Fast lookup child

L is not a so diligent girl. But she sometimes finishes the exercise very quickly. Although I found she repeatedly mistook the same problem. For example, in one exercise, 4+7 was always 12. I asked what is equal to 4 + 7. She used fingers, took a little while, but she answered correctly at the end. Then I noticed, ``Wait, if she solve the problem with this speed, how she can finish the exercise so quickly?'' So, I sat down next to her and watched how she solved the problem.

Surprise! She doesn't calculate at all. She just looks up the last pages and copies the answers. What I was impressed was she was so fast. She was faster than me to look up the questions. I understood why she made the same mistakes again and again. I learned a new German sentence, ``Bitte nicht angucken!'' This means, ``Don't cheat by looking.'' though it's hard to translate to English.

Her speed of cheating is impressive, this might be her talent. I don't know that I shall discourage her talent(?). That's a tough call for me, but I believe it is better to know to calculate numbers. So, today, I said ``Bitte nicht angucken!'' to her again. But I still don't know. Teaching is a tough business.

Comments

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

Geometric Multiplicity: eignvectors (2)

If eigenvectors of a matrix A are independent, it is a happy property. Because the matrix A can be diagonalized with a matrix S that column vectors are eigenvectors of A . For example, Why this is a happy property of A? Because I can find A's power easily. A^{10} is not a big deal. Because Λ is a diagonal matrix and power of a diagonal matrix is quite simple. A^{10} = SΛ^{10} S^{-1} Then, why if I want to compute power of A ? That is the same reason to find eigenvectors. Eigenvectors are a basis of a matrix. A matrix can be represented by a single scalar. I repeat this again. This is the happy point, a matrix becomes a scalar. What can be simpler than a scalar value. But, this is only possible when the matrix S's columns are independent. Because S^{-1} must be exist. Now I come back to my first question. Is the λ's multiplicity related with the number of eigenvectors? This time I found this has the name. Geometric multiplicity (GM): the number of in...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Lx=d: SundayResearch

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