# Does a joint numerical magnitude code underpin the processing of symbolic and non-symbolic quantities?

André Knops, Alexis Garsmeur

► To cite this version:

André Knops, Alexis Garsmeur. Does a joint numerical magnitude code underpin the processing of symbolic and non-symbolic quantities?. 2024 Experimental Psychology Society Meeting, Experimental Psychology Society, Jul 2024, York, United Kingdom. hal-04935777

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https://hal.science/hal-04935777v1

Submitted on 14 Feb 2025

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Does a joint numerical magnitude code underpin the processing of symbolic and non-symbolic quantities?

I'm going to present you the study that André Knops and I did on the numerical distance and size effects correlation in unfamiliar Indo-Arabic number comparison.

In the field of numerical cognition there is two main effects that have been observed since a long time. The first one is the Distance effect. Number comparison are easier when the two compared numbers are numerically distant than when they are numerically close together.

For example, the distance between 5 and 10 is of 5 unit and it's therefore easier to discriminate these two numbers than 9 versus 10, where the numerical distance is of only one unit.

The second main effect, is the size effect. When we compare two numbers that have the same numerical distance, it is more difficult to discriminate between large numbers than small ones.

For example, 5 versus 6 is easier to discriminate than 8 versus nine even if the distance in these two comparisons is exactly the same, namely one unit.

Two propositions have been made to give an explanation of these two effects: The approximate number sense (or analogue magnitude system) and the discrete semantic system.

The analogue magnitude system use spatially organize representations one axis, some sort of horizontal line (the mental number line). These representations follow a logarithmic scale. Therefore, because of this logarithmic organization, size and distance effect are products of the ratio between two numbers along this axis.

The discrete semantic system is non spatial in nature. Number concepts are nodes connected to each other's by semantic relationships. Semantic relations are not restricted to numerosity and con also include parity, being a prime number etc.

In this view the size effect is the product of the frequency of encounter with a given number and the distance effect is caused by the semantic distance between two numbers.

In 2017, Krajcsi proposed that dots comparison (non-symbolic comparisons) are the product of the ANS (analogue magnitude system) solely, and made a study to disentangle between these two hypotheses in the case of symbolic quantities.

The two hypothesis make different predictions concerning the link between the distance and size effects. In the case of the ANS, both effects are supposed to be by-product of one effect the ratio effect due to the logarithmic organization of the number line. Therefore, there should be a strong correlation between the two effects among individual.

Contrarily, in the case of the discrete semantic system, both effects are supposed to be produce by different variable: the semantic distance for the distance effect and the frequency of encounter with a given symbol for the size effect. With this model, there is no particular correlation that should be expected between the two effects.

In his study, Krajcsi used only single-digit numbers between 1 and 9. All possible pairs (72) were created and each pair was presented 3 times to each participant.

There were two conditions, a symbolic comparison task with indo arabic digits that I just described, and a non-symbolic comparison task with dots arrays.
Two numbers were presented on each side of the screen (left and right) and participants had to choose by pressing a response key which of two number was the largest.

For every participant, in a given condition, authors calculated the slope of the regression between distance units and the average reaction time per unit and the slope of the regression between units of size effect and the average reaction time per unit.

That is to say for each increase in distance between two numbers what was would be the decrease in reaction time or for each increase in size of two numbers what was would be the increase in reaction time. If you have a high increase or decrease of reaction time, then the slope of your distance or size effect would be large.

After that, authors calculated the regression coefficient between the slopes of the mean RT on unit of distance effect and the slopes of the mean RT on unit of size effect.

Here are the results, there was a significant correlation for non-symbolic task and no significant correlation for the symbolic task with indo Arabic digits.

We have few criticisms to this study and in particular to the symbolic task.

First, authors used digits too familiar to participants and therefore too automated (we always encounter indo Arabic digits from 1 to 9 in our everyday life, and therefore we are too trained to respond to them).

Second, there was no sufficient variance in the size units. Number ranged from 1 to 9 as I said, which is maybe not sufficient to see the effect of the logarithmic compression on the reaction on small versus large pairs of numbers.

In our study, we first replicated Krajcsi results both on the symbolic and non-symbolic task, but we also and more importantly extended krajcsi results with new stimuli sets. In a third task, we used less automatized stimuli (decimal numbers between -0.99 and 0.99). Doing so we used less familiar stimuli and also obtained more variance in our size units.

Our study was separate in 4 different tasks that used different stimuli sets that were more or less automatized: positive intra-decade trials (were the decade digit of both pairs is the same), positive inter-decade trials (were the decade digit is different between both pairs), negative intra-decade and negative inter-decade trials.

We performed the same analysis as in the Krajcsi's 2017 study and we also runned the analysis using only the participants that had both effects that were significant.

We obtained a significant correlation between the two effects replicating Krajcsi findings. We obtained a non-significant correlation in the symbolic integer task also replicating Krajcsi findings. There was a significant correlation among positive decimals and selecting only participants that had both significant distance and size effects, we obtained a significant correlation between in the integer task.

In conclusion, our findings go in the direction that the ANS take care of non-symbolic comparisons, that symbolic integers are treated by both ANS and DSS (among different individuals at least) and that symbolic positive intra-decade numbers are processed by the ANS.

That's all, thank you for listening.
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