How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics

Round 1

Reviewer 1 Report

- The introduction contains personal statements that are not checked against the research literature, for example, “That young children often struggle with the notion of coinage is well attested by visiting any primary school “ or “Fractions are also a topic which is taught year after year because learners do not fully 35 comprehend fraction arithmetic, much less appreciate its use.” It is recommended that these statements be checked against research literature or reworked from a research point of view.

- It would be interesting to extend the theoretical underpinning to the concept of didactic relationship, including the knowledge used to knowledge taught

Method:

What is missing for the reader is the context in which the tasks have been carried out. For example, what is the nature of the “Early tasks” to be different from the “later task”

Task1 & task2

The analysis of the task is very interesting, but some aspects of counting already developed by Gelman and Gallistel (1978) in their model are left out. It would be interesting for the reader to complement it with some of the characteristics developed by these authors.

Task 3&4

children are also using a multiplicative proportional distribution in these tasks. That gives support to the work with fractions. It would be interesting to highlight this possible connection.

Task 5&6&7

It is possible that the notion of “didactic variable”

Task 11:

What happens when you subtract from X in fractions? It would be interesting to develop this aspect.

Task 19:

The potential of working with the area as summation has remained underdeveloped.

Conclusions:

In trying to achieve such an apparently well-defined goal, you are sure to come across facts not obviously akin to it, but which in a deeper study will show their intrinsic relatedness. It would be interesting if you could develop these studies in the future.

Author Response

please see attachment

Reviewer 2 Report

The proposed manuscript presents a comprehensive and complex phenomenological analysis of 23 mathematical tasks.

The introduction provides the essential insight into the problematic, mainly based on author's previous work.  Different kinds of mathematical binary operations are didactically transposed using the 'doing&undoing' theme. And the approach is manifested in several different contexts and contents. What I miss in the manuscript are the weaknesses of the approach, or when it would be counter-productive to use it.

Overall, the manuscript is well written, easy to read and presents very useful ideas.

Author Response

please see attachment (same document for both reviews)