I think there is a misunderstanding here. When you say "definition" of an operation I think of a sentence or two that distinguishes the operation from everything else. Just enough to do that and nothing else. Simple and to the point. I don't include in that definition the whole essence of the operation. That is the part I teach and that includes the other points you make regarding units, proof etc.
I like to start with a simple definition, like that of a fraction, a quotient of two numbers, written one on top of the other, and then through lesson, application and practice, the student recognizes just how significant something so simple is, a quotient of two numbers, written one on top of the other.
On Nov 4, 2012, at 7:06 PM, Jonathan Crabtree <firstname.lastname@example.org> wrote:
> Hi Bob, > > I agree this is an interesting point because it highlights how western mathematicians have failed. > > Mathematics exists to simplify our life. Our life is made simpler by solving problems. For every problem there is a question and the answer to that question always has a unit attached to it. > > Unfortunately the track record of western mathematicians translating life into symbols is woeful. It's like many are deaf from birth yet insist their chords, and keys produce beautiful music. They don't, they sound ghastly and by the way, we don't yet know the notes in your chords! > > Many mathematicians make the mistake of failing to define the four operations of arithmetic. They fail to teach the diffent kinds of numbers, nominal, ordinal and cardinal. Then they fail to teach the structure of expressions and equations while Ms Jones next door teaches the formal parts of grammar very well. > > So teach the parts of mathematics and don't overlook it because of the lastest distracting shiny toy! (iPads, apps, smart whiteboards, software games) > > For example, the multiplicand always pairs with the unit/thing. the multiplier is the adverbial that describes the action going on in the story about the multiplicand. > > Oh, do you just talk about factors being interchangeable in your lessons? Do you teach logic via arrays/area models to prove 4 x 3 = 3 x 4? Do you point out you cannot multiply two multiplicands? Do you point out you cannot multiply two multipliers? > > Oh just make the bigger number the multiplicand and the smaller number the multiplier because their are fewer partial products to add that way! Bad teacher! > > In the real world you cannot multiply debt times debt just as you cannot multiply 1/3 of a pizza times 1/3 of a pizza. > > (Rap lyrics) > "Know that a root is the side of a square > Swapping factors, they just don't care! > One pill a day for a week ain't even close > Because seven pills right now is an overdose" > > Q1) What is 2? > Q2) What is 2? > Q3) What is 2? > Q4) What is 2? > > A1) A product! > A2) A quotient! > A3) A difference! > A4) A sum! > > Q5) What is a product? > A5) The result of an operation between two numbers. > Q6) What is a quotient? > A6) The result of an operation between two numbers. > > Source: http://mathforum.org/kb/message.jspa?messageID=7914031 > > Aha! The four operations are all the same! The only thing that separates them is the name of the result! > > Bob your approach to defining operations appears the same as the dictionary approach. > > Multiplication is defined as: > "The process of combining matrices, vectors, or other quantities under specific rules..." > > Addition is defined as: > "The process of combining matrices, vectors, or other quantities under specific rules..." > > Division is defined as: > "The process of dividing a matrix, vector, or other quantity by another under specific rules..." > > Subtraction is defined as: > "The process of taking a matrix, vector, or other quantity away from another under specific rules..." > > Sources: > http://oxforddictionaries.com/definition/english/multiplication?q=multiplication > > http://oxforddictionaries.com/definition/english/addition?q=addition > > http://oxforddictionaries.com/definition/english/division?q=division > > http://oxforddictionaries.com/definition/english/subtraction?q=subtraction > > The solution to the problem embedded in your interesting point means you DO need to define the operations otherwise they will not be recognised in word problems. > > Once you correctly define the operations then the child can recognise them in stories! > > Simply provide a template or process map and then the child can translate the story into the symbols and solve the problem. > > Even if your definitions are as simple as addition is combining two numbers into one or division is how many times a number goes into another number, until you define the operations in a concrete method that can be recognised and replicated, word problems will contine to be badly taught. > > Jonathan Crabtree > > P.S. Since the evolution of number theory ended a century ago, the teaching of arithmetic has become truly sloppy and increasingly meaningless in the west. The Chinese are the latest to adopt the Hindu Arabic number system during the late 1800s and early 20th century. So for them, the clear original discipline of Indian logic persists. The west hasn't updated its arithmetical logic in 1400 years since zero.