Interview with Professor Gu

Professor Gu: Because the primary education in China was experiencing the most difficult era at that time, we investigated into high school graduates in Qingpu in 1977. Nearly 1/4 of the 4373 sample students could not solve questions of adding and subtracting fractions. They could not work out basic questions like calculating time with given distance and speed either, not even mentioning solving geometric questions.

The reason was there was no substantial mathematic classes. The mathematic concepts were ambiguous, and the mathematic training was in confusion. As an effective teaching method was in urgent need to change the situation at that time, Teaching with Variation became one of the important methods. Then, at the early stage of implementing Teaching with Variation, there was a summary article presented at annual conference of which was ‘The visual effect and psychological implications of Shanghai Mathematics Association in 1981, which was ‘The visual effect and psychological implications of transformation of figures on teaching geometry’. For the first time, the article distinguished variation in mathematics into two general categories, which are conceptual variation and procedural variation.

We applied these two categories of variation from experimental schools to all the schools in Qingpu, and we saw significant improvement in mathematics teaching quality. Later we have researched on Teaching with Variation for over 30 years. From 1981, we started to explore the psychological characteristics of learning via variation. The psychological characteristics revealed emotional willingness. It means, among the characteristics, we found emotional willingness is the most important motive. And there was accumulation step by step, because mathematics is systematic and in logic sequences. Another feature was trial activities, emphasising the inquiry spirit of students. Finally, based on students’ feedback we adjusted our ways of teaching in time.

Since 1990 we explored the ‘core connection’ between knowledge levels and students’ potential, and discovered findings including step-by-step teaching method from operation, understanding, mastery to exploration. These finding have been widely promoted in Shanghai and in the whole country.

Professor Fan: Thanks, could you briefly introduce what is Teaching with variation?

Professor Gu: Conceptually, Teaching with Variation is a teaching methods, which applies different forms of materials, examples and variation processes to develop a profound understanding of a specific concept, or to solve related questions. This is the general variation application in education. In mathematics, there are two main categories of variation. One of them is conceptual variation. Conceptual variation refers to extracting the essential attributes of concepts from a variety of forms of mathematic materials, and deciding the accurate extensions of the concept. This is because mathematic concepts have numerous extensions. Procedural variation targets at solving problems. It develops mathematic questions based on concepts, and aims at solving the problems via various processes like logical reasoning or modelling etc.

These two variations are the fruits of mathematic teachers’ collective wisdom that have been accumulated in classroom teaching improvement for long.

Professor Fan: Thanks. Is there any example to demonstrate the application of the variation theory in mathematic classrooms?

Professor Gu: This is a good question, because examples are the best illustration to explain the essence of the variation theory. My example is the modelling of division with remainders. For primary school students, division requires comprehensive calculation skills, especially division with remainders. The teacher will instruct like this. Division is like allocating beans. We allocate 7 beans into 3 plates, and how many beans can be allocated to each plate? Let’s try. One bean in each plate. There could be more. Then two beans. Three beans won’t do. So we put two beans in each plate. In this case, we have allocated 6 beans out of the 7, and what should we do with the one left? This is the remainder. Putting it aside.

We can connect bean allocation to the model of equations and remainders. The dividend is the number of beans, while the divisor is the number of plates. The number of beans in the plates is called quotient. The beans left outside the plates is the concept of remainders. With the understanding of the acquired knowledge, we can create new mathematic questions. What is the relationship between remainders and divisors? In our experiment, students could clearly answer that the remainder beans must be fewer than the plates, that the remainder beans must be fewer than the plates, otherwise, we can put one more bean in each plate. Therefore, we extract an abstract concept. The remainder is less than the divisor.

In the process, first, in the classes with modelling, the following behaviours decreased to zero, such as quietness due to students could not understand, the teachers compellingly asking students questions, or criticising students as not smart. Second, the teachers’ lecturing and students’ expressions as commanded by the teachers considerably reduced as commanded by the teachers considerably reduced in comparison with traditional lectures. in comparison with traditional lectures. Additionally, students actively expressed their own discoveries, for example, the remainder beans were fewer than the plates. The quality of the teacher-raised questions was also much higher. Teachers’ expressions of acknowledging students’ responses were more than traditional lectures. were more than traditional lectures.

These led to the changes of teachers’ ideology and behaviours in the classrooms, which indicated the effectiveness of the variation theory on empirical education.

Professor Fan: If new teachers would like to learn and apply the variation theory, what suggestions would you provide for them?

Professor Gu: My first suggestion is to start with conceptual variation for the new teachers. Mathematic concepts are highly abstract, and therefore their extensions vary a lot. There are many variations, which can be confusing, but concepts are the basic knowledge of student learning. Also, as the first step, it is practical for new teachers to start. This is the first step. Then, we go to the second step of the basic approach of solving mathematic problems. For example, we can choose common mathematic questions, and provide scaffolding for students with procedural variation. The scaffolding refers to the step-by-step reduction from the unknown to the known, and from the complicated to simple knowledge, which is Pudian in Chinese.

It is crucial for the teachers to stimulate students’ problem-solving abilities. Upon this, the third step is to ask students to try to solve some new mathematic questions which need further exploration. The second suggestion is based on a problem we met before. It is to avoid cramming in disguise of variations. Teaching with Variation should not be misunderstood as mechanic training of solving problems. While it looks like flexible variations, it becomes more robotic and tedious indoctrination. The third suggestion is variation surrounding core connection. The variation does not mean ‘the more, the better’, especially in the actual classroom teaching. It does not mean ‘the more difficult, the better’ either.

The most important thing is to carefully consider the principle and the level of variations. The principle of mathematic study is to stimulating students’ advanced cognitive thinking and conceptual understanding of mathematics. The level of variations depends on the achievement of students’ different learning goals. Too many variations will only decrease students’ learning results. These are the suggestions I provide for new teachers.