Objectives

At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:

• The teaching of basic numeracy skills to all pupils. The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry) to most pupils, to equip them to follow a trade or craft. The teaching of abstract mathematical concepts (such as set and function) at an early age.
  
• The teaching of selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and a model of deductive reasoning
  
• The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world
  
• The teaching of advanced mathematics to those pupils who wish to follow a career in science
  
• The teaching of heuristics and other problem-solving strategies to solve non routine problems.
  Methods of teaching mathematics have varied in line with changing objectives.
  

Standards


Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to and realistic for their pupils.
In modern times there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England, while Scotland maintains its own educational system.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels. [2]
In North America, the National Council of Teachers of Mathematics (NCTM) has published the Principles and Standards for School Mathematics. In 2006, they released the Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are not nationally enforced in US schools.

Content and age levels

Different levels of mathematics are taught at different ages. Sometimes a class may be taught at an earlier age as a special or "honors" class. A rough guide to the ages at which the certain topics of arithmetic are taught in the United States is as follows:

• Addition: ages 5-7; more digits ages 8-9
  
• Subtraction: ages 5-7; more digits ages 8-9
  
• Multiplication: ages 7-8; more digits ages 9-10
  
• Division: age 8; more digits ages 9-10
  

The ages at which other math subjects (rational numbers, geometry, measurement, problem solving, logic, algebraic thinking, probability, statistics, reasoning skills and so on) are taught vary considerably from state to state.
Elementary mathematics in other countries is similar, though fractions (typically taught from 1st grade in the United States) are often taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.[3]

A typical pre-college sequence of mathematics courses in the United States would include some of the following, especially Geometry and Algebra I and II:

• Pre-algebra: ages 11-13 (Pre-Algebra taught in schools as early as 6th grade as an honor course. Algebraic reasoning can be taught in elementary school, though)
  
• Algebra I (basic algebra): ages 12+ (Algebra I is taught at 9th grade on average, or as early as 7th or 8th grade for an honors course)
  
• Geometry: ages 13+ (Geometry taught at 10th grade on average, or as early as 8th grade as an honors course)
  
• Algebra II: ages 14+; usually includes powers and roots, polynomials, quadratic functions, coordinate geometry, exponential and logarithmic functions, probability, matrices, and basic trigonometry
  
• Trigonometry or Algebra 3 or Pre-Calculus: ages 15+
  Statistics: ages 15+ (Probability and statistics topics are taught throughout the curriculum from early elementary grades, but may form a special course in high school.)
  
• Calculus: ages 16+ (usually seen in 12th grade, if at all; some honors students may see it earlier)
  

Mathematics in most other countries and in a few US states is integrated, with topics of algebra, geometry and analysis (pre-calculus and calculus) studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses à la carte as in North America. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16-17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions and infinite series their final year of high school