## France: Description of the Advanced Mathematics Programs and Curriculum

The Grade 11 and 12 scientific track offers robust mathematical knowledge and skills to students aiming for careers in science, technology, engineering, and mathematics (STEM). The mathematics curriculum is meant to develop students’ scientific thinking and strengthen their interest in and affinity for scientific research. Together with introducing new mathematical knowledge and content, the curriculum targets developing students’ skills and mathematical faculties in these areas:

- Implementing mathematical investigations and employing a variety of problem solving strategies
- Mastering a wide range of reasoning processes
- Interpreting and validating mathematical results
- Communicating mathematics both orally and in writing

Mathematical activities assigned to students both in class and for homework are focused on intra-mathematical or contextually diverse problem solving situations. Students are trained in:

- Searching for information, experimenting, and modeling, all using technology
- Choosing and executing calculation techniques
- Implementing algorithms
- Reasoning, proving, and validating results
- Explaining an answer, communicating a result

The mathematical content is organized in three parts: Analysis, Geometry, Probability and Statistics. About half of class time should be devoted to Analysis, one quarter to Geometry, and the last quarter to Probability and Statistics. The topics included in each content area are listed below.

Content Area | Topics |
---|---|

Analysis | Quadratic functions: solving quadratic equations, sign of a quadratic function
Sequences: arithmetic and geometric sequences, induction, finite or infinite limits, bounded sequences Function limits: finite or infinite limits, limits of a sum, product, quotient or composite functions, asymptotes Continuity on an interval, including the Intermediate Value Theorem Differentiation: calculating derivatives, including the derivatives of common functions, derivatives of sums, products, and quotients of functions, and applications of derivatives, including the relationship between the intervals over which a function increases or decreases and the value of its derivative on those intervals and function extrema Sine and cosine functions Exponential functions Natural logarithms Integration on an interval, including the relationship between the definite integral and the area under a curve, notation, the antiderivative of a function, linearity, and the additive property of definite integrals |

Geometry | Complex numbers, including the algebraic form, conjugate, geometric representation, and polar form of a complex number; the sum, product, and quotient of complex numbers, complex solutions to quadratic equations
Euclidean vectors, including the characterization of a line and a plane, scalar product, coordinates, equation of a plane Trigonometry, including trigonometric functions defined on the unit circle, radian, the sine and cosine of supplementary and complementary angles |

Probability and Statistics | Descriptive statistics, including variance, standard deviation
Conditional probability, independence Probability density functions, including discrete and continuous random variables, probability distributions (normal, Bernoulli, binomial, uniform, exponential), variance, standard deviation Confidence intervals Sampling, confidence interval for a proportion |

Starting in Grade 10, scientific track students continue to develop and implement algorithms. Students are trained to:

- Describe algorithms in natural or symbolic language
- Devise basic algorithms using spreadsheets, calculators or specific software programs
- Interpret complex algorithms

Algorithms fit naturally in all mathematical fields. Algorithmic problem solving in each content area is situated in contexts related to academic subject areas and contexts from real life. Students learn how to implement elementary instructions, loops, and conditional instructions as well as to implement validation and control steps in their programs.

Students learn how to use formal mathematical notation (e.g., for functions, derivatives, and integrals) as well as notation for number sets and intervals.

Students learn elements of formal logic, such as the logical operators for “and” and “or”; the concepts of the contrapositive, the converse and the negative of a conditional statement; logical equivalence; types of arguments, such as the counterexample, the logical disjunction, and the contrapositive; and proof by contradiction.