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edit approved Aug 20, 2024 at 10:41

Adding another view to the point, one could view a fraction $\dfrac{a}{b}$, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $7$ mean? It is $7$ of some unit/unitary quantity. For instance, $7$ cats, or $7$ pies etc. This can lead us to introduce the notion of $\dfrac{7}{1}$, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to $\dfrac{7}{2}$, meaning that we talk about 7 objects of a unit seperatedseparated in two parts. Similarly, we can write $\dfrac{7}{3},\dfrac{7}{4}$, etc.

In this context, a fraction is not seen as the result of division but more as a shorthand for a process - changing measurement units. Under this view, division is "hidden" as a process within the fraction symbol.

One advantage of viewing fractions as shorthands for the "measurement unit shift" process is that it this definition stands in the middle of a purely procedural and a purely declarative perception of the fraction. It is neither entirely a process nor entirely a number - so, it remains easy to lean towards one of the two views of the fraction (number or division operator) whenever needed.

Adding another view to the point, one could view a fraction $\dfrac{a}{b}$, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $7$ mean? It is $7$ of some unit/unitary quantity. For instance, $7$ cats, or $7$ pies etc. This can lead us to introduce the notion of $\dfrac{7}{1}$, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to $\dfrac{7}{2}$, meaning that we talk about 7 objects of a unit seperated in two parts. Similarly, we can write $\dfrac{7}{3},\dfrac{7}{4}$ etc.

Adding another view to the point, one could view a fraction $\dfrac{a}{b}$ as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $7$ mean? It is $7$ of some unit/unitary quantity. For instance, $7$ cats, or $7$ pies etc. This can lead us to introduce the notion of $\dfrac{7}{1}$, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to $\dfrac{7}{2}$, meaning that we talk about 7 objects of a unit separated in two parts. Similarly, we can write $\dfrac{7}{3},\dfrac{7}{4}$, etc.

One advantage of viewing fractions as shorthands for the "measurement unit shift" process is that this definition stands in the middle of a purely procedural and a purely declarative perception of the fraction. It is neither entirely a process nor entirely a number - so, it remains easy to lean towards one of the two views of the fraction (number or division operator) whenever needed.

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answered Jan 21, 2020 at 19:09

Vassilis Markos

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Adding another view to the point, one could view a fraction $\dfrac{a}{b}$, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $7$ mean? It is $7$ of some unit/unitary quantity. For instance, $7$ cats, or $7$ pies etc. This can lead us to introduce the notion of $\dfrac{7}{1}$, as the symbol that denotes exactly the above; 7 instances of some (whole) unit (1). Then, we extend this notation to $\dfrac{7}{2}$, meaning that we talk about 7 objects of a unit seperated in two parts. Similarly, we can write $\dfrac{7}{3},\dfrac{7}{4}$ etc.