PROPER PLACE PROBLEMS SCOTT MacKENZIE University of Rhode Island, Kingston, Rhode Island
EVERY chemist has either answered or asked, a t one time or another, questions which call for the arrangement of a number of items in correct order. These questions might be termed "proper place problems" since they demand tabulation of things in a correct or proper place. A typical example might well require the examinee to "arrange in order of decreasing acidity the substances water, ammonia, hydrogen fluoride, and methane." Does the student who writes
know anything about the ionic character of bonds? If it is decided that he does, what should be the part credit assignment?
Questions concerning boiling points frequently take t,his form since these can be difficult to ask in some of the other ways. "Arrange the following substances, all of which have approximately the same molecular weight, in order of decreasing boiling point: I-butene, n-propyl alcohol, methyl formate, n-butane, acetic acid, and acetone." 1-Butene n-Propyl alcohol Acetic acid Methyl formate Acetone %-Butane
X
X X X
Does the above response deserve any credit? It should be pointed out that such "proper place" questions invite grading by a detailed mathematical system that bears lit,tle relation t o the more frequently
AUGUST, 1954
employed percentile calculations. Consideration of TABLE 1 this system by others is hereby welcomed. Tabulation of Z,, the N u m b e ~of Ways of Arranging N Items with X Incorrectly Placed The person who selects the uniquely correct answer from among many deserves, naturally, the best score. Items arranged N Z, Z , Z, Za Z. ZS Zr However, a fact frequently not appreciated is that Zi many answers, even those with several items misplaced, are very difficult to come by through chance and therefore demonstrate knowledge. Table 1 reflects the ease of obtaiming any given answer; the values therein have been calculated by the customary combinatorial methods.' Note that the sum of each horizontal line is N factorial. It is now suggested that the examinee be given credit for any answer in accordance with the difficulty of obtaining the answer by chance, i. e., be given credit for with two correct merits 50 per cent, not 33 per cent. the extent to which he defies the laws of probability. The more customary percentile assignments have been Such defiance represents, after all, prior knowledge. reversed. Initially surprising, this turn of events need If credit is to be proportional to that distance trav- not find its sole support in combinatorial intricacies. eled on the road to the uniquely difficult answer, such Thoughtful consideration of two questions aids claricredit can be calculated from the expression fication: did not each student correctly place two items only? Did not he who faced six items skirt successfully many more pitfalls? In conclusion, i t might be pointed out that several Table 2 makes this suggestion. suggestions made by Table 2 are a t variance with cusAccording to the system suggested by Table 2, the tomary practice in grading "proper place problems." student who arranged four acids with two correct does These suggestions merit emphasis. not receive 50 per cent but only 33 per cent, while he When a long series of items is involved, no (1) who tabulated six substances according to boiling point credit should be given for the act of arranging - one item The calculation of Z, was made using the expression in its proper place. (2) When a long series of items is involved, the act of la cine two items correctlv merits fiftv Der cent. {3) i v e r y substantial portion of crehft (SO per cent in which ( N / z ) represents or over) should be given for the act of arranging any long series of items with more than two placements correct. in accordance with the notation used by W. Feller in "An Introduction to Probability Theory and Its Applications," John Wiley & Sons, Inc., New York, 1950. Qz should be read as "the number of way8 of arranging z items with none in its proper place" and can be calculated from either of two expressions:
Q.
=
+
zQ=-I (-1)"
or
This latter expression can be read as "the number of ways of arranging z objects with all items misplaced is equal to the total number of ways of arranging them less the number of ways of arranging them with less then z misplaced."
TABLE 2 Tabulation of Suggested Partial Credit To Be Given for Any Answer t o the Problem of Arranging Items i n Correct Order
Items arranged
0
1
Number of mistakes S L
B
6
6
7
