On Oct 28, 2012, at 4:00 PM, Jonathan Crabtree <email@example.com> wrote:
> "I would not start teaching fractions until multiplication and division (with whole numbers) are firmly established." > > Thank you for this post Bob. > > If we are to truly benefit from exploring and discussing this topic, we should start off with accepted definitions relevant to children. > > So what is your definition of multiplication? > > What is your subsequent definition of division?
Here they are...
Multiplication is an operation, like addition and subtraction, between two numbers and produces a result, called the product of the two numbers.
Division is an operation, like addition, subtraction and multiplication, between two numbers and produces a result, called the quotient of the two numbers.
Definitions, other than being syntactically correct, are not important to children. What I said was "firmly established". By firmly established I mean that the student can multiply and divide whole numbers readily and it is apparent through their use of multiplication and division in varying contexts and problems that they are familiar with the relationship between the two and their basic properties (like commutativity or lack thereof).
> > What is your subsequent definition of fractions? >
A quotient between two numbers, the top most being called the numerator and the bottom most the denominator. What follows this definition is familiarity through discussion and usage and at some point we would say that fractions have been firmly established, which is not the same as finished. The only way I know of defining "firmly established" is through use, with problems and contexts, and once you list the problems in succession it seems stupidly simple. Unrepresentative of the actual effort involved in stepping a student through those stages.