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Underpinning Variation theory with research

In this video we will show how Variation theory is underpinned by research from both East (Asia) and the West.
So welcome to this video on the research basis for variation theory. I assume you’ve watched the previous video by Professor fan, and you’ve got some basic idea what variation theory is and what the role of variation can be, especially in practicing materials for example. Of course this again sounds perhaps quite like common sense, and quite logical. But just like all the other topics, they actually have quite a sound research basis as well. I will discuss some of these articles or these studies that actually are about variation. And first example comes from the book by Professor Fan. In this book he mentions the origins of variation theory. An important name in variation theory is Marton.
With his colleagues, he promoted this theory and explained what it is. It was later extended by Watson and Mason, who are based in the United Kingdom. And in this theory he proposes that learners must experience variation in the critical aspects of a concept within a limited space and time in order for the concept to be learnable. The idea behind this is that invariance structures. So let’s say you have a sequence of tasks, and some elements vary. They change in the sequence of tasks that they can actually aid the learner, and they can actually help make a concept well known.
The idea that invariance structures during changing phenomena often denote the present knowledge acquisition is an essential part of something called phenomenology. There are two types of variation, conceptual variation and procedural variation. Conceptual variation as a starting point, that concepts can be understood from multiple perspectives. Variation is created in several ways the first way is called standard concept variation. And this happens by varying the concept in a standard way by inducing concept by varying visual and concrete instances. We will see some examples later on as well. The main purpose of this type of variation is to help students establish the connection between concrete experiences and abstract concepts.
So hopefully you can again recognize some of the wording from for example the CPA and model method. The second way is something called non-standard concept variation, highlights the essence of a concept by contrasting the concept with a non-standard example. So for example if I have some triangles and I put a square in the middle of it. It is quite clear hopefully that the square is not a triangle. So this variation will aid the learning. This stresses the teaching strategy that examples should not only be the normal ones, but also the non-standard ones. Then finally the third or non-concept variation uses non concepts for example counter examples to reinforce a concept.
Then there’s also procedural variation which progressively involves unfolding mathematical activities in procedural variations. Students can arrive at a solution to a problem and form connections among different concepts step by step from multiple approaches. The word procedural here is quite important. This type of variation is also created in several ways. The first way addresses the formation of concepts, and the process of unfolding concepts. The second way uses scaffolding for problem-solving. Multiple variations analyses of the configurations of a problem do not only help students clarify the process of solving the problem and the structure of the problem, but they also are an effective way of experiencing problem-solving and enhancing the competency of problem solving.
The third way establishes a system of mathematical experiences. You can see overlap again with the themes that we covered in previous weeks. You have heard the word scaffolding which is actually quite well known from Bruner’s work, and we mentioned Bruner when we were discussing the week about concrete pictorial and abstract. You can see some examples of each type here as well. There is an example where there’s one key point but there are different methods in which it is solved. Or there is one key point but there are different numbers. You can see the variation directly on the screen here. because the numbers are changed, the numbers are varied. And this is part of this procedural variation.
And then you can also have one key point with different applications. I think the underlying theme here is that you vary the ways in which you present the tasks. You sequence them in a cunning way. You scaffold them and that helps the learner to learn the concepts. So I think this also shows how procedures go hand in hand with understanding, something we covered in the last lecture. We can go into a little bit more detail. And Marton, Runesson and Tsui actually gave four concepts that underpin variation theory. One of them is contrast. In order to experience something a person must experience something else to compare it with. Think of the example with the squares and the triangles.
The second one is generalization. In order to fully understand what “three” is, we must also experience varying appearances of three. A third point is separation. In order to experience a certain aspect of something and in order to separate this aspect from other aspects, it must vary while other aspects in variant. And then the fourth one, fusion. If there are several critical aspects that the learner has to take into consideration at the same time, they must all be experienced simultaneously. So for example, perhaps you have a graph of something and you have an image of something and you present them both at the same time.
If you have a collection of triangles, you can hear I love triangles, I’ve given several examples with them, you can apply these four aspects as well. You have triangles that actually have the same structure, for example equilateral triangles at the bottom right and the top left. You’ve got right-angled triangles. You see one on the left and at the bottom left. But there also are differences. You could argue there are differences in color of course. There are differences in angles. There are differences in acute and obtuse angles. So taking into account this variation between the different objects will actually help you implement and clarify the concepts.
I hope these examples have further emphasized what variation theory is and what research underpins it. It is important to realize that variation is not only integrated in asian lessons, but also in their textbooks. And you can seem for more examples in one of the interactive tasks that we have included in the course.

This video explains how Variation theory fits with contemporary research on learning.

As explained in the previous video by Professor Fan, Variation theory states that learners need to experience variation in the critical aspects of a concept, in order for the concept to be learnable. Varying each task within a sequence of tasks will help learning. Variation focuses on concepts in conceptual variation and on procedures in procedural variation.

Examples, non-standard examples and counterexamples play a crucial role in Variation. As with CPA, there is a role for multiple representations (see CPA). Marton, Runesson and Tsui (2004) formulate the functions of variation like this:

  1. Contrast. “… In order to experience something, a person must experience something else to compare it with…”

  2. Generalisation. “… In order to fully understand what “three” is, we must also experience varying appearances of three …”

  3. Separation. “… In order to experience a certain aspect of something, and in order to separate this aspect from other aspects, it must vary while other aspects remain invariant.”

  4. Fusion. “…if there are several critical aspects that the learner has to take into consideration at the same time, they must all be experienced simultaneously.” (p. 16)


Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton and A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). New Jersey: Lawrence Erlbaum Associates, INC Publishers.

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