# Mathematics Instruction

Mathematics Instruction is the practice of teaching and learning mathematics, along with associated scholarly research. Mathematics instruction attempts to achieve a variety of different objectives, including:

- Teaching basic numeracy skills to all pupils.

- Teaching practical mathematics e.g. arithmetic, elementary algebra, plane and solid geometry, and trigonometry to most students.

- Teaching abstract mathematical concepts at an early age.

- Teaching selected areas of mathematics e.g. Euclidean geometry as an example of an axiomatic system and a model of deductive reasoning.

- Teaching selected areas of mathematics e.g. calculus as an example of the intellectual achievements of the modern world.

- Teaching advanced mathematics to those pupils who wish to follow a career in Science, Technology, Engineering, and Mathematics (STEM) fields.

- Teaching heuristics and other problem-solving strategies to solve non-routine problems.

Methods of teaching mathematics have varied in line with changing objectives and are largely determined by what the relevant educational system is trying to achieve. Methods of instruction include:

- Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. This approach begins with arithmetic and is followed by Euclidean geometry and elementary algebra.

- Classical education: the teaching of mathematics within the quadrivium, part of the classical education curriculum of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm of deductive reasoning.

- Rote learning: the teaching of mathematical results, definitions and concepts by repetition and memorization typically without meaning or supported by mathematical reasoning. Rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics.

- Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving quadratic equations.

- Problem solving: the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students open-ended and sometimes unsolved problems. Problem solving is used as a means to enhance mathematical knowledge.

- New Math: a method of teaching mathematics which focuses on abstract concepts such as set theory, functions and bases other than ten.

- Historical method: teaching the development of mathematics within a historical, social and cultural context. This method provides greater human interest than the conventional approach.

- Standards-based mathematics: a vision for pre-college mathematics education in the US and Canada, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.

There was very little research in the years prior to 1960 into how students learn mathematics and how teachers teach it. The teacher's job had previously been a simple matter of telling students what they were expected to learn. As research in mathematics education increased and matured, questions arose about how students learned the subject. During the 1960s and 1970s educators began to see great value in studying the teaching and learning of mathematics. This research created an interest in developing a psychological basis for understanding how and why some students improved but others did not, and what kind of teaching methods and curricula could contribute to or stifle student learning. Contemporary teacher training focuses on enabling teachers to create and use such questions so that they can analyze their students' understanding of mathematics.

In order to teach mathematics at elementary level, undergraduate students take two math courses that are either part of the institution's core program or are specifically designed for elementary teaching majors. It is also likely that prospective elementary teachers will be expected to complete one or two courses that deal specifically with the teaching of elementary school math. Concerns that teachers at elementary level need more background in the subject have resulted in recent trends toward upgrading the mathematical education of prospective elementary teachers.

Teachers training for a career at high school level are typically expected to have completed a major in mathematics, or in some cases, a closely related field with an additional mathematics course alongside. Smaller programs are more likely to offer only a single course in mathematics education. Training for a middle school teacher varies from institution to institution. In some schools, middle school teaching follows a program similar to elementary level training, but with extra courses in math. In others, they take a program for secondary school pre-service teachers specializing in mathematics, with one or more additional courses in middle school education. Larger universities provide programs specifically designed for prospective middle school math teachers.