How can I help the children's learning? My current method.

I think mathematics and computer science are fun. I can explain some example cases, but, why do I think these are fun in general? The same as why am I interested in novels, movies, climbing, or teaching? I don't have a clear answer. How do children think about mathematics? If I could understand a part of that, maybe I could help them. Therefore, I would like to learn how the children think rather than teaching. I learn how the people think about mathematics.

I am interested in artificial intelligence. Therefore, I am also interested in how the people's intelligence develop. Children learn speaking, counting numbers, logical thinking, then develop their intelligence further. I should have experienced these process, but, I totally forgot how to learn my native language, how to learn counting numbers.

Because of that, I would like to ask children to teach me them. Though, I found it is not easy. Children hardly teach me how they think. I am not sure, but maybe children themselves don't know that.

Therefore, I first observe children. How children reacts the inputs. I would like to know what kind of model they have. It is a bit similar to debug a program. For example, a child can not process specific inputs, another child always makes a specific mistake. Then, I think, how this happens? What kind of mental model they have? This is very hard problem, but it is quite interesting. I think if I could understand their model (how they think), and if I could see the problem of the model, I might help them. This is just my hypothesis, but this is usually my case. I hope this works. So, what I do in the class is the following loop: observe a child, learn their model, think it, try to teach based on my model, fail, observe the child again, ...

I would like to record my those experiences.

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

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

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) .

Lx=d: SundayResearch

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