Interview with Professor Gu

Professor Gu: Because primary education in China was experiencing the most difficult era at that time, we investigated high school graduates in Qingpu in 1977. Nearly 1/4 of the 4373 sample students could not solve questions involving adding and subtracting fractions. Neither could they work out basic questions like calculating time with given distance and speed, or solving geometric questions.

The reason was that there were no substantial mathematics classes. Mathematics concepts were ambiguous, and the mathematics training was confused. An effective teaching method was needed urgently to change the situation. Teaching with variation became one of the important methods. Then, at the early stage of implementing teaching with variation, there was a summary article at the annual conference of the Shanghai Mathematics Association in 1981 which was called ‘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: conceptual variation and procedural variation. We applied these two categories of variation, which had been trialled in experimental schools, to all the schools in Qingpu, and we saw significant improvement in mathematics teaching quality. We then researched 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. There was accumulation step by step, because mathematics is systematic and takes place in logical sequences. Another feature was that we trialled activities that emphasised the inquisitive nature of students. Finally, based on students’ feedback we adjusted our ways of teaching over time. From 1990 we explored the ‘core connection’ between knowledge levels and students’ potential.

Among other things, we discovered the step-by-step teaching method: students move from operation, understanding, mastery to exploration. These findings have been widely promoted in Shanghai and in the whole country.

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

Professor Gu: Conceptually, teaching with variation is a teaching method, 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 finding the essential characteristics of concepts from different forms of mathematic materials, and deciding the accurate extensions of the concept. This is because mathematic concepts have numerous extensions. Procedural variation targets solving problems. It develops mathematical questions based on concepts, and aims at solving the problems through various processes Such as logical reasoning or modelling, etc.

These two categories of variation have been gathered from mathematics teachers’ collective wisdom, looking at improving classroom teaching improvement over a long time.

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

Professor Gu: This is a good question, because examples are the best illustration to explain the variation theory. My example is the modelling of division with remainders. At primary school, division requires comprehensive calculation skills, especially division with remainders.

The teacher will say something like this: “Division is like sharing beans. If we divide 7 beans over 3 plates, how many beans will end up on each plate? Let’s try. One bean on each plate. There could be more. Then two beans. Three beans won’t do. So we put two beans on 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. Put 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 on the plates is called the quotient.

The beans left outside the plates is the concept of remainders. With the understanding of this new knowledge, we can create new mathematical 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, otherwise, could put one more bean on each plate.

Therefore, we extract an abstract concept: the remainder is less than the divisor. During the process,

the following behaviours decreased to zero: silence due to students not understanding, teachers forcing students to answer questions, or criticising students for not knowing answers. In addition, the teachers’ teaching and students’ expressions as commanded by the teachers considerably reduced in comparison to traditional classes Students actively expressed their own discoveries, for example, the remainder beans were fewer than the plates. The quality of the questions the teacher raised was also much higher. Teachers acknowledged students’ responses more than in traditional classrooms. These led to changes in teachers’ ideology and behaviour 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 for new teachers is to start with conceptual variation. 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 it is the first step, it is practical for new teachers to start here. Then, we go to the second step of the basic approach of solving mathematics problems. For example, we can choose common mathematical 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 teachers to stimulate students’ problem-solving abilities.

Therefore, the third step is to ask students to try to solve some new mathematics questions which need further exploration. The second suggestion for new teachers is based on a problem we have met before. It is to avoid misunderstanding variations as spoon-feeding education. Teaching with variation should not be misunderstood as mechanical training of solving problems. Although it looks like flexible variations, it is actually robotic and tedious indoctrination. The third suggestion is variation surrounding core connection. Variation does not mean ‘the more, the better’, especially in 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 mathematical study is to stimulate students’ advanced cognitive thinking and conceptual understanding of mathematics. The level of variation depends on the achievement of students’ different learning goals. Too many variations will only decrease students’ learning results. These are my suggestions for new teachers!