(4) Without further suggestion, give the young discoverer suitable opportunity for finding the sum of 3/8 and 1/8. In the act of discovering, an implicit _hypothesis_ takes form in the mind through analogous reasoning. This point marks the climax of the lesson――the supreme moment, when the skill and tact of the teacher is assessed to the limit. Just here the child must have a comfortable environment where perfect concentration is possible. Nothing must be forced; and there should be nothing suggestive of disgrace or shame, if the youthful Columbus is unsuccessful. The first attempt should be without books. If more help is needed, access to books may be given. If the investigation is still without definite result, then _as a last resort_ the teacher may, in the presence of the child, _add fractions_, solving _with deliberation_ example after example, until the child believes he has discovered the process. (5) Stimulate a desire to _verify the facts discovered_. Suggestions leading to verification: Afford opportunity for mathematical demonstration. Illustrations: The fractions 1/4 and 3/8 have been added in this way―― 1/4 = 2/8 3/8 = + 3/8 ――― 5/8 Use is now made of the crucial fact, when the example assumes this form―― 2 eighths + 3 eighths ――――――――――― 5 eighths _Or_ if the class has been trained in the use of the diagram the following may be the form of proof: { ━━━━━━━━━━ 1/4 { ────────── { ━━━━━━━━━━ ────────── ━━━━━━━━━━ { ────────── 3/8 { ━━━━━━━━━━ { ────────── { ━━━━━━━━━━ Explanation from diagram. I see that 1/4 equals 2 parts and 3/8 equals 3 parts; the sum of 3 parts and 2 parts are 5 parts. But the name of the part is eighths; hence the answer 5 parts may be written 5 eighths, or 5/8. Thus the final form is 2 parts + 3 parts ───────── 5 parts = 5/8 Give opportunity to consult answers in text books as further verification. The _summary_ and _application_ of adding fractions according to the “discoverer’s method,” are virtually the same as the corresponding steps in the “school room method.” _Second Illustration of Discoverer’s Method._ David P. Page in his Theory and Practice of Teaching well illustrates the discoverer’s method in conducting a general exercise in nature study. We cannot do better than to quote from him: