Procedure. Draw on a piece of paper a horizontal line six or eight inches long. Above it, an inch or two, draw a short horizontal line about an inch long and parallel to the first. Tell the subject that the long line represents the perfectly level ground of a field, and that the short line represents a cannon. Explain that the cannon is “pointed horizontally (on a level) and is fired across this perfectly level field.” After it is clear that these conditions of the problem are comprehended, we add: “Now, suppose that this cannon is fired off and that the ball comes to the ground at this point here (pointing to the farther end of the line which represents the field). Take this pencil and draw a line which will show what path the cannon ball will take from the time it leaves the mouth of the cannon till it strikes the ground.”
Scoring. There are four types of response: (1) A straight diagonal line is drawn from the cannon’s mouth to the point where the ball strikes. (2) A straight line is drawn from the cannon’s mouth running horizontally until almost directly over the goal, at which point the line drops almost or quite vertically. (3) The path from the cannon’s mouth first rises considerably from the horizontal, at an angle perhaps of between ten to forty-five degrees, and finally describes a gradual curve downward to the goal. (4) The line begins almost on a level and drops more rapidly toward the end of its course.
Only the last is satisfactory. Of course, nothing like a mathematically accurate solution of the problem is expected. It is sufficient if the response belongs to the fourth type above instead of being absurd, as the other types described are. Any one who has ever thrown stones should have the data for such an approximate solution. Not a day of schooling is necessary.
(b) Problem as to the weight of a fish in water
Procedure. Say to the subject: “You know, of course, that water holds up a fish that is placed in it. Well, here is a problem. Suppose we have a bucket which is partly full of water. We place the bucket on the scales and find that with the water in it it weighs exactly 45 pounds. Then we put a 5-pound fish into the bucket of water. Now, what will the whole thing weigh?”
Scoring. Many subjects even as low as 9- or 10-year intelligence will answer promptly, “Why, 45 pounds and 5 pounds makes 50 pounds, of course.” But this is not sufficient. We proceed to ask, with serious demeanor: “How can this be correct, since the water itself holds up the fish?” The young subject who has answered so glibly now laughs sheepishly and apologizes for his error, saying that he answered without thinking, etc. This response is scored failure without further questioning.
Other subjects, mostly above the 14-year level, adhere to the answer “50 pounds,” however strongly we urge the argument about the water holding up the fish. In response to our question, “How can that be the case?” it is sufficient if the subject replies that “The weight is there just the same; the scales have to hold up the bucket and the bucket has to hold up the water,” or words to that effect. Only some such response as this is satisfactory. If the subject keeps changing his answer or says that he thinks the weight would be 50 pounds, but is not certain, the score is failure.
(c) Difficulty of hitting a distant mark
Procedure. Say to the subject: “You know, do you not, what it means when they say a gun ‘carries 100 yards’? It means that the bullet goes that far before it drops to amount to anything.” All boys and most girls more than a dozen years old understand this readily. If the subject does not understand, we explain again what it means for a gun “to carry” a given distance. When this part is clear, we proceed as follows: “Now, suppose a man is shooting at a mark about the size of a quart can. His rifle carries perfectly more than 100 yards. With such a gun is it any harder to hit the mark at 100 yards than it is at 50 yards?” After the response is given, we ask the subject to explain.
Scoring. Simply to say that it would be easier at 50 yards is not sufficient, nor can we pass the response which merely states that it is “easier to aim” at 50 yards. The correct principle must be given, one which shows the subject has appreciated the fact that a small deviation from the “bull’s-eye” at 50 yards, due to incorrect aim, becomes a larger deviation at 100 yards. However, the subject is not required to know that the deviation at 100 yards is exactly twice as great as at 50 yards. A certain amount of questioning is often necessary before we can decide whether the subject has the correct principle in mind.