A scrutiny of the bonds now formed in the teaching of arithmetic with questions concerning the exact service of each, results in a list of bonds of small value or even no value, so far as a psychologist can determine. I present here samples of such psychologically unjustifiable bonds with some of the reasons for their deficiencies. (1) _Arbitrary units._--In drills intended to improve the ability to see and use the meanings of numbers as names for ratios or relative magnitudes, it is unwise to employ entirely arbitrary units. The procedure in II (on page 84) is better than that in I. Inches, half-inches, feet, and centimeters are better as units of length than arbitrary As. Square inches, square centimeters, and square feet are better for areas. Ounces and pounds should be lifted rather than arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and cubic inches are better for volume. All the real merit in the drills on relative magnitude advocated by Speer, McLellan and Dewey, and others can be secured without spending time in relating magnitudes for the sake of relative magnitude alone. The use of units of measure in drills which will never be used in _bona fide_ measuring is like the use of fractions like sevenths, elevenths, and thirteenths. A very little of it is perhaps desirable to test the appreciation of certain general principles, but for regular training it should give place to the use of units of practical significance. [Illustration: FIG. 3. A ---------- B ------------------------------ C -------------------- D ---------------------------------------- I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3? _A_ and _C_ together equal what line? _A_ and _B_ together equal what line? How much longer is _B_ than _A_? How much longer is _B_ than _C_? How much longer is _D_ than _A_?] [Illustration: FIG. 4. A ---------- B ------------------------------ C -------------------- D ---------------------------------------- II. _A_ is 1 inch long. Which line is 2 inches long? Which line is 4 inches long? Which line is 3 inches long? _A_ and _C_ together make ... inches? _A_ and _B_ together make ... inches? _B_ is ... ... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ... ... longer than _A_?] (2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as single connections should be left for the pupil to acquire by himself as he needs them. These connections interfere with the process of learning two-place multiplication. The manipulations of numbers there required can be learned much more easily if 11 and 12 are used as multipliers in just the same way that 78 or 96 would be. Later the 12 × 2, 12 × 3, etc., may be taught. There is less reason for knowing the multiples of 11 than for knowing the multiples of 15, 16, or 25.