– Mathematically correct definitions. Examples of ill-defined objects

– Set theory – basic concepts

– Proof methods (direct/indirect proofs, proof by induction) – review

– Basic algebra, equation solving techniques

– Functions – basic knowledge, graphing in the coordinate system

– Inequalities – different solving methods

– Basic geometric constructions, simple geometric loci

– Triangles and quadrilaterals: summary of what has been taught so far

– Number sets (natural numbers, integers, rational, irrational numbers, the notion of a

real number), basic operations, decimal fractions

– Non-repeating and repeating variations, permutations and combinations. Introducing

Pascal’s triangle

– Combinatorial geometry – basic problems of (number of intersections and areas on the

plane)

– The pigeonhole principle – simple application

Elements of logic

– Pigeonhole principle – use in solving problems

– Method of proof – using separation of cases (built into other subtopics)

**Number theory**

– Divisibility – concept. Composite and prime numbers. Divisors – the number and the

sum of them. Perfect numbers

– The fundamental theorem of number theory

– Number of primes and their distribution, twin primes. Finding prime numbers. The

Euclidean algorithm

– Different number systems, the importance of the binary system. Divisibility rules.

Basic arithmetic operations in other bases

– Identities – reviewing of already covered ones

– Algebraic fractions – operations

– Irrational and real numbers – review of concepts. Examples for irrational numbers.

Proving that if a positive number is not a complete square, then its square root is

irrational

– Irrational numbers – decimal form. Real numbers and the number line

– Identities of square roots – reviewing and proving

– Nth root and its identities

– Rational exponent, identities of exponentiation. Illustration of the concept of irrational

exponents

– Extending exponentiation to irrational exponents – the principle of permanency.

Monotonicity requirements

**Algebraic expressions**

– Review of already learned basic identities: e.g. sums/differences of nth powers

– Simple algebraic fractions – operations

Equations, inequalities

– Linear equations with parameters, including solving practical problems

– Equations containing algebraic fractions, solving inequalities

– Equivalence of equations, spurious roots

– Equations containing absolute values

– Systems of linear equations in several unknowns – introducing a new variable

– Quadratic equations – solving by factorization. The discriminant and the quadratic

formula

– Quadratic equations – problems and applications that can be solved by them

– Equation solutions and systems that can be reduced to quadratic equations. Practical

applications

– Solving equations of higher degree by reducing to quadratic ones

– Solving practical and physical extremal problems by using quadratic equations

– Viete’s formula. Roots and coefficients of the quadratic equation – relationship.

– Quadratic inequalities – solving

– The concept of geometric mean, comparing the geometric and arithmetic mean of n

positive numbers

Functions, including radical functions and exponential functions

– Functions – concept and basic properties – review

– Basic functions – graphs and properties (linear, quadratic functions, the absolute

value, the integer and fractional parts, fractional linear functions)

– Monotonicity, extrema and boundedness

– Composite functions

– Equations and inequalities – solving using graphs

– System of equations in two unknowns – graphical solution

– Linear transformation of functions and their graphs

**Shapes, constructions**

– Geometry – basic concepts, the concept of axiom and theorem

– Triangles – lines, points and circles, including excircles

– Quadrilaterals – classification, relations among special quadrilaterals

– Euclidean construction. Construction – different steps.

– Non-Euclidean constructions, Bolyai-Lobachevsky geometry

Geometric transformations

– Congruence transformations

– Reflection about an axis and about a point, rotation and translation

– Transformations – composition

– Isometries as composition of reflections

– Isometries – application in geometric constructions and proofs

– Vectors. Addition and subtraction. Multiplying vectors with numbers

– Theorem on parallel lines and similar triangles. The converse of the theorem

– Central similarity – concept and properties. Constructions

– Similarity transformation, similarity of figures. Similarity of triangles – basic cases

Figures, geometric measures

– General angle (when the measure of the angle is any real number). Degrees and

radians. The length of an arc. The area of a sector

– Inscribed and central angles

– Cyclic and tangential quadrilaterals – theorem

**Combinatorics**

– Non-repeating and repeating variations, permutations, combinations

– Construction of Pascal’s triangle, binomial coefficients – properties and symmetry,

subsets of a finite set – counting their number, finding a general formula

– Combinatorial problems involving lottery and pools. Finding the number of integers

with a given set of digits

– Using induction as a method of proof

– Combinatorial problems on the plane

– Sums of squares and cubes

– The binomial theorem

– Further combinatorial problems, repeating combinations, the concept of one-to-one

mapping for counting the elements of finite sets

**Statistics and probability theory**

– Collecting and systematizing data. Histogram, median, mode, average and variance

– The algebra of events, with simple proofs

– Relative frequency

– Probability as a measure

– Computing probability using combinatorial methods. Examples

**Year 10**

**Elements of logic**

– Further applications of the pigeonhole principle

– Indirect and other types of proofs. Induction – some more applications

– Logical value. Propositions. The language of logic

– The logical “and”, “or”, “not”, “implication” and “equivalence” as operations

– Statements and their converses. Examples and counterexamples

– Necessary and/or sufficient conditions. Examples from number theory, geometry and

combinatorics

– Some unsolved problems of mathematics

**Combinatorics**

– Permutations, variations and combinations (repeating and non-repeating)

– The binomial theorem and binomial coefficients. Basic properties

– Graph theory – basic concepts (vertex, edge, degree, connected graphs, complement

graph, trees, cycles)

– Special graphs (complete graphs, subgraphs, trails, Hamiltonian circuits)

– Planar graphs, Euler’s theorem on convex polyhedra

**Algebra, basic operations, number sets**

– Exponentiation, the definition of rational exponents – basic identities

– Laws of indices, inverse operations

– Logarithm and its identities

– Calculators. Approximate computations. The world before calculators

– Mathematica/computer algebraic systems.

– Polynomials. Polynomials with integer coefficients. Relations between roots and

coefficients. The Horner scheme

– Integral and rational roots of polynomials with integer coefficients

**Equations, inequalities**

– Equations containing fractions and logarithms

– Trigonometric equations and inequalities

– Higher order polynomials – factorization

**Logarithmic and trigonometric functions**

– Inverse functions

– Exponential and logarithmic functions are the inverses of each other

– The logarithmic function – elementary transformations

– Exponential and logarithmic inequalities – graphical solution

**Shapes, constructions and measure in geometry**

– Unit vectors in the unit circle. The sine and cosine, tangent and cotangent functions.

Basic properties (periodicity, roots, local extrema, parity, monotonicity and

boundedness). Graphs and their linear transformations. The Pythagorean theorem in

trigonometry

– Basic trigonometric equations and their solution by definition or by graphs. Checking

infinitely many solutions

– Similarity transformations – review

– Vectors. Sums and differences. Multiplication by a constant

– Applications of similarity transformations: medians, centroid and bisectors of the

triangles. Proportions in triangles

– Euler-line and the nine point circle

– Ratio of areas of similar plane figures and ratio of volumes of similar solids

– The power of a point with respect to a circle – basic properties

– The decomposition of vectors in 2D and 3D. Uniqueness. Basis vectors and

coordinates

– The Pythagorean theorem in 2D and 3D and its converse

– Trigonometric functions in the right-angled triangle

– Relations among trigonometric functions in the right-angled triangle

– trigonometric functions – practical applications

– Orthogonal projection of a line segment – length. Computations with vectors

– Distance and angle in 3D

– The cube, cuboid, parallelepiped, tetrahedron, pyramid, prism

– Polyhedra and their regular counterparts

– The sphere

– Geometric loci in the space

**Statistics and probability theory**

– The proof of Chebyshev’s inequality.

– Discrete distributions: binomial, negative binomial, Bernoulli, Poisson, geometric and

hypergeometric. Continuous distributions: uniform, exponential, normal.

– Some applications of the geometric probability

– Central limit theorem

**Year 11**

**Sets, elements of mathematical logic**

– Set theory, properties of set operations. Boolean algebra

– Equivalence of finite and infinite sets, cardinality of sets

– Cardinality of infinite subsets of an infinite set

– Cardinality of the sets of rational and real numbers

– Paradoxes in math. Naïve set theory. Russell’s paradox

– Comparison of Boolean algebra and the number field

– Conjunction, disjunction, implication and equivalence

– The inclusion-exclusion principle

– Necessary and/or sufficient conditions

– Universal and existential quantifiers

**Combinatorics and graphs**

– Proofs of Euler’s polyhedron theorem

– Colouring problems. The five colour theorem. The four colour theorem

– Different types of sorting (bubble sort, comb sort)

**Equations, inequalities, identities**

– Exponential, logarithmic, trigonometric equations that can be reduced to linear and

quadratic equations

– Cubic equations and Cardano’s formula

– Trigonometric identities. Addition formulae

**Linear algebra**

– Unary and binary operations and their properties – summary of those covered

– Matrix, square matrix, unit matrix, zero matrix, inverse matrix

– Matrix and vector operations

– The determinant and its properties

– Linear programming problems

– Rotation about the origin – finding the matrix

– Solving linear systems of equations with Gaussian elimination

**Sequences**

– Arithmetic and geometric sequences. Formula for the general term and for the sum of

terms

– Fibonacci sequence, recursive sequences

– Comparing arithmetic, geometric, harmonic and power means

– Boundedness and monotonicity of sequences

– Limit

– Properties of convergent sequences. Methods for calculating limits

– Convergence of sequences

– The number e

– The infinite geometric series

**Calculus**

– Continuous functions – continuity and properties

– Limits of functions (finite and infinite, left and right), properties. Methods for

computing limits. Limit theorems

– Differentiability and the derivative. Basic differentiation rules (algebraic operations,

composition and inverse functions)

– Derivatives of elementary functions

– Convexity and second derivatives

– Sketching graphs of functions using derivatives. Local and global extrema,

monotonicity and convexity

**Vectors and trigonometry**

– Different coordinate systems. Cartesian and polar coordinates. Vectors in 3D

– The scalar and vector products of vectors. Properties and determination

– The law of sines and cosines

– Practical applications

**Complex numbers**

– The complex plane. Basic arithmetic operations. De Moivre’s theorem

– Roots of complex numbers. Roots of unity

– Solvability of higher order algebraic equations. Irreducible polynomials

– Geometric properties of complex numbers. Transformations

**Coordinate geometry**

– Different equations of a line in 2D and 3D. Orthogonal and parallel lines

– Equation of a plane

– Distance of two points. Different equations of the circle

– Conic sections – definition, properties and examples. Tangents

– Sections of cylinders and cones.

– Applications

**Statistics and probability theory**

– Random variables, expected value

– Conditional probability. The total probability law

**Year 12**

**Calculus**

– Indefinite integrals and basic properties

– Definite integrals. Lower and upper approximations

– The Newton-Leibniz formula

– Approximate integration

– Some simple improper integrals

**Spatial geometry**

– Parallelepiped, pyramid, prism, frustum of a pyramid, the regular solids, cylinder,

cone, frustum of a cone, sphere, ellipsoid, hyperboloid, solids of revolution

– Using vectors and coordinates to solve geometric problems

**Geometry involving calculus**

– The notion of area and its properties. Determining the area of basic geometric figures

– The definite integral as a general tool for finding areas

– Volume and its properties. Determining the area of basic solids

– The volume of regular solids

**Groups**

– The axioms of a group {G,∗}.

– Abelian groups.

– The groups:

– R, Q, Z and C under addition

– matrices of the same order under addition

– 2×2 invertible matrices under multiplication

– integers under addition modulo n

– groups of transformations

– symmetries of an equilateral triangle, rectangle and square

– invertible functions under composition of functions

– permutations under composition of permutations.

– Finite and infinite groups. The order of a group element and the order of a group.

– Cyclic groups. Proof that all cyclic groups are Abelian.

– Subgroups, proper subgroups. Use and proof of subgroup tests.

– Lagrange’s theorem. Use and proof of the result that the order of a finite group is

divisible by the order of any element. (Corollary to Lagrange’s theorem.)

– Isomorphism of groups. Proof of isomorphism properties for identities and inverses.

**Statistics and probability theory**

– Expected value and its determination

– Trials with infinitely many outcomes

– Basic concepts of polls and statistics

– Samples and relative frequency

– Confidence intervals

– Significance testing for a mean. Significance testing for a proportion. Null and

alternative hypotheses H0 and H1. Type I and Type II errors. Significance levels;

critical region, critical values, p-values; one-tailed and two-tailed tests.

– The chi-squared distribution: degrees of freedom, υ. The χ 2 statistic. The χ 2

goodness of fit test. Contingency tables: the χ 2 test for the independence of two

variables.

**Discrete mathematics**

– Linear diophantine equations ax+by=c .

– Modular arithmetic. Linear congruences. Chinese remainder theorem.

– Fermat’s little theorem.

– Graphs, vertices, edges. Adjacent vertices, adjacent edges. Simple graphs; connected

graphs; complete graphs; bipartite graphs; planar graphs, trees, weighted graphs.

Subgraphs; complements of graphs. Graph isomorphism.

– Walks, trails, paths, circuits, cycles. Hamiltonian paths and cycles; Eulerian trails and

circuits.

– Adjacency matrix. Cost adjacency matrix.

– Graph algorithms: Prim’s; Kruskal’s; Dijkstra’s.

– “Chinese postman” problem (“route inspection”). “Travelling salesman” problem.

Algorithms for determining upper and lower bounds of the travelling salesman

problem.

**Year 13**

**Series**

– Convergence of infinite series. Partial fractions and telescoping series (method of

differences). Tests for convergence: comparison test; limit comparison test; ratio test;

integral test. The p-series. Use of integrals to estimate sums of series.

– Series that converge absolutely. Series that converge conditionally. Alternating series.

– Power series: radius of convergence and interval of convergence. Determination of

the radius of convergence by the ratio test.

– Taylor polynomials and series, including the error term. Maclaurin series for ex , sin x,

cos x , arctan x , ln(1+ x), (1+x)p . Use of substitution to obtain other series. The

evaluation of limits.

– using l’Hôpital’s Rule and/or the Taylor series.

**Ordinary differential equations**

– First order differential equations: geometric interpretation using slope fields;

numerical solution using Euler’s method. Homogeneous differential equation using

the substitution y=vx. Solution of y′+P(x)y=Q(x) , using the integrating factor.

– Second order linear equations – homogeneous and non-homogeneous equations.

Solution techniques – variation of parameters and the method of undetermined

coefficients

**General topology and metric spaces**

– Definition of a topological space

– Open and closed sets

– Interior, exterior, boundary, closure

– Neighbourhoods

– Compactness, path-connectedness

– Dense sets. The set of rationals is dense in the set of reals

– Convergence of sequences

– Continuity of functions

– Limit of a function

– Definition of a metric space

– Different metrics, examples

– Metric spaces as topological spaces

– Diameter of a set, boundedness

– Cauchy sequences

– Completeness – The rational numbers as a metric space is not complete in the set of

reals.

– The Cantor set

**Computer programs – structure and interpretation**

– Building abstractions with procedures. The elements of programming. Procedures

and the processes they generate. Formulating abstractions with higher-order

procedures.

– Building abstractions with data. Introduction to data Abstraction. Hierarchical data

and the closure property. Symbolic Data. Multiple representations for abstract data.

Systems with generic operations.

– Modularity, objects and state. Assignment and local state. The environmental model

of evaluation. Modeling with mutable Data. Concurrency: Time is of the essence.

Streams.

– Metalinguistic abstraction. The metaciricular evaluator. Variations on a scheme –

lazy evaluation. Variations on a scheme – nondeterministic computing. Logic

programming.

– Computing with Register Machines. Designing register machines . A register

machine simulator. Storage allocation and garbage collection. The explicit control

evaluator.

I am happy to see the syllabus of mathematics. Training of mind is important and there is in the syllabus a place for the logic of mathematics. Congratulations to the teachers and directors.

Mathematics has gone a long way. Our efforts should be to gear the study of Mathematics to life experience.

Keep it up!

Dr.Ivo da C.Souza

Comment by Ivo da C.Souza | 2008/08/08