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Professor Gu:  Because the primary education in China

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was experiencing the most difficult era at that time,

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we investigated into high school graduates in Qingpu in 1977.

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 Nearly 1/4 of the 4373 sample students 

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could not solve questions of adding and subtracting fractions. 

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They could not work out basic questions 

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like calculating time with given distance and speed either, 

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not even mentioning solving geometric questions. 

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The reason was there was no substantial mathematic classes. 

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The mathematic concepts were ambiguous, 

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and the mathematic training was in confusion. 

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As an effective teaching method was in urgent need

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 to change the situation at that time, 

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Teaching with Variation became one of the important methods. 

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Then, at the early stage of implementing Teaching with Variation, 

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there was a summary article presented at annual conference of 

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which was ‘The visual effect and psychological implications of 

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Shanghai Mathematics Association in 1981, 

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which was ‘The visual effect and psychological implications of 

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transformation of figures on teaching geometry’. 

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For the first time, the article distinguished variation in mathematics into two general categories, 

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which are conceptual variation and procedural variation. 

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We applied these two categories of variation 

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from experimental schools to all the schools in Qingpu, 

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and we saw significant improvement in mathematics teaching quality. 

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Later we have researched on Teaching with Variation for over 30 years. 

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From 1981, we started to explore 

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the psychological characteristics of learning via variation. 

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The psychological characteristics revealed emotional willingness. 

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It means, among the characteristics, 

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we found emotional willingness is the most important motive. 

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And there was accumulation step by step, 

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because mathematics is systematic and in logic sequences. 

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Another feature was trial activities, emphasising the inquiry spirit of students. 

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Finally, based on students’ feedback 

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we adjusted our ways of teaching in time. 

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Since 1990 we explored the ‘core connection’ 

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between knowledge levels and students’ potential, 

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and discovered findings including step-by-step teaching method

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 from operation, understanding, mastery to exploration. 

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These finding have been widely promoted in Shanghai and in the whole country. 

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

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Professor Gu:  Conceptually, Teaching with Variation is a teaching methods, 

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which applies different forms of materials, examples and variation processes

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 to develop a profound understanding of a specific concept, or to solve related questions. 

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This is the general variation application in education. 

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In mathematics, there are two main categories of variation. 

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One of them is conceptual variation. 

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Conceptual variation refers to extracting the essential attributes of concepts

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 from a variety of forms of mathematic materials, 

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and deciding the accurate extensions of the concept. 

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This is because mathematic concepts have numerous extensions. 

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Procedural variation targets at solving problems. 

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It develops mathematic questions based on concepts, 

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and aims at solving the problems via various processes

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 like logical reasoning or modelling etc. 

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These two variations are the fruits of mathematic teachers’ collective wisdom

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 that have been accumulated in classroom teaching improvement for long.  

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Professor Fan:  Thanks. Is there any example to demonstrate

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 the application of the variation theory in mathematic classrooms? 

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Professor Gu:  This is a good question, 

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because examples are the best illustration to explain

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 the essence of the variation theory. 

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My example is the modelling of division with remainders. 

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For primary school students, 

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division requires comprehensive calculation skills, 

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especially division with remainders. 

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The teacher will instruct like this. 

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Division is like allocating beans. 

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We allocate 7 beans into 3 plates, 

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and how many beans can be allocated to each plate? 

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Let’s try. 

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One bean in each plate. There could be more. 

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Then two beans. Three beans won’t do. 

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So we put two beans in each plate. 

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In this case, we have allocated 6 beans out of the 7, 

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and what should we do with the one left? 

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This is the remainder. 

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Putting it aside. 

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We can connect bean allocation to the model of equations and remainders. 

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The dividend is the number of beans, 

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while the divisor is the number of plates. 

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The number of beans in the plates is called quotient. 

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The beans left outside the plates is the concept of remainders. 

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With the understanding of the acquired knowledge, 

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we can create new mathematic questions. 

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What is the relationship between remainders and divisors? 

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In our experiment, students could clearly answer

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 that the remainder beans must be fewer than the plates, 

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 that the remainder beans must be fewer than the plates, 

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otherwise, we can put one more bean in each plate. 

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Therefore, we extract an abstract concept. 

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The remainder is less than the divisor. 

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In the process, first, in the classes with modelling, 

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the following behaviours decreased to zero, 

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such as quietness due to students could not understand, 

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the teachers compellingly asking students questions, 

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or criticising students as not smart. 

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Second, the teachers’ lecturing and students’ expressions

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 as commanded by the teachers considerably reduced 

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 as commanded by the teachers considerably reduced 

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in comparison with traditional lectures. 

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in comparison with traditional lectures. 

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Additionally, students actively expressed their own discoveries, 

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for example, the remainder beans were fewer than the plates. 

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The quality of the teacher-raised questions was also much higher. 

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Teachers’ expressions of acknowledging students’ responses

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 were more than traditional lectures. 

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 were more than traditional lectures. 

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These led to the changes of teachers’ ideology and behaviours in the classrooms, 

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which indicated the effectiveness of the variation theory on empirical education. 

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Professor Fan:  If new teachers would like to learn and apply the variation theory, 

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what suggestions would you provide for them? 

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Professor Gu:  My first suggestion is 

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to start with conceptual variation for the new teachers. 

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Mathematic concepts are highly abstract, 

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and therefore their extensions vary a lot. 

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There are many variations, 

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which can be confusing, but concepts are the basic knowledge of student learning. 

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Also, as the first step, it is practical for new teachers to start. 

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This is the first step. 

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Then, we go to the second step of the basic approach of solving mathematic problems. 

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For example, we can choose common mathematic questions, 

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and provide scaffolding for students with procedural variation. 

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The scaffolding refers to the step-by-step reduction from the unknown to the known, 

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and from the complicated to simple knowledge, 

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 which is Pudian in Chinese.

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It is crucial for the teachers to stimulate students’ problem-solving abilities. 

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Upon this, the third step is to ask students to try to solve

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 some new mathematic questions which need further exploration. 

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The second suggestion is based on a problem we met before. 

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It is to avoid cramming in disguise of variations. 

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Teaching with Variation should not be misunderstood

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 as mechanic training of solving problems. 

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While it looks like flexible variations, 

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it becomes more robotic and tedious indoctrination. 

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The third suggestion is variation surrounding core connection. 

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The variation does not mean ‘the more, the better’, 

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especially in the actual classroom teaching.  

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It does not mean ‘the more difficult, the better’ either. 

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The most important thing is to carefully consider the principle and the level of variations. 

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The principle of mathematic study is to stimulating students’ 

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advanced cognitive thinking and conceptual understanding of mathematics. 

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The level of variations depends on the achievement of students’ different learning goals. 

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Too many variations will only decrease students’ learning results. 

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These are the suggestions I provide for new teachers.