Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » Education » math-teach

Topic: Some important demonstrations on negative numbers
Replies: 70   Last Post: Dec 8, 2012 1:28 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Clyde Greeno @ MALEI

Posts: 220
Registered: 9/13/10
Re: Some important demonstrations on negative numbers
Posted: Nov 27, 2012 4:11 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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" <pduveen@yahoo.com>
Sent: Tuesday, November 27, 2012 9:12 AM
To: <math-teach@mathforum.org>
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.



Date Subject Author
11/27/12
Read Some important demonstrations on negative numbers
Peter Duveen
11/27/12
Read Re: Some important demonstrations on negative numbers
James Elander
11/27/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/27/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/27/12
Read Re: Some important demonstrations on negative numbers
Clyde Greeno @ MALEI
11/27/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/27/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/27/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/27/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/27/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/27/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/28/12
Read Re: Some important demonstrations on negative numbers
GS Chandy
11/28/12
Read Re: Some important demonstrations on negative numbers
GS Chandy
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/28/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/29/12
Read Re: Some important demonstrations on negative numbers
kirby urner
11/28/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/28/12
Read Re: Some important demonstrations on negative numbers
syeds
11/28/12
Read Re: Some important demonstrations on negative numbers
GS Chandy
11/28/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/28/12
Read Re: Some important demonstrations on negative numbers
Paul A. Tanner III
11/28/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/29/12
Read Re: Some important demonstrations on negative numbers
kirby urner
11/28/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Paul A. Tanner III
11/28/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
11/28/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/28/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/29/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/29/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/29/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Peter Duveen
11/29/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/29/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
11/29/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/29/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/29/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Clyde Greeno @ MALEI
11/30/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
11/30/12
Read Re: Some important demonstrations on negative numbers
Clyde Greeno @ MALEI
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
12/1/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Clyde Greeno
12/1/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
11/30/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
12/1/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
12/1/12
Read Re: Some important demonstrations on negative numbers
Robert Hansen
12/8/12
Read Re: Some important demonstrations on negative numbers > a MACS syllabus
Clyde Greeno @ MALEI
12/1/12
Read Re: Some important demonstrations on negative numbers
Joe Niederberger
12/2/12
Read Re: Some important demonstrations on negative numbers
Jonathan Crabtree
12/2/12
Read Re: Some important demonstrations on negative numbers > Wallis/Pycior
Clyde Greeno @ MALEI
12/3/12
Read Re: Some important demonstrations on negative numbers > Wallis/Pycior
Robert Hansen

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.