On Sun, Oct 28, 2012 at 3:26 PM, Robert Hansen <email@example.com> wrote: > > On Oct 28, 2012, at 4:00 PM, Jonathan Crabtree <firstname.lastname@example.org> wrote:
<< snip >>
>> 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. > > > Bob Hansen
There's some ambiguity here in that in some contexts "fractions" are treated strictly as rational numbers Q, meaning (p/q) where p, q are members of Z (integers).
More informally though, we say any a/b is "a fraction" where a and b are any two entities for which the division operator is defined.
The ambiguity comes in when it comes time to say whether we're "done evaluating".
pi / e could be converted to some decimal approximation or perhaps some infinite series, but don't we lose more information than we gain? pi / e looks "done".
5/10, on the other hand, is a member of an equivalence class of fractions that "all mean the same thing" but with 1/2 the canonical "lowest terms" delegate of this class of equal numbers.
You may lose information by reducing to lowest terms though, especially in situations of mixed units, e.g. if your private jet goes 600 miles on 200 gallons of fuel, that's an indication that your tank might have room for 200 gallons. 600/200 -> 6/2 = 3 miles / gallon doesn't tell you as much about capacity and range.
It's interesting how much is done with the division symbol in grade school, a symbol that's not in ASCII and not on the standard US keyboard.
Then, in the higher grades, using a bar or line to indicate division becomes the most natural, with the division symbol hardly used.
The minus sign is likewise cast in a confusing manner, at first as a small superscript in the upper left (the so-called negative sign), but then more as a unary operator, or a binary one with 0 implicit i.e. -3 = 0 - 3.
Why I like bringing in more computer languages is here we get more math notations with real world currency that demonstrate their own internal consistency, their own rules. Students develop a feeling for mathematical concepts more independently of a specific typography or symbol set.
I also consider it essential that we have plenty of non-numeric or semi-numeric operations in the picture, such as concatenation: "abc" + "def" = "abcdef"
Fine if you want to use ++ instead, or some other operator. Constructionism or constructivism has its place here, in that we go back and show how all the great mathematicians were constructivists.
The ones who did everything by rote and recitation have been forgotten by history as mere wannabes.
A key point is we (the curriculum designers) want students to talk about rules and rule-following (big in Wittgenstein, a philosopher of mathematics), deriving outputs in a deterministic fashion given specific inputs.
How does one evaluate expressions? Lots of practice, lots of anticipating what the answer is before hitting the enter key. But the rules are dependent on the namespace (the language game).
We don't want to foster the impression that any one particular notation is the "real" notation.
Rather, they see that notations are multifarious and always changing.
That's why I was suggesting to Paul we do more to introduces prefix notation, where the operator comes before the arguments in a left to right scan.
Don't get students habituated to "just one way to write things" too early, or they'll get hardening of the mental arteries too early and become straitjacketed adults before their time. A mind is a terrible thing to waste in that way.