(8) _What armament of satisfiers and annoyers, of positive and negative interests and motives, stands ready for use in the formation of the intrinsically uninteresting connections between black marks and meanings, numerical exercises and their answers, words and their spelling, and the like?_ School practice has tried, more or less at random, incentives and deterrents from quasi-physical pain to the most sentimental fondling, from sheer cajolery to philosophical argument, from appeals to assumed savage and primitive traits to appeals to the interest in automobiles, flying-machines, and wireless telegraphy. Can not psychology give some rules for guidance, or at least limit experimentation to its more hopeful fields? (9) _The general conditions of efficient learning are described in manuals of educational psychology. How do these apply in the case of each task of the elementary school?_ For example, the arrangement of school drills in addition and in short division in the form of practice experiments has been found very effective in producing interest in the work and in improvement at it. In what other arithmetical functions may we expect the same? (10) _Beside the general principles concerning the nature and causation of individual differences, there must obviously be, in existence or obtainable as a possible result of proper investigation, a great fund of knowledge of special differences relevant to the learning of reading, spelling, geography, arithmetic, and the like. What are the facts as far as known? What are the means of learning more of them?_ Courtis finds that a child may be specially strong in addition and yet be specially weak in subtraction in comparison with others of his age and grade. It even seems that such subtle and intricate tendencies are inherited. How far is such specialization the rule? Is it, for example, the case that a child may have a special gift for spelling certain sorts of words, for drawing faces rather than flowers, for learning ancient history rather than modern? * * * * * Such are our problems: this volume discusses them in the case of arithmetic. The student who wishes to relate the discussion to the general pedagogy of arithmetic may profitably read, in connection with this volume: The Teaching of Elementary Mathematics, by D. E. Smith ['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic, by the author ['21]. THE PSYCHOLOGY OF ARITHMETIC CHAPTER I THE NATURE OF ARITHMETICAL ABILITIES