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Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance.

Notations play a crucial role in the development and advancement of mathematics and science. Some of the oldest records of human writing are notches on bones, a simple notation to keep track of quantities. Today, it is difficult to imagine a world without numerals: we use the decimal place-value notation based on the ten digits from ‘0’ to ‘9’ in our daily lives and they form the basis of our communication and commerce. But what features do

The American philosopher, logician, and mathematician Charles Sanders

Our current notations for natural numbers and basic arithmetic operations, such as addition, subtraction, multiplication and division, seem to use only symbolic signs for the operators (e.g. ‘+’, ‘-‘, ‘×’, ‘/’) . Could the use of more iconic signs for mathematical operators contribute to the effectiveness of notations? Our goal in this paper is to investigate this possibility. We begin by reviewing one early articulation of this hypothesis, due to Christine Ladd.

Christine Ladd (1847-1930) was the first woman to complete all requirements for a Ph.D. in mathematics and logic at Johns Hopkins University in 1883, where she studied with J.J. Sylvester and C.S. Peirce. However, since the university did not officially admit women at the time, she was not granted her Ph.D. until 1926. From 1884 onwards she was known as Christine Ladd-Franklin. In addition to mathematics and logic, she also worked on experimental psychology and the theory of color vision (

Our interest here concerns Ladd’s views on notational choices in mathematics. While the choice of individual signs in a symbolic notation (which we refer to as ‘symbols’) is purely conventional, the question arises whether some symbols are better suited than others for expressing certain contents. In other words, can we find some effect that the shape of a symbol has on the way it is used? If so, such cognitive considerations could be used to guide the choice of individual symbols.

Ladd's hypothesis regarding the relation between a symbol and its meaning was first articulated in the early literature on notations for logic. In a discussion of different systems of logic, she compared Peirce’s symbol for implication, ‘

Although Ladd made her remark in the context of symbols for logical relationships such as implications, biconditionals, and so on, the issue also applies to binary operations, which are perhaps more pertinent to school-level mathematics education. Binary operations can be understood as mathematical rules for combining two elements of a given set to produce a third. For example, the addition of integers is a simple binary operation: two integers are combined to produce a third. Moreover, addition is a symmetric, or

In view of Ladd's remark concerning logical notation, it seems reasonable to suppose that she would have applauded the symmetrical nature of the addition symbol, but criticised the symmetry of the symbol for subtraction. In sum, in the context of binary operations we can formulate the following principle that underlies Ladd’s assessment:

It is notable that symbols typically used to represent binary operations do not always follow Ladd’s principle. While the choice of our symbol for addition, ‘+’, does (the symbol is symmetric and the operation is commutative), the symbol for subtraction does not. Subtraction is not commutative but the symbol ‘−’ is nevertheless symmetric. Similarly, some of the symbols we use to represent the (non-commutative) division operation are asymmetric, whereas others are not (compare ‘10/2’ with ‘10÷2’). Analogous observations can be made about advanced mathematics. For instance, introductory abstract algebra textbooks typically represent an arbitrary group operation (which is not in general commutative) with a symmetric symbol such as ‘✭’ or ‘•’ (e.g.,

Indeed, an analysis of mathematical typesetting practice reveals that the majority of symbols used to represent binary operations are symmetric. Of the 59 primary symbols used to represent binary operations in LaTeX, a common mathematical typesetting language, 45 (76%) are symmetric along their vertical axes (we considered those symbols available in LaTeX by default or within the American Mathematical Society’s amssymb package,

Ladd did not elaborate what advantages she hypothesised would be gained by following the principle that commutative operations should be expressed by symmetric symbols. Neither did ^{th} century Charles

To our knowledge, despite the long history and apparent plausibility of Ladd’s hypothesis, it has never been empirically tested. This is what we set out to do in the experiments reported here. We derived two research questions from Ladd’s hypothesis: (1) Are symmetric symbols intuitively associated with commutative binary operations? and (2) Is the use of iconic symbols advantageous for students’ mathematical engagement with binary operations?

In this first study, we investigated whether the use of vertically symmetric symbols in mathematical statements depicting binary operations would bias people to endorse the commutativity of the operation. By comparing whether participants endorsed commutativity more often when symbols followed Ladd’s hypothesis than for the symbols with horizontal symmetry which did not follow Ladd’s Hypothesis, we were able to test whether vertically symmetric symbols are interpreted iconically, in the sense that they are intuitively associated with commutative binary operations.

We designed a simple task to test people’s intuitive judgement about the mathematical properties of a binary operation while we varied the visual properties of the symbol used. We used a range of arbitrary symbols to depict a binary operation (i.e. ‘3 ♠ 4’). Participants were asked whether they believed a statement about the commutativity of these operations to be true or false (‘3 ♠ 4’ is equal to ‘4 ♠ 3’ ?). We presented the operations once with the symbols having a vertical axis of symmetry (symmetrical as understood by Ladd) and once with the symbols rotated by 90° to change the symmetry axis to be horizontal.

The intended sample size and analysis plan was preregistered prior to data collection (see

Thirty undergraduate and postgraduate students (14 male, 16 female, mean age = 26.23) participated in this study. Participants were invited to the Cognition Laboratory at Loughborough University where they completed the computerized task individually. The full duration of each session was about 20 minutes and participants were compensated for their time with £4.

The study task was programmed using the PsychoPy 3 software (

Participants judged 210 trials in an individually randomized order. A trial consisted of a binary operation, represented by a non-mathematical symbol, and a written statement about its relationship to the inverted form. An example trial is shown in

Out of all 210 trials, 140 asked participants for a judgement about the commutativity of the operation. Those trials stated that the operation ‘is equal to’ its inverse. The statements about the relationship in the remaining filler trials were either ‘is larger than’ or ‘is smaller than’ and responses to those trials were not analysed.

We constructed the trials from seven pairs of numbers between 1-99. Each pair was presented with each of ten different non-mathematical symbols (shown in

As shown in _{10} = 7824.42. A robustness check revealed that this Bayes factor was not substantially affected by the choice of prior width.

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We followed this main analysis up by calculating a generalized linear mixed model with a binomial link function to account for the random effects of participant and symbols. The aim of this analysis was to model random intercepts for each participant and symbol. This technique controls for the variation in the data that is introduced by participants differing in their general tendency to endorse a statement and some of the symbols generally being endorsed more frequently than others; both regardless of the symmetry axis of the symbol. The model was specified with an unstructured covariance structure and parameters were estimated using Laplace Approximation. Using this model, we calculated that statements with vertically symmetric symbols had significantly higher odds to be endorsed as commutative than statements with horizontally symmetric symbols,

In sum, we found very strong evidence that the notation for binary operations influences how that operation is intuitively interpreted. Specifically, in line with Ladd’s hypothesis, using symbols with a vertical axis of symmetry seems to be intuitively associated with commutativity. In other words, the visual properties of symbols for binary operations influence the mathematical properties that are intuitively attributed to these operations, as if the symbols were indeed iconic.

Given the findings from Study 1, it is natural to ask whether using symbols consistent with Ladd’s hypothesis would be advantageous for student’s mathematical learning. To our knowledge, no research has directly investigated this issue. However, it is known that learners do often inappropriately assume that binary operations are commutative. For example,

The proposal that using symmetric symbols to represent non-commutative operations might cause problems for learners is consistent with recent work that has found that the visual properties of notations are cognitively relevant. As an example,

We sought to test Ladd's principle by using an artificial symbol learning paradigm. Specifically, we asked whether it is better to associate vertically symmetric symbols with commutative operations and vertically asymmetric symbols with non-commutative operations (the congruent condition) than it is to associate asymmetric symbols with commutative operations and symmetric symbols with non-commutative operations (the incongruent condition).

The intended sample size and analysis plan were preregistered prior to data collection (see

Fifty-eight undergraduate mathematics students (34 men, 24 women, mean age = 19.9 years) participated during a statistics lecture at Loughborough University. Participants were randomly assigned to either the congruent or incongruent conditions based on the parity of their student ID numbers (which are assigned pseudo-randomly upon the students' enrolment at the university). Participants worked through booklets individually in silence. The experiment consisted of three parts. First, participants read an information sheet about the experiment, gave consent for their data to be used in the analysis, and self-reported their gender and age. In the second part, the learning phase, participants were given three minutes to read about, and learn, a set of novel symbols to represent addition, subtraction, multiplication, and division.

In the congruent condition participants were taught to associate symmetric symbols (◇, ◆) with addition and multiplication, and asymmetric symbols (▷, ▶) with subtraction and division. In the incongruent condition these symbols were reversed so that participants associated asymmetric symbols with the commutative operations and symmetric symbols with the non-commutative operations. The full text of this section, for those in the congruent condition, are shown in

After reading these instructions for three minutes participants were asked to turn over the page. This revealed the test phase, which consisted of a simple fluency arithmetic task. Specifically, participants were asked to solve as many simple two-term arithmetic problems as they could in three minutes.

The booklet contained a total of 396 arithmetic problems, split equally between the four operations, and were presented in a different randomised order for each participant. The problems were designed so that participants could not infer the operation from the identity of the numbers in the problem (for instance, if we had asked a participant to solve 36 ◆ 41 they might reasonably have inferred that ◆ did not represent division).

For example, a participant may have seen:

Solve: 40 ▷ 2 =

Solve: 18 ◆ 2 =

Solve: 2 ◇ 2 =

Solve: 24 ▶ 1 =

And so on. All 396 problems, together with example test booklets for each of the conditions, are provided in the

Two participants failed to meet our preregistered inclusion criteria (their fluency scores were not within 3

The mean numbers of problems correctly answered in each condition are shown in

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In sum, in line with Ladd's principle, we found that participants in the congruent condition outperformed those in the incongruent condition. A caveat to note here is that we did not assess or control for potential arithmetic fluency differences between the congruent and incongruent conditions. However, those participants who were randomly assigned to learn to associate vertically symmetric symbols with commutative operations and vertically asymmetric symbols with non-commutative operations could perform arithmetic operations more fluently than those participants who learned to associate asymmetric symbols with commutative operations and symmetric symbols with non-commutative operations.

Christine Ladd formulated her hypothesis that commutative operations should be expressed by symmetric symbols almost 150 years ago. In two studies we have provided what we believe to be the first empirical test of her hypothesis. We demonstrated that the binary operation symbols that have a vertical axis of symmetry are more likely to be intuitively associated with commutativity than those with a horizontal axis of symmetry. We further found that Ladd's hypothesis was supported in the context of basic arithmetic where the use of iconic notation proved advantageous for students when solving problems.

These results suggest that despite the seemingly arbitrary nature of a symbol's visual appearance, it may nevertheless have some iconic features that influence the manner in which that symbol is processed. Specifically, our studies support Ladd’s hypothesis that it is advantageous if symbols have iconic aspects that connect in some way to the represented mathematical concept. This observation extends

But what mechanism underlies our results? Whereas

There are theoretical reasons to suppose that congruent symbols might be more memorable than incongruent symbols. Consider onomatopeias: words such as ‘buzz’, ‘snap’ or ‘whack’ whose pronunciation is related to their semantic meaning. Phenomena of this kind have been studied under the label of ‘sound-symbolism’ and ‘ideophones’ (

But do linguistic shape-semantic associations, analogous to the sound-semantic associations involved in onomatopeias exist? Several sources of evidence suggest that they do. It has been known since the early work of gestalt psychologists (e.g.,

More directly related to our work, the early gestalt psychologists also demonstrated that shapes can spontaneously generate semantic associations. For instance,

These classic findings that demonstrate relationships between visual properties and meaning converge with recent work on the association between shape, space and mathematical concepts. When thinking about numbers, people seem to generally associate smaller numbers with the left side in space and larger numbers with the right side as well as expansion in space (for overviews see

The symbols of advanced mathematics are often introduced very consciously to satisfy certain desiderata, thus they provide a rich resource for the study of cognitive and practical advantages of notations (e.g., see

Finally, it is natural to ask whether our findings have any practical implications for mathematics teaching and learning. An obvious suggestion would be that, all things being equal, we should follow Ladd's principle and favour choosing symmetric symbols for commutative operations and asymmetric symbols for non-commutative operations. This practice would certainly seem to be entirely feasible in many contexts. In advanced mathematics for instance such notational choices are largely a matter of convention (there is no reason why an introductory group theory course could not favour a symbol like '▷' to represent an arbitrary binary operation). Similarly, it would seem possible for teachers to introduce division with the '/' symbol rather than '÷'. Using a notation that iconically represents the concepts of commutativity might improve students’ learning and understanding of the concept. A possible benefit of supporting the understanding of commutativity could lie in the faster retrieval of arithmetic facts: If a student understands commutativity and remembers ‘5 + 6 = 11’, they also know ‘6 + 5 = 11’. These suggestions should perhaps be productively tested in a more ecologically valid context before strong conclusions are drawn: whereas the dependent variable in our Study 2 was fluency, this is rarely the outcome prioritised in more advanced settings, and it is unknown whether our results would generalise from fluency outcomes to measures of conceptual understanding. Nevertheless, there seems to be no obvious downside, and a plausible upside, to following Ladd's principle when choosing which symbols to use when teaching mathematics.

We want to thank Fenner Tanswell for valuable discussions on the topic of mathematical notation.

We declare that no competing interests exist.