Children, especially the less gifted intellectually, need more experiences as a basis for and as applications of an arithmetical abstraction or concept than are usually given them. For example, in paving the way for the principle, "Any number times 0 equals 0," it is not safe to say, "John worked 8 days for 0 minutes per day. How many minutes did he work?" and "How much is 0 times 4 cents?" It will be much better to spend ten or fifteen minutes as follows:[14] "What does zero mean? (Not any. No.) How many feet are there in eight yards? In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt. 5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour when he works. How much does he receive when he works 3 hr.? 8 hr.? 6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he receive? How much is 0 times $600? How much is 0 times $5000? How much is 0 times a million dollars? 0 times any number equals....
232 (At the blackboard.) 0 time 232 equals what? 30 I write 0 under the 0.[15] 3 times 232 equals what? ---- 6960 Continue at the blackboard with
734 321 312 41 20 40 30 60 etc." --- --- --- --
[14] The more gifted children may be put to work using the principle after the first minute or two.
[15] 232 30 If desired this form may be used, with the appropriate --- difference in the form of the questions and statements. 000 696 ---- 6960
Pupils in the elementary school, except the most gifted, should not be expected to gain mastery over such concepts as _common fraction_, _decimal fraction_, _factor_, and _root_ quickly. They can learn a definition quickly and learn to use it in very easy cases, where even a vague and imperfect understanding of it will guide response correctly. But complete and exact understanding commonly requires them to take, not one intellectual step, but many; and mastery in use commonly comes only as a slow growth. For example, suppose that pupils are taught that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03, etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean 1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal fractions. They may then respond correctly when asked to write a decimal fraction, or to state which of these,--1/4, .4, 3/8, .07, .002, 5/6,--are common fractions and which are decimal fractions. They may be able, though by no means all of them will be, to write decimal fractions which equal 1/2 and 1/5, and the common fractions which equal .1 and .09. Most of them will not, however, be able to respond correctly to "Write a decimal mixed number"; or to state which of these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which are decimals; or to write the decimal fractions which equal 3/4 and 1/3.