During the recent EARCOME 7 conference, one of the exhibitors was a group of Japanese who showed participants how to create a lot of concrete learning materials (manipulatives) such as origami sculptures. I assume that they are members of The Association of Mathematical Instruction (AMI) because they gave me a complimentary copy of AMI’s Principles of Mathematics Education.
The 52-page (including the covers) booklet has a size of 5 inches by 7 inches, is perfect bound (bound using adhesive), and has been published in at least seven languages. (Mine is in English.) It seems to have been written by Kô Ginbayashi in 1984. Aside from a short description of AMI and its guiding principles, the book has three parts: Theory of Quantity, The “Suido Method” (Water Supply Method), and Enjoyable Lessons. The writing is very clear and succinct. The 20 pages on the theory of quantity have so many important ideas that I’m having difficulty writing this blog post. (I prefer to keep my blog posts as short as possible.) My discussion below is not just a summary of what is actually in the book, but also includes some of my ideas that were inspired by my reading.
I found the emphasis on applied (instead of pure) mathematics very interesting. Page 4 of the book states:
The “Theory of Quantity” not only clarifies the meaning of numbers and the four arithmetic operations, but also makes it possible to apply mathematics to the real world, and to connect elementary mathematics with the Calculus.
As its name implies, the theory of quantity (proposed by Hiraku Tôyama) uses quantity (instead of pure numbers) as the basic concept. (Although it doesn’t seem to be explicitly stated in the book, a quantity is a numeral value and a unit.) To represent the concept of the magnitude of a set (the number of its elements), each element is represented by a (flat, square) tile. Although each element of a set can differ from the others, the representation of each element (in this case, by a tile “which has no other attribute than magnitude”) is the same. The representation of ten tiles as a bar, and of ten bars as a sheet aids in the understanding of positional notation (of decimal numbers). Aside from representing natural numbers, tiles can also be used to represent decimals and fractions.
Classification of quantities
A quantity is either discrete or continuous. A discrete quantity is the magnitude of a countable set (one whose elements are “mutually separated and individually distinct”). Its numeral value is a natural number (“division into a quantity less than a unit cannot be considered”) and its unit is clear at the start. An example of a discrete quantity is “three boys.”
A continuous quantity is the magnitude of a “continuum” (“a continuous entity which can be divided into any number of smaller parts” such that “any two such entities can be combined into a larger one”). Its numeral value (a decimal or a fraction) and its unit “have not been determined a priori.” An example of a continuous quantity is “three dollars.”
A continuous quantity is either extensive or intensive. The former expresses breadth or magnitude (such as area or weight); the latter expresses quality or intensity (such as density or speed). An extensive quantity has additivity: the attribute of the union of two bodies is equal to the sum of the attributes of the two bodies. An intensive quantity does not have additivity. For example, the weight of two bodies is necessarily the sum of their weights, but the speed of two bodies is not necessarily the sum of their speeds.
The introduction of basic unit for extensive quantity
The representation of an extensive quantity involves four steps (which should be taught in order).
- Direct comparison: an immediate judgment using perception can be used to determine which of two quantities is greater. For example, two objects may be placed on a scale balance to determine which is heavier.
- Indirect comparison: if the two objects cannot be directly compared, a third object may be used. For example, if the first object is found to be heavier than the third, and the third object is found to be heavier than the second, then it can be concluded that the first object is heavier than the second.
- The use of a particular unit: multiple copies of an object having a lesser quantity (where the copies have exactly the same quantity) are used to compare two objects.
- The use of a universal unit: a unit which is common to members of a wider society.
Representation of extensive quantity by numbers
When an extensive quantity is measured using a given unit, there are two ways to deal with the remainder.
- Measure the remainder using smaller units, say one-tenth of the given unit. If this yields a remainder, then measure it using much smaller units (one-tenth of the previous unit), and so on. This leads to a decimal. (The case of non-terminating or non-repeating decimals is treated separately.)
- Find a smaller unit that can measure both the given unit and the extensive quantity to be measured. This leads to a fraction. (The case of irrational numbers is treated separately.) For example, if n of these smaller units make up the given unit and m of these smaller units make up the extensive quantity to be measured, then the extensive quantity is m/n of the unit.
Note that decimals involve smaller units that are known in advance, and fractions involve smaller units that are known only after the extensive quantity to be measured is given. Thus, the concepts of decimals and of fractions are different (the concept of a decimal is not a particular case of the concept of a fraction).
Representation of intensive quantity by numbers
Intensive quantities may sometimes be directly compared. If this is not possible, then the intensive quantity may often be represented as a quotient of two extensive quantities (the measurement of which was previously discussed): a distributive quantity divided by a basis. The unit of the intensive quantity may be obtained from the units of the extensive quantities.
The basis may be a spatial quantity (such as length or mass) or it may be time. The distributive quantity may be existential (a single attribute such as amount of money) or it may be differential (a difference of two values of a point function such as electric potential). (This distinction is not absolute.)
Examples of intensive quantities expressed as ratios of extensive quantities are density (existential/spatial), gradient (differential/spatial), flow rate (existential/time), and speed (differential/time).
Quantity and operation of numbers
Quantities may be added or subtracted, but only if they have the same unit. Different quantities may be multiplied or divided; the result is a different quantity. The most typical cases are when the ratio of two extensive quantities is an intensive quantity (such as the ratio of mass and volume resulting in density) and when the product of an intensive quantity and an extensive quantity is an extensive quantity (such as the product of speed and time resulting in distance). (There are other cases such as the product of two extensive quantities and the product of a pure number and an extensive quantity.)
Addition and subtraction are to be taught first in connection with extensive quantities, then multiplication and division are to be taught next in connection with intensive quantities.
Quantity and proportion
There are at least three ways to find the missing quantity in a proportion. For example, given that a wire of length a in meters has a mass of b in grams, we are to find the weight x in grams of a wire of length c in meters.
- Express x [g] as a product of c [m] and b [g], which in turn is divided by a [m]. This is problematic because the product of a length and a mass “cannot be interpreted.”
- Express x [g] as a product of c/a (a pure fraction) and b [g]. However, the concept of fractional multiples may be difficult to understand.
- Express x as a product of b/a [g/m] (an intensive quantity) and c [m] (an extensive quantity). Here the concept of linear density as the weight per unit length presents the solution in a natural way.
Proportional function and intensive quantity
The proportion in the previous example involves an intensive quantity readily obtained from two extensive quantities that belong to the same object; this may be called a quantitative proportion.
If two extensive quantities x and y are related by a function f from x to y (that is, y=f(x)) such that y=ax, then we call the function a proportional function. Here, a=f(1)=y/x is an intensive quantity called the proportional constant that is deduced only after the proportional function is considered. The proportion in this case is called a functional proportion.
The concept of a proportional constant (an intensive quantity) as a ratio of extensive quantities may be extended to the concept of a differential coefficient of f. Here, f is a differentiable function from x to y such that the derivative at a point x=k is f'(k)=dy/dx, where x and y are real numbers and f'(k) is the differential coefficient.
It may also be extended to the concept of a representative matrix of f. Here, f is a linear mapping from a linear space E to F such that y=Ax, where x and y are vectors and A is the representative matrix.
It may also be extended to the concept of a Jacobian matrix of f. Here, f is a mapping from a manifold E to F such that J=∂y/∂x, where x and y are vector-valued functions and J is the Jacobian matrix.
The basic concept of quantity leads to the basic concept of proportion, which can be applied to differential calculus, linear algebra, and vector analysis.