External representations are used by scientists to communicate with one another and with the non-scientist population. Therefore, the meaning that is embodied in a representation by its creator should be understood by all those who perceive it [1
]. Furthermore, using external representations in a flexible manner allows scientists to manipulate those representations, thus creating a working model that can be updated when new experimental results are obtained or a new understanding is reached [2
]. Thus, it seems that the power of representations is rooted in the ability to transform them, an ability that is reserved for those who can comprehend the semiotic system.
We focus in this article on visual representations, i.e., an external representation perceived by the visual senses, which is the result of an attempt to communicate a mental model [1
]. It has been claimed that just as visual representations are important to the work of scientists and the development of science, they are also important in teaching the subject [3
]. However, students at various ages experience difficulties in interpreting and using scientific representations [5
]. They often misinterpret them, for example, they interpret an arrow in a scientific representation as a movement instead of a process or vice versa [7
]. In addition, they sometimes think that a diagram representing a structure actually represents a complex process [8
], and they often apply the subscript of an element in a chemical formula to the following element rather than the preceding one [5
Fluent performance in the transformation of visual representations has been described as requiring and demonstrating meta-visualization, namely the ability to acquire, monitor, integrate and extend learning that involves both internal and external representations [9
]. Furthermore, this ability can foster the idea that distinct representational levels of scientific knowledge exist [3
]. This means that the ability to use visual representations is supported by knowledge of the nature of visual representations and by metacognitive skills.
Visual representations in science can be organized in various ways. One method of organization, born out of the need to identify and address students’ challenges in the science classroom, is Johnstone’s [10
] distinction between three organizational levels. The first level is at the macroscopic one, that of the phenomenon being studied, as it is directly perceived. The second is at the submicroscopic level, a representation of those entities that are thought to underlie the properties being displayed. The third is the symbolic level, the representation of the identities of entities. Biology is often addressed as unique in that there are four levels of representations: the macroscopic (visible biological structures), the microscopic (cellular level), the submicroscopic (molecular level) and the symbolic (symbols, formulas, pathways, etc.; [11
At each of these levels, the same object or process can be represented by various types of representations (e.g., graph, table, chart, symbol or formula). In some cases, these representations complete or constrain the interpretation of their represented object or process [12
]. While the literature usually refers to the ability (or inability) of learners to translate information from one type of representation to another [3
], it could just as well be argued that the translation and manipulation of representations among the same types of representations should be performed and mastered by students. As an example, Gilbert [9
] points out the intellectual demands and confusion that can emerge from using several symbol systems in the classroom, such as zinc + hydrochloric acid, Zn + HCl, and Zn(s)
. While these are all visual representations of the symbolic type, each of them follows its own code, compelling the student to learn how to translate one into the other and when to use each of them.
As demonstrated above, one type of visual representation is symbols. However, while representations at the macroscopic, microscopic and submicroscopic level represent the real phenomenon on different scales, the symbolic level is a representation using formulas, equations, and diagrams. This usually makes symbols less depictive and more reliant on means of convention, rather than visual similarities. Thus, these symbols are only understood when the observer is familiar with the objects and the convention underlying the symbolic representation. This foundation might lead to a construction of an internal mental representation of the content described by the symbol [13
As in the translation and manipulation of one type of representation to a different type of representation, the translation and manipulation of one symbol to another can be viewed as both a prerequisite for understanding the symbol and a way to establish the learners’ understanding of that symbol’s interpretation. An example of this can be seen in elementary-school students’ difficulties in translating fractions to decimals and vice versa [16
]. As we illustrate in this article, the translation of information between the same types of representations cannot be considered too simple a task for the novice high-school biology student.
In biology, symbols of alleles are used by experts to depict the genetic combination of a certain phenotype or to calculate, using a Punnett square, the progeny probabilities of a certain cross. In accordance with the three levels of external representations, the macroscopic level in this case is the representation of phenotypes—the appearance and disappearance of traits in a given organism. Hidden from the learner’s eyes and quite abstract for the novice high-school biology student [18
] are both the microscopic and submicroscopic levels. According to Marbach-Ad and Stavy [11
], the microscopic level is the representation of alleles, whereas the submicroscopic level is the representation of DNA. The fourth level is the symbolic level of alleles, used to depict the genetic combinations in the individual with letters, and termed genotype.
Studies have shown that students are able to apply algorithms and use allelic symbols to solve problems in classical genetics [21
]. However, they have difficulty interpreting these allelic symbols and referring them to biological entities and processes [21
]. These obstacles in understanding and using symbolic representations might well be expected, as the difficulties are obviously greater in domains in which the represented objects or processes are not clear to the learner. Many difficulties have been recorded in understanding the concept of genes and their different versions, termed alleles [11
]. Alleles, in general, have been shown to be a challenging concept in genetics, as students seem to confuse between the terms ‘allele’, ‘gene’ and ‘chromosome’ [25
]. Even more fundamental is students’ inability to distinguish between a gene (a DNA sequence) and the genetic information encoded by that gene (which determines the precise nature of the gene product). Without this understanding, the whole concept of alleles is meaningless [27
]. In addition, when students try to explain the progeny probabilities for a specific cross, they use alleles and their symbols in their explanations, but cannot explain the underlying biological processes in which these alleles participate [22
While a vague understanding of the concepts of genes and alleles is an obstacle to the use of their symbolic representations, we have postulated that another variable affects the misuse of these symbols. As the semiotic representation is crucial for the manipulation of allele symbols (e.g., Punnet squares), the symbol must not be confused with the true object that it represents, i.e., the allele [2
]. This, claims Duval [2
], is a cognitive conflict for the learner, especially when shifting from one representation system to another. Therefore, changing from one symbol system of alleles to another might lead to confusion between alleles and their representations, especially when students have no access to the concept of alleles.
Curiously, this is exactly the case in genetics education. As an object, the allele is inaccessible to the student. In addition, while working with genetics textbooks, we noted that alleles are represented by several sets of conventions, which we term symbol systems. By this term, we mean a set of conventions that represent the method by which objects are represented by symbols. For example, in the Aa symbol system, the set of conventions includes using a consistent letter for the gene, and denoting the dominant allele by an uppercase letter and the recessive allele by a lowercase one. On the other hand, in the A1A2 system, the set of conventions includes using a consistent letter for the gene, and denoting each different allele with a different superscript number.
We wondered whether these different symbol systems are used in the same manner worldwide, and why these various systems are used. While many studies in genetics education have emphasized the difficulties in connecting a macroscopic phenomenon with the microscopic and submicroscopic levels [11
], less attention has been paid to difficulties in understanding how the symbolic level and symbol system represent the other organizational levels. In this study, we focus on this latter aspect.
The significance of multiple symbol systems is apparent when reviewing studies in mathematics education. According to Duval [2
], symbols in mathematics allow for the substitution of one sign with another, thus enabling the manipulation of representations and the transformation of symbols across a set of symbols. It might be argued that experts in mathematics can flexibly turn one symbol into another, allowing them to demonstrate their grasp of the mathematical objects. This is not the case for novices, who may encounter many difficulties in understanding and justifying the use of a specific symbol in some cases, but not in others [16
For students to be able to use representations, they should be able to understand the ‘convention of the representation’ and to construct a representation of any appropriate type for a given purpose [9
]. Thus, in genetics education, lack of a clear convention of symbols for alleles may further confuse the learner.
Considering the reported difficulties experienced by students in understanding what alleles actually are [18
], and the fact that there is no conventional symbol system representing alleles in genetics education, it is reasonable to suggest that inconsistent use of allele symbols may affect students’ ability to understand and use them. In this paper, we sought to better understand the existing symbol systems used in genetics textbooks and high-school students’ conceptions of them.
Since textbooks provide a structure for classroom activities in science [29
], we chose to focus, in the first stage, on mapping the symbol systems that are used in genetics textbooks and understanding the reasons for moving from one symbol system to another in each given book. In the second stage, we identified the meanings attributed by students to the differences in symbol systems used in genetics. Accordingly, our research questions were:
In this study, we aimed to learn more about the symbol systems used to represent alleles in genetics textbooks, and to understand how students justify their use. We found several symbol systems for alleles coexisting along and across genetics textbooks. We also found that the transition from one system to another is not clearly explained in those books. After mapping the symbol systems and the context in which each system was presented, we found that the symbol system changes when the presented phenomenon exemplifies different allelic relationships or when there are more than two alleles for one gene.
Finding the contextual reasons for changing the symbols allowed us to understand how the limitations of a particular symbol system require shifting to another system: in all the textbooks, the first symbol system presented was of uppercase/lowercase letters to symbolize alleles in a dominant/recessive relationship. The dominant allele was always marked by the uppercase letter and the recessive one by the lowercase letter, indicating that this system is one convention across all textbooks. Using this system, it is not possible to symbolize more than two alleles because there are only two possibilities, uppercase or lowercase. Similarly, it is not possible to use this system to symbolize two co-dominant alleles of the same gene since both would be marked by the same symbol, making the two symbols indistinguishable. Considering that this is the first system presented in all textbooks, it is clear why another system is needed to symbolize cases of co-dominance/partial dominance or multiple alleles.
This ‘birds’-eye view’ of the symbol systems may not be accessible to novice students without explicit explanations. Understanding the meaning of a symbol is not only knowing which object it represents, but also acknowledging its limitations in representing the object [3
]. To enable communication using symbols, the meaning that is embedded in a symbol by its creator must be understood by all those who perceive it [1
As already noted, the field of genetics is challenging for students, who exhibit difficulties in understanding the concepts of genes and their different versions, termed alleles [11
]. Consequently, they use the symbols of alleles as an algorithm to solve problems without necessarily knowing the underlying concepts that teachers expect them to know after instruction, and they do this while conveying incorrect conceptual knowledge [21
]. Allele symbols, on the other hand, are a main means of communication for displaying and explaining processes and models of allelic relationships. As in the past, lack of convention regarding visual representations gave rise to doubts as to their effect on the understanding of the object being symbolized [31
], we could only suspect that adding confusion to the already apparent difficulties for novices might lead to complications in communication using these allelic symbols.
Indeed, this was apparent in our examination of the justifications provided by students for the existence and use of two different symbol systems. Findings indicated that some students (which we refer to as type 1 students) can link the change in the inheritance paradigm to the change in the symbol system used. However, several students (which we refer to as type 2 students) could not do so. Unable to explicitly name this link, these students relied on an external authority (such as the teacher or the Internet) that arbitrarily determined which symbol system should be used in each case. Without understanding the rationale behind the use of multiple representations, the students justified the existence of any representation with non-genetic justifications. In almost none of the cases did this lead them to the justification of two complementary representations.
It might be argued that some of the interviewed students interpreted our questions differently. Following the logic of this argument, higher-level students were able to understand our intention to talk about the inheritance paradigm, whereas lower-level students thought that we were talking about their personal disposition or the identity of the entity responsible for symbolic conventions (i.e., the teacher or the scientific community). However, even students who insisted on ignoring the allelic relationship as a guideline for choosing the symbol system were prone to point out the gene itself (i.e., the object) as the reason for the constant use of the letter ‘d’ (i.e., the symbol). Furthermore, as the pattern of answers surfaced repeatedly in each group when faced with different questions, and as all students displayed this pattern almost exclusively, we cannot ignore the fact that some of the interviewed students relied on the link between the inheritance paradigm and the symbols to justify the use of the symbols and some of them were unable to do so.
Overall, our results suggest that the absence of a clear conventional symbol system and the lack of explanations concerning the reason for shifting from one symbol system to another might hinder the understanding of the connection between the symbol and the object that it symbolizes. Learners might give an incorrect meaning to the switch between one symbol system and another or even ignore the meaning altogether, missing an important concept (such as the inheritance paradigm) embedded in the rationale behind the switch.
These results support the “deep-level” and “surface-level” reasoning described in previous studies with respect to the use of representations by novice students [6
]. When comparing the way in which experts and novices implement conceptual knowledge, it has been found that the latter use surface features of the visual representation when assessing them, as opposed to the former who are able to use deeper conceptual knowledge when given a sorting task. For example, when novices were asked to sort visual representations from different media (videos, graphs, images, etc.) they tended to group them either by media or by surface features of the phenomenon, such as “molecules moving around”. Experts, on the other hand, tended to group more cross-media examples together and used deep conceptual knowledge such as “precipitation” or “equilibrium” to sort visual representations [6
]. We suggest that students who linked the symbol system to the inheritance paradigm and valued the use of both symbol systems were able to display deep-level reasoning. On the other hand, our findings suggest that students who displayed a preference for one of the symbol systems based on graphical features, such as the addition of numerals, were unable to use deep-level reasoning and resorted to the evaluation of surface features of the symbol system.
While students who relied on surface features of the symbol system displayed a preference for one of the systems, we also identified one type 2 student who did justify the switch between the two symbol systems, yet was unable to link this switch to the inheritance paradigm. This student justified the use of two systems based on the existence of two different tissues in the displayed phenomenon. Results from a previous study have demonstrated that novice students tend to understand and focus more on observable structures than on more complex features, such as behaviors and functions [33
]. The authors of that study concluded that experts, on the other hand, view the behavioral and functional understanding of the system as a deep principle, which facilitates the organization of their knowledge of the system. Unable to link the A1
system to co-dominance, the interviewed student linked the use of a different symbol system to a surface feature that we did not expect to be significant: tissue type. As such, we find that student’s justification and analysis to be consistent with these previous findings. We suggest that while the student was able to judge the alleles by a simple property (location), she was unable to properly analyze and organize the information according to the allele’s behavior. Our results suggest that just like structure, some entities’ properties are easier to understand and therefore easier to link to features of the visual representation for the novice learner.
We view our students’ justifications as evidence for the use of surface features, of either the symbol or the object represented by it, for the evaluation and processing of visual representations. We suggest that as type 2 students lacked the tools to use deep-level reasoning, they resorted to any surface feature available to them. Consequently, while these students had no apparent trouble identifying crucial features of the visual representation itself (letters, numbers, etc.), their focus on surface features constrained their ability to link the representation’s features to meaningful biological concepts.
Another possible interpretation to our data can be suggested. As the evidence clearly shows a correlation between the level of the students (as reported by the teacher) and the type of arguments they provided, it could be argued that students with lower academic performance who lacked content knowledge turned to surface explanations to meet the expectations of the interviewer. It is important to point out that while we did record several instances in which type 2 students could not recall key terms (such as co-dominance) and made improper use of others during the interview, they did make proper use of several terms which are considered a source of confusion in genetics education. For example, we could recognize in all students’ discourse a clear separation between genes and alleles, a known hurdle in the biology classroom for both students and teachers [25
]. Furthermore, we do acknowledge that we have no ability, in this study, to determine whether the use of several symbol systems in the classroom confused students and prevented them from using deep-level reasoning. However, it was quite apparent to us that for some of the students, the introduction of several symbol systems to the classroom was not a strong enough instigator to promote deep-level reasoning concerning the difference between the two inheritance paradigms they represent.
While using a small sample of 8 students allowed us to deeply probe and focus on meaningful justifications in students’ reasoning, we acknowledge this small sample as a limitation of our study, as this method prohibits us from claiming that our results can be generalizable. Thus, we cannot claim that all classes display the same type of students as we have observed, and we do not claim any knowledge as to the distribution of arguments within the classroom we observed or any other classroom for that matter. We do claim, however, that our results provide some insight into the complexity of introducing multiple visual representations into the classroom. This study supports the suggestion that further attention to this issue is required.
While studies on the use of multiple representations in the classroom exist [12
], they all focus on the pros and cons of multiple visual representations. However, whereas all these aim to guide the teacher in choosing representations for his or her students, there seems to be a question that garners less attention: “Does the student understand why we use multiple representations in the classroom?”
We wish to emphasize that students who preferred the Aa system were showing a preference for the very first symbol system they had encountered in their introduction to Mendelian genetics in the classroom. As we elaborated earlier, considering that in all textbooks, this is the first symbol system presented, it is clear why another symbol system is needed to symbolize cases of co-dominance/partial dominance or multiple alleles. While these are just causes to switch to a different symbol system, none of them are explicitly stated in the students’ textbooks. It was quite apparent to us that some of the students were unable to understand the reason for switching to a different visual representation. As novices tend to focus on the surface features of the representation, some students viewed the added representations as redundant, differing from the previous ones only due to superficial characteristics. While it could be postulated that the introduction of a new visual representation will trigger an assumption in the learner of the introduction of a new phenomenon or concept, this assumption was not triggered in some of the students that we interviewed.
A similar phenomenon was observed by Gericke, Hagberg, and Jorde [38
] as they introduced students with different texts containing incommensurable models of gene function, both of them present in the genetics classroom. From this study’s data, it would seem that students fail to notice any contradictions between the texts or any of the different epistemological features presented by them. Similarly, some of them find it difficult to justify the co-existence of these two models. One of the conclusions of the authors was that this difficulty, combined with the absence of an explicit embedded explanation regarding the variation in models, might be the source of hybrid models, created by the students. These hybrid models might lead to student difficulties in learning.
As the effect of a missing explanation regarding variation in models or symbols becomes clearer and as the knowledge pertaining to the reason for using each of the different symbol systems seemed flawed while interviewing some of our students, we first questioned the need for two symbol systems in genetics teaching. Although the Aa system has its constraints, the A1
system can be used as a ubiquitous default visual representation for any of the inheritance paradigms. While we do acknowledge that some scientific communities use uppercase letters to denote dominant alleles [39
], if truly needed, teachers can introduce this information to the learners later in the process, after they have established a better understanding of the term allele and its symbols.
Whereas we view the use of one symbol system as convenient, the use of multiple symbol systems is currently common practice, as seen through the analysis of the genetics textbooks. As a conclusion from our study, we recommend that this practice be used with care and thought. While the textbooks that we reviewed bore no explanation for the reason to switch between symbol systems, we wish to point out the merits of creating such a discussion on the reason for this switch and mainly the constraints of the Aa system. This discussion might allow students to understand the relationship between the two systems, strengthen the connection between the symbols and the models that they symbolize, and lead to a better understanding of the relevant concepts. A previous work on “models of modeling” demonstrated this process, as students who were tasked with the analysis of the relations between representations acquired a more fluent visualization capability [3
]. Similarly, we view one of our students’ interviews as evidence of such a process, as a dialogue with her during the interview was followed by a change in the type of justifications that she provided (Hila, see Section 3.2.2
In conclusion, we show that textbooks in genetics education use different symbol systems to represent alleles. Each system that we surveyed is linked, in the textbook, to a different inheritance paradigm or to a unique phenomenon. However, this switch between symbol systems is not explained in the reviewed textbooks. While we, as researchers, were able to speculate about the reason for the switch and while some students that we interviewed were able to link each symbol system to the correct inheritance paradigm, some of the interviewed students were unable to do so. These latter students justified the application of two different systems using surface reasoning and related them to surface features of the phenomenon.
While educators might use different visual representations to elaborate and represent different features of an object, process or system, the use of multiple visual representations should be practiced with care. We propose that any addition of a new visual representation should be accompanied by a discussion, both in the classroom and in the textbook, about the flaws of the previous representation and the need to introduce the new one. Without such a dialogue, which includes the affordances and constraints of the representation, students are left with routine tasks, stripped of context and devoid of deep conceptual understanding, especially when a different, known and usable visual representation has already been presented to them. After all, for the novice student, it is never as simple as that.