The Psychology of Arithmetic

by Edward L. Thorndike

Denote by a + b, a + g, a + l, a + q, a + v, and a + B certain situations alike in the element a and different in all else. Suppose that, by original nature or training, a child responds to these situations respectively by r₁ + r₂, r₁ + r₇, r₁ + r₁₂, r₁ + r₁₇, r₁ + r₂₂, r₁ + r₂₇. Suppose that man's neurones are capable of such action that r₁, r₂, r₇, r₁₂, r₂₂, and r₂₇, can each be made singly.

Case I. Varying Concomitants

Suppose that a + b, a + g, a + l, etc., occur once each.

We have	a + b	responded to by	r1 + r2,
	a + g	" "	r1 + r7,
	a + l	" "	r1 + r12,
	a + q	" "	r1 + r17,
	a + v	" "	r1 + r22, and
	a + B	" "	r1 + r27, as shown in Scheme I.

Scheme I

	a	b	g	l	q	v	B
r1	6	1	1	1	1	1	1
r2	1	1
r7	1		1
r12	1			1
r17	1				1
r22	1					1
r27	1						1

a is thus responded to by r₁ (that is, connected with r₁) each time, or six in all, but only once each with b, g, l, q, v, and B. b, g, l, q, v, and B are connected once each with r₁ and once respectively with r₂, r₇, r₁₂, etc. The bond from a to r₁, has had six times as much exercise as the bond from a to r₂, or from a to r₇, etc. In any new gross situation, a 0, a will be more predominant in determining response than it would otherwise have been; and r₁ will be more likely to be made than r₂, r₇, r₁₂, etc., the other previous associates in the response to a situation containing a. That is, the bond from the element a to the response r₁ has been notably strengthened.

Case II. Contrasting Concomitants

Now suppose that b and g are very dissimilar elements (e.g., white and black), that l and q are very dissimilar (e.g., long and short), and that v and B are also very dissimilar. To be very dissimilar means to be responded to very differently, so that r₇, the response to g, will be very unlike r₂, the response to b. So r₇ may be thought of as r_{not 2} or r_-2. In the same way r₁₂ may be thought of as r_{not 12} or r_-12, and r₂₇ may be called r_{not 22} or r_-22.