[case]
"How many angles does the five-pointed star have?"
It was a "observation and discovery" mathematical inquiry activity in grade one, and the children were communicating around the corner of life. "Mr. Zhang-"Just a few minutes after class, Rui Hao Ming came to see him in a hurry. "I found that there are five angles on the five-pointed star, and these five angles are all the same size." "Does * * * have five corners?" I emphasized the word "* * *". When I didn't like it, I refuted it. He suddenly perked up: "Why not five? Look, one, two, three, four, five. " As he spoke, he counted. "Take a good look again and don't jump to conclusions easily." Seeing that I was so stubborn, he didn't argue again and reluctantly returned to his position. In that class, his confused eyes are still clear in my mind.
Two years later, when I formally studied the content of "knowing the angle", Rui Haoming came to tell me: "Teacher Zhang, now I understand that there are not only five angles on the five-pointed star, but there should be ten, because there are five obtuse angles that open outward, right?" I smiled at him. "Maybe."
In a blink of an eye, it was the sixth grade again. Maybe I came into contact with more graphic knowledge, and he found me again. "Teacher Zhang, I finally understand this time. Scientifically speaking, there are countless angles on the five-pointed star. Because there are many angles greater than 180 degrees or 360 degrees, and we have not paid attention to these angles before. "
To tell the truth, I only had a strong feeling at that time, which was why, at the beginning, I didn't even have the courage to be sure in the face of such an inaccurate but amazing discovery by a first-year child. Just to defend the accuracy and scientific dignity of mathematics?
[thinking]
When mathematics gradually changes from static to dynamic, from certainty to variability, from accuracy to fallacy, the sacred aura of mathematical science gradually fades. Mathematics is not so much like a saint who stepped down from the altar, but more like an approachable mortal. In a sense, mathematics is no longer a collection and embodiment of unchanging truth, on the contrary, it is more like a moving body that is constantly developing, evolving and updating. In this case, what reason do we have to ask those children who have just come into contact with mathematics to complete the accurate construction of mathematical knowledge in one step? What reason do we have to reject the fuzziness and intuition of mathematics?
The five-pointed star has five angles. What's wrong with a first-grade child? In other words, even if this expression is not scientific enough, it is only our judgment from the perspective of adult mathematics. For the first-grade children, there is no more understanding of obtuse angles and rounded corners, and making such a judgment just reflects his understanding level. For them, maybe this is the real "accuracy". If this is an accommodation for children, don't think that such accommodation will miss our children. On the contrary, children's mathematical development itself is a gradual process of spiral rise. From vagueness to clarity, from chaos to order, should be the only way for children's mathematics development. The children's understanding of the angle in the above cases fully illustrates this point.
In fact, it is not only a "cognitive perspective", but this phenomenon can be found everywhere in mathematics education. Only when we really look at mathematics, mathematics education and children's mathematics growth from the perspective of development and change can our mathematics education really promote the healthy and sustainable development of students.