As another instructive topic in the constitution of arithmetical abilities, we may take the case of the reasoning involved in understanding the manipulations of figures in two (or more)-place addition and subtraction, multiplication and division involving a two (or more)-place number, and the manipulations of decimals in all four operations. The psychology of these is of special interest and importance. For there are two opposite explanations possible here, leading to two opposite theories of teaching. The common explanation is that these methods of manipulation, if understood at all, are understood as deductions from the properties of our system of decimal notation. The other is that they are understood partly as inductions from the experience that they always give the right answer. The first explanation leads to the common preliminary deductive explanations of the textbooks. The other leads to explanations by verification; _e.g._, of addition by counting, of subtraction by addition, of multiplication by addition, of division by multiplication. Samples of these two sorts of explanation are given below. SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION MULTIPLICATION is the process of taking one number as many times as there are units in another number. The PRODUCT is the result of the multiplication. The MULTIPLICAND is the number to be taken. The MULTIPLIER is the number denoting how many times the multiplicand is to be taken. The multiplier and multiplicand are the FACTORS. Multiply 623 by 3 OPERATION _Multiplicand_ 623 _Multiplier_ 3 ---- _Product_ 1869 EXPLANATION.--For convenience we write the multiplier under the multiplicand, and begin with units to multiply. 3 times 3 units are 9 units. We write the nine units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869. SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION 1. The children of the third grade are to have a picnic. 32 are going. How many sandwiches will they need if each of the 32 children has four sandwiches? _Here is a quick way to find out_:-- 32 _Think "4 × 2," write 8 under the 2 in the ones column._ 4 _Think "4 × 3," write 12 under the 3 in the tens column._ -- 2. How many bananas will they need if each of the 32 children has two bananas? 32 × 2 or 2 × 32 will give the answer. 3. How many little cakes will they need if each child has three cakes? 32 × 3 or 3 × 32 will give the answer.