What Is Variation Theory? A Guide For Teachers

This blog will look at the concept of Variation Theory and attempt to demystify the confusing language surrounding it. It is hoped that this will make Variation Theory an accessible and exciting tool that you can begin to use in your math lessons, or give you an understanding of what is happening if you recognize the principles, despite never having heard of Variation Theory before!

What is Variation Theory in elementary math?

Variation Theory is a general theory of learning that is used predominantly in mathematics teaching and pedagogy. It emphasizes a concept’s essential features by focusing on what is kept the same and what changes, which offers the opportunity to make meaningful connections.

Origins of Variation Theory

The study of Variation Theory arises mainly from work across two groups of researchers who arrived at similar, but not exactly the same conclusions.

One was Swedish researcher Marton, F. and colleagues who worked on the concept of phenomenography, which developed into Variation Theory; the other was Gu Ling Yuan and his team in Shanghai who called it Bianchi (teaching with variation).

As one would expect, the emergence of two similar theories has caused confusion in some areas of the sector and educational research. In the context of the US, the distinction between them is rarely made and so aspects of Bianchi are used under Variation Theory.

They both agree that ‘the central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features’. (Gu, Huang & Marton, 2004). This blog will mainly draw on the learning study of Gu and his team.

The application of Variation Theory is much wider than just the domain of mathematics teaching, but rather is a theory that underpins how students learn anything. The principle that underpins the theory is that new learning occurs when learners understand how a critical aspect of a new concept changes (varies) against something that does not change (invariance).

By adjusting their perception of what has remained the same and what has changed, students are more likely to learn the object of learning and understand the underlying mathematical structures being studied.

In other words, it is when an educator carefully constructs questions, considers the order in which they appear, and brings the thinking process between questions to the forefront for the learners, so that meaningful connections to the mathematical structures can be made from the task.

It is unlikely that a worksheet on its own will bring about the required cognitive process in students needed to benefit from the advantages that Variation Theory offers students in their conceptualization of mathematical structures.

Do not be deceived into purchasing paid resources that promise to bring about Variation Theory with just a worksheet, unless you can specifically identify how the pattern of variation enacts the principles of the object of learning, and you feel confident that you can bring out these mathematical structures to the students through your questioning in a very deliberate way.

Indeed, as you will see through some examples, attempts at Variation Theory through only a worksheet approach and no dialogue risks student learning less than not trying to use Variation Theory.

What are the types of Variation Theory?

There are two types of Variation Theory; conceptual variation and procedural variation.

What is procedural variation in math?

Procedural variation is a process for the formation of mathematical ideas stage by stage, in which students’ experience in solving problems is manifested by the richness of varying problems and the variety of transferring strategies.

It is derived from three parts:

1. Varying the problem
2. Multiple methods for solving a problem
3. Multiple applications of a methodology to similar problems

As with conceptual variation, the key to success is to teach students to look for connections within the mathematical structures from an early age.

What is conceptual variation in math?

Conceptual variation is a process that looks at providing students with multiple perspectives and experiences within a given concept or object of learning. It refers to the strategies a teacher may employ to bring to the forefront critical aspects of the desired learning outcome, and increase their understanding by demonstrating it from different perspectives.

Marton and his team of researchers concluded that students notice difference between different objects of learning before they notice sameness.

When employing conceptual variation pedagogy in classroom discourse, it is best to provide examples that are totally different before looking at examples that are similar. For example, if you wanted to demonstrate to students what a triangle was, it would be best to present a triangle along with other shapes that are not triangles to help students distinguish what it is that makes a triangle.

Is this an elephant?

A common non-mathematical approach to explain this is to look at the animal kingdom.

How is it that we are acutely aware of what an elephant is? We are able to distinguish it quite easily from the countless other members of the animal kingdom. If pressed to answer how we know the picture above is an elephant, the first two features that people describe to explain their certainty are the trunk and tusks.

To which my reply is this: is this therefore NOT an elephant? In order to be aware of certain aspects of a phenomenon, we must experience variation.

Clearly, it is an elephant. But this is a tuskless elephant, and tusks are one of the distinguishing features of an elephant that many people claim. Perhaps we could go on to explain that the size of an elephant is a distinguishable feature, or the gray color. An elephant must be gray, have a trunk, be big, and can have tusks, but this is not a requirement.

My reply is: if that is the case, is this an elephant?

Patently it is. Yet, this is hardly a big animal and so would not meet the conditions placed upon it to characterize it as an elephant from above. Therefore, an elephant can be large or small, have tusks or not have tusks, be gray and have a trunk. The necessary conditions of what constitutes an elephant become ever clearer.

We also know, instinctively, that this picture is not an elephant despite it being small, gray and not having tusks. We know it cannot be because of the ears, the shape of the head and the lack of trunk. And here, through our visual perception, the idea of Variation Theory comes to life.

By seeing what it is to be something in as many different forms as possible and in forms that are related, be it closely or otherwise, we are able to see that concept from multiple perspectives and experiences to truly know it. At its most basic level, this is conceptual variation.

Examples and non-examples of Variation Theory

As mentioned above, we tend to recognize sameness before we recognize difference. It is from this principle that we can consider introducing critical aspects of mathematical structures through examples and non-examples that we share with students.

In this instance, to remain true to Marton’s findings, the examples that demonstrate the critical object of learning that we want students to learn will be invariant, but the background in which we teach will vary. Subsequently, when we show non-examples they act, as one might imagine, to define what the critical aspect is not.

For example, if teaching the meaning of the equal sign, you could have examples of different equations using all four operations where the answer was correct and examples using all four operations where the answers were incorrect and so the equivalence sign could not be used.

Standard and non-standard in Variation Theory

Once students have developed a sense of what the critical aspect is and is not, it is important to then demonstrate this aspect in both standard and non-standard forms.

I remember the first time that students I was teaching came across an equation on where the equal sign came before any digits and other symbols ‘__ = 8 + 9’. There were looks of bewilderment as if they had been tasked with the impossible. The fault of this confusion was entirely my own as every question I had ever asked focused on the standard form when writing equation ‘8 + 9 =__’.

This meant that students had not fully grasped the concept of the symbol and its critical feature; that it is there to represent equivalence, not just as a symbol that is used to write down an answer afterwards.

In my teaching sequence for teaching equivalence and the equals sign, after looking at examples and non-examples, I should have then looked at standard and non-standard forms like those below:

__ = 8 + 9

10 + 7 = 8 + 9

This graphic might be useful in highlighting how these components fit together.

Procedural variation: what’s the same, what’s different and what do you notice?

We can use procedural variation to highlight patterns within a set of questions that bring about a critical learning point or object of learning. Say we wanted to draw students’ attention to one of the field axioms of multiplication – distributivity – so that they can employ it at will when dealing with any multiplication. We may present this in the following way:

The incorrect way to go about this would be to treat this as a worksheet, in which you deliver an input and tell the students to go ahead and complete it. There would be no method to that madness, and the opportunities for bringing out new and previous learned relationships between the numbers would likely fall by the wayside for all students, or would only be found by those who are high attaining in mathematics.

Rather, a more controlled approach should be taken. You should ask students to complete the first two questions and then ask them, ‘what is the same, what is different and what do you notice?’ This will provide the opportunity to pick up on what the design of these minimally different questions is attempting to draw out.

Procedural variation steps

Firstly, you would want students to notice the relationship between the multiplicand from the given statement (12) and the combination of the multiplicands from the first two equations (11 + 1 and 10 + 2). Students will be drawn to the fact that they all total the same and that the product is also going to be 96.

From here, students can see if that holds true for the next two questions. The danger of letting students complete all 4 questions first without discussing it could result in students seeing that the product is the same, but not understanding the relationship between the multiplicands.

Notice that the 5th question in the sequence does not maintain the same pattern exactly. The multiplicands total 12, but the multiplier is now half of the original. This is to stop students aimlessly writing 96 down without giving any thought.

Crucially though, the statement at the beginning can still be used to solve this as the answer will be half of 96. This relationship between the multipliers can also be brought out. Question 6 allows the commutative property of multiplication to come through, as well as the associative property, and the final question reinforces division as the inverse of multiplication.

At all points of getting students to answer these questions in a controlled environment, the questions, ‘what is the same, what is different and what do you notice?’ are the ones used most, so that students can attend to those differences and similarities. Indeed, completing the above exercise could take up the vast majority of the lesson and students would get more from completing it than a whole worksheet.

Problem solving in procedural variation

At the heart of procedural variation are three forms of problem solving:

1. Varying a problem – extending the original problem by varying the conditions, changing the results and generalization;
2. The use of multiple methods of solving a problem by varying the different processes of solving a problem and associating different methods of solving a problem;
3. Multiple applications of a method by applying the same method to a group of similar problems.

1. Procedural variation: varying the problem

This form of variation aims to consolidate a concept by varying certain conditions to bring information, often information that is in conflict with students’ current understanding of the concept, to light. Look at the example below:

The variation within these problems should be self-evident. The amount of apple juice remains a constant, with only the amount that each jar holds getting varied question by question. Students are often comfortable with division as sharing into equal groups (especially at elementary school), where students may generalize that dividing results in making something smaller.

It is this misconception that the example above attempts to exploit through the last three questions. Indeed, we see that dividing by a whole number and decimal produces a quotient greater than the dividend.

2. Procedural variation: varying the method

This type of variation is one that many teachers do, but may not be aware of. Simply teaching and demonstrating different methods in solving a problem is the first step in this type of variation, but its success lies in being able to draw out what students notice about the similarities and differences between each representation. Here are a few worked examples to vary the method for solving 12 × 5.

3. Procedural variation: multiple application of a method to similar problems

This final example looks at bringing about variation, not through changing the method, but with the application of a method or representation against a group of varied yet similar problems.

Variation Theory examples for elementary school

In elementary school, this may look like the following:

1) James has 8 flowers and Julie has 5 flowers. How many flowers are there altogether?

2) James has 5 flowers and Julie has 8 flowers. How many flowers are there altogether?

3) There are 13 flowers, some are red and some are blue. 8 are red. How many are blue?

4) There are 13 flowers, some are red and some are blue. 8 are blue. How many are red?

Here the problems are varied, yet placed within a similar context, and the numbers used are the same, yet what each question is asking students to do differs slightly.

For example, for question one, students may use Cuisenaire rods to visualize the problem. They would lay down the yellow rod and the brown rod to make a train and then the orange and the green above (or below) to help solve the total.

Question two enables students to experiment with the commutative property of addition. Notice how all the rods used are the same, but the order in which the bottom train has been constructed has changed. The underlying structure remains unchanged.

Question three tells us that there are thirteen flowers altogether and 8 of them are red. The next step would be to see how many more are needed to take 8 up to 13. Students may initially use white rods before exchanging it for a yellow rod of equal value. Again, the underlying structure of the problem remains unchanged.

The final question is merely the inverse of question 3. We know the total amount of flowers and that a certain subset of them equals 5. Therefore, to find the missing amount, we need to count up from 5 until we get to 13. Again, students may need to use 8 white rods first, but they will see that this can be exchanged for a brown rod.

By the end of this exercise, students will be able to see that the 4 questions have similar underlying structures despite the variation within the problems.

Variation Theory worked examples

As mentioned above, just providing students with questions that contain aspects of Variation Theory will not bring along the benefits of the approach. Therefore, if you do not understand the variation taking place, it would probably be best not to use those examples.

For each of these, we need lots of examples of what the type of variation you’re referring to looks like ‘in practice’. In other words, with reference to regular sorts of questions and learning objectives from your school’s curriculum, so teachers can read it and see how they could try implementing it in their own lesson.

If you can pull from different topics in your curriculum that would be great, but it’s really just a case of showing the wide variety of ways people can think about including this practice in their teaching.

Number examples: number and place value

The examples below vary the representations to give access to the same idea. This approach can be easily transferred across all grade levels to look at place value objectives. Using this in an elementary class would involve looking at each representation and guiding the students through the process of asking ‘what is the same, what is different, and what do you notice?’

Introducing students to these questions early on will hopefully result in them becoming internalized automatically and so students will come to naturally ask themselves these questions.

The use of color is particularly useful here as it supports connections between the place value counters, cards and base 10, but the presence of the Cuisenaire rod and the Numicon (whose colors do not conform to this) stops the association of ‘tens’ being green etc.

A crucial element to draw out of students is the unitization of 2 in some representations or where the 2 has been conserved as 2 ones. The number grid provides an opportunity to link these representations within the abstract number system.

Number examples: addition and subtraction

563 – 100 = ___ – 200

387 – 100 = ___ – 200

983 – 100 = ___ – 200

___ – 100 = 874 – 200

This question focuses on subtracting by multiples of 100 and students’ understanding of the equal sign. It is the subtrahend that remains invariant while the minuend varies. This is done so that generalizations can be brought about in regard to the subtraction of multiples of 100. The minuends themselves play no major role in bringing about the effects of Variation Theory.

Here, we would want students to understand that ‘equals’ does not mean ‘answer’, but rather that two mathematical statements are balanced and produce a product that is the same as each other.

We can also bring about the relationship between the two minuends, one of which is unknown, when compared to the effect of subtracting the first one by one hundred. Namely, that the tens and ones do not change and that as the subtrahend has increased by 100 from 100 to 200. In order for the proportionality to remain constant between each statement, the minuend must also increase by 100.

It would be in the interest of students to quickly subtract a multiple of 100 from a 4-digit number where the difference is a 3-digit number in subsequent lessons so then they can see the impact this can have on the generalizations they are producing.

Number examples: multiplication and division

What is the same/different and what do you notice?

__ × 5 =__

5 × __ =__

__ =__ × __ 

__ + __ + __ + __ + __ = __

---

__ × 7 =__

7 × __ = __

__ =__ × __

__ + __ + __ + __ + __ + __ + __ = __

Write what you notice below.

This task links multiplication to the array/area model of multiplication, the understanding of the equal sign, the commutative property of multiplication and multiplication as repeated addition.

Students will recognize that the answer is 35 and that in the first model they can see 7 groups of 5 when looking vertically across from left to right. When looking at it horizontally from top to bottom, 5 groups of 7 can be seen.

They can use these facts to write a multiplication equation where the product is written first to bring about their understanding of the equal sign, and relate the multiplication of each vertical column to repeated addition of the number of columns in total.

The opposite is the case with the second representation where there are 5 groups of 7 on the horizontal and 7 groups of 5. Such activities will aid in students’ understanding of the difference in visual representation when the same numbers are used as multiplicand and multiplier across two different equations.

Number examples: fractions, ratio and proportion

This series of questions is designed to link all the above areas of fraction as well as ratio. This is important as the connections between percentages and ratio can be missed and so more time than needed is used to teach the concept of ratio.

Here the methods used vary but the question is exactly the same, albeit in the context of the concept that it is drawing out.

As such, the answer will be exactly the same for each question (24), but the methods and mathematical thinking behind each approach will draw out the connection between the question, as well as the different representations that have been used, such as the numerator and denominator of a fraction. Notice how all of the above are either concrete or pictorial.

Measurement examples

The following task comes from @J0shMartin and his contribution to Craig Barton’s Variation Theory website. Although this is predominantly a high school-based website, there are some examples that are relevant to elementary school. The following task requires students to notice patterns when converting between measurements (something students should have some understanding of doing BEFORE they embark on this task).

The answers are provided below:

1. 3 m
2. 0.3 m
3. 0.3 m
4. 0.03 m
5. 4500 mm
6. 450 cm
7. 45 cm
8. 5 cm
9. 50 mm
10. 15000 m
11. 1500 m
12. 100 m
13. 50 m

As this is a context that many teachers are familiar with, here it would be best for you to try and consider how the questions have been carefully crafted to bring about variation, and how you might go about using this sequence within your own lesson.

Consider the questions you would ask, the number of questions you would get the students to answer before stopping them to discuss the task so far, and how you would explain the patterns to the students.

Geometry examples: properties of shapes

Property of shape is a particularly useful curriculum area to deploy Variation Theory and can be used from kindergarten onwards to great effect. Below is a sequence of slides used in a lesson on describing rectangles with some kindergarten students.

The rotation, color and the type of shape have all varied so that students can generalize about the properties of rectangles (many believed that the ones on slide 17 and 19 were not), so they can really see the distinguishing features of the shape and not get caught on elements that are not important.

This is the conceptual variation as outlined at the start of this blog in action. It uses examples and non-examples, standard and non-standard examples to draw out what it means for a shape to be a rectangle.

Enjoyed this? Why not try our ‘How I Wish I’d Taught Math’ series, inspired by Craig Barton’s bestselling book:

• Cognitive Load Theory 
• Direct Instruction And Rosenshine Principles of Instruction 
• Goal Free Problems And Focused Thinking 
• Most Effective Teaching Strategies

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The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza