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The following example she did yesterday mentally, without external help, after it was communicated to her by manual alphabet:

- - +

In this example the mind had to retain its grasp of the terms, as it was performing, in proper succession, seven or eight steps; then, taking up eight new-found terms, had to combine them in pairs, factor the result, and then cancel between numerator and denominator.

In four months more, I am confident, she could, with lessons only once a week, master the whole of those parts of Algebra required for admission to Harvard College. Already the foundations have been laid and enough of the superstructure raised to assure the harmonious and perfect completion of the work.

In Geometry it seemed to me necessary not only to begin anew, but even to undo. Starting with very elementary concepts about space, points, lines, angles, etc., we have traversed carefully the matter usually contained in the first book of Plane Geometry, together with many "originals." I have forbidden the doing of any proposition by memorizing what has been told her. She has been taught to work out originally everything possible. For instance, all the theorems about quadrilaterals were reasoned out carefully by her in my presence, her previous reading having been only the necessary definitions.

Although her Geometry has given us more trouble than any other subject, she has shown herself able to do the work in the proper methods and spirit. And I have no doubt that before long she will revise her estimate of the value of mathematical studies.

Truly yours,

MERTON S. KEITH.

46 Irving street, Cambridge, Mass.

Now compare these statements about Helen's proficiency in Algebra and her estimate of Mathematics written at the same time. She evidently had not yet seen much beauty, or good of any kind, in Arithmetic, which she had studied in the Cambridge School, or in the Algebra. Much lower was her estimate of Geometry. Yet she could then do mentally, without hint from me, after I had twice read them to her, examples such as these:

1) a2 - -b-2a2 -{3c2 + 4b - (3a2 - 2b +c2)}- }-

2) x / (x3 + y3) - y / (x3 - y3) + x3y + xy3 / x6 - y6

3) I / a - -a2 -I- /a + (I / a - 1)

Indeed, the first example is like those she did in the way described at the third lesson we had. The others she did in like manner within three months after we began.

I had found her mental condition in Algebra very much confused. She was then working on simultaneous equations, and yet I found her unable accurately or understandingly to work simple equations of one unknown quantity; even transposition of terms was for her a stumbling-block. She had hardly any knowledge of factoring, or of fractions. Least Common Multiple and Greatest Common Factor were practically impossibilities for her. She had had no drill by means of the Braille type-writer in doing even rather simple examples in long division, or in multiplication. She was very inaccurate in addition and subtraction. This condition of things was discovered at the first talk I had with her. The text-book was changed, and I began the subject as if she knew nothing of it. Of course, she had some ideas about the meaning of the negative sign, although even here I found it necessary to discuss the subject thoroughly, that her knowledge might be more than merely mechanical. Indeed, we discussed together carefully and with many practical illustrations the nature and use and value of Algebra in general compared with Arithmetic. In our second lesson we took up simple examples like No. 1. I set out to discover how far I could rely on her eager attention, tenacity of mental grasp, and memory to save her the time and weariness of paper work. I was delighted to find her able, after some practice, not only to carry in her mind the example with its complications of letters and coefficients and exponents and various signs, but to perform mentally the operations through to the solution. And from that day on she has ever been taught to rely on the Braille, the raised points punctured in paper, only in cases of very great complexity, or when she is anxious to avoid every chance of error in examination. Not only in removing brackets, but in factoring, in multiplying binominals, and raising binominals to powers, and in much of division she has used no Braille. In this way much time was saved, and very thorough, permanent impressions were made upon her mind. We had a long tussle over long division and Greatest Common Divisor by long division, chiefly because of the difficulty of producing the work in Braille, and arranging the work within the limits of the sheets of paper. But we succeeded in devising short cuts even there, and after four or five lessons devoted to these topics she became quite expert. Now she is so sharp at factoring that she often surprises me with the little written work used in arriving at the answer.