Peter is right: to "not demonstrate" is to fail to disclose the common-sensibility of curricular mathematics. But *poor* demonstrations can do even much worse.
Did Peter's new student actually need to experience a formal proof? ... or, instead, to personally perceive the common-sensibility of "multiplication" and "division" of signed numbers?
All too often we believe that our instructional formalities have convinced students of truths ... when in fact, those formalities further convince students that math is too mysterious for them to own as conceptual understanding. So, they acquiesce ... while their suffering worsens ... and their "math anxiety" escalates ... until they can somehow escape.
"Multiplication" and "division" of signed numbers rely on per-(+1)-unit rates . At the per-unit rate of m-per-(+1), what is the (y?) result of going forward or backward (x?) at that rate ... when m is positive? ... m is negative? ... m is zero? In the other context: ... at the rate of y-per-x, what is the m-per-(+1) unit rate?
As Joe hinted, there are many real-world contexts in which both kinds of operations are common-sensible. Formal proof is not one of them.
Cordially, Clyde - -------------------------------------------------- From: "Peter Duveen" <firstname.lastname@example.org> Sent: Tuesday, November 27, 2012 9:12 AM To: <email@example.com> Subject: Some important demonstrations on negative numbers
> I have a new student, who was confused about negative numbers, both > multiplication and division of the same. I wanted to demonstrate to him > that a negative times a negative is a positive, but got flustered because > I could not produce the demonstration. I promised to show him the > demonstration at our next weekly meeting. > > Later, I sat down and derived the following demonstration: > > Proof that the multiplication of two negative numbers is a positive > number: - 1 + 1 = 0 (Definition of -1); -1(-1 + 1) = 0 (0 times any number > is 0); -1x-1 + -1x1 = 0 (distributive law of multiplication); -1x-1 + -1 = > 0 (1 x any number is that number itself); -1x-1 = 1 (definition of -1; the > same quantity added to equal quantities produce equal quantities). > > Main point is, it is not particularly self-evident that a negative number > times a negative number yields a positive number. Some will undoubtedly > argue that to demonstrate this property rather than to just state it will > confuse the student. At the same time, I would argue that to not > demonstrate it will confuse the student. > > I'm just throwing this out for discussion, but other properties that ought > to be demonstrated are that 1/-a = -(1/a), another property that many may > feel does not need to be demonstrated. On the contrary, I believe that not > demonstrating it will lead to confusion.