Date: Oct 29, 2012 12:24 AM Author: kirby urner Subject: Re: The Teaching of Fractions On Sun, Oct 28, 2012 at 3:26 PM, Robert Hansen <bob@rsccore.com> wrote:

>

> On Oct 28, 2012, at 4:00 PM, Jonathan Crabtree <sendtojonathan@yahoo.com.au> 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.

Kirby