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Presenting teaching with Variation

In this video we will introduce teaching with Variation.
Teaching with Variation is being used in mathematics classrooms for a long time. Based on the previous experiences and his own experiment in Shanghai mathematics classrooms, Professor Gu who’s one of the most well known mathematics educators in China, systematically analyzed and synthesized the concept of Teaching with Variation. You will hear what he has to say later in this course about Teaching with Variation. And according to Professor Gu, there are two forms of variations, conceptual variation and procedural variation. Conceptual variation is about understanding concepts from multiple perspectives. As Professor Gu said, so it is to illustrate the essential features by demonstrating different forms of visual materials and instances, or highlight the essence of a concept by varying the non-essential features.
So here is one example to help students understand the concept of the height, or altitude of a triangle from the top vertex. So let’s look at the first triangle. Normally we will use this kind of position, so it is easy for student to understand the concept of the height. So this is the height. Then in this triangle, the height and one side of the triangle are the same. So that is height, so that is a bit more challenging. And the following three are variations to understand the concept. So the third one is this one. So you can see some students will make mistakes like this.
They draw a horizontal line, and then they draw a vertical line from the top vertex of the triangle to this horizontal line. Of course this is incorrect. The correct one is this. Okay, the next one is another variation. The situation for students to learn the concept. So you should extend this side beyond this point. Then you draw a height, that is a perpendicular line from the top vertex of this triangle to this horizontal line. So that might not be so difficult for students. So the most difficult one is this one. So many students will likely draw a horizontal line passing through this vertex of the triangle.
Then they will draw a line from the top vertex here, perpendicular to the horizontal line. So that is the height they might get, but this is incorrect. So the correct one should be this one. You extend this side of the triangle, go beyond this point. Then you draw a perpendicular line to this side, the extension of the side from the top vertex of the triangle, so that is height. The next is procedural variation. It means to progressively unfold mathematical and activities. The variations for constructing a particular experience system are derived from three dimensions of problem solving.
First, varying a problem which means varying the original one as a scaffolding or extending the original problem by varying the conditions changing the results and generalization. The second is multiple methods of solving a problem by varying the different processes of solving a problem and associating different methods of solving a problem. So you vary the process of solving the problem. And the third one is multiple applications of a method by applying the same method to a group of similar problems. So in Chinese we say you use one method to solve three problems or even more. So here is an example, and you will think it’s a very simple example. So we ask students to calculate 2 plus 8 equals how much.
Of course student will know it’s 10. In terms of variation theory, we can vary the problem into different scenarios. So variation one is 2 plus what equals 10. And variation two is what plus 8 equals 10. Variation three is more difficult. So what number plus what number equals 10. So students are asked to find two numbers whose sum is 10. And variation four is 10 equals 2 plus what. So I’m sure you can find more ways to vary the problems. According to researchers in this area, that Teaching with Variation is a main feature of Chinese mathematics classroom.
And by adopting Teaching with Variation, even with a large classes like in Shanghai and in many other Asian classrooms, students still can actively involve themselves in the process of learning. And they can achieve excellent results.

This video explains what teaching with Variation is and demonstrates the principle with some concrete examples.

In China, teaching with Variation has been used in classrooms for many years.

Professor Gu is one of the most well-known mathematics educators in China. Based on previous experience and his own experiments in Shanghai maths classrooms, he systematically analysed and synthesised the concepts of teaching with Variation (1981). According to Gu, there are two forms of Variation: conceptual variation and procedural variation.

According to researchers (Gu, Huang, & Marton, 2004), teaching with Variation characterises mathematics teaching in China. By adopting teaching with Variation, even with large classes, students can actively involve themselves in the process of learning and achieve excellent results.


Gu, L. (1981). The visual effect and psychological implication of transformation of figures in geometry. Paper presented at annual conference of Shanghai Mathematics Association. Shanghai, China.

Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In Fan, L., Wong, N-Y., Cai, J., & Li, S. (Eds.) (2004). How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific.

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