Cartesian axes and parallel lines
I always feel apprehension on seeing the number line or x-axis introduced as a horizontal line, with the numbers increasing left to right. When the time comes, the y-axis is commonly introduced as perpendicular to the x-axis and hence vertical. Seldom there is enough time for a trifle observation that the direction of a number line is irrelevant and that the choice of the orientation of the system of axes in the plane is a matter of convenience. The trouble comes when there is a need to change variables or, say, introduce the inverse function.
It is a long time since I last taught Calculus but the images came back to me when I began to browse my new online acquisition. I have recently purchased at amazon.com two books by John Mason and one by Robert Simson, see below. J. Mason's books are directed to a learner, a teacher, but mostly to student teachers. The book on geometry starts with an account narrated by an adult - probably a primary school teacher - of experiences with children that took place in school.
The class had been studying the concept of parallel lines and I asked one boy if he could show me some examples of parallel lines in the classroom. He very quickly pointed out many examples to me including window and door frames, some panelling on the ceiling and the drawers of a desk. I was impressed and thought that he appeared to have a good grasp of the concept. I then happened to notice that the class had a display of flags of the world on the wall, arising from a project they had done. One of the flags was the Tanzanian national flag.
I pointed to this flag and asked if it had any parallel lines. He looked a bit puzzled and asked if I meant the edges of the flag. I agreed they were parallel but asked if he could see any others. He shook his head. I said ‘What about these lines?’ pointing to the yellow (diagonal) lines. He shook his head vigorously, so I asked him why not. "Well, because they’re not straight!" he replied forcefully.
I sat with two girls who had been learning to name some basic geometric shapes. They had a box of flat wooden shapes and I took out a triangle, placed it on the desk between them and asked one of the girls to tell me the name of the shape. From the girl’s position the triangle in the diagram.
She frowned and seemed not to be struggling with her answer. At last she said "I'am not sure, but it is a triangle for her", pointing to friend on the other side of the desk.
J. Holt devotes much of his book How Children Fail to this phenomenon of children's apparent understanding of the material. For example, on p. 231 (October 30, 1958):
Everyone around here talks as if, except for a few hopeless characters, these children know most of the math they are supposed to know. It just isn't so.
And earlier in the book (pp. 177-178) but chronologically later (June 20, 1960) while talking of the difference between real learning as opposed to apparent learning:
There are many, of course, who say that this distinction does not exist. It's their handy way of solving the knotty problem of understanding ...
There is a serious difficulty in articulating what real learning or real understanding are. J. Holt himself on p. 169 (March 2, 1960) suggests that
A child who has really learned something can use it, and does use it. It is connected with reality ion his mind, therefore he can make other connections between it and reality when the chances come.
Note that while Holt's point is a reasonable one, it is certainly at odds with the previous examples.
References
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