I'm mostly happy with the revision, however, I do have a few points to note.

In the table in Appendix C some of the hypotheses that have been stated are explicitly directional and some are not explicitly directional when they should be. 

For example, RQ3b: "This means we will expect to see a decrease in performance in the non-symbolic
primary task in the VSSP dual-task condition in compared to the standalone condition for large quantities, but no decrease in performance for small quantities."

But then "If t-tests for both small and large quantities are significant (p < .05)" Surely this should be directional. 

or in RQ1b there's a mix of "If t-test is significant at p < 0.05, and indicates that performance is lower" (which is directional) but then "but the t-test indicates that there is no significant difference between the VSSP dual-task" (which is non-directional). If the hypotheses are directional then they should be clearly stated as directional. Furthermore, since two-tailed tests are being used, it should be clarified whether "non-significant" means two-tailed non-significant or whether it just means a result NOT "significant in the specified direction" (that is, one-tailed non-significant: i.e., t(10) = 2 gives a two-tailed p of ~.07, but a one-tailed p of either ~0.036 or ~0.96 depending on the tail that one looks at). For example, what if there is a significant difference, but just not in the predicted direction. For this reason, I think it might be easier to use one-tailed tests (at an alpha of 0.025) rather than two-tailed tests (at an alpha of 0.05) when the direction actually matters and only use two tailed tests when the direction doesn't matter. 

Thanks a lot for implementing our comments so rigorously, especially specifying the analysis decision tree.

I have only one more comment on data pre-processing regarding the following sentence in the introduction of Appendix C:

“Outliers will be examined for each condition (all combinations of primary and secondary task) and extreme outliers (> 3.29 SD, Field, 2016) will be removed from the analysis for that condition.”

As it makes a difference whether you look at group means or individual means per condition, I would like you to clarify this here. Furthermore, I do see it critical to look at outliers per condition, as Andre (2021) outlined in a recent simulation study that data pre-processing needs to be blind to the research hypothesis and addressing outliers separately per condition can artificially induce effects.

André, Q. (2021). Outlier exclusion procedures must be blind to the researcher’s hypothesis. Journal of Experimental Psychology: General.

The authors conceptualized a thorough study, investigating whether adults can automatically process and translate between numerical representations. Their research can be an interesting foundation for further studies, as they’re aiming at deepening the understanding of the link between numerical entities coded in different modalities within human cognition. They use the experimental design of a dual-task set-up, with different secondary tasks to assess whether working memory is a necessary prerequisite for (non-)symbolic magnitude comparisons and cross-modal translations.

Overall remarks:

I would like to acknowledge the authors’ intention to make their raw data publicly available and would like to stress the importance of well-documented meta-data with accessible descriptions of the respective variables (following the FAIR criteria).

I feel like there are some small language errors in your manuscript – please revise.

Introduction

Page 5, last paragraph: I wouldn’t agree that the first numerical representation children acquire is the verbal one, as studies implementing the looking time paradigm already show an increase in precision of the ANS between the ages of 6 to 10 months (e.g., Brannon, Suanda, Libertus, 2007; Lipton, Spelke, 2003, Wood, Spelke, 2005; Xu, Spelke, Goddard, 2005).

Page 5: “The ANS is assumed to provide estimates of the numerosity of a given set. Repeated presentations of the same numerosity result in varying points of activation and consequently representations of quantity in the ANS are approximate.” -> it’s not quite clear to me what you’re trying to say here. Can you reformulate?

I would probably introduce the role of inhibition and visuo-spatial skills for performance in ANS tasks when describing the association of ANS and arithmetic on page 6. This already gives a hint for your later research question that domain-general cognitive abilities may play a role for the associations between the different representations.

Can you clarify why working memory is the only domain-general cognitive function you expect to modulate the relationship between the different numerical representations? Considering the literature in this field, I do see several other candidates such as e.g., inhibition, verbal skill, or visuo-spatial skill.

Methods

What is your rationale for such a broad age range? With working memory and executive functions decreasing with age, I would suspect that you might find differences between age groups. Will you correct for these (for example adding age as a covariate in case it is significantly correlated with your outcome measure)?

Thanks for sharing your sample size considerations so transparently in the Supplementary. I’m just not quite sure how you come to the conclusion that expected decrease in performance in dual-task condition (number of trials difference or increase in RT) is 5 trials or 2 sequences. Could you elaborate on that a bit more in the manuscript?

Will you apply some kind of manipulation check that participants actually tried to stick to the secondary task? One could imagine some participants to only focus on the primary task resulting in low performance in the secondary task but a low effect on working memory load as well. You state that you will record accuracy of recall for both secondary tasks, but are you planning to control for this by excluding participants not reaching a certain level of performance regarding the secondary task?

Do I understand your design correctly: one participant will have both secondary tasks alongside each primary task so that the effect of the respective secondary tasks can be compared within participants?

In my view, the section on planned analyses lacks specification: Could you please lay out a decision tree, starting with testing the assumptions of your planned statistical analysis and then elaborating which changes you will make if the assumptions are not met (e.g., data transformation, robust hypothesis tests, etc.). Generally, I miss a section on your planned pre-processing pipelines, as these can heavily influence your results and should therefore be pre-registered as well. Will you only consider reaction times or accuracies as well?

Please consider possible limitations like the fact that you’re only using single-digit material which doesn’t allow for generalizations to multi-digit material. What if your material is too easy, resulting in ceiling effects and low variance?

As I believe the PCI design table to be very helpful in thinking through possible outcomes, it would be great if you included these considerations not only in the Supplementary but in an anticipated discussion section as well. One point regarding the design template: I think it would be more straightforward if you enumerated the three RQ3 as RQ3a, RQ3b and RQ3c as I was a bit surprised by the repetition of RQ3 which is only explained by the differences in the hypothesis column.

I’m looking forward to seeing the results of your interesting study.