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I read about the Theory of Quantity in the booklet Principles of Mathematics Education written by Kô Ginbayashi in 1984. (A summary of it is here.) Instead of using pure numbers as the basic concept, it uses quantities (numeral values and units). So, for example, instead of teaching "1 + 1 = 2," you would teach "1 apple + 1 apple = 2 apples," or "1 inch + 1 inch = 2 inches."

I think this makes a lot of sense. Note that only quantities with the same units can be added or subtracted. Quantities that are multiplied or divided yield new quantities. For example, "1 m / 1 s = 1 m/s," that is, a length divided by a unit of time is a speed.

This leads to ratio and proportion as the next basic idea. From here, it is easy to find similar relationships in

  • differential calculus ($f'(k)=dy/dx$, where $x$ and $y$ are real numbers and $f'(k)$ is the differential coefficient)

  • linear algebra ($\mathbf{y}=\mathbf{Ax}$, where $\mathbf{x}$ and $\mathbf{y}$ are vectors and $\mathbf{A}$ is the representative matrix)

  • vector analysis ($\mathbf{J}=\partial\mathbf{y}/\partial\mathbf{x}$, where $\mathbf{x}$ and $\mathbf{y}$ are vector-valued functions and $\mathbf{J}$ is the Jacobian matrix.)

Note that these are applied mathematics fields.

Are you familiar with any other book that uses quantity (instead of pure number) as the basic concept?

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    Most of the ancient geometrical texts, such as the geometrical books of Euclid. Later books, too. Newton's Principia, for instance, does analysis (what we now tend to conflate with "calculus") in terms of quantities, not numbers. I'm not sure when they gave up writing power series in terms of $x^n/a^n$, where $a$ represents some sort of unit quantity. – user1815 Aug 18 '15 at 17:33

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The Elkonin-Davydov mathematics curriculum from Russia "is characterized by the early development of quantitative and algebraic reasoning through measurement of continuous quantities" (source).

I was lead to it by a recent article (link, access to which might need a subscription) in the October 2015 (Vol. 22, Issue 3) issue of Teaching Children Mathematics.

The article ("Units Matter" by Ji Yeong I, Barbara J. Dougherty, and Zaur Berkaliev) has the subtitle "The meat and potatoes of fraction multiplication is the change of units."

Here are some relevant quotes from it:

The distinguishable characteristic between additive and multiplicative reasoning is the change of unit of measure (Berkaliev 2008).

Also,

[...] A unit cannot be separated from a number, and a number is dependent on its unit. This explicit use of unit is usually observed in scientific measurement activities, but units are also used implicitly in mathematics.

In the measurement perspective, all numbers involve the concept of unit. Berkaliev (2008) indicated that “a number itself is not just an absolute and final entity, but is only an expression of a relationship between two different quantities” (p. 9). When multiplying two numbers, one number is the unit of measure and the other is a quantity to be measured [...]. [...] It also exemplifies the difference between addition and multiplication: Addends in addition simply use one common unit, and multiplication involves more than one unit.

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