It is of paramount importance that after new processes are taught and drilled upon in the first learning, systematic drill be pursued to maintain that skill which has been built up. Much of the loss of skill in the simplest operations is due to the fact that not enough opportunities have been provided for using them.
The second aspect of elementary algebra to which one will wish to refer is the use of the equation as the mode of expressing mathematical relations. It is the handy tool for solving equations. It has been called the “backbone of algebra.” To handle this tool skillfully, it is necessary to learn certain fundamental principles governing the formation of equations and changes made in them. The power to interpret the meaning of problems varies so greatly in individuals that a large part of the work in elementary algebra consists in trying to gain improvement in reading comprehension. The next step is to gain power to express the data of the problem in algebraic language. This is a most difficult phase of the work, for the correct solution of any problem depends on an accurate statement of the conditions of the problem and the expression of those conditions in correct mathematical language.
Transposition and clearing of fractions, two of the most fundamental processes in the solution of the equation, must be approached through simple problems where a need for these processes will be apparent. The terminology can be gradually introduced, but the fundamental principles that like changes must be made in both members of the equation to keep the balance must he adhered to. These processes are made very graphic by means of the balance or scale, and axioms and principles derived by the pupils themselves.
Generalizations is derived are of more permanent value than “cut and dried” facts handed out to pupils. If beginners in the study of algebra could f eel that they were investigators in a new field of activity, and felt the realization of accomplished aims, we would have more cooperation between teacher and pupil in working out desired ends and necessarily more satisfactory results.
One of the important concepts in arithmetic is place value in our number system. Many students reach high school without a complete understanding of this basic characteristic of our system of numeration. That their understanding of place value is incomplete is shown when pupils have unusual difficulty with digit problems in algebra. To understand digit problems and their solution, the student must have a clear understanding of place value.
Before introducing digit problems, a brief summary of the history of our number system is needed. In the discussion, one can point out the major developments of our number system from the earliest tracings of the Babylonians down to the Hindu Arabic system, in the perfected form which we use today. Information on the development of our decimal system can be found in any good history of mathematics. When students realize that the Egyptians, Romans, and other early scholars did not recognize the idea of positional value to express numbers, they usually have more respect for the inventors of our system of numeration.