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By July, 1898, she was ready for examination in Algebra except-in the following parts: Quadratic Equations, Surds, Ratio and Proportion, and part of the Theory of Indices. Resuming work in October she mastered the theory and method of solving Quadratic Equations by applying the factoring method first, as in the example (x -- a) (x -- b = 0, and never resorting to the methods of completing the square, unless the task of factoring was very difficult or impossible. Being taught how we obtain, from the equation ax2 + bx + c = 0, x = -b + or - square root of b2 - 4ac / 2a it was remarkable how soon she became expert in solving all numerical quadratics, and how quickly she could tell whether the values of were exact or surd. But in literal quadratics we had something of a tussle. For here again she had to resort much to the Braille in the more complex examples, and errors crept in very often. But, after a time, here again she became accurate and rapid in her work.

In the Theory of Exponents and in Surds and Ratio and Proportion she had no great difficulties. In examples containing fractional and odd exponents she was rather subject to error, apparently on account of the mechanical troubles in recording her work.

And of what use to her has the Algebra been? Is it simply that her ambition has been gratified? or that she has experienced the exhilarating joy springing from successful effort? This fruition of work she has indeed had during the last three quarters of the time devoted to Algebra; and, more, she has taken delight in the work for itself. She has done it not as a mere task. Clearness and definiteness have been added to many a mental picture once obscurely outlined, or dimly colored. Her mental vision has been sharpened to discern relations unseen before. She has seen order and simplicity and neatness and rigid exactitude issue from confusion and complexity. She has acquired new qualities of mind, or at least developed or strengthened latent ones. She has seen new beauty and heard new harmony.

In Geometry I found that about one-half of Book I had been studied and repeatedly reviewed since the previous October. But Miss Sullivan's laments over Helen's frequent failures with propositions reviewed many, many times showed that either our task was impossible, or the methods must be changed. Here too, as in the case of the Algebra, the subject was taken up anew, as if she had never studied it. I talked with her about the simplest elements of Geometry; such as points, lines, angles, surfaces. By means of the cushion and the wires with bent and pointed ends, I gave her much tactical practice in judging position, distance, direction, parellelism, angular quantities, etc. I found her defective here, and constantly in need of patient and persistent practice. Her judgment of position, size, and form in Geometry seemed poor. Her imagination in geometric fundamentals needed stimulation. I found, too, that many misconceptions stood in the way of her getting right ones, and had to be detected and removed.

Accordingly, I was so careful about methods of statement and of proof, that for the first two or three months I did not allow her to take any new steps except under my immediate direction. She studied between recitations only in review of what I had done with her. Propositions which she had studied with others, but which I had not yet done with her, I tried to have her forget, and go over with me afresh, while I led her along by question and hint, to the end that she might get some insight into the meaning of the subject, and especially into the method and logic of it. Every proposition wholly new to her was done originally. The first statement of it she had ever heard was from me. She repeated the proposition until she stated it correctly, and then she drew the figure on the cushion. If this was correct, she lettered it and stated algebraically the hypothesis and conclusion. She was then asked to analyze the requirements of the conclusion and the facts given in the hypothesis, to find relations between them suggesting the proof, or suggesting facts already proved leading to the proof. If this analysis suggested new construction, she drew it and lettered it, and renewed her analysis. Of course, there is no originality in this method. All good teachers of Geometry follow this method except where synthesis may lead more directly to proof. I simply had to be more careful and patient with Helen at first than would be necessary in teaching her other subjects; and this, either because her intuitive powers were weak here, or her mind had been confused and misdirected in her previous study of the subject. I am very sure that both reasons are necessary to an explanation of her early difficulties. The second reason I am positive holds. For she certainly had somehow got the notion that memorizing the proofs of others was the chief secret in Geometry. She seemed bewildered in a mass of words and disconnected ideas of no value or interest, and not even of much meaning. To tell her the method of work, to show her how to help herself to the use of facts or ideas already familiar and admitted to be true, in order to come at new truths not before acknowledged, was the real task. To get one proposition proved by herself in the proper way, even if it required days of teaching and study, was worth more than her ability to rehearse another's proofs of a hundred propositions.