Conventional Methods In Teaching Math

New Mathematics, also new math, name given to a variety of topics in mathematics that have recently been introduced in many primary and secondary schools throughout the United States and in other countries. The term is usually applied to subject matter that is new for those grades, but it also refers to a shift in pedagogic stress and intent. The changes of new math reflect university studies begun in the early 1950s in reaction to scientific and technological advances and to the greatly increased use of mathematics in the physical, biological, and social sciences as well as in industry and commerce. Because of these studies, both the mathematics curriculum and teaching methods were revised so that the average citizen could learn more mathematics sooner.
The main change was a shift from rote memorization (such as of the multiplication table) to emphasis on meaning and concept, in order to give pupils insight into what they are doing and why. The student is taught, for example, why 3 + 5 is the same as 5 + 3, the reasons for the various steps in long division, and why it is appropriate to multiply in order to find the total cost of a dozen rolls that cost nine cents apiece. When asked, “What number must be put into the box to make 2 + □ = 7 a true mathematical sentence?” the pupil is introduced to subtraction and simultaneously given a taste of algebra.
Many educators believe that a child who studies the new math under competent teachers will not only be able to do routine arithmetic as well as a child trained in the traditional manner, but will also be far ahead in understanding and in preparation for advanced mathematics.
The new math is new only in that the material is introduced at a much lower level than heretofore. Thus geometry, which was and is commonly taught in the second year of high school, is now frequently introduced, in an elementary fashion, in the fourth grade—in fact, naming and recognition of the common geometric figures, the circle and the square, occur in kindergarten. At an early stage, numbers are identified with points on a line, and the identification is used to introduce, much earlier than in the traditional curriculum, negative numbers and the arithmetic processes involving them.
The elements of set theory constitute the most basic and perhaps the most important topic of the new math. Even a kindergarten child can understand, without formal definition, the meaning of a set of red blocks, the set of fingers on the left hand, and the set of the child’s ears and eyes. The technical word set is merely a synonym for many common words that designate an aggregate of elements. The child can understand that the set of fingers on the left hand and the set on the right hand match—that is, the elements, fingers, can be put into a one-to-one correspondence. The set of fingers on the left hand and the set of the child’s ears and eyes do not match. Some concepts that are developed by this method are counting, equality of number, more than, and less than. The ideas of union and intersection of sets and the complement of a set can be similarly developed without formal definition in the early grades. The principles and formalism of set theory are extended as the child advances; upon graduation from high school, the student’s knowledge is quite comprehensive. The amount of new math and the particular topics taught vary from school to school. In addition to set theory and intuitive geometry, the material is usually chosen from the following topics: a development of the number systems, including methods of numeration, binary and other bases of notation, and modular arithmetic; measurement, with attention to accuracy and precision, and error study; studies of algebraic systems, including linear algebra, modern algebra, vectors, and matrices, with an axiomatic as well as traditional approach (see Matrix Theory and Linear Algebra; Vector); logic, including truth tables, the nature of proof, Venn or Euler diagrams, relations, functions, and general axiomatics; probability and statistics; linear programming; computer programming and language; and analytic geometry and calculus. Some schools present differential equations, topology, and real and complex analysis.