Here is a link to an old encyclopedia called The Cyclopedia of Education. Although it was published over 100 years ago, there are a number of articles worth reading — they offer a different perspective from some of the modern encyclopedias. In particular, read the articles “Arithmetic” and “Arithmetic, Hygiene of.”
Below is a quote from the article “Arithmetic, Hygiene of”:
ARITHMETIC, HYGIENE OF. — The hygiene of instruction in its positive aim of developing habits of healthful activity scrutinizes all the processes of learning and all methods of instruction from the point of view of healthful development. Even in a subject like arithmetic, which represents par excellence an intellectual branch of instruction, the demands of hygiene are important.
In the first place arithmetic is of ancient origin, and from its history an undue importance has often been assigned to it. A feeling of the transcendent importance of this subject has prevailed in modern times even among everyday people as well as among philosophers. In the minds of many both instinctive and practical interests are associated with it. Hence it has come to pass that in the history of the schools an undue amount of time and attention have been devoted to the subject; and a considerable ballast of unessential or extraneous material has crept into the textbooks. To-day in many urban schools probably from one sixth to one fifth of the total time in the classroom is devoted to this subject, and in the rural schools probably often considerably more. Besides this, pupils are very apt to spend considerable time in home study in arithmetic. Recently voices have been raised against exaggerated views of its value. According to Sir William Hamilton, mathematics is not useful for training the powers of observation, nor for cultivating the reasoning power; so that even the traditional significance of the subject is questioned by competent authorities. As an instrument of mental culture mathematics can pretend to but a single benefit. This study “if pursued in moderation, and if efficiently counteracted, may be beneficial in the correction of a certain vice and in the formation of its corresponding virtue. The vice is the habit of mental distraction; the virtue, the habit of continuous attention.” This beneficial effect of mathematics in training attention is recognized by hygiene and strongly commended. The exaggerated ideas of the efficacy of arithmetic in the cultivation of the mind and the resulting over-pressure and premature training are strongly condemned by hygiene. With the many subjects that always crowd the curriculum, the question whether too much time is not spent on arithmetic and whether improper material is not often presented, although primarily pedagogical, becomes hygienic also.
Again, certain individuals seem to have little ability for work in mathematics, and others seem to be in special danger of nervous overstrain from work in this subject. An English physician, Dr. Sturgis, has studied chorea in children, and many of these cases he has found due, as he thinks, to causes connected with the school work, and arithmetic he deems an especial factor in producing the disorder. In case of a nervous child he maintains that working sums is liable to cause chorea. In the case of some children, as pointed out by General Walker, work in arithmetic is a frequent cause of worry and interference with sleep. When children do sums in their dreams, this is a danger signal. All such obvious causes of injury to health are condemned by hygiene, but it demands special attention to some less obvious but more general results of certain methods of instruction.
Pedagogy is concerned with the direct results of instruction. But besides the primary results of instruction in any subject, there are, as has been pointed out by Dr. Baade, certain secondary effects of instruction, certain by-products, to use the language of industry, which are often of great importance. In arithmetic there is a good opportunity to study the latter. Certain habits of interference of association, certain arrests, as they have been called by Dr. Triplett, illustrate very well these secondary effects of certain methods and processes of learning.
Number forms sometimes illustrate the secondary effects of instruction. Such habits represent not only so much mental ballast, but usually also interference of association and often the germs of pathological neuroses. They are probably pretty common. The counting habit, arithmomania, so-called, is likely to have several representatives in each class, according to Triplett’s investigations. This is a real handicap, filling the mind with quantitative ideas to the exclusion of causal relations.
Hygiene is especially concerned with the problem of the age when work in arithmetic should be begun. In order to answer this question it is necessary to consider briefly the mental operations involved in arithmetical work. In the simpler study of number and number relations, in addition, subtraction, and the rest, the process of learning is chiefly one of acquiring habitual associations. What hygiene demands here is that these should be formed naturally and that interference of association or mental confusion shall be avoided.
Again, in teaching arithmetic, to very young children all sorts of objective methods and devices have been developed, and these are deemed necessary in such instruction. Still further, it appears that the number forms and the like which are common in adults are developed in the early years of instruction. From these are likely to develop artificial and grotesque habits of thought, as illustrated by Dr. Triplett’s so-called arrests and by some of the number forms.
The problem of the proper age for beginning arithmetic is then something like this. At what age can a child be drilled in arithmetical processes without the aid of artificial devices and the like which are likely to persist as arrests or habits of interference of association; and at what age should the study of logic be begun; at what age does the child have a nascent interest for arithmetical work? We have at present no adequate data for answering these questions, but until further investigations have been made the verdict of hygiene is that ordinarily formal instruction in arithmetic should be postponed until at least the age of 8 or 10. The Italian physiologist, Mosso, President G. Stanley Half, Professor Patrick, and others agree in condemning formal instruction in this subject before this age. “Mathematics in every form,” writes Professor Patrick, “is a subject conspicuously ill-fitted to the child mind. It deals not with real things, but with abstractions. When referred to concrete objects, it concerns not the objects themselves, but their relations to each other. It involves comparison, analysis, abstraction. . . . The grotesque number forms which so many children have, and which originate in this period, are evidence of the necessity which the child feels of giving some kind of bodily shape to these abstractions which he is compelled to study.”
The practical teachings of the hygiene of instruction as regards arithmetic may be summed up in the light of our present knowledge somewhat as follows: the formal instruction in this subject should not be begun before the age of 8 or 10. Arithmetical work before this should be spontaneous activity on the part of the child. By postponing arithmetic until this age, it is possible to do away for the most part with artificial devices and methods which may lead to arrests or interference of association later on. The work in arithmetic should be simple, and the complex examples in logic and the like should be eliminated. In the case of nervous children special care should be taken to avoid worry and the development of neuroses like chorea. And, in general, special attention should be given in this subject to the secondary effects which are important from the point of view of mental hygiene. W. H. B.
Baade, Walter. Experimentelle und krilieche BeitrOge zur Frage nach den sekundaren Wirkungen da Unterrichit insbesondere auf dieEmpfanglichkeit des Schalere. (Leipzig, 1907.)
Bailey, M. A. The Teaching of Arithmetic in Elementary Schools. Proc. of JV. E. A., Asbury Park, N. J.. July, 1905, pp. 380-387.
Browne, Charles E. The Psychology of the Simple Arithmetical Processes: A study of Certain Habits of Attention and Association. Amer. Jour, ofPsych., Jan., 1906, Vol. 17, No. 1, pp. 1-37.
Saffohd, T. H. Modern Teaching of Arithmetic. Atlantic Monthly, May, 1891, Vol. LXVII, pp. 668-675.
Triplett, Norman. Pedagogical Arrests and Peculiarities. Ped. Sent., June, 1905, Vol. 12, pp. 141-157.
Walker, F. A. Arithmetic in the Primary and Grammar Schools. In his Discussions in Education, pp. 209-232. (New York, 1899.)