Exposition and illustration in teaching . s is beyond him. Anyone who has not given thematter attention will be surprised at our general weak-ness in estimating area. We are all singularly feeblein the matter of comparing the relative sizes of surfaces,and in particular in correlating lengths with can compare two lines with each other with a fairchance of justly estimating their ratio, but few amongus can make even a reasonable guess at the relativeareas of two given circles or squares. To prove howeasily we may be misled in comparing lines with areas,ask any friend who has not had th
Exposition and illustration in teaching . s is beyond him. Anyone who has not given thematter attention will be surprised at our general weak-ness in estimating area. We are all singularly feeblein the matter of comparing the relative sizes of surfaces,and in particular in correlating lengths with can compare two lines with each other with a fairchance of justly estimating their ratio, but few amongus can make even a reasonable guess at the relativeareas of two given circles or squares. To prove howeasily we may be misled in comparing lines with areas,ask any friend who has not had the experiment alreadyimposed upon him how many cent pieces or ^pennieswe can place flat on the surface of a silver dollar without 358 EXPOSITION AND ILLUSTRATION IN TEACHING any one of them overlapping, however slightly, the cir-cumference of the larger coin. The ordinary answervaries from three to five. The fact, however, is thateven two of the little copper coins are more than thedollar can receive on its surface under these Fig. 5. Take two small coins, say quarters, and place them on atable at such a distance from each other as you thinkwill leave room for one other quarter to fit in exactlybetween the two. When you have tested your result,you will probably find that you are considerably wrongin your calculation, and that any friend with whomyou experiment goes wrong in the same direction asyourself. You are really trying to determine the lengthof the diameter, but the area of the coin leads you into THE DIAGRAM 359 error. Our weakness is shown also in our inability toguess correctly without previous practice the height ininches of a silk hat resting on its crown. Further, taketwo pieces of paper and, putting the one above the other,cut out in duplicate the shape indicated in figure will then have two pieces of paper of exactly equalarea, but if you place them one below the other, as infigure 5, you will find it very difficult not to maintainthat the lower
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