CHAPTER III

HOW ARITHMETIC HAS DEveloped

Reasons for studying the subject- The historical development of the reasons for teaching arithmetic has already been considered. For the well-informed teacher there remain two other historical questions of importance. The first relates to the development of the subject itself, and the second to the methods of teaching it.

There are good and sufficient reasons for considering briefly the history of arithmetic. In the first place, the child learns somewhat as the world learns.1 "The individual should grow his own mathematics, just as the race has had to do. But I do not propose that he should grow it as if the race had not grown it too. When, however, we set before him mathematics, be it high or low, in its latest, and most generalized, and most compacted form, we are trying to manufacture a mathematician, not to grow one." 2 This does not mean that the child must go through

1 Cette longue éducation de l'humanité, dont le point de départ est si loin de nous, elle recommence en chaque petit enfant. — Jean Macé, L'Arithmétique du Grand-Papa, 4ième éd., p. II.

2 Jas. Ward in the Educational Review, Vol. I, p. 100.

all of the stages of mathematical history—an extreme of the "culture-epoch" theory; but what has bothered the world usually bothers the child, and the way in which the world has overcome its difficulties is suggestive of the way in which the child may overcome similar ones in his own development.

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In the second place, the history of the subject gives a point of view from which we can see with clearer vision the relative importance of the various subjects, what is obsolete in the science, and what the future is likely to demand. Sterner1 has compared the teacher of to-day to a traveller who by much toil has reached an eminence and stops to take breath before attempting further heights; he looks over the road by which he has journeyed and sees how he might have done better here, and made a short cut there, and saved himself much waste of time and energy yonder. So one who considers the historical development of arithmetic and its teaching will see how enormous has been the waste of time and energy, how useless has been much of the journey, and how certain chapters have crept in when they were important and remained long after they became relatively useless. He will see the subject as from a mountain instead of from the slough of despond which the textbook often presents, and he will be able, as a result, to teach with clearer vision, to emphasize the impor

1 Geschichte der Rechenkunst.

tant and to minimize or exclude the obsolete, and thus to save the strength of himself and of his pupils. He will also learn that some of the most valuable parts of arithmetic knocked at the doors of the schools long centuries before they were admitted, and that teachers have had to struggle long and persistently to banish some of the most objectionable matter. As a result, while he may condemn the conservatism which excludes the metric system and logarithms and certain of the more rational methods of operations today, he will have more faith in the ultimate success of a good cause and will see more clearly his duty as to its advocacy.

Extent of the subject-It is manifestly impossible to give more than a glimpse at the history of arithmetic. The simple question of numeration, discussed with any fulness, would fill a volume the size of this one. 1 De Morgan's masterly little work, "Arithmetical Books," hardly more than a catalogue (with critical notes) of certain important arithmetics in his library, fills one hundred twenty-four pages. 2 For the student who cares to enter this fascinating field some suggestions are given in a subsequent chapter. But for the present purpose it suffices to consider merely a few important events in the general development of the subject.

1 See, for example, Conant, L. L., The Number Concept, New York, 1896. 2 London, 1847.

The first step-counting - The first step in the historical development of arithmetic was to count like things, or things supposed to be alike; in the broad sense of the term this is a form of measurement.1 Arithmetic started when it ceased to be a question of this group of savage warriors being more than that, and began to be recognized that this group was three and that two; when it was no longer a matter of a stone axe being worth a handful of arrow heads, but one of an exchange of one axe for eight arrows. How far back in human history this operation goes it is impossible to say, just as it is impossible to say how far back human history itself goes. Indeed, counting is not limited to the human family, for ducks count their young and crows count their enemies. Any discussion of the nature of this animal counting must lead to the broader question of the ability to think without words, a matter so foreign to the present subject as to have no place here.3

The race has not, however, always counted as at present. It was a long struggle to know numbers up

1 In this connection the teacher should read, though he may not fully indorse, Chap. III of McLellan and Dewey's Psychology of Number, New York, 1895.

2 This subject of animal counting has often been discussed. It is briefly treated in the chapter on Counting in Tylor's Primitive Culture, and also in Conant's Number Concept mentioned on p. 44.

3 For Max Müller's side of the case see his lecture on the Simplicity of Thought.

to ten. The primitive savage counted on some low scale, as that of two or three. To him numbers were "I, 2, many," or "1, 2, 3, many," just as the child often says, "1, 2, 3, 4, a lot," and somewhat as we count up very far and then talk of "infinity."

It is evident that there must be some systematic arrangement of numbers in order that the mind may hold the names. For example, if we had unrelated names for even the first hundred numbers, it would be a very difficult matter to teach merely their sequence, to say nothing of the combinations. But by counting to ten, and then (or after twelve) combining the smaller numbers with ten, as in three-ten (thirteen), four-ten (fourteen), twice-ten (twenty), and so on, the number system and the combinations are not difficult.

We might take any other number than ten for the base (radix). If we took three we should count,

one, two, three, three-and-one,

three-and-two, two-threes, ...,

and (with our present numerals) write these,

I, 2, 3, II (i.e., one three and one unit), 12, 20,

1

But most peoples, as soon as they were far enough advanced to form number systems, recognized the

1 A brief but interesting summary of this subject is given in Fährmann, K. E., Das rhythmische Zählen, Plauen i. V, 1896, p. 21. It is also treated in numerous text-books and elementary manuals in English.