Discussion: Separation and similarity – the forgotten patterns of variation Free

Separation and contrast are distinct patterns of variation. Contrast focuses on differences between categories, such as comparing linear and quadratic functions. Separation, on the other hand, focuses on differences within a single category, allowing learners to distinguish between aspects of the same concept – for example, identifying the slope and the intercept within a linear function. Marton and Pang (2013) describe the necessity for learners to discern an aspect by varying it against a background of invariance. This description, though not explicitly labelled as separation, points directly to its function. Without separation, learners risk conflating related aspects, such as the numerator and denominator in fractions, or the variable and constant in algebraic expressions. For example, in algebra, students often fail to distinguish the role of the coefficient from that of the variable in expressions like 3x. If separation is not foregrounded, teaching may inadvertently reinforce misconceptions – students may see 3x as 3 plus x or as a number whose digits are 3 and x rather than 3 multiplied by x. Only by separating the aspects of the number of groups and what is being grouped can learners grasp the multiplicative meaning of the expression.

Empirical classroom research supports this claim. For example, Kullberg et al. (2017) demonstrate how, in early algebra lessons, the distributive property was not made accessible for learning because it was never explicitly separated from other operations. Only when later lessons deliberately isolated and varied this aspect against a background of invariance did the distributive property become discernible. This example demonstrates why separation must be explicitly recognized: otherwise, key mathematical structures risk remaining invisible to learners. The study by Mårtensson and Ekdahl (2021) also provides empirical support for the importance of separation in mathematics teaching. The results show that pre-service teachers modified tasks to make key aspects of concepts easier to notice, for example, by separating addition from subtraction or by explicitly emphasizing particular elements of a problem. These redesigns allowed students to focus on one aspect at a time, avoiding confusion and helping them to distinguish related elements – exactly what the separation pattern prescribes. Without such deliberate separation, students risk conflating related features, as often happens in algebra when coefficients and variables are misunderstood.

Similarity is an important pattern of variation, even though it is often hidden under other terms. Marton and Pang (2006) emphasize the importance of experiencing the same meaning across different contexts. Lo (2012) similarly points out that learners need to recognize the stability of a concept's meaning. However, neither study presents similarity as a distinct pattern of variation. Olteanu (2015, 2017, 2018, 2022) explicitly defines similarity as a pattern of variation that helps learners recognize the meaning that remains the same across different examples or representations. In mathematics, this pattern underlies generalization. For example, understanding that an equation represents a balance is true whether it is written as 2 + 3 = 5 or 3x – 4 = 11. Without similarity, learners may see each equation as completely separate, failing to grasp the underlying principle.

In several studies, researchers have used similarity as a pattern of variation, although often implicitly. For example, in Kullberg et al. (2017), the results showed that in the first cycle of teaching, concepts were introduced in isolation, and students struggled to make connections between them. In later cycles, when multiple representations of equations were presented together, students could see that the underlying structure of balance remained constant across different forms. In other words, what supported learning was not only contrast but also the experience of similarity – the preservation of meaning across variations. Another example of similarity is provided by Mårtensson and Ekdahl (2021). In their study, pre-service teachers used the same metaphor and representation – such as balances – to represent equations across multiple tasks. This variation helped students discern that the underlying mathematical principles remain the same, even when the examples appear different. This example illustrates the use of similarity and demonstrates that the use of this pattern of variation helps learners generalize meaning across different examples. Without attention to similarity as a pattern of variation, students' knowledge can become fragmented, making it difficult for them to see the shared principles underlying various cases. The pedagogical risk of ignoring similarity is clear: teaching can produce fragmented knowledge, where students cannot connect experiences across different contexts. By naming and theorizing similarity, teachers gain a tool for deliberately designing opportunities where students encounter variation yet maintain conceptual stability.

Much of the literature collapses separation and similarity under the broader heading of contrast. While contrast is powerful, it cannot carry the full explanatory load.

1. Contrast highlights differences between phenomena.

2. Separation highlights differences within phenomena.

3. Similarity highlights constancy of meaning across variation.

4. Generalization highlights the necessary conditions for the perceived constancy of meaning to hold in new and unknown instances.

Research has demonstrated the drawbacks of superficial treatment of separation and similarity in different ways (cf. Kullberg et al., 2017; Mårtensson and Ekdahl, 2021). While it may be possible to overlook these two patterns of variation when working with basic objects of learning in mathematics, this is not the case when dealing with more complex objects of learning. We illustrate the consequences of this briefly with mathematical series (cf. Baskoro, 2021). Significantly, we speculate on the potential of experience with patterns of variation for arriving at a suitable solution. Given:

$pq=p+q:$ $∀p,q≠0;q−1≠0$⁠, find the value of the following series:

One of the solutions to this task is:

thus,

Since

$pq=p+q,$ we obtain

thus,

Several aspects need to be visible for students to appreciate the solution to the task below. Using the patterns of variation mentioned earlier, it is demonstrated how these can be applied to make the underlying mathematical relationships discernible. In this example, the variation patterns’ similarity and separation are both evident. The pattern of similarity is demonstrated through different algebraic expressions that represent the same underlying mathematical structure. Initially, the infinite geometric series

is presented. This structure is then extended by introducing the factor $1p.$

It is possible to talk of different layers of contrast depending on the object of learning, e.g. in delineating different mathematical series, then it is possible to contrast arithmetic and geometric series, divergent and converging series, etc. Specific to this task, providing a counterexample, i.e. $pq≠p+q$ might not be sufficient, probably leading to questions such as “why does the rule only apply in certain cases?”. Students need separation to see how the unique solution depends on the condition. Both expressions are based on the same geometric pattern, even though they involve different parameters. This similarity enables students to discern the invariant structure of the geometric series across varying forms and to understand its general applicability. The pattern of separation becomes visible through the contrasts established between quantities and algebraic transformations within the derivation. Initially, q functions as the variable in the geometric series, while p is treated as a multiplicative constant. When the relationship pq = p + q is introduced, the expression transforms as follows:

This step separates the roles of p and q, clarifying how their relationship affects the overall expression. Furthermore, the process distinguishes between summation, substitution, and simplification, thereby making each step's contribution to the final equality visible. Through these variations, critical aspects of the reasoning process – such as the interaction between factors and the transformation of expressions – are brought into focus. In this way, the task exemplifies how similarity helps reveal what remains constant across different representations, while separation highlights the distinct elements that together lead to a coherent understanding of the solution.

Assuming students can contrast arithmetic and geometric series, it is expected that the structure of the task might, on the strength of similarity, point to a kind of geometric series with $(a=1p,r=1q)$⁠. However, this might be difficult to perceive if the task is not further simplified – that is, separated. Thus, separation is needed here to reveal similarity. Only after simplifying does the geometric structure become apparent, and similarity across series can be perceived. With this insight, students can reflect on and apply the general case

and thereby engage in generalization. The task can therefore be seen as highlighting, or separating, the general geometric series from a special case.

The findings of Kullberg et al. (2017) also show that simply introducing contrast is insufficient. Even when variation was present, students sometimes failed to notice the critical aspect, because it was neither clearly separated nor explicitly linked through similarity. If we only speak of contrast, we fail to account for why learners confuse related aspects, or why they fail to see the general across the particular. By restoring separation and similarity as independent patterns, we gain both analytical clarity and pedagogical guidance. The findings of Mårtensson and Ekdahl (2021) show that separation and similarity, as patterns of variation, have direct implications for teaching. Separation helps students avoid confusion by distinguishing critical aspects, while similarity supports them in generalizing across different examples. Suppose these patterns are treated only as part of contrast. In that case, it becomes difficult to explain why students struggle with both making internal distinctions within a concept and applying principles across contexts.

The international research community has already provided evidence that separation and similarity are important patterns of variation. Marton and Pang (2013) and Lo (2012) describe these patterns implicitly, while Olteanu has defined and operationalized them explicitly. Kullberg et al. (2017) present classroom data demonstrating how the absence of these patterns limits what can be learned. The current task is therefore not to invent new patterns, but to recognize and legitimize what is already present in practice. Since different examples of contrast afford different possibilities for discerning critical aspects, the use of contrast alone may not necessarily lead to separation. Moreover, separation is a prerequisite for generalizing the meaning of mathematical concepts to other examples. Students' ability to generalize depends on the extent to which they have been able to distinguish the relevant general categories from one another. Thus, the necessity of separation is emphasized; it can be understood both as a state, in which generalization can be achieved, and as a process, through which meaning is created.

If variation theory is to remain a sharp and valuable tool for understanding learning and teaching of mathematics, it must evolve. The established patterns – contrast, generalization, fusion – are not sufficient on their own. By bringing separation and similarity out of the shadows, we equip educators and researchers with a more complete set of analytical tools for designing powerful learning environments.