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Topic: new tutor here
Replies: 151   Last Post: Jul 17, 2013 4:09 AM

 Messages: [ Previous | Next ]
 Robert Hansen Posts: 11,345 From: Florida Registered: 6/22/09
Re: new tutor here
Posted: Jul 10, 2013 12:37 PM
 att1.html (4.6 K)

I like this quote...

"A heavy warning used to be given that pictures are not rigorous; this has never had its bluff called and has perma- nently frightened its victims into play- ing for safety."

I do not claim that a visualization of a proof or of a mathematical conclusion is less rigorous. In fact, my claims have nothing to do with rigor at all. I claim that when you show students a visualization they will see either the visualization or the mathematical reasoning and the only way to know which it is they see is to ask them to explain the reasoning behind the visualization.

This could probably be easily proven. Show students two "visually appealing" proofs, one mathematically correct and the other not, even though it "visually" looks correct. I suspect that you will find that the students lacking mathematical reasoning sense will judge both to be correct while the students that have the sense will see the flaw. I am not saying that the first set of students can't be taught mathematics or will never get the sense. I am only pointing out that a visualization of a mathematical argument still requires the same sense of mathematical reasoning to see it as does a mathematical argument in any other medium.

Another example of this is data visualization. Data visualization appeals to our non mathematical senses. These are the senses Dehaene writes about and confuses with mathematics. When I can create a data visualization I can create it in such a way that it is mathematically correct but invokes a response completely at odds with the data being represented. For example, Richard posted some time ago a graphic of a bag three times as large as another bag, by volume (or something to that effect). Not a good representation to feed to our Dehaene rat-sense of spatial area. Our rat-sense does not recognize it as three times as big. It requires actual mathematical sense to make sense of this visual contradiction. Obviously, when you visualize data you strive to avoid this situation. You choose graphical methods that look "rat-sense" correct rather than mathematically correct.

The bottom line is that rat-sense is not mathematical reasoning nor is it a possible substitute for mathematical reasoning. Visualizations appeal to rat-sense. The mathematical argument behind the visualization can only appeal to the sense of reasoning. Saying that a visualization is appealing does not tell me which occurred.

Bob Hansen

On Jul 10, 2013, at 10:15 AM, "Dave L. Renfro" <renfr1dl@cmich.edu> wrote:

> Joe Niederberger wrote (on 21 June 2013):
>
> http://mathforum.org/kb/message.jspa?messageID=9142053
>

>> But onward and upward -
>> Here's a nice book, but I won;t be paying \$159 for an book copy!
>> It's a collection and has essays on various sides if the issue.
>> The entire introduction, which I just read, seesm promising.
>>
>> "Visualization, Explanation and Reasoning Styles in Mathematics"

>
>
> For those who might be interested, I came across something this
> morning that fits well into this discussion:
>
> Bill Casselman, Review of "Visual Explanations" by Edward R. Tufte,
> Notices of the American Mathematical Society 46 #1 (January 1999), 43-46.
> http://www.ams.org/notices/199901/rev-casselman.pdf
>
> Dave L. Renfro

Date Subject Author
6/12/13 Michael Mossey
6/12/13 Wayne Bishop
6/12/13 Michael Mossey
6/13/13 Wayne Bishop
6/13/13 Dave L. Renfro
6/13/13 Joe Niederberger
6/14/13 Robert Hansen
6/14/13 Joe Niederberger
6/14/13 Robert Hansen
6/14/13 Joe Niederberger
6/14/13 Robert Hansen
6/14/13 Joe Niederberger
6/14/13 Robert Hansen
6/15/13 Louis Talman
6/14/13 Michael Mossey
6/15/13 Robert Hansen
6/14/13 GS Chandy
6/16/13 GS Chandy
6/15/13 Louis Talman
6/15/13 Joe Niederberger
6/16/13 Robert Hansen
6/15/13 Pam
6/16/13 Robert Hansen
6/15/13 Pam
6/16/13 Wayne Bishop
6/16/13 Joe Niederberger
6/16/13 Robert Hansen
6/16/13 Joe Niederberger
6/17/13 Robert Hansen
6/16/13 Michael Mossey
6/17/13 GS Chandy
6/17/13 GS Chandy
6/17/13 Pam
6/17/13 Robert Hansen
6/17/13 Pam
6/17/13 Robert Hansen
6/17/13 Pam
6/17/13 Pam
6/17/13 Joe Niederberger
6/17/13 Robert Hansen
6/17/13 Joe Niederberger
6/17/13 Joe Niederberger
6/18/13 Robert Hansen
6/17/13 GS Chandy
6/18/13 GS Chandy
6/18/13 GS Chandy
6/18/13 GS Chandy
6/18/13 Joe Niederberger
6/18/13 Robert Hansen
6/18/13 Joe Niederberger
6/18/13 Joe Niederberger
6/18/13 Robert Hansen
6/18/13 Joe Niederberger
6/18/13 Joe Niederberger
6/18/13 Robert Hansen
6/18/13 Robert Hansen
6/18/13 Joe Niederberger
6/19/13 Robert Hansen
6/19/13 Robert Hansen
6/21/13 Wayne Bishop
6/21/13 Wayne Bishop
6/19/13 Joe Niederberger
6/19/13 Robert Hansen
6/19/13 Joe Niederberger
6/19/13 Robert Hansen
6/19/13 Joe Niederberger
6/19/13 Robert Hansen
6/19/13 Joe Niederberger
6/19/13 Robert Hansen
6/19/13 Robert Hansen
6/19/13 Joe Niederberger
6/19/13 Joe Niederberger
6/19/13 Joe Niederberger
6/19/13 Robert Hansen
6/21/13 Joe Niederberger
6/21/13 Robert Hansen
6/21/13 Joe Niederberger
6/21/13 Robert Hansen
6/21/13 Joe Niederberger
6/21/13 Robert Hansen
6/21/13 Joe Niederberger
6/21/13 Joe Niederberger
6/21/13 Dave L. Renfro
6/22/13 Robert Hansen
6/22/13 Robert Hansen
6/22/13 Joe Niederberger
6/22/13 Robert Hansen
6/22/13 Robert Hansen
6/23/13 Joe Niederberger
6/23/13 Robert Hansen
6/23/13 Robert Hansen
6/23/13 Joe Niederberger
6/23/13 Robert Hansen
6/24/13 Robert Hansen
6/24/13 Joe Niederberger
6/24/13 Joe Niederberger
6/24/13 Joe Niederberger
6/25/13 GS Chandy
7/10/13 Dave L. Renfro
7/10/13 Robert Hansen
7/10/13 Joe Niederberger
7/10/13 Robert Hansen
7/10/13 Robert Hansen
7/10/13 Robert Hansen
7/11/13 Joe Niederberger
7/11/13 Robert Hansen
7/11/13 Joe Niederberger
7/11/13 Robert Hansen
7/11/13 Robert Hansen
7/11/13 GS Chandy
7/11/13 Joe Niederberger
7/11/13 Robert Hansen
7/11/13 Robert Hansen
7/11/13 Louis Talman
7/11/13 Robert Hansen
7/12/13 Wayne Bishop
7/12/13 Louis Talman
7/12/13 Wayne Bishop
7/12/13 Robert Hansen
7/12/13 Gary Tupper
7/12/13 Wayne Bishop
7/12/13 Louis Talman
7/11/13 Joe Niederberger
7/11/13 Robert Hansen
7/11/13 Joe Niederberger
7/11/13 GS Chandy
7/12/13 Wayne Bishop
7/12/13 Joe Niederberger
7/12/13 Wayne Bishop
7/12/13 GS Chandy
7/12/13 Joe Niederberger
7/13/13 Robert Hansen
7/13/13 GS Chandy
7/13/13 Joe Niederberger
7/13/13 Wayne Bishop
7/13/13 Robert Hansen
7/13/13 Robert Hansen
7/13/13 Peter Duveen
7/14/13 Joe Niederberger
7/14/13 Joe Niederberger
7/14/13 Joe Niederberger
7/14/13 Robert Hansen
7/14/13 GS Chandy
7/15/13 Wayne Bishop
7/15/13 Joe Niederberger
7/15/13 Robert Hansen
7/15/13 Peter Duveen
7/15/13 Joe Niederberger
7/16/13 GS Chandy
7/16/13 Wayne Bishop
7/16/13 GS Chandy
7/17/13 GS Chandy