Across the mathematical sciences, the ability to proficiently use symbols to represent concepts and processes is important. The fluidity of either a single symbol being used to represent multiple concepts, or multiple symbols being used to represent the same concept requires a flexibility of thought from practitioners to contextually identify the correct meaning of a symbol each time it is encountered. In addition, font and style applied to similar notation may vary according to the medium of communication. Bardini & Pierce (2015) drew attention to the challenge for students learning mathematics as they transition from secondary school to studying university mathematics, where new symbols are introduced, and familiar symbols may be used with changed or extended meanings. This paper explores the difficulties experienced by physics students navigating this symbolically dense and potentially confusing language. We investigated whether first-year undergraduate students studying physics perceive any difficulties or differences related to the use of symbols during their transition to university. Do they experience difficulties with the diversity and/or duplication of symbols used within their physics studies or between mathematics and physics? How do these students manage these difficulties? First, we review key related literature.

Literature

Mathematics derives much of its power from the use of symbols (Arcavi, 2005), but research over a long period of time has shown that their conciseness and abstraction can be a barrier to learning (MacGregor & Stacey, 1997; Pierce et al., 2010). Symbols also have a significant role in physics, with increasing usage and importance at the tertiary level. In a study involving first-year university physics students (Torigoe & Gladding, 2007), it was found that students’ performance is highly correlated to their understanding of symbols. Our focus in this article is on students’ understanding and interpretation of symbolic expressions that are used in first-year undergraduate physics, including those that are unique to physics, as well as symbolic expressions that overlap with the mathematics that students have previously studied or are studying concurrently. We are not focusing on students’ skills in calculation or manipulation of symbolic expressions.

Previous published research identifies four key issues in this regard: students’ difficulty dealing with the number and variety of symbols; the need to be cognizant of unwritten conventions; the benefit of recognizing meaning in common symbol templates; and the need to be aware of the epistemological difference in working with and interpreting symbols in mathematics and in physics.

Firstly, as Bailey (1999) points out, physics uses many symbols. Roman and Greek letters are both used with lower- and upper-case letters being employed with different meanings. Not only must students be alert to this detail, but the same letter may be used with different meanings, for example as a pronumeral to indicate a quantity and as the abbreviation for a unit of measurement. A typical example is the use of m for the quantity of mass and m for meters. Here, a hint of the different meaning is given by using italics for the variable quantity. Bailey (1999) suggests that students may be helped by providing them with a list of symbols with their respective meanings and explaining to students that they will need to work at memorizing symbols as they are introduced during their course.

Torigoe and Gladding (2006; 2007) report on their investigation of physics students’ numeric computation versus symbolic representation. In the initial phase of their study, in which they gave students parallel numeric and symbolic versions of examination questions, they found the mean score on numeric questions was 50% higher than the matching symbolic questions. Based on their first study, Torigoe and Gladding hypothesized that the discrepancy was due to students’ misunderstanding of variables rather than skills in calculations. Their extended study, using further parallel questions, yielded additional insights: working with numbers means each step may condense to a single value while symbolic expressions must be carried forward from step to step. Torigoe and Gladding (2007) then hypothesized that working with symbols leads to greater cognitive load especially in multiple-step problems.

Giammarino (2000), a student writing on behalf of a group of students, gives voice to the anxiety they feel when faced with what they perceive to be different symbolic notations representing the same variables. He notes variety from teacher to teacher and textbook to textbook. Students worry about which symbol to use and which ones they will be faced with in examinations. While other authors stress the need for students to be flexible, these students make a plea for more consistency.

Certain unwritten conventions need to be observed when constructing and writing symbolic equations. These conventions make it easier to recognize the patterns in expressions and hence aid interpretation. Bailey (1999), De Lozano and Cardenas (2002), and Moelter and Jackson (2012) each draw attention to these subtleties. As mentioned above, italics are typically used for variables and Roman letters for units. Constants, parameters, and variables are written in that order when they are multiplied together in an expression; where there are fractions, this order is followed separately in the numerator and in the denominator.

A step further in considering the setting out of symbolic expressions was introduced by Sherin (2001) and taken up by Redish and Kuo (2015). Sherin highlights the importance of what he terms “symbolic forms” and their interpretation or conversely, in modelling, identifying that a physical phenomenon may be represented abstractly by a particular symbolic form. For example, if a “whole” is made up of parts, then symbolically it may be represented by an expression that will take the form □+□+□… or if one variable is proportional to another then an appropriate form will be prop- […/(….x….)] (Sherin, 2001, p. 590).

Choosing appropriate symbolic forms requires the student to consider not only variables and constants but also the mathematical processes and their meaning in a specific context. Students need to think, for example, about the real-world impact of adding rather than multiplying, adding rather than subtracting etc. They need to make an informed, thoughtful choice, not just hope to have memorized a formula correctly.

De Lozano and Cardenas (2002) take up the theme of an epistemological lens with which symbols in mathematics and physics must be viewed. Mathematics, they contend is theoretical, built in steps from axioms, with a focus on being logically consistent. Physics on the other hand is always to be viewed and interpreted in the light of real-world constraints, which may or may not be made explicit. They suggest that while in mathematics “=” indicates a relationship that has properties of symmetry, reflexivity, and transitivity, in physics, it may be used to more loosely indicate “corresponds to.”

Methodology

Students studying first-year physics, mathematics, or statistics were recruited from two Australian universities with which the researchers had existing collaborations. Participants we focused on in this report were lecturers of first-year physics (who were also the subject coordinators)and 187 first-year physics students recruited via advertisements posted on their subject’s online learning system noticeboards. Forty three (24%) of these students were enrolled in engineering studies with physics studies forming an element of the course. The remaining 134 (76%) were enrolled in a variety of undergraduate single and double-degree courses, with a decision regarding whether physics would become a final-year undergraduate major in their course still to be made. Of these students, 19 agreed to meet with us individually for a 30–40 minute semi-structured, audio-recorded interview, in addition to responding to an online survey. Survey responses were categorized by common response patterns and interview transcripts analyzed to identify repeated themes in students’ comments.

Findings and discussion

Do first-year undergraduate students studying physics perceive any difficulties or differences related to the use of symbols during their transition to university?

Symbolic convention changes. Symbols that are used to represent a physics or mathematics concept at secondary school can, for a student new to university physics (illustrated by S13 below), appear to change unexpectedly and without explanation, with the new symbolic representation incorporating unfamiliar syntax such as subscripts and superscripts.

S13: I just found it a really kind of very off-putting at the start…Like, s we used in Specialist [Year 12 mathematics]—it is now delta x. Instead of u and v for initial and final velocities, we have v i [v_i] and v f [v_f]. I mean like, the physics notation makes more sense and is more explicit, like it is better, but why are they different at all?

Students mentioned a range of changes in notation (as opposed to new terminology) compared with prior studies after attending seven weeks of classes in first-year undergraduate physics. These have been summarized in Table 1. It is interesting to note that the difficulties which students highlighted regarding changed notation for unit vectors from î, ĵ, and k ̂ to x̂, ŷ, and ẑ was observed in the reverse by Lecturer 1, where they use î, ĵ, and k ̂ in first-year physics, but noticed students with prior experience using x̂, ŷ, and ẑ. Lecturer 2 noted that “you’ve got the i, j, k people and I guess one of the things…probably between academics, is there’s not going to be a lot of consistency, because they’re from a lot of different backgrounds.” This illustrates the difficulties both students and lecturers may have in written communication with each other.

One contributing factor to students’ confusion, exemplified by S21 below, is that changes in notation from secondary studies were not always explicitly highlighted by lecturers or tutors, leaving students to decipher the equivalence of the symbols being used. Students interpreted this silence to mean that their instructors were unaware of the notational changes, and for example, commented:

Student S21: I don’t think they realized, because I was just sitting in one of the workshops and the girl next to me had s and I was like, I’ve never seen that equation before I don’t recognize it as something that I’ve seen before, and then she goes ‘s means displacement.’ Oh, okay then.

Within the subject of physics, a broad variety of what we term as “symbolic synonyms” (i.e., multiple symbols being used to represent a single concept) are used. International standards for notation in physics, based on the Système International (International System of Units) have been published since 1961 (see Cohen, 1987), however, they are not always used or adhered to even in commonly prescribed textbooks (for example, Walker et al., 2011).

The physics lecturers we interviewed were aware of the inconsistency of notation in physics and the impact it can have on students’ understanding. They each aimed for consistency within their own teaching materials. For example:

Lecturer 2: I’ve deliberately changed all the symbols over and I make sure when we’ve changed textbooks or we’ve changed references, I use the same symbols as in the textbook...[because] when they’re at the beginning, it becomes a barrier to entry to the topic in the first place, if you’re confusing them with a mixture of symbols.

However, this was undertaken in isolation, with policy still to be developed to ensure consistency of notation within the physics departments (particularly given students can be taught by multiple physics lecturers throughout the course of each year) or with the Australian school curriculum.

Do students experience difficulties with the diversity and/or duplication of symbols used within their physics studies or between mathematics and physics?

Physics uses a very broad variety of symbols to represent constants, parameters, and variables in the equations, with the number and variety of symbols used increasing during undergraduate studies. This was noted multiple times through interview:

S02: Like it’s...especially cos they’re almost like running out of symbols, so they’re just chucking Greek letters at us…

S06: There’s a lot more variety of symbols and things this year. In high school, they were strict that this is this, and I think also with physics it’s just because there’s more equations to use, there’s more symbols and variables.

Students (e.g., see S20 below) noted that some symbols are used to represent multiple concepts (which we describe as “symbolic homonyms”) both from newly introduced concepts at university and also for concepts met at school.

S20: Oh well even just in physics, omega is about three different things. It’s angular velocity… [and] it’s angular frequency as well… So they’re the same for a circle, I think that’s why it might have the same symbol but it’s different for everything else, which is annoying.

The large number of terms requiring a symbolic representation in physics may make this inevitable, but the varied meanings for a single symbol added a layer of deciphering, which students sometimes found difficult. Cohen (1987) indicates that in standard conventions the symbol that S20 highlighted, lower-case omega (ω) is used to represent each of a solid angle, angular frequency, and angular velocity, all within the area of mechanics, with five additional uses of lower-case omega listed using subscripts throughout various sections of physics. It is not difficult to see how students might be confused by this.

Conversely, within undergraduate physics, students also encounter situations where multiple symbols were used to represent a single concept.

S05: So, it turns out we have three different systems and it’s like really chaotic sometimes. Because I converted to the version that the textbook uses but my friends all use the ones in high school, and the lecturer uses a different one.

Physics lecturers acknowledged the diversity and complexity of the symbolic aspect of the subject, highlighting even that “notation convention actually changes based on situation or context as well” (lecturer 2) with bespoke notation being used for emphasis at times.

Students also encounter duplication or changes in conventions between physics and mathematics, for example:

S22: Physics often uses a k and math often uses a c, although I have seen a bit you know, one borrowing the other” [in relation to general constants].

S13: Physics will use delta t in the constant acceleration formula, which is better than d [in mathematics], because that kind’ve indicates a change over a time interval.

Other topics where students highlighted that symbolic synonyms are pervasive between subjects include vectors (i, j, and k versus x, y, and z, and tilde versus arrows or bolding), motion (s - displacement, u - initial velocity, v - final velocity, a - acceleration, t - time: mechanics notation colloquially known as suvat formulae versus other notation), forces (N versus R for normal reaction forces) and different conventions with SI units.

Physics lecturers also commented on the differences in notation between their subject and mathematics, for example, Lecturer 1 stated that “I would say that the use of the same symbol for different things is the biggest one [symbol difficulty]. It’s the difference in how certain mathematical concepts are presented. For example, differentials are presented in a number of different ways.”

Aligning physics and mathematics curricula

Added to this is a complexity in scheduling the timing of physics topics to ensure required university mathematics knowledge has already been studied by their students as exemplified by the following feedback:

S02: Ah Physics, yeah… One of the other things is like... vectors came into it [physics] very early and I’m only doing [introductory first-year calculus subject], and we’ve only just finished vectors. So like now it all makes sense, but back then it was... what’s going on?

How do first-year physics students manage difficulties with symbols?

When asked whether the inconsistencies and differences caused difficulties, students felt they could accommodate the changes, but the associated additional load was noted as making the tasks harder.

S02: …But, just like... working through it has confused me a bit. I’m getting a bit more on top of it going over it all a few times, but yeah, it’s a lot harder to pick up the first time than it was for me in school.

Some students were also anxious as to whether the notation they had learned in their previous studies would be understood by examiners. This was a driver for them to change nomenclature (such was the disconnect in notational use from secondary to tertiary physics studies). For example:

S05: Actually, I was quite concerned about my use of symbols as well. I wasn’t sure if my—the teaching associate who marks my assignments will understand my... the symbols I used in high school. That’s why I converted to the new one, and I did struggle in the start.

Some students commented, as we might expect, that they worked out the meaning of the symbol using the context of the situation. However, other students had more difficulty; they chose to perform calculations using the symbols familiar from school then “translated” to the symbols used in their university physics class.

S13: Mainly [it] just slowed me down. … I’m debating... should I use his notation or my notation? … I find it weird when the lecturer’s lecturing in a particular notation, to use different notation when I’m writing it… I’m actually trying to write in my own notation. But then your brain will be trying to listen to him at the same time and then you end up just writing what he writes and it’s like half in one half in the other ….

This error-prone method adds complexity and cognitive load for these students. Nevertheless, some students felt more confident with this approach rather than working with the new notation.

Discussion and implications for teaching

Our findings, and the earlier research literature discussed above, suggest that students may experience difficulties coping with the increased variety, duplication, and reliance on symbolic notation in undergraduate physics. They are trying to make sense of newly encountered symbols or contexts, in part by using prior conceptions that spring to mind when they see a familiar symbol and in part struggling to match new notation symbolizing a familiar context.

The difficulties reported by students who encountered unexpected and at times, unexplained notational changes during their transition from secondary to undergraduate physics studies, were not unforeseen by the experienced physics lecturers. However, these senior staff were glad of research evidence to share with their colleagues. Given the variance in symbolic standards within the subject, a disconnect in symbol usage would seem almost inevitable for some students, as they transition to university physics studies. We would advocate for lecturers and tutors to maintain familiarity with the content and conventions in the secondary national curriculum, but perhaps a useful approach is for tutors to ask students to read symbolic statements aloud and then explain what the notation means. This will both determine knowledge and preconceptions. Differences or changes in notation need to be explicitly addressed.

In addition to being alert to and explicitly highlighting the many symbolic synonyms and homonyms that students encounter after transition to university physics and mathematics studies, additional measures can be taken to scaffold a student’s competence in using physics’ symbolic language. For example:

• Taking note of and specifically highlighting a symbol’s form, noting if it is italicized, capital or lowercase, which alphabet it is from, are there subscripts or superscripts and what they each mean. For example, the transition from s to x_i in the topic of mechanics is symbolically more complicated as a result of the subscript, but x_i conveys additional important information.
• Explicitly teaching conventions such as ordering the symbols in equations with constants first, parameters next, and then variables as suggested by Moelter and Jackson (2012) may also be helpful.
• Setting tasks to increase students’ familiarity with the varied symbols they encounter independent of context. For example, Greek letters are ubiquitous in this field, but many students have little or unrelated prior experience with them. Introducing students to the entire Greek alphabet including case, phonetics, and how to write them can help to make these symbols less of an obstacle for students, when they encounter them in formulae.
• Tutors querying students to identify new, changed, or conflicting use of symbols within and between the subjects they study.

Regardless of internationally published standards, the notational inconsistencies in the physics domain, though not new (Bailey, 1999), remain an obstacle impeding some students’ learning. This complexity cannot—and arguably should not—be avoided with students, if they are to navigate their pursuits in physics successfully. It is necessary for students to develop a flexible mindset regarding physics notation. However, lecturers and tutors need to be alert to likely areas of confusion. Physics and mathematics educators should be aware of the extensive interdisciplinary use of the same symbols, and the potential for confusion that the different uses of the symbols both within and between the subjects can cause for students, particularly as the format and information conveyed by each symbol increases in complexity as studies progress. Contemplating a novice’s perspective can influence pedagogical techniques, with a view to helping to improve a student’s experience and success in the subject.

Acknowledgments

The authors wish to thank Dr. Caroline Bardini for her development of this research project, and to the physics lecturers who generously gave us their time and insights on the challenges faced when teaching undergraduate physics. The authors also wish to thank the students, lecturers, and tutors across multiple universities who gave their time, knowledge, and course materials for the benefit of this research. This research has been funded by the Australian Research Council: DP150103315.

Meredith Begg (meredith@meric.id.au) was formerly a research assistant and Robyn Pierce is an associate professor, both in the Melbourne Graduate School of Education at the University of Melbourne in Carlton, Australia.