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One very important pedagogic principle, familiar to all teachers, but very often disregarded, has been deeply impressed upon my mind in my experience in teaching Miss Keller. Apparent stupidity (not habitual) on the part of a pupil is much more often due to wrong preconceptions than is commonly realized. Reiteration of the proper view in all possible variation of language is of no avail as long as the wrong impression is present in the pupil's mind. During the first few months of our work together, there were occasions when Helen's mind seemed impervious to statements which seemed to me absolutely clear and simple, about facts or ideas equally simple. Work came to a standstill, or got into a tangle. She seemed obstinately stupid. When the trouble was over, it was always discovered that a misconception had thwarted our efforts. By questionings, by asking her freely to give her ideas on the subject in hand, or by going back to the primal elements underlying it, I always succeeded in finding the false notion. Sometimes I had been at fault in some ambiguous or inaccurate form of statement, sometimes her own imagination had distorted the facts by some strange association of ideas; sometimes it was hard to see the cause. But with the wrong idea removed, the right one usually came to her like a flash of light. By care and deliberation in the method of presenting a subject, and above all, in the choice or order of words, cases of misunderstanding became very few. During the last part of our work together the rapidity and smoothness of the progress of instruction were remarkable.

Doubtless with Miss Keller, more than with the seeing and hearing pupil, it is incumbent on the teacher to convey the correct impression first. With her avidity for knowledge, vividness of imagination, rapid and complex association of ideas, and her tenacity of mental grasp, the first impression is very strong and persistent. To correct or remove it and replace it by another is difficult, chiefly for these reasons. One of the most remarkable facts about her seems to me to be that she so rarely gets first impressions wrong, and arrives at conceptions which are on the whole so sane and perfect.

By July 1, 1898, we had finished Book I in Geometry, with about fifty "originals," and made a beginning on Book II. From October 15, 1898, to February 1, 1899, we finished the Geometry, with frequent reviews and many originals. In doing the latter Books she had the use of a book of raised letters and figures, which rendered her progress more rapid. But it would have been unwise to use it from the beginning, or to use it in all the propositions of the later Books.

It must be noted that Miss Keller labored under much greater disadvantages in Geometry than in Algebra, or in any other subject. For she used no Braille to record her own work, having only a figure, constructed often with much trouble, to refer to. She had to carry in her mind the lettering of the figures, the hypothesis and conclusion, the construction, and the process of proof. To keep the mind clear under such conditions is very hard. After seeing through the proof, perhaps by many trials, she had to record the results by her typewriter for the examiner. This in itself is a great task to do accurately, as was often shown by misprints, or mistakes about the naming of lines, or by the omissions of steps in proof which had been really in her mind. All these things required patient practice, and consumed much time. To give her practice with economy of time, we sometimes did theorems wholly in the mind, even the figures with the lettering being only mentally pictured. We were very successful here. I give one instance of a problem thus solved by her correctly in about five minutes.

Problem: The diameter of a circle is 20 inches. Through a point in the diameter four inches from one end a chord is drawn perpendicular to the diameter. What is the length of the chord? and what are the lengths of the chords drawn from the ends of this chord to the ends of the diameter?

CE = square root of (AE x EB) = square root of 16 x 4 = 8

triangle CD = 16Ans. (I)

AC = square root of AB x AE = square root of 20 x 16 = 8 x square root of 5 Ans.

CB = square root of AB x EB = square root of 20 x 4 = 4 x square root of 5 Ans.

From February 1, 1899, till June 1 Miss Keller did about fifty examination papers in Geometry, consisting of review book-work and originals; and also, about twenty Harvard Admission Examination Papers. These were almost wholly written out for me to examine, and were carefully reviewed and discussed. During June I did very little work with her in Algebra and Geometry, but kept her occupied with originals. Most of this she did so well that I needed but little time with her.

Miss Keller had had something more than two months' Greek when I began my work with her in February, 1898. My article of May, 1898, quoted already, describes the early stages of the work. In five months we had practically completed a very full course in etymology and syntax in a very thorough manner, and applied it all in the translation of Greek into English, and English into Greek. She had by July, 1898, translated nearly two Books of the Anabasis, mostly by herself, as I had had but very little time to hear her recite much connected narrative, perhaps not more than twenty pages. During the summer vacation of 1898 she finished two more Books of the Anabasis by herself. She had the first Four Books in Braille.